Quintessence from nothingness, 9986171, 978-0-599-93154-1

INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bieedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. QUINTESSENCE FROM NOTHINGNESS: ZERO, PLATONISM, AND THE RENAISSANCE A DISSERTATION SUBMITTED TO THE DEPARTMENT OF COMPARATIVE LITERATURE AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Eugene Ostashevsky April 2000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 9986171 ___ __® UMI UMI Microform9986171 Copyright 2000 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Copyright by Eugene Ostashevsky 2000 All Rights Reserved u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Hans Ulrich Gumbrecht, Principal Adviser I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Patricia Parker, Co-adviser I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Timothy Lenoir Approved for the University Committee on Graduate Studies: iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. QUINTESSENCE FROM NOTHINGNESS: ZERO, PLATONISM, AND THE RENAISSANCE Eugene Ostashevsky, PhD Stanford University, 2000 Principal Adviser: Hans Ulrich Gumbrecht Co-Adviser: Patricia Parker Quintessence from Nothingness: Zero, Platonism, and the Renaissance explores the role played by Hindu-Arabic numerals, and especially the zero, in the Renaissance and early modem restructuring of the concepts of number, God, self and thing. My first chapters focus on the classical concept of number as plurality, the effect it has on Plato’s ontology, and extent to which it survives in the writings of such Renaissance Platonists as John Dee. Although, in a decisive break with the past, Simon Stevin bases his overall understanding of numbers on the functioning of Hindu-Arabic numerals, he also performs the strange operation of assigning to zero nearly all the characteristics held by the one in the concept of number he critiques. Stranger still, similar usurpation of the one's traits by zero occurs even earlier in theology, where the transcendent God's relationship to things moves from being interpreted as the relationship of one to numbers, to being intepreted as the relationship of zero to numbers by the Salem Codex and Charles de Bovelles. Finally, in texts as divergent as Shakespeare's Henry V and the philosophical treatises of the Ukrainian hermeticist Hryhorii Skovoroda, the zero also replaces the one in the concepts of self and thing. Analyzing the unanimous insistence of early arithmetics that the zero "signifieth not,” I argue that the fundamental change underlying these replacements occurs in the understanding of how signs function. The Platonic model of signs pointing to discrete, real, and ontologically prior entities yields to a model in which, due to its stress on syntax, the signified is held to be posterior to the signifier. The classical number concept, with its accompanying representations of numbers by means of counters and Roman numerals, accords with Plato's understanding of signs. Hindu-Arabic notation, the value of whose signs depends upon their position, offers a key example of the syntactical model, with zero providing the ocular proof that Plato's treatment of signification is wrong. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The most beautiful is the object which does not exist Zbigniew Herbert "Study of the Object" O B bl, H yjIH MOH H HOJTHKH, R B a c j u o 6 h j i , a B a c jik > 6 jiio ! H H K O JiaH OjieHHHKOB „ 0 H y /ia x “ V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A C K N O W LED G M EN TS Before I go to meet Stanford's Registrar, Roger O. Printup, I have some last words: Had it not been for my principal adviser. Sepp Gumbrecht, I never would have finished. Had it not been for my parents, Helen and Joseph Ostashevsky, I never would have started (or finished, for that matter). I would like to thank my co-adviser Patricia Parker and my reader Tim Lenoir for providing the transportation. I would like to thank my sister Luba, my cousin Anna and my friends for providing the impetus. And, O yes. To my tier of knots An attire of noughts, As promised. e /q vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TA B LE O F CON TENTS CHAPTER O INTRODUCTION 0. Introduction I 1. Signifying Nothing I 2. Cipher 3 3. Essence vs. Appearance 8 4. Signification 10 5. Isomorphisms 11 CHAPTER I NOTATIONS 0. Introduction 14 1. Roman and Greek Notations 1.1 The Imitative Principle 14 1.2 Greek Herodian and Alphabetic Numerals 15 1.3 Roman Notation 16 2. Counter-Casting 2.1 Historical Background 18 2.2 Calculating with Counting Boards 19 2.3 Counter-Casting and Positional Notation 20 3. Hindu-Arabic or Positional Notation 3.1 The Imitative Principle vs. the Symbolic 23 3.2 Historical Background 24 3.3 Positional Notation in Fifteenth-Century England 26 3.4 Sixteenth-Century English Arithmetics 30 CHAPTER 2 THE CLASSICAL CONCEPT OF NUMBER 0. Introduction 33 1. Number as Multitude of Units 34 2. Number as the Unity of a Multitude 37 3. Greek Arithmetic 39 4. Number in Magnitude 43 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 NUMBERS AND PLATO’S ONTOLOGY 0. Introduction 47 1. Pythagoras: “Numbers are the substances of all things" 48 2. "The Actual Object of Knowledge... is the True Reality" 51 3. Distinguishing Types of Number and their Characteristics 52 4. Platonic Signification and the Doctrine of Ideas 4 .1 Number as Evidence of Non-Sensual Being 56 4.2 Ideas as Meanings 57 4.3 Signification and the Problem of Nothing 60 4.4 Otherness and Ideas in The Sophist 62 5. Ideas as Ideal Numbers 64 6. The Generation of Ideal Numbers 67 CHAPTER 4 NUMBER IN JOHN DEE'S MA THEM A TIC ALL PRAEFACE 1. Number in Christian Platonism 70 2. A Mathematical Preface to The Mathematicall Praeface 11 3. Arts Mathematical Derivative in Dee's Mathematicall Praeface 81 4. Dee's Concept of Number 85 5. Conclusion. What is Arithmetic? 93 CHAPTER 5 ZERO AND THE CONCEPT OF NUMBER IN SIMON STEVIN 0. Introduction 96 1. Algebra 1.1 Signification in Rhetorical and Syncopated Algebra 98 1.2 Signification in Symbolic Algebra 100 2. Stevin 2.1 Decimals 101 2.2 General Concept of Number in Stevin's Arithmetique 106 2.3 Specific Types of Number in the Arithmetique 115 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6.1 GOD AS ONE AND GODAS ZERO; PART I: THE TRANSCENDENT ONE 0. Introduction 120 1. Being and the One in Plato 121 2. Plotinus 2 .1 The Transcendental One 125 2.2 The One as One and as Nothing 129 3. Augustine 131 4. Pseudo-Dionysius 4 .1 God as the Transcendent One 134 4.2 Knowing God in a Floating World 136 4.3 God as Nothing 138 5. Eriugena 5.1 The Dilemma of Creatio ex Nihilo and the Eternity of Ideas 142 5.2 The Model of the Monad and Numbers 144 5.3 Nihil in Creatio ex Nihilo as Nothing by Transcendence 148 CHAPTER 6.2 GOD AS ONE AND GOD AS ZERO; PART H: GOD AS ZERO 0. Introduction 154 6. Imagery of the Circle 6.1. Archetypal or Historical? 155 6.2 The Circle as Generated by the Transcendent Center 156 6.3 Circle as Circumference: Construction 160 6.4 The Circular Motion of Divinity 163 6.5 Geometry of the Infinite in Nicholas of Cusa 169 7. God is Zero 7.1. Theology Before Arithmetic 171 7.2 The Salem Codex 173 7.3 Bovelles; Nothing in the Liber de Nichilo 176 7.4 Bovelles: Zero is Like God 181 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 CROOKED FIGURE: CIPHER AND SEQUENCE IN SHAKESPEARE'S HENRY V 0. The Chorus and Numerical Increase 189 1. Humility and Sequence 194 2. Place Makes the Man 200 3. The Genealogical Line, Right and Sinister 203 4. Bloody Constraint 212 5. Crooked Figures and Treacherous Crowns 215 6. The Count 218 1.0 PerSe-O 226 8. Accompts and Reckonings 230 9. Judgment 236 CHAPTER 8 SKOVORODA, CARTESIAN PLATONIST 0. Introduction 239 1. Figurative Speech 240 1.1 The Hermetic Hypothesis 241 1.2 Biblical Figures in Skovoroda 244 2. Figure in the Book of Nature 2.1 Parallel Scripture and Positionality 247 2.2 Figure as Geometrical Aspect of Things 249 2.3 Figure as Thought 251 2.4 Second Nature 252 2.5 Figure as Unity and as Point 254 3. The Book of Number 3 .1 Figure and Number 259 3.2 Circles 261 3.3 Figure Arithmetical 265 4. Skovoroda and Seventeenth-Century Philosophy 266 BIBLIOGRAPHY 272 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER O INTRODUCTION 3HaneHHe 6y6nHKa HaM HenoHHTHO1 0. Introduction I started thinking about the topic of this dissertation after reading Brian Rotman’s Signifying Nothing: The Semiotics o f Zero several years ago. At the time I was looking for a way to allow me to work on English and Russian seventeenth-century poetry simultaneously. In a stanza of a piece by Stefan Iavorskii, the Place-Holder of the Patriarchal See under Peter the Great, I found a conceit of a pair of compasses strikingly redolent of Donne's "Valediction Forbidding Mourning.” Iavorskii, needless to say, didn’t speak English; my search for third-party sources eventually brought me to the themes of emblematics and geometry in Renaissance Platonism. After reading Rotman, and, at his book's prompting, Jacob Klein’s Greek Mathematical Thought and the Origin o f Algebra, I switched to something yet more fundamental — the Platonist understanding of arithmetic. For the Florentine Platonist Marsilio Ficino, contemplation of mathematical entities leads one to abandon the world appearances, and to spiral into the rarefied air of objects of pure intellect. My research suffered the parodic version of the same process. First the Russians went the way of all flesh; then emblematics; then Donne; then geometry itself. In the following pages, too much attention shall be paid to the Greeks. I say "too much” because I have next to no training in Greek philosophy, nor mathematics Greek or otherwise. However, I thought that if I were to study Renaissance Platonism, I should start with Plato, and so I did. The result perhaps accords with that well-known American rule of business, in which one gets promoted to the first level of one’s incompetence. 1. Signifying Nothing Rotman’s Signifying Nothing argues for the isomorphism, on the level of semiotics, among the restructuring of mathematics by zero and Hindu-Arabic numerals, painting by perspective, and economic exchange by "imaginary money," i.e. notes of credit. In all three cases what Rotman calls a "meta-sign," i.e. a sign indicating the absence 1 Nikolai Oleinikov, "Bublik," in Oleinikov 123. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of other signs, establishes a code whose claim to represent things anterior to it (what I shall refer to as prior signifieds) can be deconstructed by pointing to the purely semiotic nature of the meta-sign itself. Furthermore, Rotman argues that in all three cases, each of the meta-signs of zero, vanishing point, and imaginary money, engenders a secondary construction: the algebraic variable, the multi-perpectival punctum, and paper money. The secondary meta-sign is characterized by its detached and free-ranging nature: instead of pointing to things or even to elements of a code, it is, fundamentally, an operation performed in the code. Rotman regards each of the two types of code, the primary and the secondary, as inscribed with its own species of semiotic subject. In the case of zero, the vanishing point, and imaginary money, the subject appears in the exterior and authorial relationship to the code: he counts, he sees, he trades. In the case of the algebraic variable, the multi- perspectival punctum, and paper money, subjectivity is a lot more complex: the code is seen as defining the subject as much as the other way around. This second type of subject — Rotman's meta-subject — performs acts that he realizes have no meaning outside the code they are performed in. "I have no more made my booke than my booke hath made me," Rotman quotes Montaigne. The meta-subject constantly deconstructs himself, and his acts are acts "about" acting in a given code. I found Rotman's thesis of isomorphism in the semiotics of Renaissance and early modem mathematics, painting and finance tremendously — I believe the American word is "exciting." Rotman was focusing on what I identified with real history: the history of the structures of human thought, of the generation and corruption of mental paradigms by means of which we interact with the world, and which never remain bounded to any one discipline. In addition, as somebody for whom the period Rotman describes is of immense personal importance, I could not fail to appreciate his thesis of the meta-subject, which seemed so applicable to the phenomena. Consider, for instance, Donne's elaboration on the self as nothing in "A Noctumail upon S. Lucies Day," where the I exhibits the changes wrought in it by the death of its beloved. "I am re-begot, Of absence, darknesse, death; things which are not;” complains the I, and "I [...] am the grave, Of all, that's nothing;” and "I am by her death [...] Of the first nothing, the Elixer grown" [II. 17-8, 21-2, 28-9]. Whatever may be the metaphysical connotations of Donne's "quintessence [...] from nothingnesse" [1. 15], it also has to do with writing. Anyone who has ever noticed Donne's puns on "grave"2 knows this to be no : E.g.. "All-graved tome In cypher writ," of "Valediction to his booke," II. 20-1; or "Loves graves" of "Elegie IX: The Autumnall." 11.13-6. For Donne love is nothing primarily by the virtue of its being no thing of sense. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all-purpose postmodernist potshot, or motivated only by the fact that "nothing" and "noting" were homophones in Renaissance England. "A Noctumall" presents itself as an utterance that is identical with the speaking subject, whose name, incidentally, curls in another typical Donnean flourish: "But I am None" [1. 37]. Such disembodiment of the self in its own portrayal to me appears completely congruent to Rotman's readings of Vermeer's The Artist in his Studio and Velasquez’s Las Meninas, with Donne’s idea of the writing I as a secondary nihil ex nihilo fit embodying Rotman’s formation of the secondary meta-sign from the primary, e.g. of the algebraic variable from the Hindu-Arabic zero. "A Nocturall" is of course also "about" lots of other things; I use it merely to show that Rotman's ideas apply to fields and persons not mentioned in Signifying Nothing. 2. Cipher Everyone always makes fun of Isidore of Seville for thinking that etymologies — and bad etymologies at that — inform us of the essence of things signified. Even Plato has his fifteen minutes of gloating over Isidore’s ancestor, Cratylus, in a dialogue of the same name. Poor Cratylus! In the end, losing faith that things have any being at all, he renounced their signifiers — words — altogether [Aristotle, Metaphys. I0 l0 a l2 -1 5 ; for Cratylus as Plato's first teacher, see ibid., 987a33]. It is therefore with no small degree of trepidation that I set out to make certain historical insinuations based on the etymology and history of the word cipher. If I do dare the attempt, it is only because today, April 4, is St. Isidore's feast-day. The English cipher ultimately derives from the Arabic word for zero, gifr, itself a translation of Hindu sunya, empty. The derivation for the word zero is the same, albeit by a different route, and therefore cipher and zero are cognates. Introducing Hindu-Arabic numerals in his Liber abaci, Leonardo of Pisa or Fibonacci (c. 1175 - c. 1240) transliterated the Arabic word as zephirum. As it traveled through Italian dialects, fast-talking Venetian merchants contracted the word to zero, and hawked it about in France and Spain [Menninger 401; Smith and Karpinski 57-58; Jordan 190]. An alternative version of gifr in thirteenth-century Latin was cifira, whence the French chiffre: the new numeral, writes one Renaissance arithmetic, "s'appelle chiffre vulgairement: les vns I'appellent zero" [Smith and Karpinski, 62 n. 3; Jordan 191]. Both of these forms reached English, with cipher arriving earlier (first OED quote of cipher from 1399, and zero from 1604), and becoming overwhelmingly predominant. Unlike zero, the various versions of cipher acquired a plethora of other denotations, and what these are strikes me as quite significant. In Latin, Italian, French, and English 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (among other languages), cipher expanded its application to designate all other Hindu- Arabic numerals, also known as figurae arithmeticae, in opposition to the litterae arithmeticae or Roman numerals. So, in 1542, Recorde writes that "O [...] is called priuately a cyphar, though all the other [figures] somtyme be lykewyse named" [Grounde, Bvv], Cotgrave’s A Dictionary o f the French and English Tongues (1611) translates "Chifre" as "A Cypher; a figure, or number; the figures o f 1, 2, 3, 4, & c." In addition, the verb to cipher (Fr. chiffrer) acquired the meanings not only of expressing numbers by means of Hindu-Arabic numerals, but also of calculating. Cotgrave renders "Chifrer” as "to calculate, or examine, an accompt, by cyphering," with the meaning of "accompt" not restricted to the keeping of business ledgers [see OED "account," n. 1, where arithmetic is defined, in 1570, as "the arte of accompts and reckoning"]. As we shall see in my first chapter, "to cipher" acquired the sense of calculating because calculation was performed in writing only in Hindu-Arabic numerals, whereas users of Roman numerals had to employ their fingers and other, more artificial aids. The expansion of cipher did not stop at Hindu-Arabic numerals and their uses, but carried over into writing by means of any "strange, unknowne characters" [Cotgrave, "chifrer"]. Two general types of these might be indicated, the first being ideograms like (supposedly) Egyptian hieroglyphs, and Chinese characters: "Yeat ware not their Letters facioned to ioyne together in sillables like ours, but Ziphers, and shapes of men and beastes" [OED, "cipher," n. 4; example from 1555]. The second type of cipher in this sense applies to cryptography: the OED illustrates it with a reference to the Roman monoalphabetic substitution cipher, where "hee putteth .b. for .a., .c. for .b., and so forth" ["ciphering" 2; from 1606]. Why does cipher acquire these meanings? Although Renaissance cipher alphabets did sometimes consist of Hindu-Arabic numerals, they usually employed Latin letters. Still, the process of substitution generally relied on the order of these letters, whether standard or devised for the occasion, and therefore treated them as if they were numbers. More importantly, then as now, all Renaissance codes where characters or groups of characters stood for letters were broken mathematically, by matching the frequencies of signs in text with the frequencies of letters in the language. This is why cryptoanalysts were, as they still are, mathematicians: Francois Viete, for instance, whom we shall encounter in his capacity as the inventor of the letter sign,3 broke the cipher of Philip II for Henri IV [Kuhn 84-85], Still, this cannot explain why all "strange and unknowne characters" were referred to as "ciphers": if the writing and breaking of alphabetic ciphers depended on numbers, Egyptian hieroglyphs 3 Descartes in his Geometrie refers to them as chiffres, 299. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. allegedly signified not letters or even words, but concepts [see my chapter 8]. Their "shapes of men and beastes" were interpreted not mathematically but as ideograms.4 The primary unifier of the two types of "strange" writing is not the manner in which characters were read, but their shared resistance to reading. Not only cryptographers, but also scholars of Egyptian hieroglyphs stress the concealing nature of their "ciphers." Let us quote Blaise de Vigenere, member of the Platonist-Kabbalist circle at the court of Henri IH and a contributor to the Polyglot Bible, as well as inventor of a poly alphabetic cipher whose substitution deploys an arbitrary remembered key word, and which was therefore virtually unbreakable. Vigenere defines his ciphers as 1'occulte secrete [escriture], qu'on desguise d’infinies sortes, chacun selon sa fantasie, pour ne la rendre intelligible qu'entre soy & ses cons^achans [...]; anciennement les Hebreux, Chaldees, Egyptiens, Ethiopiens, Indiens, ne s'en servoient, que pour voiler les sacresecretes de leur Theologie, & Philosophic [3V].S To write in either ideograms or cryptography is to write "darkly by way of ciphring" [OED "ciphering" 2], with "darkly" employed in its then much more current sense of "obscurely," as in Paul's "through a glass darkly," i.e. "per speculum in aenigmate" [/ Cor. I3.I2].6 What do Hindu-Arabic numerals have to do with difficulty of interpretation, with resistance, so to speak, to decipherment? Why would the word cipher apply to cryptography on that account? One might claim that the usage arose when Hindu-Arabic notation was first introduced, and therefore seemed as abracadabra to the populace. One can imagine this populace (circa 1540), ambulating the countryside in their smelly jerkins, dismissing anything they do not understand with "it's all ciphers to me, Jack!" because the dactylographic Eos of humanism had already made them embarrassed of invoking the Greeks in that capacity, but was herself as yet embarrassed of "the number of Merchants 4 Russians were more prone than Westerners to use Hindu-Arabic numerals as characters in theis codes. since Russian letters had number values [Soboleva, 35]. Seventeenth-century Russian split up its borrowing of the Polish cyfra into tibicjipa, meaning zero, and Ubi4>Hpb, meaning both Hindu-Arabic numerals and cryptography. Today, Qbi(|)pa means Hindu-Arabic numeral, and cryptographic code is designated by its French-derived cognate uiH(i>p. 5 In referring to the "Hebreux," Vigenere is not thinking of the Hebrew language, but o f the (Cabbalistic methods for intepreting it, such as gematria. He continues: "Nam aliud Cabalista profert <k scribit. aliud subintellegit & legit.” Vigenere's Traicte des Chiffres ov Secretes Manieres d'Escrire, which I am quoting, expounds not only his very modem invention, but also ciphers in the general sense of secret writing. Same stress on exclusivity and difficulty for Egyptian hieroglyphs may be found in, e.g., Spafarii 127; it appears as early as Clement of Alexandria [Iversen 45]. 6 Compare with Spenser’s Faery Queene. "being an Allegory, or dark conceit," whose "general intention and meaning" he must expound in the letter to Sir Walter Ralegh, printed in the first edition of the work, in order to avoid "misconstructions" [1], 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. buying, & selling" [Agrippa 170], Fine — although I know no evidence to that effect — but why does cipher jump from zero onto all Hindu-Arabic numerals in the first place? Yes, of course it is because they include zero, and Roman numerals do not. Yet zero, as we shall see in our first chapter, is more than just one numeral among many. Its introduction permits the employment of the principle of place-value, thereby rendering the value of "arithmetical figures" not fixed like that of "arithmetical letters," but, in the words of Robert Recorde, "uncertaine," dependent on "the seate or roume that a fygure standeth in" [Grounde, Bvf]. While people who come to the Hindu-Arabic system from the Roman see zero as the cornerstone of the new dispensation, they also find it extremely perplexing since it does not stand for anything they consider to be a number. Unlike the numerals I through 9, "The Ciphra O [...] of himselfe signifieth not" [Digges 1]. I shall spend many pages relating zero's incapacity to "signify" to how both signification and numbers were conceived. For now, let us note the following. Within the paradigm of signification then current, the cornerstone of Hindu-Arabic notation does not seem to signify anything. And a signifying system whose, to borrow Rotman's term, meta-sign cannot be understood, is rendered incomprehensible as a whole: a house built on a weak foundation cannot stand, etc. So when all Hindu-Arabic numerals assume the moniker of "ciphers," one can taste a bit of terror in that. Terror in general is an inchoate phenomenon, but this particular instance can be interpreted in two ways. One is that ciphers, including zero, do signify in some manner which we do not understand. The other is that they simply don't signify at all, that they are arrows pointing to nothing, and that behind their illusion of sense chums absence. Now, while the second alternative is unarguably awful, the first is none too comforting either. Let us again consider the transfer of cipher to cryptographic writing and other "strange and unknowne characters." A ciphered text appears suspicious if only for the fact that the encoder must have some reason to hide his meaning. Even in the case of commercial or private uses, such texts are duplicitous by nature, for their characters signify something other than what they habitually signify, nor can you know what that something is. You read "boat,” but maybe it really says "sheep"; you see the hieroglyph of a hawk, but it really stands for velocity. Such is the ominous nature of cryptography in general. Add to it the fact that the art provides the common and natural idiom for espionage, treachery, sedition, and the like. Communications between "intelligencers" and their base, between ambassadors and their courts, among members of dissident underground groups, among participants in periodically erupting plots, etc. — all of these were conducted in ciphers, with both external and internal state security requiring codewriters and codebreakers like Thomas 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Phelippes, Walsingham's decipherer of the Babington letters [Kuhn 85]. Ciphering was also associated with magic, whose practice employed not only "strange and unknowne characters," like alchemical signs or John Dee's Enochian language, but also the same alphabetic squares as those of more mundane codewriters. The connection between magic and cryptography can be seen in the activities of Johann Trithemius (1462-1516), the one­ time host and subsequent enemy of Charles de Bovelles (the latter’s writings on God, zero, and nothing appear in my sixth chapter).7 So in any of its applications, the word cipher connotes a menacing instability and a deficiency of the beholder's knowledge. It is, I think, the threatening taste of zero, the sign we do not understand as a sign, that bears the greatest responsibility for the word's transfer to Hindu-Arabic numerals and then to other "Secretes Mcmieres d'Escrire" [Vigenere].3 As we shall see, Digges and all other writers on Hindu-Arabic numerals claim that "The Ciphra O [...] signifieth not" because were it to signify in the manner of all other numerals, its signified would be nothing. And this cannot be, for nothing is no object, and therefore cannot serve as an object of signification. Yet in early modem literature we witness a veritable eruption of nothing: the nothing that, upon its appearance, redoubles still and multiplies, ceaselessly producing itself from itself, as if to equivocate on the "eternal truth" that ex nihilo nihil fit.9 This eruption of nothing accompanies an epistemological anxiety of the highest magnitude, the lingering fear that, as one thinker put it, "nothing is known," for, in the echo of another, "While we look up to heaven, we confound Knowledge with knowledge" [Sanches; Webster, White Devil 5.6.262]. 7 When, to give the most exotic illustration I can think of. the Moscovite noble Artemon Matveev was accused of conjuring evil spirits to help him create drugs in order to harm the Tsar, the investigators were far more interested in the boyar's "ciphered" medical book, stolen and burnt by a suspicious servant (Jew with the improbable name "little Ivan"), than in his argument that another witness, a dwarf, could not have seen him talking to evil spirits because they are invisible. Matveev's instructor in black magic from ostensibly the same "ciphered" book was alleged to be Nicola Milescu Spatar, or Spafarii, a Moldavian whose nose had been cut off for sedition, after which he fled his homeland, translated the Bible into Rumanian and wrote in Russian on the subject of allegories and Egyptian hieroglyphs; he was then, conveniently, leading an embassy to China. We have no reason to doubt Matveev's claim that the "ciphered" book that figured so prominently in the case was a banal catalogue of pharmaceuticals, whose ingredients were indicated by their alchemical signs, with a price-list [Matveev; Russkii Biograf. Slovar', v. 19. "Spafari"]. 8 By an unrelated but poetic coincidence, what can be thought of as an example of ciphered, or at least "dark," writing in the Bible fuses resistance to reading with a message of doom delivered by means of numbers. I am, of course, talking about "mene, mene, tekel, upharsin," inscribed by a disconnected hand on Belshazzar’s wall in Daniel 5. The king summoned all the wisemen of Babylon, "but they could not read the writing, nor make known to the king the interpretation thereof." Daniel deciphered it as "numbered, numbered, weighed, divided," and foretold the end of Belshazzar’s kingdom, which came that very night {mene, teke and upharsin or peres are probably Aramaic renderings of Hebrew weight units). * This sentence cites Donne, "Exstasie" 1.40, Shakespeare, Ham. 5.1.134, Lear 1.1.90, Descartes, Principles o f Philosophy, 1.49, not necessarily in that order. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The word cipher, as it applies to natural languages, can sometimes be used to stand for not only cryptography, but any writing [OED, "cipher," v. 2]. Does this indicate that the signifying problem presented by zero touches upon a doubt regarding the nature not of some, but of all signification? That, in other words, all signs become "unknowne characters"? The pages to come should not be read as claiming that zero creates the doubt, but that it feeds into the doubt, and provides it with one of its best expressions. 3. Essence vs. Appearance Several years ago I had the good luck of being asked to lecture on early modem intellectual culture in Estonia, or rather (luckily for me), to lecture in Estonia on early modem intellectual culture. One of the lectures had to do with the dependence of meaning on syntax, while another tried to relate anatomy to English theater, treating both in light of epistemology. Although repeating what one had said in an earlier work strikes me as being in very bad taste, the earlier work in this case was oral rather than written, and I have not been able to think of a better example of the two methods of defining things that were at war in the said period. So here goes. Imagine an extraterrestrial spy arrives on earth and assumes the shape of a chair. The shape he (she? it?) assumes is both outer and inner: the chair is a chair in every way possible, except that's actually an extraterrestrial spy. We do not know, nor have we any way of knowing, that it's an extraterrestrial spy. For some unearthly reason, suppose we still want to know whether the chair is a chair. What do we do? Look inside it, of course! We take an ax and hack the chair in half. The spy does not give himself away. He looks exactly like a sundered chair. So we keep chopping and chopping, breaking and breaking, until all we have left is a pile of wood chips, some springs, horsehair, cloth. What did we just destroy? Well, since what the chair really is, is a spy, we really destroyed the spy. We never took the ax to the chair; there was never a chair before us at all. But we don't know that. We think the chair is a chair. Why? Well, because it looks like a chair. And now that we've destroyed it, we are ever so much more sure that it was a chair. Why? Because now we've got more to look at! We've quadrupled, decupled, n- tupled its surface area. It's now nearly all appearance. The moral of the story is: a thing defined according to being and a thing defined according to appearance is not necessarily the same thing. And there might not be a way of getting from appearance to being. And all we got to go on is appearances. Then I said that the first definition (spy) may be roughly associated with Platonism, and the second (chair) with such seventeenth-century discourses as physics. For what 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. physics operates with, after all, are dimensions of appearance. It defines objects according to how they show up in its discourse. This is why its discourse, and any definitions one can make with it, is posterior to the objects themselves. The change between the two types of definition, the two paradigms of knowledge, the former relying on what I shall call the prior signified, and the latter on what I shall call the posterior signified, takes place during the early modem period. Let us think about two nearly identical sentences. The first is "the invisible things of [God] from the creation of the world are clearly seen, being understood by the things that are made." It appears in Romans 1.20. The second is "That which is vnsensible by the immediate iudgment of ye sense; may by consequence or effecte be made very sensible." It is from the ballistics researches of Thomas Harriot, and dates from the 1590s. Harriot’s biographer interprets it as "unobservable forces may be determined by observing their effects" [Shirley 252]. If the first was used by Christian Platonists to justify the existence of Ideas, the second comes from one of the earlier applications of mathematics to physics. If the first allegedly tries to understand things "as they really are," the second tries to understand the way they behave with respect to one another. The period in which the first paradigm of knowledge breaks down, and the second has not yet formed, is the period of illegibility. One its loudest voices belongs to the Montpellier surgeon Francisco Sanches (1551-1623). In his elegant and virulent tract Quod Nihil Scitur (That Nothing is Known; 1581), which prefigures both Montaigne's Apologie de Raimond Sebonde and Descartes, Sanches launches an attack on the syllogistic method. Definitions which demonstrate the natures of things are impossible, fumes Sanches, for definitions are words which point to other words and not things: You will say that what you define by the terms "animal, rational, mortal" is a thing (namely Man), not a verbal concept. This I deny; for I have further doubts about the word "animal", the word "rational", etc. You will further define these concepts by higher genera and differentiae, as you call them, until you arrive at the thing's "Being." I will ask the same question about each of these names in turn. Finally, I will ask it concerning the last of them, namely Being; for you do not know what this term signifies. You will say that you will not define this Being, for it has no higher genus to which it belongs. This I do not understand; nor do you. You do not know what Being is; much less do I.10 ‘“English translation in Sanches 175. The Latin, on p. 95-6 [ed. princ. 2] is more vigorous: "Dices definire te rem quae est homo hac definitione, Animal rationale mortale, non verbum. Nego. Dubito enim rursus de verbo animal. & de rationale, & alio. Definies adhuc haec per superiora genera <4 differentias. ut vocas, usque ad Ens. Idem de singulis nominibus quaeram. Tandem de ultimo Ente: nec enim scis quid significet. Non definies, quia non habet superius genus, dices. Non intellego hoc. Nec tu. Nescis quid sit Ens. Minus ego." 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although Sanches's scepticism is particularly directed against Aristotle, buying into it will force us to regard Plato’s system to be as bankrupt as the Stagyrite's. Sanches declares an epistemological crisis, from which he sees no way out. Denying that signs refer to pre-existent things, he still believes the world of such things (discrete and independent of signs) to be the proper object of knowledge.11 The result is an utter divorce between the world of signifiers and the world of their would-be signifieds. Yearning for sure and certain knowledge of the latter, Sanches claims that the former cannot be of any help in bestowing it. Therefore, as his title proclaims, nothing is known. 4. Signification My second and third chapter deal with the classical concept of number ( = plurality), which remains fairly static from the Greeks to the sixteenth century. Focusing on its Platonist12 interpretation, I shall attempt to show the role number played in Plato's ontology. Here I shall heavily rely on Jacob Klein's Greek Mathematical Thought and the Origin o f Algebra. Unlike Klein, however, I shall attempt to examine Plato's treatment of number and of the relationship between objects and Ideas in terms of his treatment of the relationship of signifiers to signifieds. Plato’s interpretation of signifying systems does not approach them as systems, i.e. it disregards the syntax according to which signifiers are concatenated. Instead, it seeks after the identities in signifiers, after that which is context-invariant in them. It breaks the signifiers into groups, and explains the similarities of each group by positing a real, immutable, and discrete signified as prior to this group. As one such signifying system, the world is composed of sensible things pointing to their nonsensual and reified Ideas. Plato's interpretation of the Greek concept of numbers as pluralities is of such importance to this undertaking that, according to Aristotle, late Plato identifies Ideas with numbers. The Platonist model of signification offers no difficulty when we try to explain how Roman numerals work, but when early arithmetics attempt to apply it to Hindu-Arabic numerals, they stumble over zero. Furthermore, Hindu-Arabic notation, whose principle of " 'Knowledge can be knowledge of one thing only, or rather knowledge is only of each individual thing, taken by itself, not of many things at once, just as a single act of seeing relates only to one particular object"; "Vnius enim rei solum scientia esse potest. Imo unius cuiusque rei p er se solum est scientia. nec plurum simul: quemadmodum <4 unius solum cuiusque obiecti visio una." [ed. princ. 14-15; Lat. 104-5, Eng. 190J. 121 use the word "Platonist" in this dissertation when I do not think that a distinction between the thought of Plato, the "Platonism" of immediate followers, and Neoplatonism whether original or Christian is in order. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. place value gives syntax the importance it lacks in Roman numerals, permits calculation to be carried out in writing. Due to their commercial nature, early arithmetics forego Greek attempts to study numbers as separate entities, and instead focus on teaching mathematical operations. In analyzing a 1570 text by John Dee, my fourth chapter shall explore the discrepancies between his very modem mathematical practice and his theory, which still relies on the Platonist conception of numbers and signs. While Dee’s theory and practice generally diverge, I shall argue that in certain places the former already feels the effects of the latter. Dee provides us with a detailed illustration of the fact that the development of new modes of writing does not change concepts automatically — for, if it did, the work of Simon Stevin on the concept of number, which I talk about in my fifth chapter, would have been carried out hundreds of years earlier by Al-Kworezmi or Fibonacci. No: the events that occurred in mathematics towards the close of the sixteenth century, including the emergence of symbolic algebra, were spurred by and participated in what can only be called the early modem semiological revolution. Shortly after Dee's Mathematicall Praeface saw print, mathematics swung towards a more modem understanding of signification. My fifth chapter will examine the reshaping of the concept of number by Simon Stevin (1548-1620). Brushing off the view shared by Dee of numbers as separately existing, discrete entities, Stevin regards them as the effects of notational and operational syntax. Such, at least, is his overall strategy; but in fact it is restricted to positive numbers, numbers that may be construed as expressing magnitudes. Hence, although the novel character of his number concept may be seen in his proclamation that the zero, and not the one, is the origin of numbers, this same proclamation still leaves zero out from membership in their genus. My dissertation never reaches the point where zero becomes a number like any other. 5. Isom orphism s One of the striking aspects of Stevin's Arithmetique (1585) is the way the zero takes over many of the traits formerly assigned to the one. In their respective concepts, the one and the zero are the origin of number, but no number; they are also declared to be the arithmetical counterpart of the geometrical point. The particular character of their being "origin" is symptomatic of how other numbers are conceived, and therefore to the understanding of signification proper to each case. My sixth chapter traces the same replacement in descriptions of divinity. The Neoplatonist identification of God with the One rests on the distinction between the one and 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. other numbers in the classical number concept: as the one is the origin of numbers but no number, so God is the origin of things but no thing. The convenience of the analogy lies in its ability to illustrate the transcendent aspect of God's relationship to things; elaborations on how numbers come from the one also offer a refined understanding of God's immanence. The paradox, however, is that by virtue of becoming transcendent, the one becomes no thing, and therefore in a sense nothing. At the same time, the Platonist concept of signification is thrown into disarray: the signified must be an object; all things in some way signify God; but God is not an object. In pseudo-Dionysius, the problem presented by the transcendent God to signification renders the existence of other purely intelligible entities extremely tenuous, whereas Nicholas of Cusa declares all things to be ultimately unknowable. My sixth chapter will also devote considerable attention to the concepts of nothing. While Plato interprets nothing or nonbeing as merely a way of talking about Otherness, i.e. of being-not-this-thing-but-another, and Augustine regards nonbeing as the effect of a thing's ontological distance from God as Being, and therefore a privation, the transcendentalist systems of Plotinus, pseudo-Dionysius, Eriugena, etc., endow nothing and nonbeing with other meanings entirely. As I already intimated, they interpret God’s being nothing through his being no thing at all; furthermore, he is no thing not because of a defect, but because of an excess (suberabundance) of being. This is why the association of God with zero by the Salem Codex (c. 1200) and by Charles de Bovelles (1479-1553) may be regarded as essentially erroneous, a mistaking of one kind of nothing for another. Bovelles’ argument is drawn in part along the traditional numerical analogy, save that he too substitutes the zero for the one: as God is to things, so zero is to numbers. Apart from this substitution, which precedes Stevin by 75 years, Bovelles also offers the earliest instance I could find of the Hindu-Arabic numerals set out with zero in the first position; early arithmetics, on the other hand, present their sequence as 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Since I regard the association of God with zero as confusion of nothings, my sixth chapter also includes a section on the circle as the symbol of God, which presents, as it were, another motive for making the error. There is of course no better medium than theater for portraying things as devoid of prior signified, devoid of essence. In theater, what you look like and how you behave determines what you are: put on a different costume and you are a different person. All judgments are made on the basis of appearances only, and all signifieds are therefore posterior. My seventh chapter traces the substitution of the one by zero in the concepts of self and thing in Shakespeare's Henry V. The "crooked figure" conceit of the Prologue emphasizes not only zero’s peculiar character as a sign, but Hindu-Arabic notational syntax 12 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. in general, with its "sinister," i.e. leftward action. Attempting to arithmetize Patricia Parker's notion of the Shakespearean preposterous, I argue that the "crooked figure" conceit foreshadows the play’s peculiar representations of sequence, such as the genealogical sequence, in which the descendants "create" their ancestors. I go on to claim that the same reversal of priority occurs in the relationship between signified and signifier, essence and appearance, meaning and word: the first of the pair is always the effect of the second in its syntactical arrangement. My basic argument is that, for Shakespeare, any signifier is a zero: nothing in itself, something in operation. My final chapter brings together virtually all of these strands by analyzing the philosophical writings of Hryhorii Skovoroda (1722-94). the extreme particularity of whose Platonism I ascribe to the influence of seventeenth-century natural philosophy, notably that of Descartes. Like Stevin and Descartes, Skovoroda bases his system on his refusal to address certain ontological questions. Merging the hermetic concept of thing as rhetorical figure with the Cartesian concept of thing as figure subject to mathematical handling, he argues that it does not matter what a figure signifies as long as it signifies. Skovoroda's God is not the end-result of signification but its very fact, common to all things in equal measure; he is therefore not reified like Plato’s Being, but present in all things as the fact of their existence. Because Skovoroda's concept of figure implies the new number concept, his God, as the fact of existence void of any particulars, is symbolized by zero as the origin of numbers but no number. Hence the proper way to read the Scriptures, according to Skovoroda, consists of seeking out circular objects and placing them in pairs. Each member of the pair can be alternately nothing or all, depending on how we look at the other. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 NOTATIONS X B a j i a H3 o o p e T a T e jiH M , n o n y M a B U iH M o m c j i k h x h C M eu iH b ix n p H c n o c o o jie H H J tx , O uiHiiHHKax m ix caxapa, o MyHUHiryKax juui namtpoc, XBajia TOMy, k to npezmoacHJi nenaxH craBHTb b yuocTOBepeHHax, K t o k n aH H H K y n p n a e j i a j i K p b im e H K y h h o c . Inter narrantes Chilfram iuvat esse figuras.1 0. Introduction This chapter will contrast the Roman and early Greek system for writing numerals with the European variant of the Hindu-Arabic system. Because Roman and Greek notations are not suited for performing calculations, we shall also look at way calculation was performed in Europe before Hindu-Arabic numerals acquired hegemony. Notations are like all systems: because they work according to different principles, they put forth some things, and obscure others. Roman and Greek notations cannot articulate anything like the zero, whereas in the Hindu-Arabic system the zero becomes not only possible but indispensable. How we define what number is depends in part on the notation we use to write numbers. Nonetheless, a new notation does not automatically bring in new concepts. The last part of this chapter shall focus on medieval and Renaissance attempts to explain the way Hindu-Arabic numerals signify with the help of concepts drawn from the Roman system. We shall look at how and why their attempt fails. 1. Roman and Greek Notations 1.1 The Imitative Principle The general principle by which the Roman and the so-called Greek Herodian notations signify numbers may be thought of as imitative. In imitative systems, notation reproduces counting. To write the number three, for instance, we put down three signs for one; to write the number thirty, we put down thirty signs for one, or six signs for five, or three signs for ten, and so forth. Our writing hand re-enacts the counting process either in a 1Nikolai Oleinikov, "Khvala Izobretateliam." Poety gruppy OBERIU, 396; Alain de Lille in Du Cange, "cifrae." 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one-for-one (thirty as thirty ones) or in an abbreviated, one-for-group correspondence (thirty as three tens). According to Karl Menninger’s Number Words and Number Symbols, notations of this type ultimately descend from tally-marks, such as those of cattle-herds [240-6]. Let us imagine ourselves counting heads of cattle: we make them file past us, and make a notch on a stick corresponding to every one cow or bull. Sooner or later, we start breaking these notches down into groups, for instance by making every fifth stroke a crossing-out of the four previous. Such tally-marks lie at the origin of the ancient Egyptian, Etruscan, Greek Herodian, and Roman notations. 1.1 Greek Herodian and Alphabetic Numerals The earliest Greek notation is called Herodian after a Byzantine grammarian who lived circa 200 AD. It was in official use in Attica until the first century BC, when the alphabetic system, described below, was adopted; however, alphabetic notation started competing with Herodian already in the classical period. Herodian notation functioned very much like the Roman, and we shall omit a close description of its principles for that reason. Its signs for 1,10, 50, 100, 500, 1000, 5000, and 10,000 were combined in the Roman fashion. The only major difference between the Roman and Herodian notations is that, except for I (one), the Greek signs were either abbreviations of number words - A for ten (deka), H for hundred (hekaton) —or ligatures thereof, the sign for five combined with the sign for ten being fifty, for instance [for details, see Menninger 268-70; Cajori, Notations 22, 25]. Roman notation, on the other hand, abbreviates tally-marks more directly. The alphabetic system that replaced the Herodian assigns numerical value to letters based on their place in the alphabetic sequence: a = 1, (3 = 2, y = 3, etc., adding three more letters to the 24 already present in the Greek alphabet in order to provide the user with 27 numerals: digits 1-9, tens 10-90, and hundreds 100-900. To signify thousands, a short downward stroke precedes the digit: ,a; other conventions denote several of the higher ranks. To prevent confusion between letters intended as numerals and simply letters, a horizontal line is drawn over the former: a. Alphabetic notation does not descend from tally-marks and it is much more removed from the imitative principle than systems originating in tally-marks are. Nonetheless, it does exhibit some imitative traits. In it, juxtaposition (writing of several numerals side by side to make one number) is the same as addition: the number 1453 is written as ,aUvy, or 1000+400+50+3, whereas 808 is (off, or 800+8. Because the numeral 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. already contains information about the rank (or power of 10) of the number it signifies - for instance 5 is €, 50 is u, 500 is <p —each numeral has one and only one value (as opposed to three values of 5 in 5, 50, and 500). Since each numeral contains information about rank, there is no need for zero in the writing of integers: cUff = 808. For the same reason, the ordering of numerals is, strictly speaking, void of semantic function, serving only to reproduce the order of the number words. When the order of the number words is reversed, the order of numerals often repeats it, as for example in the standard Russo- Slavic ai, B i, ri for 11, 12, 13, etc.2 Alphabetic notation has several drawbacks when we compare it to the Hindu- Arabic. First of all, there are three times as many numerals to remember. Second, one needs to keep coming up with ways to signify numbers of higher ranks. In positional notation such inventiveness is not necessary —any number, no matter how great, can be expressed by a combination of 10 numerals. Third, the alphabetic system is not suited for calculation (compare $ • I = v with 4 10 = 40; S • p = u with 4 • 100 = 400). Thus, even if Archimedes could make grand calculations on a shred of papyrus, an ostrakon, a sandbox or even his oiled thigh [see Plutarch's "Life of Marcellus," 19], lesser mortals required the aid of counters. Alphabetic notations include the Greek, the Phoenician, the Hebrew, the Gothic and the Slavic.3 1.2 Roman Notation The Roman numerals we are familiar with are actually not Roman numerals at all, but their early modem "edition" so to speak, certain aspects of which have suffered the influence of positional notation. In their notation, the Romans counted by quinary groupings: I, II, IH, IHI digits 1-4 v, vi, vn, vm, vim digits 5-9 X, XX, XXX, XXXX tens 10-40 L, LX, LXX, LXXX, LXXXX tens 50-90 C, CC, CCC, CCCC hundreds 100-400 I) or D, DC, DCC, DCCC, DCCCC hundreds 500-900 (I), (IHI), (1)0)0), 0)0 )0 )0 3 , thousands 1000-4000 I)), I)) 0), 13)0)0), DHIHDO), 13)0)0)0)0) thousands 5000-9000 (0)) ten thousand : In Russian numbers 11-19, the digit is pronounced before the -teen-, eleven is odinnadtsat', or "one-on- ten”, twelve is dvenadtsat', and so on. 3 For alphabetic notations, see Menninger 260-78; Cajori 18-22; for Slavic, see R. A. Simonov, ch. 1 and 3. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The symbol M for thousand appeared only under Augustus [Cajori, Notations 30], became common in the Middle Ages [Meninger 2 8 1], and standard only in the seventeenth century. The fact that 100 in Latin is centum served to fix C as sign for 100 at the expense of competitor signs [ibid. 243]. Historically, Roman numerals are Etruscan number signs as interpreted by the Latin alphabet. Thus, the symbol D for 500 is really an alphabetic assimilation of I), or half of the symbol for 1000, Q), in the same way as 5,000 is I)), half of the symbol for 10,000, (CD), and 50,000 —I])); the latter two numerals, being less common, never became letters. Signifying half of the first number of a rank by halving its sign must be recognized as an application of the imitative principle. Numerals that are used with an even greater frequency that those for 500, i.e. the smaller numerals I, V, X, L and C, became assimilated into the alphabet earlier. The same application of the imitative principle as we see in I) being half of CD, is present in the sign V as the upper half of the sign X, and probably even in L as an alphabetization of half of an ancestor of C [Cajori, Notations 30, Menninger 243]. The sign X appears very archaic. If the sign I clearly derives from tally-marks, as the alphabetic adaptation of a single notch or scratch, X may come from the crossing of out nine such scratches. Boethius (480-524) in his De Institutione Arithmetica (generally known, and henceforth referred to, as his Arithmetic) does not derive X, as the sign for ten, from the crossing-out of nine tally-marks, nor does he derive V from X. These signs, to him, are the results of custom; wishing to signify numbers ''naturally," he inadvertently falls back on the forgotten origins of the system: when we wish to demonstrate five, we will make five strokes, and lay them out this way: HHI. And when we wish to make seven, likewise, and when ten, no less, because it is very natural that whatever the number, so many elements it contains in itself and so many designations of unity are assigned to it [2:4, 129].4 The imitative nature of Roman notation may be seen in other of its features as well. In alphabetic notation juxtaposition as the same as addition, i.e. the values of numerals writing together to make up the number add up to the number. It is also the rule in Roman notation, although less so in the version we use today (see below). Thus, (D.U.CXXXVTI, * The Arithmetic of Boethius is an adaptation - translation of the Arithmetic of Nicomachus of Gerasa (1st - 2nd c. AD), who writes that "the natural, unartificial, and therefore simplest indication of numbers would be the setting forth one besides the other of the units contained in each," and then proceeds to build his numbers from Greek ones, i.e. alphas. 17 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. or 1637, equals CD + D + C + X + X + X + V + I + I, or 1000 + 500 + 100 + 10 + 10 + 10 + 5 + 1 + 1. In fact, the Latin word additio means both juxtaposition and addition, the latter meaning derived from the former. Even early modem arithmetic textbooks understand addition as "bringing together 2 sums or more into one" [Baker 9], the word "sum” in Recorde and Baker meaning simply "number”. The zero in Roman notation is not only unnecessary but counter-intuitive. A cowherd counts cows only where there are cows to be counted, and where there are no cows, there is no count. Nor is there need for zero as placeholder; where we write 808, the Romans wrote DCCCVIII. To a lesser extent than in the alphabetic notation, the Roman suffers from the need to invent more and more signs to designate numbers of higher rank. Even millions the Romans spoke of and wrote as tens times hundreds of thousands, because, as Pliny explains, "the ancients had no number higher than a hundred thousand" [non est apud antiquos numerus ultra centum milia; in Menninger 43-44]. The reason we think of Roman numerals as an entirely letter-based and self-sufficient system is that we never employ them to express numbers of more than four ranks. Although the Roman system is based on the principle of addition, it is not suited for complicated calculations - Rotman, in a fine example of academic irony, describes the process as "byzantine" [10]. Both in the classical and the medieval period the operations of arithmetic were carried out with counters; numerals only recorded the results. As with alphabetic numerals, since classical Roman numerals indicate the number's rank and treat juxtaposition as addition, the right-ward ordering of numerals within a number is due solely to the higher ranks coming first in the number words. Hence, because Latin words for 11-17 place the integer before the ten - undecim, duodecim, tredecim, etc. - there are documented cases of m X meaning 13 and VIX - 16. Conversely, the now- standard Roman forms like IV, IX, XL, XC for 4, 9, 40 and 90 do not appear until the Renaissance [Cajori, Notations 31, but see 32]. These latter operate by a different principle, the principle of subtraction, where "if a letter is placed before one of a greater value, its value is to be subtracted from that of the greater" [Cajori, Notations 31]. The principle of subtraction in written numerals has nothing to do with the morphological ordering of number words (here quattuor, novem, quadraginta, nonag into). Since this principle gives the order of some numerals a significance independent of language, all the while violating the principle of juxtaposition as addition, it seems safe to view IV, EX, XL and XC as the influence of by-then-dominant Hindu-Arabic numerals. My impression is that even in the early seventeenth century these forms are far less common than mi, VLLLI, XXXX and LXXXX. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The pressure exerted by positional notation upon Roman numerals in the late Middle Ages and the Renaissance is most evident in the many examples of non-standard forms where Roman numerals are partially arranged according to positional syntax. Thus, we encounter "vij. c. and XL" instead of DCCXL for 7 4 0 ,I.V^V instead of (I).I).V for 1505, and so on [Menninger 284-285]. We also encounter numbers written with numerals from both notations, as on a Dirk Bouts altar: MCCCC4XVII [Menninger 287, see also Smith and Karpinski 122-123]. These non-standard forms offer an extremely valuable insight on the overlapping of the two systems in the popular mind in the transitional period; in our discussion of early textbooks of positional notation we will also show an overlapping on the theoretical level. 2. C ounter-C asting 2.1 H istorical Background I briefly mentioned the inadequacy of Roman and alphabetic numerals for the purposes of calculation. In Europe, calculations were generally carried out with the help of the counting (or reckoning) board (or table). The earliest known counting board is the so- called Salamis tablet from ancient Greece; the Romans also knew the abacus, which is still in use in China and Russia. The two instruments work according to the same basic principles, save that the counters on the board are not attached, whereas on the abacus they are strung together in rows. Menninger cites a charming description of Charles the Bold entering his Chamber of Finance: Luy mesmes siet au bureau a ung bout, jecte et calcule comme les autres, et n'y a differance en eulx en iciluy exercise sinon que le due jecte en jectes d'or et les autres de jectes d'argent.5 Even when positional notation became standard, many North European merchants continued to rely on counting boards, as we can see from numerous German and French sixteenth-century woodcuts.6 This device left many traces in our language. The very world "calculate" (1. Lat. calculare) evolved from the Latin calculus or pebble, which the Romans used as counters. In the passage about Charles the Bold, counters are called jectes, in modem French, they : "He himself would sit at one end of the table and would move counters and calculate like the others; and there was no difference between their reckoning and his, except that the duke worked with golden counters and the others with silver ones” [Olivier de la Marche in Menninger 332]. 6 See Menninger 337, 340; Peter Bruegel the Elder’s Temperance in Crosby 4; Swetz 32: the Fugger businesses were still using counting boards in 1592. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are jetons. Both forms come from jeter as parallel to the English "to cast" (the Romans said calculos ponere), which the American Heritage Dictionary still lists as a synonym for compute. Iago contemptuously refers to Cassio as "counter-caster" [1.1.32], and the Clown in The Winter’s Tale wants counters to solve a problem typical for textbooks of commercial arithmetic [4.3.32-35]. Counter-casting has a rather complicated history, for which the reader should consult Menninger. Gillian R. Evans gives a bibliography of published eleventh and early- twelfth-century treatises on the art [n. 6]. The chapter "Accomptynge by counters" in Robert Recorde’s The Grounde ofArtes (first ed. 1542) gives all sorts of sixteenth-century operational details. Here I shall outline only the basic principles. 2.2 Calculating with Counting Boards The surface of the board bears several parallel lines. The lines indicate ranks (powers of 10) or currency denominations; for simplicity's sake, let us for now ignore the intricacies of currency. To represent a number such as 1234 upon such board, counters are placed in the following manner: four on the lowest line, three on the next, two on the one after that, and one upon the last. --------- O----------- thousands OO---------- hundreds OOO tens OOOO ones As with the Russian abacus, a line on a counting board often holds only four counters. Five is designated by placing a counter above the line, so that 17, for instance, is represented thus: --------- O----------- tens O OO---------- ones To add 1234 and, let us say, 73, we represent both numbers in the same place. Should any of our lines exceed the permitted number of counters (four upon the line in question and one over it), we take away its counters and place one more counter upon the next line: O------------------------------- O- thousands OO----------------------------- OOO---- hundreds —OOOOOOOOOO— =>--------------------------------------------- tens - O OOOQOOO O O -........— ones 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The process is simpler than calculating with Hindu-Arabic numerals, because we don't have to remember what separate digits add up to. Hence there is no calculating involved, only arranging and counting. Subtraction on a counting board is equally elementary, and it is also possible to multiply and divide, as the many sixteenth-century arithmetics that have sections on the art instruct us [see e.g., Recorde's section on counter-casting, also reprinted in Steele]. 2.3 Counter-Casting and Positional Notation Without a doubt, my reader has already noticed the similarity between positional notation and how numbers are represented on counting boards. Lines indicate rank, thereby fulfilling the function of place. As a counter changes lines, it changes in value, in the same way as 2 alone signifies two, but twenty when followed by one other digit.7 In both Roman and alphabetic notation, as we remember, rank is demarcated by the numerals themselves. Naturally, in that two is represented by two counters and not a single symbol, counting boards retain some imitative tendency. As may be seen in the above depiction of 1307 as the sum of 1234 and 73, an empty line indicates the absence of digits of its particular rank. It thus carries out the place- holding function of our zero. There are, however, two vital differences. Counter-casting is not a system of notation in the full sense of the term: it was never used to represent numbers for any aim other than calculating. The results were inevitably translated into another notation. It is physically possible, of course, for someone to write 1307 not as CH.CCCVn but in circles and lines. To do so, however, one must make quite an intellectual leap, for counters in counter-casting must be mobile, and calculation is conceived as the writing and rewriting of numbers in the same location. The conceptual principles of counting-boards exclude recording. As far as the empty line is concerned, it is a zero only passively, by default. It indicates the absence of digits of its rank by the absence of counters. The presence of digits, on the other hand, is represented actively, through the presence of the requisite number of counters. Both instances are imitative, that is they demonstrate absence by a real 7 A curious metaphor based on counters: the Greek historian Polybius (second century B. C.) writes that "The courtiers who surround kings are exactly like counters on the lines of a counting board, for, depending on the will of the reckoner, they may be valued either at no more than a mere chalkos, or else at a whole talent!" The chalkos and the talent are, respectively, the lowest and the highest currency denominations. Polybius here works off the fact that a counter's value depends on its place; no such metaphor is possible in either the Roman or alphabetic notation [Polybius in Menninger 300; see 366 for other examples]. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. absence, and presence of number by the real presence of units adding up to that number. However, since there is no sign that like our zero actively stands for absence, signification by the absence and presence of counters differ in nature. Signification by absence is, strictly speaking, only absence of signification, that is to say, not signification at all (see discussion of zero in early arithmetics below). Let us return to the number 1307 represented by counters as --------- O----------- thousands OOO--------- hundreds ----------------------- tens O OO----------- ones. We may try to record it not as (D.CCC.VII, but by listing the number of counters on each line in the descending order of ranks. Since Roman notation has no equivalent to the empty line, we would get I.III.VII, i.e. either 137, or 1037, or 1307, or 1370, etc. It might seem natural to represent the empty line by a blank space, as in: I.1H..VII, but this is hard to read and would never survive the copyist.8 The representation of number by counters, representation that is essentially positional, can be recorded only in notation that has a graphic counterpart of an empty line, i.e. a sign like zero. There is, however, one all-important difference between the zero sign and the empty line or blank space. The very fact that the zero is written just like all the other numerals makes it an active signifier - a sign of the same nature as other signs, and not an absence of signs that signifies by default. Yet representation of absence by presence is clearly a trespass upon imitation; according to the latter, signs can signify only existent entities, which absence is not. With historical hindsight, the counting board appears as a transitional stage between the Roman and positional notations. The invention of the zero sign - and also the re- figuring of numerals 1-9, for the strokes of Roman digits tend to blend together, even if, as above, ranks are separated by periods - is all that is necessary to convert counter-casting into a system of writing numbers far more economical and efficient than the Roman, a system that could represent any number no matter how large, and in which one would be able to calculate. Such a system was introduced into Europe in the thirteenth century. Why then did it take positional notation over 300 years to become more popular than Roman 8 As early as the twelfth century, one O'Creat proposed to employ nine Roman numerals with t as a sign for zero according to positional principles. Not surprisingly, the proposal was in the form of a letter to Adelard of Bath, a writer on counter-casting [Smith and Karpinski 120]. For attempts to put the Arabic gobar numerals 1-9 on counters, see Smith and Karpinskii 110-8. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. numerals? Moreover, why, when positional notation was already in universal use, did it need another 250 years to fully obliterate counter-casting? Several answers have been proposed to these questions, especially the former [see Swetz 182; Menninger 400, 422; Smith and Karpinski 133, 136-137]. To me, it seems that some operations are simply easier in counter-casting, which fact I hope I have demonstrated above. In my example, I added only two numbers; let my reader, however, imagine him or herself an accountant having to tally up a hundred entries of varying amounts - here it is much easier to cast and move counters than to add in columns! In fact, in Russia positional notation never did replace the abacus, and the instrument remains in use in many stores. The other reason why counter-casting survived for so long is currency standards. Lines on counting boards usually stood not for powers o f ten, but for various coin denominations. Our money is decimal, but this, of course, is a fairly recent development. The English pound, for example, at one point equaled 20 shillings, a shilling equaled 12 pence, a penny 4 farthings. Each of these denominations could be represented by a counting table line; the farthing line held no more than three counters, the pence - no more than 11, the shilling - no more than 19; decimal ranking started with pounds. To prevent confusion, the medieval English treasury used checkered counting tables, whence our words "Exchequer" and "cheque/check" [Menninger 347; see illustration 348], The rules for addition were as above: if I had to add a counter to a line that was already full, I took off all its counters and placed a counter onto the next line. This is incomparably more efficient than writing the process out in positional notation. 3. Hindu-Arabic or Positional Notation 3.1 The Im itative Principle vs. the Symbolic The notation we call positional, place-value, Arabic, Hindu or Hindu-Arabic is so familiar to us that we never think about its workings. We shall do so now, contrasting the imitative principle present in the Roman notation (and to a lesser extent in the Greek alphabetic) with what we shall call the symbolic principle of Hindu-Arabic numerals. As we have seen, the primary characteristic of Roman notation is that the writing of numerals physically reproduces the counting process. To write three, we thrice repeat the sign for one: HI. In Hindu-Arabic notation, on the other hand, the form of the numeral is fully conventional and has nothing to do with counting. The symbol for three is 3. It is not composed of three units and has no formal relation to the shapes of 1, 2 or 4. It might as well look like a dragon or the letter P. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Each Roman numeral corresponds to one and only one number. Each numeral carries the information concerning rank. In Hindu-Arabic notation, each numeral can stand for an infinity of numbers, since its value is determined by place. Place provides the numeral with its rank. The 2 in 20 stands for twenty, whereas in 200 it stands for two hundred. Depending on how many places lie to its right, the 2 can stand for 2 of any rank. In imitative systems, juxtaposing numerals within one number is the same as adding them to make that number; in positional notation it is not. MDCVII = M + D + C + V + I + I, whereas 1607 ^ 1 + 6 + 0 + 7. While in imitative systems number is felt to be the sum of the values of numerals, the symbolism of positional notation lets a group of numerals signify a number only in their totality, as a whole. In other words, the syntax of positional notation is extremely pronounced. What number a group of numerals signifies depends on their order. Positional notation needs a place-holding sign to signify ranks which are not filled by units. In 805 the rank of tens is, so to speak, empty, yet if we wrote the number omitting the sign for that absence, we would write 85, not 805. The origin of zero therefore lies in the syntax of Hindu-Arabic notation.9 Finally, Hindu-Arabic numerals are easy to use in calculations, unlike the Roman or the Greek alphabetic. As we shall argue in our chapter on Simon Stevin, positional notation conceives of calculating as writing. The rules for calculation may themselves be thought of as syntactic or morphological rules: in the same way as juxtaposing c and h gives us ch, adding 7 and 2 gives us 9. Thus symbolic, Hindu-Arabic notation signifies number in a manner totally different from the Roman, the Greek Herodian or the Greek alphabetic. In subsequent chapters we shall see that symbolic and imitative significations in fact result in two totally different conceptions of number. 3.2 H istorical Background Hindu-Arabic notation was invented in India; according to Smith and Karpinski, the earliest surviving inscription to contain these numerals, including zero, dates from 876 AD [52]. By then, the system had already been adopted by the Arabs [92], Its graphic forms underwent several transformations. The European forms descend from the now-obsolete West Arabic gobar, or "dust", numerals, which themselves evolved from the Hindu in two stages: the first without, and the second with a symbol for zero [Menninger 415-417, also 9 For zero among the Maya, see Menninger 404-5. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Smith and Karpinski 65-66, Cajori, Notations 50, 53-54]. The numerals standard in Arabic countries today have a very different appearance. Medieval Europe generally called the art of calculating in positional notation by some version of algorism (e.g., Middle English augrym). The word is a barbarization of Al-Kwarizmi, the name of the Bagdad mathematician whose book on the Hindu numerals with zero (c. 820) was several times translated into Latin in the 1100s [Karpinski 53, Menninger 411], Italians, however, called such calculation by the term abbaco or abaco: the modem student is apt to confuse this with the abacus and its proponents, called abacists, as opposed to the proponents of the new numeration, i.e. of the abaco, called algorists. Full-fledged positional notation appears in the West in the thirteenth century. For the standpoint of mathematics, the most important text on the matter was the Liber abaci (1202, 1228) of Leonardo of Pisa, also known as Fibonacci [see e.g. Smith and Karpinski, ch. 8]. However, Fibonacci's sophisticated treatise proved commercially unviable. A different fate awaited the later Algorismus Vulgaris of John of Hollywood or Sacrobosco (c. 1250) and the slightly older Carmen de Algorismo by Alexander de Villa Dei. Both were frequently copied, commented on and used in instruction all the way up to the Renaissance; their translations are the only extensive arithmetics we have from fifteenth- century England. Who needed Hindu-Arabic numerals? The teaching of arithmetic in medieval universities focussed on Neoplatonic ideas of number as passed down by Boethius; to this pursuit practical calculation is unimportant. Scholarly literature is unanimous in associating the appearance and spread of the new numerals with expansion in trade. This is why they caught on in Italy much earlier than elsewhere. As early as 1338, Florence had six calculating "shops" (botteghe d'abbaco), which doubled as schools of arithmetic [Swetz 17]. Why did Italian merchants embrace what was known as the figure mercantesche [Swetz 328] and not continue to rely on counting board and Roman numerals? As I understand it, there are two reasons, the mathematical and the quotidian. The positional system is more efficient for complex operations. Since, unlike the Roman, it can be used without the counting board, it fulfulls both the function of recording numbers and of calculating with them. The latter is an enormous boon for merchants, for counting boards are not portable like the Russian abacus, whereas positional notation could be employed anywhere and any time: in the marketplace, on the road, and so on.10 10 One might argue that calculating with the new numerals requires a lot of scrap paper, which was not the cheapest of commodities [see Smith and Karpinski 136-137]; however, calculations could be carried out with chalk [see Menninger fig. 253] or even, as with gobar figures, on sand. It should be noted that some 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These are not immediately obvious advantages. But aside from counter-casting, the superiority of Arabic numerals to the Roman is clear: the Arabic system is more economical both in the quantity of signs in its "alphabet" (0-9), as well as in the quantity of numerals in its numbers (1637 vs. MDCXXXVTI). In addition, new signs do not have to be invented to signify larger ranks: "any number can be written with these nine figures and the sign O, which the Arabs call zephirum" Initially even in Italy the new notation had little legitimacy. Laws passed in Florence (1299) and elsewhere against the keeping of records in "abaco m odemo" in favor of words or Roman numerals [see Swetz 182-183, Menninger 426-427]. Yet, in the 1400s, positional notation gained the upper hand: for instance, again in Florence, Medici account books gradually switched to Hindu-Arabic numerals [Swetz 182], while the supreme court of Genova ruled against the argument that accounts acceptable as evidence must be in words [Murray 151]. In the fifteenth century, positional notation also started making significant advances in the rest of Europe, its spread congruent to the growth of commercial activity.11 Its dissemination - and the standardization of its graphic forms - was greatly aided by the printing press. The first Italian printed arithmetic dates from 1478, German 1482, French and Spanish 1512, Portuguese 1519, English 1537. As in Italy, the audience for these textbooks is clearly businessmen, and the word problems focus on buying and selling. The spread of the numerals met with a good deal of resistance. Woodcuts in German arithmetics of 1508 and 1529 show the users of counting boards (abacists) and of positional notation (algorists) in competition [see Swetz 29-33 and Menninger 431-438]. Zero was judged to be of particular difficulty, and one fifteenth-century Burgundian writer billed it as "une chijfre dormant umbre et encombre" [George Chastellain in Jordan 170, Menninger 422]. In northern Europe, Hindu-Arabic notation becomes hegemonic in the first half of the sixteenth century. sixteenth-century arithmetic texts, Robert Recorde's The Grounde ofArtes among them, teach the casting of portable counters - counters without tables. 11 "Cum his itaque figuris et cum hoc signo O. quod arabice zephirum appellatur, scrititur quilibet numerus” [Fibonacci in Jordan 173, see also 159], 12 If Florence forbade it in 1299, Frankfurt forbade it in 1494 [Menninger 427]. The prohibition did not last long. In German popular almanacs, "Calendars of 1457-1496 have generally the Roman numerals, while Kobel's calendar of 1518 gives the Arabic forms as subordinate to the Roman” [Smith and Karpinski 133]. See Cajori 50 for earliest North European coins dated with Hindu-Arabic numerals: Swiss 1424, Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Positional Notation in Fifteenth-Century England Two of the English fifteenth-century algorist textbooks to have survived in manuscript are The Crafte o f Nombringe and The Art o f Nombryng [both reprinted in Steele]. The former is a commentary on the Carmen de Algorismo by Alexander de Villa Dei, the latter - a translation of the Algorismus of Sacrobosco. Their originals, as I mentioned above, date from the 1200s. By the time they were rendered into English, they had already been surpassed by arithmetics elsewhere. This is not to say that the history of positional notation in England starts with these two texts: the first mention of the "numbers of augrim" cited in the OED dates from 1230. Notation or, as these texts call it, numeration is regarded as the first operation of arithmetic.13 Sacrobosco's Art defines it thus: "numeracioun is of euery numbre by competent figures an artficialle representacioun" [34]. The Crafte introduces the "teen signys of Inde” as follows: ye must vndirstonde that in this craft bin vsid teen figurys, as here bene writen for ensamul, 0 9 8 7 6 5 4 3 2 1 [...] the first signifiyth one, the seconde signifyeth tweyne, the thryd signifyith thre, & the fourthe signifyith 4 [szc/]. And so forthe towarde the lyft syde of the tabul or of the boke that the figures bene written in, til that thou come to the last figure, that is called a cifre [3-4]. The order in which the figures are presented in both manuscripts is the same: ”0.9.8.7.6.5.4.3.2.1.” This is quite striking. Sacrobosco and Villa Dei, we remember, are among the first writers introducing Hindu-Arabic numbers into Europe. This is why they still list numerals according to the Arabic fashion of writing right to left. When done within an Occidental text, the ordering of the numerals in the number sequence contradicts the ordering of the surrounding words, which are of course written left to right. Arabic habits persist in the texts' directions on how to write numbers. Arabs write numbers right to left in the ascending order of rank, i.e. with the lower ranks first. Our numbers look the same way, except that since we write everything left to right, the order is a descending one, i.e. we set down the higher ranks first. Both manuscripts, however, write numbers as the Arabs do, commencing with lower ranks: when thou schalt rede a nombur of figure, thou schalt begyne at the last figure in the lyft side, & rede so forth to the right side [...] But when thou schalle write, thou schalt be-gynne to write at the right side [4-5; commentary to Villa Dei]. 13This is the case for early modem arithmetics as well. In fact, notation stops being classed among operations only in the early 1800s [Karpinski 100], 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We writene in this art to the lift side-warde, as arabiene writene, that weren fynders of this science, othere for this resoun, that for to kepe a custumable ordre in redyng, Sette we alle-wey the more nombre before [35; Sacrobosco].14 The order of reading is therefore the reverse of the order of writing ("thou schal write bakeward” [5]): I write I first, 3 to the left of it, and then 4 to the left of 3, and pronounce the result as four hundred and thirty one. Before reading left to right, I count the places occupied by the numerals right to left, in order to assign proper values to them.15 The principle of positional notation is explained in the following manner: Euerry of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele [...] If it stonde in the secunde place of the rewle, he betokens tene tymes hym selfe [...] ffor euery figure, & he stonde aftur a-nother toward the lyft side, he schal betokene ten tymes as mich more as he schul betoken & he stode in the place there that the figure a-fore hym stondes [4; commentary to Villa Dei 4]. euery figure sette in the first place signifyethe his digit; In the seconde place . 10. tymes his digit; In the .3. place an hundrede so moche; in the .4. place a thousande so moche [35; Sacrobosco]. As we have seen, a system where a numeral's value depends on place requires a symbol indicating the absence of digits of a particular rank, otherwise numbers like 101, 1307, etc., will collapse on themselves, becoming 11, 137, etc.. This symbol is clepede theta, or a circle, other a cifre, other a figure of nought for nought it signyfiethe. Nathelesse she holdyng that place givethe others for to signyfie [34].16 Here Sacrobosco describes what we now call the place-holding function of zero. The Crafte s commentary to Ville Dieu's "Nil cifra significat sed dat signare sequenti” is, as usual, more detailed and primitive (by "sequens" Ville Dieu means the numeral of the next highest rank): A cifre tokens noght, bot he makes the figure to betoken that comes aftur hym more than he schuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure by-fore hym he schuld token bot one, for than 14 "Sinistrorsum autem scribimus in hac arte more Arabum hujus scientiae inventorum, vel hac ratione ut in legendo, consuetum ordine observantes numerum majorem proponamus" [Sacrobosco in Halliwell 5]. 13 "Questio. In quych syde sittes the first figure? Solucio, forsothe loke quich figure is first in the right side of the boke or of the tabul, & that same is the first figure, for thou schal write bakeward, as here, 3.2.6.4.1.2.5. The figure of 5 was first write, & he is the first for he sittes on the right side [...] The last figure schal token more than alle the other afore, thought there were a hundryth thousant figures afore" [4-5; commentator to Villa Dei]. 16"Decima ftgura dicitur theta, vel circulus, vel cifra, vel figura nihili quia nihil significat, sed locum tenens dataliis significare" [Sacrobosco in Halliwell 3], 28 Reproduced with permission of the copyright owner. Further r e p r o d u c tio n prohibited without permission. he schuld stonde in the first place. And the cifre tokens nothyng hym selfe. for al the nombur of the ylke too figures is bot ten [5]. When these manuscripts say that the zero, unlike numerals 1-9, does not itself signify or "token," they are reiterating a common formula. Here is another English example, dating from 1400: "Although a sipher in augrim have no might in signification of it selve, yet it yeveth power in signification to other" [Thomas Usk in OED, "cipher," n.].17 To differentiate zero from numerals 1-9 by saying that "a cifre is no figure significatyf ’ [5], is to understand signification exclusively in terms of pointing to positively existent entities. Numerals signify a number only when there can exist entities - let's say, marbles - that, if counted, will add up to this number. A "figure significatyf," therefore, always refers to the potential presence of some number of things. But zero, pointing to no units at all, cannot be conceived in such a way. At this point, Sacrobosco, Villa Dei and their translators encounter a paradox: while, unlike the empty space on the counting board, the zero is a sign like the others for it exists positively and communicates information, it is not a sign like the others since it does not communicate information about anything that exists positively. This is why the sequence of numerals as given by both The Art o f Nombryng and The Crafte o f Nombringe places zero not where we are accustomed to see it - before 1 - but after 9: "O.9.8.7.6.5.4.3.2.1." Zero is not thought of as a number: there is nothing preceding 1 in the progression 1, 2, 3, 4..., and zero is not used alone in operations.'8 Rather, it is an aid towards the writing of other numbers.19 Before zero becomes a number, one must obliterate the distinction between "figures significatyf' and "no figure significatyf' —but this can only be accomplished if signification is re-conceptualized. One must cease to envision the signified number as a positively existent something. However, in texts introducing Hindu-Arabic numerals, this step is not taken and therefore symbolic practice coexists with a theory of signification that does not fit. 17 The phrasing is international: "zeueroperse solo non signified nulla ma e potentia difare significare" (1307) or ”Sono died le figure con le quali ciascuno numero si pud significare: delle quali n'e une che si chiama zero: et perse sola nulla signified'(14,91), "decima figura... ratione sua nihil significat" [Smith and Karpinski 58, n. 1; 59, n. 5; 61, n. 1], 18 Calculations with zero are of course unavoidable when it is used in representing a larger number (that is to say the texts do not multiply 6 by 0 as such, but they do in cases like the multiplication of 60 by 10). 19 Since zero is not conceived as a number, the field of numbers is limited to positive integers and fractions. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 Sixteenth-C entury English Arithm etics One of the earliest English printed arithmetics (the first came out in 1537), Robert Recorde's The Grounde o f Aries (1542) served as the standard introductory text for all of the sixteenth and even much of the seventeenth century. Going through 27 editions up to 1699 [Karpinski 60], it was revised and augmented by a number of mathematicians including John Dee. Some of its competitors, such as Humphrey Baker's The Well-spring o f Sciences (first ed. 1562), copied from it verbatim. In sixteenth-century texts the numerals 1-9 are presented in the modem fashion, left to right: 1, 2, 3. 4, 5, 6, 7, 8, 9. Groups of numerals making up numbers also tend to be written in the same manner, with the upper ranks noted first, although Recorde's Maister advises the Scholar to begin with the lowest ranks until he has learned the system perfectly. However, since places are counted from the back or the rightmost side of the number, the rank of ones is said to occupy the first place, of tens the second, of hundreds the third, and so on.20 Zero for Recorde is no more a number than it is for his predecessors. To prevent confusion between it and the "signifying figures”, he introduces them separately: there are but, x, figures, that are used in arithmetike, and of those, x, one doth signyfie nothing, which is made lyke an ,0, and is called priuately a cyphar, thoughe all the other somtyme be lykewyse named. The other, ix, are called signifiyng fygures, and be thus fygured. 1 2 3 4 5 6 7 8 9 And this is their value. i. ii. iii. iiii. v. vi. vii. viii. ix. But here must you marke, that euerie fygure hath two values: One alwaies certayne that it signifyeth properly, which it hath of his forme. And the other uncertaine, whiche he taketh of his place [Bvv- vir]. The concepts of certain and uncertain value provide Recorde with a useful pedagogical tool to explicate the signification of numerals: a numeral's uncertain value comes from its rank, determined by place, whereas its certain value is the number of units of that rank. Thus, the numeral 9 in 90 refers to nine units (certain value) of tens (uncertain). Why does Recorde express his "certain" values in Roman notation? It appears that Recorde's understanding of the way Hindu-Arabic numerals signify is dependent on the :o "M[aister]: now expres you this summe. Fiue thousande, two hundred fyftie and seaven. S[choiar]. This troubleth me nowe, whether I should begyn at the first fygure or at the laste. For reason (me thinketh) shoulde cause me to begynne at the first, and yet y f I write it as you spake it, I must begin at the laste. M[aister], When you knowe your places perfectly, you maie begynne where you lyste. but the most ease for youre hand is, to beginne with the last, that is to say. as I did speake them. But for the most suertee, a whyle you may be gyn with the first” [Ciiv]. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. imitative principle. The Hindu-Arabic sign "7", for example, refers to the number "7" because this sign is only a shorthand way of writing the Roman "VII"; it is upon the Roman "VII" that the burden of signifying number ultimately rests. The Roman "VII” can carry out the job because it, like the signified numbers, is a multitude of units (slashes). Recorde's thinking is still conditioned by the principles inherent in the Roman system. The problem comes with the second step in the signification process, the step of uncertain value. The Roman sign expressing the certain value of the Hindu-Arabic sign can express its uncertain value only symbolically, because uncertain value depends on place. The Roman "VII", as the certain value of the Arabic "7", will at various times signify seven ones, seven tens, seven hundreds, and so on. At the level of uncertain value, the Roman sign acquires an infinity of signifieds and only its context can determine which signified it currently points to. certain value: Arabic numerals 1-9 -> Roman numerals I-EX uncertain value: each Roman I -> unit of appropriate rank Recorde’s understanding of Hindu-Arabic numerals is only cosmetically different from the twelfth-century attempt to write Roman numerals positionally, with, let's say, 1307 written as I.m.T.Vn ( t here is our 0; see footnote 5). Interpreting Hindu-Arabic signification through Roman numerals permits Recorde to retain the concept of number as the number of something. For him as for his medieval predecessors, "to signify" means to point to positively existent units. Since this state of affairs is inapplicable to zero, it "doth signyfie nothing." Hence the quandary of Recorde's Scholar: when I count the places of a number, do I note the "significant figures" only or zeroes as well? for I remember, you tolde me that they do signifie nothyng, and therefore I dout whether I shuld recken them for a figure in settyng of the prickes.21 Conceptually, the notational section of Leonard and Thomas Digges’ Stratioticos (1572) does not greatly differ from The Grounde ofArtes. Here is how Leonard Digges defines number and introduces the Hindu-Arabic numerals: :I Bviiiv; "prickes" to Recorde, like our commas, separate groups of three. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Number is the multitude of Unites set together, as 2. 3. 4. &c. All numbers may be expressed by these characters following. 1 2 3 4 5 6 7 8 9 0 whose simple ualue by themselues considered, you may heere-under behold. i ij iij iiij v vi vij viij ix 0 The Ciphra 0 augmenteth places, but of himselfe signifieth not [1], The "simple" value of Digges is, like Recorde's certain, conceived of in Roman notation. Digges too interprets how numerals signify with the aid of the imitative principle. Hence his employment of the classic Greek definition of number as multitude of units [Euclid 7 def. 2, Aristotle Metaphysics 1053a30], to which we will return in the next chapter. The striking aspect of this passage lies in its treatment of zero. Although for Digges "the Ciphra 0 augmenteth places, but of himselfe signifieth not," he lists it not only with the Hindu-Arabic numerals - in its then-standard place after 9, thus keeping it out of the number sequence - but also in his row of simple values, otherwise represented in Roman notation: " i, ij, iij, iiij, v, vi, vij, viij, ix, 0". The result is "modem" if probably unintentional: whereas 1, 2, 3 signify i, ii, iii; 0 signifies 0. It cannot signify anything else, since what it "signifies" is not a multitude of units, nor can it signify (a) nothing, for all signification has been defined as a signification of something. It must then signify itself. However, signifying oneself is not signifying either, for signification can occur only when the signifier be other than the signified, representation other than being. This is why Digges states that "the Ciphra 0 [...] signifieth not.” And yet the sign zero obviously has some meaning, some value. It must then have it in another way, a way unforeseen by the given theory of signification. "To signify a thing" and "to have meaning" cannot be absolute synonyms. If the simple value of zero is, as Digges would have it, zero, his zero becomes, by necessity, symbolic: as a tautology, it points at nothing but itself; as a sign, it indicates itself as a sign and not a thing; therefore, when it changes value (e.g. in 20 as opposed to 202), it does so without indicating any multitude of units. Would it not have made more sense for Leonard Digges to have portrayed the signified of 0 in the same way as it is appears on the counting board - by an empty space? That, too, would have brought forth a paradox: there would be signification, because the signifier would be other than the signified, but there would at the same time be no signification, for the empty space is not a something, but a nothing. And you can’t signify nothing. 32 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 THE CLASSICAL CONCEPT OF NUMBER Numerus autem est multitudo ex unitatibus constituta. Nam unum semen numeri esse, non numerum. Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number..1 0. Introduction While discussing Roman notation in the previous chapter, I mentioned the “natural” way of writing numbers according to the Arithmetic of Boethius, where five is represented as TTTrr and seven as >111111. such “natural” numerals reproduce tally marks, the historical origin of Roman, Etruscan, Egyptian and other similar systems. The decapitated philosopher’s historicism is unwitting: he does not claim that this is how numbers were once written, only that his “natural” numerals are the best graphic approximations of numbers as they really are. Boethius conceived of numbers in the fashion I shall describe in this chapter. His thought comes from the Greeks; in fact, the Arithmetic confesses to being an abridgement of the Arithmetic of Nichomacus of Gerasa, a first-century AD Platonist [epistle ded., 67]. Boethius was instrumental in passing the Platonist take on number to the Middle Ages and the Renaissance; these periods also knew of Aristode’s more nominalist approach to the subject. The foundation of what may be called the classical number concept, the understanding of number prior to Simon Stevin, is well expressed by Robert Recorde’s imitator Humphrey Baker: Number is as much to say, as a multitude, composed of many unities, as two is composed of two unities, three is composed of three unities, foure of foure unities, five of five unities, ten of ten, fourteene of fourteene, fifteen of fifteene, twenty of twenty, unities, &c. [1], ! Isidore of Seville, Etymologiae, 3.3; Bertrand Russell, 11. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sacrobosco represents this same basic idea along with its more complicated Platonist inference when he claims that "Sothely .2. manere of nombres ben notifiede; Formalle, as nombre is vnitees gadrede to-gedres; Materialle, as nombre is a colleccioun of vnitees'' [in Steele 33]. My description of the classical number concept focuses on Greek sources, although I do incorporate supporting citations from later, especially sixteenth-century, texts. It does not seem wrong to do this because, as I shall demonstrate at length in my chapter on John Dee, Greek ideas about number were extraordinarily long-lived. In the next chapter, we shall move on to the complex and creative relationship of the classical number concept with Plato’s ontology, the way his metaphysical system defines and is defined by number. I shall argue that one of the most constructive elements of Platonist thought, Plato’s understanding of how signs signify, is extremely similar to the imitative principle we have encountered in Roman and Herodian notations. I certainly will not argue that the classical concept of number derives from Herodian numerals. The Academy probably designated numbers using the alphabetic notation, which is too different from the classical number concept to have served as its origin. O f course, the preceding chapter has shown on the example of Hindu-Arabic numerals that newly appeared systems tend to be interpreted in terms of the systems being replaced. Still, whatever notations the Greeks employed to write numbers, they performed calculations with pebbles. If we are looking for an intuitive notational origin of the classical number concept, pebbles appear to be the most qualified candidate. As far as the persistence of the concept into the Middle Ages is concerned, I do believe it is legitimate to indicate Roman numerals and certain aspects of the counting board as encouraging and cementing it. I do not pretend to any real expertise regarding Plato. I do not understand Greek and have waded only ankle-deep into the vast sea of contemporary Platonic scholarship. What I say about Plato’s number and its ontological ramifications comes primarily from Jakob Klein’s Greek Mathematical Thought and the Origin o f Algebra, and secondarily from my attempts to reconcile Klein’s difficult text with those of Plato’s earlier interpreters: Aristotle, Proclus, Boethius, Nicholas of Cusa, Charles de Bovelles, John Dee, and others. Were I writing about what Plato said, such a procedure would be unpardonable; I am, however, at least just as interested in how Plato was understood. 1. Number as Multitude of Units The Greeks thought of numbers, Jakob Klein writes, as always the numbers of something. The tendency is natural; it originates in reflections upon the act of counting. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. What do we do when we count things in the world, things that are always in some way dissimilar? We presuppose a certain homogeneity among them: the fact that each of them is one thing. Counting three lions, for example, we ignore their particularities and focus on the fact that each of them is one lion. A lion and a lion and a lion gives us three lions. When we compare three lions to other groups of threes - eagles, rocks, cigars, whatever - we see that all groups of three are three in exactly the same way, i.e., in the fact that each thing being counted is, in exactly the same manner, one thing. Hence the way Euclid phrases his definition of one: "An unit is that by virtue of which each of the things that exist is called one" [VII. 1], To say that what makes things in the world countable is the fact that each of them is one thing, however, does not yet yield us what Tobias Danzig calls the homogenous abstract number concept [6]. In his Physics, Aristotle distinguishes between numbers of things in the world - for example, three lions - and number as such, for example three. A triad of sheep, by virtue of being a three o f sheep, differs from a triad of dogs, but the number three in each case is the same, "since it is [...] rightly said that the number of sheep and dogs is the same if each is equal to the other" [224a, qtd. Klein 47]. But what is this number three, this number that is one to the triads' many? It must be exactly what these groups share as regards their countability. If we take that being-one-thing which makes sheep, dogs, etc., countable, and subtract it from all of their other characteristics (including being things of sense, since concepts too can be counted), what remains is a group of identical ones or units. Since these units come from the being-one of each thing, as opposed to its being-many (parts), we may think of them as unities and monads (from G. monas = unity). The closest we can get to visualizing the pure number three, then, is as three monads apprehended at once. Explaining number in his Mathematicall Praeface to the first English translation of Euclid (1570), John Dee makes all the classical distinctions: Three Lyons, are three: or a Temarie. Three Egles, are three, or a Temarie. Which Ternaries, are eche, the Vnion, knot, and Vniformitie, of three discrete and distinct Vnits. That is, we may in eche Temarie, thrise, seuerally pointe, and shew a part. One, One, and One. Where, in Numbryng, we say One, two, Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pure Arithmetike, principally considered) no man is ignorant [*jv-*ijr]. Dee's "Indiuisible Vnits" are what composes "pure" number for Plato: a discrete group of units each of which is "equal to every other without the slightest difference and admitting no division into parts" [Rep. 526a], The Greek "pure" number is a number o f monads', the fact that the monads are homogenous grants us the possibility of calculating with "pure" 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. numbers. It is because the three monads that make up the number three are identical to each other, and also to the four monads that make up the number four, that we can put them together to make the number seven. If the monads in different numbers were different, then three plus four would not be the same as two plus five: we would be adding, as the saying goes, apples and oranges. Calculation here is conceived as an extension of counting. Such a concept of number clearly excludes zero: if number is defined as a group of monads, where there are no monads, there is no number. A far less expected consequence is how the system conceives of the one. According to the unanimous opinion of writers before Simon Stevin, one "is not a number" [Metaph. I088a7\ Dee *jr; Klein 49, 53]. This is necessarily so if number be conceived as a multitude, as many ones, for one is assuredly not many ones. "In numbers," writes Aristotle, "the one is opposed to the many.” "Plurality," he continues, "is as if it were the genus of number, for number is a plurality measurable by the one." And elsewhere: "'one' signifies a measure of some plurality, and 'a number’ signifies a measured plurality" [Metaph. I056a33, I057a2, 1088a5 respectively]. The concept of measure in this context is, I think, quite clear: all numbers are composed of ones, and each number is the number of ones that compose it. Another strange consequence is that the Greek concept of "pure" number excludes fractions, for, since it is based on counting, the monads must be indivisible. ‘Those things which cannot be divided, insofar as they cannot be divided, are said to be one," says Aristotle [Metaph. I0l6b5\. If we split a monad into halves, for example, what we get are two potential objects of count; if we split it in thirds, we get three, and so forth. “If anyone attempts to cut up the ‘one’ in argument,” says Socrates, mathematicians “laugh at him and refuse to allow it, but if you mince it up, they multiply” [Rep. 525e; see discussion in Klein 37-45, 53]. The Renaissance magus Agrippa of Nettesheim (1486-1535) writes that if one, which he calls the Unity, seem at any time to be divided, it is not cut, but indeed multiplied into Unities: yet none of these Unities is greater or lesser than the whole Unity, as a part is less than the whole [174]. If fractions are not "pure" numbers, what are they? And, if they are not "pure" numbers, what kind of entities do we engage in calculations that use or result in fractions, as in, for example, 4 -r 3 = 1 V4? Such entities - 4, 3, 1, 73 - must all be of the same kind, otherwise the calculation would be impossible, another case of apples and oranges. Jakob Klein argues that, since "pure" numbers cannot include fractions, and since calculation cannot forego them, the Greek concept of number introduces a sharp distinction between 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. arithmetic, the theory of "pure" numbers, and logistic, the art of practical calculation, which is what "arithmetic" means to us [43, 60]. I shall discuss Greek arithmetic, as well as the problem of fractions, to a greater length below. If for the Greeks neither the one nor fractions are, strictly speaking, numbers, the one is no number in a very different way from that of the fraction. "To be one,” writes Aristotle, ” is to be a principle [arche] of some number. For the first measure is a principle" [Metaph. 1016b17]. Aristotle's phrase in its context applies to the enumeration of anything: the measure of three eagles, for example, is one eagle. In a different section of the Metaphysics, Aristotle focuses specifically on the monads: unity is not a number; for neither is a measure measures, but a measure is a principle, and so is unity. But the measure must be something which is the same in all [1088a7; see also I052b22]. For Aristotle, the one's being the arche of number - its what has been variously translated as "source," "principle," or, as Stevin would have it, "beginning" - is limited to the thought that any number is the number of monads, i.e. ones, composing it.2 2. Number as the Unity of a Multitude For the Platonic tradition - which historically has been the tradition responsible for texts on arithmetic in the Greek sense of the word - one's being the arche of number goes much further. The Platonic tradition exhibits what may seem to be a certain confusion between the one as the homogenous building block of any number, and the fact that every number is so-and-so-many of these building blocks apprehended not consecutively, but simultaneously and in isolation from other monads. In other words, the Platonists perceived a deep-seated affinity between the one and any number inasmuch as it is itself and no other. On the level of counting things, we have already met this affinity: each thing can be counted as one precisely because it is a unity. Now this also applies on the level of numbers as multitudes: 2 3 4 5 6 7 8 9 monads 1 1 1 1 1 I 1 I unity (number) : The French philosopher and mathematician Charles de Bovelles (14787-1553) expresses this view when he writes that, "there is nothing in numbers apart from the one and its repetition," “In numeris aliud nichil est quam aut unum aut unius resumption [ Liber de XII Numeris 149r]. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Any number, to be a number, must be one number. Or: any collection of monads becomes a number only when its monads are apprehended as a unity, that is to say: as one.3 Confronting number in this manner, the Platonists had in mind what Jakob Klein calls the fundamental problem of Platonic philosophy: how it is that "a multitude, a number of objects can be grasped as one number, that the many can be 'one'" [49]. As it applies to numbers, the problem lies in the difference between one and one monads, and two monads; one and one and one monads, and three monads, and so forth. Is this difference real? It must be, for the properties of a multitude of monads are not the same as the properties of each constituent monad. Let us consider the number two. Individually, each of its monads has the property of being one, while together they have the property of being two [for passages in Plato, see Klein 82]. Being two means, for instance, being divisible into two equal parts; being able to equal four when squared, and so on. None of these are proper to either monad, but only to their, as Dee would say, "Vnion " and "knot" [see Klein 79-81 on Hippias Major 300a- 2b]. Thus the unity of the two monads, their joint being-one that makes the number two possible, is not merely the effect of our apprehending them together, but has something like real being. Let us try to mentally distinguish between the two monads in the number two and their unity. Coming together in the number two, the monads form a single dyad. A dyad means that the monads are not totally separate (1 and 1), nor yet absolutely fused into one. They are rather two-as-one, a unity of multiplicity. We recall the description of Helena and Hermia during their friendship in A Midsummer Night’s Dream: Like to a double cherry, seeming parted, But yet a union in partition, Two lovely berries molded on one stem; So, with two seeming bodies but one heart. Two of the first, like coats in heraldry, Due but to one and crowned with one crest [3:2:208-14; “first” = bodies].4 3 The Greek identification of one and unity has left traces in our languages. The English word "unit," which most accords with the one as merely the building block of number, is a neologism coined by Dee in The MathematicalI Praeface: "Note the worde, Vnit, to expresse the Greke Monas. &. not Vnitie: as we have all, commonly, til now, used'[*j ]. Yet, in the same essay. Dee also applies the word to something obviously having parts (passage on Number Numbered, *j ). In Latin and French, a word like "unit” did not emerge, and writers, including Boethius and even Stevin and Descartes, refer to the one as unitas or unite instead of unum and un. x Bovelles explains this view of number as well: “Just as the dyad is something else than the one [...], so each number is something other than solely the iteration of the unity," “Sicut dualitas aliud aliquid est quam ipsum etperse vnum: ita et numerus omnis aliquid quiddam est: quam sola vnitatis iteratio, ” [ Liber de XII Numeris 149r]. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Furthermore, particular unities of multiplicities, like the dyad, the triad, the tetrad, and so forth, differ with respect to the number of their parts, but are alike inasmuch as each of them is a unity, a one. Extracting this single being-one from the particularity of the unities gives us the concept of "pure" unity as it pertains to number. To be the pure unity of number means being unity, but not this or that unity, not this or that number. Such a unity not only can be thought as separate from parts, but cannot be thought to have any parts at all. For if this unity had parts, how many parts would it have? Curiously, this unity which, since it is "unity in the true sense and rightly defined [,] must be altogether without parts" [Sophist 245a], is not only the common aspect or measure of all particular unities, but it also turns out to be identical to the system’s building block, i.e. the monad. For we can think of the monad as a unity only if it too be a unity so abstract and separate as to be void of parts. Thus, according to Aristotle, the Platonists say the one is the arche of number because it is indivisible both as a unity and as a building block [Metaph. J084bl4-21]. At this point, we have arrived at a three-tier concept of number: at the bottom are ones, in the middle are their collections apprehended as ones, at the top is the one of their collections. The monads at the bottom are homogenous and indivisible, the collections are divisible as collections but indivisible and heterogeonous as unities, the one or pure unity is again indivisible.5 Aristotle is dismissive of such Platonist speculations. He himself does not assign the unity of each collection of monads a real being: "why is a number, taken as a whole, one?" [Metaph. 992a2]. In fact, he regards even the monads themselves as only abstractions, lacking in an separate reality: "A thing can best be investigated if each attribute which is not separate from the thing is laid down as separate, and this is what the arithmetician and the geometer do" [Metaph. J078a22, see Klein 103-5]. Due to this view of arithmetical entities, number in Aristotle's philosophy does not play the role it does for Plato. 3. Greek Arithmetic I said that the problem of fractions led the Greeks to distinguish between arithmetic and calculation (logistic). It is of course possible to calculate with integers only, but then we must agree that some instances of division and extraction of roots know no solution. 5 Since number has been defined as the exemplar of unity in multiplicity, and since the pure unity has no parts, we once more ascertain that pure unity - whether thought of as monad or as what the particular unities of number have in common - cannot be considered a number. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, if we consider numbers as an endless collection of unique unities, a strange problem follows. Being alike in that they are unities, but not being homogenous, particular unities can be objects of count, but not objects of calculation. The unity of seven monads is not the unity of three monads plus the unity of four monads: all unities are to each other as apples and oranges. Yet that we can add three to four monads and get seven monads is beyond dispute; therefore also beyond dispute is the fact that we have in some way acted upon two dissimilar unities to replace them with a third, dissimilar to either. But are these unities entirely dissimilar? Some have the combined property of their numbers being divisible into two equal whole number multiples, while others do not; the halves of some of the former can again be divided into two equal multiples, whereas some numbers have no multiples at all apart from the themselves and the one. Greek arithmetic occupied itself with patterns such as these. Being, in the words of Dee, "the Science that demonstrateth the properties, o f Numbers" [ijr], Greek arithmetic appears to us a restricted and non-algebraic version of number theory. Starting from the Pythagoreans [see Metaph. 986a22], the Greeks separated number into the two most basic kinds, the even and the odd; every number belongs to either one or the other [see Metaphys. 986aI8-20, Boethius 1:2, 76, and Klein 17 on Gorgias 45lb-c\. The sub-kinds as defined by Euclid are: even-times even (2n); even-times odd (divisible by two but not by four); odd-times odd (divisible only by odd); prime (divisible only by itself), composite (not prime), square (number whose root is an integer), cube (number whose cube root is an integer) and perfect (sum of its multiples, including one but excluding itself, like 6 = 1 + 2 + 3). For the purposes of brevity, I have explained these in the modem fashion.6 Greek arithmetic aimed at the classification of positive integers. I have not found explicit statements to the effect, but its ultimate desire appears to have been a classification tree of numbers much like the classification trees in Plato's Sophist and Statesman. Investigating the kinds of number, which of course sometimes intersect (for instance 4 is even, even-times even and square), is the preparatory labor necessary for creating a tree with a location for every number. Any unity, being unique, will manifest a unique set of properties; we should be able to find any number, therefore, by questions like: "Is it X or Y ?" - "X." - "Is it X, or X2?" - "X,." - "Is it X Ia or X2a?" - "XIa," and so forth, until we reach the number. The classification tree, ultimately, would also be a history of how any number emerged from the one. In texts such as the Arithmetic of Boethius, different sub-kinds of numbers create different number series, such as that of the even (2, 4, 6, 8...), of the even-times-even (2, 6 For more information and additional kinds of numbers, see Danzig, chap. 3; Boethius; Heath's comments to Euclid VII. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4, 8. 16, 32....), of the square (1, 4, 9, 16, 25...) and so forth. A curious method of generating and analyzing such series are the so-called numerifigurati, as described in Boethius 2.6-19 [see also Heath on Euclid 7, def. 16; Danzig 41-43]; it obviously emerges from calculating with pebbles. Let us take square numbers. We may envision them as oooo ooo oooo oo ooo oooo o oo ooo oooo 12=1 22 = 4 3:=9 42= 16, The series of square numbers may also be arranged in the manner of Russian matrioshkas [see Shichalin's commentary to Proclus, Prokl 215-216]: o o Oi o 12= 1 o o o o 22= 4 o o o o rl 9 m rlTf II II o o o o 16 This is called a gnomon. It portrays square numbers as literally squares (hence the name) whose sides are their square roots. The number series of the sides (1, 2, 3, 4) produces, as it were, the number series of the squares (I, 4, 9, 16); the gnomon allows us to easily see the difference between any two consecutive squares, which, in fact, is determined by the odd number series: 3, 5, 7, 9, etc. (the first odd number to the Greeks is the three, "princeps imparium" [Bovelles, X II Numeris, 150r]).7 An arithmetic of the Greek type would present the relationship in a grid like the following, which is a simplification of the one found in Charles de Bovelles’ Liber X II Numeris. While Bovelles uses Hindu-Arabic numerals, the Greeks obviously do not. 7 For Plato, one is odd [Hippias Maj. 302a], for Aristotle it is both odd and even [Metaphys. 986a20\. Other writers start the odds with three. That three is the first odd number is implicit in Boethius' definition of the odd: "the odd cannot be [...] divided [into two equal halves] unless a unity comes between the halves" [1:13, 89], See also Euclid VII:7, which appears to admit either one or three as the first odd, and Heath’s commentary to it. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. natural numbers l8 2 3 4 5 6 7 8 9 square numbers 1 4 9 16 25 36 49 64 81 odd numbers 3 5 7 9 11 13 15 17 19 sum sq and odd 4 9 16 25 36 49 64 81 100 A grid makes the pattern, well, visible: adding any square number and odd number that hold the same position in their respective number series will give us the next square number. Confronted with such a grid, the modem person (at least this modem person) blanks, and then reaches for his or her pencil. The modem ’’natural" instinct is to express the pattern of the grid in algebraic notation: only at that point will it be "understood.” Let us call any natural number n: then the n h square will be n~ and thesubsequent squarewillbe (n + 1)2. The difference between them will be (n + 1)2—n1 = n~ + 2/i + 1 - n = 2n + I. Since the first odd number to the Greeks is three, 2n + 1 is, infact,the / I th odd numberfor any n. Or, as a Greek-type arithmetic would have it. natural numbers 1 2 3 4 5 6 7 8 9 dyad 2 2 2 2 2 2 2 2 2 even 2 4 6 8 10 12 14 16 18 unity I 1 1 1 1 1 1 1 1 odd numbers 3 5 7 9 11 13 15 17 19 Yet the algebraic expression of the gnomon of the squares obscures at least as much as it explains. For the Greeks, the gnomon shows the relationship between sub-kinds of number: here between the squares and the odds. This relationship is the effect of a property of unities of odd numbers of monads, and of a property of unities of square numbers of monads. If we think of the odd and the square as different, although sometimes overlapping, kind of number, the fact that such a relationship exists is striking, to say the least. At the same time, even our focusing on the relationship is a modem habit: for the " Although the one is, strictly speaking, no number, it appears in the different series of sub-kinds of number. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aims of Greek arithmetic, it is not the relationship that matters, but what it contributes to the understanding of the odd and the square. Algebraic notation not only shifts the object of inquiry, but also implicitly changes the nature of the elements involved. While, with n being an integer, any 2n + 1 is bound to be odd, and any n2 bound to be what the Greeks called a square, the algebraic expression of the relationship defines the odd and the square solely in terms of n. In other words, the relationship is formulated as occuring not among different sub-kinds of number, but within a single domain whose elements are identical in kind. So the first result of introducing algebraic notation is that it takes us outside the Platonic consequences of the definition of number as multitude, outside, that is, of the search to classify the unities of each multitude. This is because algebraic notation treats all numbers as if they were the same type of thing. The second result is even more startling. If we take the equation that expresses the relationships of the gnomon, n2 + 2 n + 1 = (n + 1) 2 will it not equally work if n is not the kind of number we encounter in arithmetic, but the kind we encounter only in logistics - for example, a fraction? The equation itself will be just as true. But then 2 n + 1 will no longer be an odd number, n2 will no longer be a square in the Greek sense of the word, and all or some of the elements of the equation, not being positive integers, will be inconceivable as multitudes.9 4. Number in Magnitude Failing to develop a theory of pure number that would include fractions, the Greeks grounded fractions in the divisibility of the one: not of the monad, but of the body in the world [see Klein 39-40]. We have seen that the idea of number as a multitude of monads is essentially an abstraction from counting bodies in the world: lions, eagles, etc. As Aristotle says in Metaphys., I088a5, the "measure" of such a count is the one. Yet bodies in the world, taken as wholes, also permit a different sort of measuring: the measuring of their 9 Other numeri figurati according to Boethius are: the triangular (I, 1 + 2 = 3, 1 + 2 + 3 = 6, I + 2 + 3 + 4 = 10, etc.), the pentagonal (I, 1 + 2Z= 5, 1 + 2 + 32 = 12, 1 + 2 + 3 + 42 = 22), the hexagonal and the heptagonal [see II.6-19, 131-142]. Some numbers can assume various shapes, like the 6 which can be either triangular or hexagonal [Boethius 1:17, 141]. This indicates that numeri figurati do not reproduce the actual arrangements of monads within their numbers: monads, after all, are incorporeal and therefore cannot be arranged at all. Rather, the arrangements of numeri figurati isolate some property shared by a set of unities: that of being a square, for example. We may think of them as visual aids; we may also think of them as the first level of the spatialization of number, i.e. of its turning into "copy". 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. length, width, or breadth, in terms of conventionally chosen units. This type of measuring implies that the body in question is not a collection of discrete units, but - at least so far as the measuring is concerned - a continuum. If we isolate the measurability of bodies in the world in the same way as we earlier isolated their countability, we get magnitude. When distinguishing between plurality and magnitude, Aristotle applies the term “measure” (metron) only to the latter: A plurality is a kind of quantity that is numerable, a magnitude is a kind of quantity that is measurable. It is called "a plurality" if it is divisible potentially into parts which are not continuous, but "a magnitude" if it is divisible potentially into continuous parts [Metaphys. 1020a8-12]. The fact that the numbers we get when we measure magnitude are not “as much” numbers as pluralities may be seen the fact that only pluralities are, for Aristode, “numerable.” Since magnitudes are divisible, the numbers of magnitude admit fractions. Let us refer to numbers measuring magnitudes and numbers enumerating pluralities as geometrical and arithmetical respectively: this, in fact, is the traditional terminology. Like the arithmetical, geometrical numbers are, in Klein's terms, still "numbers of something", i.e., of length, width and breadth. That is to say, geometrical numbers are numbers o f magnitude in the same way as arithmetical numbers are numbers o f monads. Visually and as magnitude is how number appears in the proofs of Book VII of The Elements. The assumption there is that any number can be represented as a straight line of so many units. In addition, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another [VII: 16]. when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another [17]. If we think, for a minute forgetting about fractions, of geometrical numbers solely as a way of representing numbers as multitudes, we shall see that this method allows far more variation than that of the numeri figurati. Any number that has a pair of multiples is a plane, any that admits more is either a plane or a solid. Or rather, any number can be portrayed as either a line or a plane or a solid, since any number can be thought of as itself, itself multiplied by one, and itself multiplied by one and by one. In other words, the differences among various geometric versions of a single plurality depend not only on the arithmetical 44 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. properties of the number, but also on what type of geometrical unit measures it: whether the unit is of length, of area or of volume. Let us take the case of the so-called linear number, number represented as line. To be linear number x means to be x times measurable by a unit. We shall later see that the numerical one is thought by the Greeks to correspond to the geometrical point; not so, however, with the unit that measures linear numbers. That type of unit is a positive magnitude, a line segment. Let us imagine it as line AB. A__________________________ B AB = I To us it appears that the distance from A to B equals 1, with B corresponding to 1 and A to 0. Euclid’s habit of representing numbers by lines should lead to the development of the concept of zero. Yet it does not; why? There seems to be a dual reason. Geometrical number is not the subject of the science of numbers; the science of numbers occupies itself with pluralities. Therefore the Greek concept of “measure,” even when it is applied in geometry, still implies a measuring by ones, by wholes; this is why Aristotle can speak of the one as the “measure” of number. We must conclude that Euclid regards the entire line segment AB as a one, even if divisible (by, for instance, bisection), rather than as a continuous movement from zero to one. We must also consider the case of incommensurables. The Pythagorean theorem implies incommensurability, and Plato’s Theatetes seems to have been the one to prove that the diagonal of a square is not commensurable to the side [Heath on Euclid 3, prop. 4; see also Theatetes 147d-8b\. However, to think of Greek incommensurables as irrational numbers is extremely misleading. Incommensurability is a relation between magnitudes. If the concept of geometrical number is based on measure, all incommensurability means is that two magnitudes cannot share units, that their units are, so to speak, mutually untranslatable.10 Seeing that geometrical numbers include both integers and fractions, whereas arithmetical numbers are integers only, one might suppose that the proper object of calculation would be geometrical numbers. Such a supposition would be wrong. Klein's thesis that Greek number is always the number of something means that the existential aspect of calculating with numbers is specific to the kind of number involved in the 10 While Euclid’s definition of the topic in book X, props. 6 and 8 uses the word “number” in such a way as cannot be interpreted to refer to irrational numbers, his book V, containing the doctrine of proportions ascribed to “Eudoxus, the teacher of Plato” [see Heath on Euclid v. 2, p. I l l ], is phrased in a much more general manner. Eudoxus’s theory of proportions has aspects formally similar to Dedekind’s definition of irrationals [see Heath on Euclid V, props. 3-5]. However, if the Greeks did think of incommensurables numerically, we would expect more evidence to the effect, as well as a revised conception of number. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. operation. Adding horses, in other words, is not the same as adding feet of cloth. Adding horses is an extension of counting pluralities, whereas adding feet of cloth is an extension of measuring: in this case, of the continuous piece of cloth. Therefore the calculator must be always aware of the kind of unit he is operating with. The "practical" science of calculation in Antiquity and the Middle Ages almost always involved real or simulated situations, expressed in words, and required a reality check: the question whether the numbers of the result make sense in the situation. I am reminded of a Russian poem from my childhood, where a lazy student solves a word problem, getting two and two-thirds of a ditch-digger as his answer: 3anany 3aaajm y Hac. Ee peuiaji a uejiBiH Hac, H bbiui.no y Mena b oTBeTe: JXB a 3 e M J ie K o n a h U Be T p e T H ." The implication is that he did not do his math correctly: this is true only because he is operating in terms of not numbers, but numbers of ditch-diggers. 11 “We were assigned a word problem. I solved it for an entire hour. And the answer I got was Two ditch- diggers and two thirds.” Samuil Marshak, “Pro odnogo uchenika i shest' edinits" [Marshak 208]. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 NUMBERS AND PLATO’S ONTOLOGY neTbipe - — O h h h - 3TO a , HBa - s t o TejieHOK, TpH - s t o K o p o B a , 3TO Ebnc, IIHTb - 3TO K oH b, UieCTb - 3TO CBHHbH, CeMb - 3TO K oT, BoceMb - 3To Ilec, aeBHTb - s t o Eapan h aecan. - s t o rieTyx. — Ypa! Ko3JieHKy! Y pa-a-a! — 3aKpHHaiTH TyT Bee b o z l h h roiioc. Number must exist if anything does.1 0. Introduction This chapter is about Aristotle's assertion that for Plato, as for the Pythagoreans, the world is number. The fact that number here is conceived in the terms described by the previous chapter means that the role allotted to number is ontological rather than scientific in the modem sense of the word. The world is not regarded as the object of calculation; quite the contrary. In the previous chapter I referred to Klein's argument that the problem of fractions and of the indivisibility of the one separated calculation from arithmetic as the science of number of pluralities. In this chapter we shall risk Ockham's opprobrium by regarding the autonomy of arithmetic as dependant upon the way Greek philosophy generally inquires into things. Turning to the Sophist's discussion of how words mean, we shall look at Plato's handling of the relationship between signifier and signified. Plato considers the act of signification to occur cardinally, without reference to the syntax in which the signifier participates. This is the approach the Pythagorean-Platonic tradition uses when it inquires after the being of number as free from mathematical operations. It is not interested in what number does, but in what it is. The very separation of "does" and "is," and the predicating of the former upon the latter, to a large measure determines its answer. Treating numbers severally, it regards their whatnesses as the identities shared by the many varied and changing examples of the given number in the world, and assigns them a separate existence. Numbers in this sense become autonomous, immutable, non-sensual "things," 1.e. Ideas. 1 First quote from Prioison; second said by Theatetes in Sophist 238b. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I shall attempt to show that the act of assigning Ideas a separate existence depends on Plato’s understanding of signification. In the Sophist, a signifier is said to have meaning by pointing to a positively existent signified; hence the question of "what does this signifier mean?" can be answered only by discussing the ontological nature of the signified thing. Given being in such a way, Plato's Ideal Numbers may be thought of as really existent signifieds to which many varied and changing signifiers (mathematical numbers and numbers in the world) point. The Phaedo is unambivalent in that numbers conceived in such a way make calculation possible, and not the other way around. Thus, the statement that the world is number regards the world as existent in the same way as various ontological strata of number exist. Number here provides evidence of the existence of Ideas. Moreover, if one believes the portrait of Plato's ontology in the Metaphysics, this structural similarity is also an ontological identity. The world is number because Ideas are numbers, and sensible objects are also numbers by virtue of being Ideas reimprinted upon the material cause of difference and multiplicity. I must admit that, for me, this is rather hard to understand: I do not see how the Greek concept of number and its ontological ramifications explain the fact that I have eyelashes. Perhaps the fault is mine; perhaps it is Plato's; perhaps it is Aristotle's. The Stagyrite's account of Ideas as numbers is fragmentary and full of contradictions, and at a certain point we must just throw up our hands and walk away. 1. Pythagoras: “Numbers are the substances of all things” Aristotle reports that the Pythagoreans identified numbers with the being or principles of things [986al5-17; 987aI9]. Modem readers of Plato are often surprised that the Metaphysics sees so little difference between this ontological aspect of the Pythagorean doctrine of number and Plato’s doctrine of Ideas. Renaissance Platonists, however, or at least the mathematically inclined among them, use Pythagoras’s words as one of their chief slogans; they also believe Aristotle’s claim of kinship between the Pythagorean and Platonic ontologies. Nicholas of Cusa, for instance, asserts a single tradition passing from the Pythagoreans to the Platonists to Boethius, each of whom allegedly held number to be “the first exemplar of things in the mind of the Creator.”2 Aristotle reads the statement that numbers are the principles of things in two senses, the first of which is things' being literally composed of numbers [986al5-7]. The Pythagoreans, he writes, : “Primum rerum exemplar in animo conditoris numerum esse," Idiota de Mente <5,ed. princ. 94-95: 5.140. See also Docta Ignorantia 1:1. Nicholas is thinking of Boethius 1:1-2, 74-75. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. posit the whole universe as constructed out of numbers, but not numbers consisting of units in the usual sense, for they believe the units have magnitude [Metaphys. 1080b 17-20]. For Pythagoras the monads composing numbers are the atoms of things. He identifies the numbers of magnitude and plurality by equating the indivisible one with the point, and defining the latter as a "unity in position" [De Aninia 409a6\ Heath on Euclid / def. 1]. "Behind this flowery verbiage," comments Danzig, "we detect the naive idea of a line as made up of a succession of atoms just as a necklace is made up of beads" [99]. Although the discovery of incommensurability shattered the identification of the one with point and the atom, it survived as an important analogy. We shall return to it in our chapters on Dee and Stevin. The second sense of Pythagoras’s claim for Aristotle is that numbers determine things’ properties [“affections and possessions,” 986aI5-7]. What this means is unclear. Are we to suppose that the total number of ones/points/atoms constituting a thing3 is responsible for its qualities, like hardness, whiteness, and so forth? If so, in what way might this be possible? We have spoken of Greek arithmetic as breaking down numbers into sub-kinds such as even, odd, even-times even, etc.. Should we postulate that qualities such as hardness and whiteness arise from what sub-kind the number of atoms belongs to? The possibility, while not void of beauty, is untenable. If what a man is and does depends on the exact amount of atoms he has, gaining weight, for example, makes him an entirely different entity with different properties —and so does cutting his nails. Aristotle’s account of Pythagorean teaching is full of contradictions. In one passage, for example, he reports the Pythagoreans say numbers "do not exist separately, but that sensible substances consist of' them [I092bl0]; in another, "Pythagoreans say things exist by imitating numbers" [987bl2]. It cannot be both: either things are numbers or things imitate numbers. The only way we may believe the two claims is by viewing the second as correcting the first: numbers are initially equated with things and then, when the unit ceases to be identifiable with the point and the atom, given autonomous being which things “imitate.” Aristotle errs not in his report of what “the Pythagoreans” said, but in that he presents as static the tradition that, by the time of writing of the Metaphysics, saw two 1 Aristotle mentions one Eurytus, who “used to assign a certain number to a certain thing, for example, this number to a man and that number to a horse, like those who were arranging numbers in the shape of a triangle or a square and in this manner were producing by the use o f pebbles likenesses o f the shapes of plants” [1092b 11-14: note the curious use of numeri figurati]. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. centuries of active development, from the oral teachings of Pythagoras (6th c.) to the writings of Philolaus (second half of 5th) and Archytas (mid. 4th).4 Considering Plato’s ontology to be the heir to the Pythagorean tradition, the Metaphysics has Forms or Ideas assume the place of, and be equated with, numbers. In saying that the One is a substance, [...Plato] spoke like the Pythagoreans, and like them he believed that Numbers are the causes of the substance of all other things [!987b22-24]. The only change he made was to use name “participation”; for the Pythagoreans say that things exist by imitating numbers, but Plato, changing the name, says that things exist by participating in the Forms. As to what this imitation of or participation in the Forms might be, they left this an open question [987bI0-4]. Aristotle's account of Plato is a notorious gargoyle. Its identification of numbers and Ideas bears little resemblance to what Plato himself says in the Dialogues, or rather, whatever resemblance there be between Plato-as-presented-by-Plato and Plato-as-presented-by- Aristotle may be glimpsed in the writings of Plato only after one reads Aristotle. The work on “Plato’s late ontology” performed by Jakob Klein and Kenneth M. Sayre relies on the Metaphysics in just this way. The situation is compounded by two further problems. Aristotle’s reference to Plato’s “unwritten teachings” [Physics 209bl5], such as his lecture on the Good [see Sayre 76-8 and Metaphys. 988al4-l7 ], seems to be confirmed by the Seventh Letter if its authorship is indeed Plato’s, which is far from certain. If Plato indeed had a doctrine which could not have been put in writing “like other studies” [Let. VII 241c\, perhaps this doctrine is what the Metaphysics explains and argues against. After all, having been for many years a student at the Academy, Aristotle got his information straight from the horse's mouth. The second problem is that the Stagyrite shows so many contradictions in what he expounds, that we wonder whether he understands it himself, or whether his report of it is not a straw-man fallacy. As a historian of philosophy, meticulous he isn’t. Therefore opinions on Aristotle's portrait of Plato vary enormously, ranging from claims that he got everything wrong to claims that he extracted his version of Plato’s ontology from the Dialogues themselves [see Sayre 76-84 for resume of positions]. For the purposes of my dissertation, however, the historical accuracy of Aristotle’s report is not of the utmost importance. I am much more interested in how Plato was 4 For attempt to separate the earliest Pythagorean doctrines from later ones, see Zhmud’. For the influence of the harmonics of Architas on Plato, see Bumyeat; I shall omit harmonics entirely, focusing only on the problem of the being of number. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. perceived by later thinkers. None of them, from Plotinus to Nicholas of Cusa, appear to have doubted the portrait of the Metaphysics. What Aristotle said about Plato fashioned Plato for succeeding generations to the same extent as the Dialogues did, if not greater. Cases in point are Nicholas’s tracing of the "Pythagorean" tradition; or Proclus’s characterizations of Ideal, mathematical and sensible entities; or Plotinus’s identification of Ideas and numbers: none of these could have arisen from the reading of Plato himself, without Aristotelian prompting. This chapter, then, is devoted to how one might understand Plato's late ontology if one believes Aristotle’s assertion of the essential synonymity of the Pythagorean and Platonic views. We shall make use of those passages in the Dialogues that seem to prefigure or be consistent with Aristotle’s account. The claim that numbers are the being of things shall be examined in terms of Plato’s “grammar of inquiry” (the phrase is Russell’s, 11), i.e. the assumptions that make his doctrine of Ideas possible and even necessary. 2. "The Actual Object o f Knowledge... is the True Reality"5 In the Republic Socrates claims “the real object” of geometry to be “pure knowledge,” defining pure knowledge to be "the knowyng o f that, which is euer: and not that, w h ic h , in tyme, both is bred and brought to an end."' Agreeing, his colloquitor Glaucon calls geometry '"the knowledge o f that which is euelastyng" [Republic 527a-b as quoted by John Dee *ijv]. What does this mean? Whether one studies the laws or the figures of geometry (and the Greek geometer studies the figures, with the laws being conceived as their properties), the objects of study are immutable. Neither the circle nor the Pythagorean theorem change from one moment to the next; neither depends on who studies them, when or where. The characteristics of objects of knowledge determines the characteristics of knowledge: geometry is immutable by virtue of being knowledge of immutable things. Let us contrast such knowledge with the “knowledge” o f where a weathervane is pointing, now West, now East. If I say, “now the weathervane points West,” my statement is true one moment and false the next. If “pure knowledge” is like geometry in that it is immutable, knowledge of where the weathervane points to is not knowledge. For the latter faculty the Republic reserves the term “opinion” [476b]. “Knowledge pertains to that which is” [477a], whereas opinion pertains to that which changes. 5 Letter VII, 342b. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The conception of knowledge as "the knowledge o f that which is euelastyng " leads to the famous distinction between being and becoming. Objects of pure knowledge must “be,” i.e. remain identical to themselves now and forever. The problem here is that none of the things we see, hear, taste, touch, smell, “are” in this sense of the word, for they all change with time. They are continuously becoming other than what they were. According to Aristotle, Plato having in his youth become familiar first with Cratylus and the Heraclitean doctrines (that all sensible things are always in a state of flux and that no science of them exists), [...] continued to believe these even in his later years [Metaphys.987a32-b l ]. Everything that is available to the senses is becoming and therefore unknowable. Are there “things” that do not change? Does being exist? If it does, how would we know it? May it be known from the sensibles or are being and becoming hopelessly and absolutely severed? It is extremely important that the problem of “pure knowledge” comes up in the context of a mathematical discipline, even if that discipline be geometry and not the science of number. Nonetheless, we have earlier seen that geometry is dependent on number. Perhaps inquiring into numbers will shed more light onto objects of pure knowledge, objects which, if they exist at all, must have a being whose ontological status for now appears incomprehensible.6 3. Distinguishing Types of Number and their Characteristics Platonist analysis of number starts with the distinction between the numbers in our mind and the numbers in things. Let us recall John Dee's passage on counting as the source of number: Three Lyons, are three: or a Temarie. Three Egles, are three, or a Temarie. Which Ternaries, are eche, the Vnion, knot, and Vniformitie, of three discrete and distinct Vnits. That is, we may in eche Temarie, thrise, seuerally pointe, and shew a part, One, One, and One. Where, in Numbryng, we say One, two, Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pure Arithmetike, principally considered) no man is ignorant [*'-*jr]. 6 The place accorded to geometry by the Republic may be responsible for the anecdote, first related by the Byzantine historian Joannes Tzetzes, that Plato "wrote this sentence on his schole house dore: [...] Let no man entre here [...] without knowledge in Geometry" [Recorde, Pathway. £.ii'; Tzetzes’ Variorum Historiarum Liber viii: 249 in Henninger, 34-35 n.5]. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When we point at an eagle and say "one," we are, in fact, dealing with two ones: the one in the world (eagle) and the one in our mind. When we then point at a lion, and say "one", the one in the world is different inasmuch as one lion is a different animal, but no different inasmuch as it is a one. The one in our mind is the same in both cases, and it is also the same one as the one of the eagle and the one of the lion, inasmuch as these ones are ones. Like any thought, the one in our mind is not apprehendable by the senses, and it does not take up any real space. Unlike other thoughts, it is not changed by time. Most importantly, it is not unique to our mind, but is a thought that can be thought uniformly by anyone. The ones of eagles and lions exist in an impure manner, in attachment to eagles and lions, but they too certainly exist, for each thing is one thing whether or not I am counting it. The fact that each thing is one prevents it from turning into a mist of atoms. But even such atoms would be little ones. So the one also exists in the world. And if we take the one, as it exists in the world, and "purify" it - i.e. abstract from it the particular something it happens to be one of - what we get is identical to the intelligible one, the one in our mind, a unity which is "altogether without parts" {Soph. 245a]. What can be responsible for the identity of the ones in our mind and the ones in the world, the identity that enables us to count the latter? Plato assumes the cause of such an identity - of the “oneness” of both kinds of one - is the One itself, unique, endowed with being and separate from both the mind and sensibles. This One is immutable (i.e. eternal), since, if it changed, all the ones in our mind and in the world would no longer be ones. As the cause of ones both in our mind and in the world, it must exist prior to both, and therefore cannot be located at any specific point of space and time ("For time consists of number, & all motion, & action, & all things which are subject to time, & motion" [Agrippa 171]). How such a thing might exist is unclear; nonetheless, since it cannot be grasped by the senses, but can be grasped by reason, we might characterize its being as "intelligible" (or to use a word with a Greek root, noetic). The intelligible One appears to be identical to the pure unity underlying all numbers and responsible for their being-one as described in the previous chapter. We must also postulate the separate and intelligible existence of other numbers, for otherwise how would we know that one and one and one eagles are equinumerical to one and one and one lions? For Socrates these separate numbers are what makes calculation possible: Suppose [...] we add one to one. You would surely avoid saying that the cause of our getting two is the addition [...]. You recognize no other cause for the coming into being of two than participation in duality [Phaedo I01b-c\. 53 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. As we can see from Socrates’s reference to “duality,” the other separate and immutable numbers are not composed of monads, but are the kinds of unities that presuppose particular numbers of monads appearing in the mind or in the world. We must therefore identify them with the particular unities of the previous chapter. Let us refer to both the pure unity of the Intelligible One and the particular unities of numbers existing separately from our minds and things, as Ideal or Formal Numbers. The latter combination was preferred in the Renaissance; both, of course, stem from Plato's interchangeable or near-interchangeable use of "idea" {idea) and "form" (eidos). If, in what follows, I shall stick to "Idea" and "Ideal", it is because they seems to me less misleading, since the word "form" can be applied to the magnitudinal aspect of a thing's appearance, and since it also sounds free from considerations of quality, whereas "idea" does not. Let us therefore attempt some brief remarks on other differences between Ideal Numbers, numbers in the mind and numbers in things. The separate existence of Ideal Numbers is dependent upon that of the Intelligible One, since the One allows each of them to be a unity. Heterogenous and therefore not subject to calculation, Ideal Numbers are separate from one another and singular: there is only one Ideal Two, one Ideal Three, and so forth.7 If Ideal Numbers “cause”8 arithmetical and geometrical numbers in the mind, they also cause them in things: as the one of the eagle, and the measure of his wingspan. As we have seen, numbers in the mind are composed of non-sensual and homogenous monads, and thus can serve as the objects of calculation and count. While they are changeless, they are also many for every one of Ideals, for in 2 + 2 = 4, there are two 2s, but only one Ideal Two. Differing from their “Forms in that there are many alike whereas the Form itself corresponding to these is only one,” numbers in the mind or “mathematical numbers” are located “between Numbers as Forms and sensible numbers” [Metaphys. 987b 17-8, 1090b 34-6).9 Sensible numbers, i.e. numbers as physically present in things, are subject to things’ change, generation and corruption (for instance, if I have five oranges and eat two, 7 If there were more than one Ideal Two there would have to be another Ideal Two responsible for their identity. 8 In the way that one and one eagles are two because they “participate in duality." Ideal Numbers must be considered “causes” of number as appearing not only in plurality but also in magnitude, although fractions pose a surmountable, and incommensurability an insurmountable, challenge to this statement. 9 The being of mathematical numbers is problematic, since their eternity and changelessness seems to warrant them a being separate from the mind, which they do not have. The Platonists, claims Aristotle, “neither said nor could say how Mathematical Numbers exist or from what principles they are composed” [1090b34-5]. Their being is generally referred to as “intermediate.” 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or if I suddenly grow a foot). Numbers of things are inextricable from things in the same way as numbers of monads are inextricable from monads. They are therefore heterogenous and do not serve as objects of calculation. The various types of Platonic number can be arranged in this hierarchy: The Intelligible ONE (pure unity) IDEAL numbers (particular unities) N u m b e r s in m in d , or MATHEMATICAL numbers a) arithmetical numbers (monads) b) geometrical “numbers” (magnitudes) Numbers IN THINGS a) pluralities of things b) sensible magnitudes We have just performed an intellectual experiment that yielded us different ontological kinds of numbers, some of their characteristics, and relationships. Little of this is explicitly in the Dialogues, which certainly give no systematic treatment of the problem of number, approaching it rather in starts and snatches. While the Metaphysics does provide more information, that information is 1) subject to the provisos indicated above; b) not systematic either. However, as I said, the historical image of Plato’s thought is more important to us that the thought itself: the organon by means of which subsequent generations made sense of the Dialogues is the Metaphysics. Let us therefore second our thought experiment with reading the way Proclus (5th c. AD) treats the ontological problem of mathematical entities in the first pages of his commentary to Euclid. Proclus is clearly working with those passages in the Metaphysics in which Plato is said to have assigned the objects of mathematics a being between that of Ideal Numbers and Forms in general on the one hand, and that of sensible numbers and things in general on the other [987bl5-8, I090b33-6]. For Proclus, the opposition between the being of Ideas and of things occurs along axes such as immutable-mutable, eternal- temporary, simple-complex, indivisible-divisible, one-many, nonsensual-sensible, discrete-continuous, not having vs. having magnitude. Mathematical being necessarily belongs neither among the first nor among the last and least simple of the kinds of being, but occupies the middle ground between the partless realities - simple, incomposite, and indivisible - and divisible things characterized by every variety of composition and differentiation. The unchangeable, stable, and incontrovertible character of the propositions about it shows that it is superior to the kinds of things that move about in matter. But the discursiveness of [mathematical] procedure, its dealing with its subjects as extended, and its setting up of different prior principles for different objects - these 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. give to mathematical being a rank below that indivisible nature that is completely grounded in itself [3], Proclus, it should be remembered, is introducing a text about geometry. What he says about propositions, the discursiveness of mathematical procedure, and its dealing with mathematical entities as extended should be taken in the context of Euclidean method. How much simpler and closer to the first kinds of being are mathematical numbers, which are extended only in the sense that they are numbers of monads, whose divisibility stops at the one and whose study involves classification rather than calculation [see 48]\ As forms of knowledge differ from one another, so also are their objects different in nature. The objects of intellect surpass all other things in the simplicity of their modes of existence, while the objects of sense-perception fall short of the primary realities in every respect. Mathematical objects, and in general all the objects of understanding, have an intermediate position. They go beyond the objects of intellect in being divisible, but they surpass sensible things in being devoid of matter. They are inferior to the former in simplicity yet superior to the latter in precision, reflecting intelligible reality more clearly than do perceptible things. Nevertheless they are only images, imitating in their divided fashion the indivisible and in their multiform fashion the uniform patterns of being [4\. Thus Ideal Numbers are, numbers in things become, and mathematical numbers - the numbers we calculate with - are somehow located upon the threshold between becoming and being. 4. Platonic Signification and the Doctrine of Ideas 4.1 Number as Evidence of Non-Sensual Being Let us backtrack a little and defend our speculations regarding various ontological levels of number and their characteristics by examining the general framework of Platonic ontology. We will start with Klein’s thesis that numbers offer evidence for the existence of immutable and non-sensual being, i.e. the type of being proper to Ideas. We will then analyze those of Plato’s presuppositions which we believe instrumental to the genesis and form of Ideas, namely his understanding of the process of signification. A very important part of Jakob Klein's Greek Mathematical Thought and the Origin o f Algebra is devoted to the role played by mathematical number in forming Plato's ontology. According to Klein, for Plato the act of counting stumbles across a kind of being that is always already known, and that exists, albeit differently, both in our mind and in the world of sense; in our mind, in addition, it possesses the kind of permanence that always allows us to recognize it in any disparate and non-identical objects in the world [49-51]. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let us pause on the momentous ramifications of such thinking about number: 1) By erecting a bridge between the mind and the world of sense, numbers prove that the latter is, at least to some extent, knowable to the former. 2) Since everyone can count, everyone can see - "if only it has been emphatically enough pointed out" [Klein 51] - that non-sensual things exist and that at least some of them are not subject to change. 3) The fact that everyone can learn how to count in the same way,10 means that non-sensual units are always in some way already known to all minds. 4) Finally, the universal accessibility of counting makes whatever logical inferences we draw from the above three points equally accessible to everyone. For Klein, the role of numbers in the Platonic inquiry is not limited to their providing evidence for the existence of some form of eternal and non-sensual being. Klein goes on to analyze the passage devoted to arithmetic in the Republic, where numbers lead us to Ideas by the virtue of every sensible being at once a one and a many. Since in every sensible we may glimpse the relational coexistence of opposing dualities (my index finger is longer than my pinky but shorter than my ring finger), we are led to ask after the separate being of each term in the particular duality [523c-5a; Klein 72-79; see also Sophist 257b]. For the purposes of this dissertation, however, it seems better to momentarily place the problem of number on the back burner, and to focus on the relationship between Ideas and signification. 4.2 Ideas as Meanings One of Aristotle’s summaries of Plato’s development regards the doctrine of Ideas'1 as issuing from concerns that are best characterized as linguistic or, more accurately, semantic: Socrates was engaged in the study of ethical matters, but not at all in the study of nature as a whole, yet in ethical matters he sought the universal and was the first to fix his thought on definitions. Plato, on the other hand, taking into account the 10 People do learn how to count; however, what does it mean to "learn" how to count? If it involved only memorizing number words in a proper sequence, it could not be proven that, for example, "seven" precedes "eight.” But it can be proven that "seven" precedes "eight" —because seven monads are less than eight monads. Therefore, "learning how to count" means learning how to apply words and how to discriminate between things we already in some way know. Even in the pseudo-Platonic Epinomis, where human beings learn to count from the alteration of night and day, the capacity to do so has been given to them by God [978b-979], 11 We should note that Plato's doctrine of Ideas or Forms is not, like the Renaissance called it, a "doctrine" but a working hypothesis. As such, it is extremely fluid and incomplete —hence its difficulty. It is never clear of what things there are Ideas, and of what there aren't [see e.g. Farm. 130c-e]; the exact mechanism by which things relate to Ideas goes under a barrage of terms which are only more or less synonymous. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. thought of Socrates, came to the belief that, because sensible things are always in a state of flux, such inquiries were concerned with other things and not with the sensibles; for there can be no common definition of sensible things when these are always changing. He called things of this other sort ’Ideas’ and believed that sensible things exist apart from Ideas and are named according to Ideas. For the many sensibles which have the same name exist by participating in the corresponding Forms [987bl-10J.12 Aristotle's testimony is very much supported by the development of the dialogues: Ideas are first posited for things like justice and holiness and only then for things like man or horse. Let us therefore try to envision Ideas of things as meanings of words. We must contrast the multitude of forms in which the word "justice" can reach our hearing - the many voices, accents, intonations, as well as the many semantic contexts, - to the single meaning "justice" that all these pronunciations indicate. They do indicate a single meaning, although at the same time this meaning is shared: how else could I recognize the noises you make when you say “justice”? Since the meaning is not to be equated with pronunciation, and since there is no physical object that is justice, the meaning must be non-sensual. Since it is separate, it does not, like the pronunciations, have a beginning, middle and end: i.e., it is indivisible. Finally and perhaps most importantly, it also must be granted a certain permanence of being, for the meaning is no different at any pronunciation, and it is also prior to any meaningful pronunciation of the word.13 It is the particularity of his "semantics" that Plato approaches each of his words in isolation. The term he uses for noun is "name" [onoma]; and he conceives of each name as the name of something that exists. The Sophist describes names and verbs [see 262a] as "signs we use in speech to signify being" [261e]\ that is to say that they point at something real, with "something real" in the passage below referring to sensibles: STRANGER: I will make a statement to you, then, putting together a thing with an action by means of a name and a verb. You are to tell me what the statement is about. THEATETES: I will do my best. STRANGER: 'Theatetes sits' [...] Now it is for you to tell me what it is about - to whom it belongs. THEATETES: It belongs to me [262e-263a], '* i: Socrates “was the first to seek universal definitions." Aristotle says elsewhere, but he "did not posit the universals as separate, nor the definitions" [1078bl8-9. 30-1], 13 Should my reader object that language does not work like this, that meaning is posterior to syntax, that the view I just expressed precludes evolution, the reader will be right. But my topic is not language, but Plato's approach to it. 14 The fact that Plato is building his theory of signification solely from statements that invite a reality check, that refer to real things and can be judged true or false, without a doubt forces his conclusions. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus the first step in assuming names are names of something is that they are names of sensible things. If I say "I go to the store", the meaning of my sentence is composed of the denotations of the two names (me and the store) plus the action indicated by the verb. The strange consequence here is that the denotations have to be particular to be meaningful: it's not "I go to the store” as much as "I, Eugene Ostashevsky, go to such-and- such store." To really understand what I mean, you would have to know both me and the store in question very well. This is extremely important. An inquiry into the meaning of the word is always an analysis of the signified thing - after all, the meaning is the thing. Hence Socrates' question to Theatetes: "You do not suppose a man can understand the name of a thing, when he does not know what a thing is?" [Theat. 147b\ see also Sophist 2I8c]. To answer what the word "store" means, I must ask what makes a store a store. Signification for Plato runs into and blends with ontology. Contemplating "what a thing is" exposes Eleatic Stranger’s claim that names point to sensible objects to be somewhat preliminary. For the statement "I go to the store" is particular when it applies to me, Eugene Ostashevsky, but it also has the general property of applying to any number of such particular statements. It can point at any number of particular me's and stores and goings, but it cannot point to tables and chairs and flyings. Names treat things as belonging to kinds or species. Our ability to name is impossible without the pre-existent classification of things in the world of sense. Therefore, while names receive meanings from pointing to things, these "things" cannot be only particular things of sense. They must be the "things" that are responsible for "what a [sensible] thing is." Thus, the word "lion" can apply to a sensible lion only by pointing to whatever it be that makes this lion a lion. Let us provisionally refer to this "whatever" as the Idea of a lion. While sensible lions are subject to change, the Idea must be immutable - all lions are the same qua lions. Does this Idea also exist separately and non-sensually, in the same mode as that of Ideal Numbers? Or is there an Ideal Lion somewhere, physically roaming an Ideal Savannah? Let us now note a striking parallel between naming and counting. Three lions, three eagles, etc., are all instances of the number three in the same way as all eagles are instances of eagle, and all lions instances of lion. All eagles and all lions are different but each of them is equally eagle or lion; all triads in the world are different, but each of them is equally a three. Due to this fact, I think, Plato's concept of Ideas of species is formed by his concept of Ideal Numbers: the immutable Idea of Lion is just like Ideal Three in their autonomy, discreteness and non-sensualness. The characteristics of Ideal Numbers are regarded as the characteristics of Ideas in general. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let us return to the analogy between counting and naming. We recognize all triads as instances of three by means of the three monads in our mind, and, if we don't necessarily recognize lions and eagles by means of the name "lion" and "eagle," we must have some conception of what a lion or an eagle is, in order to be able to name them correctly. There must be some connection between our minds and the classification of things in the world. The problem with taking the analogy further is that we cannot prove our conceptions of lions and eagles to be as innate as are our conceptions of numbers underlying the ability to count. Here Plato's positing of Ideas of species again straddles his positing of Ideal Numbers: it is because we have unimpeachable proof of the existence of the latter, that we can come to terms with the existence of the former. Then, just as the numbers in the mind and the world are “caused” by Ideal Numbers, so the names and the individuals of the species are “caused” by the Idea of that species. Or rather, in Plato’s own words, "The reason why [...] things are called after the forms is that they participate in the forms" [Phaedo I02b\ my italics]. 4.3 Signification and the Problem of Nothing In previous chapters we saw that conceiving and representing numbers as numbers o f something - whether of tally-marks or of cows - makes zero, as the sign for nothing, impossible. In language, there are nouns that one would be hard put to interpret in terms of pointing to positively existing signifieds. One might expect that such nouns would cause Plato to revamp his understanding of signification; he, however, fits the meaning of the former to concord with the latter. Assuming that all names are names of something produces the primary problematic of The Sophist. If "whenever there is a statement, it must be about something" [262e], how can we say things that are not? In other words: if meanings of names are the things named, how can we make false statements? The Eleatic Stanger solves the problem by claiming that false statements are still composed of signifiers whose signifieds are things, although these signifiers are combined in a manner other than how the things are combined: STRANGER: The true [statement] states about you the things that are as they are. THEATETES: Certainly. STRANGER: Whereas the false statement ['Theatetes, whom I am talking to at this moment, flies', 263a] states about you things different from the things that are. THEATETES: Yes. STRANGER: And accordingly states things that are not as being. THEATETES: No doubt. STRANGER: Yes, but things that exist, different from things that exist in your case [263b]. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, according to Plato's understanding of signification, "there could not be a statement that was about nothing" [263d]. All statements (i.e. syntactically joined signs, see 262b-c) that seem to signify nothing, signify concrete and discrete somethings but represent the jointure of these somethings in a false manner. But, in that case, how can the terms "nothing" and "non-being" have any meaning whatsoever, since they appear to indicate that which is not something, that which by definition does not exist [237a-239b]2 To speak of what is not 'something' is to speak of no thing at all [...]. Must we not even refuse to allow that in such a case a person is saying something, though he may be speaking of nothing? Must we not assert that he is not even saying anything when he sets about uttering the sounds a thing that is not'? [237e]. To answer the paradox, Plato is forced to introduce a very important distinction. The terms "nothing" and "nonbeing" receive their meanings not from pointing at the concrete entities Nothing and Nonbeing, but from indicating a type of being, namely being- other-than, being-different-from ("When we speak of ’that which is not,’ it seems that we do not mean something contrary to what exists but only something that is different," 257b). While no such thing as Nonbeing can be, the Idea of difference or otherness, as the signified of the words “nothing” and “nonbeing,” turns out to be its ontological stand-in [see Klein 96], 15 Let us now summarize Plato's conception of signs. Platonic signification treats signifiers of class A as pointing to their single signified A severally and irrespective of syntax. This signified A is separate and endowed with being. It is, so to speak, a “thing.” Even the words “nothing” and “nonbeing,” which do not seem capable of having a signified that is a “thing,” are reinterpreted in such a way as to provide them with a signified that may exist without contradicting its own nature. As words, so objects in the world are conceived as signifiers pointing to separate, non-sensual and immutable signifieds, i.e. Ideas. It is the Idea of lion that allows a lion both to be and to be appropriately-called a lion. 15 In the same way as “being” means being this or that thing, not-being means not being this or that thing, but being some other thing. To understand the ramifications of this, let us for a moment imagine what our sign for nothing - the zero - would signify according to the Sophist. What it would signify would not be nothing in our sense of the word, but difference; i.e. the zero would be positive integer inasmuch as it not any other positive integer; 1 inasmuch as it is not 2 or 3 or 4, etc., 2 inasmuch as it is not 1 or 3 or 4, etc., 3 inasmuch as it not I or 2 or 4, etc. Thus the zero would be the symbolize the uncontrolled proliferation of things continually differentiating themselves from themselves, of Chaos resulting from the absolute absence of Sameness, of the reign not of nothing, but of no-thing. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Otherness and Ideas in the S o p h ist If Plato initially thought of Ideas as absolutely simple and autonomous, the Sophist shows they must be in some way complex and interrelated. This insight, Klein argues, led to the arithmetical structuring of the Ideal realm. While contemplating the Ideas of rest and motion, the Eleatic Stranger concludes that, while Rest and Motion are mutually exclusive, since each of them is, they cannot be isolated from the Idea of being. In other words, if Rest and Motion exist, therefore Rest, Motion and Being exist [250a-b], In addition, since Rest and Motion are the same as themselves but different with respect to each other, Rest and Motion imply not only the existence of Being, but also of Same and of Other. The dyad of Rest and Motion implies three other things apart from itself. For Klein, who devotes considerable attention to this passage, it is indicative of Plato’s new views concerning blending or methexis. Let us clarify some terms. We have quoted the Phaedo as claiming that things “participate” in the Ideas, and also Aristotle as claiming that Pythagorean “imitation” and Platonic “participation” are synonymous, although neither is sufficiently defined. Whatever the exact ontological meaning of “participation” may be, it is quite clear that a lion, if it be a lion, must participate in the Idea of lion. Since the lion is one, it must also participate in the Intelligible One. Since it is taller than some lions but shorter than others, it must participate, as the Republic says, in “the great and the small” [524c]. How far this ought to be extended is unclear: Parmenides smashes young Socrates with questions like whether there ought to be an Idea for hair [Parmenides I30c-e\. Blending, on the other hand, is the coexistence of various Ideas in the lion: of Lion, Animal, Great, Small, Rest, Motion, Being, One, and others. Now, whereas in Plato’s earlier theory the Ideas were unblended, Klein points out that in the Sophist Rest and Motion are unblended with respect to each other, but both are equally blended with Being, Other, and Same [see 25lc-253d\. In fact, Being, Same and Other pervade all things: the first inasmuch as they are, the second inasmuch as they are themselves, the third inasmuch as they are not other things. If we erect an ontological hierarchy of 1) Being Itself, 2) Ideas and 3) sensibles, we note that the further we move from Being, the greater the degree of blending of higher within lower. If, in the realm of Ideas, opposites such as Rest and Motion cannot be blended with each other, they are certainly blended in our everyday experience. Blending and participation are made possible by the interplay of Same and Other. Rest and Motion cannot be blended with each other because they are total opposites. A lion, on the other hand, participates in the Idea of lion by being to some extent same, and to 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. some extent other than it. Or, on a more general level, all beings other than Being Itself are the same as Being Itself inasmuch as they are beings, but different from it inasmuch as they are not Being Itself. Because they are not Being Itself, they can be said to be, in this way, nonbeing. Hence we can say that "what is not, in some respect has being, and conversely, that which is, in a way is not" \24ld\. The enormous problem arising here is whether we shall treat Being, Same and Other as the products of the existence of any thing, or whether we shall posit a separate Being Itself, Same Itself and Other Itself? What is prior: that which participates, or that which is participated in?16 While elsewhere in Plato the latter is the case - and therefore Being Itself, Ideal Numbers, Same and Other exist separately - the Eleatic Stranger entertains questions that make the entire edifice quite audibly creak. For the Eleatic Stranger points out that Being too "'is not' in as many respects as are other things, for not being those others, while it is its single self, it is not all that indefinite number of other things" [257a], We could speculatively posit a Being Itself that is beyond Being as mixed with Same and Other; however, Plato does no such thing. I shall return to the problem o f Being in the chapter on God and the One. As Klein points out, the fact of blending (as interplay of Same and Other) makes the nature of the reality dual. The dual nature of reality allows it to be interpreted in terms of image or copy, an image being like but not the same as its original.17 Thus, for instance, Proclus refers to mathematical numbers as “images” of Ideal Numbers [Proclus 5 'eikones', Dee *v]; in the same way, all lions and all pronunciations of the word "lion" are images of the Idea of lion as well as of Being Itself. The differences between Ideal Numbers and their "eikones" offers us an excellent model for comparing the characteristics of image or copy with those of its original in general. Let us recall the passages we quoted from Proclus. The copy is many to the original's one: an infinity of sensible two's or lions partake in the single Ideal Two or Idea of lion. The copy - two things or two meters or lion —is divisible, the original Two or Lion are not. The copy is mixed with things whereas the original is more simple. The copy is perceptible to the senses; the Ideal Two or Idea of lion, on the other hand, cannot even be imagined to take up space. Finally, originals are changeless and eternal, whereas sensible copies are subject to change, generation and corruption. 16 Or, to make an analogy with numbers, do we really have particular unities of monads, i.e. Ideal Numbers, existing separately? 17 "The fact that everything which is can be "duplicated" by an image (cf. Timaeus 52 C ), an image which is, in some enigmatic way, precisely not which it re-presents, so that it is at once this being and "another," is ultimately founded in the "mirror-like nature" of being itself: being itself has within itself the possibility of acting as the source of repetition, the ability locounterfeit itself, to con-front itself." Later, Klein adds that an image "can only 'be' if'non-being' and 'being' can 'mix’ with one another” [Klein 82, 86]. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The word "image" exhorts us to think of the relation between thing and Idea as that between signifier and signified. Participation is the ontological parallel to signification as Plato understands it: all things which participate, qua images signify that which they participate in. The world of the sensibles is a field of discrete signifiers, whereas all non- sensuals are at once discrete signifiers (e.g. of Being and Same and Other) and discrete signifieds - of things. Despite of the fact that the higher orders receive their greatest possible blending in the sensibles, the participation of the lower orders in the higher is still accomplished immediately, separately, and asyntactically. A lion and an eagle point at the Ideas of lion and eagle with no reference to the surroundings and with no reference to the onlooker. 5. Ideas as Ideal Numbers What of Aristotle’s claim that Plato’s Ideas are numbers [987b22, 1086a. 12]! How can Ideas be numbers? And which Ideas does Aristotle mean —those like Rest and Motion or Ideas of species? It is clear that, at least in Aristode’s mind, the latter are numbers, for his critique of Plato’s theory includes dismissive questions like, "If the Triad is Man Himself, what Number will be Horse Itself?" [Metaphys. 1084al4-15]. For Klein, Aristotle mistakes late Plato’s arithmetical structuring of the realm of Ideas for its arithmedcal nature, and thereby inappropriately critiques Ideas as if they were objects of mathematics. Klein’s own interpretation regards the arithmedcal structuring of Ideas as arising from the scenario of blending, which takes as its model the fact that, say, two monads joined together have the combined properties of being two and even. Applying this insight to the structure of the realm of Ideas, Klein regards Ideas as heterogenous "monads" [Philebus 15a-b] gathered together in "numbers" composing classes. Thus the Idea of man, of horse, of dog, etc., will be gathered together in the class "animal." The property "animal" will not come from a separate Idea, instead belonging to the Ideas of this class by virtue of their jointure. In the same way, the Sophist’s Rest and Modon conjointly give rise to Being [89-93]. Whether or not Klein is right, however, matters very little for our purposes. We are more concerned with how Aristode’s account might be read by somebody who does not think it errs in saying that Ideas are numbers. The quesuon such a person would pose is, what type of numbers - Ideal, mathematical or sensible? The only possible answer here seems to be the Ideal. In that case, the entire arithmetical basis for classification must have been interpreted in a manner very different from Klein’s. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Philebus does refer to Ideas as "monads" [15a-b], but we must remember that "monad" means not only unit, but also unity. As the building blocks of number, the monads are absolutely simple, indivisible and homogenous; the particular unities that are Ideal Numbers can also be thought of as "monads" for they are indivisible - but, like the Idea-monads in the Philebus, they are heterogenous, too. In this respect, the most curious feature of Ideal Numbers is their combination of simplicity and complexity. We have already shown how the unities of particular numbers of monads are approached by Greek arithmetic: it treats them as at once simple and unique but also as complex and interrelated in the sense that they can be broken down into sub-kinds of even, odd, even-times-even, odd-times-even, etc. It is certainly inviting to envision the Ideas of the species in such an arrangement; this too would define the Idea "ontologically as a being which has multiple relations to other [Ideas] in accordance with their particular nature and which is nevertheless in itself quite indivisible" [Klein 91]. For Klein class is the shared property of the Idea-monads of that class. WTiile for him there is no such Idea as that of animal, Aristotle believes there is. His critique of Plato several times returns to the hypothetical dependence of Ideas upon other Ideas: "There will have to be many [Ideas] of the same thing [...]: of a man, for example, there will be Animal, Two-footed, and, at the same time Man Himself' [991a28-29; also 1079b32-36\. Later Platonists like Boethius also posit the separate Idea of animal before the Idea of man: "if you take away 'animal', immediately also is the nature of 'man' erased. If you take away 'man', 'animal' does not disappear" [1:2, 74; used as example of ontological priority; 1:4 in the Arithmetic of Nicomachus]. In his critique of Plato, Aristotle claims that "if Four Itself is the Idea of something, say, of horse or of whiteness, 'Man' will be a part of the Idea of a horse if, say, 'Man' is a Dyad"’ [1084a24-25]. This, as Aristotle immediately points out, is absurd, but he is clearly thinking of Ideal Numbers as subject to partition by virtue of being numbers of monads, which Ideal Numbers are not. As particular unities, Ideal Numbers can in no way be considered "parts" of one another.18 Let us then think of the series Animal, Biped, Man in terms of Ideal Numbers. Let us, for example, say that Animal is like two, the first even, Biped is like four, the first even-times-even, and Man is like eight, the second even-times-even: could this not model 18 We should note that the error - if it be an error —repeats: "In Ten Itself, for example, there are ten units, and Ten Itself consists of ten units as well as of the two 'Fives'" [Metaphys. 1082al-3\. Aristotle immediately shows this to be absurd, which of course it is: 10 + 5 + 5 = 20. If this is indeed a straw-man fallacy, Aristotle conflates the Ideal Ten, which presupposes ten units in mathematical or sensible tens, with these other types of ten. Since Aristotle himself does not believe in the existence of separate Ideal Numbers, such conflation is entirely possible. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. how Man participates in Biped and Biped participates in Animal? Quadrupeds, on the other hand, could be even-times-odd, and so forth. This is a simple illustration; a more complex one would surely occur in higher numbers, taking advantage of the fact that numbers in Greek arithmetic tend to be overdetermined: thus, all even-times-evens are also squares and evens (e.g., all men are bipeds and animals), whereas not all squares are even-times-even or even (e.g., not all beautiful things are animals). The classification of Ideal Numbers in terms of sub-kinds as performed by Greek arithmetic seem capable of accounting for the complexity of the realm o f Ideas. This scenario then may be the way in which Ideas are. or at least are arranged in the manner of, numbers. My alternative to Klein is extremely speculative.191 bring it us because it appears to agree with the remarkable claim in the Arithmetic of Boethius that "God the creator of the massive structure of the world considered [arithmetic] as the exemplar of his own thought and established all things in accord with it" [/:/, 74].20 What does the phrase "God's thought" refer to? I believe that in the context of Creation the answer is "Ideas," for Ideas were regularly interpreted by Christian and some pre-Christian Platonists to be the thoughts of God: "Numbers," writes Agrippa of Nettesheim, have "the greatest, and most simple commixtion with the Idea's in the mind of God" [171].21 If this be the case, we can see why Greek arithmetic should survive for almost 2,000 years: its object was nothing less than the structure of divine thought and of the principles of creation. If in some of Aristotle's examples Ideal Numbers appear monadic, for Renaissance Platonists numbers in the mind of God are certainly not pluralities, but the objects of arithmetic, Ideal Numbers. Thus Nicholas of Cusa speaks of "the number that proceeds from the mind of God and of which mathematical number is the image." These Ideal Numbers for him are particular unities: "if you say that the triad is composed of three unities, you are like someone who says separate walls and roof make a house" [Idiota: de Mente. 6:88, 90, 5.133-134].22 !V It is rendered even more problematic by Aristotle's assertion that for Plato ten is the last Ideal Number, and there are no Ideas "of eleven or the numbers that follow" [Metaphys. 1084a26-27]. See Nicholas of Cusa attempting to work with this claim in De Conjecturis 1.3, and Agrippa in Occult Philosophy. 173; also Charles de Bovelles in my sixth chapter. :o Like the rest o f Boethius's Arithmetic, this sentence is but an abridgement o f Nicomacus of Gerasa. Nicomachus says; "Arithmetic [...] existed before [music, geometry, and astronomy] in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations and makes them attain their proper ends" [1:4], If we read Boethius's words literally, arithmetic is external to God's thought and prior to it; not so with Nicomacus. 21 This contradicts Plato's assertions of their autonomous existence; see Jones and Rich for history of Ideas as thoughts of God. 22 "Numer[us] qui ex divina mente procedit, cuius mathematicus est imago.” "Si dixeris temarium ex tribus unitatibus compositum, loqueris quasi si quis diceretparietes et tectum separate facere domum.” 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As we have seen, the arithmetical structuring of the realm of Ideas is motivated by their mixture of simplicity and complexity in that each Idea is homogenous with respect to itself and heterogenous with respect to others; at the same time, Ideas are interrelated. Ideal Numbers for Nicholas are simple and complex in just this way,23 for he regards number as composed from itself, i.e. from the Even and the Odd.24 Nicholas probably got these principles from Boethius [1:2, 76] and Aristotle. For the latter they represent the Pythagorean point of view, the odd being equated with the limited and the even - with the unlimited [to peras and to apeiron respectively, 986al7-19].25 In the extant fragments of the Pythagorean Philolaus, all things are said to be composed of limit and the unlimited, while numbers are composed of the odd and the even; things can be known only through number [in Muilachius 1]. For the Neoplatonist Proclus, the limit and the unlimited are the principles of number and of all Ideas, the former being responsible for all identities and the latter for all the differences among them [1.5-7]. In the next and final section of this chapter, I shall talk about the generation of Plato's Ideal Numbers from their two principles, the One and the Indefinite Dyad, and of Aristotle’s claim that the One and the Indefinite Dyad are the principles not only of number but of the world, since Plato's Ideas are numbers. 6. The Generation of Ideal Numbers As reported by Aristotle, the doctrine of Plato concerning Ideas and numbers is based on the Pythagorean, with the significant exception26 that the unlimited becomes "not one principle but a Dyad, consisting of the Great and the Sm all" [987b25-7\ this Dyad is called "the Indefinite Dyad" to distinguish it from the Ideal Two].27 Aristotle explains the Indefinite Dyad as the material cause of number, whereas the One is its formal cause, "for n "Immo dum in numero non nisi unitatem conspicio, video numeri incompositam compositionem et simplicitatis et compositionis sive unitatis et multitudinis coincidentiam. Immo si adhuc acutius intueor. video numeri compositam unitatem," [Idiota: de Mente, 6:91, 5.135]. u "Nam numerus est compositus et ex se ipso compositus — ex numero enim pari et impari est omnis numerus compositus — sic numerus est ex numero compositus" [Idiota: de Mente, 6:90, 5.134], 25 The reason for the identification appears to be that the even is capable of being "divided into equal parts without the one coming between the two parts,” and the odd is not [Boethius, 1:3, 76-77]. The Pythagoreans also associated the limit and the unlimited with the one and the many respectively [Metaphys. 986a24-27]. 26 Other exceptions are differentiating between numbers and sensibles - but those Pythagoreans who said things exist by imitating in number surely must have done that already - and positing of mathematical numbers between the Ideal Numbers and the sensibles [987b25-29\. 27 For Sayre’s use of the Philebus to support Aristotle’s claim that the Pythagorean unlimited gives rise to Plato’s Indefinite Dyad, see his chapter 3. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from the Great and the Small and by participation in the One come the Forms, and these are Numbers" [987b20-22; see also 987b34-8al].2% It is of the utmost importance for the understanding of Ideal Numbers as particular unities studied by Greek arithmetic that their generation is not accomplished by means of successively adding ones, and that the sequence of natural numbers is merely its effect. Instead, our last passage from Aristotle presents the series of Ideal Numbers as images of the One upon the Indefinite Dyad. These images differ from one another because the Great and the Small participates in the One in conjunction with previous images, i.e. previous numbers. For instance, "according to these thinkers Four was generated from the first Dyad [i.e. Ideal Two] and the Indefinite Dyad" [1081b21-22\. The latter remains the sole material cause of any Ideal Number (e.g. Four), since other Ideal Numbers (e.g. Two) are also composed of it. Aristotle's example of the generation of Ideal Four implies that Ideal Numbers are generated according to sub-kinds: qua even, odd, even-times-even, odd-times-even and so forth. Among the passages in the later Dialogues that recall Aristotle's picture of Plato's doctrine concerning numbers is the hypothetical generation of numbers in the Parmenides. Here, Zeno is working not with the One and the Indefinite Dyad, but with the three notions of "being," "one" and "different." Nor is there any talk of participation, imaging, etc.: the numbers one through three are created by combining any two of them into a pair, and then adding one. However, once we get three and two, the addition stops: Three is odd, two even. Now if there are two, there must also be twice times, if three, three times, since two is twice times one and three is three times one. And if there are two and twice times, three and three times, there must be twice times two and three times three. And, if there are three which occur twice and two which occur three times, there must be twice times three and three times two. Thus there will be even multiples of even sets, odd multiples of odd sets, odd multiples of even sets, and even multiples of odd sets. That being so, there is no number left, which must not necessarily be [I43d-I44a]. If we do accept most of what Aristotle says about numbers in Plato as true for Plato's last period, we must regard passages such as this as preliminary sketches, tests of thought. And the problem with this test, as Sayre points out [91; see Metaphysics 987b34] is that Zeno leaves the primes out in the cold. 28 "Since the Forms are the causes of all other things, he [Plato] thought the elements of the Forms are the elements of all things. As matter, the Great and the Small are the principles; as substance [ousia], it is the One. For from the Great and the Small and by participation in the One come the Forms, and these are Numbers [987bI8-22]." 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. So far we have spoken of Ideas being structured like numbers, but Aristotle reports them to be numbers. If we trust his claim that Plato, like the Pythagoreans, "believed that the Numbers are the causes of the substance of all other things" [987b23-24\, everything said so far about the generation of Ideal Numbers applies to "the Forms, and these are Numbers" [987b22]. According to the Metaphysics, Plato uses only two causes, the cause of the whatness and the cause according to matter (for the Forms are causes of the whatness of the other things, and the cause of the whatness of the Forms is the One). It is also evident what the underlying matter is, in virtue of which the Forms are predicated of the sensible things, and the One is predicated of the Forms; this is the Dyad, or the Great and the Small [988al 1-14], The ontological picture in this extraordinary passage, and also in Metaphysics 987bl8-22, is, in Sayre’s intepretation, as follows: I) Ideas, as Numbers, come into being by interaction between the One as formal cause and the Indefinite Dyad as material cause; 2) The sensible things come into being by the interaction between the Ideas/Numbers as formal causes, and once again the Indefinite Dyad as material cause [89-95, 14-15]. Sayre also points out the following. In this scheme, the only ontological difference between Ideal Numbers and sensibles is that the former are a single and the latter a double imprinting of the One upon the Indefinite Dyad. Hence Ideal Numbers and things are ontologically homogenous ("the elements of Forms are the elements of all things," Metaphysics 987b 19). But this means that all things - both Ideas and sensibles - are, as the Pythagoreans opined, nothing but number.29 This conclusion is, of course, beautiful, but it is problematic in equal measure. If Ideas are to be equated with Ideal Numbers, how do we explain the extent of the heterogeneity of the cosmos? How do explain qualities? How do we explain abstractions like justice, wisdom and so forth, which caused Plato to postulate Ideas in the first place? The last question is not so hard as it seems, if we remember the Pythagorean attempts to numeralize abstractions, with, for instance, 2 being the number o f opinion, 4 of justice, 5 of marriage [see Metaphys. 985b29-30, 1078b22-23, Danzig 40]. In his unrecorded "Lecture on the Good," Plato did in fact assign good to the One and evil to the Indefinite Dyad \Metaphys.988al4-l7; Sayre 77-78]; this must arise from the notions of harmony as unity and discord as disunity. And yet these scattered bits of information are terribly insufficient. :9 If, for Boethius, arithmetic is the exemplar of the realm of Ideas, when God created things, "number was the principal exemplar in the mind of the Creator (principale in animo Condiioris exemplar)" [I: 2, 76: qtd. in Agrippa, 170; Dee *j'; compare with Nicomachus 1:6]. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 NUMBER IN JOHN DEE’S MA THEMA TICALL PRAEFACE B C M a ip H B a io c b b B a c , o HHCJia, H Bbi MHe BHiieTHCb o n eT b iM H b 3 B e p n , b h x u iK y p a x , P y K o ii o irap aio iU H M H C Ji H a B b ip B aH H b te jiy 6 b i. T aM BepTeJIHCb BKOCb H BKpHB HHCJia no .M b i h m o p a b H ecym ecT B eH H O M ox K p b iB cym ecT B yxom ee 3 p a ‘ 1. Number in Christian Platonism B o e th iu s 's L a tin a d a p ta tio n o f th e N e o p la to n ic a r ith m e tic o f N i c o m a c h u s o f G e r a s a tr a n s m itte d G r e e k a r ith m e tic t o th e M id d le A g e s . T h u s th e m e d ie v a l u n d e r s ta n d in g o f n u m b e r c a m e o u t o f th e G r e e k m o d e l d e s c r ib e d in m y s e c o n d c h a p te r , a n d v ie w e d its o b je c ts a s p lu r a litie s b e l o n g i n g to s u b - k i n d s s u c h a s th e e v e n a n d t h e o d d . M e d ie v a l d e s ire to s tu d y n u m b e r c a m e f r o m r e lig io u s a n d c o s m o lo g i c a l c o n s i d e r a t i o n s . A s is w e ll k n o w n , P la to 's Timaeus, w h o s e d e m iu r g e s h a p e s m a t t e r b y d i v i d in g it a c c o r d in g to h a r m o n ic r a tio s , e x e r te d a s t r o n g in f lu e n c e u p o n th e i n te p r e ta tio n o f th e C r e a tio n a c c o u n t in Genesis.2 Genesis d o e s n o t m e n tio n a r ith m e tic , b u t in Wisdom o f Solomon 11:20 G o d is s a id to h a v e "Created all thynges, in Number, Measure, and Waight" [a s q td . b y D e e , A iiijv]. T h is v e r s e b e c a m e th e m a in S c r ip tu r a l j u s t i f i c a t i o n f o r th e q u a d r iv i u m .3 B o th P la to a n d th e B ib le a p p e a r to s a n c tif y th e c l a i m , p u t f o r th b y th e C h r is tia n "All thynges (which from the very first originall being o f thinges, P la t o n is t B o e th iu s , th a t haue been fram ed and made) do appear to be Formed by the reason o f Numbers. For this was the principall example or pattem e in the minde o f the Creator" [1:2, 75, a s q t d . b y D e e , * jr].4 B o e th iu s 's w o r d s a s s u m e th e p r id e o f p l a c e in n e a r ly e v e r y m e d i e v a l a n d Velimir Khlebnikov, "Chisla" in Khlebnikov, 79; Aleksandr Vvedenskii, "sneg lezhit,” in Poety gruppy OBERIU. 124 : See e.g. Pelikan 35-37. For the tie between medieval mathematics, music and architecture, see Simson. ch. 2. and Male 5-14. 3 In the seventeenth-century (but very archaic) Arifinologia of Nikolai Spafarii, Arithmetic justifies herself by God's command in Gen 1:28 to "Be fruitful [i.e. increase] and multiply," crescite et multiplicamini [37]. i "Omnia quaecunque a primaeva rerum natura constructa sunt, Sumerorum videntur ratione formata. Hoc 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Renaissance defence of mathematics, including texts teaching the use of Hindu-Arabic numerals [Agrippa 170, Swetz 41, 326, Evans 116]. Sacrobosco very indicatively expands upon them, echoing a parallel claim by the Pythagorean Philolaus (preserved by Johannus Stobaeus), that "all things which can be known have number; for it is not possible that without number anything can be conceived or known” [Danzig 43; original in Mullachius I]: Boys seying in the begynnyng of his Arsemetrike: Alle thynges that bene fro the first begynnyng of thynges have proceded, and come forthe, And by reasoun of nombre ben formede; And in wise as they bene, So owethe they to be knowene; wherfor in vniuersalle knowlechyng of thynges the Art of nombring is best, and most operatyfe [in Steele, 33]. Nothing is what it seems. The "Boys" of this passage is actually Boethius (Fr. Boece), "arsemetrike" - arithmetic, and the "vniuersalle knowlechyng of thynges" enabled by the "Art of nombring" does not look forward to the Scientific Revolution. In the thirteenth century the words "vniuersalle knowlechyng" could not apply to knowledge of the sensible world in itself; no one would call knowledge that ignores God and angels "vniuersalle." Rather, Sacrobosco’s claim is fundamentally Platonic;5 it rests - whether consciously or not - on Plato's notion of knowledge, which, while encouraging study of number, made applying number to the study of the sensibles in themselves theoretically impossible. We have seen this in the preceding chapter, while discussing the argument put forth in Republic 527a-b that geometry is "the knowledge o f that which is euerlastyng" [Dee a.ijv].6 Both arithmetic and geometry "force the soul to tum its vision round to the region where dwells the most blessed part of reality" [526e]. The movement here is ontological rather than mathematical. Numbers in things and mathematical numbers lead enim fuit principale in animo Conditoris exemplar. “ An abridgement of Nicomachus, 1.6. 5 Sacrobosco is clinging to the Greek definition of number —numbers as collections of monads and numbers as particular unities: "Sothely .2. manere of nombres ben notiflede; Formalle, as nombre is vnitees gadrede to-gedres: Materialle, as nombre is a colleccioun of vnitees" [33; "formal" = Ideal]. Sacrobosco also gives an alternative Aristotelian definition, where number is only a collection of unities. 6 If the Republic indicates geometry's importance for philosophy, another Platonic passage taught Renaissance Platonists how geometry may be used to prove the truths of religion. In Meno 82b-86b. a slave-boy finds the side of a square whose area is doubled - at Socrates' "prompting," of course; since the slave boy had never studied geometry in this life, Socrates proposes geometry as evidence for anamnesis. and therefore the immortality of the soul. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. us to inquire after their signifiers, Ideal Numbers, whose existence is no less, and in fact more real, than that of sensibles. Syntactical jointure of numbers (mathematical operations) becomes a distinctly secondary procedure. To focus on calculation at the the expense of philosophy may be likened to studying the phrase "the dog barks" without knowing what the words "dog” or "barks” mean. The employment of mathematics in modem science, on the other hand, rests on the refusal to contemplate the ontology of number. The modem number concept has to do not with things, but relations; number therefore depends on operations, on syntax. As Jakob Klein argues, modem physics becomes possible only when Descartes identifies "by means of 'methodological' considerations, the 'general' object of [his algebraic] mathesis universalis - which can be represented and conceived only symbolically - with the 'substance' of the world, with corporeality as ’extensio"' [197]. For us, numbers are posterior to things and to the laws of algebra. An aspect of this difference may be spoken of as the orientation of number. In Republic 525c-e, the role of arithmetic in "facilitating the conversion of the soul itself from the world of generation to essence and truth" is explicitly presented as arising from studying "the nature of number" or "pure number" as opposed to calculation, which deals with numbers of sensibles, and therefore keeps the soul in the world of objects, below. This orientation is implicit in all Greek-type arithmetics, including that of Boethius with its division of number according to the sub-kinds. The same orientation acquires a specifically religious form in Augustine's dialogue on free will, De Libero Arbitrio. For Augustine as for Plato, numbers "transcend our minds and remain unchangeable in their own truth," at the same time as making themselves "present to all who think" [2.11, edit, princ. 126; 2.8, 80]. Curiously, Augustine is more syntactically minded than his predecessor: his argument that numbers both really exist and are not first known through the senses rests on the immutability of the number sequence and of calculations: "seven and three are ten, not only now, but forever" [2.8, 55]. If for Plato, the presence of number in things instigates philosophical activity, for Augustine it shows that all things remain in God: Look at the sky, the earth and the sea, and at whatever in them shines from above or crawls, flies, or swims below. These have form because they have number. Take away these forms and there will be nothing. Whence are these except from number? Indeed, they exist only insofar as they have number [2.16, 164]. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus number is the divine and eternal stamp upon the face of otherwise absolute mutability. Not only does it permit the existence of something instead of nothing ("If [number] is removed, the mutable dissolves into nothing"), but it also provides us with ample evidence for the existence of God himself: "do not, then, doubt that there is some eternal and immutable Form [...through which] every temporal thing can receive its form and, in accordance with its kind, can manifest and embody number in space and time" [2.16, 171]. Number in things is the channel bearing us back to their - and our —Creator; this is its prime "scientific" purpose: When you begin to slip toward outward things, wisdom calls you back, by means of their very forms, so that when something delights you in body and entices you through the bodily senses, you may see that it has number and ask whence it comes [2.16, 163]. The less mathematically minded Renaissance Platonists like Marsilio Ficino (1433-1499) retain the orientation of Plato's number, but treat the presence of number in things solely as an invitation to a speculative voyage. In his Letter on Plato, Ficino affirms mathematics to be the stepping-stone to the study of philosophy precisely and exclusively for the reason that mathematics separates our thought from the senses and directs it to the eternal non-sensuals. Ficino is interested in the ascent and nothing else: to practice mathematics, as to practice philosophy, is to apprendre a mourir by rehearsing abandonment of the sensibles.7 Other thinkers, like Ficino's pupil and friend Pico della Mirandola (1463-1494), go further. Pico combines Ficinian Platonism with the Kabbalah as well as with the long medieval exegetical tradition, stemming from the Church Fathers and including Augustine, of interpreting numbers in the Scripture as symbols of divine truth [see Isidore of Seville, Etymologiae 3.4, Rabanus Maurus De Universo 18.3, Donne, Essays 52-55, Bovelles' Duodecim Numeris, Spafarii's Arifmologia, Male 10-2, Crossby 45-6]. According to John Donne, "the super-seraphical John Picus” in his "Judeao-Christian ' "Quoriiam vero numeri <Lfigurae, <&motuum rationes ad cogitationem potius, quam ad sensus exteriores pertinent, horum studio animus non modo ab appetitu corporis, sed ab eius quoque sensibus separatur & ad interiorem cogitationem se confert. Quod quidem mortem commentari est (quod in Phaedone Plato scribit esse philosophantis officium) per quod &. similes Deo reddimur ut ex Phoedro Theatetoque dicitur" [Opera, 1.762; contrast this with Henry of Ghent's earlier and non-Platonic distinction between metaphysics and mathematics, explaining why mathematicians are always melancholy, in Klibansky, Panowsky and Saxl 338-9, n. 187], 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pythagoras" actually proved "the numbers 66 and 99 to be identical if you hold the leaf upside down” [Essays xix].8 When he was 23, Pico published his sensational Conclusion.es nonagentae or the 900 Theses, which he wished to defend publicly but Innocent Vin just said no. The 900 Theses also quote Philolaus to the effect that "By Numbers, a way is had, to the searchyng out,and vnderstanding o f euery thyng, hable to be knowen" [108, qtd by Dee *jv]. Most of the74 questions that Pico promises to defend "per viam numerorum" seem to engage ontology rather than properties of particular numbers or, God forbid, calculation. As an example, let us translate the first six: 1) Whether God exists. 2) Whether He is infinite. 3) Whether He is the cause of all things. 4) Whether He is the simplest (i.e. as opposed to compound). 5) Whether He thinks. 6) How He thinks.9 If, using numbers to prove that God exists, we procede in the manner of Augustine's De Libero Arbitrio, we shall talk about the being of number in general. We may also employ a different strategy of identifying things with numbers and then examining the former as if they were the latter. In the second scenario, God presumably is the one,10 the necessity of his existence following from the necessity of the existence of the one if there be any number at all. In this latter method we perceive the direct dependence of medieval and Renaissance number symbolism upon the Greek concept of number. Pico’s other 8 Donne vaccilated in his opinion of Pico, whom, in a lovely act of self-portraiture, he calls "a man of an incontinent wit, and subject to the concupiscence of inaccessible knowledge and transcendencies" [Essays 13]. In the Essays he decides that, "Because Cabalistick learning seems to most Occupatisima vanitas," he shall forbear relating what Pico says and merely refer to him (and to Reuchlin and Zorgi, his followers), but. really, Pico's philosophy offers so "many delicacyes of honest and serviceable curiousity, and harmless recreation and entertainment"... [10]. For Pico's influence on Donne's poetry, see Freccero. 9 " 1. Utrum sit Deus. 2. Utrum sit infinitus. 3. Utrum sit causa omnium rerum. 4. Utrum sit simplicissimus. 5. Utrum sit intelligens. 6. Quomodo Deus intelligat" [108]. 10 "Numerorum enim ratio in multis juxta allegoriam typica significatione mysterium nobis venerandum ostendit. Unde primus numerus, hoc est, unus ad unitatem deitatis refertur. De quo in Exodo scriptum est: Audi, Israel, Dominus Deus tuus, Deus unus est (Deut. VI)", Rabanus Maurus, De Universo 18.3. For God as the one, see my chapter 6. 74 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. questions presuppose Greek arithmetical concerns: 68, for instance, refers to Augustine’s opinion that God created the world in six days because six is the first perfect number, whereas thesis 40 promises to explain whether Arius, Sabellius, Wycliffe, or the Roman Catholic church is right concerning the nature of the Holy Trinity, presumably by analyzing the nature of the three as union of three monads.11 There is a fundamental problem in such reasoning. When we explain the nature of the divine realm, including that of God and the Trinity, by means of numbers, are we claiming that a) the divine realm is structured numerically, and therefore numbers are prior to God; or b) that numbers are not prior to God, and therefore all our explanations are symbolic? To a Christian the first alternative is impossible, yet should numbers be posterior to God, God is not one, nor do the Father, the Son and the Holy Ghost form a trinity. The only escape-route seems to lie in predicating numbers on God in such a way that they become part of him. While Plato's Ideas are autonomous, later intepreters locate them in the mind of God as his thoughts. The Middle Platonist Albinus writes, "if God is mind, he has thoughts and these are eternal and immutable; if this is so, Ideas exist” [qtd. in Rich 129]. So Plotinus, who places the Ideas in the Nous; so also his followers. At the same time, Neoplatonist thinkers accept Aristotle's identification of Ideas and Ideal Numbers. Nicomachus of Gerasa and Boethius locate arithmetic and number in the mind of the Creator as "the exemplar of his own thought" as well as of "all things which have been created" [Boethius 7.7-2, 74-75; Nicomachus 1.4 and 7.6 offers greater detail]. Therefore, Ideal Numbers are the thoughts of God.12If this be so, there is a way in which God can both think and be one. To pursue the self-referential aspect of divine thought further would call for too long of an excursus (but see my sixth chapter for Eriugena on God willing himself into being). Our analysis of Dee's Mathematicall Praeface will consider the relationship of number (as divine thought) to things. Before we pass to Dee, however, let us note the same relationship in the form given to it by De Occulta Philosophia of Agrippa of Nettesheim (1486-1535), Pico's follower and Dee's predecessor in the field of magia.'3 The second book of the Occult Philosophy commences with this claim: " Thus Sir Thomas Browne contemplates "the Trinitie of our soules" and "the Triple Unity of God" as "the distinct number of three, not divided nor separated by the intellect, but actually comprehended in its Unity" [73], 11 Male attributes this belief to Augustine, without, however, giving a citation, 10. i3 For the influence of the Florentine academy on Agrippa, see Klibansky, Panowsky and Saxl. 350-60. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Doctrines of Mathematics are so necessary to, and have such an affinity with Magick, that they that do profess it without them, are quite out of the way, and labor in vain, and shall in no wise obtain their desired effect [167]. For Agrippa, the benefits of mathematics for "Magick” seem to be dual. First, there is the none-too-supematural manufacture of automata - like the mechanical flying scarab John Dee was to make for an Aristophanes production while a student a Cambridge - which involves mathematics in the sense of calculation [Agrippa 168-9; Dee., Autobiographical 5-6], The second use of mathematics focuses on the properties of specific numbers as objects of Greek arithmetic. Here Agrippa appears as a Platonist, distinguishing between the "rationall, & formall number" and that which is "materiall, sensible, or vocall, the number of Merchants buying, & selling" (he is paraphrasing Rep. 525c). The first kind of number has "the greatest, and most simple commixtion with the Idea’s in the mind of God." For this reason, "all things that are, and are made, subsist by, & receive their vertue from numbers.” Thus, for instance, "the hearb which is called Cinquefoil [...] resists poysons by vertue of the number of five”: i.e. by having five leaves, the plant participates in the Ideal Five, which endows it with its curative properties [171-2]. Agrippa does not state the reason why five is an antitoxin: perhaps it lies in five's being the first circular number (i.e. number that ends in itself when squared) or perhaps in some other definition of five in Greek arithmetic. In the cinquefoil and other examples, we see Agrippa fusing Plato’s putative doctrine of Ideal Numbers with number symbolism in the exegetical and encyclopedic tradition of medieval arithmologies, i.e. lists of things which come in a certain number, such as four Evangelists, four elements, four parts of the world, four humors. Four Horsemen of the Apocalypse, etc. Agrippa's attempt to employ Greek arithmetic in order to explain the properties of things in this manner seeems extremely curious and worthy of consideration. We may regard it as the perfect and perhaps the only possible form of studying the world by means of number while preserving its Platonic, ascendent orientation. When I first read Agrippa, his seemed the incipient stage of a project which would induce the properties of all Ideal Numbers by classifying all things in the world in the manner of arithmologies. My interpretation was, I think, wrong, since Platonism cannot recognize inductive procedures of this kind. We know a thing only through its causes, not its effects. For all their linguistic similarity, the Philolaus thesis of Renaissance Platonism ("By Numbers, a way is had, to the searchyng out, and vnderstanding ofeuery thyng, 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. hable to be knowen") and Galileo's famous slogan that the universe can be understood once we learn its language, the language of mathematics,14 have entirely different meanings. On close examination, even their phrasing turns out to be not all that similar: Philolaus speaks of numbers as discrete entities, whereas Galileo speaks of language as a whole. The discreteness of Philolaus’s numbers is fully in line with Plato's understanding of signification as described in the preceding chapter, whose main characteristic lies in refusal to assign any meaning-producing function to syntactical ties. In referring to language as opposed to linguistic entities, on the other hand, Galileo intends mathematical operations first and foremost. We know this because his mechanics focuses not on the "nature" of things, but on the relationships between them. Although the orange-and-cannonball anecdote is spurious, it nonetheless illustrates the fact that, for the purposes of Galileo (or at least late Galileo), the "whatness" of bodies is irrelevant. The move from the Platonic orientation of number to its ignoral by early modem science in favor of calculations conducted with "the number of Merchants buying, & selling" cannot be accomplished by means o f a continuous conceptual evolution, but requires - literally - a volte-face, a one-eighty. In this my fourth chapter, I shall be considering John Dee's Mathematicall Praeface to the Elements of Geometrie o f Euclid o f Megara, composed in 1570. What I find so curious about Dee's text is its lack of agreement between theory and practice, same lack as we find in early texts on Hindu- Arabic numerals. Although Dee's scientific work employed mathematics in the modem manner, when attempting explain number, he lapses into Pico’s version of ancient pieties. The result, I think, is a build-up of tension, in which appear the first signs of a revolution: omissions and inconsistencies. In the case o f Dee, these may be seen in his classifying numbers not according to sub-kinds (in a preface to Euclid!), as well as in his view of arithmetic as the science not only of number, but also of operations. The waters of practice start ever so subtly undermining the limestone of theory. u In II Saggiatore (1623), he writes that “Philosophy is written in this grand book, the universe, which stands continually open to our gaze, but the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures without which it is humanly impossible to understand a single word of it; without these one wanders about in a dark labyrinth." [Galileo 237-8]. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. A Mathematical Preface to The Mathematicall Praeface At the same time as philosophers were thinking of number platonically, a quiet revolution was underway at the introductory level of mathematical literature. Greek-type arithmetics are all very interesting, of course, but they are of no use whatsoever when it comes to running a castle or buying wool. The expansion of trade and industry in the twelfth century and again in the Renaissance created a pressing demand for a different type of textbook, one that taught calculation.15 Now calculation, as we saw in my previous chapter, cannot be carried out in Roman numerals; its media were first counters and then Hindu-Arabic notation. One may think of textbooks of calculation as writing manuals, presenting their subject as a way of manipulating counters on the table or numerals on the page. Devoted to mathematical grammar (the morphology of notation and the syntax of operations), these aids to "Merchants buying, & selling" gave the Platonic orientation of number only the most rudimentary lip-service - of the kind we have just seen in Sacrobosco - in the introductory paragraphs, to forget about it in the main and functional body of the text. Their only other engagement with theory occurred when explaining how Hindu-Arabic numerals signified, which they attempted, and ultimately failed, to do within the classical paradigms of signification. Although the new textbooks taught what the Greeks called "logistic", and not "arithmetic," at a certain point - no later than the fifteenth century - they started calling themselves "arithmetics" as well. The name is often modified by the adjective "practical," to distinguish them from the Greek-type arithmetics (first and foremost that of Boethius), called "speculative." Speculative arithmetics were studied in the universities as the numerical part of the quadrivium, whereas practical arithmetics addressed themselves to merchants and professionals. The former tended to be in Latin, the latter - in the vernacular. Between 1478 and 1500, 30 practical and 26 speculative arithmetics saw print [Karpinski 66]. The first printed edition of Euclid appeared in 1482, only four years after the first printed arithmetic, the practical arithmetic of Treviso (translated in Swetz). Geometry never experienced the magnitude of arithmetic's divorce between theory and practice. The "speculative" geometry of Euclid explains why the theorems used in the "practical" geometry of land-surveying or architecture are true, whereas speculative arithmetic performs no such service for the laws of calculation. Hence both geometry and calculation have a relation to the world of the sensibles that Greek-type arithmetics do 15 For bibliography of textbooks on counter-casting, see Evans, 129, n.6, also Swetz 32; on Hindu-Arabic numerals, Evans, n.13; for definitions of number in the latter see Swetz 41. 78 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. not, and both "practical" geometry and calculation rest on the assumption that they can offer "knowledge" about the sensibles - albeit knowledge of the limited and utilitarian sort, that says nothing about what the sensibles actually are or how they came to be, and therefore is not knowledge in the Platonic sense. But it does build houses and fill pockets. A third kind of mathematics, algebra, came to Europe at the same time as Hindu- Arabic numerals; its Muslim source was again Al-Kworezmi, and one of its first western texts was again the Liber Abbaci of Fibonacci. It was in little use before the end of the fifteenth century. Even then, early algebra is impeded by its lack of symbolic notation on the one hand, and its attachment to real-life problems on the other. Initially expressed by words pointing to things and therefore entirely referentialist, and then by syncopations like p for plus, "more," it would not have recognized a solution ending at an indeterminate. Its very function was conceived not as a theory of equations, but in terms of "findyng an vnfcnowen number, by Addyng o f a Number, & Diuision <&aequation " [Dee, *ijv]- Tellingly, this concrete unknown was in Latin called res, "thing," and in other European languages accordingly. Hence another name for algebra was the Cossic Art or the Rule of Coss, from the Italian word for thing, cosa. The determinate nature of the algebraic res renders my calling algebra "a third kind of mathematics" anachronistic, for it remained a type of calculation (practical arithmetic). In the sixteenth century, algebraic notation moves towards the symbolic, progressively shedding the trappings of referentialism. Our most basic operational signs appear: thus the "+" sign evolves as the syncopation of et, and the sign from the the tilda ~ over the m for minus, "less." More importantly, various signs emerge to represent various powers of number, including powers of the unknown. These so-called "cossick numbers" are quite difficult for the modem reader accustomed to a, a2, a 3orx, x2, x 3, etc., difficult because a) they look weird, if you excuse the colloquialism; b) the exponents and the signs for unknown quantities of that power are not distinguished, so even in Stevin 42 can be mistaken for 4x2; and c) the same unknown with different exponents is represented by different signs.16 Despite the clunky and sporadic nature of early symbolism, mathematicians start directing their attention towards the general properties of the equation itself, as may be seen in the solution of cubic equations by Tartaglia (1499-1557) and Cardano (1501-1537). Yet, technical accomplishments notwithstanding, the theoretical understanding of number and the conception of the unknown as "res” remain unchanged [Klein 148]. Operations are conducted in terms of 16 In it only with Viete (1591) that a sign, both separate and consistent regardless of its powers (expressed rhetorically as "plane", "square”, "cube", etc.), appears for designating the unknown. 79 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. real-life problems, and sixteenth-century algebra is still referentialist in its outlook. Thus, even though technical advances bring mathematicians face to face with radical and negative numbers, the former are admitted in the solution only if the res in question is a magnitude, whereas negatives are not admitted into solutions at all. In England in 1570 there were few, if any, people as aware of the state of mathematics as John Dee. At the University of Paris twenty years earlier, he "did undertake to read freely and publiquely Euclide's Elements Geometricall, Mathematice, Physice, et Pythagorice\ a thing never done publiquely in any University of Christendome." Although he was only 23 at the time, his lectures caused such a sensation "that the mathematical! schooles could not hold them; for many were faine, without the schooles at the windowes, to be auditors and spectators, as they best could helpe themselves thereto" [Autobiographical 7], His subsequent mathematical work was conducted in the fields of navigation, cartography, and astronomy: the incomplete lists of his writings compiled in 1592 contains many items in these disciplines [Autobiographical 24-7, 73-8; fuller bibliography in Clulee 302-9]. Dee was by far the most important Englishman of his generation to work on the mathematical aspects of navigation. An account of the contributions of this personal friend and scientific correspondent of all the important cartographers of his day, teacher of all major English explorers [lists in Taylor 76], and mathematical advisor to the Muscovy Company, occupies the major part of E. G. R. Taylor's seminal Tudor Geography 1485-1583. One of the mathematical aspects of navigation is, of course, astronomy; Dee's work in this field, where he showed himself a proponent of the Copemican theory (the debate revolved around whether mathematical models reflect the actual state of the celestials), is covered by Francis R. Johnson's Astronomical Thought in Renaissance England. So Dee is professionally at the forefront of applying mathematics to the study of the sensibles, of assuming the homogeneity of space as object of mathematical operations. At the same time, Dee is, famously, a magus, that is to say in the same line of work as Agrippa. Frances Yates and her students have devoted considerable attention to this aspect of his endeavors. A search for Dee on Web will yield information like "He put a hex on the Spanish Armada which is why there was bad weather and England won" (http://www.johndee.org/). I don't know about no hex, but Dee did serve as the astrological adviser to Elisabeth, and his experience in alchemy was by no means limited to the theoretical expostulations of the Monas Hieroglyphica (1564), of which "her Majesty had a little perusin" [Autobiographical 19]. Although alchemy and especially astrology involved a good deal of calculation, we can rest assured that the theory behind them was Platonist in the fashion of Agrippa. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As a Platonist magus on the one hand, and as a pure and applied mathematician on the other. Dee operated with two contradictory approaches towards number in things. For me, the most fascinating aspect of his Mathematicall Praeface is to what extent he manages - or even tries - to reconcile them. 3. Arts Mathematical Derivative in Dee's Mathematicall Praeface John Dee's Mathematicall Praeface to Sir Henry Billingsley's translation of Euclid's Elements was instrumental in sparking interest in mathematics among the Elizabethans.17 Prior to the 1540s proselytizing of Robert Recorde, the teaching of even the most basic mathematics was minimal (one noted pedagogue said it overtaxeth the memory [Ascham qtd. in Sanford 378]); Billigsley's Elements ushered in the second and much more extensive level of England's contact with the discipline. The Mathematicall Praeface is, essentially, a Renaissance defence of its subject, although it is also much more than that. It is composed of three parts: a theoretical section, a section arguing the usefulness of mathematical study to various professions and a section describing the branches of mathematics pure and applied. The middle section, where mathematics is shown to be useful to merchants, goldsmiths, military officers and lawyers, is quite traditional: we find the like in the prefaces to almost any Renaissance textbook of practical arithmetic. To this section, whose topic is the utility of mathematics in general, we should add the last two pages of the Praeface, which defend the act of translating Euclid into "our vulgare Speche," and thereby taking theoretical geometry outside university walls into the hands of "Scholer[s]" (i.e. schoolboys), "good and pregnant Englishe wittes" (keeps 'em off the streets, says Dee), and, last but not least, "many a Common Artificer [...] in these Realmes of England and Ireland, that dealeth with Numbers, Rule, & Cumpasse” [a.iijv-a.iiijr]. As is well known, Euclid’s importance to the Renaissance lies not only in the multi-faceted and encyclopedic nature of his work, but also - primarily, for some - in its teaching of the geometric method, "I'ordre en methode d'escrire les Mathematiques' [Stevin qtd. in Klein 189]. By fulfilling and surpassing the needs of mathematical practitioners - after all, why does a pilot need to know the proof of the Pythagorean theorem? - the 1570 edition of The Elements self-consciously transformed this class of people into practicing mathematicians. 17 In its article on Dee, the Encyclopedia Brittanica attributes all or part of the translation to Dee himself. Dee, however, claims merely to have annotated and added to it [Autobiographical 24], 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The third section of Dee's Mathematical Praeface is devoted to pure and "vulgar” mathematics as well as to the branches of "Artes Mathematicall deriuatiue" [a.iijr]. Afterwards, Dee was to claim that many of the latter were by him "wholy invented (by name, definition, propriety and use)" [Autobiographical 73], but, aside from the naming, this is quite an overstatement. Descending from the first of the "principall artes mathematicall," i.e. the arithmetic of natural numbers, are the "vulgar" arithmetics of fractions, irrationals, and "cossike Numbers" (i.e. algebra). From the other of the "Mathematicall Fountains," Euclidean geometry, comes "Geometrie, vulgar: which teacheth Measuring" [a.iijr'v], and, from that, the "Artes Mathematicall deriuative" of geography, "hydrographie," chorography (all types of cartography), stratarithmetrie (mathematical treatment of troop arrangement)18 and others. As one would expect, many are in some way related to navigation, the art that "demonstrateth, how, by the Shortest good way [...] a [...] Shippe, between any two places [...] may be conducted" [d.iiijv]. Since the ship travels the surface of a sphere, unless it is moving along a meridian or the equator, "the Shortest good way" is not sailing straight ahead. Sixteenth-century navigators generally sailed along loxodromes or rhumb lines, which cross the meridians at a constant angle. Rhumbs, however, do not make for mathematically shortest distances between two points; rather, "when we come to measure distances [...], Onely great circles, then, can be our scale" [Donne, "Obsequies to the Lord Harrington" 1. 114, 117]. Following the work of his friend the Portuguese mathematician Pedro Nunez, Dee invented the Paradoxal compass, a device enabling his pilot-pupils "to lay a course along a succession of rhumbs which would make an approximation to great circle sailing" [Taylor 96; for Nunez, see ibid., 84-5]. This was done at the request of the Muscovy Company [d.iiijv], since sailing by rhumbs is far more misleading at higher latitudes (at the poles, rhumbs become infinite spirals) and, to make matters worse, there is an growing increase "Of the Variacion of the Compas, from the true North" [a.iiijv]. Also following the work of Nunez, Gerardus Mercator, another friend and correspondent of Dee, designed a map projection representing rhumbs by straight lines (world map published in 1569). The immense problem of mathematical cartography lies 18 "Stratarithmetrie" is set out at length in Leonard and Thomas Digges' Stratioticos, parts of whose arithmetic section is discussed in my first chapter. A sample "stratarithmetrical" problem for the office of the Sargeant Major is: "There are 3. Regiments and in the first regiment there are but 280. Pikes. In the second 320. In the third Regiment 600. Pikes. The Generali commaundes them to be put in Bataile of Frount triple the Flack. I demaund how many Pikes must be in a Ranck. and how many Rankes, and also in what sort the Sargeant Maior may most sodainly & readily imbattel them" [47]. Recorde's Whetstone o f Witte also presents many problems of this sort. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the impossibility of making an exact "projection o f a Sphere in plaine" [a.iiijv] while keeping the scale constant. The familiar Mercator projection keeps meridians parallel to one another and perpendicular to the lines of logitude, with zero distortion of spatial relations along the equator and infinite distortion at the poles. Among other arts applying the principles of "Geometrie vulgar" to real space, Dee describes astronomy (in 1573 he was to publish a work on determining the stellar parallax, Autobiographical 24), "zographie" (perspectival drawing and painting)19 and perspective (optics). His introduction to Euclid, however, goes beyond the benefits of "deriuatives" of geometry: he also discusses, for example, "Statike [...] which demonstrateth the causes of heuynes, and lightnes of all thynges : and of motions and properties, to heauynes and lightnes, belonging" [b.iiijr]. Dee argues against the Aristotelian thesis that "Of any two bodyes, the heauyer, [is] to moue downward faster than the lighter," claiming rather, with Giovanni Battista Benedetti (1530-1590), that If there be two bodyes of one forme, and of one kynde, aequall in quantitie or vnaequall, they will move by aequall space, in aequall tyme: So that both theyr mouyings be in ayre, or both in water: or in any one Middle [c.jr]. Like Benedetti and Dee, Galileo initially applied his arguments only to those bodies that are made of the same material.20 In addition, Nicholas H. Clulee sees Dee's statics take the far more general step of rejecting Aristotle's prohibition of metabasis, the mixing of disciplines, as the result of which "mathematically derived conclusions, because they deal in things abstracted from matter, have no bearing on physics" [163]. In contradicting Aristotle thus, Dee makes way for the emergence of physics as a mathematical discipline [see Gaukroger 87-8 for same on Galileo and Descartes]. Also among the arts mathematical derivative Dee lists astrology, "anthropographie," and "thaumaturgike." This neighborhood strikes us as strange, to say the least: we hold such disciplines to be mysticist, irrational, proverbially anti-scientific. 19 Dee’s zography inciudes the effect of distance on colors: "Zographie, is an Arte mathematicall, which teacheth and demonstrateth, how, the Intersection of all visuall Pyramides, made by any playne assigned, (the Centre, distance, and lightes, being determined) may be, by lynes, and due propre colours, represented" [d.ij.v] 20 In his anti-Aristotilean De Motu (1590), Galileo writes that if a piece of wood and a piece of lead "are let go from a high tower, the lead precedes the wood by a long space: and I have often made test of this... Oh how readily are true demonstrations drawn from true principles!" In the Two New Sciences (1638) Galileo took up the view that bodies of any nature would fall with the same velocity [Crombie, 2: 149-150]. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Under Dee's pen, however, they seem nothing of the sort —if we remember what constitutes state-of-the-art science in 1570. Dee defines astrology as an Arte Mathematicall, which reasonably demonstrateth the operations and effectes, of the naturall beames, of light, and secrete influence: of the Sterres and Planets: in euery element and elementall body, at all times, in any Horizon assigned [b.iijr*]. The general thesis here is that changes in the celestial world affect the sublunary; fair enough: the only suspicious phrase is "secret influences." For Dee it includes, for example, the fact that "the Sea & Rivers (as the Thames)" do not "ebbe, and flow, run in and out, of them selues, at their owne fantasies" [b.iiijr]; presumably, the "secret influences" of the heavenly bodies are also responsible for changes in weather, as well as fertility periods and other seasonal activity of plants and animals (see his references to Aristotle, b.iijv). In his criticism of "Light Beleuers" who " think the Heaven and Sterres, to be answerable to any their doutes or desires" and also of the "vulgar Astrologien" who exploits this popular credulity [b.iiijr], Dee seems to rank among the astrological moderates for whom the stars propose and man disposes, for otherwise man would not have free will. Related to astrology is what Dee calls "Anthropographie [...] the description of the Number, Measure, Waight, figure. Situation, and colour of euery diuerse thing, contayned in the perfect body of MAN" [c.iiijr]. This science includes not only anatomy, zography and the like, but also chiromancy and physiognamy; its mathematical aim seems to be measuring and interrelating parts of the microcosm (with the practical benefit of expanding our knowledge of their correspondences to the macrocosm for the use of iatromathematical medicine). In other words, anthropography is primarily concerned with proportions - hence the references to Durer's De Symmetria Humani Corporis - interpreted through Pythagorean harmonics [see Heninger 187-94]. Overall, Dee's astrology and anthropography are, if not scientific, then scientistic. There is absolutely nothing in his exposition to hint at any difference between them and other arts mathematical derivative: his discussion of astrology is preceded by those of music and of cosmography, whereas anthopography is followed by "Trochilike [...], which demonstrateth the properties of all Circular motions, simple and compound." Between astrology and anthropography comes statics. Dee's "thaumaturgike" is, despite its name, even less magical, being defined as the art of manufacturing self-moving mechanical toys, and of creating optical illusions by means of lenses. The chief 84 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. practitioner here is Archimedes, whose thaumaturgic fame derives from the anecdotes preserved in Plutarch's "Life of Marcellus.” Ingenious machinery was, of course, a Renaissance craze, and I already mentioned one of Dee’s accomplishments in this area, a mechanical flying scarab designed for a theatrical production. Although the mathematically inclined among Renaissance intellectuals often occupied themselves with similar feats (it was a relatively painless way to make a buck), Dee's forays into thaumaturgy certainly contributed to the unsavory part of his reputation, for the page devoted to the art is followed by a three-page diatribe against "my vnkinde Countrey men" who confuse it with conjuring [A.jr-A.iijr]. True magic emerges only in the last of the entries, the art of "Archemastrie," the culmination of all mathematics and natural philosophy, which also has its own method, that of experience, whereby "it is named of some, Scientia Experimentalis” [A.iijr-A.iijv]. These words seem to look forward to the scientific method, but in fact, as Clulee notes, for Dee "experience" or "experiment" do not mean "the controlled testing of hypotheses as in its modem connotation" [171]. Unlike modem experiments, the "experiences" of "Archimastrie" are neither measurable nor reproduceable: in this they are like love. Clulee's analysis of the entry shows that the art includes "a form of natural magic for the manipulation of hidden virtues of things" as well as "divination by means of reflecting surfaces" [167-8]. Dee's description is here purposefully cryptic and truncated, for he does not wish to sully the public image of Euclid with the negative associations of magic, even if he himself does believe the former to be a necessary step to the latter. Unfortunately, since the contents of Archimastrie are only hinted at, it is impossible to tell how they relate to Dee’s Platonist understanding of the nature of number. 4. Dee's Concept of Number Dee's arts mathematical derivative assume that some type of knowledge may be had - mathematicaly - about the world of the sensibles. Is, then, his quoting Plato on geometry as "the knowledge o f that which is euerlastyng" a) a mindless genuflexion [a.ijv]? Or b) an attempt to boost the esteem of the discipline by any means necessary? Or does Dee equate "that which is euerlasting" with c) the laws of nature, of the type indicated in his discussion of statics? Paradoxically, the answer is d) none of the above. The first words of the Mathematicall Praeface are "Divine Plato," and the opening anecdote refers to Plato's famous lecture on the Good, which, identifying it with the One, treated of the entire mathematical-ontological scenario of the One, the Indefinite 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dyad, and, presumably, Ideal Numbers.21 Although Dee’s use of Plato's lecture is not mathematical in any way, he follows it with an entirely orthodox Platonic distinction between the sensibles and non-sensuals: Thinges Supematurall, are, of the minde onely, comprehended: Things naturall, of the sense exterior, ar hable to be perceiued. In thinges Naturall, probabilitie and coniecture hath place: But in things Supematurall, chief demonstration, & most sure science to be had [*v]. What does this classification of types of knowing mean for Dee? Does he regard his work on rhumb-line navigation and the stellar parallax as "probabilitie and coniecture"? One thing is clear: the Republic's "Geometria, eius quod est semper, Cognitio est" [a.ijv] is quoted in awareness of its philosophical consequences. "Betweene thinges supematurall and naturall,” Dee places things "of third being" - "Thynges Mathematicall." These are immateriall: and neuerthelesse, by materiall things hable somewhat to be signified. And though their particular Images, by Art, are aggregable and diuisible: yet the generall Formes, notwithstandyng, are constant, vnchangeable, vntransformable, and incorruptible. Neither of the sense, can they, at any tyme, be perceiued [...]. A meruaylous newtralitie haue these thinges Mathematicall. and also a straunge participation betweene thinges supematurall, immortall, intellectual, simple and indiuisible : and thynges naturall, mortall, sensible, compounded and diuisible [*v]. This passage affirms both the orientation of Plato's number and its character as a sign. Dee divides number ontologically into three kinds: 1) numbers in things, 2) mathematical numbers, and 3) Ideal Numbers ("Formes [...] vncangeable, vntransformable, and incorruptible"). Mathematical numbers, which are "by materiall things hable somewhat to be signified," themselves signify Ideal Numbers as their "aggregable and diuisible Images": the three ontological kinds are also stages of a unidirectional signification process. The consciously Platonist nature of all this is beyond dispute: Dee closely paraphrases the opening salvoes of Proclus, his famous predecessor when it comes to :1 Dee's anecdote ultimately comes from Aristotle, who used to tell it at his lectures; it was preserved by the Elementa Harmonica of Aristoxenus. 2.1 [qtd. in Bumyeat 66]. For more on lecture on the Good, see Sayre 76-8 and Metaphys. 988aJ4-17. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. introducing Euclid [5-5]. "Things Supematurall," therefore, must be understood as Proclus' first genera, the realm of Ideas. Through Proclus but also independently, Dee is pointing towards Metaphysics 987b15 and I090b33-6 (discussion of the so-called "intermediate being" of things mathematical in Plato), as well as 987a35 (according to Plato, no science of sensibles may be had). We have treated all these issues in the previous chapter. Dee continues by describing the elements of his things mathematical, whose being is. as we have seen, intermediate between that of Ideas and sensibles: Of Mathematicall thinges, are two principall kindes: namely, Number, and Magnitude. Number,we define, to be, a certayne Mathematicall Summe, of Vnits. And. an Vnit, is that thing Mathematicall, Indiuisible, by participation of some likenes of whose property, any thing, which is in deede, or is counted One, may reasonably be called One. We account an Vnit, a thing Mathematicall, though it be no Number, and also indiuisible: because, of it, materially, Number doth consist: which, principally, is a thing Mathematicall [*jr]. This paragraph hits all the salient buttons on the Greek concept of number. Number is defined as a multitude ("Summe") of units [Euclid 7, def. 2], while the indivisible unit is regarded as both "no Number" and as the building block of which "Number doth consist." Dee even renders Euclid's definition of unit as "that by virtue of which each of the things that exist is called one" [7.7] more Platonic by infusing it with the language of likeness and participation. To apply the passage to the part of the Mathematicall Praeface with which I started my discussion of the Greek number concept, each of three lions or eagles is a "visible One" "by participation of some likeness" to these "Indiuisible Units" [*.jv- * .in . The same language is used in defining magnitude: "Magnitude is a thing Mathematicall, by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke." While Dee cannot and does not identify the unit with the point, he does set up the traditional analogy between them: as the unit is indivisible and no number, so the point, "though it be no Magnitude" "is a thing Mathematicall, indiuisible" [*jr]. Several pages later, Dee emphasizes the differences between the terms: the point has both situation and motion, whereas the unit has neither; "A line, though it be 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. produced of a Point moued, yet, it doth not consist of pointes: Number, though it be not produced of an Vnit, yet doth it Consist of vnits, as a materiall cause" [a.ijr].“ I treated the Pythagorean roots of the unit-point analogy in my second chapter, and, in the next, shall demonstrate how Stevin transforms it by claiming the zero, and not the one, is the counterpart of the point. The fact that Dee repeats the analogy in its Greek form, i.e. as applying to the one, becomes particularly striking if we remember his mathematical work in cartography. Any unit in cartography (or geography), no matter how miniscule, is a magnitude, and not "like" the point at all. To equate the one with the point, therefore, is to turn a blind eye towards what, to us, seems clear as the summer's sun. Is it clear, however? In drawing the parallel between the one, the origin of number but no number, and the point, the origin of magnitude but no magnitude. Dee regards the use of numbers in measuring magnitudes as having no bearing on the inherent nature of number, which here is a plurality. The great geometer Euclid, after all, commits the same "error" as Dee. Dee's first mention of the point-unit analogy, grounded in the view of number as a multitude of ones, is followed by a remarkable chain of statements. "Neither Number, nor Magnitude, haue any Materialitie," starts Dee. Fair enough - but then he treats this lack of "Materialitie" in a reifying manner: as if it were Materialitie after all, but one so refined, that our selves know not what it is!23 How Immateriall and free from all matter, Number is, who doth not perceaue? yea, who doth not wonderfully wonder at it? For, neither pure Element, nor Aristoteles, Quinta Efientia, is hable to serve for Number, as his propre matter. Nor yet the puritie and simplenes of Substance Spiritual! or Angelicall, will be found propre enough thereto [*jr]. Dee's assumptions appear thus: number and magnitude exist and therefore are "things"; all things are made of something; if the material cause of mathematical objects seems to be nothing, it is a nothing that must in some way be a something. The grammar of inquiry is perfectly Platonic, and equivalent to proceding from the fact that Socrates is good to asking what Good itself is. “ Euclid defines a point as that "which has not part" and the line as "breadthless length" [/ def.1-2]. The idea of a line as the path of a moving point is reported by Proclus in his commentary on Euclid / def.2 and also in De Anima 4Q9a4. See Heath's commentary to Euclid / def. 1-2. 3 I am, of course, quoting Donne's "A Valediction Forbidding Mourning." 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Questions always determine their answers. In becoming the (grammatical) object of Dee's contemplation, the "matter" of numbers and, in fact, of other non-sensuals, turns into a real object. What are non-sensuals made of? "Bah; of nothing; that's why they are non-sensuals." To answer thus but begs the next question of what their "nothing" is. The "immaterial matter" of number seems to be most like thought - and yet is so superior in its properties: thought is mortal, changeable, unique, local, whereas number is immortal, changeless, universal and omnipresent. (Number, the good Socrates would remark at this point, must be the thought of some god.) Extremely telling is Dee’s assertion that contemplating the marvellous "matter" of number is what led Boethius to claim all things to have been "Formed by the reason o f Numbers [...] For this was the principall example or patteme in the minde o f the Creator' [Boethius 1:2, 75-76, Dee *jr]. In expanding upon Boethius, Dee both embraces the Platonic orientation of number leading us to the Ideal realm, and appears to follow Neoplatonists like Agrippa, who attribute the properties of things to their participation in Ideal Numbers: By Numbers propertie therefore, of vs, by all possible meanes, (to the perfection of the Science) learned, we may both winde and draw our selues into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and Formes [*jr]. For Dee as for Ficino, the study of number separates the soul from the senses, and directs it towards the non-sensuals. At the same time, in the manner of Agrippa, number in things is the conduit to their Ideas, whether these Ideas are separate and use numbers to effect their properties, or whether, as Aristotle would have it, the Ideas are to be identified with Ideal Numbers.24 Either way, knowing the world through number means knowing "that which is everlastyng": "Thinges Supematurall," only of which "most sure science may had" [*v]. The "search" here is "inward” and not inductive, and precedes by reason and dialectic. The mathematician’s speculative joumey begins with number in things and mentally leads us up the staircase of causes until it arrives at the first: "the 24 This depends on whether we think God created the world by numbers exclusively. The preceding sentence invites us to think so. The science of number deals with a subject that was "vsed of the Almighty and incomprehensible wisdome of the Creator, in the distinct creation of all creatures: in all their distinct panes, properties, natures, and vertues, by order, and most absolute number, brought, from Nothing, to the Formalitie of their being and state." The sentence we are analyzing, however, does speak distinctly of the "Formes" of things as if they were not necessarily Idea1 Numbers. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Forme of Formes, the Exemplar Number of all thinges Numerable" [*jr v]. In this Eriugenist formula, we recognize the One Itself, as identified with God [see my chapter 6], This is the point when The Mathematicall Praeface recalls the Philolaus-Pico claim that "By Numbers, a way is had, to the searchyng out, and vnderstanding ofeuery thvng, hable to be knowen" [qtd. *jv]. Dee refers his reader to the 900 Theses and its version of "the wonderfull mysteries, by numbers, to be atteyned vnto," which he refrains from expounding only because "loannes, Picus, workes, are commonly had." For Dee, as for Pico (and Agrippa), numbers not only carry us to pantingly contemplate the divine, but also permit us to use the knowledge acquired in the ascent to make rational statements about the sensibles. Hence the example, lifted from Pico, of how things may be known by numbers: the prophecies of the twelfth-century Gioacchino da Fiori by means of "Numbers Formall, Naturall and Rationall," i.e. the three ontological kinds of number as described above [*jv, Pico, 108]. Hence also the claim that, in "the conceiuying of Numbers, absolutely," we shall "finde the number of our own name, gloriously exemplified and registred in the booke of the Trinitie most blessed and aetemaH" [*ijr]. In this line. Dee obviously intends a type of gematria. Its relevance to an introduction to Euclid may lie in the fact that Dee's new and improved Kabbalah employs an invented language whose letters are shaped from points, lines, and circles, which determine their numeric value [for Dee's study of Hebrew, kabbalah, and gematria, see Clulee 85-8, 102], Making occult promises, Dee stays understandably cryptic. As in his discussion of "Archimastrie," he does not wish to scare the general reader with the mysteries to be ultimately "atteyned vnto" by the study of mathematics -- mysteries that the lay person considers illicit and confuses with conjuring -- so he limits himself to feeding sapientibus their parum. The exact relationship between Dee’s mathematics and his Enochian language remains unclear. How can number permit us to know and foreshow "great particular euents, long before their comming” [*jv]? Let us return to the marvellous immateriality of the matter of which number does consist. We recognized that such "matter" seems to be most like thought, except that it is eternal and immutable. What number is, for Dee, is the thought of God, the presence of number in things being the mode of His presence in them. Dee locates the three ontological strata of number in this manner: 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Number hath a treble state: One, in the Creator: an other in euery Creature (in respect of his complete constitution:) and the third, in Spirituall and Angelicall Myndes. and in the Soule of man. In the first and third state, Number, is termed Number Numbering. But in all Creatures, otherwise, Number, is termed Number Numbered [*jv]. Clulee identifies Number Numbering in the mind of God with the Ideal Numbers of the earlier passage [*v]; in the minds of angels and souls of men with mathematical numbers; and the Number Numbered in every creature with numbers in things [Clulee 150]. This identification of the three epistemological kinds of number with those ontological seems extremely reasonable, albeit not devoid of diffulties, of which below. Both types of "Number Numbryng” are described as "the discretion discerning, and distincting of thinges," whereas Number Numbered is having distinctions and being discrete. Number Numbering in God differs from its counterpart in angels and men in that "our Seuerallyng, distinctyng, and Numbryng, createth nothing: but of a Multitude considered, maketh certain and distinct determination," But in God the Creator, This discretion, in the beginnyng, produced orderly and distinctly all thinges. For his Numbryng, then, was his Creatyng of thinges. And his Continuall Numbryng, of all thinges, is the Conseruation of them in being: And, where and when he will lacke an Vnit: there and then, that particular thyng shalbe Discreated. [*jv]. What are we to make of this remarkable passage? Numbering here is counting. It is by His semipitemal counting of all things that Deus conservat omnia. He counts each thing ("Vnit") as a one ("in respect of [its] complete constitution"), thereby differentiating it from any other thing. In other words, every single thing’s participation in the One - the participation that makes each thing a thing and (via the Ideas’ participation in the One) this and not some other thing - is reinterpreted here as God's including this thing in His count. Dee takes the phrase "I am" and reads it as "I am counted." Each thing exits, and exists in its nows, because God wills it to exist in every now it exists in: the existence and duration of the world depends on God directly and immediately. In fact, the existence of things as themselves is synonymous with the presence of Divine Will in them. But, as we know from Augustine, what makes things exist as themselves (and not no-things) is the presence of number. Therefore Divine Will in things is present as number: God’s counting of a thing is its being-one, and, by 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. extension, God's counting of parts of things is their being-one (e.g. my losing a hair equals God's omitting it from his count). The beauty of this scenario is astounding. If God is indeed "the Forme of Formes, the Exemplar Number of all thinges Numerable" [*jr], i.e. the One Itself, then each thing, by virtue of being a thing, that is one, is His likeness. The world is an agglomeration of different images of the One, images whose individuality is formed by unique arrangements of numbers. It seems extremely plausible then, that Dee buys into Aristotle's claim of the identity of Ideal and Ideal Numbers for late Plato. Dee says that God counts each thing as one ("an Vnit"). What does this mean? Since God is ontologically prior to and outside of time, his thought differs from ours in that he thinks all things simultaneously. Strictly speaking, his counting is not counting, but the application of the principle of correspondence, cardinality. Laid upon the temporal axis, such "counting" would read : one, one, one, one, etc. Our numbering, on the other hand, is counting: it is sequential, and this is why we form progressions, why our ones are distinguished from one another ordinally, becoming one, two, three, four, etc. Dee's saying that God counts things as " Vnits" throws a wrench into Clulee's identification of Number Numbering in the mind of God with Ideal Numbers. To put it succinctly, if the divine Number Numbering consists of "one, one, one,” where do we locate Ideal Two, Ideal Three, Ideal Four, and so forth? Do they too dwell in the divine mind as a lesser kind of Number Numbering, since each of them is a particular kind of one? As we remember from my second chapter, the concepts of Ideal Numbers (reified unities holding together particular numbers of mathematical monads) and of being-one (being a unity of parts) depend on the fact that Greek does not distinguish between "unit" and "unity." Any "proper" arrangement of parts is a one and therefore, at least conceptally, number. Dee, however, does distinguish between the unit and unity; "Note the word Vnit," he says in a marginal note, "to expresse the Greke Monas, & not Vnitie: as we had all commonly till now, used " [*jr]. The OED quotes this passage as the first appearance of the word "unit" in English. Thus Dee retains the Platonist thesis that things, by virtue of being ones, participate in the One Itself at the same time as he abolishes the idea of the one as a unity of multiplicity. This may look forward to Stevin’s refusal to think of numbers as pluralities, but it primarily indicates a key tension in the text which seems to indiscriminantly combine Platonic and Aristotelian views concerning Ideal Numbers (Aristotle, as we remember, denies they exist, holding unity to be mere abstraction). At one point, Dee claims that "formally, Number, is the Vnion, and Vnitie of Vnits." Then 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. he unexpectedly clarifies: "Which vnyting and knitting, is the workemanship of our minde: which of distinct and discrete Vnits, maketh a Number” [a.ijr]. Now, it is either fish or flesh: either Ideal Numbers exist and due to them we can call three ones a "temarie” or a "tryad," or the fact that we can call three ones "three” is the "workemanship of our minde.” To this problem we might add another. In describing number, The Mathematicall Praeface to the Elements of Geometrie o f Euclid ofM egara omits to classify number according to its subkinds of even, odd, and others. I argued in both my second and my third chapters that such classification forms the necessary component not only of the Platonist concept of number, but also of its application to ontology. At least historically, it lies at the bottom of Dee's claim that "By Numbers propertie [...] we may [...] vew, of all creatures distinct vertues, natures, properties, and Formes" [*jr]. Yet Dee offers absolutely no discussion of number as composed of the even, the odd, and their permutations. This omission is all the more significant, since the key source of such division is Euclid himself, in whose seventh book it is set out at length; Boethius, whom Dee quotes, is another, and even Proclus speaks of number in a similar manner [see my chapter 3]. The fact that Dee disregards the even and the odd, presents an even stronger symptom than his strange and irreconcilable juxtaposition of Platonic and Aristotelean views of Ideal Numbers that some change is afoot in his concept of number -- a change that he himself does not appear to realize. 5. Conclusion. What is Arithmetic? The assumptions about the relation between numbers and sensibles that the practical parts of the Mathematicall Praeface rest on, are disregarded in the theoretical section of the work. While, in the former, knowledge bestowed by number can be had of the sensibles, in the latter the only knowledge that can be had is the Platonic knowledge of things that are. The role given to numbers in the "chief demonstration, & most sure science to be had" of such "things Supematurall" [*v] refuses to view number as the subject of calculation. The path to knowledge proceeds from number in things towards the One Itself without any recourse to what we call "mathematics", by means purely philosophical. In Cesare Ripa's Iconologia (first ed. 1593, second and illustrated ed. 1603), the allegorical figures of Theoria and Prattica each have a pair of compasses, the former on her head and pointing to heaven, and the latter holding it feet down to earth. Such, overall, is the case with the Mathematicall Praeface. Perhaps the discrepancy between its 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. theory and practice is due to Dee's being "pinched with straightness o f tyme", with the printer hovering over his shoulder [A.iiijv]. But that cannot be the only or even the primary reason. The primary reason must be the fact that the new mathematical techniques were still only techniques - they had not acquired a theoretical backing. As with attempts to explain Hindu-Arabic numerals by using the Platonist understanding of signification. Dee has not formally recognized that classical theory does not fit modem mathematical practice, even if the instances of this divergence today appear quite astounding. And yet there is one place in the speculative portion of the Mathematicall Praeface where the practice, so to speak, seeps into the theory. I have just mentioned Dee's omission of the classification of number according to sub-kinds. In concluding this chapter, I shall argue that the omission also has to do with Dee's treatment of fractions and the consequent definition of arithmetic. Greek-type arithmetics, as was said in my second chapter, found no place for fractions, for their focus of interest was particular unities, the latter conception based on the indivisibility of the monad.25 Medieval and Renaissance textbooks of Hindu-Arabic numerals, on the other had, although they had eventually appropriated the name "arithmetic," were interested only in the practice of calculation. As a consequence, when they taught, for example, division, their results treated remainders - fractions - as exactly the same kind of thing as integers. The way Dee speaks of fractions is extremely interesting: vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sundry Considerations: and extend their namer farder, then to Numbers, whose least part is an Vnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth them Fractions. Dee's fractions are imagined whereas his "Indiuisibie Vnits" are. Still, he does classify "vulgar" and "common" operating with fractions as "an other kynde of Arithmetike," to distinguish it from what he names "Arithmetike by it selfe," that o f "working, in whole numbers onely: where, of an Vnit, is no lesse part to be allowed" [*ijr]. So, for Dee both whole numbers and fractions belong to the single art of arithmetic, although this art may 23 Since for Aristotle mathematical numbers were abstractions, his fractions could have been considered numbers - but Greek arithmetics were not written by Aristotelians. 94 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. be broken into the "principall" arithmetic of whole numbers and the vulgar arithmetics of fractions, radicals and "cossick numbers." 26 One inadvertent result of the new textbooks was to tie the concept o f number to mathematical operations. Whereas the Greeks assigned positive integers and fractions to different disciplines because they conceived of them as different entities, Dee not only assigns fractions to "an other kynde o f Arithmetike,'' but also justifies himself by the fact that operations with fractions "do, for the most part, of their execution, differre from the fiue operations of like generall property and name, in our Whole numbers practisable." Dee's justification makes what number is dependent on what number does-f1this, to the Greeks, is putting the cart before the horse, making the cause depend on the effect. His definition of arithmetic attempts to incorporate both views of the science: "Arithmetike [...] is the Science that demonstrateth the properties, o f Numbers, and all operations, in numbers to be performed" [*ijr]. But the Mathematicall Praeface omits the description of "the properties, o f Numbers” i.e, of the sub-kinds of number as classified by Greek arithmetic, leaving room only for "operations in numbers to be performed." We might therefore claim that a certain plane in Dee’s work holds natural numbers, fractions, and radicals, as existing in the same way; as numbers; as what is subject to addition, subtraction, multiplication, division -- but this plane is fragile and momentary, a trepidation rather than a solid presence. :6 Dee also describes the astronomical arithmetic of sexigesimal fractions, i.e. trigonometry, which he calls Arthmetike Circular. In the Arithmetikeof Radicall numbers, numbers, as the Greek tradition, noteably Euclid's book 10, dictates, are imagined geometrically: "And, here. Numbers are become, as Lynes, Plaines and Solids" [*ijr’”]. :7 Note his referring to mathematical numbers as "the Mercurial fruite of Dianoeticall discourse,” i.e. the effect of the enchainments of reason [*ijrJ. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 ZERO AND THE CONCEPT OF NUMBER IN SIMON STEVIN K o r n a y M p y , t o He K Jian H T e, He n o K y n a H T e M He b c h o k , A jiy H L iie h o j i h k n o n o 5 K H T e Ha moh n e n a jib H b iH o y r o p o K . Why, these are very crotchets that he speaks. Note, notes, forsooth, and nothing.1 0. Introduction The Flemish mathematician Simon Stevin (1548-1620) may be called the initiator of the modem symbolic number concept. To call him that is, perhaps, misleading: modem philosophical reflection on what number is takes place in the late nineteenth and early twentieth centuries, and occurs primarily in the context of logic and set theory [see e.g. Russell, ch. 2]. Whether the modem interpretation of numbers as classes of sets proposes another form of referentialism, as Rotman claims [30-31], is irrelevant for our purposes. Stevin's achievement lies on a different plane: he does not ask what number "is," and resigns himself to defining it as "that, by which the quantity of each thing is explained." Facile and unspecified, his "that, by which" shrugs off ontology, bracketing, as it were, the autonomous being of numbers as unknowable. Stevin is, of course, not a phenomenologist, and a great stretch of imagination is needed to ascribe philosophical foresight to his position. He is, first and foremost, a mathematical practitioner, an engineer, an eminently practical man. His chief desire is to facilitate calculation by simplifying and homogenizing mathematical notation. Calculation, for him, is the indissolubly linked with writing. It is his focus on modes of writing - as seen in his advocacy of Bombelli's exponents and, in particular, in his work on decimals - that pushes him to reformulate the number concept. One imagines his prime psychological motivation to have been impatience with inefficiency. The old number concept is totally besides the point when it comes to putting numbers to what Stevin must have regarded as their proper use: building bridges, feeding troops, buying foreign goods. 1 Nikolai Oleinikov, "O nuliakh," Potty gruppy OBERIU, 422; Shakespeare, Much Ado About Nothing, 2.3.55-6. 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equally besides the point is the question of whether and how numbers exist prior to the syntactical operations of notation, addition, subtraction, multiplication, division.2 Stevin's "definition" of number treats the latter (usually but not always, as I shall show) as posterior to syntax, and determined by it. In other words, Stevin reasons about numbers primarily (but again not exclusively) as effects of Hindu-Arabic notation and mathematical operations. This is why, to a Platonist, Stevin's numbers would appear to be not numbers as such, but secondary formations, images of images. Stevin's identification of the being of numbers with Hindu-Arabic notation [Klein 192] is the first step in the emergence of the symbolic number concept; the second is associated with symbolic algebra [Danzig 76-98; Klein 163-185; Rotman 28-53]. Algebraic symbolism permits us to treat expressions generally and collectively, to deal not with the individual x + 1 or 2x +5 but with species: ax + b, cue + bx + c, and so forth. The algebraic letter signs employs an entirely different model of signification, intending number in general rather than some concrete and determined entity. If sixteenth-century algebraists stumble across negative and even imaginary numbers, but do not admit them into solutions, their seventeenth-century colleagues follow through on Stevin's identification to expand the number domain. Stevin's writings on algebra do not take the decisive step of representing known and unknown quantities by letters. This step was made six years later by Francois Viete (a.k.a. Vieta). The chronological and the conceptual neighborhood of Stevin's number concept and Viete's algebraic variable suggests that both are symptomatic of a more general change in understanding of how signs signify. In subsequent chapters I shall attempt to demonstrate that symptoms of the same change also occur in theology (much earlier), and in literature or, at least, in Shakespeare (a few years later). The penetrating analyses of both Klein and Rotman focus on the novel aspects of Stevin's thought. I, on the other hand, am interested in the incongruous combination of the old and the new. What particularly strikes me in Stevin's declaring zero no number but the beginning thereof is how zero takes over the place allotted to the one in the classical number concept. We shall see the exact same substitution in the arithmetical symbols of God and the self. Adding the sign <S>to Bombelli's exponential notation, Stevin does not always employ the resulting characters as clear-cut exponents. Especially problematic is the mathematical meaning of his ©, which in some contexts equals 1, in others 0. There is nothing self-evident about our a0 = 1; it may be established only with the aid of symbolic : Notation was considered an operation until about 1800 [Karpinski 100]. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. algebra. In trying to gauge to what extent Stevin's zero is or is not a number, i.e. may or not be operated upon, I try to determine if an when Stevin treats <S>as an exponent. This procedure is anachronistic and just plain wrong, since the formal possibility of potentiation by zero does not occur to Stevin, who recognizes only positive powers. The discrepancies exposed by it, however, do enable us to get a better grasp of Stevin’s employment of his <6>, as well as of the incomplete and transitional nature of his number concept. 1. Algebra 1.1 Signification in Rhetorical and Syncopated Algebra Early modem algebra's move from rhetorical to symbolic reproduces the change in signification that makes the modem concept of number possible. The change in the mode of writing the discipline results in an entirely different discipline: one with its own rules, assumptions and even content. Algebra, or the "Rule of Coss," emerges not as the general theory of equations, but as the technique for finding specific unknowns, tellingly called res in Latin and cose in Italian - i.e., "things." Such focus on the positively existent is already evident on the institutional level: the public contests which led to the solution of cubic equations involved specific real-life problems, whereby the contestant was actually discouraged from disclosing his method o f finding, in order that he may go on to win other contests [Parshall, sec. 3]. Attention here is paid not to laws, but to their real and isolated instances. The same may be seen on the level of language. While the "Rule of Coss" evolved towards greater syncopation - i.e., substituting abbreviations for words - it still remained essentially rhetorical, that is expressing itself not so much in mathematical symbolism as in human speech. Rhetorical algebra, of which the syncopated is only an "evolved" sub-class, differs from the symbolic in kind: its words a) represent pre-existent entities, and b) are not the medium of calculation. Each of its expressions stands for a particular "real" situation - and not only because it focuses on real-life problems of the kind posed by early textbooks [see e.g. Digges]. Its calculations occur, so to speak, "elsewhere": like with Roman numerals and counting boards, the best its signs can do is represent the progressive stages and the solution of an operation. Thus rhetorical algebra understands signification in the manner we encountered in Plato's Sophist, with the phrase "Theatetes sits" fingering the real Theatetes. For instance, when we say that a square number multiplied by a linear number yields a cubed number, what we mean is that a two-dimensional surface "sprouts" a third dimension. When, on the other hand, we write a x a2 = a3, we have conducted an operation symbolically, without expostulating whatever we might mean by a or by the 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. power sign. Rhetorical (and syncopated) algebra is severely restricted by its referentialism - for example, if a3 is a cubed number, what sort of an entity is a6? How do we imagine it? Does it even exist? Since the expressions of rhetorical algebra represent "real" situations, its numbers are always literally numbers o f something - just like the numbers of the Greeks. Furthermore, its referentialism judges equations and their solutions as "possible" (i.e. representing the "real” state of things) or "impossible": the rhetorical counterpart to ourx - 5, for instance, is possible only if x > 5, for five oranges cannot be subtracted from, say, four. A further restrictive impulse is introduced by the real-life problems: e.g., a problem concerning pluralities of oranges admits radical numbers into neither the equation, nor the solution. Negative and imaginary numbers, of course, are always impossible, since neither can be the numbers of any existing thing. As a consequence, rhetorical algebra cannot produce a general theory of equations; on the contrary, it breaks the equations of each degree down into several classes. Thus, in the discussion of quadratics in his Ars Magna (1545), Gerolamo Cardano considers X.quad.aeq.\Q.pos.p. 144 o rx 2 = 10x + 144 144.aeq. lO.pos.p. I.quad. or 144 = lOx + x2 To us, the two equations are almost identical: x2- 1O x - 144 = 0; (x -1 8 )(x + 8) = 0; x = 1 8 ,-8 x2 + l O x - 144 = 0; (x + 1 8 )( x -8 ) = 0; x = -18, 8 To Cardano, however, "each of these equations represented a different way of partitioning up a square into rectangles of smaller areas" and had to be solved so that "the sides of the smaller rectangles [did] not have negative length" [Parshall, sec. 3]. Needless to say, the solution too had to be positive, with x = 18 in the first case, and 8 in the second. During moments of giddiness, when Cardano was willing to toy with negative solutions, he still referred to them as "fictional" - as opposed to "true," that is positive [Parshall, sec. 3].3 3 Negative numbers offer a concise example of the transition from referentialist to the modem, symbolic signification. We have managed to referentialize the concept, treating it - at least in my schooling experience - as numbers of absences: we think of -2 as something like the absence of two oranges, which, when added to five present oranges, leaves three. But to strictly obey the dictates of referentialism, we must ask: can the absence of two oranges exist in itself? - and we are forced to answer in the negative, since absences are by definition devoid of existence. Thus, although in the sixteenth century negative numbers appear in calculations, they must wait for Stevin's pupil Albert Girard (1S95-1632) to become legitimate in 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Signification in Sym bolic Algebra The founding text of modem, symbolic algebra, Francois Viete's Isagoge (1591), may be thought of as the culmination of sixteenth-century reforms in the writing of mathematics. Viete's innovation was to designate unknown quantities by vowels, and the known by consonants. While Viete's algebra operated with letter signs, these functioned as variables; when, however, Viete "plugged in" numbers in the last step, the equations again became specific and the variables reverted to res. At the moment when he operated with letter signs, Viete focused not on particular problems but on the laws governing relationships between quantities as yet undefined. "Meaning" in symbolic algebra - i.e., how (letter) signs intend their numerical values - is constructed in a manner quite foreign to the classical understanding of signification. The algebraic letter sign stands for all quantities, for any number inasmuch as it is number: the sole restrictions upon it are syntactical, determined by the particularities of the equation. An a by itself is meaningless - the smallest unit of value in algebra is the equation. To ask what a is, we must really inquire after the solution of a = a, which is any and all numbers, if a be defined as member of their set. Operations with letter signs do not intend concrete entities for precisely this reason; a = a is good for any value of a, and therefore, if it intend any particular value, it must intend all of them equally and simultaneously. But to say even this is problematic. For the letter sign does not intend its values severally, one by one; it is rather that the expression, as a whole, represents a particular relation, a certain transformation of an unspecified quantity, and it is the relation that determines the "signified," i.e., the field of possible values for the letter sign. Whereas rhetorical algebra records calculations, the symbolic performs them literally on the page. To fully comprehend what this means, we must give some thought to how meaning works in the case of operational signs. I have already mentioned the emergence of the +, - , and = signs from the Latin et, the tilda over the syncopated m (for minus, meno, moins), and parallel lines ("bicause noe .2. thynges can be moare equalle" - Recorde in Sanford 377).4 Although there is, so to speak, a natural reason for the form of each of these - i.e., they did not emerge ex nihilo - they are nonetheless quite independent of language. Let us, for example, picture the addition of two large numbers as a column with the + sign before it: solutions. Nor did Girard’s book resolve matters entirely: "J'en sais qui ne peuvent comprertdre que qui de zero ote quatre reste zero" says Pascal [Pensees, 72 Lafuma, 199 Brunschvicg ed.]. 4 Compare with parallel lines as symbols of perfect, and therefore unrequited, love in Marvell’s "Definition of Love." 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34,567 + 98,765. The function of the + sign is to symbolize and invoke the syntax of the conversion of these sets of numerals into another, third set. It informs us what numeral will stand under any two numerals; that, for example, the last numeral of the third number will be 2, with the 1 to be added to the result of the penultimate numerals, and so forth. Thus, operational symbols point to syntaxes under which solutions are written: a minus sign would "cause" the writing of an entirely different set of numerals. This is what we mean when we identify calculation with writing. What we have demonstrated with numerals can be equally done with letter signs. The plus sign in the expression a + a indexes which "spelling" rule should be used to generate the solution. In this case it instructs us to write a prefaced by the number of a's being added - i.e., 2a. In symbolic algebra the classical order of ontological priority is reversed: the signifier determines the signified, and not the other way around. The signified, therefore, is subject to syntax - i.e., it is the laws of combination (fundamental operations, rule of three, transposition, etc.) that "create" both the values of the algebraic letter signs and numbers in Hindu-Arabic numerals. This is very different from the traditional model of signification that we encounter in Plato, with its autonomous and discrete signifieds —in fact, here the traditional model is placed topsy-turvy. 2. Stevin 2.1 Decimals Frangois Viete was building on the sixteenth-century algebraic tradition that was greatly transformed by the discovery of the syncopated Arithmetic of Diophantus (third c. AD). Diophantus shirked from proving his equations geometrically and did not limit himself to what I called "real-life problems" - problems arising from real, generally commercial, situations. Instead, his problems often focused on relationships between numbers, for instance: "To find three numbers such that the square of any one of them minus the next following gives a square" [Parshall, n. 76].5 5KIein s h o w s that Diophantine algebra still presupposes the Greek conception of number; hence an equation with negative or irrational solutions is considered impossible, and calculations ending at the indeterminate are regarded as ancillary to the task of finding results in concrete numbers [130-131, 138, 134], Nonetheless, Diophantus understands monads in the Aristotelian fashion - as abstractions - and admits 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Diophantus entered the history of European algebra when Raphael Bombelli (1526- 1572) was shown the manuscript of his Arithmetic in the Vatican Library. Bombelli's prime interest lay in simplifying the language of Cardano; thus, for example, Bombelli rejected the earlier "cossick" notation where the sign for every an or x11was utterly different for every n, in favor of circled exponents: ©, ©, ©, ®, and so on. After his study of Diophantus, Bombelli also started posing indeterminate problems of the type we saw above (in fact, he simply used those of his source), solving them, again like Diophantus, without recourse to geometric proofs. Problems couched in the traditional realism of practical arithmetics, however, he solved according to the geometrical method of Cardano [Parshall, sec. 5]. Simon Stevin continued the work of Bombelli, publishing his own translation of the first four books of Diophantus together with the Arithmetique, and in the same year as the work on decimals (1585). Accepting Bombelli's notation for powers, as well as Diophantine analysis of indeterminate problems, Stevin added <S>to Bombelli's ©, @, ©, ©, etc. [Stevin 527], and updated the style of his Diophantus by infusing it with symbolism. Like Bombelli, Stevin was concerned with simplifying mathematical exposition, and his great innovations have to do with the writing of mathematics, and with the implications of such writing for the theoretical "content” of the discipline. Stevin's expansion and homogenization of the concept of number is indebted to what he does with Bombelli's power notation - applying it, for instance, to decimals. Stevin's brochure on the subject came out in two editions - as the Flemish De Thiende and as the French La Disme (both by Plantin in Leyden, 1585); I shall be quoting from a 1608 English translation entitled The Disme: the Arte o f Tenths [Stevin, vol. 2a, 387-455; "disme" as in ten cents]. This was the first work devoted to decimal fractions.6 The practical benefit of decimals is that they make calculating with "mixed" numbers - numbers that contain both an integer and a fraction - a single and a continuous act. + 123.5 456.67 580.17 calculations with fractions [132-133]. ‘Admittedly, there were sporadic experiments with something like decimals before, chiefly in the extraction of square roots [Cajori, History o f Mathematics, 147-148; Stevin, 2a, 375-6], nor did Stevin's system survive for very long: modem decimal notation was invented by John Napier in 1617. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The drawback - and the reason we still use fractions - is that many fractions, for instance '/3tcannot be represented by finite decimals without some degree of approximation. In the preface to The Disme, Stevin promises to teach the easy performance of all reckonings, computations, & accounts, without broken numbers [i.e. fractions], which can happen in a man's business, in such sort as that the four principles of arithmetic, namely addition, subtraction, multiplication, & division, by whole numbers may satisfy these effects, affording the like facility unto those that use counters [397]. Note that Stevin claims any computation can be performed without fractions, "by whole numbers" only (he repeats himself in the first definition of The Disme). This is a somewhat strange statement: .5 is not a whole number, and decimals are, after all, fractions. What Stevin means is that any operation can be performed in numerals 0-9 without modifying their value by the value of the numerals in denominators, as is the case with the non­ decimal way of noting fractions. But to call numerals 0-9 "whole numbers" is to equate the writing of number with the being of number - something that, as Klein shows [192], underlies the reasoning of Stevin's Arithmetique. The same identification is again made in the first definition of The Disme, where decimal arithmetic is said to "consist in characters of ciphers."' We also note Stevin's parallel between calculating in positional notation (including decimals) and counter-casting —in both of these, calculation is identical to manipulating signs.8 1 mentioned that one of the reasons why counter-casting was not obliterated by Hindu-Axabic numerals is that money, weights and measures, etc., were not decimal, and rows on counting boards could be made to fit any base. Stevin’s Disme is, inasmuch as I know, the first text to argue that a decimal system should be instituted for all measurements, thus avoiding "the troublesome multiplications of rods, feet, and oftentimes of inches" [395; also see his appendix]. There would then be no reason to perform mathematical operations by means other than base-10 notation on paper. The Disme is a slim booklet, limited to several definitions, some examples of operations, including the extraction of square roots, and an appendix on weights and 7 "The Disme is a kind of arithmetic, invented by the tenth progression, consisting in characters of ciphers, whereby a certain number is described and by which also all accounts which happen in human affairs are dispatched by whole numbers, without fractions or broken numbers" [pt. 1. def. 1]. * Stevin's work on decimals is motivated by his desire to facilitate calculation; for this reason. The Disme is not addressed to theoretical mathematicians, but "To Astronomers, Land-meters, Measurers of Tapestry, Gaugers, Stereometers in general, Money-Masters, and to all Merchants” [2a, 389]. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measures. The second definition assigns to the integers, or rather to natural numbers,9 the name of "commencement." The third definition is of decimals themselves: And each tenth part of the unity of the COMMENCEMENT we call the PRIME, whose sign is thus ©, and each tenth part of the unity of the prime we call the SECOND, whose sign is ©, and so of the other: each tenth part of the unity of the precedent sign, always in order one further. The numbers in the circles signify the order of places left to right. Thus, 7©5<D6® means 7 units in the first place, 5 in the second, 6 in the third - or, .756. But Stevin is not simply labeling places in due order -as I said, his notation is Bombelli's method for noting exponents, and this is how Stevin employs it in the Arithmetique. In positional notation, places represent powers of ten: 123 = 1 x 102 + 2 x 101 + 3 x 10°; when it comes to decimals, however, places represent negative powers of 10: The Disme's 1®5®6® or our .756 = 7 x 10-* + 2 x 10-2 + 3 x 10*3. While Stevin does regard each decimal place as having a tenth of the value of the preceding, the precise meaning of his circled signs is unclear. He does not put a minus sign before these "exponents" - in fact, as we shall see, he is rather ambivalent on the question of negative numbers. The very idea of negative powers is, for the sixteenth century, outlandish, and the Arithmetique does not admit negative exponents [513-8; see below]. And yet the fact remains that Stevin uses Bombelli's notation - the same notation that in the Arithmetique stands for powers - and it's not as if Stevin could not come up with a different way of representing decimals, at least, let's say, by enclosing the exponent in a square instead of a circle. It appears then that, when Stevin is dealing with decimals, he counts the "tenth parts" of his "tenth progression" - in other words, that the numbers in his circles are, strictly speaking, not exponents with the minus sign left out, but ordinals. The already-mentioned second definition of The Disme designates natural numbers as the "COMMENCEMENT" of decimal notation, "whose sign is thus ®." For example, our 1,098.756 would be written l,098®7©5@6<a). What are we to make of Stevin's "commencement" sign? We shall deal with its possible character as exponent later, when we examine the Arithmetique; for now, let us look at The Disme's zero ordinally. As we have seen, early arithmetics gave sequences of numerals with 0 following 9, thereby stressing its essential difference from the "signifying" figures. It seems as if the sequence 0, 1,2, 3, 4, 5, 6, 7, 8, 9 would be one of the symptoms for zero becoming a number as 9 We shall later see that Stevin does not fully recognized negatives as numbers. This is why I generally say “natural number" where Stevin says "whole number,” since the former is what the latter really means for him. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. opposed to a mere writing aid. By placing his <S>literally before the "ordinals" ©, CD, CD, ©, etc., Stevin appears to present us with such a sequence - i.e., his counting of places in decimal notation starts with zero. However, in a note appended to definition XXX of the Arithmetique, Stevin compares his version of Bombelli’s power notation with Hindu- Arabic numerals: the "marques"or '"charaderes" of the former are presented as "<8>, CD, CD, CD, ®. & c.”, those of the latter as "1. 2. 3. 4. 5. 6. 7. 8. 9. 0." With respect to both cases Stevin speaks of "naturel ordre." 10 As if this discrepancy weren't enough, definition HI of the Arithmetique does place 0 before 1 in its sequence of "les characteres par lesquels se denotent les nombres" [503]. We conclude the section on The Disme with some general remarks on the relationship between decimals and the concept of number. Decimals are a simplification of the Hindu-Arabic system as it applies to fractions. In a number like I l/io, the values of the two 1’s are determined differently: the first depends on rank only, whereas the second also depends on the denominator. In the decimal l.l, however, the value of the l’s depends on rank only. The notation, thus, grows more uniform. For this reason, the "mixed” number 1.1 appears, in comparison with 1Vjo, like a single and a homogenous entity. At the same time, decimal notation can express not only all fractions, but also —and for the first time - all radicals, albeit only as approximations; therefore, it encourages thinking of different types of numbers as having the same nature. Perhaps the practice of converting fractions and radicals into what Stevin, after all, regards as "whole numbers" also contributes to this effect. Finally, decimal notation also provides a convenient standard for determining which fractions or irrationals are greater and which are lesser (e.g. 123/456 > 246/923, since .2697 > .2665; V2 > 579/456 , since 1.4142 > 1.2697; the values given are approximate). The latter property enables us to think of the various types of number not only as homogenous, but also as existing in a single, unambiguous and determinable progression - like numbers on the number line. It is no coincidence that Stevin published a work on decimals in the same year as he published the first work rethinking the traditional concept of number. 10 "Car comme les characteres I. 2. 3. 4. 5. 6. 7. 8. 9. 0. (en respect de plusieurs autres marques signifians nombres) ne sont seulemtn brieues, mais necessaires: voire il semble que sans leur conuenable <3c naturel ordre, il eust este impossible d I'homme de paruenir aux secrets d'Arithmetique qu'il a acquis; Et de mesme sorte entendra on que ceci sont les characteres [of power notation] qui au naturel ordre sont requis; lesquels aux quatre numerations generates, <&.principalement aux rompuz des mesmes qui souuentesfois se rencontrent, voire par toutes computations algebraiques, donnent telle facilite, que ce qu'a plusieurs feront autrement impossible de comprendre, leur sera facile, mettant le tout au iugement de ceux qui entendent la chose" [2a, 527], 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 General Concept o f Number in Stevin's A rith m e tiq u e Perfectly aware that the definition of number proposed in his Arithmetique is revolutionary,11 Stevin starts the book with a considerable amount of fist-waving. On the question whether "unity is a number," he claims to have consulted "tous les Philosophes anciens & modemes, que ie trouuois traicterde ceste matiere," and to have also ''communique de bouche auec quelques doctes, certes de ce temps pas des moindres." Not that he doubted being right - no, he was as certain of that as if he had heard it straight from Nature’s own mouth! He just wanted to be prepared against whatever objections may come [496]. Number according to the A,rithmetique [def. II] "est cela, par lequel s'explique la quantite de chascune chose." EXPLICATION. Comme l’vnite est nombre par lequel la quantite d'vne chose expliquee se diet vn: Et deux par lequel on la nomme deux: Et demi par lequel on 1'appelle demi: Et racine de trois par Iequei on la nomme racine de trois, & c [495], Stevin here is projecting Peter Ramus' definition of number as "that whereby any thing is numbered" [Ramus 1] onto Euclid 7, def. 1: "A unity is that by virtue of which each of the things that exist is called one.” After locating the one in the unity of each thing, Euclid goes on to construct pure numbers as multitudes [VII, def. 2]. Ramus (1515-1572), on the other hand, does not locate number in things, instead associating numbers with the act of counting things. Perhaps John Donne has this position in mind when he writes: "Numbering is so proper and peculiar to man, who only can number, that some philosophical Inquisitors have argued doubtfully, whether if man were not, there were any number" [Essays, 55]. Ramus, however, still regards numbers as pluralities, thereby restricting the number field to natural numbers.12 Stevin follows Ramus in that his myth of the origin of the concept of number grounds it in the act of "numbering" things, instead of the Platonist internal being-one of these things. The distinction between Stevin and Ramus lies in how Stevin conceives of this "numbering." According to Stevin, when human beings learned to speak, they named 11 Conservative revolutionary, rather. For Stevin, the Hindu-Arabic system is not a new development, but the "natural" way of writing numbers practiced in the Golden Age of wisdom, which the Greeks and the Romans stupidly replaced by their newfangled and inefficient systems. See Klein 186-190; Stevin's hermeticism is worthy of a separate inquiry. "A number is that whereby any thing is numbered. And it constisteth either of an unitie. which is the least number, or else of a multitude, whereof there cannot be so great but that there may be a greater," [Ramus 1]. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. each thing "one," when to this thing they joined another, they named the assemblage "two," when they divided the thing into two equal parts, they named each "half." . Puis considerans que vn, deux, trois, demi, tiers, & c. estoient noms propres, & conuenables, pour l’explication de ladicte quantite. ils ont veu qu’il estoit necessaire de comprendre toutes ces especes soubs vn genre (car telle est leur maniere de faire en tous autres semblables comme bled, orge, auoine, ils la nomment en genre Grain; aigle, tourterelle, rosignol, en genre Oiseau) lequel genre ils appelloient nombre [497]. According to the quoted definition of number and its explication, Stevin's quantite can be represented ("expliquee") by one, by two or any other natural number, by the half or any other fraction, by the square root of three or any other radical. The origin myth implies why Stevin's number is so inclusive. Since it is not grounded in the indivisible unity or being of thing (for half an orange is not half of the being of an orange), Stevin’s quantite - and consequently number - need not be limited to pluralities. Whichever "quantity" of an orange we choose to express - including, ultimately, the ratio of the circumference to the diameter - is a number to the same extent as anything answering the question, "How many?" The origin of Stevin's quantite is, without a doubt, Aristotle's category of quantitas, which "means that which is divisible into constituents, either of which or each of which is by nature one and a this." For Aristotle, plurality and magnitude are the two primary types of quantitas: "A plurality is kind of quantity if it is numerable, a magnitude is a kind of quantity if it is measurable" [.Metaphysics I020a7-9]. Unlike Stevin, however, Aristotle formulates his concept of number not with respect to quantitas itself, but only with respect to its subclass, plurality. For this reason, his numbers expressing magnitude are not, strictly speaking, "as much number" as the former. Since Stevin does formulate his general concept of number with respect to quantity itself, he avoids Aristotelian cavils and restrictions. As we remember, because Aristotle defines number as plurality, he declares the one to be "no number." Since number for Stevin is not limited to plurality, he insists that, on the contrary, "I'unite est nombre" advancing a syllogism according to which, even if number is regarded as a "multitude of unities," one is nonetheless a number. This syllogism, analysed by Klein 191-192, is, in my opinion, less consequential than Stevin's subsequent argument, a reductio ad absurdum based on operations. "If from a given number no number is subtracted, the given number remains," writes Stevin. "Let three be the given number and let us subtract from it the one, which as you insist is no number. Therefore the given number remains," i.e. 3 - 1 = 3 , "which is absurd" [496-497], In a 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. decisive break with the classical tradition, Stevin resolves the question of what is and is not a number from mathematical considerations, and not ontological. In other words, Stevin does not examine the being of number before declaring whether something possesses that being; the criterion is whether the something in question is subject to mathematical operations. Whatever you can calculate with is a number.'3 As Stevin sees it, at the source of the classical claim that the one is no number lies the Pythagorean analogy of the unit to the point: We account an Vnit, a thing Mathematicall, though it be no Number, and also indiuisible [...]. A Point we define, by the name of a thing Mathematicall: though it be no Magnitude, and also indiuisible [...]. Of Number, an Vnit, and of Magnitude, a Poynte, doo seeme to be much like Originall causes" [Dee, *j, ajv-aijr].u While acknowledged as imprecise, the analogy was still articulated frequendy enough to bear the brunt of Stevin's attack ("O heure infortunee," exclaims Stevin, "en laquelle fu t premierement produicte ceste definition du principe du nombre!" [498]). Stevin breaches it in several places. The first proposition of one of his arguments, that the one is divisible whereas the point is not, certainly would not have been admitted by his opponents. The first proposition of another, that "I'unite est partie du nombre, le poinctn'est pas partie de la ligne" [499], he employs in the first syllogism proving one to be a number. The greatest dissimilarity between the one and the point, however, is related to Stevin's second syllogism: adding one to a number increases the number, whereas adding a point to a line does not alter the line in any way. Stevin, of course, is not the first to indicate this: Boethius, for whom one is no number like the point is no magnitude, qualifies his analogy by the fact that adding ones does produce number, while "a point put upon a point [does not] bring about [a line], any more than if you joined a nothing to nothing" [2.4]. How does the Boethian caveat become Stevin's definitive difference? Preceding generations could overlook the dissimilarity in addition because they were more concerned with the ontological nature of the one as pure unity - which does offer significant parallels to the indivisible point. Only when the one is predicated upon mathematical operations and not on the inherent being-one of things does the dissimilarity in addition prove that "Vvnite [...] n'est point telle en nombre comme le poinct en ligne" [499]. I3This, as we have seen while discussing Dee's views on fractions and his definition of arithmetic, runs absolutely contrary to Greek procedure, deriving what should be prior from what should be posterior. The Greeks view calculation as a syntactical enchaining of signifiers only (mathematical numbers), to which discrete and individual signifieds (Ideal Numbers) are not subject; the latter therefore cannot be defined by combinations of the former. 14 See also Boethius: "unity has the potential of a point, the beginning of interval or longitude, just as the point is the beginning of the line and the interval, although it is itself neither interval nor line” [2.4, 129]. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Boethius compares addition of points to addition of nothings. He did not have a mathematical symbol for nothing, but Stevin does. Stevin’s next step is at once predictable and rife with bizarre consequences. He asks: If one is not the numerical counterpart to the point, what is? - and blithely answers: "le di que cest o (qui se diet vulgairement Nul[...])" [499]. Here are his arguments [499-500]. As my reader can see, in some of them zero's replacing the one in the analogy with the point is justified by re-examining the traditional topoi of the same analogy. It's as if Stevin's parvenu took over not only one's office, but also its personal history: 1) As the addition of points does not make a line, so the addition of zeroes does not result in any number; as the addition of a point to a line does not extend the line, so the addition of zero to any number results in the same number. Here the similarity between the zero and the point rides on the dissimilarity between the one and the point as noticed by Boethius. 2) As the point is indivisible, so is the zero. This used to be a shared trait between the point and the one, as long as the one conceived as the indivisible monad, "that by virtue of which each of the things that exist is called one" [Euclid 7, def. 1] - and in Stevin it is not. Does Stevin call the zero indivisible because zero divided by any number yields zero? He does not present mathematical examples of this; instead, his zero seems to be indivisible "by nature." Other arguments presented by Stevin appear strange at first glance, because they have to do with notation rather than operations: 3) As the point is on a line ["est aioinct de la ligne"] but is itself no line, "ainsi est o aioinct du nombre" but is itself no number. The strict meaning of this "jointure," I must admit, escapes me: is Stevin referring to zero at the beginning of the number series, or to zero as place-holder in, e.g., 60? I think Stevin is talking about the latter, for his next argument does also treats zero’s placeholding function in a very peculiar manner. 4) If one concedes that line segment AB can be extended to point C to create segment AC, then, for example, 6 can be extended to "o, ainsi que [...] 60 soit vn continue nombre faisant soixante" [499]. According to Klein, "Here [...] the continuous extension of a line is compared to an arbitrarily continuable lining up of 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ciphers which yield continually 'new' numbers" [Klein 194]. Unlike extending line segment AB to C, going from 6 to 60 by "adding" a zero implies a "real" jump over numbers from 7 to 59. Stevin, however, seems to identify the writing with the being of number to such an extent, that for him the "real" discontinuity in going from 6 to 60 as if does not exist. It is the numerals and their laws of combination that create numbers, and not the other way around. Stevin's zero steps into the shoes of the classical idea of the one and is, in fact, its revised version. Just like a man who, encountering a mirror in a dream, not only looks at himself in it but looks at himself looking at himself in it, Stevin’s zero is not content to replace the one in being the numerical counterpart of the point, but must assume a far more general and a far loftier role. Stevin repeatedly calls it "le vrai et naturel commencement [...] du nombre " [500-501]. Here the zero obviously usurps from the one the title of arche or principle of number. However, just as the zero's correspondence to the point in geometry occurs along different lines from that of the one, so does Stevin's "principe ou commencement du nombre" [498] mean something rather different from what it did to the Greeks. As has been said, "pure" number for the Greeks is a plurality. For Plato, the priority of the one lies in the oneness of the particular multiplicity of ones that compose any particular number, whereas for Aristotle, one is the principle of number solely because any number is a number of ones. Stevin absolutely refuses the meaning "arche" holds for Aristotle: if Greek number is composed of ones, Stevin's number is certainly not composed of zeroes. Stevin also refuses the ontological direction of Plato's argument: if the Greeks derive number from the being-one of things, Stevin certainly does not derive number from their being-zero, whatever that may mean. As has been said, ontological considerations are here entirely absent: Stevin's number does not derive from the being of things, but from how humans conjoin and divide them. Why then does Stevin call zero ”commencement du nombre ”? Nothing would be easier than seeking the reason in the role zero plays in Hindu-Arabic notation by making otherwise "unoccupied" places count. We have shown that Stevin identifies the being of number with its writing; furthermore, we have also shown that he does not distinguish zero as placeholder from zero as number. Thus, zero would be the commencement of number because Stevin's idea of number - integer and decimal - derives from the Hindu-Arabic system; and the latter is made possible by zero. We can also ascribe zero's uniqueness among numerals to the fact that it cannot possibly be thought to point at something positively existent: 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As a numeral, the mathematical sign zero points to the absence of certain other mathematical signs, and not to the non-presence of any real 'things' that are supposedly independent of or prior to the signs which represent them [Rotman 1 2 ];15 this is why early writers differentiate zero from other numerals, claiming it "signifyeth not." Since by deriving numbers from notation without reference to ontology, Stevin effectively conceptualizes them as symbolic, and not referential, entities, all numerals become as zero: not signs indicating things, but signs generating them; correspondingly, numbers cease being "numbers o f something." As Brian Rotman comments on Stevin: To make zero the origin of number is to claim for all numbers, including the unit, the status of free, unreferenced signs [...]. In the language of Saussure's distinction Stevin rejected numbers as signs conceived in terms of positive content, as names for things' supposedly prior to the process of signification, in favour of signs understood structurally, as having meaning only in relation to other signs within the sign system of mathematics [...]. In effect, Stevin was insisting on a semiotic account of number, on an account that which transferred zero’s lack of referentiality, its lack of "positive content", to all numbers [29]. Stevin's calling zero the "commencement of number" might then be taken as signaling such "emptying” of numbers: all numbers are as zero in that they do not "signify" (i.e., point at presence of real ’things') either. So Stevin’s zero is the principle and commencement of number emblematically: because the fact that it signifies nothing is apparent, and the fact that other numbers do is not. Without any doubt, both of these arguments are in some way true; but they are also easy, and therefore suspicious. The fact is, nothing forces Stevin to talk of numerical counterparts, commencements, principles, and so forth; he could have just dropped the matter entirely, or simply limited himself to showing that the one is not like point, and is not arche. But no - he does not wish to drop the "commencement" formula, he just wishes to reapply it, and in re-applying it he changes its philosophical implications - some on purpose, others not. What we have here is an exemplary case of cultural innovation, "A horse's neck joined with a human head," to steal out of Horace's Ars - and only hindsight allows us to see the emergence of man from the horse, for hindsight is, contrary to public opinion, never 20/20. The very fact that Stevin recycles the Greek notion of the arche of number ought to alert to us to the lack of a clean break with the past. And a clean break there isn't: Stevin identification of number with numeral only goes so far, for, as he 15 On anteriority of signs to things, see Rotman 27-28 and passim. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. repeatedly claims, his zero is no number. The "no number" idea is, of course, lifted directly from the old notion of the one - but, again, infused with a different and modem meaning. Let us return to Stevin's arguments that one is a number. The first syllogism regards number as a multitude of ones; no syllogism based on this premiss can prove zero to be a number. The second syllogism is a reductio ad absurdum based on the premiss that "if from a given number no number is subtracted, the given number remains;" Stevin’s example is 3 - 1 * 3. We noted that Stevin here defines number from calculation and not ontology; the conclusion seems to be that anything subject to calculation is a number. And yet this ought not be the conclusion, for Stevin's second syllogism should not prove zero to be a number - since, for Stevin, it decidedly is not. Does Stevin think that anything with the power to transform other operands is a number? Zero is perfectly capable of doing so in multiplication (a - 0 * a)\ however, Stevin does not seem to take multiplication into account, although there is absolutely no reason to ignore it. What we have is a mess - either Stevin is on the nod or his preconceived notion of zero is artificially exempt from being mathematically tested for membership in the genus Number. Stevin claims that zero is no number in two already-mentioned places, both of which compare the zero to a point. The first is a strange sentence that might or might not refer to zero's notational role as placeholder: "Comme le poinct est aioinct de la ligne, & lui mesme pas ligne, ainsi est o aioinct du nombre, & lui mesme pas nombre." The second lies in the argument that, as many points do not make a line, “ainsi beaucoup des o [...] nefont nul nombre" [499], i.e. zero itself is not a number. That the zero is called "no number" in the context of its being compared to the point indicates a relationship between the two concepts. To remind the reader, the classical analogy between the one and the point rests on two shared traits: 1) The one and the point are both indivisible; 2) The point is no magnitude but the beginning (arche) thereof, and the one is no number but the beginning (arche) thereof.16 I already noted that the shared property of indivisibility is re-applied to zero by Stevin in "Comme le poinct ne se diuise pas en parties, ainsi le o ne se diuise en parties" [499]. In being declared "no number," zero again replaces the one in its role as the numerical counterpart of the point. Stevin's reasoning seems to be: as the point is to magnitude, so the zero is to number; the point is no magnitude but the beginning thereof, and therefore, zero is no number but the beginning thereof. 16 Plato defines the point as the beginning (arche) of a line [Metaphys, 992a20\, and the line is universally considered to be the first magnitude; a certain Herundes and Simplicius define the point as the beginning of magnitudes [Heath's Euclid, v. 1, 156-157]. Dee: "Of Alumber, an Vnit, and of Magnitude, a Poynte, doo seeme to be much like Originall causes” [ajv-aijr]. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. So far we have treated Stevin’s claim that zero is no number as if it were an anachronism, dictated not by the inner exigencies of Stevin's concept of number, but by the external fact of cultural inertia. Stevin takes an element from a system of thought to which he stands heir, and revises it instead of discarding it outright. If the general trend for Stevin is to idenity number and numeral, his treatment of zero is a screaming anomaly. However, there is another side to the matter. Stevin's calling zero "no number" is, so to speak, overdetermined: he is motivated to do so not only because he is caught in a traditional trope, but also because zero's being no number satisfies the way he envisions number in general. The word "envision" here is key. Although Euclid, in book 7, defines number as plurality, his theorems represent numbers as lines. This tradition was inherited by the algebraists, with their pre-Diophantine, geometrical methods of proof - although the representation of number was never identified with the being of number. Stevin, typically, does make this identification, once again refusing to distinguish between the writing and the being of the thing: "la communaute & similitude de grandeur & nombre. est si vniuerselle qu'il resemble quasi identite" [498]. The identification of number and magnitude reverses the traditional relationship between pluralities and magnitudes: "pure" number is conceptualized in terms of one case of the latter - linear numbers. Stevin, as it were, takes the lowest, "impure” species o f number, and uses it to express the genus of quantitas, thereby identified with his own quantite. He can do so since the types of numbers proper to magnitude include those of plurality, i.e. natural numbers. The identification of number and magnitude goes hand in hand with Stevin's insistence that zero is the proper counterpart to the point. When the Greeks employ linear numbers to represent pluralities, the regard each of their units of measure as a whole. Stevin, however, seems to think of lines representing numbers in a novel way: by placing, for instance, the mark for one after one unit has been measured off. At the beginning of the line is nothing - and thus Stevin's calling zero the "commencement du nombre" can be understood literally: zero is the point A of line AB representing number B. Zero is "no number" in the same way as the point is "no Magnitude", but "the bound of a line" [Dee, *jr]. All number so conceived equals the measured distance between the end of a line and its beginning, whereas the beginning itself is at no distance from itself - there is nothing to measure, no quantite which needs to be explained. And no quantity means no number. In the case of zero, one may say, Stevin's identification of number with magnitude overpowers his identification of number with Hindu-Arabic numerals. Moreover, in Stevin the Euclidean habit of representing numbers by line segments appears to also transform itself into the general positive number line from 0 to infinity - 1 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. say "appears" because this model is never mentioned explicitly. Still, the model of the general number line represents the continuity of the field of real numbers (the so-called arithmetic continuum), for any point upon it can be expressed numerically, whether by a rational or an irrational. When, in the explication to his second definition, Stevin claims fractions and radicals are just as much "number" as natural numbers, he introduces all types of number expressible by such a line - all, that is, but transcendentals, i.e. the yet- undiscovered irrationals that cannot serve as roots of an algebraic equation. This would be a trivial coincidence, were Stevin not conscious of further ramifications. The Arithmetique devotes considerable effort to arguing that "nombre n'estpoinct quantite discontinue" [501], obviously aiming at the classical view to the contrary.17 Now, the concept of number as plurality implies two types of discontinuity - the discontinuity within each number as plurality of ones (e.g. of sixty as sixty separate units), and the discontinuity between any two adjacent numbers (that of sixty as discontinous with fifty- nine on the one hand, and sixty-one on the other). Stevin takes on both. If you think number is discontinuous because you can imagine sixty as sixty ones, says Stevin, you can equally easily imagine it as thirty twos, or twenty threes; and, anyway, if your imagination's making number discontinuous means that number is discontinuous, why, your imagination can break magnitudes down in the same way [502]. As concerns the discontinuity of numbers with respect to one another, well, if fractions and radicals are also numbers, there is an infinity of continuous numbers between any two natural numbers that thereby become continuous between themselves. The continuity of the field of numbers arises, says Stevin, from the already-mentioned "la communite & similitude de grandeur & nombre" [498]: a vne continue grandeur, correspond le continue nombre qu'on lui attribue, & telle discontinuite que puis apres recoit le grandeur par quelque diuision, semblable discontinuite regoit aussi son nombre [...] le nombre est quelque chose telle en grandeur, comme I'humidite en I'eau [502]. If, following Euclid, we represent numbers by line segments, to represent all positive numbers (as I shall show below, Stevin limits his definition of number to positives) we need only to draw a ray - a ray, whose extent is infinite, and at whose beginning lies the zero. l7Aristotlc. continuing the already-mentioned passage on kinds of quantity, defines magnitude as quantity divisible into continuous parts, whereas those of plurality are not continuous [Metaph. 1020al0-l/]. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 Specific Types of Number in the A rithm etique Starting from definition VI, the first part of the Arithmetique deals with specific types of number: the arithmetical, the geometrical, and the algebraic. Some of the restrictions it lays down are incongruous with the concept of number expressed above, and must be judged as other examples of cultural inertia. In fact, the impression one gets is that Stevin. when describing arithmetical and geometrical numbers, falls more or less into the traditional mode, and only when contemplating algebra does he correct - usually implicitly - his previous utterances. For starters, why does Stevin need arithmetical numbers at all? It seems as if the general concept of number in the first five definitions would do the job perfectly. But he does define arithmetical numbers, as numbers expressed without "adjectifde grandeur" such as square, cube, root, etc [def. VI]. Thus "nombres arithmetiques" include natural numbers {"unite, ou composee multitude d'unitez" [sic!; def. VII] and fractions. Geometrical numbers, on the other hand, are numbers expressing magnitudes, and include radicals, as well as numbers to the first (line), second (square), third (cube) and other powers [defs. XV-XVUI]. In fact, geometrical numbers for Stevin are the magnitudes they express - for instance, our a 1 for him "est une ligne droicte nombre expliquee" [def. XV], Geometrical interpretation of powers is a clear sign of referentialism: here numbers correspond to and signify certain pre-existent figures. Earlier, I mentioned that the problem with such interpretation is that it stops at the cube, unable to envision figures corresponding to any n greater than 3.18 Stevin ingeniously puts forth a series of three-dimensional figures for n >3 which he terms "docides," arguing that "ces formes ci sont les vraies & naturelles" [513], In treating geometrical numbers Stevin re-introduces Bombelli's notation which we already encountered in The Disme. Several pages later, he also applies it to algebraic numbers. The three uses unfortunately do not match, thereby making the Arithmetique - to speak unacademically - a big headache. The fundamental difference between Bombelli's notation in geometry and algebra is as follows: in geometry, the circled number is an exponent, while in algebra it is a variable under an exponent. Thus, in decimals 2® = 2-2 = .02 in geometrical numbers 2® = 22 = 4 in algebraic numbers 2 ® = 2x2 18 It is also problematic from the point of view of arithmetic numbers in that they are forbid potentiation, for 2^ is not thought of as 4, but as 4 "square" units. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The biggest problem is the "commencement of number" in each category, marked by ©. In The Disme, ©stands for natural numbers: 2©2© = 2.2. In algebraic numbers, it is anynatural number, fraction or radical that is not a coefficient; in the problems of book 13,Stevin refers to such numbers as"quelque ©" or "le ©" [608-609]. Therefore, in decimals 2© = 2 in algebraic numbers 2© = 2x° = 2 - 1 = 2 The identification of x° with 1 is a clear instance of 0 being treated like a number. It is, however, a development much subsequent to Stevin: a0 equals 1 as a corollary of the rule for division of exponents.19 Is Stevin making this identification? In The Disme, no: if the sign © were the exponent 0, any integer a (written as a©) would be converted to 1, whereas it remains a. In the case of algebraic numbers, on the other hand, it sure looks so, but this is a case where appearances can be deceiving, and probably are. For in the case of geometric numbers a© does not equal I ; nor does it seem to equal a, if the result be thought of as geometrical numbers. Stevin's definition XTV proclaims the commencement of quantity to be any arithmetical number or radical, its "character" designated as ©. Since Stevin is talking about geometrical numbers here, we assume he means geometrical quantity: and yet, since the numbers in question are without "adjectifde grandeur"20 they cannot express any magnitude - for otherwise they would be lines, whose character is © [def. XV], In other words, the "commencement du quantite" here is again zero. All natural numbers, fractions and radicals that do not represent magnitudes appear on magnitudes as points; and, in fact, Stevin's example deals with points. Thus, from the perspective of geometry, a© = 0. I would thus claim that in no single case does Stevin think of © as an exponent; that our trinomial ax2 + bxl + cx°, which by him might be written as ©+ © + ©(although I have not noticed this exact form), for him has an exponential nature only in the case of © and ©, whereas © is, as in The Disme, more or less an ordinal. I do not claim this with absolute certitude, and yet, if Stevin thought of the algebraic sign © as a0, and "knew" that a° = I for all a * 0, why would he not have explicitly stated so? After devoting some space to the fundamentals of algebraic numbers, Stevin returns to the geometrical, redefining them and, implicitly, algebraic numbers as well. The series of explanations following definition XXXI overturns the restrictions of previous definitions. 19 The rule is: to divide powers of the same base, subtract their exponents. Thus, a" r a" = a0*" = a°= 1; a * 0. :o This definition "sera seulement vse quand les nombres Arithmetiques ou radicaux ne seront pas absolument descripts" [514], 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The identification of geometrical numbers and shapes is, for all practical purposes, lifted: geometrical number becomes "nombre expliquant la valeur de quantite geometrique [...//] obtient le nom conforme a I'espece de la quantite qu'il explique"[def. XXXTJ. In the appended articles, this return of geometrical number to the role of explaining quantity is placed in the context of relativizing all numbers, thereby reinscribing them into the general number concept of the book's earliest definitions. Number once again becomes a symbolic entity. First thing that Stevin does is unmoor the distinction between line, square, cubic and "docid" numbers: any number, he writes, can be square, cube, and so forth; as a corollary, any radical is just as much number as any natural number (we remember that definition VI excluded radicals from arithmetic numbers). 8 is considered a cubic number when it represents a cube (x3 or © = 8) in which case the side or cube root is two (:to r© = 2). But 8 can also express the area of a square (x2or ® = 8), in which case the side or square root is V8 (x o r© = V8). Here Stevin argues against those who say that a radical "nest pas nombre": La partie est de la mesme matiere qu’est son entier; Racine de 8 est partie de son quarre 8: Doncques V8 est de la mesme matiere qu'est 8: Mais la matiere de 8 est nombre; Doncques la matiere de V8 est nombre: Et par consequent V8, est nombre [530]. We should note that this essentially repeats his first syllogism proving one to be a number, except that the "matter" here is unequivocally a continuum and not, as in the case of the one, potentially a plurality. By saying that 8 is not by nature a cube, and can be the result of any potentiation, Stevin in effect undoes his earlier geometric interpretation of powers in favor of the arithmetical. Potentiation is now conceived as an operation between numbers on the par with addition, subtraction, multiplication and division; like these four it becomes a set of relations between members of the number sequence. The ramifications of this for the concept of number are as follows. All numbers - including natural numbers and fractions —can be the square roots of other numbers, writes Stevin; and therefore, all numbers when regarded as "explicant la valeur de prime quantite" [i.e., n = ©] "s'appellent [...] nombres radicaux" [528]. Stevin's argument for the annihilation of the "natural" difference between integers and irrationals is clearly inspired by the doctrine of Eudoxus in Euclid's book 5. For Eudoxus, the incommensurability of the side of a square to its diagonal is relative: in the instance of a square whose side is 1, the V2 of the diagonal is not incommensurable perse; rather, 1 and 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vl are incommensurable with each other [see Heath’s Euclid, v. 2. 116-125]. So, indicates Stevin, the difference between 4, 9, 16, etc. and 5, 6, 7, 8 as square numbers (i.e., ©) is only that the former are commensurable with their roots (© = 2, 3,4) and the latter not (© = V5. V6, V7, V8). Since the integers 5, 6. 7, 8 are as "guilty" of the incommensurability with their roots V5, V6, V7, V8 as the roots themselves are, the roots are as much number as the integers, and ought not be considered literally "absurdes, irrationels, irreguliers, inexplicables, ou sourds" [532]. This expansion of the concept of number, I should note, occurs several definitions before on the level of the exponent: exponents, writes Stevin in his commentary to definition XVTH, need not be natural numbers, but can be also fractions and radicals [513]. Therefore, 1 is not the least exponent, "mais comme ily a vn infini maieur progres des quantitez depuis I'vnite [...], ainsi y a il semblable infini moindre progres de la prime quantite [i.e., CD] en descendent" [517-518; Stevin identifies e.g. the exponent lh with the square root]. It should be stressed, however, that Stevin's "infini moindre progres" is asymptotic to the zero: neither 0 nor negative numbers are admitted among exponents. Stevin's treatment of negative numbers is standard for algebra of the period: they are admitted in the process of solving equations but not in the solutions themselves [see rules for multiplying negative numbers on p. 560, other example on pp. 608-609]. Such restriction is referentialist: negative numbers cannot be admitted into solutions only if the numbers in solutions be thought of numbers of something. It is not surprising, however, that negative numbers were fully legitimized for the first time by Stevin's prize pupil, Albert Girard, who reissued the Arithmetique after Stevin's death.21 Making radicals numbers from the point of view of notation means not only that whatever can be expressed by numerals (except for 0) is a number, but also whatever can be expressed by numerals and operational signs. Definition XXVT even calls algebraic expressions like the polynomial ax2 + bx + c "number" ("multinomie algebraique est vn nombre consistent de plusieurs diuerses quantitez"). This usage is exteremly illuminating, for it calls attention to the final and most radical change in the concept of number possibly enacted by Stevin. Let us return to the consequences of Stevin's identification of number and magnitude. One of the chief differences between plurality and magnitude is that in the former the unit is natural, whereas in the second it is arbitrary. By basing his concept of number on linear magnitude instead of plurality, Stevin in effect defines the natural in terms of the arbitrary, and as a particular instance of the arbitrary (which to the Greeks, :iFor imaginary numbers in Stevin. see pp. 615-620. The first person to confront the imaginaries was Bombelli. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. incidentally, is ever so much nonsense). Stevin can do that only because he does not ask after the whatness of number, but the result of this "oversight" is that his number no longer reflects natural divisions, instead becoming autonomous. Its autonomy is pronouncedly algebraic: since the value of the one is indeterminate, we think of linear numbers not as concrete entities, but in their simplest form as a series of relationships to the one, with the natural number series, for instance, becoming effectively a, 2a, 3a, 4a, etc. The algebraic intepretation of Stevin's arithmetical numbers, however, brings us back to the exponential nature - or the lack thereof - of the sign ©. I have said that Stevin's definition XIV proclaims the commencement of quantity to be any arithmetical number or radical, its "character" designated as ©. From its context we gather that Stevin is talking about geometric quantities, in which case any <£>is analogous to the point, i.e. zero. However, in the explication Stevin follows a geometrical example with the clause that he is talking about "tout nombre Arithmetique ou radical quelconque, qu'on vse en computation algebraique comme 6 ou V i ou 2 + V3, &c.” [514], As I also mentioned, in the problems of book II Stevin refers to the c of the polynomial ax2 + bx + c (i.e., ax2 + bxl + cx°) as "quelque ©" or "le <8>" [608-609]. Thus definition XIV could be, at least in part, talking about about algebraic quantities, in which case any integer, fraction or radical a is conceived of as a®, with <S> = x° = 1. If this be the case, the sequence of natural numbers could be represented algebraically as 1®, 2<S>, 3<S>, 4®, 5®, 6®, 7<J>, 8®, 9<S> or lx°, 2x°, 3x°, 4x°, 5x°, 6x°, lx ° , 8x°, 9x° i.e., 1, 2, 3, 4, 5, 6, 7, 8, 9, while the values of 1©, 2©, 3©, 4©, 5 © , 6©, 7©, 8©, 9© (or Ix1, 2x*, 3x', 4x!, 5 X 1, 6 x l , 7x‘, 8 X 1, 9x‘); or 1©, 2®, 3®, etc., or 1®, 2®, 3®, etc., remain indeterminate. Again, I insist that the exponential nature of Stevin's sign <S>is extremely ambiguous. It is also unclear whether the algebraic intepretation of arithmetic is possible if a<§> = a be only an orthographic convention, a way of writing. In either case, we have a new reason why Stevin calls his zero "commencement du n o m b r e it is so since any integer, fraction or radical, when placed before the sign <S>, remains itself in value, whereas any other member of Bombelli's notation changes that value (geometrical numbers) or makes it indefinite (algebraic numbers). If, however, in the case of algebra the sign <8> does function as an exponent, zero is the arche —or commencement - of number because numbers are nothing but relations between indeterminate quantities. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6.1 GOD AS ONE AND GOD AS ZERO PART I: THE TRANSCENDENT ONE God [...] is like the unit of number. For the unit, being the source of all numbers, and the root of them all, contains any number within itself, and is contained by none of them; it generates every number, and is generated by no other number. Now everything which is generated is incomplete, and divisible, and subject to increase and decrease; but that which is complete is subject to none of these things. TaM t o m h jic x b KJieTKe E o r 6 e 3 o'teft 6 e 3 pyK 6 e 3 H or1 0. Introduction We have seen how in Simon Stevin the zero usurps the place of the one as the "commencement" of number. In doing so, it takes on many of the traits formerly allotted to the one, for instance that of being no number but the origin thereof, or being the arithmetical counterpart of the point. This chapter will trace a parallel development in conceptualizations of divinity, where God goes from being thought of as one to being thought of as zero. Strangely, the theological change happens before the arithmetical. The second part of this chapter will end with a text which presents the sequence of numerals as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, justifying it by the ontological priority of God (whose symbol is zero) to creation (numbers). This text dates from 1508; arithmetic textbooks contemporary to it unanimously present numerals in the old fashion, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Did theology have a hand in forming or at least facilitating the change in arithmetic? We cannot answer this question. The first and present part of this chapter will trace the history of God as One or the Monad. I shall argue that God's transcendent relation to things is modeled upon the relation between the one, as no number but the origin thereof, and numbers within the classical number concept. If one is no number, the transcendent God is no thing. We shall see how by virtue of being such a kind of One, God becomes a kind of nothing. 1 Corpus Hermeticum 4.10-1 as translated by Scott, 1:155-7; Vvedenskii "sneg lezhit" in Poety gruppy OBERIU 124. For Corpus Hermeticum, see my chapter 8. According to the Platonist Hippolytus, the “Egyptians [...] said that God is an indivisible monad, that it begot itself, and that all things were built up from it" [Scott, 2:151, as quoted by Copenhaver 138]. This belief is certainly not pharaonic Egyptian; for Greek Egyptology, see Iversen, ch. 2. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A substantial portion of what follows will be devoted to discussing what various writers meant by referring to God and/or the One as "Nothing." We will see that the word has many meanings which do not coincide: it may refer to the Platonic Otherness or the Indefinite Dyad; it may refer to absence or privation; it may refer to that which is more than a thing and therefore no thing. We shall see what relation these meanings have to the thesis that the one is no number, and how some concepts of nothing are engendered by what seems to be its opposite, i.e. the concept of the one. Stevin's pronouncing the zero to be the origin of number goes hand-in-hand with his proposing a concept of number that, in its major features, is not based on referentialism. In this chapter, we will also consider what divine transcendence does to referentialism, and to the doctrine of Ideas that depends upon it. I shall argue that once we introduce a signified to which all signifiers ultimately point but which itself has no proper signifier, referentialism falls apart. We can no longer no what any single thing is, only that it is. The best we can do therefore is study the relations between things, relations which might have no being of themselves. 1. Being and the One in Plato As my reader no doubt remembers, the classical concept of the one is derived from the being-one of each thing: "An unit," quoth Euclid in a line we have many times encountered, "is that by virtue of which each of the things that exist is called one" [VII, def. 1]. All things share two and only two features - that each of them is, and that each of them is one. Abolish either of these and you abolish the thing. According to the logic of Plato's system (which explains how one thing can have many features by relating each feature to its cause or Idea, and views this relationship through the Platonic paradigm of signification), there then must be a One Itself and a Being Itself in which all things that exist as ones participate. Lovely, but problematic. For such system begs the question of the relationship between the One Itself and Being Itself, or, to refer to them in a less bulky manner, the One and Being: are they one thing, or two things? Before we tackle this question, I would like to do away with the impression my readers might have that it is moot, irrelevant, or inconsequential. As the ultimate foundation of things, Being and the One in Plato need to be intelligible: both are conceived in terms of immutability, which, for reasons I explained in an earlier chapter, makes them into objects of sure and certain knowledge. Incomprehension may be had of snails and tadpoles, but to 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. have a paradox materialize on this level is nothing short of catastrophic —it pulls the rug out from under everything. We might propose that the One and Being in Plato are the same, that the highest step of the Platonic ladder is occupied by the One that is Being, One that has being, Being that is one - or something of the sort. But a conscientious Platonist will immediately object: "What do you mean by 'the same'? If One and Being are one thing, is 'the same' the same thing, or yet another? For if it is the same thing, we are talking about a three-in-one, and not just a two-in-one; and, if not, we are talking about two separate things, and not one. And, incidentally, in either case we are already assuming the existence of number."2 This is the unfortunate result of the Platonic paradigm of signification: it isolates its signifieds, concretizes, reifies [from res +facio] them. Etienne Gilson, whose overview of the Being vs. One problem inspired some of what follows, argues that "les expressions eivai, etre; eivai r t , e T i / a t n t o j v o v t o j v , etre quelque chose, etre I'un des itres, sont des expressions equivalents dans Vesprit de Platon" [74],3 This is because in Greek the word o v refers to both "thing" and "Being as such” - rather like in English, with the proviso that the English noun "(a) being" means something animate. Being as such is, so to speak, a thing, an incorporeal “thing” to be sure; still, this impedes us from reducing it to the simple fact of the existence of the One. A section in the Sophist may be thought of as the preliminary run-through of the Being/One problem. Here the question of Being is posed with respect to the existence of the dyad of rest and motion: the Eleatic Stranger points out that rest and motion have being, but are not themselves Being, for then rest would be motion, and motion, rest. Instead, rest and motion participate in Being (i.e., are blended with it), whereas Being itself must be ontologically prior to the blending [methexis] —as it also must be according to the signification paradigm: "You conceive of reality [ousia, i.e. being]," says the Stranger to Theatetes, "as a third thing over and above these two [rest and motion], and when you speak of both as being real, you mean that you are taking both movement and rest together as embraced by reality [...]. In virtue of its own nature, then, reality is neither at rest nor in movement" [250b-c]. Stymied by the possible existence of something neither at rest nor in motion, the Stranger lets this train of thought drop [250e]. What the exchange does make clear, however, is that the uppermost step of the ontological ladder —whether it be occupied by Being or One - cannot itself know blending. It must needs be absolutely pure. 1 1 am avoiding the problem o f “the same” here: Is “the same” of the Sophist the same as the One of late Plato? An aporia identical to that of Being and One. 3Not accidentally, the phrasing resembles Klein's point that number for the Greeks in always the number o f something, thereby reminding us that the question of the being of number is posed according to the same grammar o f inquiry as the question of Being in general. 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The problem of the relationship of the One with Being becomes all the more acute in the doctrines of late Plato as reported by Aristotle, which I summarized in my third chapter. If Aristotle's account is faithful, Plato explicitly recognizes the One as one of the two principles of things - in Aristotelian terms, their formal cause; the other and material cause being the Indefinite Dyad or the Great and the Small [Metaph. 987b18-22, 988a8-14], Aristsotle's Plato joins the motley crew of those who argue over the number of principles, over "how many real things there are and what they are like," a controversy which the Plato of the Dialogues considers misguided [Sophist 242c-243a\. The basic fault of the debate, argues the Eleatic Stranger, is that the participants wholly ignore the question of Being: Let us put this question [to the debaters]: 'you who say that hot and cold or some other such pair really are all things, what exactly does this expression convey that you apply to both when you say that they both are "real" or each of them is "real"? How are we to understand this "reality" [i.e., being] you speak of? Are we to suppose it is a third thing alongside the other two and that the all is no longer, as you say, two things but three? For surely you do not give the name "reality" to one of the two and then say that both alike are real, for then there will be only one thing, whichever of the two it may be, and not two [Sophist 243d-e], It is certainly not unjust to put the same question to Plato himself: If the One and the Indefinite Dyad are the two principles of all things, where do we place Being? Five possibilities arise: a) Being is to be identified with both the One and the Indefinite Dyad: they both are, and there is no Being that is not them. This is impossible: if we identify both the One and the Indefinite Dyad with Being, we must also identify the One with the Indefinite Dyad, in which case we are dealing with one and not two principles. b) Being is to be identified with the blending of the One and the Indefinite Dyad (Ideas and objects are), or rather, since objects by virtue of their mutability are not. Being is to be identified with the uppermost level of the blending (the Ideas are). But then the One and the Indefinite Dyad by themselves are not, and this too is impossible. c) Being is be identified with neither the One nor the Indefinite Dyad, but is prior to both. Yet in this case, as the Eleatic Stranger points out with respect to rest and motion, we have three principles and not two; furthermore, Aristotle's account of late Plato does not mention a Being prior to the One and the Indefinite Dyad. Finally, - and this parallel to the stumbling block of the Stranger is the starting 123 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. point of Plotinus —if Being is prior to the One, Being is not one (and not many, since it is also prior to number), and thus there cannot be a "one-thing" referred to as Being itself. d) Being is to be identified with the Indefinite Dyad. This is clearly not the case, for then the immutable One would have no being. e) Being is to be identified with the One. This proposition makes the most sense; at the same time it agrees with Aristotle's calling the One ''the cause of whatness" of the Ideas [Metaphys. 988a J I] and the ousia of all things [987b21]. We might see a potential problem in that the Indefinite Dyad is thereby relegated to the status of Nonbeing [confirmed by Aristotle in Physics 192a6-10], since, if the One is Being, the Indefinite Dyad is literally not-Being. However, since the Sophist interprets Nonbeing as being-other (see chapter 3), the Indefinite Dyad can be identified with Nonbeing, for it is certainly both other than the One, and posterior to it.4 (Curiously, as being-other, the Indefinite Dyad, also known as the Great and the Small, is - as we can tell even from its names - not in all senses immutable; is then knowable? This is a problem to which we will return.) So Being appears to be the same as the One, and Being should be same as the One (if we, once again, ignore the problem of “the same”). For, if Being is not identical with the One, then: a) the One participates in Being - since it is; and b) Being participates in the One, since there is only one Being. But I have already stated the argument that in the Platonic hierarchy of things that are, there must be at least one thing - the apex - which remains unblended. Therefore, if Being is not identical with the One, either Being is prior to the One or the One is prior to Being. In classical Platonism, both propositions appear absurd: given the former. Being is not one (and not many); given the latter, the One does not exist. So is Being the same as the One? No. Plato is very explicit about this fact when he treats the relationship of language to Being and the One in the Parmenides and the Sophist. Using the reified, immutable concept of Being, Parmenides argues that If a one is, it cannot be, and yet not have being. So there will also be that being which the one has, and this is not the same thing as the one; otherwise that being 4If the Indefinite Dyad is being-other, it has being, but is not Being; and yet it is not Being in such a way that the problem o f how it comes to be from being does not arise: since its being is being-other, it is by nature other than Being. (On the essential duplicity of Being and being-other in Plato, see Klein 86, 89.) 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. would not be its being, nor would it, the one, have that being, but to say ‘a one is’ would be tantamount to saying ‘a one [is] one' [142b], which it is not. Both of Parmenides’ distinctions - between the One having its own being and being Being as such and between “a one is” and “a one [is] one” - are grounded in a certain view of language. What underlies them is the assumption that linguistic distinctions reflect real distinctions, that every separate signifier has its own signified. The Sophist, which in many other respects offers a correction upon the Parmenides, encounters the same obstacle to identifying Being with the One, the problem of signification. Is Being "the same thing as that to which you give the name one?" asks the Eleatic Stranger [244c]. This question must be considered in light of the more general dilemma of how one thing can be called by many names, discussed later in the dialogue [251a-b]. According to the Eleatic Stranger, several signifiers can indicate a single physical object because of blending: a good man is both good and man because he participates in both the Ideas of Man and the Good. Therefore, if the names “Being” and “One” refer to the same object, say “the-one-that-is,” that means that Being and the One are blended in “the-one-that-is,” and that “the-one-that-is” participates in both Being and the One. In other words. Being and the One must be separate and prior to “the-one-that-is.” We are back where we started from. Thus, the relationship of Being and the One in Plato remains an insolvable and catastrophic aporia. 2. P lo tin u s 2.1 The One in the Plotinian system One of the essential differences between the Neoplatonism of Plotinus (c. 205-270 AD) and the, so to speak. Platonism of Plato is that Plotinus confronts the aporia of Being and the One and resolves it. This of course is not to say that the Plotinian alternative is perfect - no philosophical system can be; that’s why philosophy has a history. However, the aporias which must arise in Plotinus to make up for his solution of Plato’s do not concern us here. Nor is it important that Plotinus did not view his relationship to Plato as critical and corrective - as in fact he did not, holding himself to be an exegete rather than critic or innovator [see Gatti in Gerson 17-22]. A few remarks about his exegetics. Plotinus’ treatment of the One is glaringly beholden to the paradoxes of Plato’s Parmenides. The difficulty of this dialogue lies not so much in that it offers a lengthy and painful critique of the theory of Ideas, but in its second 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. half, devoted to the ramifications of the statement that there is a One, i.e. that the One exists. The main speaker is Parmenides himself, and the thesis he explores appears to be the historical Parmenides' claim that "the all is one” [Parm. 128a]. However, the logical consequences of the claim are shown to be utterly absurd, or at least in absolute contradiction with the evidence of both the senses and human reasoning. For instance, Parmenides demonstrates that the One cannot be anywhere, neither in another nor in itself [138a]; that it is neither at rest not at motion [139b]; that it is neither like nor unlike, neither same nor other than itself [140b]. In fact, immediately prior the passage quoted above, which implies the One's dependence on a separate Being and hence paradoxically gives the lie to his claim that "the all is one",s Parmenides argues that the One "neither is one nor is at all" [141e\. "consequently, it cannot have a name or be spoken of, nor can there be any knowledge or perception or opinion of it” [142a]. But we are speaking about it, aren't we? asks Parmenides. The dialogue seems to offer us an unpleasant choice: either human logos is insufficient, or the Parmenidian thesis that "the one is" is false; that is to say, either reality is irrational, or there is no reality at all. In Plato’s oeuvre the paradoxes of the Parmenides carry a destructive function; the Sophist shows a way out by introducing the concept of blending. Blending [methexis] answers the Parmenidian critique of Ideas, as well as provides us with the tools with which to rationalize the existence of the One by erecting a hierarchy of reality based on blending. The only thing that resists rationalization is the One’s relationship with Being as such. Plotinus’ reading of the these dialogues, however, commits the brilliant error of treating the paradoxes of the Parmenides constructively, as effects and indicators of the transcendence of the One. In other words, Plotinus takes the suggestion that reality is not wholly rational and runs with it. And the outcome is his resolution of the Being/One problem. Plotinus chooses to sacrifice the being of the One in order to preserve the oneness of Being. Perhaps the word "sacrifice" is misleading: on the contrary, Plotinus chooses the One over Being, making the former prior to the latter. The clearest justification of his move may be found in the Sixth Tractate of the Sixth Ennead, although the thought recurrs throughout the corpus: the One is the principle of Being, and the being of Being rests upon this; for otherwise it would be scattered; but the One does not rest upon Being; for then Being would be one before attaining the One [6.6.9.40-42]. sIf the all is not one, then it must be many, but the dialogue starts with Zeno's proof that the all cannot be many [127e]. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In other words - and as I’ve already indicated - if Being were ontoiogically prior to the One, and the One acquired its being from participating in Being Itself [Parmenides 142b], Being Itself could not be one - but neither could it be many, since the many derive from the One. So it really could not be at all. The One, on the other hand, if placed prior to Being, not only remains one —here Plotinus capitalizes on the distinction in Parmenides 142b between “a one [is] one” and “a one is” - but also cannot be ascribed to Nonbeing in our sense of absolute absence, since it does show at least one trait of "presence," viz., the fact that it remains one. As we have already noted, each thing in the world exists by virtue of its being-one: its oneness is derived from its participation in the One, whereas its being is derived from its participation in Being Itself, which in its own turn also draws its oneness from the One [6.6.10.11-12],6 Therefore the One also exists in the sense that it is an object - really the object - of participation. At work in Plotinus are the same reifying assumptions as in Plato. As the One is the foundation of things by virtue of participation, so it is of signs by virtue of signification; as things evidence the existence of the One, so signs: How is it possible for that not to exist without which it is not possible to think or speak? For it is impossible to say that something does not exist of which, since it does not exist, you cannot think or say anything at all. But that which is needed everywhere for the coming into existence of every thought and statement [e.g. the One] must be there before statement and thinking: for this is how it can be brought to contribute to their coming into existence [6.6.13.41-49]. At the same time, the One as such is, as in Parmenides 142a, unknowable and ineffable: "it is false even to say of it that it is one, and there is 'no concept or knowledge' of it" [5.4.1.9-11 quoting Parmenides 142a; see also 5.3.13-14],We refer to it as "one" only provisionally and by analogy [6.9.5.31-34], Even to say that it is the cause [of all] is not to predicate something incidental of it but of us, because we have something from it while the One is in itself; but one who speaks precisely should not say "that” or "is"; but we run around it outside, in a way, and want to explain our own experiences of it, sometimes near it and sometimes falling away in our perplexities about it [6.9.3.49-55; see also 6.7.38]. How is this possible? How can something be beyond Being, beyond language, and still get talked about? Plotinus’ ontology rests on a curious and an un-Platonic cavil. We 6"It is by the one that all beings are beings, both those which are primarily beings [i.e., the immutable Ideas] and those which are in any sense said to be among beings [i.e.. things subject to mutability]. For what could anything be if it was not one?" [6.9.1.1-3]. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. remember the question Socrates puts to Theatetes: "You do not suppose a man can understand the name of a thing, when he does not know what a thing is?" to which Theatetes replies in the negative [Theat. 147b]. For Plotinus, however, at least with respect to the One, matters stand otherwise: we know that it is [in a sense, see 6.7.38.1], we can even deduce some of its properties - hence the propositions of the Enneads - but we cannot know what it really is [see 6.7.38 ]. It is as if we knew a man was coming towards us but we did not know that the man was Pierre ("Pierre" because example borrowed from Gilson 84). In other words, we can know of the existence o f the One, but not its essence (in fact, being above Being —esse —it has no essence, see below). Plotinus is not so much telling us about the One, as instructing us how we may behold it ourselves; like grace or divine presence, the One "is always present to anyone who is able to touch it, but is not present to one who is unable" [4.9.7.4-5; see Bussanich in Gerson 41, 55-57], "Whoever has seen, knows what I am saying" [6.9.9.46-7]. Transcendent, the One can and cannot be known. It is beyond the reach of discursive logic [episteme], which manipulates enchainments of signs, and even of intuition [noesis], which delves deep into discrete signs.7 The One can be known only ecstatically, in bypassing episteme and noesis, when the latter annihilates itself, losing all objects of vision [6.9.4.2-10]. One's encounter with the One occurs in a site with no signs. Thus the chief signified of the Plotinian system is radically different from other signifieds: while the One is represented by all things that are one (the hierarchy of being may be thought of as hierarchy of signs approximating the One with increasing accuracy), it itself is void of proper signifier. Is this still referentialism? Certainly, for the ineffability of the One rests on its being above Being: because it is a "thing" that is not, it has no sign. "No one could either think or say 'what is not"' [Sophist 260d\. But it is referentialism with a singularity, since the ineffable “thing” that is not, is, at the same time, the ultimate signified of all things. Equally paradoxal is the One’s relationship to beings or things. It is beholden to the Parmenides, with the exception of the admitted transcendence of the One, which fact allows things a measure of reality. Needless to say, there is no big sphere called "the One" at some point within or without space: the One is omnipresent in the sense that "though he is nowhere, there is nowhere where he is not” [5.5.8.24] - it is "everywhere", in all things, "and again it is nowhere" [6.8.16.2], in no particular thing as opposed to another. Void of ubi (Lat., "where"), the One is ubiquitous. The One is in all things and all things are "in" the One, though [5.2.1.6] only in the sense that they come - not "came" - from him [for 7 When we must use signs, we can proceed only provisionally, relying on an embryonic form of the pseudo-Dionysian method [see 6.7.36.7-8 for comparisons and negations; also Bussanich in Gerson 38-42]. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the intricacies of the Plotinian processus, see Gatti and Bussanich in Gerson 29-33,45- 57]. As the positive denominations of the One - which we must use "for the sake of persuasion" [6.8.13.5] - have it, the One is "existence," "actuality," "being,” "substance," "life" and so forth [see Bussanich in Gerson 59 for quotes; compare to pseudo-Dionysius' Divine Names], because it is the cause of all these [but see 6.9.3.49-55]. At the same time, the One itself transcends them, is not them, and remains in splendid isolation, For since the nature of the One is generative of all things it is not any one of them. It is not therefore something or qualified or quantitative or intellect or soul; it is not in movement or at rest, not in place, not in time, but "itself by itself of single form," or rather formless, being before all form, before movement and before rest; for these pertain to being and are like what make it many [6.9.3.40-45 quoting Symposium 21 lb]. 2.2 The O ne as O ne a n d as Nothing It is glaringly obvious that Plotinus’ concept of the One rests squarely upon the Greek concept of the one as number or rather “no number.” As the “numerical” one is no number, so the Plotinian One is no thing or being. As the “numerical” one is the arche of number, so the Plotinian One is the arche of Being and beings. If, with the Greeks, we think of the natural number sequence as composed of the one and the pluralities that follow it (i.e., numbers properly speaking), so the hierarchy of being is composed of the One and the things that follow it (i.e., beings properly speaking). The first term of each sequence, being its arche, is in fact outside the sequence, differing in essence from the subsequent members at the same time as generating and conserving them. In fact, the resemblance is even closer in that every entity but the One is, for Plotinus like for late Plato, number.® It can also be added that the Plotinian One’s lack of “where” and “when” brings to mind the alternative Pythagorean definition of the numerical one as “point with no position” [Metaphys. I084b26], In previous chapteres I described the Platonist interpretation of monadic numbers as composed of pure unities whose assemblages form particular unities; furthermore, I claimed that the monad, as the building block of number, very much resembles the unity of all particular unities. This too greatly informs the Plotinian notion of One. Like the indivisible monad and the unity of particular unities, the One, “unity in the true sense and 3 “For number will be either the substance [ousia] or the actual activity [energeia] of being [...]. Is not Being, then, unified number, and the beings number unfolded [...]? Since, because Being came into existence from the One, as that One was one. Being must also in this way be number: this is why they called Forms henads and numbers. And this is substantial number, but the other, which is called monadic, is its image” [6.6.9.26-36]. Plotinus' doctrine of upper genera as numbers is extremely complicated (devoted to the question. 6.9 is by no means exhaustive). 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rightly defined,” is “altogether without parts” [Sophist 245a]. Being without parts, it is not a one o f something —which perfectly accords with the fact that it is above Being. Its being responsible for the being-one of each thing is also a concept that Platonism confronts on the ground of number, explaining the distinctness of each monadic number by its participation in the Ideal Number proper to it. Finally, the One’s role as the cause and origin of things parallels the general unity’s relationship to numbers, with the proviso that the One is transcendent. In late Plato, the generation of things as numbers from the One is accomplished by combining the One with the Indefinite Dyad. Plotinus is, famously, a monist; that is to say, his processus of things from the One does not involve the mediation of anything like the Indefinite Dyad. While he does operate with the concepts of Nonbeing and Otherness, his Otherness is not reified but instead seems to be the difference among Ideas and their and our distance from the One [see 6.9.8.30-36], The concept of Nonbeing, on the other hand, may be applied to the One, but in a restricted manner. The One is Nonbeing, or rather Not-Being, very literally: it is not Being, it is above Being, and it does not in any way include or rely upon Being. It may also be spoken of as Nothing, but also literally: resisting total reification, it is not a thing and no thing. Again, neither Nonbeing as absence nor as Otherness apply here. Rather, the One is Nothing/Nonbeing because it is the arche of things/beings, just as the numerical one is “no number” because it is the arche of numbers. At this point, Plotinus injects the concept of Nonbeing with new and startling content. The One’s priority to Being lets its "no-thingness" be seen not as dearth but as excess of being. The One, as it were, has not less but more being than Being. Plotinus refers to this as superabundance. Superabundance is, for Plotinus, the effect of the One’s identification with the Good, first made by Plato in his famous and unrecorded Lecture on the Good [see Sayre 77-78; Metaphys 988al4-I5]. Plotinus uses the identification of the One with the Good to explain not only why the One should generate the universe, but also why the One itself should exist and why it should exist as it does [see Gatti in Gerson 28 for this expansion of philosophical inquiry and for its pre-emptive critique by Aristotle]. Summarizing Enneads 6.8, Maria Luisa Gatti writes: “while other being are satisfied with themselves only because they participate in the Good, in the Good is contained the choice and the will for its own being. The first principle posits itself and creates itself as well, and is self-productive activity" [29]. ‘T he good,” says Plato, “is the end of all actions” [Gorgias 499e]. While all things strive towards the Good, the One has nothing to strive for but itself. Because it is thus free and self-contained, the One is also perfect; hence a contingent reason for both its 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. existence9 and for its generation of the universe: "All things when they come to perfection produce; the One is always perfect and therefore produces everlastingly; and its product is less than itself' [5.1.6.37-9; see Bussanich in Gerson 47]. Thus the One produces all things by the necessity stemming from its self- sufficiency and aloofness (hence Plotinus is not an emanationist, see Gatti in Gerson 29- 31). For if, apart from its self-sufficiency, the One did not also produce, it would not be perfect, not be the Good, not be itself. Being perfect, the One is not exhausted by its activity - “behold, the bush burned with fire, and the bush was not consumed” [Exod. 3.2]. Plotinus describes the mode of existence of the One, the mode of existence that is at once superabundant and superessential (in the sense of being above Being), as pure activity, as energeia without ousia [6.8.20.10-11], i.e., activity or existence without essence.10 If, because of this, we call the One "nothing," thereby conflating all terms, it is only "our darkness which can make us think it dark."11 3. Augustine It barely needs saying that early Christianity was forced to wrestle with the Greek philosophical heritage; one of the Jacobs in this match was Aurelius Augustinus. Augustine too walked away with a limp: Gilson speaks of the "insuffisance philosophique de son platonisme chretien" [194], For Gilson, as for us, the limp consists in the conservative nature of Augustine’s Platonism: though bom 75 years after the death of Plotinus (himself a fellow student of Origen), Augustine identifies God with Being, thereby wholly ignoring the Platonic aporia of Being and the One. Limp, yes, but the move itself is rather glorious —even if Augustine is not the first to make it. In appropriating Greek philosophy, Christian Platonists transformed Christianity into something intellectually consequential; at the same time, the tenets of religion received their "scientific" - in the sense of scientia as episteme - basis and proof. The contrast with classical mythology is quite sharp here: despite the pagans’ bandying about with the names of Zeus & Co., the misbehaving denizens of Olympus were, for anyone philosophically inclined, never more than allegories and symbols. To be sure, and as Christians never tired of pointing out, pagan Platonists too had their very non-Olympian 9 Since the perfection of the One depends on its identification with the Good, this is not an ontological proof of the kind we find in Anselm. 10 The contrary, being without activity, would be imperfect. See Bussanich in Gerson 48. 11 Herbert. Lord of Cherbury, "Sonnet of Black Beauty,” 1. 14. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. God [eternal in Phaedo I06e, immutable in Republic 380d], but this God was hardly the object of religion in the Christian sense: he was an Idea or rather an aspect of Ideas. Or rather, not an aspect but - the very fact that it is so difficult to describe Plato's God attests to the philosopher’s utter disinterest in what we call "theology." Agathon's beauty made his admirers call him "divine" - how much more divine is Beauty itself? But what can we do with such Divine apart from contemplating it? Can we pray to it? What's the point? Even if Ideas had ears - and hearts - what have they to do with the crooked timber of humanity? The identification of the Old Testament God with Being made God into, so to speak, the real thing; at the same time, it paradoxically rendered Being not in all ways impassable. For here was Being that cared, Being that was present in all things not only as their ontological principle, but also ethically, inasmuch as they strove towards the Good - the Good understood much more personally than the Good of Plato and Plotinus. This conception, as curious as it is overwhelming, in Augustine rested on the following extremely problematic premises. Augustine inherited the Platonic definition of Being as what is, i.e., what is immutable, and the correlative thesis that the mutable becoming is only to an extent. Thus, if the world was a hierarchy of intensities of being, being at 100%, i.e., Being as such, was God [Gilson 75, see also 193]. The key definition of God was provided by God himself at Mount Horeb: And Moses said unto God, Behold, when I come unto the children of Israel, and shall say unto them, The God of your fathers hath sent me to you; and they shall say to me, What is his name? what shall I say unto them? And God said unto Moses: I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me to you [Ex. 3.13-4].i: Augustine interprets God’s words to mean, "I am the immutable being, that which always is and never changes" [qtd. in Gilson 191]. According to Gilson, when Augustine claims God is Being, his notion of Being is, as in Plato, reified: wherever there is a fact of esse it can be made possible only by an essentia; in fact, true Being is essentia, essence. Thus, if it is God that causes the being of all things, he must be Being Itself. If God is Being, evil is defect or want of being. If there were such a thing as pure, unadulterated evil, it would have to be the same as Nonbeing itself on the one hand, and total mutability on the other. But having no being at all, none of these three suppositious entities can have a part in existence. I’ll return to this in my section on pseudo-Dionysius. i: "Ego sum qui sum. Dices filiis Israeli: Qui est misit me ad vos." 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The prime point of identification of God with Being in the oeuvre of Augustine is, for Gilson [76], De Trinitate 5.2.3. Referring to the Exodus passage above, Augustine writes: God is without doubt a substance, or perhaps essence would be a better term, which the Greeks call ousia. For just as wisdom [sapientia] is so called from being wise [sapere], and knowledge [scientia] is so called from knowing [scire], so essence [essentia] is so called from being [esse]. [...] All other things that are called essences or substances are susceptible of accidents, by which a change, whether great or small, is brought about in them. But there can be no accidents of this kind in God. Therefore, only the essence of God, or the essence which God is, is unchangeable. Being is in the highest and truest sense of the word proper to him. There are many problems with such a view. While above I say that God's identification with Being means that Being is, in a way, not impassable, it is still rather hard to imagine Being jealous, or angry, or experiencing any other emotion attributed to God in the Old Testament, or as a historical agent at all. At the same time, as soon as we think of God as immutable, we cannot imagine him going through any of the above experiences either. Yes, God does surpass human understanding - but why should Being or first essence be exempt from it as well? And if we do claim that Being is incomprehensible, is it, for Platonists, Being at all? For Being and all other "things that are," are supposed to serve, by virtue of their immutability, as objects of sure and certain knowledge. Conversely, if Being is comprehensible, God must be comprehensible, which the very paradox of impassability shows not to be the case [see Oxford Dictionary o f Christian Church, "impassability"]. In fact, such paradoxes can be solved only through transcendence, but there does not seem to be a radical enough discontinuity between Being and other beings, first essence and other essences, in order for Being - and therefore Augustine's God - to be defined as in ontologically transcendent. Augustine's inability to account for divine transcendence leads to further problems, this time in the domain of theology rather than philosophy. The first concerns creation. If God is Being, all that the creation is, is the creation of being by Being. This means that there always was, in some way, being. The resultant picture is pretty unremarkable, and I think flies in the teeth of creatio ex nihilo. Or, to put matters differently, eternity of being, even if that being is God, smacks of the Aristotelian eternity of the world. The second problem concerns immanence. If God is Being and therefore the being of things, the world in fact is God: the necessary outcome of such insufficient differentiation is pantheism. And if God is Being, what do we make of the One? 133 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 4. P seudo-D ionysius 4.1 God as the Transcendent One The Iate-fifth-century anonymous who wrote under the name of Paul's student Dionysius the Areopagite, and who, after his sixteenth-century "unmasking," became known as pseudo-Dionysius, placed God above Being, identifying him with the transcendent One of the Plotinian tradition. The Scriptures, writes pseudo-Dionysius in his tractatus On Divine Namesine Names, "celebrate the Supreme Deity by describing it as a monad or henad [i.e., one], because of its supernatural simplicity and indivisible unity" [Divine Names 1.4, 589d], Here, pseudo-Dionysius is clearly capitalizing on the Platonist understanding of the numerical one as “pure” unity, a unity that paradoxically does not imply parts. God is called the One, adds the Areopagite’s seventh-century Byzantine commentator Maximus the Confessor, not only because he is indivisible, but also because, having created the multiplicity of the world, he himself remained himself and integral [for Maximus' commentary, see the Russian-Greek edition of pseudo-Dionysius in works cited]. This ambivalence, of course, is very much in line with the Plotinian One, unaffected by and separate from the many that come to be from it.13 Also Platonist is the One’s connection with the many, as evidenced by pseudo- Dionysius' calling God the "henad unifying every henad" [Divine Names 1.1, 588b]. God as the One is the cause of the being-one of each thing: When things are said to be unified, this is in accordance with the preconceived form of the one proper to each. In this way the One may be called the underlying element of all things. And if you take away the One, there will survive neither whole nor part nor anything else in creation [Divine Names 13.3, 980b]. Since pseudo-Dionysius, like Plotinus, conceives of the numerical one in the Greek manner, he regards it not only as a unity, but also as no number. The essential difference between the one and numbers, I think, is what serves as the model for the Plotinian and pseudo-Dionysian notion of the One’s transcendence to Being and beings.14 While the unity of each thing and number stems from its participation in the One Itself, the nature of this participation15 is different from what it is in Plato: l3We may glimpse its aloofness in the arithmetical truth that, while all pluralities are composed of ones and may be broken down into them, the one, as Maximus the Confessor says, generally cannot be divided [comment to 13.2, 977c\ see my second chapter], u For example: "Without the One there is no multiplicity, but there can still be the One when there is no multiplicity, just as one precedes all multiplied number" [DN 13.2, 980a]. 15 "Nothing in the world lacks its share of the One. Just as every number participates in unity - for we refer to one couple, one dozen, one-half, one-third, one-tenth - so everything, and every part of everything, 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The name “One” means that God is uniquely all things through the transcendence of one unity and that he is the cause of all without ever departing from that oneness. [Divine Names 13.2, 977c; my italics]. The phrase "transcendence of one unity" is rather knotty. Its meaning may be deciphered from the passage that follows: "The One cause of all things is not one of many things in the world but actually precedes oneness and multiplicity and indeed defines oneness and multiplicity [...] the transcendent unity defines the one itself and every number" [Divine Names 13.2, 977c-d; 13.3, 980d].16 Thus the doctrine of the transcendent One lead to a positing of another, subordinate and a numerical one, the first member in the sequence Ideal One, Ideal Two, Ideal Three, etc. We might surmise that the Ideal One is, as far as participation is concerned, a stand-in or viceroy for the transcendent One of God, and that, by participating in the Ideal One, things really participate in its cause, the transcendent "oneness beyond the source of oneness" [2.4, 641a]. Whether things receive their being- one from the transcendent One directly or from it by the mediation of the Ideal One, however, is not that important: in either case, there can be no "direct" participation in the transcendent One. The reason for this is as for Plotinian remove of the One (the hierarchy of being is does not start with the One, for the One is above Being), but reformulated in theological terms: God is not continuous with the world - for then we would lapse into emanationism.17 That the One is transcendent literally means that it is prior to Being. Although "Being" is one of the names of God and the Divine Names devotes a chapter to it in that capacity, God is "beyond being" [Divine Names 1.1, 588a; Myst. Theol., 1.1, 997b].iS Rather, he is called "supra-existent" or, more commonly, "superessential" [hyperousion], "not essence but above all essence" [Maximus on 1.1, 588b], In fact, the existence of an independent Being, a thing apart from all other things, is, in pseudo-Dionysius, a major problem. Being may be identical with the Ideal One, since the oneness of either is ensured by the transcendent One: pseudo-Dionysius does refer to the Ideal One as t o kv ov, vs. the uTtepoucaov kv of the transcendent One [Divine Names 13.3, 980d\. In general, Being is participates in the One" [DN 13.2, 977c]. It appears that pseudo-Dionysius thinks of numbers as both pluralities and magnitudes. See Maximus' comment to the quoted passage and to DN 13.2, 980a. 16 Maximus comments: "God, of course, exists before the one, since he is the origin of all. Therefore he is the creator or inventor of number." 17 "In reality," writes pseudo-Dionysius, "there is no exact likeness between caused and cause [...]. The fire which warms and bums is never said itself to be burned and warmed" [2.8, 645c]. 1SAs Gilson points out, what for Augustine was the name of God - "I am that I am" interpeted as "I am being" - is, for pseudo-Dionysius, one name among many [Gilson 199; DN 1.4, 596a]. The anonymous author of Divine Names clearly prefers "the Nameless One" as much more accurate, as the "name which is above every name" [DN 1.4, 596a; reference to Philippians 2.9 where the name is "Jesus"]. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not separate and dissolves in the immediate being of all things: in one passage, pseudo- Dionysius denies it altogether [Divine Names 5.8, 821d], Maximus, commenting on the adjective "superessential," writes that "essence” [ousia] derives from the verb "to be" [einai], and that "to be" implies change [comment on L I, 588a] - a want of distinction that must have made Plato spin in his grave. Elsewhere, however, both texts oppose eternity, immutability and Ideas to time, change, and the sensibles, with God prior to all of the above [Divine Names 5.10, 825b]. Like Augustine, pseudo-Dionysius is not motivated by philosophical concerns. My attempt to deduce a clear-cut ontology from the Divine Names is therefore rather Procrustian; in addition, the language of the text is used dialectically, so that the meaning of every passage is provisional, approximate, apt to be contradicted by other passages. Yes, of course, all literature is like that, etc., etc., - but it is a question of degree.19 The linguistic strategies of pseudo-Dionysius, and to a slightly lesser extent of Plotinus, are less referentialist than those of Plato —precisely because Plato tends to think that the world is rational, i.e., can be expressed in words [but see his Letter 7, 341c, qtd. by Plotinus in 6.9.4.12], whereas the former two talk about the Absolute, and the Absolute can only be talked about, i.e., around [see Plotinus 6.9.3.49-55 quoted above]. At the same time, the presence of the Absolute distorts all other (non-transcendent) relations: once we ask what is God?, the ontological question what is the world? becomes what world? -but God is no what, and therefore both objects of reification vanish. This psychological event also has a rational parallel: introducing the Absolute throws a wrench into the entire paradigm of signification, for all signs or things, by virtue of their being-one,"transcendentally" point to the Absolute - which means that, as Nicholas of Cusa was to insist, no thing is ultimately knowable [Docta Ignor. 1.1], i.e., no signifier has its proper and exclusive signified. Pseudo-Dionysius stresses this to a far greater extent than Plotinus - probably because he is far less shackled by the philosophical tradition, because his aim is figuring out God and not understanding the world - and yet the difference is of stress, not nature. For our purposes, pseudo-Dionysius as it were brings Plotinus to his logical conclusion. 19Pseudo-Dionysius' language is so complicated that the latest Russian and English translations often seem to have different originals. 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Knowing God in a Floating World It barely needs saying that, prior to Being (whatever we may mean by this word), the God of pseudo-Dionysius is also "gathered up by no discourse, by no intuition, by no name" [Divine Names 1.1, 588b]; in fact, It is at a total remove from every condition, movement, life, imagination, conjecture, name, discourse, thought, conception, being, rest, dwelling, unity, limit, infinity, the totality of existence. And yet we must talk about it, and so " since [...] by merely being there it is the cause of everything, to praise this divinely beneficent Providence you must turn to all of creation" [Divine Names 1.5, 593c], Pseudo-Dionysius’ method for talking about divinity is certainly the most celebrated aspect of his thought, and we shall therefore touch on it only in passing. The method rests on the two elephants of assertion (kataphasis) and denial (apophasis). According to the former, the hierarchy of being offers us progressively more accurate "hypothetical expressions" [Myst. Theol. 1.1, 1000d\ of God: thus God is more like an animal than a vegetable, more like a vegetable than a mineral, and so forth. This is a semiotic of relations, where the system of things is construed as a system of symbols but no symbol signifies anything by itself, for, according to the other and weightier side of the coin - apophasis - God is not like a mineral, not like a vegetable, not like an animal [see Divine Names 1.4, 592c-593a; Myst. Theol. 2-3; comp, to Plotinus 6.7.36.7-8, see also Busanich in Gerson 41, 55-57]. Signification here is drifting, unmoored - although not entirely, for the facade of hierarchy still remains. In the world of floating signs, the pseudo-Dionysian ecstasy*0 derives exclusively from negation. As in Plotinus, the ecstatic person must gradually abandon all objects of knowledge (first episteme then noesis) to plunge into the truly mysterious darkness of unknowing [...]. Here, being neither oneself nor someone else, one is supremely united by a completely unknowing inactivity of all knowledge, and knows beyond the mind by knowing nothing [Myst. Theol. 1.1, 1001a]. Pseudo-Dionysius is working with - and violating - the Platonic framework. All knowledge is knowledge of an object, but God is no object, no being, and therefore the closest we can get to knowing God is by "unknowing."21 Since God has a certain :oLovely combination of words for the classically minded, isn't it? : '"If all knowledge is of that which is and is limited to the realm of the existent, then whatever transcends being must also transcend knowledge” [DN 1.4, 593a.]. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "presence," this unknowing is a directed act: it is surely not like the not-knowing of God we indulge in when we forget him by concentrating on some object of sense, sniffing a rose for instance. The way to God is not outward, but inward. Whether in pseudo-Dionysius one ever really "knows" God, even by unknowing, is a moot point, since unknowing is not knowing, and therefore God cannot be known by unknowing. Or, to put it differently, "the One who is beyond all things" "dwells" in "darkness" [Myst. Theol. 1.1, 1000c commenting on Ex 20.21]. Because unknowing too is darkness, when we un-know we become like the place "where he dwells" [Myst. Theol. 1.1, 1000d\. But God himself, of course, is not darkness [Myst. Theol. 5, 1048a]; thus even during "the supreme union" that is ecstasy, he must needs remain aloof. Or, if we forget that he is not darkness: let’s grant that absolute unknowing is possible - a tall order, if you ask me - we would still know the "what" of God no more than a raindrop knows the "what" of the ocean. But neither does the ocean know the raindrop! In the litany ending the Mystical Theology, a litany which denies anything attributable to God, pseudo-Dionysius asserts: "Existing beings do not know [Godhead] as it actually is and it does not know them as they are." It is superbly comic to watch Maximus scurry about picking up the pieces - but he does do a very good job of it: "God himself does not know Being as it is, i.e., he cannot approach the sensible as a sensible, or beings as a being, for that is not proper to God" [Myst. Theol. 5, 1048]. 4.3 G od as N othing The least complicated aspect of God’s being Nothing lies in divine immanence, whose description, like that of the Plotinian One, ultimately stems from the paradoxes of the Parmenides. God is no thing and in no thing, but is "everywhere" and "is in nothing" [Divine Names 5.10, 825b; also Maximus on 1.5, 593b]; not limited by a "where" and a "when", "he does not possess a this kind of existence and not that" [Divine Names 5.8, 824b; translator's italics; for God as outside time, see Divine Names 5.4, 8J7d]. At the same time, he is transcendent, which is to say that the divine names of "Life", "Being," "Good," "One,” and so on, in the end do not apply, for God is not life, not being, not goodness, not one or oneness [Myst. Theol. 5]. Moreover, it is even inaccurate to say that God transcends them: not only because he "is beyond assertion" but also because he is "beyond [...] denial" [Myst. Theol. 5 , 1048b]. Divine transcendence transcends itself, so to speak. We remember that Plotinus characterizes the One as activity without being. While pseudo-Dionysius insists that God "is not" [Divine Names 5.4, 817d], that does not mean 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that God does not exist. He is not only to the same extent as he is, since he "falls neither within the predicate of nonbeing nor of being" [Myst. Theol. 5, 1048a]. God does exist - I mean for pseudo-Dionysius —in the same way as the Plotinian One: through an "unfilled overfullness" or superabundance [Divine Names 2.11, 649d] escaping reification. This superabundance comes from the fact that, like Plotinus, pseudo-Dionysius identifies the transcendent One with "the inexpressible Good" [Divine Names 1.1, 588B; for "Good" as the name of God, see Divine Names 4], Because God is the Good, he, unlike all things, causes his own existence, as well as that of all else [for Good as source of superabundance, see Divine Names 5.1, 816b]. If God did not exist, things would lack not only their first cause, but also their final cause, their motive for any change whatsoever - since all things strive towards the Good. Therefore God is Nothing not only because he is no particular thing, but also because as superabundant he exceeds essence and so is Not-Being or Nonbeing. Again echoing Plotinus [5.2.1.5], Maximus predicates the emergence of things from the One on these very facts [comment to Divine Names 1.5, 593c], However, the terms "nothing" or “nonbeing” have a much greater range in the Dionysian corpus than what we have explored so far. Let us, for instance, take the just- quoted assertion that God "falls neither within the predicate of nonbeing nor of being." What pseudo-Dionysius means by the "predicate of nonbeing" is rather difficult to fathom. It cannot be limited to God's transcending the statement "God is not," for "nonbeing" [me on] in the Divine Names has an ontological aspect, or rather aspects, not at all in line with the Platonist "Other." According to Maximus, the meaning of me on in the Divine Names shifts, sometimes "indicating the divine, as that which is not any of the things having [i.e., participating in] being, sometimes matter, and sometimes for various reasons - evil" [comment to Divine Names 4.7, 704b]. "Nonbeing [panta ta ouk onta]," writes pseudo-Dionysius, "is transcendentally in the Beautiful and the Good" [Divine Names 4.10, 708a]; the same is said using the term me on [Divine Names 4.18, 716a], "because nonbeing [me on], when applied transcendentally to God in the sense of denial of all things, is itself beautiful and good" [Divine Names 4.7, 704b]?1 Pseudo-Dionysius seems to be ontologizing the “not-this-thingness” of apophasis and locating its source in God himself. But is not denial merely our way of approaching God? Or should we think that the apophatic method reproduces God’s mode of being? At the same time, denial cannot be ontologized as Otherness, for to place Otherness in God is both blasphemous and counterintuitive, since then the One would no longer be one. In his “ Maximus comments: "nonbeing exists by the cause o f being and super-being: God who is All as the Creator, is Nothing as He who surpasses all, or rather is supersurpassing and supra-essential.” 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. comment to the last of the above passages, Maximus defines "nonbeing" as "an absence, since it exists due to the lack of something having being" [my italics]. However, this "absence” cannot be understood in the Augustinian mode as defect of being and ontologically subsequent to being, for God does not lack being - rather, he exceeds it. In addition, whatever “the nonbeing” that “is transcendentally” in God may be, pseudo- Dionysius defines it as the only real nonbeing: "nothing is completely a nonbeing [me on], unless it is said to be in the Good in the sense of beyond-being" [Divine Names 7.19, 716d}. The term "nonbeing" also extends to non-transcendentals. The Good generates not only being, but also "nonbeing," "things that are not" [Divina Nomina 5.1, 816b]; Maximus interprets the phrase to mean "things that are not yet come into being but will." Here the distinction between being and nonbeing seems to be the Aristotelian distinction between actual and potential existences, the in actu vs. the in potentia. We are thus not very surprised when "nonbeing" is said by Maximus to indicate matter in general, in his comment upon pseudo-Dionysius' claim that "even that which is not wishes for a place in [the Good] since even in the unformed it evokes form, which is why nonbeing [me on] is said to be transcendentally in it" [Divine Names 4.18, 716a; for Aristotle and Plotinus on matter as nothing that is something, see Wolfson 360, nn. 20-21]. Thus, if we believe Maximus, matter, which is not a being and therefore has no being-one, also transcendentally participates in the One, due to its desire to be one. But how can nonbeing desire and participate?23 Is it that matter, unlike the "nothing" that is "transcendentally in" the One [see comment to Divine Names 1.5, 593c], is not "pure" nonbeing [see Divine Names 7.19, 716d]l As I've mentioned, Maximus claims that the term "nonbeing" also sometimes refers to evil. The discussion of evil in the fourth chapter of the Divine Names seems polemically directed against both Augustine and the Gnostics. For Augustine, if God is Good and Being, then evil is identified with privation of being and ultimately with nothingness as non esse; for the Gnostics, evil, as lack of being, is identified with matter [see references in Dictionnaire de Theologie Catholique, "mal," v.9, cols. 1693-4 and cols. 1686-7 respectively; see Divine Names 4.28 for argument that matter is not evil]. Since pseudo- Dionysius distinguishes between the Good and Being as between transcendent cause and its effect, his evil is opposed to the Good but not to Being. Thus, although evil is a being inasmuch as it is in beings and is the opposite of Good [Divine Names 4.19, 717a], "evil 23 Actually, if we define nonbeing as privation of what should be there instead, it is, of course, pure desire. But this brings up the problem of what we mean by “there.” On the Aristotelian argument against nonbeing that rejects that nonbeing can take up space, see Grant, p. 5-8. 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [...] in itself has no being" [Divine Names 4.19, 717a], for all that has being comes from the Good [Divine Names 4.20, 720b]. Therefore evil merely "occupies" being,24 and exists only when it is not pure evil [Divine Names 4.18, 716a]. In contrast to Augustine, the occupation of being by evil does not lessen the being; in accordance with Augustine, evil serves Good in the long run [Divine Names 4.20, 717c].15 While the fact of its —literal - insubstantiality means that evil is not a being [Divine Names 4.31, 732c], evil "has a greater nonexistence and otherness from the Good than nonbeing has," "for nothing is completely a nonbeing unless it is said to be in the Good in the sense of beyond-being" [Divine Names 4.19, 7l6d; also c]. The reader will have noticed to what extent my teasing out of the meaning of "nonbeing" depends on Maximus; the reader will also rightly conclude that if these distinctions could have been made with the text of the Divine Names only, I would not have turned to the commentator except for an occasional confirmation. But the text itself seems to use the term "nonbeing" in so general a manner, that what stands behind it - if anything - is extremely perplexing. Thus the fact of divine transcendence brings pseudo-Dionysian signification in flux; from this flux emerge concepts of "nonbeing" that can be defined only by their opposition to being, to thingness, and to the transcendental itself. At the same time, this flux of signification goes hand-in-hand with the absence of a coherent ontology, with the text's evident disregard of Platonic Ideas. Not that such disregard is purposeful, nor does pseudo-Dionysius propose an alternative system for understanding why, say, a raven is a raven and not a writing desk. Rather, although pseudo-Dionysius gives lip-service to Paul s claim that the invisible things of God are known through things visible [Rom. 1.20; Divine Names 4.4, 700c], he forces its meaning to do a volte-face: a) the invisibilia Dei are not-known; b) the visible things aid us only because we look at them and say, "Not this." Such negation both is and is not otherness. It is otherness in the sense that through it we know God to be Nothing as no-thing, as other to things. It is not otherness in the Platonic sense - it ascends instead of descending; it is ontologically unspecified except as "participating" in the nonbeing that is said to transcendentally be in the One which itself contains no Otherness. Therefore not only does the world come from the One "because :'Through a "weakness and deficiency of the Good" [DN 4.30, 732b] in things due to "their remoteness from the Good" and "in proportion with it” [DN 4.20, 720b]. This Augustinian definition of evil as privation of Good, however, is not accompanied by the Augustinian equation of Good and Being. The presence of evil in things does not affect their being at all. :5This is another way in which evil is a being: "through the workings of the Good it is a being, a good being, and confers being on good things" [ibid.] 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. there is nothing in it" [Plotinus 5.2.1.5], but our contemplation returns us to the One for the same reason. God is the One and God is Nothing. We, whose conceptual arsenal includes the zero, feel this to be as an acute paradox: I = 0. Pseudo-Dionysius, however, is operating with the Greek concept of number, where the one is both unity and no number. In a sense, the paradox of 1 = 0 is already present in the Greek system. It is the Greek opposition of the one to number that gives the God of pseudo-Dionysius his particular traits. Like the arithmetical one, the transcendent One, as pure unity, unity "altogether void of parts" [.Sophist 142b], is prior to number and therefore form [for form as number see e.g. Augustine, De Libero Arbitrio 2.16, 164; also Nicholas of Cusa, Docta Ignorantia, 1.1]; it is also void of where and when [the unit as "point with no position," see Metaph. 1084b26]. So formless, boundless, indivisible, the One is no thing. In a system of thought like that of pseudo-Dionysius, which opposes being as thingness to nonbeing as not-being- a-particular-thing (and not to Plato's ontologized not-this-thing, nor to Augustine’s defect of being), the One which is no thing but nonetheless exists, exists as nonbeing, nothing. 5. Eriugena 5.1 The Dilemma o f Creatio ex Nihilo and the Eternity o f Ideas The Corpus Dionysiacum exerted an immense influence. Since prior to the sixteenth century no one doubted but that its author was St. Dionysius, first Bishop of Athens, the Athenian philosopher converted by Paul in the Acts of Apostles, its doctrinal authority had few equals.26 In this section I would like to explore an important, if eccentric, interpretation of pseudo-Dionysius’ concept of nonbeing and of God as Nothing in the work of his first Latin translator John Scottus Eriugena or Erigena (9th c.). While many of the distinctions Eriugena makes are normative, his conclusion decidedly is not, and as such was condemned at Paris in 1210 and by the papacy in 1225. Nonetheless, Eriugena’s subordinate ideas shouldered their way into mediaeval philosophy, proving especially vigorous in the thought of Nicholas of Cusa [see Moran 269-281]. What matters for us is Eriugena’s treatment o f nonbeing, and in this field, which Eriugena scholar Dermot Moran sharply calls “meontology,” he is one of the clearest voices in the Western philosophical tradition. Starting with the claim that “nature is the general name [...] for all things, for those that are and those that are not” [441a, referring to Divine 16 He was thought to have inspired the pagan Neoplatonists and not the other way around. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Names 5.1, 816b\ for an earlier source, see Moran 214-215], Eriugena’s Periphyseon (later known as De Divisione Naturae) treats of nonbeing in two places: at the outset of Book I, and in the famous Quaestio de nihilo of Book HI [634a-687d\. In his analysis of being and nonbeing in Book I, Eriugena takes the general relativistic and dialectical tendency of pseudo-Dionysius’ thought and systematizes it. As a result, the concept of being is de-ontologized: the reason we refer to something as “being” or “nonbeing” is that we refer to what we fo r now consider to be its converse as “nonbeing” or “being”; any object from different perspectives may be “being” and “nonbeing.” While the details of Eriugena’s many modes of being need not concern us here, we should mention some examples. According to the first mode, things that fall within the reach of sense or reason are said to be being, while things that transcend them - i.e. God - are said to be nonbeing. According to the fourth mode, things that are immutable - like God - have being, whereas things that are not —like objects of sense - don’t. The most radical of all modes is the second. According to it, if any level in the hierarchy of being is said to be, the others are said not to be: so if man is, angels are not, and vice versa. Thus, in Eriugena’s list of the modes of being, the terms “being” and “nonbeing” are not referentialist at all. There is no such “thing” as Being, nor is the quantity of being assigned to any creature in any sense absolute. The Quaestio de nihilo in Book HI approaches nonbeing with a different set of problems in mind; the central problem is from what sort of “nothing” God is said to have made all things. Eriugena starts off with contemplating whether this “nothing” can be understood in the traditional Augustinian way, which in its own turn contests the identification of “nothing” with the matter of the Timaeus, uncreated and co-etemal with the Creator.27 Eriugena predictably argues that “He Who made the world from unformed matter also made unformed matter out of nothing at all” [636d]. If this nothing is not matter, what is it? The answer Eriugena considers is that “nothing” in the phrase creatio ex nihilo stands “for the total privation of the whole of essence and, to speak more accurately [...], for the absence of the whole of essence; for privation means the removal of possession [habitus]” [634d]. For the moment he takes that to mean that things “were not before they came into being” [634c], And yet this thesis comes into conflict with Eriugena’s eccentric version of Platonic Ideas, which he terms primordial causes of all things. In the Christian Neoplatonist fashion, he locates these causes in the Wisdom or Word of God [636b and passim. 11 This opinion, held by Justin and Clement of Alexandria, also has biblical backing: Wisdom 11:17 proclaims that God’s “almighty hand [...] made the world out of matter without form” [see Oxford Dictionary o f Christian Church, “creation”]. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. identified with Christ as Logos in 642b-c]. Since, for Eriugena, “every caused thing always subsists in the cause” [639c], i.e., the cause is the being of the effect [640a], the primordial causes must needs themselves have been made in order for the Creation to have occurred at all. However, Eriugena regards them as eternal. Can something eternal be said to have begun in time? Eriugena works with two versions of temporality, one of them being eternity and the other time proper. a) He conceives of time proper as a succession of moments that may be represented as a line. Succession permits change: the state of an object at point A differs from the state of the same object at point B. b) Eternity for Eriugena is the "contraction of all times," [contractio totius temporis, in Poulet 45, n. 2] into a single now, temporal development taken as a simultaneity. If time is a line, then eternity is a point. If temporal “nows” are points upon the line of time, the eternal present is extrinsic to it. It is the entire line collapsed upon itself, as it were.23 The two temporalities are mutually exclusive. According to the second paradigm, nothing eternal - including the primordial causes - can not be at one moment and be in the next, since all of eternity is only one moment. Eternal things are things that always have and always will exist in the same state. Therefore the primordial causes cannot have had a temporal beginning [see 636b]. Hence, 1) Creatio ex nihilo cannot mean that things did not in any way exist before they came into being, since “every caused thing always subsists in the cause” and causes are eternal; 2) Since God did create the world, i.e., things and their causes, “being made” and “being eternal” must in some paradoxical way be synonymous.29 5.2 The Model of the Monad and Numbers To explain how it is that the primordial causes are “eternally created” in the Logos, and therefore “how all things [qua effects - EO] are at one and the same time both eternal and made” [638b], Eriugena employs the model of the relationship of the one and numbers. :s For these in Boethius and Bonaventure. see Poulet 26-7; popular Christianity generally understands eternity as an endless form of time, i.e., of time that goes on and on. For line of time as circumference around the center point of eternity, see part 2 of this chapter, section 1.3. 29 This is evident from Psalms 103:24, Omnia in sapientia fecisti. “Omnia, " says Eriugena, has to mean both causes and effects [666c]. For Wisdom = Word of God, i.e. Logos, see e.g. 635c. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which he got from pseudo-Dionysius [Divine Names 5.6, 820d-821 a; see pt. 2 sec. 1.2 of this chapter]. This model is all the more apt since Eriugena follows pseudo-Dionysius (and of course Plotinus) in considering God as the One. “that Monad which is the sole Cause and Creator of all things visible and invisible” [659b], which presumably means that God as the transcendent One (monas creatrix) exhibits some likeness to the numerical one (monas creata). Moreover, Eriugena follows Pythagoras, late Plato and Plotinus in believing that causes and things are themselves categories of number.30 Thus the relationship of God to his creation is like that of the monad to numbers. Eriugena calls the one or the monad or “unity” “the beginning and the middle and the end of numbers”: “they are in it causally because it subsists as the beginning of all numbers, and in it all are one and simply indivisible” [652b]. When numbers flow out of the one31- whether into human or angelic intellects or into things [661c] - “they do not cease to be in it” [653d], presumably since the one remains their cause, and since each number is, as we have so often repeated, a kind of one. The procession of numbers is also said to return to the one: Numbers which proceed from their beginning proceed from nowhere else than their end - for their beginning is not one thing and their end another, but they are one and the same unity and therefore it must be concluded that if they extend to an infinite end their extension must begin from an infinite beginning. But the infinite end of all numbers is unity; therefore the infinite beginning of all numbers is the same [653c-d]. This rather difficult passage claims a certain identity between the monad and the totality of numbers forming an infinite progression. Both the “end” of numbers (by which Eriugena means the infinite progression) and the “beginning,” i.e. the monad, are each a kind of infinity. a) The monad is infinite in the sense of “boundless” or “undifferentiated” by virtue of being a pure unity “altogether devoid of parts” [Sophist 245a]. Such an entity cannot be limited or circumscribed by anything, for then it would have parts (border vs. not-border), nor would it remain the sole origin of number but must share the honor with whatever delimits it. 30 He thinks the Bible [ Wisdom 11:20, “thou hast ordered all things in measure and number and weight”] confirms the statement of “the supreme philosopher Pythagoras" that “intellectual numbers are the substances of all things visible and invisible” [652a; Meiaphys. 987al9 read through its portrait of late Plato with formula for things derived from Romans 1:20]. For Eriugena, “substances” means “primordial causes”: since the cause is the being of the effect, physical things are numbers as well. 31 For the generation of numbers and the resultant subkinds of number, see 654a-b For Eriugena's elaborate theory of the eight ontological orders of number, see 65Ia-660c, 731c-732a. 145 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. b) When numbers reach the end of their emergence from the monad, they form an infinite progression (in modem terms: 1, 2, 3,.... °°). Eriugena refers to this progression as a unity because it contains an infinite (i.e. uncountable and therefore indivisible) number of parts, each of which is finite in itself. The parts, i.e. numbers, are finite because they are conceived as aggregates of monads.32 The two infinities are the synchronic counterparts of eternity as the eternal now and of time envisioned as duration without terms. They too may be represented by a point and a line respectively [see pt. 2, sec. 6.2 of this chapter]. Eriugena predicates numbers’ capacity for flowing out in infinite progressions upon their pre-existence in the monad [653a], where “all numbers are at once together and no number precedes or follows another since all are one” [654c\. He defends this view of the essential eternity of numbers with the argument that “an infinite progression” cannot “arise out of a finite beginning” [652c; this is why causes cannot have been temporally made]. Thus the one both is and is not simple, in that it contains the causes of all numbers but in an undifferentiated state. Itself having neither beginning, middle, nor end, it is also the beginning, middle and end of numbers proceeding out of it. This is a paradox and certainly cannot be imagined. Numbers in some senses do have a beginning and end; in others, they do not. Causally, they do not have a beginning in that they always subsist in the monad, but they do have it in that they proceed from the monad as their cause; same for the end [655a]. Temporally, Ideal Numbers are eternally in the monad, whereas the lower genera of numbers - those in things or those we count with - may be said to have a beginning and end in time, inasmuch as the things and our counting begin and end in time. On the other hand, since the lower genera of numbers do not have their own independent being [see 646b] and subsist in Ideal Numbers, they are also in this way eternal. Eriugena’s versatile paradigm of the monad and numbers maps the relationships between la) the monad and Ideal Numbers; lb) God and primordial causes; 2a) a particular 3: For the Pythagoreans, the one and the many are associated with the Limit and the Unlimited respectively [Metaphys. 986a24-27], Limit being responsible for all identities and the Unlimited for all differences in number [Philebus 24e-25a; Proclus 5-7], By saying that the infinite many is the same as unity, Eriugena appears to read the Limit-Unlimited opposition in a monistic manner, by pointing out that Limit itself cannot be limited by anything else. Thus the one as the Limit is in a sense the same as the Unlimited many. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ideal Number and the same number in lower genera; 2b) a particular primordial cause and its effects. All four uses of the paradigm rest on an important Aristotelian distinction made by Eriugena that he, even more importantly, immediately overcomes. The numbers in the monad exist in potentia; numbers that form an infinite progression exist in actu (in fact, the two infinities described above are now called the potential and the actual). However, they are one and the same: “The same numbers are eternal there where they are potentially in their cause, that is, in the Monad, but where they are understood to be actually, there they are made” [657b]. Eriugena’s distinction between the in potentia and in actu is general: it applies to Ideas and physical objects as much as it does to numbers. It is because the actual (effects) and the potential (causes) are essentially33 the same, that we can think of the effects as not only made but also eternal, and of the causes as not only eternal but also made. To claim that causes and effects are essentially the same means that everything that exists in potentia at some point becomes actualized. To defend this unorthodox view, Eriugena identifies the primordial causes with God’s volitions [673c]. Causes are that which God wills to be causes. Divine Will differs from the human in that, for God, willing is the same as having: “all things which He has willed to make He always had in his volitions. For in Him the will does not precede that which He wills to be made” [675a]. The same must apply to the relationship between God and the effects. If God willed you to be made, you are; conversely, if you are not, therefore he has not willed you to be, and so your cause is not in the Logos. Seen in this light, it becomes ridiculous to assume that not all causes descend into actuality, for why should God will to be that which he does not will to be? What will never happen is not among the potentials. The world cannot be otherwise than as it is. The concept of primordial cause for Eriugena extends far beyond Plato’s Ideas of the species. Causes are individual as well as specific, or rather - presumably - the specific cause contains the causes of all individual actualizations in the same way as the monad contains the causes of all particular numbers. The primordial cause or Idea of a duck, for instance, consists of the “ideas” of all individual ducks that ever were, are, and will be, and of nothing more. We must conclude that the difference between the causes and effects, Ideas and physical things, is only one of perspective. This startling doctrine has two implications: 1) 33 I.e., in their essence or being, with the causes as the essentia of the effects. Since, in this way we are not one thing and our causes another [640a], “every creature whatsoever considered through itself is nothing” [646b]. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the causes are not only eternal but also, as their effects, made; 2) the Divine Logos and the world are essentially identical. 5.3 N ih il in Creatio ex Nihilo as Nothing by Transcendence If the primordial causes are made, what are they made from? The traditional answer, based on 2 Maccabees 7:28,34 is “from nothing.” But what does this mean? We have already shown that, if “nothing” be understood as privation, it cannot temporally apply to primordial causes, for the causes are eternal. Can nothing as privation be the material cause of the primordial causes? No, for in this case “nothing will not be nothing, but it will be a cause” [663c\. In fact, for this reason “nothing as privation” cannot be any cause at all.*5 When Eriugena returns to this issue several pages later, he refines the meaning of “privation” in a manner recalling Rabelais’ drunks: “privation is the privation of possession” [686a; Gargantua 5; “privatio presupponit habitum”; the “bien yvres" debate what comes first, drinking or thirst]. “It is impossible that there should be privation where there is not [prior] possession of essence [...] and therefore where” - as with causes - “possession does not precede privation does not follow” [686a]. With physical things, on the other hand, ex nihilo may be used with the meaning of temporal privation, for “there was a time when they were not” - in the sense that they existed only as causes, in potentia, not in actu - “and therefore we are not unreasonable in saying: ‘They were always; they were not always’” [665a]. However, to limit the meaning of creatio ex nihilo only to the temporal privation of effects whose causes always are in possession would be to enormously trivialize the text, since what each effect is, is its eternal cause. And the question we are trying to solve is from what eternal causes were - or are - made. The answer is “from nothing” since any other answer would imply that there was a something coetemal with God. Yet this “nothing” must be capable of being a cause, nor can this “nothing”, as the cause of causes, be external to God, for then it would again be something - in fact, the All - coetemal with God [663d]. Therefore the word “nothing” must in some way refer to God himself. In what way is, for the translator of pseudo- M”1 beseech thee, my son., look upon the heaven and the earth, and all that is therein, and consider that God made them of things that were not," Donne, arguing that the nothing in ex nihilo isn't really nothing, comments: "Only it is once said. Ex nihilo fecit omnia Deus [Machab. 2.7.28]; but in a book of no straight obligation (if the matter needed authority), and it is also well translated by us. O f things which were not" [Essays 28]. 35 In fact, “no place is provided for nothing [as privation] either external or internal to God” [665a], so the concept is void of referent. Nothing as privation should be used only to refer to the absence of effects whose eternal causes are already in possession. 148 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Dionysius, clear as day: as nothing by transcendence, nothing by virtue of God’s being prior to Being:36 I should believe that by that name [of “Nothing”] is signified the ineffable and incomprehensible and inaccessible brilliance of the Divine Goodness which is unknown to all intellects whether human or angelic - for it is superessential and supernatural which while it is contemplated in itself neither is nor was nor shall be, for it is understood to be in none of those things that exist because it surpasses all things [680d\. “Nothing” for Eriugena is a Divine Name and he claims that the Bible uses it all the time [685a]. The propriety of the name rests on the fact that “on the authority of St. Dionysius negations are more apt for divine knowledge than affirmations” [685a]. However, as in pseudo-Dionysius, negation is not just a method of knowing: it is a mode of God’s real relation to things. In this sense, transcendence implies a kind of ontological privation, for “Nothing” as a Divine Name refers to “the total negation of possession and essence or of substance or of accident or, in a word, of all things that can be said or understood” [686c- d], and yet God paradoxically also has these things as causes in his possession. As we have seen, everything in Eriugena’s system hinges on his identification of being or essence of a thing and its cause.37 For thus a thing and its cause become co­ essential, or, to apply a later philosophical distinction, their only difference is a difference- in-identity [Moran 236-237]. Eriugena’s notion of co-essentiality is clearly beholden to, or at least identifies itself with, the consubstantiality of God the Father, Christ (who, as we remember, is the Logos [642b-c]) and the Holy Ghost as proclaimed in the Creed. However, by interpreting consubstatiality in a general manner, i.e. as relationship between any cause and its effect, Eriugena broadens the concept almost to the point of non­ recognition. This leads him to make the following identifications. We have already seen that, since in God willing equals having, his Will is no other than his volitions [675a-b], which are no other than the primordial causes and their effects, “the things which God has made” [673d]. At the same time, as Eriugena’s translator Sheldon-Williams points out, “God’s Will is not different from his Essence, or rather, Superessence” [Eriugena vol. 3, p. 9; in reference to 675b]. Therefore we must conclude not only that the Logos is the beginning, middle and end of things,38 but also that, since what “is made in the Son [i.e., the Logos] 36 There is no separate Being in Eriugena and he quotes pseudo-Dionysius to the effect that “being itself is never bereft of all things that exist” [682c; DN 5.8, 821d\. 37 “I find no reason why that which is predicated of the cause should not also be predicated of the caused” [646c\. 38 Eriugena calls the Logos as “the Being of the things that exist and their Motion and the Distinction of 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is made in the Father and the Holy Spirit” [672d\, then “the One God is all things in all things” [unus deus omnia in omnibus, 675c\ for this in Eckhart and others, see Poulet 33, 40]. This amounts to saying that God and the created world are essentially the same. But if God and creation are the same, he not only “makes all things” but also “is m a d e in a ll things” [671b, my italics]. To ask. Who makes him? but begs the question. If "God and His volitions and all the things that He made are one and the same, [...] the conclusion [...] will be: Then God made Himself' [674a; my italics].39 God’s self-creation is not limited to the historical in principio; rather, it happens in every now [for essential identity of eternity and the moment, see Poulet 28-29; compare with Dee’s "Numberer Numbering” in my chapter 4, section 4], His creation of the world and of himself may be represented as the atemporal conversion of a single, transcendent and infinite nothing into an infinite multiplicity of limited somethings: The Divine Goodness which is called ‘Nothing’ for the reason that, beyond all things that are and that are not, it is found in no essence, descends from the negation of all essences into the affirmation of the essence of the whole universe; from itself into itself, as though from nothing into something, from non-essentiality into essentiality, from formlessness into innumerable forms and species [681b-c\. In this atemporal “descent”, nothing is not one thing and something another: “both the creature, by subsisting, is in God; and God, by manifesting Himself, in a marvelous and ineffable manner creates himself in the creature” [678c], Rather, God and the creature are two aspects of a single Reality, two sides of a single coin in whose metal being and nonbeing coincide: The Divine Goodness, regarded as above all things, is said not to be, and to be absolutely nothing, but in all things it both is and is said to be, because it is the Essence of the whole universe and its substance and its genus and its species and its quantity and its quality and the bond [copula] between all things and its position and habit and place and time and action and passion and everything whatsoever that can be understood by whatever sort of intellect in every creature and about every creature [681d-682a]. This is how, considered from two different vantage points, God remains “superessential in essences, supersubstantial in substances” even as he, by virtue of being things that differ, and the indissoluble Continuity of the things that are mingled” [67Ib-c]. He says of God that “of all things He is the Beginning and the Middle and the End, and their limit, and their circuit and their going forth and their return" [675d\. 39 Astounding and gorgeous as it is, Eriugena’s conclusion is not altogether without precedent, if we remember that Plotinus’ identification of the One with the Good not only causes the One’s creation of all things, but also renders the One's own existence self-caused. Unlike Plotinus, Eriugena does not stress the One’s aloofness: God creates himself as he creates the world. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all essences, constantly creates himself “within all creation [...], moving Himself through Himself, and moved towards Himself’ [677c-d]. This circulation of divinity shall persist until, at the end of time (breathlessly adds Eriugena), “all things shall be converted into God as the air into light” [683c], We should note that God as the monas creatrix is here behaving in exactly the same way with respect to creation as the one or monas creata behaves with respect to numbers. Different things are God in the same way as, say, 17 or 23 are different versions of the one: they come from the one, they subsist in the one, and, as parts of the infinite progression, to the one they return. In their fully differentiated monadic state, they also consist of ones. The world then is God in the same way as an infinite progression is a unity. Because God is omnia in omnibus, “therefore every visible and invisible creature can be called a theophany” [681a], For everything that is understood and sensed is nothing else but the apparition of what is not apparent [...], the affirmation of the negated, the comprehension of the incomprehensible, the utterance of the unutterable [...], the body of the bodiless, the essence of the superessential [...], the number of the unnumbered [633a-b]. Innumerabilis numerus, number unnumberable: in the finite numbers of creatures we stumble upon the undifferentiated infinity that is God as the Monad.40 Such an infinity is incomprehensible; it “seems to the intellect to be nothing” [Nicholas of Cusa, Docta Ignorantia, 1:17], and this is why by means of its theophanies the superessential Logos affirms only “that it is” but “not what it is” [665d]. Eriugena tries very hard not to slide into pantheism. It might seem that he fails, and yet to think so is to be misled.41 For Eriugena’s notion of identity includes difference; this difference on the epistemological level is perspectival, while on the ontological level it is grounded in God’s superessentiality. The world is made out of God as Nothing; and the world is God as Nothing; but this is a nothing by transcendence. Eriugena quotes pseudo-Dionysius to the effect that “the being of all things is the divinity that is beyond being,” esse enim omnium est super esse diuinitas [644b; Celestial Hierarchy 4.1, 177d], As the transcendent nothing, 40 We may think of the world as a nothing delimited into an infinite multiplicity of somethings. At the same time, the limit that converts nothing to something is itself not limited by anything else and is not therefore a something. That which limits is therefore the same as that which becomes limited. Considered in this way the world becomes a division of nothing by nothing. 41 For instance, in his identification of the Logos with the nature of all things, “which is as yet infinite and common to all and not yet distinguished by any sure form or species” [685b], thereby making the latter, like the former, “consubstantial with the Father” [685d]. 151 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. God, while he is the same as the world and identical with it, is also more than the world and other to it. The “Creative Nature” of God, writes Eriugena, “contains within itself [...] everything which it has created and creates [...], but in such a way that it itself is other, because it is superessential, than what it creates within its e lf [675c; my italics]. The superessential cannot be utterly co-essential with anything —for the exact reason that it is sz<per-essential. Thus Eriugena paradoxically reintroduces a version of Otherness into nothing. When talking about Plato’s Indefinite Dyad earlier in this chapter, we made the side-swipe that it ultimately cannot be immutable and palpable, but Plato must keep this fact concealed if his Dyad remain a principle. Eriugena’s nothing, on the other hand, makes full use of the impalpability of Otherness: the absolute Other can never be any thing, for it is always other than itself. While everything we see, hear, touch, smell, taste is God, he is at the same time slippery - protean - always evading our grasp, always elsewhere, always retreating into himself. Were Eriugena’s vision not so optimistic, I would compare a man looking for God in things to Tantalus in Hades. Yet it is optimistic, for Eriugena’s version of Otherness also accounts for God’s presence in things - and the general co-essentiality of causes and effects symbolized by the copula. If it lies in the nature of the Other to be other, it also lies in its nature to be other than itself, and therefore “not other,” non aliud. Such “absolute” other is paradoxically the same as the same42 —while remaining not the same as the same. This is how God’s superessentiality grounds Eriugena’s use of co-essentiality as difference-in-identity [see Moran 236-7]. This is also how the monad is and is not everything in all numbers. Some finishing comments. We can call any object both “being” and “nonbeing,” “something” and “nothing,” because each object unites both being and nonbeing. But our word “unites” may be misconstrued. Objects are not blended of Being and Nonbeing in the Platonic sense of methexis. There is no Being as such and Nonbeing as such. As transcendent, God is the absolute Nonbeing in that his Nonbeing transcends itself and becomes Being, and he is the absolute Being in that his Being transcends itself and becomes Nonbeing. Since its being is God, the world too is transcendent in just this manner. As principles, Being and Nonbeing are de-ontologized; they are ontologized in that we can call any object “being” and “nonbeing” because “the being of all things” is the “super esse diuinitas” [644b], i.e. Being that is other than itself and therefore is Nonbeing, Nonbeing that is other than itself and therefore is Being. 11 This is Eriugena's parallel to pseudo-Dionysius’ assertion that God is beyond assertion and beyond denial [MT5, 1048b], which, as we have said, absolutizes divine transcendence as transcending itself. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is profoundly incomprehensible. Eriugena’s prepares the way for Nicholas of Cusa’s assertion that no thing is knowable [Docta Ignor. 1.1,4). All knowledge, as Aristotle says, is knowledge of causes; the cause, says Eriugena, is the being of the effect. God is the cause of all and the being of all. But he himself has neither cause nor being. Therefore he is unknowable; and therefore nothing can be known. And yet such essentialist conception of knowledge is clearly incomplete. We might not know what anything is, because we do not know what God is. But we do know that God is, that he exists - because we know that these inexplicable things we call “objects.” whatever they may be, exist [see 665d]. We know it because we see them, hear them, touch them, smell them, taste them. And so they are our theophany [see 681a). Once again: Eriugena’s claim that God’s making the world from nothing means he made it from himself was condemned by the Church. The standard exegesis of ex nihilo became that of Aquinas: ex nihilo refers to privation, both in the sense of absence of material cause, and in the sense of “after non-existence” interpreted without reference to a prior time, since time began at creation [see Wolfson 364-5]. Yet the standard doctrine developed in opposition to Eriugena, whose ideas, as I said, influenced subsequent philosophy. Perhaps there is a greater allegorical truth in William of Malmesbury’s account of the philosopher’s end: his students stabbed him to death with their writing implements, “because,” as they later said, “he forced us to think” [Sheldon-Williams in Eriugena, vol. I, p. 5]. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6.2 GOD AS ONE AND GOD AS ZERO PART II: GOD AS ZERO Omnis theologia circularis et in circulo posita exsistit EOr EOr r,OE)KET bI EOr EOr R O/JHH1 0. Introduction We have seen how the analogy between God and the one conceived in terms of the classical number concept suggests the transcendence whereby God becomes a kind of nothing. We will now examine the next stage, where God is no longer compared to the one but the zero. In a sense the story I am telling is the story of an error, of mistaking one kind of nothing for another. Zero does not signify nothing by transcendence. The sign 0 indicates the absence of numerals 1-9 in a certain place within a number in the same way as the sign 1 indicates the absence of 0 or 2-9, the sign 2 of 0-1 and 3-9, and so forth. The concept of nothing by privation might perhaps be modernized to be the “signified” of zero, nothing by transcendence may not. Since I am telling the story of an error I thought it would be rather apt to follow the analogy between God and the monad, which coins God into nothing, with the analogy between God and the circle, and then proceed to the analogy between God and zero. The assumption here is that the existence of the hollow circle as the image of God facilitates the jump between God as transcendent nothing and God as zero. My analysis of the circle as the image of God argues that the Greek plenary definition of the circle was replaced by the idea of circle as the result of geometrical construction and therefore hollow, an O. I argue that such a circle’s symbolizing God depends on Aristotle’s treatment of circular motion and its applicability to transcendence of the one hand, and to the Trinity on the other. 1Nicolas of Cusa, Docta Ignorantia 1.21; Vvednskii, "sud ushel", in Poety gruppy OBERIU, 147. 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first likening of God to zero dates from c. 1200. We shall see that it depends not only on the arithmetical properties of zero and on the rule of positional notation, but also on the circular form of the sign that signifies nothing. My chapter ends with a brief analysis of two books by Charles de Bovelles, written in the first decade of the sixteenth century. The first book defines nothing through privation. In my section on pseudo-Dionysius I argued that divine transcendence makes referentialism impossible; we see the results of this in Bovelles’ opposition of natural and logico-discursive sequencing. However, the opposition is missing from Bovelles’ second book. As a result, when Bovelles argues that zero should come in front of the number sequence 1-9 because God is ontologically prior to things, he conflates the two kinds of nothing. Had Bovelles not made that error and still likened God to zero, God would become ontologically posterior to things, the product of our discourse about them. 6. Imagery of the Circle 6.1 Archetypal or Historical? In analyzing the thought of Eriugena, we have made extensive use of his paradigm of the monad and numbers. We shall now show that this paradigm originates in the imagery of the circle; we shall also investigate some of the other applications of that figure. This sort of study is not uncontroversial. After all, the problem with metaphors, analogies, and suchlike, is that one never knows when they end, and when the dissimilarity between the signifier and the signified begins. If we compare pages of writing to veined leaves, does that mean that literature too has its autumn? The poet nothing affirmeth and therefore never lieth. The image of the circle which I shall explore in this section is, however, entirely consistent in its applications. This is because its extremely simple form has been used over and over to represent a set of related concepts, ultimately becoming not their passive receptacle, but agent, as it were. The very precision with which it fits so many Neoplatonist conceptions suggests that it helped form them. The metamorphoses of the circle have been admirably explored by George Poulet in a study of the same name. This section, however, is not so much a synopsis of Poulet as an attempt to understand his and other examples in light of specific mathematical, ontological and theological concepts. Unlike Poulet, I am only interested in the circle as standing for God; while I do mention the soul and eternity, I do so only in passing, and confine other and related meanings to the footnotes at best. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 The Circle as Generated by the Transcendent Center In previous sections I have shown how God is “like” the one or the monad. Now I would like to show how God is “like” the circle. “The circle,” says Proclus, “corresponds [...] to the number one.”2 We can thus claim a sort of transitive property for analogies, according to which if God = one, and one = circle, then God = circle. But, as I said, analogies don’t always work that way. So we must proceed in a more pedestrian, roundabout manner. The semantic associations of the circle first derive from its complicated definition in Greek geometry. For Plato the circle is "the thing that has everywhere equal distances between its extremities and its center” [Letter 7, 342b\ same in Parmenides I37e\ also said of sphere in Timaeus 34b]. The definition is employed by Aristotle [see Heath on Euclid /, def. 15]. Euclid too regards the circle as a set of radii: “A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And the point is called the centre of the circle” [Euclid 1, defs. 15-6]. Plotinus makes the circle jump through a large number of interpretive hoops; it is the vehicle for imaging his ideas. A passage in Enneads 6.8.18 analyses the formal relationship of the center point to the circle. For Plotinus, “what the center is like is revealed through” the radii collectively forming the circle; the circle is in a sense the center point enlarged, “spread out.” At the same time, the center remains what it is, generating the radii and the circle “without [the center itself] having been spread out.” Plotinus calls the center point the “father of the circle,” its “archetype,” whereas the circle is its “image” [6.8.18.8-34]. In this section of the Enneads, the center point represents the One and the circle Being considered as Nous, which contains all Ideas as radii [like Eriugena’s Logos, see Enneads S.9.8.4].3 The dual nature of the relationship between the : For Proclus the one controls the middle of the procession of numbers from the unity to the decad because the numbers of this middle, i.e. Five and six, are circular, i.e. end in themselves when squared [comment to Euclid /.defs. 15-6; 147, 150-1 ]. 3 The Ideas are like radii in that they are indivisibly united in the One, and an infinite many of discrete entities in the Nous. The infinite number of Ideas is what makes the Nous a unity like the One, but a unity- in-multiplicity. The fact that an infinite many of discrete points paradoxically forms the continuum of the circumference, in an age without infinitesimal calculus and haunted by the ghost of Zeno of Elea, can only be explained by unity-bestowing participation of the Nous in the One - which is what the diagram represents. 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. center point and the circle indicates the One’s paradoxical combination of aloofness and immanence. I should like to stress that at no time does the alternative model - that of numbers - recede too far into the back of Plotinus’ mind, for the coming-to-be of the world is thought of as the emergence of the many from the One. This is no mere metaphor: the world as many appears by “the power of number” [6.6.9.26] and Plotinus takes the next and ontological step of declaring everything to be number. So the Nous “is number,” and “Being [...] unified number, and the beings number unfolded,” and so on [6.6.9.26-36; soul as number 6.6.16.44-5]. Pseudo-Dionysius too brings together the constitutive elements of the circle and of numbers to explain the relationship of the “absolutely transcendent Goodness” to beings: Every number preexists uniquely in the monad and the monad holds every number in itself singularly. Every number is united in the monad; it is differentiated and pluralized only insofar as it goes forth from this one. All the radii of a circle are brought together in the unity of the center which contains all the straight lines brought together within itself. These are linked one to another because of this single point of origin and they are completely unified at this center. As they move a little away from it they are differentiated a little, and as they fall farther they are farther differentiated [Divine Names 5.6, 820d-821a\. In this version, the monad is like the center point, whereas numbers are compared to radii. Any particular radius represents the descent of any particular number from the monad through various ontological levels. The further any number moves from the monad, the more divisible and complex it becomes; nonetheless, it remains the same number quantitatively (i.e. lies on the same radius). Thus the radii symbolize individual vicissitudes of participation. Finally, there are as many radii as there are numbers - i.e. an infinity - yet as the radii “preexist” in the center point, so do the numbers in the one. This is a pretty tight fit. We’ve presented it as static as Plato’s definition. It is obviously dynamic as well: the numbers “go forth” from the monad, the radii “move away” from the center. The model therefore is also that of procession or emanation. It shows how Being (circle) and beings (radii) both participate in and emerge from the “transcendent Goodness.” 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although Eriugena does not deal with the circle model in great detail, he too declares that “the universe is created from the One just as all numbers burst forth from the monad and all radii from the centre.”4 That Plotinus, pseudo-Dionysius and Eriugena conceive of the transcendent One as the center point is largely due to the standard analogy between the numerical one and the point as mathematical entities [see my chapters 2 and 4], As the origin of numbers, the one is all numbers in epitome: there is nothing in numbers that is not in the one. As the origin of radii, the center is the circle in epitome: there is nothing in the circle that is not in the center point. Despite being origins, the one is no number and the point no magnitude. In the same way, the transcendent One is the origin of beings but itself not being. The fact that the point lacks magnitude but is the beginning thereof places it on the physical threshold between non-sensual and sensible entities. Here, the point may be thought of as the origin of a circle in a different manner. Proclus differentiates between the Ideal, the geometrical and the sensible circles by locating our perception of the first in the mental faculty of understanding (proper to the nous), the second in the imagination, and the third in sensible things. Since “extended things exist without extension in the realm of immaterial causes, and divided things without division, and magnitudes without magnitude,” The circle in the understanding is one and simple and unextended, and magnitude itself without magnitude there, and figure without shape; for such objects in the understanding are ideas devoid of matter. But the circle in the imagination is divisible, formed, extended - not one only, but one and many, and not a form only, but a form in instances - whereas the circle in sensible things is inferior in precision, infected with straightness, and falls short of the purity of immaterial circles [54], Since the Ideal Circle has no magnitude, it appears on magnitudes as a point. This extends to all non-sensuals, God as well as Ideas. As a result, once we make the very 1 “Ab uno enim universitas creata est sicut a monade omnes numeri ec a centro omnes lineae erumpunt ” [637a]. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. non-Greek move of identifying real space with geometrical extension,5 we can “point” our finger at any “point” in space and exclaim, “God and the Ideas are here in their entirety!” This thought could become fully developed only during Scientific Revolution.6 The Neoplatonists, however, do prepare the way for it by employing the image of the circle to demonstrate the most basic and general facet of participation, applicable to the participation both in the One and in Ideas. For pseudo-Dionysius, “the entire wholeness is participated in by each of those who participate in it; none participates only in part. It is rather like the case of the circle. The center point of the circle is shared by the surrounding radii” [Divine Names 2.5, 644a]. With this model, pseudo-Dionysius as it were rescues young Socrates, stymied by Parmenides’ skeptical questioning of how a one can be present in the many while remaining indivisible [Parmenides I3Ia-c]. What pseudo-Dionysius does not do is base participation on the fact that all points of the circle are as much “point” as the center point, and that they themselves can become center points of equivalent circles. A similar omission may be seen in Plotinus, who does not use the image of the circle to argue that each Idea is, like the Leibnizian monad, a Nous (or even the One) in its entirety while also remaining only a part of the whole [5.9.8.3-8]. The closest Plotinus does get to the concept of a uniform and polyvalent space is by plotting the circular motions of souls around their particular centers upon the circumference of such centers whose center point is the One [6.9.8.1-22], He also toys 5One may interpret this move as identifying the material cause of the sensibles not with prime but with intelligible matter. For intelligible matter, see Proclus 53, derived from Metaphysics 1036a9-J2; see Shichalin’s introduction to the Russian edition of Proclus, 16-31. 6 The homogenization of real space under the aegis of geometry received some aid from the widespread thesis, attributed to Hermes Trismegistus but actually dating from the twelfth century, that Deus est sphaera cujus centrum ubique, circumferentia nusquam. In the seventh chapter of his Liber de Nichilo, Charles de Bovelles uses it to describe the omnipresence of God. For Bovelles, God is present in any point in space not only as the center of an infinite sphere, but also as the intersection of the axes of longitude, latitude and depth, which appear to be identified with the Trinity. In this way God is “the true and consummate plenitude” of the real space we imagine to be infinite and empty [92], The phrase that God is a sphere whose center is everywhere and circumference nowhere was extremely popular in the early modem period [see, e.g. Browne, 71]. Pascal applies it to nature [Pensees, Brunschvicg 72-199, 65], which is curiously reminiscent of Eriugena’s definition of nature as the omnipresent Logos [685b-d], For other instances of the thesis see Poulet, for whom it represents primarily the decentering of the world in the Renaissance. For space, not an Aristotelian category, as a continuum in Patrizi, see Grant, 206. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the image of concentric circles representing the essential unity of the levels of reality expanding out of the One [e.g. 4.3.17]? 6.3 Circle as Circumference: Construction The reader will surely have noticed that the Greek definition of the circle differs from ours. For the Greeks, the circle is a plenary figure, the set of all points enclosed by the circumference. For us, the circle is the set of all points equidistant from the center, i.e. just the circumference. In later medieval and Renaissance imagery of the circle we witness the radii growing less relevant, and the circle itself becoming defined by the practice of construction - as either center point and circumference, or merely the circumference (qua path of “th’other foot” that ‘‘must [...] obliquely runne”).8 Envisioned in such a way, the circle derives its semantic associations not from Euclidean geometry, but from how Plato [Timaeus 34b-37c] and especially Aristotle characterize circular motion. The new version of the circle diagram emerges from attempts to graph the linear time of Christianity onto the diagram of the circle. Plotinus and pseudo-Dionysius represent emanation by dilation of the center point. The circumference here appears all at once. They lay no stress on the fact that when the circle stands for the emanation of numbers from the monad, numbers that unfold along the circumference form a sequence - even if, in the center point, they are present as one and without sequencing. Things 7 Proclus' argument concerning the Ideal Circle as lacking in magnitude implicitly identifies the point with the Idea or essence of a circle. This Neoplatonist identification does not initially seem to be prefigured in Aristotle. On the contrary, Aristotle claims that “the essence of a circle and a circle are the same, and so are the essence of a soul and a soul.” Yet Aristotle is quite explicit in that he is not here talking about geometrical or sensible circles, both being composites of form and matter, but of only of the circle as form and object of definition [Metaphys. I036al-I3]. In other words, the circle itself contains no accidents but is an indivisible whole. But that which is indivisible can be neither a magnitude nor a plurality[1020al 1-2]: the simple circle is therefore like the point or the one. In other words, even for Aristotle, what the circle is, is again the point. (Aristotle's De Anima 409a6 defines the point as a unit which has position. But the point existing in the same way as the Ideal Circle has no position.) We might add that the Aristotelian analogy of the circle to the soul on the grounds that both are the same as their own essence, may with equal success extended be to God. 8 Donne, “A Valediction forbidding mourning,” 1. 34. For Donne’s compasses, see primarily Freccero, but also Cunnar, Lederer... For Donne’s use of the Greek model, see ‘T o the Countesse of Bedford” (“Honour is so sublime perfection...), 11. 46-8: “In those poor types of God (round circles) so Religious tipes, the peeceless centers flow. And are in all the lines which always goe.” 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stand otherwise in mediaeval uses of the diagram to represent the relationship of time to eternity. The origin of this conception lies in Plato’s Timaeus, which calls time “a moving image of eternity [...], moving according to number while eternity itself rests in unity” [Timaeus 37d\. Let us remember Eriugena’s opposition of eternity as the eternal present to time as duration or succession of presents. We have said that Eriugena imagines time as a line; it would have been more accurate to say that the line is a curve, the circumference of a circle whose center is eternity: Time (writes Eriugena) unfolds from eternity as circle from the center, and the eternity is in time as the center is in the circle [...]. For eternity is nothing other than the contraction of all times beheld by him, who sees all things as one present.9 The diagram draws attention to the paradoxical status of both eternity and the now, neither of which has any duration and is therefore best represented by that no-magnitude, the point. Moreover, it displays eternity’s being “the contraction of all times” by representing the equivalence between all temporal nows taken together and the center point as the epitome of the circle. In exactly the same way, the infinite progression of all numbers, as we remember, is equivalent to the monad. Eriugena is not alone in this conception [see e.g. Aquinas in Poulet 28].10 ‘ “Nam, ut a centro circulus sic ab aevo deducitur tempus; et idem est in tempore aevum quod est in circulo centrum... Nihil est enim aliud aevum quam contractio totius temporispraesentialiter habita in conspectu omnia videntis. ” In Poulet 45, n. 2. 10The fact that God, as Eriugena says, sees all times as one present, i.e. as a point, permits inversions of the paradigm where eternity becomes the circumference. The most famous of these is the Renaissance ouroboros. a serpent devouring its own tail and thereby forming a circle [see e.g. Vaenius. emblem 1], While the origin of the emblem is thought to be the Hieroglyphica o f HorapoIIo, a pseudo-Egyptian text discovered in 1419 [items 1.1-2], its interpretation was read into rather than in Horapollo's extremely muddled comments, superimposing Horapollo’s serpent upon the following ideas. Time is the unfolded version of eternity; when time “perfits its circle", i.e. when the circumference closes, everything that was prefigured in eternity becomes figured in time (comp, with Eriugena’s statement that whatever infinite progression has in actu, the monad has in potentia.). Thus eternity is the same thing as time taken in its totality, i.e. the circle as well as the point (comp, with Proclus’ argument that what the circle really is, is a point). This inversion of the paradigm is additionally justified by the fact that duration is ultimately finite, and everything finite is but a point from the do Ice naufragar, as Leopardi says, of the infinito [see Bonaventure in Poulet 27]. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Unlike Plotinus, whose employment of the circle emphasizes the timeless character of emanation, Christianity stresses the diachronic character of God’s making of things. Its prime concern is not ontology, but salvation, which occurs in and depends on history. As far as the beginning of history is concerned, it needs to downplay emanationism with its aspersions concerning evil and freedom of the will, and to portray God rather anthropomorphically as Creator, as Artifex mundi. So it hands him a pair of compasses. Thus he sometimes appears in medieval Bible illumination [Funkenstein, frontispiece; also see Klibansky, Panowsky, Saxl, pi. 105-6, 108]. The justification for these images is Wisdom of Solomon 11:20 (“thou hast ordered all things in measure and number and weight”), as well as Plutarch’s claim that Plato’s God is a geometer [Table- Talk 8.2]. We encounter this as late as Milton, whose Christ took the golden compasses, prepared In God’s eternal store, to circumscribe This universe and all created things [Paradise Lost, 7:5-7], The image of God with compasses can be applied not only to the Creation, but also to the course of human history as a whole. Here history is regarded as Creation by Divine Providence. I must admit, however, that the only example I’ve come across dates from the early modem period. Donne writes: The Body of Man was the first point that the foot of Gods Compasse was upon: First, he created the body of Adam: and then he carries his Compasse round and shuts up where he began, he ends with Body o f man againe in the glorification thereof in the Resurrection” [Sermons v. 7 p. 97]. We have demonstrated that the circle of divine praxis is conceived in terms of movement along the circumference. When the movement ends, that is to say when the curve closes, we get a circle that is not, like the circle of Euclid, a plenum filled by radii, but one that is hollow inside, an O. Now we must show that such a hollow circle can and does signify God himself. The locus classicus for circular motion and its application to God lies in Aristotle’s Physics and De Caelo [264b-265a and269a-b respectively]. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 The Circular Motion of Divinity For Aristotle, all motions consist of the circular, the rectilinear, and combinations of the tw o .11 Circular motion is single and continuous: for in it “that which is in motion from A will in virtue of the same direction of energy be simultaneously in motion to A” [Physics 264b 10-1]. This fact also makes it “eternal,” “regular,” and changeless. Rectilinear motion, on the other hand, is compound, divisible and so forth. The chief reason for the latter’s inadequacies lies in Aristotle’s denial of infinitely extended space, and his consequent claim that all lines are finite. Therefore “rectilinear motion on a finite straight iine is if it turns back a composite motion, in fact two motions, while if it does not turn back it is incomplete and perishable.” When rectilinear motion turns back, it again stops “and with the occurrence of rest the motion has perished” [Physics 265a20- <5]. Moreover, In rectilinear motion we have a definite starting-point, finishing-point, and middle-point, which all have their place in it in such a way that there is a point from which that which is in motion can be said to start and a point at which it can be said to finish its course [...]. On the other hand in circular motion there are no such definite points: for why should any one point on the line be a limit rather than any other? [Physics 265a28-33]. For all of the above reasons, circular motion is judged to be prior to the rectilinear. Aristotle argues that circular and rectilinear motions are “natural” only to those entities that share their respective characteristics: circular to those that are unique, simple, untouched by “alteration” and “increase,” rectilinear to those that are many, complex, and perishable. Motion gets more circular as we move up - so the eternal and pure heavenly bodies move in circles, but, as Love said to Dante, tu autem non sic [see Freccero 335-7, 42]. Even the motion of the supralunary world is, strictly speaking, not absolutely circular - the planets, Ptolemy was to show, have their epicycles, the stars have their precession. The sole entity so perfect that his motion is absolutely circular is Aristotle’s prime mover, whom Christian readers identified with God, and whom Aristotle locates upon the circumference of the heavens, pushing them in his spin. As one might expect, the subject of absolutely circular motion is unique, simple, eternal, changeless, and so forth. He is 11 Circular motion in Plato’s Timaeus is classified as “motion of the same” [36c, as opposed to the rectilinear “motion of the other”] and belongs to the stars inasmuch as they are perfect in the sense of changeless. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. also indivisible and without magnitude [Physics 267b18-26], a point, as it were, moving along a circumference. If any motion is proper to the Christian God, it is of course the circular.12 Yet God circles in a much more sublime manner than Aristotle’s prime mover. Charles de Bovelles in his Liber de Nichilo (1509) writes of God’s orbicularem motum and gyrum prior to the Creation [58]. Does this gyrating have a spatial aspect? No, for then space would be coetemal with God; and anyway, space is still thought of as a plenum, non­ existent when things have not yet been created to fill it. Rather, Bovelles imagines a “divine circulation of divinity” emanating from itself, moving within itself, and returning to itself [58] - what we saw in the movement of numbers from, through, and back to the monad, but as yet occurring entirely within God, and implicating him alone. This “divine circulation” is extremely important. It is how the One may also be conceived as the Trinity. “We use the names Trinity and Unity,” writes pseudo- Dionysius, “for that which is in fact beyond every name, calling it the transcendent being above every being” [Divine Names 13.2, 981a]. The Three Persons of the Trinity are consubstantial, with the Father generating the Son and both generating the Holy Spirit (in Orthodoxy, the latter task is performed by the Father only). According to the Councils of Nicea and Constantinople, the generations are eternal, i.e. atemporal. In De Trinitate, Augustine compares the generation of the Son to an act of thought on the part of the Father, and the generation of the Holy Ghost to an act of mutual love between Father and Son. Yet such acts in the case of the Trinity do not occur between the self and the other, but within the single self [see Oxford Diet. O f Christian Church, “Trinity”]. The Trinity, to alter a line of Donne’s, is a trialogue of one. Pseudo-Dionysius’ discussion of it slips into the Neoplatonic vocabulary of the tripartite process of origin, procession and return [Divine Names 4.2, 641a, see n. 114 in edition cited]. This vocabulary describes not only the relationship of the monad to numbers, or of the One to things, but also of the One to itself.13 The most succinct statement of the Trinity as circular motion on the part of the one is the first thesis of the Liber XXIVphilosophorum, a twelfth-century text ascribed to Hermes Trismegistus [see Poulet 25-6]. The thesis proclaims, “God is a monad begetting a monad and reflecting its ardor back onto itself’ (Deus est monas monadem gignens et in 12 “God himself is so much a Circle, as being every where without any comer,” writes Donne, and “he is no where any part of a straight line,” Essays 38-9. 13 In Plotinus, its generation of the Nous, and the generation of the Soul by the Nous [see 5.1.8; comp, to circle image in 6.8.18.8-34, described above, where the unmentioned Soul would be the radius from the circumference to the center]. Thus Plotinus may be said to prefigure the Orthodox take on the Trinity. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. se reflectens suum ardorem). Poulet reads it in the traditional Augustinian manner: "Le Pere se reflete dans le Fils, le Fils se reflete dans le Pere, et cette reciprocity d'amour n 'est rien d ’autre que la troisieme personne de la Trinite, le Saint-Esprif' [26].14 Thus in Bovelles the “divinus [...] orb is, divina circulatio deitatis, in sese conversio" intends the fact that in the beginning “God, being trine, was fecund and numerous all by himself, and existed in a relation of commonality, knowledge and reciprocity with him self’ [Liber de Nichilo, 56].15 God’s self-engendering, with its paradoxical combination of circular movement and lack of space and time16 to move in, proceeds along and forms not a physical but “an intelligible circumference, without beginning and without end” [Bonaventure in Poulet 27; said of eternity]. Two aspects here demand further attention: a) the fact that God at once moves and does not move; b) the formula concerning the beginning and the end. Both again come from the same Aristotelian passages on types of motion. As it applies to geometrical shapes, Aristotle understands motion as a body’s displacement from a separate beginning point, through separate middle points, and to the separate end point. We have already quoted Physics 265a28-29, which defines rectilinear motion in just this way. As “motion from [one] place to another” [Physics 264b20], rectilinear motion is the paragon of motion as displacement. This is not the case with circular motion. An entity engaged in such motion “is also in a sense at rest, for it continues to occupy the same place.” This occurs because in the circle the centre is alike starting-point, middle-point, and finishing-point of the space traversed; consequently since this point is not a point on the circular line, there is no point at which that which is in process of locomotion can be in a state of rest as 14The first thesis of the Liber XXIVphilosophorum receives a curious treatment from Nicholas of Cusa. Nicholas explains the Trinity by indicating the one's dynamic and tripartite “eternal generation” of itself from itself. For Nicholas, “generation is the repetition of oneness.” However, oneness presented once is for him always already repetition, the begetting of equality from oneness, whereas the one-time nature of the "repetition" begets union from both oneness and equality. Thus the one for Nicholas is also trine: oneness, equality, union or “this," “it," “the same." Nicholas argues that oneness, equality, union are eternal, since they are prior to their opposites of otherness, inequality, separation; they are also one, since, if there were many eternals, oneness would precede eternity. Oneness for Nicholas stands for the Father, Equality for the Son, and Union for the Holy Ghost [Dl 1:7-9, 18-26]. 15 “Erat tamen in seipso fecundus et numerus quippe trinus, sibi quoque ipsi communis sibimet cognitus ac pervius. ” 16 For Aristotle, time is the “number [i.e. enumeration] of motion with respect of ‘before’ and ‘after’," Physics 2I9b2. 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. having traversed its course, because in its locomotion it is proceeding always about a central point and not to an extreme point: therefore it remains still, and the whole is in a sense always at rest as well as continuously in motion [Physics 265b I -8). In this passage, “place” is conceived as the place of the path around the center point. If the center point remains stable, so does the path. Since circular motion is exempt from the mutually exclusive categories of motion and rest, it best reproduces the “non-motion” of divinity, for the transcendent God, as pseudo-Dionysius writes, is neither moving nor at rest [MTS. 1048a; same of the One in Enneads 6.9.3.40-45 and Parmenides 139b]. While Aristotle’s idea of motion as displacement requires the beginning, middle, and end points to be distinct, continuous motion along the circumference exhibits no such partition. Upon the circumference any one point as much as any other is alike starting-point, middle-point, and finishing-point, so that we can say of certain things both that they always are and that they never are at a starting point and at a finishing point [Physics 265a33-bl]. So both the center and any point on the circumference can in different ways be considered as “alike starting-point, middle-point, and finishing-point” of circular motion. It therefore can be said to have neither beginning nor end, or to have its end in its beginning. Its nature is simple and eternal. The Aristotelian idea of the circumference as the figure having neither end nor beginning developed into one of the most common ways of describing God. Bovelles explains his “sine principio” and “sine fine deus" [38] as follows: God is the maker of all things. He himself is the beginning and the end of all things. He was before all things and after them he shall remain. The beginning and the end of all things lacks beginning and end. For of the beginning itself there is no beginning and of the end itself there is no end [Liber de Nichilo 56].17 The frequency of the formula is due no doubt to its matching Revelations 22:13, where Christ calls himself simultaneously “Alpha and Omega, the beginning and the end, the 17 "Deus auctor est omnium. Idem ipse est omnium initium et finis. Ante omnia erat et post omnia futurus est. Omnium autem initium et finis initio et fine caret. Nam principii nullum est principium. Ipsius item finis nullus finis. ” 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. first and the last.” This confluence of the Bible, Aristotle, and geometrical construction permits us to envision God as a hollow circle, a circumference which, for its “indefiniteness having no special place of beginning nor end, beareth a similitude with God and etemitie” [Puttenham 8 1].18 As such hollow circles, God and eternity are opposed to created things, for “all that is finite and bounded [i.e. creatura] has a beginning point and an end point” [Nicholas of Cusa, Docta Ignor., 1:6]. The hollow circle symbolizes God in a very overdetermined manner. I hope the reader will forgive me if I add one more vicissitude —that of divine perfection. We know that God is perfect from the New Testament: in Matthew 5:48, this is how the Son speaks of the Father [see also Heb. 2:10, 5:9, 7:28]. For pseudo-Dionysius, “Perfect” along with “the One” is “the most enduring” of divina nomina by virtue of its implying divine transcendence [Divine Names 13:1, 977b]. When we symbolize God by the circle, we indicate his perfection first and foremost as it concerns his indivisibility and his identity with his own essence. Aristotle calls the circle “a perfect thing” [De Caelo 269a20].19The circle’s perfection arises from its being entirely and indivisibly whole: take away a part -- but where do you see parts? -- and it is no longer a circle. So, in a beautiful and historically fruitful analogy, Aristotle says: “the essence of a circle and a circle are the same, and so are the essence of a soul and a soul” [Metaphys. 1036a2-3].:o Moreover, Aristotle also calls circular motion “the only perfect motion” since in it the beginning and end coincide [Physics 264b28]. Therefore, given the practice of construction, God as the circle becomes perfect in another, dynamic sense. The first meaning of the verb “to perfect” in the OED is the Latin “to complete, to finish.” So Donne speaks of the sun’s inability to “perfit a circle” for it rises in a different spot each day. In this it is like the stars since, due to precession, “none” of them “ends where he 18 For eternity as circle without beginning or end, see Bonaventure in Poulet 27. The Puttenham passage at length is: “The most excellent of all the figures Geometrical is the round for his many perfections [...]. He contayneth in him the commodious description of every other figure, & for his ample capacitie doth resemble the world or vniuers. & for his indefiniteness having no special place of beginning nor end. beareth a similitude with God and etemitie." “Contayneth in him the commodious description," etc., refers to inscribing regular polygons in the circle. See Michael Meier, Malania Fugiens, emblem 21, in Cunnar, pi. 6. 19 Cesare Ripa, in his elegant and influential Iconologia, calls the circle “inditio di perfectione, essendo quella da ogni parte la piu perfetta figura di tutte Valtre [Ripa, “Perfectione,” 348, source of figure of Perfection in Jonson’s Masque o f Beauty], :o Compare with the visual pun in Donne’s “O Soule, O circle” [“Obsequies to the Lord Harrington,” II. 105-6], Similar pun may be found in the last line of Crashaw’s “Bulla": “O sum scilicet O nihir. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. begunne" [“First Anniversary,” U. 268-75]. Another of Aristotle’s definitions of motion calls it “the fulfillment of what exists potentially” [Physics 20lal0], and thus “to perfect” also acquires the sense of bringing potentials into actuality. For Plotinus what the center holds in potentia is expressed on the circumference in actu. Later thought capitalizes on the curve’s return to itself, on the Aristotelian definition of circular motion as “motion of a thing from its place [back] to its place” [Physics 264b20]. Thus Eriugena’s numbers return to the monad, where they pre-exist in potentia, when they have reached the actuality of infinite progression, i.e. when the last number touches the first. God “perfits” a circle both in relation to himself —in the Father’s generation of the Son and the Holy Ghost - and also in relation to the world, since “of all things He is the Beginning and the Middle and the End, and their limit, and their circuit and their going forth and their return” [Eriugena 675a hinting at the “spiritus” of Eccl. 1:6]. While for Eriugena this thought means that the world is God, it also bears the more conventional exegesis. All things come from God, he holds all of them in being, and to him they return. The reason God can bring everything to perfection, to completion, to return to himself - both as far as the development of individual things is concerned, and the course of time as a whole - is because he is perfection, as the Good. The origin of this argument is Plotinian: The One is perfect because, as the Good, it desires itself; and “all things when they come to perfection produce” [Enneads 5.1.6.37; also 5.4.1.27-36; see Gaddi and Bussanich in Gerson 28-29; 44-45, 47]. While in Christianity creation is an act not of necessity, but of divine Will - otherwise it would be an impediment upon God - already in Plotinus necessity depends on the Will and not the other way around. The paradox is as follows: being perfect, the Perfect is absolutely free, but, since it is perfect, its Will becomes necessity.21 Thus, since “of al figures the circle is of most absolute perfection” [Billingsley’s Euclid in OED “perfection”], when standing for God it indicates (but does not resolve) mysteries as profound and unknowable as why God is as he is, and why he generates the world. Finally, God’s circling exhibits a formal uniqueness. A hollow circle is constructed by motion along the circumference; when the circumference closes, the circle becomes one, simultaneous and simple.22 Yet God’s own motion, being atemporal and :1 Delving into such things, however, shows one to be “a man of an incontinent wit, and subject to the concupiscence of inaccessible knowledges and transcendencies” [Donne, Essays 13; said of Pico's interest in numbers and the Kabbalah], “ John Donne: “This life is a Circle, made with a Compasse, that passes from point to point: That life is a Circle stamped with a print, endlesse, and perfect Circle, a soone as it begins.” Sermons, vol. 2, p. 200; in Freccero 339. 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. perpetual, combines the process and the result. He is both the movement along the circumference and its simultaneous and indivisible totality. He is not only the infinity of numbers that the center point contains in potentia, but also their infinite progression along the circumference. This infinite progression runs not on a line, but on a closed curve. Therefore it has neither beginning nor end. Therefore it is an infinity of infinite progressions superimposed upon one another in such a way that each of its points is all numbers at once, from the one to the infinitely great. This is what is meant by divine perfection - that everything which is potential in it, exists in actu. 6.5 Geometry of the Infinite in Nicholas of Cusa All I needed to demonstrate for the purposes of this chapter is that God “is like” a hollow circle, and I’ve done that. I do however desire to make several parenthetical remarks about the way God is symbolized in the De Docta Ignorantia of Nicholas of Cusa. Nicholas transofrmation of the tradition is surely worthy of notice. Nicholas’ thought derives primarily from the second thesis of the Liber XXIV philo sop ho rum. The thesis proclaims that “God is a sphere whose center is everywhere and circumference nowhere” (Deus est sphaera cujus centrum ubique, circumferentia nusquam). This claim propels Nicholas to envision God in terms of infinity. The result is stupendous to say the least: Nicholas’ imagination, as it were, glides over the realms of calculus of infinitesimals and Riemannian geometry, perhaps even of transfinite numbers, without being able to formalize them mathematically. “The great Dionysius,” writes Nicholas, “says that our understanding of God draws nearer to nothing rather than to something. But sacred ignorance teaches me that that which seems to the intellect to be nothing is the incomprehensible Maximum” [1:17]. Defining nothing by transcendence as coincidentia oppositorum, Nicholas identifies the Absolute Maximum with the Absolute Minimum, neither of which is number conceived as plurality. Rather, they are actual infinity23 and the one (i.e. potential infinity), the end and beginning of number, respectively [1:4-5]. Since all knowledge is knowledge by comparative relation, i.e. by number [/. /], and since the one and infinity are outside number [/;5], they cannot be known. The only solution is to understand God 23 Actual infinity for Nicholas is not infinite progression, “since number is finite,” and we cannot “come to a maximum number than which there can be no greater number, for such a number would be infinite. Therefore it is evident that the ascending number-scale is actually finite.” which does not mean that it ends, but only that it never includes what Nicholas regards as the infinite [DI 1:5, 13]. 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “incomprehensibly” by manipulating mathematical entities, the most accurate of all possible images of divinity [1:11]. Nicholas’ docta ignorantia is a method for understanding God via geometry of infinites. His greatest achievement arises from the thought that, since God is not a finite entity, not a being, the mathematical entities representing him must be infinite in extension. Thus, the circle representing God must have an infinite circumference, and therefore the Maximal Circle is an infinite straight line. Since this line is infinite, it is, for Nicholas, indivisible [1:17]?* Not only is the infinite circle the same as the infinite line, but the infinite line can generate the infinite circle by rotation. As the beginning and the end of the infinite circle, the infinite line contains the center point, the diameter and the circumference at once in potentia and in actu. This is how it exhibits both the potential infinity of Absolute Minimum and the actual infinity of Absolute Maximum. For Nicholas, who regards the circle as the natural symbol of oneness, the sphere is the actualization and perfection of the circle [1:13; 1:19; 1:23];25Therefore the infinite sphere is essentially the same as the infinite circle and, by extension, also as the infinite line. The infinite line is the same as the infinite triangle [1:14], i.e. a triangle of three right angles [1:12, 33]?6 Thus, in their infinite aspect, the most perfect corporeal, surface, rectilinear and straight figures (sphere, circle, triangle and line respectively) [1:10, 27] are the same. While the Maximum itself transcends them, they do indicate how it operates. For instance, they are the essence (qua both being and form) of figures of finitude [1:17, 48]. In the exact same way, God is the forma formarum and the being of each thing and of the universe as a whole. The essential identity of the four figures demonstrates the synonymity of actuality (sphere), oneness (circle), trinity (triangle) and essence (line) in God [1:19]. Although the simplest of all figures for Nicholas is the line, his philosophy derives it from the circumference of a circle stretched to infinity. This is why his description of the transcendent nothing, or the Absolute Maximum, is but the “infinitized” version of the circle as standing for God: 24 Nicholas thinks of divisibility as divisibility into a finite number of equal and finite parts. This is why irrationals for Nicholas are more pure numbers than rationals, Idiota de Mente, 6. 25 Nicholas represents actualization mathematically as revolution around own axis; thus a circle for him is the actualization of a line, see Docta Ignorantia 1:13 26 Ergo God is one and trine. A triangle of three right angles may also be plotted on the surface of a sphere, with the apex at the pole and the base on any parallel. 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the Maximum the center is the circumference. You see that because the center is infinite, the whole of the Maximum is present most perfectly within everything as the Simple and the Indivisible; moreover, it is outside of every being - surrounding all things, because the circumference is infinite, and penetrating all things, because the diameter is infinite. It is the Beginning of all things, because it is the center; it is the End of all things, because it is the circumference; it is the Middle of all things because it is the diameter. It is the efficient Cause, since it is the center; it is the formal Cause, since it is the diameter; it is the final Cause, since it is the circumference. It bestows being, for it is the center, it regulates being, for it is the diameter; it conserves being, for it is the circumference. And many similar such things [1:21]. To summarize. Nicholas attempts to rethink the transcendental relation of God to things “mathematically,” as the relation of infinity to fmitude, infinity meaning the “maximally large” or “all that which can be” (actual infinity) and the “maximally small” or “that than which there can no lesser” (one, not zero) [1:4-5]. We cannot comprehend either infinity or the fact that the maximally large is the maximally small -- for comprehension proceeds by finitudes, i.e. numbers [/:/, /. J,].27 The Maximum, on the other hand, is no number and therefore “seems to the intellect to be nothing” [1:17]. This nothing is in fact infinite oneness [/:5] whose symbol is the infinite circle, indivisible and lacking beginning and end [1:21]. 7. God is Zero 7.1 Theology Before Arithmetic So far in this chapter I have shown how God is like the monad or one, how being like one makes him into a kind of nothing, and how both he and the one are like the circle. The next and final step is to show that God is like the zero. This is the task of the present section. Here we shall look at two texts - one dating from c. 1200 and another from 1510 - that rather adamantly expound the simile. The simile seems to suggest itself. You got your transcendent God who is nothing and is like the circle, and here you got a circle that stands for nothing, so presto! God is like zero. But as soon as we think about it, problems arise. For starters, God is nothing by ” The closest we can get is by mentally subtracting “large” and “small” from “maximally large” and “maximally small” - for being greater and lesser are the traits of finitude, and this why numbers are finite. Then we will clearly see that the two maximums coincide [ 1:4, II; 1:3, 9], 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transcendence, i.e. by superabundance. His is an excess and not a defect of being. It stands otherwise with zero. Whatever nothing be signified by zero, it is definitely not nothing by transcendence. I don’t say “I own 0 cows” if the number of cows I own is too large to count. While every thing in existence points to God by virtue of its having being or of its being-one, the zero cannot be signified in such a way, whereas the one can. Nor does the zero itself signify in the way the word is used by early arithmetics. We cannot say that it signifies nothing by privation, since everything that exists is a something, and that which is not cannot be signified. Hence early arithmetics regard zero only as a writing aid. In lists of Hindu-Arabic numerals, it comes after nine like a fat and asthmatic dwarf chasing a stately cavalcade. In number mysticism, sequencing depends on the symbolic properties on numbers. Numbers strut their stuff in the fashion of Benozzo Gozzoli. One, the unity, is the symbol for God; it comes first. Two is division, plurality, emergence of number; it comes second. Three is the return, completeness: the Trinity, three faculties of cognition, three types of intelligent beings, etc. And so forth. So the zero - this dwarf, this minimus, this nihil, this insignificant scrawl - is to signify God? The last is to become the first?28 For it does become the first, thereby usurping the function of the one. The zero’s new role rests on the one’s previously being the symbol for God. The zero inserts itself into the place of the one in the structure of the symbol created. One fine morning it wakes up in the one’s bed, puts on its robes, makes mouths at its round mirror. Mirror, mirror on the wall: Am I none or am I all? Am 11 or am I role? Am I whole or am I hole?29 Probably the most uncanny event in my dissertation is that the zero usurps the place of the one both in regard to symbolizing God and - as we saw in Stevin - in being the origin of number. We can imagine God being declared similar to zero because the concept of number has been rethought, and the zero has been shown to be the origin of :i! See next chapter, section on Christian humility and Paul. 29 A Renaissance poet would rather liken it to a cuckoo - for cuckoos lay eggs in other birds’ nests; once hatched, the fledglings kill their “siblings,” eat their food. 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number. But the theological here precedes the arithmetical! The texts in this section are considerably earlier than the Arithmetique. Chances are Stevin had not read them, and even if he had come across that of Bovelles, he surely would have regarded it as pure balderdash: number symbolism was decidedly not his cup of tea. So the theological and the arithmetical substitutions appear to have occurred independently. But then why are they so much alike? For it is not just the case of the zero becoming the symbol of God and the origin of number, it is the case of the zero waking up in the one’s bed. And in both cases, many of the clothes zero puts on that morning hang like giant’s robes. The incongruities are especially striking with Stevin, for he, unlike Bovelles and the anonymous author of the Salem Codex, flaunts his non­ traditionalism, and explicitly aims to rethink everything ab ovo, from principles present only in numeration and calculation. Whether this parallel development is a coincidence, or a historical inevitability due to the introduction of new numerals on all fronts, or an indication of a more general shift in understanding of meaning, or repressed but stubborn unconscious love of mysticism on the part of Stevin, or whatever else, I leave to the reader to decide. 7.2 The Salem Codex Unfortunately I have not been able to get a hold of the full text of the Salem Codex, which is in the library of the University of Heidelberg, and must make do with excerpts in Leo Jordan’s article “Materialen zur Geschichte der arabischen Zahlzeichen in Frankreich." However, what these excerpts say is extremely noteworthy. The codex written at Salem monastery around 1200 contains one of the earliest European texts to treat of Hindu-Arabic numerals. Its arithmetical content heavily depends on the work of the ninth-century mathematician Al-Kwarizmi, translated into Latin by Robert of Chester and others in the 1100s [Menninger 411]. So, contemporaneous with the first draft of Fibonacci’s Liber abbaci, the Salem Codex appears at the point when few people in Europe had come into contact with Hindu-Arabic notation. Maybe it was the very exoticism of the subject and shock of novelty that elicited its reflections of the zero, or maybe its author, being acquainted with the writings of pseudo-Dionysius (very probable) and possibly even Eriugena (not very probable), had a mind especially pliant in the matters of both symbolism and arithmetic. He uses the zero in operations. The fact that one could employ zero in calculation with the same assurance as other numbers is by no means self-evident. There is a big theoretical gap between operating with numbers in which zero plays the role of place­ holder (i.e. 20 - 5), and operating with zero alone (i.e. 0 • 5). If we may grudgingly accept 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the first —after all, it’s just a way of writing XX five times, the second does not make sense within the referentialist system. W hat does it mean to multiply nothing by five? Is that even arithmetic? What does the “nothing” stand for? How does it respond to the question “how many”? And if we can check the former on a counting board, how do we check the latter? What do we put down, five non-pebbles? And yet the anonymous author of the Codex does use the zero alone in operations. Here is the excerpt comparing God to zero: Nor should we omit the fact that the zero is subject to the rules of algorism just as other figures are, except that it does not increase any number, and is itself increased by no number. What indeed would you call a thousand times nothing, if not nothing? Or nothing added to a thousand, if not thousand? The zero does, however, perform a certain kind of increase, but only by multiplying things tenfold: thus, by the grace of the Word, put a zero in front of30 one, and you get ten, put it in front of ten and you get a hundred, put it in front of a hundred and you get a thousand. And in this, you must know, hides a great sacrament. For by that which is without a beginning and an end, is figured he who truly is Alpha and Omega, that is without beginning and end; and just as the zero neither increases nor diminishes, so he receives neither addition nor diminution; and just as it multiplies all numbers tenfold, so he multiplies them not only tenfold, but a thousandfold - or rather, to be more precise, from nothing he creates all, conserves all, and governs all.31 The passage says nothing about the monad.32 The circular form of zero, however, is very much in evidence. We might justify the author’s claim that the zero has neither beginning nor end by more fanciful means - nothing, for instance, cannot be said to have beginning 30 “In front o f means that the numbers are written in the Arabic fashion, with the rightmost ones set down first: see chapter 1. 3: "Nec praetereundum est quod 0 per omnia omnibus algorizmi utiturlegibus quemadmodum et alia figura, excepto quod nullum numerorum multiplicat, sed et ipsa a nullo multiplicatur. Quid enim [aliud] si dixeris milies nichil quam nichil? Aut nichil ad mille quam mille? Facit tamen quondam multiplicationem, sed tandum per decuplationem: Verba gratia praepone 0 uni, et fiunt X, praepone deceno et fiunt C., praepone centeno et erunt M. Et sciendum quod in hoc magnum latet sacramentum. Per hoc, quod sine inicio est et fine: Figuratur ipse, qui est vere alpha et tu, id est sine inicio et fine; et sicut 0 non auget nec minuit, sic ipse nec recepit aucmentum nec detrimentum; et sicut omnes numeros decuplat, sic ipse non solum decuplat, sed millificat, immo ut verius dicam, omnia ex nichillo creat, conservat atque gubemat, " [Jordan 160, n. I], 32 Elsewhere the author claims that “omnis numerus ab una generatur, ipsa a nullo ” [Jordan 159, n. 4]. 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nor end. But the very formula comes from Aristotle’s description of the circle. Its suitability to God is legitimated by the reference to Revelations 22:13. Yet the author does not stop with the basics. We might imagine him slightly hunched at the scriptorium, tinkering with the sign to press our further similarities - by using it to write numbers, by using it to calculate. It works. He sees the connections. He is filled with awe.33 Let us start with the similarities discovered in calculation. A number multiplied by zero yields zero, and zero added to a number yields that number. In this, the author writes, zero resembles God, who “receives neither addition and diminution,” i.e. is aloof and immutable. Now for the way Hindu-Arabic numerals signify numbers. Although zero does not increase numbers in calculation, it does “increase” them in a different way. When we place a zero to the right of a numeral, it “multiplies” its value tenfold. So God’s presence in things “multiplies” them, whether in the sense of spiritual value, by election, or in the sense of their very existence, by creation, conservation and governance.34 The analogy initially suggests we regard every thing as the joining of a 1 (thing) and 0 (God) not as addition, but as notation. In the former case the thing remains unchanged, in the latter its value miraculously decuples. Thus the presence of God as 0 makes each thing not a 1, but a 10. The author catches himself. After all, things void of God simply do not exist; whether spiritually or physically, they are not ones but nothings. God then exceeds the analogy, for his presence multiplies things not only “tenfold,” not only “thousandfold,” but brings them from nothing to all. The author claims each number to be generated from the one, whereas it itself is generated from none (“omnis numerus ab una generatur, ipsa a nullo", Jordan 159, n. 4). 33 There’s something of Abulafia in all this, of the {Cabbalistic practice of random permutations of letters to suddenly glimpse fragments of the Name of God. Curiously, this fortuitous exactitude of result is also where the process of writing poetry is identical to the process of doing mathematics. 34 Election and existence are in a way related, for in the Old Testament election leads to multiplication of progeny. Donne writes: “In that place in Genesis, when Abram took 318. to rescue Lot [...], the Septuagint have Numeravit, and Saint Ambrose says, the Hebrew word signifies Elegit: as though it were so connaturall in God, to number and to Elect, that one word might express both.” As Simpson points out, Ambrose’s comment does not touch the Hebrew, but says, “Quid est ’numeravit'? Hoc est elegit” [Donne Essays 128, n.]. Compare Donne's thought with this thought with Dee's Numberer whose Numbering keeps things in being, in my chapter on Dee, and with e.g. Gen 17:4: “my covenant is with thee and thou shalt be a father of many nations.” Crescite et multiplicamini - “Increase and multiply,” God says to man [Gen 1:22; some writers use this verse as the justification of study of arithmetic, Spafarii 31]. 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Are we take this nullus numerus as referring to zero, and therefore positing the analogy that as God is the origin of things, so zero is the origin of numbers? This analogy would also be problematic. If God is the origin of things by virtue of his being nothing, it is a nothing by transcendence. Zero, on the other hand, cannot be construed as nothing by transcendence. In addition, God is the origin of things in a manner entirely different from that of zero, and to claim otherwise would be to lapse into Eriugenism. Let us return to the line that, comparing the multiplicative powers of the zero to those of God, breaks in mid-sentence with an “or rather, to say more truly, he creates all from nothing.” By implication, in the first half of the sentence nothing is the transcendent God, in the second half it is other to God, as the absence “from” which the world was created by divine Will. In the first half, God is zero, in the second half he is the calculator manipulating zero. It therefore does not seem that the Codex gives zero the title of origin of numbers. 7.3 Bovelles: Nothing in the Liber de Nichilo The second reference to God as zero and nothing dates from 1510. This is the period when Hindu-Arabic notation is making its greatest gains outside of Italy, so this reference is a lot less unexpected than that of the Salem Codex. We should also be in a much better position to understand what it means by “nothing,” since its author, the French philosopher, theologian and geometer Charles de Bovelles (1479-1553), preceded it by an essay on nothing entitled “Liber de Nichilo” (1509), and by a poem reiterating the claims of the essay. Liber de Nichilo is not easy to understand. Its nichil is not per transcendentiam, but per privationem. Bovelles identifies existence with having being, and his God is Being, although he also exhibits transcendent traits. Since existence is having being, Bovelles denies the existence of nothing, arguing that what has no being “is neither in the mind, nor in nature, nor in the intelligible nor in the sensible worlds, nor in God, nor outside God in any creature” [40].35 Well, we might ask, if there is nothing anywhere and no nothing at all [“exulat a cunctis nichil, hoc natura repellit," Hecatodia de Nichilo, 1.7], what is the Liber de Nichilo about? The nothing that is not anywhere and that does not exist at all. 35 “Nichil enim est nichil, nichil non aliquod est, non hoc aut illud aut aliud quodpiam ens est, sed nullum ens, nichil nusquam est neque in mente est neque in rerum natura, neque in intelligibili neque in sensibile mundo, neque in deo neque extra deum in ullis creaturis. ” My text and pagination of the Liber de Nichilo are of the 1983 French edition, since the 1510 text is much harder to read and obtain. I do rely on it for other Bovelles pieces. 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The way in which Liber de Nichilo attempts to envision what can have no ontological ground whatsoever while retaining essentialist ontology may be exemplified in Bovelles’ discussion of physical space. Here his presuppositions clash head on with the gesturing of his verbal games. Since nothing is privation, absence, the question at hand is whether there may exist a privative space, space with nothing in it. Bovelles is a believer in Aristotelian pienarism, or, more precisely, Aristotle’s rejection of the anatomists’ concept of nothing as void space. For Aristotle, possessing extension is the property of a body, and therefore if nothing had extension, it would no longer be nothing. An empty space is a contradictio in termis. The counter-claim that there is no empty space outside of bodies, but that bodies themselves are located in empty space, is not only superfluous, but also triggers an infinite regress - for this void can be placed in another void, and so on. So all space is a plenum [for references and analysis Aristotle’s argument, see Grant 5-8]. On the level of essentialism, Bovelles agrees with this. At the same time, he also entertains the idea of a plenary space that itself is located in and penetrated by nothing. He even makes the striking proposal that nothing is an immense vacuum enclosing the plenum, as well as an infinite number of minute vacua each of which encloses a particular thing or part [86]. This differs from the Aristotelian concept of place36 in that Bovelles’ nothing is not just a border, but also penetrates the thing it holds. However, here Bovelles is only - how shall I say it - punning; “nothing is a vacuum” is the same thing as saying “there is no vacuum.” The first statement is affirmative and the second negative. Bovelles appropriates affirmation and negation, the discursive tools negative theology employs to talk about God, in order to talk about nothing. The fact that he does so plays on the extreme ambivalence the word “nothing” has as a sign - the ambivalence that tricked Alcuin’s student Fridugis into claiming that nothing is something, since the word “nothing” has meaning, and therefore must refer to something! Hence the essentialism that leads him to deny nothing a share in existence, lives hand-in-hand with such extreme cases of linguistic flirtatiousness, such titillatingly heretical headings as “Whence follows that God from his substance created and produced nothing” [64: Unde fit ut deus de sua substantia creaverit protuleritque nichilj. Sounds rather Eriugenist, no? The only real reason we know it means “God did not create the world from his own substance,” is that negation is more truthful that affirmation [130]. In pseudo-Dionysius negation is more truthful than affirmation because God, by virtue of being beyond Being, is nothing. Bovelles takes the road less traveled. He makes 36 "Quodlibet enim ens in proprio recipitur vacuo, ut in peculiari et equali loco, ” 86. 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the curious argument that the opposition of God and nothing is reproduced in the opposition of natural and logical-discursive ties. While the being of every thing is predicated on God, to say that God exists does not lead one to conclude that things exist. God can exist without things, as he did before the Creation. It is the opposite with nothing. While the being of no single thing is predicated on nothing (since everything is predicated on something, i.e. God), to say that nothing exists, means that something exists, and to say that something exists means that its Creator exists. But what do we refer to we say that nothing exists? Nothing - for nothing to exist is an impossibility. Yet the fact that there is an impossibility, argues Bovelles, implies the necessary existence of the possible (world) and the necessary (God) [104-116]. Bovelles’ idea of God is a bizarre and uncomfortable marriage of the legacies of Augustine and pseudo-Dionysius. “God is a being, “ writes Bovelles, “and creatures are beings, but God is an infinite being, whereas creatures are finite beings” [80].37 Like Nicholas, Bovelles interprets transcendence as the relation between the infinite and the finite. It should be stressed that, although the totality of things before God is said to be nothing, a mere point [chap. 5, esp. p. 88], the relation is not formalized mathematically: th e world compared to God is nothing because no proportion can be established [80], not because a/oa = 0 for any finite a [see Docta Ignorantia, e.g. 1:20]. Yet, as I said, Bovelles also tries to be Augustinian. For him, nothing is lack of being and God is a being just like creatures are beings. The difference between God and creatures qua kinds of being lies in th e fact that infinite being is innumerable, and therefore unknowable.38 Since God is Being, he is the opposite of nonbeing or nothing. He exists, nothing does not. Bovelles arranges the hierarchy of all creation, from angels to prime matter, all to various degrees under the lordship of mutability, between the simple and immutable terminals of God and nothing.39 The fact that creatures in this hierarchy are not entirely being and not entirely nonbeing is Augustinian [see Confessions 8:2], except that Bovelles treats the “simple and immutable terminal of nothing” as if its non-existence translated into a kind of existence after all. As the terminals of creation and each other’s contraries, God and nothing appear strikingly similar. They are in fact mirror images of each other. I have already given an example of their oppositional symmetry on the level 3' “Nam deus ens est et creatura ens, autem deus ens infinitum, creatura vero finitum. ” 3!i “Deus ens sine numero et supra numerum, creatura vero ens in numero numeratum et mensum ” [80]. For Nicholas, the Maximum is not Being - it is Absolute Being, the being of Being. 39 “Est igitur semper deus in esse. Nichil semper in non esse. Creature autem varie, quandoque in esse et quandoque in non esse. Deus et nichil extrema sunt, simplicia et immutabilia. Creature vero medie, varie et mutationi obnoxie, ” 104. 178 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. of natural vs. Iogico-discursive predication. On the discursive level, nothing’s relationship to nonbeing replicates that of God to being. As God is Being infinite in act, so nothing is Nonbeing infinite in act [86]."*0 What does this mean? Negatively or on the level of nature, nothing. There is no nonbeing infinite in act or otherwise. Affirmatively or on the level of discourse, it means that the simple and immutable nothing non-exists infinitely, and lies at an infinite and innumerable remove from being. It can be neither exhausted nor destroyed. The finite entirety of creation that God had managed to extract from nothing is, when compared to the infinity of nothing, but a nothing, a mere point. In fact, had God one day changed all of nothing into being, there would have arisen a being infinite in act, that would be equal to, external to, and separate from God, and the omnipresence of God would have found itself limited and oppressed [89].41 Thus, just as God is capable of creating as many worlds as an infinite sphere has points, so nothing can be converted into as many worlds as an infinite sphere has points [96]. But, again, nothing does not exist, so it’s not clear what we are talking about. Since God and nothing are on the opposite ends of the hierarchy of being, all affirmations proceed from God, all negations from nothing [128]. This means that for the purposes of discursive reasoning about God, nothing is more fecund that God himself. Yet even this can be read negatively: since God is responsible for the being of all, and the possibility of discourse in the first place, nothing but God is ultimately responsible even for negation. So what do we make from such a system? What is the ontological status of the affirmative side of Bovelles’ statements about nothing? What reality do they refer to? They refer to nothing. Does this nothing exist? No. Then what is the purpose of what Bovelles says about it, for instance of the idea of nothing as nonbeing in actu, or of nothing as the receptacle and interpenetrator of the plenum? Is Liber de Nichilo just an occasion for sophistic games, for the virtuoso pyrotechnics of paradox?42 That’s certainly not the feeling one gets from it. Light-hearted it’s not. There is clearly a serious inquiry going on. 40 “Nichil est actu ininifium et in tantum non ens: quantum deus est ens. Deaus ens est actu infinitum. Nichil vero non ens actu itidem infinitum.” 41 “Quod si nichil a deo aliquando absolveretur et proferretur ad esse totum, tandem id quod factum esset a deo quodve de nichilo ortum, esset actu infinitum equate deo exterum et separatum a deo; in quod absoluta, finita et expleta esset divina omnipotentia, extero ente actu infinito ac sibimet equali propagata, explicita et consumata.” 4~ For Renaissance paradoxes of nothing, see Rosemarie Colie. 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The inquiry concerns the relationship of logical discourse and reality. It does not occur on the level that entrapped Fridugis and the hypothetical sophists whom Plato’s Sophist proves wrong, the level that asks, “If signifiers mean by indicating things, how can the word “nothing” be meaningful if there’s nothing there for it to indicate?” Rather. Bovelles opposes logical discourse {ratio) as a system to essentialist reality (natura) as a system. Even though separate signs (in ratio) refer to separate things (in natura), somehow, when logical discourse as a whole is compared to essentialist reality as a whole, logical discourse and reality end up as mirror images of each other. The Liber de Nichilo implicitly calls attention to the problem of sequence: the sequence of logical discourse does not reproduce the sequence of essentialist reality, but reverses it. In nature, God precedes all things, and all things come to be because of him [...]. In the intellect, however [...], no being can be established to be on the basis of the being of God [...]. In nature [...], nothing causes the being of no thing, and precedes no thing [...]. In the intellect, however, nothing precedes all things and everything is inferred and deduced from it [114].43 Thus, while the origin of natura is God, and everything in the sequence that is natura depends on God, the origin of logical discourse is nothing, and everything in the sequence that is logical discourse depends on nothing. Liber de Nichilo is also an inquiry into truth. In essentialism, truth is one. Truth is immutable. Truth is Being, Identity. Truth is the being of a rock as a rock, and this is why truth is God, for the being of a rock as a rock depends on God, and only he is fully identical with himself. This is also why we can never fully know truth, for we are not identical with it. We depend. We depend on causes. And the way we can understand causes is though logical discourse, whose sequence mirrors the sequence of causes, reproduces them in the opposite direction. Since everything is known by knowing its causes, the ultimate cause - God, Truth, Being, Identity, Oneness - is unknowable. Still we move through the causes, we move upstream, we move towards the source. We are like salmon. What Bovelles says about nothing is, strictly speaking, the opposite of truth. Yet if we are to understand essentialist reality - inasmuch as such reality can be understood at 43 "In intellectu autem et rationationis necessitate, nullum esse ob divinum esse statuitur in esse. Nichil autem, cum sit nullius causa, nullum natura infert esse nillive antecedit. [...] Deus igitur in natura omnium est antecedens, omnia vera sunt eius illata et consequentia. In intellectu autem nichil antecedens est omnium et cuncta ad eo illata eiusve consequentia.” 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all - we must start with the only “thing” that essentialism does not, indeed cannot, include. The road to truth, the sequence that ends at Being, originates at that which lacks being, and which is therefore utterly false. Nothing. 7.4 Bovelles: Zero is Like God Bovelles’ Book o f Twelve Numbers —Liber de Duodecim Numeris —is the earliest text I found to put zero at the origin of the number sequence: O.I.2.3.4.5.6.7.8.9. Judging by the Liber de Nichilo, we would expect its reasoning as follows. Zero is written before the sequence 1-9 because it is a sign for nothing; nothing is the origin of logical discourse about reality; by implication, this is what numbers are - logical discourse about reality. In fact, Bovelles’ reasoning is “natural,” i.e. ontological: zero precedes the number sequence because, as nothing, it is like God, and God is the origin of things. And numbers are things. Liber de Duodecim Numeris is not an arithmetic, but a book on what we now dismiss as number mysticism. It is not concerned with Hindu-Arabic numerals, but with the symbolic properties of Ideal Numbers. In it, the Monad stands for essence and point, the Dyad for intellection and line, the Triad for love and the plane (by inference, they signify the Father, the Son and the Holy Ghost respectively [see 152 r v]). The Dyad also stands for matter, appearance, memory and justice; the Triad for God, angel and man, and the three dimensions; the Tetrad for the body, for four kinds of knowing and so forth. Number mysticism is essentialist: if what God is, is not determined by the numbers one, two, and three, what memory or the body are, depends on Ideal Two and Four. The Tetrad is prior to the four kinds of knowing. The passage on zero is set not before the section on the Monad, but within the section on the Endecad, or eleven [168V-9V]. Here Bovelles wrestles with Aristotle’s enigmatic assertion that ten is the last Ideal Number, and there are no Ideas "of eleven or the numbers that follow" [Metaphys. W84a26-27], interpreting it as follows: Ten is the limit of the intellectual region, the end [...] of simple and immaterial numbers. For it is neither wholly simple, nor entirely mixed and complex. Rather, it is the point between that which lies above and that which lies below, being most 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. similar to the rational soul. It is [...] stationed on the hinges [...] of the intelligible and the sensible, and participates in both spiritual and corporal nature.44 Belonging to both the numbers of intelligibles and of the sensibles, ten can be inserted into neither order. So -- both and neither -- it stands at their jointure, on one side touching nine, the last Ideal Number of the intelligibles, and, on the other, eleven. For eleven is the first of another group of numbers - the numbers that regulate the sensible realm.45 Bovelles refers to numbers 1-9 as simple and 11-19 as mixed. Eleven, for instance, is mixed in that it consists of ten and unity. It is analogous to the one in that both start the progression of the numbers of their kind. Bovelles refers to eleven as “the first position of unity in the decad” [my italics].46 If ten. not in the ranks of numbers regulating the sensibles, is the origin of such numbers, what is the origin of simple numbers 1-9 regulating the intelligibles? The seat of simple numbers is itself something older than the monad. Nothing indeed is to be considered higher and prior to the monad than the very formula of privation, than the little circular number of negation. Just as, in any progression of numbers, in the mind it is conceived and intuited before the monad, so it is to be set down by the pen in this manner: O.I.2.3.4.5.6.7.8.9.47 44 “Si (vt Pythagoras voiuit) decas vltimus est numerus: aut sequens endecas numerus non est: aut principatum quendam inter numeros obtinet. Quod enim vltimo supremoque succedit: aut omnino non est aut inter ea est primum: quae ab huiusmodi vltimo postergant queue eidem substunt. Decas iimis est intellectualis regionis. Meta est (vt diximus) simplicium <4 imaterialium numerorum. Nam neque ipsa ex toto simplex numerus est: neque omnino permixtus atque comp lex us. sed est superiorum <4 inferiorum intercapedo: rationali persimilis anime. Hanc enim in intelligibilium <4 sensibilium cardinibus ac valuis collocatam monstratum est: atque vtriusque spiritalis & corporee nature esse participem.” 45 This does not mean that eleven itself is perceptible to the senses: numbers regulating the sensibles are also Ideal, but they do not have power over other Ideas. “Erit igitur endecas merito vt sensibilium primum vtue prima mere sensibilis creatura. illique primatus: inter complexos ac secunde regionis numeros est iure tribuendus. Decas siquidem perinde atque rationalis anima velut permixte nature: neque intellectualium neque sensibilium inseri ordinini poposcit. sed in vtriusque meditulio residens ac fortis vtriusque particeps: hanc ab ilia regione dirimit. Vnde fit vt sicut Enneas intelligilium dicta est vltima: ita <4 modo endecas sensibilium sit prima vocitanda.” 46 “Est enim endecas decas <4 vnitas siue prima vnitatis in decade positio. In simplicibus numeris: progressio fit a monade ad enneadem. In secundis vero mixtisque frequentijs: par ab endecade ad decimum usque nonum progressiose extendit." 47 “Simplicium numerorum sedes: idipsum est quicquid monade antiquus. Nil autem rimabere monade excelsius ac prius: quam ipsam priuationis formulam quam negationis numerum orbiculum. qui in omni 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This, as I said, is the earliest instance known to me of the number sequence we use today. The initial motive of the Liber de Duodecim Numeris is to give zero its position by analogy to ten: as ten precedes and is the seat of mixed numbers, so simple numbers too require an origin and a seat. Inasmuch as the origin of mixed numbers is not among mixed numbers, being more simple than they, the origin of simple numbers cannot be among simple numbers and must be simpler than they. What is simpler than simple? Nothing. Bovelles’ reasoning with respect to the sign zero recalls his treatment of the word “nothing” in the Liber de Nichilo. He both admits the non-existence of nothing and signifies it. If in the Liber de Nichilo nothing had no basis in Being, but headed the sequence of syllogisms about Being, here nothing is not among simple numbers, and the sign for nothing heads the sequence of the numerals representing them. If you will tell me that there is nothing before the monad, and therefore that no sign ought to be placed before it, to this I will in part assent and in part object. I will acknowledge it as true that nothing is before the monad. Nevertheless I will say it does not contradict reason that the very nothing that is before all numbers and after which unity emerges into light be noted down before them by the sign of privation or nullity. So before all numbers write nothing, that is, the meager little circle of privation, 0. Then write one, then, two, then three, and so all they way to nine. Rather convenient is that law, allowing you to examine the seat of simple numbers.48 Does Bovelles distinguish between numbers and numerals? If “the little circle of negation” is placed at the origin of numerals, does that mean that zero is the origin of numbers as well? In the beginning of Liber de Duodecim Numeris, the latter is clearly not the case. Bovelles’ conception of numbers is explicitly monadic. “There is nothing in numbers apart from the one and the iteration of the one. The one is the monad and the numerorum progressu: sicut mente ante monadem concipitur & intellegitur ita & stilo ante monadem exarandus erit hoc pacto. 0.1.2.3.4.5.6.7.8.9." “Quod si dixeris nichil esse ante monadem: Nullam ideo illi anteferendam notulam: & hoc ipse partim assentior partim inficiari pergo.Ac verum quidem fatebor: nichil prius esse monade. Dicam attamen haud a ratione videri alienum: si idipsum nichil quod ante cunctos numeros est post quod vnitas emergit in lucem: quadam priuationis nullitatisue notula propingatur. Scribe enim imprimis numerorum nichil: id est exiguum priuationis orbiculum o. Deinde vnum: Hinc duo rursum tria: Et ita vtque ad enneadem. Conuentientius hac lege: simplicium numerorum scrutaberis sedem.” 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. outstanding source of numbers,” he says.49 This does not mean that simple numbers are composed of monads. It is their manifestations in the mind that appear so, thereby becoming subject to mathematical operations. Simple numbers themselves are simple, i.e. indivisible. They are Ideal: they emerge from the monad. Either way, it is the monad that is “forts numerorum Considering the liminality of ten gives Bovelles second thoughts. As ten is neither a simple nor mixed number but the hinge between them, so nothing is the hinge - or rather threshold -- before simple numbers, and by extension, before all numbers. In the scenario of the section on the Endecad, one is parallel to eleven, whose position with respect to numbers of sensibles is called a “principatum.” Eleven’s principatum seems to be only the matter of position, not origin. Mixed numbers do not come from eleven - they come from combinations of ten and simple numbers. This is why in the section on the Endecad the monad does not seem to have any ontological priority to two, three, four, five, and other numbers. It is number as much as they. And it has the same origin as they - nothing. This is a huge problem. Liber de Duodecim Numeris comes out and depends upon the ontological tradition, whose assigning properties to numbers went hand in hand with deriving them from the monad. Yet it declares the origin of numbers to be zero. At the same time, it entirely omits the distinction between the natural and the logico-discursive ties so prominent in the Liber de Nichilo. If the sequence of numerals is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, then we imagine the ontological sequence of numbers, if expressed according to the lines laid down in the Liber de Nichilo, to be 9, 8, 7, 6, 5, 4, 3, 2, 1, 0. But it is not: The seat [of simple numbers] which is noted by the circle of privation is to these numbers as God is to the immaterial substances of angels. For theologians exempt God from the angelic orders. They define him by privation and negation. They attach to him, they dress him in no appearance [.speciem]. They name him the seat, support, state and position of angelic orders, and the perpetual custodian of being.50 See section on the Dyad: “in numeris aliud nichil est quam aut vnum aut vnius resumptio. Vnum monas est precipuusque fans numerorum” [I49r|. 50 “Eorum quippe sedes que priuationis orbe exaratur: ad eosdem numeros erit velut deus ad immateriales angelorum substantias. Eximunt siquidem thologi deum ab angelicis ordinibus. Ilium priuatione negationeue definiunt. Nullam illi indunt allinuntue speciem. Angelicorum ordinum sedem fulturamstatum positionem & in esse iugem custodelam appellitant." 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As there are nine simple numbers, so there are nine angelic orders.51 Angelic orders are the same as what Bovelles had earlier called “intelligences,” and should perhaps be thought of as the simplest Ideas, prima genera. The orders are regulated by numbers 1-9: “For the intelligence [angelic order] which is first and closest to God may be called the monad, the second the dyad, triad the third. And so all the way to the last and angelic mind, which is measured by the number of the ennead.”52 God precedes angels just as nothing precedes simple numbers: because “as God is the seat of nature and the support of simple and angelic minds, so in numbers the naked and empty circle is employed as the state and position of simple numbers.”53 God precedes and gives origin not only to the angelic orders but to all orders of Being, which after angels includes men, animals, plants, minerals and prime matter [Liber de Nichilo 128]. The reason “he is established to be before all Being and more eminent than all Being” is that “he is said to be nothing of the things that are.”54 Here Bovelles’ conception of God is fully transcendental. Unlike in Liber de Nichilo, God is not a being. Unlike in Liber de Nichilo, God is not the opposite of nothing but the same as nothing, if only symbolically (in the pseudo-Dionysian sense of “symbol”). But symbolically is the only way we can speak of God. He may be defined only “with the help of privation.” This is because in the Liber de Duodecim Numeris he is superabundant, exceeds Being. There is a double problem here. I) Nothing by transcendence is - ontologically speaking - the opposite of nothing by privation. When we imagine the absence of all things, we imagine pure nothing by privation, and, according to the Liber de Nichilo, from there we can prove the existence of the world and God. So the “priuationis orbiculum o” is the symbol not of God but of the origin of the logico-discursive road to God, whose sequence runs contrary to the ontological sequence. If we say it signifies God, we mean only that it signifies what lets us reason about God but is not like God at all. 2) However, the sequence O.I.2.3.4.5.6.7.8.9. is for Bovelles not just the sequence of numerals but also of numbers. Furthermore, these are ontologically grounded numbers, 31 Bovelles' orders are those of the Celestial Hierarchy of pseudo-Dionysius, they are: seraphim, cherubim, thrones, dominations, virtues, powers, principalities, archangels, angels. 32 “Prima vero intelligentia & que proximo deo adstat: d id poterit monas. Secunda dyas. Trias tertia. Et ita ad nouissintam vsque angelicam mentem: quam metitur enneadis numerus. ” 33 “Deus itaque sicut in rerum natura sedes est fultura simplicium angelicarumue mentium: ita <t in numeris nudus vacuusue orbiculus: per simplicium numerorum statu ac positione vsurpatur.” 34 “Igitur si per priuationem finitur deus: cum nichil esse dicitur eorum que sunt: cum ante omne esse & omni esse stabilitur eminentior: ” 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ideal Numbers. Their order is simultaneously the order of things and of signs. It is the same order. And it is headed by nothing —by privation that must be ontologically identical to nothing by transcendence. In the Liber de Duodecim Numeris, the superabundant God and the privative nothing that is his opposite in the Liber de Nichilo are one and the same.55 I have used the image of the mirror to describe the way in which the order of signs in Liber de Nichilo runs contrary to the ontological order. In the Liber de Duodecim Numeris, both God and zero are said to be like the mirror: “God [is] like the seat of simple numbers,” write Bovelles. “He is like the orb with no feature, simple, empty and void. He is like a mirror without image, pure and naked.”56 The simile unfolds into a scenario not unreminiscent of the generation of Ideas as numbers and of things in late Plato. While God is “vt speculum sine imagine,” angels or intelligences are “imagines [...] libere atque absolute” They are also the brightnesses of the divine face joined to the mirror, being visible and shining outside the mirror while remaining in themselves and carrying themselves[...]. They that [...] in the intellectual world are absent from themselves, they that dwell simple and separate, first come together in the decad or the rational soul.57 For Bovelles, the rational or human soul is the jointure and link of the mirror and images. Here he calls the images that are angels with the pregnant word “species,” which means images, appearances, and, of course, Ideas. The human soul reflects these images in its mirror, but in such a way that the images are “mixed with” the mirror. In the human soul, as it were, the mirror and the images form a single reflection. 55 This dilemma might be resolved by applying Nicholas’ idea of God as coincidentia oppositorum, with the one having to be replaced by zero in the role of Absolute Minimum. Then the superabundant God as Absolute Maximum will be the same as zero. Absolute Minumum, nothing by privation. Bovelles does no such thing. 56 "Erit deus vt simplicium numerorum sedes. vt orbis ipse sine specie simplex vacuus inanis. vt item speculum sine imagine sincerum ac nudum.” 57 “Sunt & ipsius diuini vultus nitores inconiuncti speculo. parentes micantesue extra speculum seipsis residentes seque ferentes. Que igitur ad hunc modum in intellectuali mundo sibimet absunt, que simplicia & separata degunt: coeunt imprimis in decade siue in ipsa rationali anima.” 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “The soul,” writes Bovelles, “is most similar to the decad, in whose little circle, as in the compass of vision, resides the appearance of the monad.”58 The ten, as we remember, is like the zero in that both are thresholds - the latter of the simple numbers, the former of the complex. This analogy between ten and zero suggests a similar analogy between the soul and God. The blank mirror that is God is present in the soul as the “priuationis orbiculum o" is present in the 10. The combination of the ten of rational soul with simple numbers as “free and absolute images” brings into being the complex numbers of the sensibles.59 It appears that for Bovelles this is also the ontological history of numbers. Numbers 1-9 emerge from nothing and are images of nothing, but their permutations with themselves and with nothing yield all other numbers. This “second imprinting” is rather similar to what we see in late Plato. There, as we remember, the impressions of the One upon the Indefinite Dyad gave us Ideal Numbers, and the impressions of the One upon Ideal numbers gave us things. The greatest differences between Plato’s scenario and that of Bovelles are the absence of the Indefinite Dyad and that the one has been replaced by nothing. What does it mean, however, to think of numbers as images of nothing? In Plato every number is a particular kind of one because every number is a unity of a particular number of monads. This cannot be so in the case of Bovelles’ origin. Is it that Bovelles, like Stevin, identifies the being of numbers with the being of numerals each of which 5S "Esc enim animus persimilis decadi: in cuius orbiculo (velut in speculari ambitu) residens monadis species.” 59 This mental process must also have a parallel in nature, since nature is not merely composed of the intelligibles but includes the sensibles. Here is Bovelles’ diagram for numbers 0-19 and their meanings: 9 Angels 19 Prime Matter 8 Archangels 18 Elements 7 Principalities 17 Elements Mixed 6 Powers 16 Plants 5 Virtues 15 Bushes 4 Dominations 14 Trees 3 Thrones 13 Serpents 2 Cherubim 12 Fish 1 Seraphim 11 Birds 0 God 10 Man Note the weird absence of mammals and the undue attention paid to the vegetable kingdom. This is due to the fact that the ennead of angels is actually three triads. The first triad for Bovelles corresponds to the animal kindgom, the second to the vegetable, the third to mineral. Still, one fails to see how the combination of seraphim and principalities results in rocks. 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. does not point to a prior signified and therefore “signifieth not”? Stevin can make such an identification because he ignores ontology; Bovelles, however, is using Hindu-Arabic notation to explain ontological matters. For the mixing of numbers for Bovelles is modeled on their being written as Hindu-Arabic numerals. Inasmuch as ten is a mixed number, it is mixed because it combines the zero and the monad —hence it is written as 10.60 The mixed numbers 11-19 are also presented as if mixing were the same as writing. True to the form so far displayed in Liber de Duodecim Numeris and contrary to Liber de Nichilo, Bovelles identifies the order of writing with the order of being. Although Bovelles’ thinking is modeled on Hindu-Arabic notation, he is also thinking in referentialist terms. If he were not, ten, written with two numerals, would entirely belong to the class of mixed numbers, and the zero, written with one numeral, to that of the simple. In the Liber de Duodecim Numeris Hindu-Arabic numerals have invaded, but have not dispensed with, the classical number concept. As a result, zero takes on characteristics of the one, replacing it in its position as symbol of God. 60 “Conflatur enim decas [...] privationis orbiculo <&monade: siue hoc pacto decadent pingis 10 posita unitate extra orbiculum, siue hoc modo enneadem exarabis (1) vbi priuationis orbis gestat clauditue monadem." The second description is an allegory of the soul itself as the inseparability of mirror and image. 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 CROOKED FIGURE: CIPHER AND SEQUENCE IN SH AK ESPEA RE’S H ENRY V Mistress line, is not this my jerkin? [...] Do, do; we steal by line and level, an't like your grace. What art thou That counterfeit'st the person of the King? The King himself.1 0. The Chorus and Numerical Increase This chapter will examine the resonance of the metaphor of zero and Hindu-Arabic notation in the Prologue to Henry V. My prime concern will be with sequences: sequences represented and sequences in the process of representing. My understanding of Henry V is heavily indebted to Patricia Parker, both her teaching and her Shakespeare from the Margins. I will argue that the Hindu-Arabic numerals of the Prologue illustrate the kind of inversion of “natural” sequences that Parker calls “the preposterous” (from L. prae + posterns, roughly: “the end before beginning”). I will also argue that the play understands Hindu-Arabic numerals in a non-referentialist manner, and that, furthermore, it employs them as the model for the workings of signs in general. Henry V is the last play Shakespeare wrote about the houses of York and Lancaster. Dating from 1599, it was preceded by, in the order of writing, 1-3 Henry VI, Richard III, Richard II and 1-2 Henry IV. The order of writing does not match the order of action, Henry V being the last in the former and holding center position in the latter. As its epilogue reminds us, this final York-Lancaster play is followed by events “which oft our stage hath shown” [5.2.13], i.e. those represented in the first plays of the sequence which in fact starts with Henry V ’s funeral. By virtue of being the end touching the beginning, Henry V is, quite self-consciously, a play not only about history, but also about the making of history - in several senses of the expression - to greater extent than the plays which came before. One of the formal features setting Henry V apart is its use of the Chorus, the classical device that for many critics stresses the epic nature of the events represented. 1Stefano and Trinculo in Tern. 4.1.235-240; Douglas and Henry IV in 1H4 5.4.26-8. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Shakespeare’s Chorus, however, differs radically from its prototypes. While a classical chorus would consist of the inhabitants of the polis, which we may willy-nilly equate with “the youth of England” [2.ch.l], Shakespeare’s Chorus does not participate in the plot but in its presentation. The presence of the Chorus forces the stage to alternate between mimesis (the historical scenes) and commentary upon the mimesis. A Russian Formalist would argue that the Chorus de-automatizes the play, yanks us out of its “reality,” reminds us that its representations are artificial. "Artificial" in the Renaissance does not carry our meaning of false or fake: rather, instead of simply affirming or negating the historical scenes, the Chorus advertises the whole play's status as representation by means of signs. The apparent primary function of the Chorus - the sensus litteralis of his words - is to encourage us to “piece out” the “imperfections” of theater “with our thoughts” [Prol.23] by “minding true things by what their mock’ries be” [4.ch.53]. The actor playing the Chorus offers an elaborate and flourished apology for the paltriness of what we see onstage compared to the magnitude of what it is supposed to represent. If only the play were the same as reality, the stage “a kingdom,” actors “princes,” and the audience “monarchs” (presumably of other European states), then would “the warlike Harry” be “like himself,” that is: like “Mars” [pr. 1-8]. In other words, “true” representation calls for absolute identity between signifier and signified.2 But no theater can do that. So, bowing upon the stage, the Chorus wonders whether the play can convey reality, and apologizes, and asks for help: But pardon, gentles all, The flat unraised spirits that hath dar’d On this unworthy scaffold to bring forth So great an object: can this cockpit hold The vasty fields of France? or may we cram Within this wooden O the very casques That did affright the air at Agincourt? O pardon! since a crooked figure may Attest in little place a million; And let us, ciphers to this great accompt, On your imaginary forces work. [8-18] Let us start with the most naive, almost literal level of meaning of the “crooked figure” lines. Here they point to the metonymic aspect of theatrical convention, where the presence of one actor ("crooked figure") may symbolically indicate ("attest") a large number of people, generally commoners, in mass scenes ("million", comp, with "the play... pleased 2 Let us note that if signifier = signified, then not only is the play reality but reality is a play. Hence “the warlike Harry” becomes himself when he becomes the representation of somebody else, i.e. Mars. For treatment of theatrum mundi employing the alleged motto of the Globe, see Burton 36. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not the million," Ham. 2.2.460). We are asked to look at one and see many: "Into a thousand parts divide one man And make imaginary puissance" [pr.24-5]. The first test of this is literally before our eyes, in the fact that we are supposed to see the one actor speaking the lines as a Chorus [pr.32], i.e. a multitude. This is another, even more striking departure from the classical tradition. As the actor, so the theater. While the “crooked figure” represents persons of greater quantity (crowd scenes) or rank (kings, etc.), the “cockpit,” “wooden O” represents places vaster and more powerful: "Suppose within the girdle of these walls Are now confined two mighty monarchies" [18-9]. To enable such increase, the Chorus asks us to employ our faculty of imagination (“imaginary forces”). The word "imaginary" is technical, and connected both to representation and numerical increase. Neoplatonic theoreticians, both late classical and Renaissance, assigned to the mental faculty of imagination the leading role in the psychology of the poetic process: so it appears in Puttenham.3 In Theseus' famous speech in A Midsummer Night's Dream, it is imagination that "bodies forth The forms of things unknown," while "the poet's pen Turns them to shapes," i.e. written words [MND 5.1.19- 22]. Apart from being the faculty "in virtue of which an image" of things that are not there "arises for us" [Aristotle, DeAnima 428b], imagination, as the spatial language of Theseus implies, is also tied to geometry. In his Commentary to Euclid, Proclus regards the imagination as the locus of geometrical figures.4 Imagination thus implies number, although number of the relative sort: unlike the Ideal Circle, the circle in the imagination can be bisected, although, unlike the sensible circle, it cannot be measured. Imagination therefore is one of the places where the many comes to be. Whether poetic or geometrical, the image arises in the imagination by a kind of a swelling that divides into parts [for division as multiplication of parts, see chapter 2]. The origin of the swelling is the point, which, as we have seen in previous chapters, is identified with either the one (Pythagorean tradition) or the zero (Stevin). As the coming to be of the many from the one, imagination permits us to "see" the one actor playing the Chorus as a multitude. Can we also interpret it as the coming to be of the many from the none? There are three instances in the Prologue of the one, that imaginary forces are to convert into many, described as zero. The actors whom we are supposed to see as either 3 "By it as by a glasse or mirrour, are represented vnto the soule all maner of bewtifull visions, whereby the inuentiue parte of the mynde is so much holpen, as without it no man can deuise any new or rare thing" [14]. 4 As Ideas, geometrical entities are single, indivisible and unextended; in the imagination, they are divisible, formed, extended; in the sensory realm, they are inexact, available to the senses, and subject to corruption f54; see my chapter 6]. 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. multitudes or persons of greater quality are called "ciphers" and likened to "a crooked figure" despite the fact that each of them is a one. "Crooked figure" here means zero: "figure" is a technical term for Hindu-Arabic numeral; in Latin, the phrase figurae arithmeticae is opposed to the litterae arithmeticae of Roman notation [OED, "figure," 19; Du Cange, "cifrae"; see "figures of Augrim" vs. letters in Recorde Bviir]. The early modem period had no separate character like our 0 and employed the letter O sometimes upper, sometimes lower-case.5 The theater incorporating the multitude of persons or the two kingdoms is described as "wooden O," which may therefore be read as zero although the theater too is a one. The fact that the origins of mathematical increase are simultaneously ones and zeroes poses a great problem, for there appear to be two mutually exclusive types of increase in the Prologue. The first is arithmetical. It multiplies numbers by expanding magnitudes or by dividing a whole into countable fractions. This is what the Chorus asks us to do: to increase (by making the “cockpit” hold “the vasty fields of France”) and to multiply (“into a thousand parts divide one man And make imaginary puissance”). The arithmetical increase is an example of many coming to be from the one. The other image is the increase of value by place in Hindu-Arabic notation. This is how we may read “A crooked figure may Attest in little place a million”: the rightmost zero, working “bakeward” [Villa Dei in Steele 5], makes the leftmost one signify a greater quantity than it would otherwise. The connection between zero, place-value and increase is explicit in The Winter's Tale: “like a cipher, Yet standing in rich place, I multiply, With one “We thank you” many thousands more, That go before it” [1.2.6-9]. Arithmetical increase is wholly legitimate within the framework of classical ontology: it produces real entities from real entities by division resulting in multiplication of parts, as described already by Plato [Republic 525e\. Notational increase, on the other hand, is not: it produces something from nothing, both in the metonymic employment of the zero to stand for the principle of place-value, and, more importantly, in the fact that Hundu-Arabic numerals do not imitate what they represent, since they lack what I earlier described as the mimetic property of the referentialist Roman notation. While arithmetical increase involves the production of things from things, notational increase involves the production of "things" from images. Paradoxically, the distinction between arithmetical and notational increase in the Prologue does not hold. The one thing which gives rise to the many - whether this thing is the player or the theater - is repeatedly described as "crooked figure," "cipher," "wooden 5 Picinelli’s Mundus Symbolicus speaks of “the circle O [...] whether regarded arithmetically or grammatically” [“O"]. 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O." Whatever lies at the origin of number is therefore a monster: both one and zero, thing and nothing. Numbers here coincide with numerals, being with seeming. This brings us to the second problem in the Chorus' request to "let us, ciphers to this great accompt. On your imaginary forces work." So far we have spoken of the role of the imagination in the poetic process, but the Chorus employs it to describe the reception of the text / performance. This too accords with Neoplatonic theory: Longinus, for instance, talks of the orator imagining what he speaks and placing it before the mental eye o f the audience [in Princeton, "imagination," # 5]. It is therefore not anachronistic to conceptualize the creation and reception of a "figurative" text in the following manner: mental image in author —> text —> mental image in recipient The process may be likened to the transmission of information by means of enciphering and deciphering. The problem consists in the fact that "imaginary forces" are singularly ill-fitted for ensuring the faithful transmission of information. Accounts of the imagination's role in creating texts stress its autonomy, its capability to make something out of nothing. Puttenham even likens the poet to God, "who without any trauell to his diuine imagination, made all the world of nought" [/]. We should not place too much emphasis on Puttenham’s imagination creating ex nihilo, for he still regards the poet as an imitator or a "counterfaitor" [/]. Shakespeare’s Theseus, with his "forms of things unknown" (creatio ex nihilo? Ex aliquo sed ignoto?) and his "airy nothing" (mental image? thing unknown?), is equally sybilline. Nonetheless, he too stresses the autarchic quality of the mental faculty: "the lover," for instance, "sees Helen's beauty in a brow of Egypt" [10]. Whether or not imagination lets us make something from nothing, it definitely does force us to see a thing as other than it is and something it is not.6 To make the decoding of text dependent on this faculty is an uncertain venture at best. We therefore must distinguish between what appears in the poet's and the audience's imaginations. In Act 4, the Chorus asks us to envision the spectacle of “four or five most ragged foils” engaged in “brawl ridiculous” [4.ch.5l-2\ as the day-long battle between 72,000 French and English knights and footmen [4.3.2]. Should we oblige, the battle in our mind will be posterior to the performance, whereas the battle in the mind of the poet will be prior to it. But why stop at this distinction when we ought to complicate it 6 We remember the mad Ophelia, whose "speech is nothing," but "may strew Dangerous conjectures in ill- breeding minds" [Ham. 4.5.7-15]. Granted, the mind of the lover and "the ill-breeding minds" that frighten Horatio suffer from proclivity to distortion, and yet the difference is only that of degree: the stimulus triggering the imagination ultimately cannot define its result. 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. further? The battle is claimed to represent not the imagined, but the real, historical Agincourt. The problematic nature of our deciphering the performance of Henry V is matched by the problematic nature of the poet's deciphering the historical texts about Agincourt, and their deciphering of the physical text of the battle itself. What, then, do the signs of the play point to? How do they signify? A word in Act 4 encapsulates the difficulty. Reiterating the requests made in the Prologue, The Chorus asks us to “mind [...] true things by what their mock’ries be” [4.ch.53]. "Mock’ries" is an extremely ambivalent way to describe what we see on stage. It may mean imitations of “true things” that are prior to the “mock’ries”: thus the “brawl ridiculous” may be said to imitate the real and absent Agincourt. However, “mock’ries” probably points to other senses of the word as well: “to mock” means to make fun of, in which case the brawl parodies the historical battle, or to gull, to fool, to con. In Henry V the verb "to mock" bears the latter meanings. Having received the Dauphin’s gift of tennis- balls, Henry promises that “many a thousand widows Shall this his mock mock out of their dear husbands. Mock mothers from their sons, mock castles down” [1.2.284-6 ]. "Mock’ries” in this sense may cheat us out of the historical Agincourt, give us a substitute, interpose themselves between us and the "original" event. The employment of zero and Hindu-Arabic notation in the Prologue, therefore, is somehow implicated in the general problem of representation, in the working of signs as such. How? 1. Humility and Sequence Metaphors employing zero and Hindu-Arabic notation provide the space for the early modem period to wonder at the creative autonomy of syntactical arrangement, at the capacity of what was called "conjunctio" to make the wholeof the sequence differ from the sum of its parts. Shakespeare's contemporary John Owen, whom Ben Jonson described as a "poor pedantique Schoolmaster, sweeping his living from the posteriors of litle children” [Jonson 21], calls attention to Hindu-Arabic syntax in one of his many epigrams: Addideris unam nihilo, nihil inde creatur; Uni addas nihilum, nascitur inde decas. If you add one to zero, one is created; If you add zero to one, ten is bom. [qtd. in Picinelli, "O."] J. H. Walter’s Arden edition of Henry V glosses the lines “a crooked figure may Attest in little place a million” by referring to Peele’s Edward /: “’Tis but a Cipher in Augrum, And it hath made of lOOOO pounds, lOOOOO pounds.” Other parallels may be 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. found in the entry dedicated to “O” by Filippo Picinelli's Mundus Symbolicus, the greatest seventeenth-century dictionary of symbolism. In this article, as well as the article on the “I” (just as O was the figure for zero, I - and generally not 1 - was the figure for one), Picinelli presents several emblems of conjunctio in which several of the least things, when joined together, grow into something large. For instance, one impresa presents a row of zeroes (OOOO) with the lemma Addito minimo, maximum fie t ("with the least thing added, becomes the greatest"). The conceit of course is that the arithmetical value of OOOO is zero; however, if one writes "the least number", i.e. one, in front of the four zeroes, "suddenly this very respectable sum is bom: IOOOO." Another example presents a IOOOOOO, that is to say a million, with several lemmas to the effect that this immense number grows from nothing. The conceit of several O's following an I exposes all the ways in which Hindu- Arabic figurae arithmeticae differ from the Roman litterae: a) that there is a numeral which signifies nothing; b) that numerals affect the signification of numerals to their left, and therefore c) the value of any numeral is relative to its place in the sequence;7 d) that the value-bestowing properties of the sequence have nothing to do with Roman numerals’ juxtaposition as addition: this last point is illustrated by Owen's pun on addere. All examples of the conceit use O and I and not numerals of higher certain value in order to stress the dependence of all value on place, since one joined to zeroes “occasions the greatest arithmetical increase” ["maximum aritmeticae incrementumfacit," Picinelli‘T ’]. The increase occasioned by Hindu-Arabic numerals is entailed by something we are extremely familiar with, but which I shall review nonetheless. A group of Hindu-Arabic numerals signifies a number by means of two orientations, i.e. of rank, which decreases to the right, and of place, which is counted/rom the right. rank------- > 1,000,000 <------- place Which of the two orientations has a greater role in the signification process? The first, i.e. leftmost, place according to the order of rank signifies the greatest quantity, but the ability to do so does not come from the numeral itself. In the example above, it is not just the numeral I that points to a million; rather, the five zeroes following it multiply what Recorde calls its "certayne value" a million times. Multiplication of "certayne value" is enacted by the other order, that of place. The numeral that is first according to place - i.e. the 7 Even Recorde's “certayne value” [see chapter 1] is relative to place, since it may be defined as the value of a numeral that has no numerals to the right of it. 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rightmost zero —may "signyfie nothyng" [Recorde, Grounde Bvv], but take it away, and the million decreases tenfold. Take away all six "crooked figures" and the value of the numeral I falls to the number one. We shall see this paradox of signification to be of the utmost significance to Henry V. The presence of two orders in Hindu-Arabic notation makes it quite evident that each numeral acquires its value only after the totality of the sequence has been apprehended. In other words, to know that the numeral 1 stands for a million we first have to count the zeroes - from the right. The role assigned to syntax in Hindu-Arabic notation assures that the signified be regarded as posterior to the signifier} This too finds its resonance in the conceptual framework of Shakespeare’s play. Early arithmetics emphasized the double orientation offigurae arithmeticae, which we are used to and therefore do not notice. The emphasis took the form of wondering according to what order numerals ought to be written: following "as arabiene writene," medieval manuals set down numbers sinistrorsum or "bakeward," i.e. right-to-left, according to place and contrary to the order of writing letters [Sacrobosco English in Steele 35; Latin in Halliwell 5; "thou schal write bakeward" in Villa Dei in Steele 5]. The language they employed was that of precedence: for instance, in IO the I was said to "follow" the O. Although as the system became habitual, numbers began to be written left-to-right, their places were still counted sinistrorsum, and sixteenth-century schoolboys still practiced "ciphering" in the old manner [see my chapter 1 for detailed analysis of sources and concepts]. In the Prologue to Henry V, however, “little place” refers to the rightmost place in the sequence, which Recorde still describes as the first. The orientation of numerals elected by the words "little place" is that of appearance (both in the sense of rank and in the sense of looks on page) and not of place: the “crooked figures” follow the one. The word "follow" here may be viewed as a pun, for the Prologue of Henry V presents itself as an apology. The argument runs something like: “Pardon our insignificance, but even things insignificant in themselves can become significant in combination (Picinelli's conjunctio). Even though we are nothing, if you apply your imaginary forces, we may convey grand historical personages and events.” To the speech genre of apology the linguistic gestures of “little place” and “crooked figure” run particularly apropos. Early modem etiquette favors what to us seem hypertrophied forms of self-deprecation, always in order when addressing social superiors (note the Prologue referring to the audience as “gentles all”). So, for instance, in his letters to Lucy, Countess s In Roman notation, on the other hand, as we saw in my first chapter, signifieds correspond to individual signifiers: MLIII = 1000 + 5 0 + 1 + 1 + 1. 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of Bedford, Donne not only calls his addressee “divinity,” “Gods masterpeece,” and so forth, but refers to himself as “nothing.”9 At the same time, calling oneself “nothing” is not only a social, but also a religious trope, encapsulating one’s proportion, or rather, as I pointed out in the section on Bovelles, lack of proportion with respect to God. An eloquent example is a prayer recommended by Thomas a Kempis in his Imitatio Christi: "Behold, Lord, that I am nothing, that I have nothing, and that I am worth nothing” [107]. "Here lie dust, ashes and nothing," the perpetually gloomy Andrea Barberini wrote on his funereal slab.10 Before the appearance of Hindu-Arabic notation, the numerical emblem for humility was the one. In article “I” of Picinelli’s Mundus Symbolicus, the one usually symbolizes what St. Jerome called "prima virtus Christianorum." Humility is the greatest virtue since all our virtues, when we have not humility, are worthless (“pretio penitus sunt nullo”). The one is an appropriate symbol for humility since the one is both the least natural number and the first in their sequence ( “inter caeteros omnes & minimus & primus"), and the humble person acquires primacy in the eyes of God by being the least in his own eyes. The emergence of Hindu-Arabic numerals allowed the zero to replace the one in its role as the symbol of humility, for the zero, regardless of whether we consider it to be the "least number," can also serve as the arithmetical counterpart to the linguistic cliche of the humble self as nothing. After the replacement, metaphors of Christian humility began to exploit the principle of place value, as we can see in the following passage from a sermon by Simeon Polotskii, Muscovite Baroque preacher-poet of the second half of the seventeenth century: Mathematicians depict humility by the letter O or the cipher, for the reason that the cipher, as number, by itself is nothing and signifies nothing, but when placed next to arithmetical letters, depicts a number ten times greater than the letters. In the same way, humility considers itself to be as nothing to the eyes of men, but when it is 9 In two poems “To the Countesse of Bedford” : “Madame, Reason is our Soules left hand...,” 11. 2 and 33, and “T ’have written thus...,” 1. 7. Since "I" in Renaissance English is the first person singular pronoun as well as the numeral I, I find it difficult to read Donne’s aforementioned words to Lucy Bedford, or to Christopher Brooke in the line “Thou which art I (‘tis nothing to be soe)," as not playing with the impossible equation I = O or 1 = 0 [“The Storme,” 1. I: comp, with his doubting “if there be such a thing as I” in “To Sir H.W. at his going Ambassador to Venice,” 1.33]. 10 Hie iacet pulvis, cinis, et nihil, in 1646, at the Capuchin S. Maria della Concezione on Via Veneto in Rome. Approach the altar, look down: you're standing on it. Barberini's plain slab is among the most memorable of Roman funeral monuments of the period, the rest of which are extraordinarily given to ostentation. Another Baroque ecclesiastic fond of the formula is Stefan Iavorskii, the Place-Holder of the Patriarchal See under Peter the Great, although this title implies the zero only in English. See Iavorskii's letters in Maslov, xxiii, xli, as well as his “Elegia supra bibliothecam suam" and hymn to the Virgin Mary. 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. placed next to the letters Alpha and Omega, that is to say when it is taken up for the sake of Christ, multiplies many God-pleasing virtues.11 Placing the zero "next to" Alpha and Omega (in the Greek system, 1 and 800 respectively) confuses notations; also. Alpha and Omega bestow value upon the zero of humility, and not the other way around as the mathematical half of the analogy holds. While translating the example from some Western source as he was wont to do, Polotskii scrambled it up. But we get the picture: humility multiplies the value of all other Christian virtues preceding it. "Humilitas," says Picinelli, "valorem add.it" ["O"]. The fact that the "crooked figure" of the Prologue presents us with a secular variant of the Christian humility trope adds resonance to Shakespeare’s zero, inscribing the passage into the general problem of appearances vs. essences, being vs. seeming. Christian humility, whose expression is "I am nothing” or sometimes, in the early modem period, "I am zero," grounds itself in the fact that things are not what they seem, distinguishing between the order of things according to God and that according to the world. The concept of order here is hierarchical: we are dealing with two competing hierarchies of value in which, like in Hindu-Arabic notation, a decreasing movement in one corresponds to an increasing movement in the other. The classical topoi for humility in the teaching of Christ are Luke 14:11 and Matthew 18:4. In the King James Bible, the latter verse reads: "whosoever [...] shall humble himself [...], the same is greatest in the kingdom of heaven." In the line that first inspired humility's being symbolized by the one, "inter caeteros [numeris] & minimus & primus" [Picinelli, "I"], Jesus says, “many that are first shall be last; and the last first” [Mark 10:31; see also Luke 13:30]. Paul takes this principle and applies it to the opposition of being vs. seeming. For him, the crucifixion contradicts the logic of the world because, for the latter, appearances represent reality adequately and transparently. The divine order is reflected by the social order which at the same time is, in American terms, a meritocracy. According to such logic, !1 „M acJ}eM aTH U bi CMHpeHHe rm c b M e ite M o m h k p o h h j i h U H ^ p o io H 3 o 6 p a 3 y io T , T o r o p a / m , h k o h m ace o o p a 3 0 M im tfcp a b tH C Jie ca.M a c o o o f o HHHToace e c T b , HHHToace 3H aM eH yeT , n p m io a c e H H a s ace n H C b M eaaM apm jjM eTH H ecicH M , b z te c jr r tc p a r h h c j io irac b M e H e H 3 o 6 p a a c a e T . T o h h i o c v m p eH H e ca.w o o c e o e h k o HHHToace b o i e c e x H ejioB enecK H X m h h t c s 6biT H , h o B H e m a npm ioacH T C H eM y k o o h b im rm cM eH e.w ajitfca h O M e ra , eace e c r b , e r / t a X p n c r a p a /m h p h c m j i h m o 6 b iB a e T , yM H o acaeT b c e 6 e M HoacecTBO 6 o r o / n o 6 e 3 H b i x z to 6 p o /te T e J ie H , h m h ace z ty rn a xpH CTH aH CK aa HewapeMeHHO x p a c H T c a h TeM b o ocH O Bbi M ecT n o /m a r a e T C « “ [O b e d D u s h e v n y i, 155; q td . in P a n c h e n k o 181]. P ic in e lli’s a rtic le “O” a ls o h a s s e v e r a l e n tr ie s f o r th e z e r o s i g n if y i n g h u m ility a s w e ll a s , c u r io u s ly e n o u g h , th e o th e r e n d o f the spectrum : v a n ity , c o r r u p tio n o f th e w o r ld , a n d th e p ro p e n s ity o f v e n ia l s in s to m u ltip ly . T h e e n trie s on sin r e m in d o n e o f Sir T h o m a s B r o w n e ’s w o r d s o n th e s a m e s u b je c t: “ a s [s in s ] p r o c e e d , th e y m u ltip ly , an d like fig u r e s in a rith m e tic th e la s t s ta n d s fo r m o r e th a n a ll th a t w e n t b e f o r e it" [113]. N o te th e u n - S h a k e s p e a r c a n u s a g e o f " la st" a n d " b e fo re ." 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "the lieutenant is to be saved before the ancient" [Oth. 2:3:102]; it is patently impossible that God, who belongs in the front of the sequence, should pop up somewhere in the middle of history, live in poverty and obscurity, and die the death of a slave. Reality here is at odds with appearances. To Paul, however, reality is at odds with appearances, for the logic of God is unknowable \j'Quis enim cognovit sensus Dei?" see Rom. 11:33-4]; in fact, the only thing one can know about it, is that it cannot be identical to the logic of the world, for then it would be knowable. So for Paul, in the "scandal" of the Cross [/ Cor. 1:24], God gives the exemplary sign of the difference between the predictable worldly and the unpredictable divine orders. This sign is precisely that of the least (according to worldly wisdom) being the greatest (according to divine wisdom).12 To signify God's supercession of worldly logic, Paul uses the shocking phrase stultitia Dei, “the foolishness of God” [/ Cor. 1:18-25]. We might call it the Christian preposterous. Paul converts it into a logical algorithm.13 His theology of the Cross sets up a series of binary oppositions such as Jew vs. Christian, Law vs. Grace, Flesh vs. Spirit, corpus animate vs. corpus spirituale, works vs. faith, and so on. The first of the pair is prior in the eyes of the world: prior in time, prior in importance. In the eyes of God, however, the last acquires precedence over the first: the "successors" preposterously are placed before the "ancestors" [comp, with MWW 1.1.12-3]. The type for this preposterous overcoming is Esau and Jacob, with the result that to each pair may be applied the words of the Vulgate that "major serviet minori"[Gen. 25:23; quoted by Paul in Rom. 9:12', King James: "the elder shall serve the younger;" see this motif in, esp., Merchant o f Venice', we shall encounter it in the genealogical backdrop to Henry V]. What does this have to do with Hindu-Arabic notation? Paul, needless to say, had no access to it, although as a Jew working in Greek he must have been rather familiar with issues of leftward and rightward writing. Yet, since in the early modem period zero became the emblem of humility, and since humility is grounded in the problem of appearances, we should see how the metaphor may be developed. Clearly, the order of rank is the order of appearances: we read left to right, with the greater ranks first. But since the last and "little" place plays the greatest role in the signification process, the left-to-right hierarchy that we see is a posterior event, enabled by the right-to-left working of the sequence of place. The '■ Note that this is a sign of difference, not the universal rule of value. Were the least always the greatest, we would have a paradise reserved for the proletariat, a purgatory for the middle class, and hell for the rich. But then divine wisdom would simply be the inverse of the worldly, i.e. knowable. IJ Making the foolishness of God into a logical algorithm is proper only if it does not apply to all uneven pairs, only if it is not, strictly speaking, an algorithm. If it is, appearances only seem to lie, for we can deduce God’s choice from them. If we can do so, we know the mind of God. 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. importance allotted to the last place can surely be put forth as illustration of major serviet minori, of the posterior superceding the prior. Can the order of place be identified with the "true" hierarchy, hierarchy secundum Deum? No. The divine order of being is ontologically prior to the order of seeming. We may think of the former as a Pythagorean taxis, wherein essences are identified with numbers, and priority among them is distributed according to the natural number sequence. Permit me the following reductionist example. If God says Lazarus is 17 and Dives is 58, Lazarus is ontologically prior to Dives. Paul points out, however, that in the world of appearances Dives may look like a 17, and Lazarus may look like a 58. Dives seems as if he were ontologically prior to Lazarus. But the principle of major serviet minori means that essential priority may belong to Lazarus after all. Hindu-Arabic numerals have nothing to do with ontology: the order of place is not ontologically prior to the order of rank. If we say the order of place is prior, drawing an analogy between it and the order of essences, it is only because the order of place enables signification, and because its workings run contrary to the order of (signifying) appearances, i.e. of rank. But both orders are merely rules for writing, and that which is written is appearances, "mock'ries.” Moreover, the process of signification in Hindu- Arabic numerals is entirely different from the one foreseen by Plato or Paul. The "crooked figure" of the "little place " does not point to a separate prior signified "nothing" which is precedes the separate prior signified of the leftmost numeral I, "million." The signified is posterior to the totality of signifiers. Numbers are posterior to their signification by numerals.14 2. Place Makes the Man To symbolize humility by zero is to promise its eventual triumph. "Humilitas," Picinelli says, "magnos facit" ["O"]. One may overhear a deep-seated disingenuousness in the "crooked figure" lines: yes, we are actors, we are nothings, but it is we - and your "imaginary forces," of course - that make the representation of the "great accompt" of history possible. This puts a modem spin on a common topos. Homer is a mere turtle before Achilles, but, as Pound-Propertius exclaims, "Small talk O Ilion, and O Troad [...] If Homer had not stated your case!" [Pound 206]. 14 I a m n o t c la im in g th a t th e r e is n o p r io r n u m b e r c a lle d " m illio n " - w e c a n h a v e a m illio n th in g s a n d p e rh a p s th e y a d d u p to a m i ll io n b e c a u s e th e r e " e x is ts " a p r i o r n u m b e r c a ll e d m illio n , o r p e rh a p s fo r s o m e o th e r r e a s o n - b u t th is p r io r n u m b e r is in n o w a y im p lic a te d in th e s i g n if i c a ti o n p r o c e s s o f th e n u m e r a ls , as it w o u ld b e in R o m a n n o ta tio n . 200 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. A much more ambivalent version of this type of irony occurs on a different level. It is thanks to the order of place that Hindu-Arabic notation presents the signifier as prior to the signified. Let us therefore attempt to read the lines with respect to social place. After all, one of the problems that the “imaginary forces” of the audience must compensate for is the social disproportion between “the flat unraised spirits” and their “so great an object” [pr. 9- II). There is already a marker of social irony in the Prologue's referring to the audience as “gentles all,” since it assuredly is not all gentles. The "crooked figure" lines fully live up to it. Socially, the only possible meaning of “A crooked figure may Attest in little place a million” is that the end of the hierarchy is the source of the power of the head. Like a group of numerals making up a number, social hierarchy has two contradictory orientations: that of having value and that of bestowing value. The rightward orientation is the apparent, the sequence of king preceding nobles who precede commoners in the decreasing order of rank: social magnitude and possession of power. The "crooked figure" lines imply, however, that the origin of power is not the king, but those in the “little place.” The orientation of bestowing power is leftward. The presence of subjects makes kings kings. Commoners make nobles nobles, and kings kings. And yet it does not look that way: letting your imaginary forces work, you shall see the rightmost figures as literally “crooked” because they bow before the greatness of the I. I understand that this reading might seem to come out of the left field. I certainly do not intend to argue that Henry V contains some “progressive” message in the vulgar sense of the word. Commoners do not make kings kings by any kind of willed action, but merely by occupation of place. As Lear was to learn, to be a king, a king has to have subjects: without them, he is a thing of nothing. Although I am trying to keep other Shakespeare plays out of this chapter, I hope I may be excused if I mention several lines in Lear because of their centrality to Shakespeare's treatment of signification. When Lear descends into folly, he occasions to see: not transcendentally, of course, but that he cannot see to see [see 4.7.60]. Yet foolishness15 also permits him to register 15 The reader no doubt recalls that Paul also uses the "foolishness of God" as the model for his own behavior, constructing his own rather paradoxical version of humility therewith (“Imitatores mei estote. sicut et ego Christi," I Cor 11:1). On the epistemological level, Paul's foolishness is a doctrine of self in the world of appearances, "mock’ries" that fool the observer by seeming to reproduce the order of essences. Because they fool, everyone is a fool; the best one can do is recognize oneself as such. The Pauline fool can see that appearances lie, and even how they lie, but not what they belie. Quite a lot may be written about Shakespeare’s employment of Pauline foolishness in King Lear, with its problem as to what constitutes a "natural" sequence of appearances, its multiplication of nothing from nothing, and its coupling of the Fool with the "O without a figure" [1.4.183-5', fool, "cipher" and appearances also juxtaposed in As You Like It 3.2.285-6]. 201 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. how appearances work: the general scandal of signification, in which the order of signifiers creates signifieds and not the other way around: See how yon justice rails upon yon simple thief. Hark in thine ear: change places and handy-dandy, which is the justice, which is the thief? Thou has seen a fanner’s dog bark at a beggar? [...] And the creature run from the cur - there thou mightst behold the great image of authority: a dog’s obeyed in office [4.6.147-55]. Lear's words bear a direct correlation to Henry's “For what have kings that privates have not too. Save ceremony, save general ceremony?” [H5 4.1.240-1]. What Lear calls "place" ("change places, and handy-dandy") manifests itself, in the case of kings, in what Henry calls "ceremony": display, appearance. We are, after all, dealing with a society where class distinctions are reproduced in behavior and dress, the latter reinforced by statutory laws. The place of commoners, for instance, shows itself in whatever common thing they happen to be wearing, or whatever common way they happen to be speaking or behaving that spells “commoners” all over them.16 Place makes itself known through the senses. Customs here are the equivalent of costumes; both are types of appearance. A metaphor using the latter may intimate the former as well. Since kings are the same as privates "save ceremony," social hierarchy turns into empty theatrical robes hanging on a line, robes that express and determine the place of their wearers. What is the essence of kingship? What makes a king a king? On the social level, we have answered that question: the presence of subjects. Let us now disregard subjects and look at it in terms of all other appearances. Lear again. After Lear has given away "rule, interest of territory" [1.1.49-50], the Fool says: "Now thou art an O without a figure; I am better than thou art now. I am a fool, thou art nothing" [1.4.183-5], Let us try to envision this other depiction of kingship employing Hindu-Arabic numerals: Lear as king IO Lear as thing O The Fool's words reproduce what may be called Lear's divestiture [see 1.1.49], with the division of the represented and real Albion, the removal of "rule, interest of territory," subjects, etc., brought to its logical conclusion in Lear's tearing at his robes: "Off, off, you lendings!" [3.4.106]. Naked, he is "the thing itself, [...] a poor, bare, forked animal" [Lear 16 Any West Country peasant who speaks the dialect Edgar puts on when he murders Oswald [Lr. 4.6] - “zir” for “sir,” “zo” for “so” and so on -- upholds the king by his very use of that “whoreson zed,” that “unnecessary letter” [2.2.62]. 202 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5.104-6].17 Lear's self - whatever we may mean by that word - remains; what is removed, and what therefore the numeral "I" symbolizes, is "mock’ries," "lendings," appearances. It is therefore they that are the root of power: they make Lear king. Do appearances seem to make Lear king? Surely not; appearances seem proper to Lear because he is a king. They seem to follow him; they make him look like the "I" and themselves as ciphers. His control over territory, his wearing a crown appear to be predicated on his being every inch a king. His kingship seems essential, they accidental. Thus the "million" of "crooked figure" lines may also be said to represent active kingship as it appears, with the king as the I, and subjects, clothes, territory, etc. as 0 0 0 , 0 0 0 . In Henry V, the trope arrives in its habitual left-to-right form, stressing the orientation of rank. In Lear, the Fool reverses its assignations, portraying in the left-to- right manner the "real" priority of appearances to self, of seeming to being. In the "O without a figure" lines, a king is he who has the appearances of a king. What happens if in Henry V we remove all that follows the king, if we divest Henry in the same way as Lear? The seeming result is not an “O without a figure,” but the figure, the one, the I. Does this mean that, even alone, Henry differs from his subjects? No. The I alone is no I. The I is only another place in the sequence. If there is no such thing as two, no such thing as three, no such thing as four, etc., in other words, if there is no sequence, there is no one. So poor, bare and bereft of all relations, the two kings are indistinguishable, each both an I and an O. As the eighth Sonnet promises, “Thou single wilt prove none." 3. The Genealogical Line, Right and Sinister Kings, needless to say, attribute their right to rule neither to ceremony nor to the presence of subjects. Characters in Henry V put forth an entirely different argument for why a particular person ought to occupy a particular throne, and it is to this argument and its "proof' that I shall devote this and the next section. In Henry V - and, of course, in other texts of the period —a king derives his right to rule from his ancestors, or rather from God through his ancestors. Both claims involve that which is prior determining that which is posterior: the former on the historical level, with the crown passing from older to younger, from father to son; the latter on the metaphysical, with divine truth reproducing itself in appearances. My claim in both sections will be that, rather than the prior choosing the posterior, it is, paradoxically, the posterior that chooses the prior. 17 We duly note the synonymity between "the thing itself' and "O without a figure." In a seeming paradox, Lear's "thing itself' is the also human body, i.e. another appearance. 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Henry V is a play concerned with legitimacy: primarily with the legitimacy of Henry’s claim to France, but also with the legitimacy of his claim to England. The latter intrudes in the shape of the revolt of Cambridge, the genealogical motives for which were expressed in the Henry IV plays, “which oft our stage hath shown” [5.ch. 13', see Parker 40-41 on revolt of Cambridge and 1H6 2.5.73-92]. The two problems are closely related in that Henry’s claim to France is rather like the Mortimers’ claim to England. Act 1, scene 2 of Henry Kestablishes Henry’s “right” to the French crown. The word “right” is quite prominent in the scene. Henry delegates authority to the Archbishop of Canterbury to determine whether his claim may be made “with right and conscience” [96]. Should Canterbury’s Salic Law speech open “titles miscreate, whose right Suits not in native colors with the truth” [16-7], the responsibility for the blood spilled in the consequent campaign will fall on the head of the Archbishop [13-32, 96-97], The Archbishop shows that Henry has the right and now must win it from the French [131]. Henry’s right depends on the rightness of the genealogical sequence, of what the Henry plays refer to as “line” [e.g. 3H6 1.1.19; 1.3.32; see also "lineal" in 1H6 3.1.165; R3 3.7.200; 2H4 4.5.46]. We may regard legal “right” and genealogical “line” as expressions not wholly divorced from mathematical imagery. The ideal, patrilineal “line” can be modeled on the sequence of natural numbers, as accentuated in cases where the king’s name bears the generic character of familial marker (Henry IV - > Henry V -> Henry VI, also Hamlet] -> Hamlet2, FortinbraS] - > Fortinbras2). It may also be envisioned as a geometrical straight line, in which the “right” to the crown is bestowed rightwards, from the ancestors to the successors.18 king]----------->king2----------->king3----------->king4----------->king5 This mathematization of the concept of “right by line,” i.e. entitlement by descent, may be justified by puns found in the play: so Henry speaks of enforcing his right to the French crown as putting forth his “rightful hand in a well-hallow’d cause” [293]. An illegitimate claim, on the other hand, is seen as a leftward movement. So Exeter shows the French King the “most memorable line,” i.e. the genealogical tables on which Henry’s claim is based, in order that the French King may see the claim is right and not “sinister,” i.e., leftward [2.4.84-95]. Henry V is of course a history play, and so we may permit ourselves to say a few words about the genealogical background of the historical personae portrayed therein. It 18 Here “line” and “right” are semantically connected, with the original meaning of right being “straight” (“right” became the antonym of “left” only in the thirteenth century; see Merriam-Webster, "right"). 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. may be safely assumed that the English genealogies were known to the audience; should there have been the clueless rara avis, Shakespeare’s other history plays would have filled in the blanks. A bird’s eye view of French genealogical history is given in the Salic Law speech, although it curiously omits the problem at hand, i.e. the Valois inheritance of the Capetian crown. Since both the French and the English genealogies are quite confusing to the modem reader, I thought it best to include charts. I should warn that they are incomplete, feauturing only those personages important to the conflicts of Henry V. Let us start with the French genealogies: CAPETIANS + VALOIS Philip III r-1270-85 Philip IV Charles, C. of Valois r. 1285-1314 d. 1325 I ____________________________________________ I Louis X Isabella Philip V Charles IV Philip VI r. 1314-16 m. Edward II r. 1316-22 r. 1322-28 r. 1328-50 of England I f PLANTAGENETS I Charles VI. r. 1380-1422 Henry’s right “to some certain dukedoms. And generally to the crown and seat of France” derives from the right of his great-grandfather, Edward HI [1.1.86-8], son of Isabella, the only survivor of the Capetian line to have male offspring. After Isabella’s brothers died leaving only daughters, the crown passed to their cousin, Philip VI Valois. The translation of the crown from older to younger brothers (Louis X and John I -> Philip V -> Charles IV) already complicates the simplicity of the line. If the older brothers are survived by female offspring, the translation is possible only if women are denied the right to rule. However, unless women are also prohibited from passing the right to their male hairs, the male progeny of the daughters of the older brother may argue that they, and not his younger brother, should inherit the crown. In this case, the line becomes a fork. The “right” to the throne of Philip VI, the first Valois king, was based upon the prohibition of matri lineal succession, adopted upon his election by the French nobles and later antedated as the Salic Law that Canterbury debunks.20 Without such law, there was no legal reason 19 Louis X did have a male heir, John I, but he died upon inheriting the throne. All other Capet children were female. :o Prohibition of matrilineal succession consisted o f two principles: that women cannot inherit the throne, and that they cannot pass the claim to the offspring. The first was used by Isabella’s younger brothers to inherit over the daughters of the older, the second was used by the Valois line to invalidate the potential claims of Capetian descendents, including Edward m. Both o f these principles were new-fangled: the Capetian line until Isabella’s siblings had a continuous succession of male heirs, and the possibility of 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. preventing the French crown from passing to the older line of Isabella’s English son, Edward III. The English situation is as follows: PLANTAGENETS Edward II r. 1307-27, m. Isabella o f France Edward HI r. 1327-77 Edward the Black Lionel, D. of Clarence John of Gaunt Edmund of Langley P. of Wales, d. 1376 d. 1368 D. of Lancaster, d. 1399 D. of York, d.1402 I I I I Richard II Philippa Henry IV Richard, E. of Cambridge r. 1377-99 m. Edmund Mortimer 1399-1413 d. 1415; m. Anne Mortimer-1 MORTIMER (+ YORK) Henry V YORK (+MORTIMER) The dynastic struggles in the history plays occur among the descendants of Edward III. Only four of Edward’s sons are important here: Edward the Black (b. 1330), Lionel, Duke of Clarence (b. 1338), John of Gaunt, Duke of Lancaster (b. 1340), and Edmund of Langley, Duke of York (b. 1341). Edward the Black died before his father, and so the throne was inherited by his son, Richard II. Richard II was deposed and succeeded by Henry IV, the son of John of Gaunt. However, the crown was also claimed by the Mortimers, matrilineally descended from Lionel, Duke of Clarence, older brother to John of Gaunt. This story is summarized in 7 Henry VI by the imprisoned and aged Edmund Mortimer to the son of Richard, Earl of Cambridge executed as traitor in Henry V: Henry the Fourth, grandfather to this king, Deposed his nephew Richard, Edward’s son, The first-begotten and the lawful heir Of Edward king, the third of that descent; During whose reign the Percys of the north. Finding his usurpation most unjust, Endeavored my advancement to the throne. The reason moved these warlike lords to this Was for that —young King Richard thus removed, Leaving no heir begotten of his body - I was the next by birth and parentage; matriiineal succession never came up. Some time after it was passed at the election of Philip VI Valois, the law barring matriiineal succession became attributed to the legendary Frankish king Pharamond, mentioned in the Salic Law speech. For more on genealogical problems behind the Hundred Years War, see Anne Curry 44-46 and tables pp.xii-xiv, on which my tables are based. 11 The great-granddaughter of Clarence through Philippa and Edmund Mortimer. Shakespeare's Mortimer is her brother. Holinshed and Shakespeare also identify him with his uncle, Edmund Mortimer, 5th Earl of March, historically Richard's appointed heir. 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For by my mother I derived am From Lionel, Duke of Clarence, third son To King Edward the Third; whereas he From John of Gaunt doth bring his pedigree. Being but fourth of that heroic line. But mark. As in this haughty great attempt They labored to plant the rightful heir, I lost my liberty and they their lives. Long after this, when Henry the Fifth, Succeeding his father Bolingbroke, did reign Thy father, Earl of Cambridge then, derived From famous Edmund Langley, Duke of York, Marrying my sister that thy mother was, Again in pity of my hard distress, Levied an army, weening to redeem And have installed me in the diadem. But, as the rest, so fell that noble earl And was beheaded. Thus the Mortimers, In whom the title rested, were suppressed [2.9.63-92]. In both the Valois and the Lancaster examples we see that the rightward movement of the genealogical line is ensured only by a historically posterior and “sinister” action, i.e. by the prevention, whether de facto or de jure, of succession by an older but matriiineal line.22 We may speak of the passing of the English and French crowns to the Lancasters and the Valois as opposed to the Mortimers and the Plantagenets with the Biblical terms of “the elder shall serve the younger,” major serviet minori. In both cases, however, the elder line a) is matriiineal rather than patrilineal; b) actively disputes the right. The result is that anyone’s claim to the crown by means of “linear descent” [1H6 3.1.165] is contestable.23 In addition to the revolt of Cambridge and Henry’s mention of his father’s “fault” in 4.2.299, another incident in the play Henry V intimates to what extent the question of Henry’s legitimacy to the English crown is central to the play. The “memorable line” borne to the French king by Exeter demonstrates not Edward Hi’s right to the French throne, which the situation calls for, but Henry’s right to the English. “Overlook his [Henry’s] pedigree,” Exeter says: “ When the historical Henry IV came to the throne, he considered outlawing female succession as the Valois were to do ten years later, but laid the idea aside in order to not compromise his claim to the French crown [Curry 92], 23 The English situation is complicated yet further by the fusion o f the York line, younger than the Lancaster, with the matriiineal Mortimer line, older than Lancaster, the York line inherits the Mortimer claim, again matrilineally. Since the York matriiineal claim to the crown is strengthened by the line’s patrilineal descent, we may say that the necessary purity and singularity o f rightward motion in the concept of “right descent” is additionally discredited. 207 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. And when you find him evenly deriv’d From his most fam’d of famous ancestors, Edward the Third, he bids you then resign Your crown and kingdom, indirectly held From him the native and true challenger \2.4.91-5\. Henry’s “demonstration” of his “no sinister nor no awkward claim” utterly ignores the Valois objection to the matriiineal claim of any king of England to be the king of France [for demonstration that does focus on matriiineal issues, see the possibly apocryphal poetic debate between Edward HI and Philip VI in Puttenham 9]. There is no reason why recognition of Henry’s right to the throne of Edward HI of England would make Charles VI Valois “resign” his “crown and kingdom” in Henry’s favor.24 What we would have expected Henry to have done is to send the French King something like the Salic Law speech. This at least would have disproved the French claim that prohibition of matriiineal succession is ancient French law, leaving the field open. That Henry does not such thing is a notable incongruity, which can only be explained in psychological terms. One may overhear a measure of irony in Exeter’s request that the French king “overlook?' the “pedigree” of his “native and true challenger” (first OED example of “overlook” as fail to see dates from 1524). But just as striking is the subsequent exchange between the French King and Exeter. “Or else what follows?” asks the former. “Bloody constraint,” replies the latter. Before we consider the ramifications of "what follows," let us attend to the Salic Law speech, the only example in Henry Vof a “right by line” scrutinized. Although the line or rather lines in question are those of the French crown, what the speech intimates about the general nature of genealogical claims also casts a shadow upon the English problems. Its is best we quote Canterbury’s words at full, so that the reader need not constantly circulate between our text and Shakespeare’s. There is no bar To make against your highness' claim to France But this, which they produce from Pharamond, In terram Salicam mulieres ne succedant, “No woman shall succeed in Salic land:” Which Salic land the French unjustly gloze To be the realm of France, and Pharamond The founder of this law and female bar. Yet their own authors faithfully affirm That the land Salic is in Germany, 24 Rather, the obvious counter-argument would be: if Henry has the right to the English throne, the Valois have the right to the French. The Valois are certainly aware of the Mortimer claims, which they support financially [2.ch.20-2]. 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Between the floods of Sala and of Elbe; Where Charles the Great, having subdued the Saxons, There left behind and settled certain French; Who, holding in disdain the German women For some dishonest manners of their life, Establish’d then this law; to wit, no female Should be inheritrix in Salique land: Which Salic, as I said, 'twixt Elbe and Sala, Is at this day in Germany call'd Meissen. Then doth it well appear that Salic law Was not devised for the realm of France: Nor did the French possess the Salic land Until four hundred one and twenty years After defunction of King Pharamond, Idly suppos’d the founder of this law; Who died within the year of our redemption Four hundred twenty-six; and Charles the Great Subdued the Saxons, and did seat the French Beyond the river Sala, in the year Eight hundred five. Besides, their writers say, King Pepin, which deposed Childeric, Did, as heir general, being descended Of Blithild, which was daughter to King Clothair, Make claim and title to the crown of France. Hugh Capet also, who usurped the crown Of Charles the duke of Lorraine, sole heir male Of the true line and stock of Charles the Great, To find his title with some shows of truth, Through, in pure truth, it was corrupt and naught. Convey'd himself as heir to the Lady Lingare, Daughter to Charlemain, who was the son To Lewis the emperor, and Lewis the son Of Charles the Great. Also King Lewis the Tenth, Who was sole heir to the usurper Capet, Could not keep quiet in his conscience, Wearing the crown of France, till satisfied That fair Queen Isabel, his grandmother, Was lineal of the Lady Ermengare, Daughter to Charles the foresaid duke of Lorraine: By the which marriage the line of Charles the Great Was re-united to the crown of France. So that, as clear as is the summer’s sun, King Pepin's title and Hugh Capet's claim, King Lewis his satisfaction, all appear To hold in right and title of the female: So do the kings of France unto this day; Howbeit they would hold up this Salic law To bar your highness claiming from the female, And rather choose to hide them in a net Than amply to imbar their crooked titles Usurp'd from you and your progenitors [1.2.35-95]. 209 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The French claim that the Salic Law against female succession was instituted by the Frankish king Pharamond and that, ostensibly, it was in continual force since then. Canterbury’s speech proves them wrong on several counts: A) The Salic land is not France at all, but “in Germany, Between the floods of Sala and of Elbe:’’ and therefore the law “Was not devised for the realm of France,” and does not apply there. B) As if that weren't enough, the French only captured Salic land under Charlemagne, “four hundred one and twenty years After defunction of King Pharamond,” and therefore the law can have nothing to do with Pharamond. C) In any case, French genealogical history has been in constant disobedience of the presumed law, and both the Carolingian and the Capetian dynasties legitimated themselves matrilineally.25 It is therefore “clear as is the summer’s sun” that if King Pharamond did bar women from the right to pass on succession, all French Kings from Pepin onwards would be usurpers. By implication, if what the Valois claimed were true, not only the Plantagenets but also the Valois themselves have no right to the French throne. At the same time, the legal justification for Henry’s claim to France provided by the Archbishop of Canterbury is “in the book of Numbers it is writ: “When a man dies, let his inheritance. Descend unto the daughter” [98-100], If this be applied to the English situation, one may argue that the crown should have gone from the stock of Edward the Black (i.e. Richard II his childless son) to the daughter of Lionel, Duke of Clarence, married into the Mortimer line, rather that to the stock of his younger brother John of Gaunt, i.e. Henry IV. This is what the Mortimers do argue: after the removal of Richard, says Edmund Mortimer in 1 Henry 6, I was the next by birth and parentage; For by my mother I derived am From Lionel, Duke of Clarence, third son To King Edward the Third; whereas he [Henry VI] From John of Gaunt doth bring his pedigree. Being but fourth of that heroic line [73-8]. 25 So “King Pepin" the Short, who “deposed Childeric,” made “claim and title to the crown of France” through matriiineal descent from an earlier Merovingian king, Clothair, thus founding the Caroiingian dynasty. So also Hugh Capet, “who usurped the crown Of Charles the duke of Lorraine, sole heir male Of the true line and stock of Charles the Great,” claimed descent from a Carolingian princess. Hugh's claim was mendacious, and so the Capetian line became “legitimate” only subsequently, by marriage of a descendent king to the heir of a female heir to the Carolingian line. Thus the legitimacy of the Carolinegian line is matriiineal and the Capetians descend from it doubly matrilineally, with kings from Hugh Capet to Phillip II or III (error in text) either usurpers or kings preposterously —through their descendants. 210 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We might object that the key legal difference between the Valois-PIantagenet and the Lancaster-Mortimer situations is that Isabella is the sister of the last Capetian kings, whereas the struggle in England occurs between cousins. Numbers 27.8 would therefore apply to the former and not to the latter.26 But Shakespeare’s history plays don’t seem to make this distinction. And so, one of the things “which oft our stage hath shown” is Henry VI, the son of Henry V, admitting that his “title’s weak” [3H6 1.1.135], Therefore, as far as the history plays are concerned, if what Henry V claimed for France were applied in England, his right to the English throne would be questionable at best. We may therefore consider it not only as historical accident, but also as incident to the history plays, that the actions of “Pepin, which deposed Childeric” and founded the Carolingian line, and Hugh Capet, who “usurped the crown Of Charles the duke of Lorraine” and founded the Capetian line, are not unlike those of Henry IV, who deposed Richard and founded the younger Lancaster line. The differences - that Pepin is matrilineally descended from the Merovingians, that Hugh Capet is an outright usurper, and that Henry IV is descended patrilineally from younger stock - pale in comparison with the constant of deposition of the rightful heir that lies at the origin of all genealogical lines mentioned in the play. The Salic Law antedates itself. The charge of antedating is implicit in Canterbury's speech, since he proves that Pharamond could not have been the author of the law; and that the two prior dynasties, the Carolingians and the Capetians, put forth matriiineal claims. Therefore the Valois legitimate themselves in a “sinister” manner. Their genealogy is not cooked up; they do descend from the Capetians. What is cooked up, however, is their “right,” their ciaim that the crown had to pass to them, and not to somebody else. Their “right” comes not rightwards from the ancestors, but leftwards from the descendants. In their "righting" of genealogy, the Valois resemble Pepin, Hugh Capet, Henry IV. The Salic Law is also - at least ostensibly - a piece of writing. We cannot bypass this connection between writing and the presentation of that which is posterior as if it were prior. Writing, after all, is appearances, and has, since Plato [Sophist 252e-3a, Statesman 278d\, been employed to conceptualize the workings thereof. Perhaps the antedating of Salic Law implicates all types of writing, and maybe all types of signs, in a sinister movement, as a kind of "righting." As an example of what I mean, let us take the French recognition of Henry’s “right” in Act 5. :6 Actually, Numbers 27:8 says, "If a man die, and have no son, then you shall cause his inheritance to pass until his daughter" [my italics]. Canterbury's omission of lack of male heir is most strange, for it would have distinguished the Plantagenet claim to France from the Mortimer claim to England. 211 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As I already mentioned, when Exeter brings Henry’s “memorable line” to Charles VI, the latter asks, if I do not recognize his right, “what follows?” “Bloody constraint,” answers Exeter. And indeed, both in the temporal and causal senses, what follows - bloody constraint - establishes what preceded it, whether “the right” had always belonged to the Plantagenets or the Valois, belonged to them prior to and independent of the constraint. Here the English bloody constraint is the exact counterpart of the French Salic Law. Both erect what I called the posterior signified, and neither has any actual relation to the prior signified: the Salic Law because it contradicts French history, and bloody constraint because the principle of the debate is not might makes right, but the reverse. At the same time, both bloody constraint and Salic Law are appearances, and the posterior signifier they establish determines the meaning of the appearances that went before them. Only then does the meaning of what precedes become “clear as is the summer’s day.” Thus the "crooked figure" lines resonate also throughout the genealogical background of Henry V, with the sinistrorsum orientation of place recalling the "sinister" movement along the genealogical "line." Assembling his ancestors and writing the rules of descent in such a way that the crown seemed destined to pass to him, the "crooked figure" of the heir [for "crooked figure" as usurper, see section 5] presents his handiwork as if it were a prior fact. His "righting" consists of a leftward creation of a line seeming to move rightwards and to culminate in his "little place." As Russian poet Alexei Parshchikov says - in a line modeled on “he who laughs last, laughs best” - “he who was bom last, makes history.”27 4. Bloody Constraint The argument of entitlement by "linear descent" [1H6 3.1.165] is not merely historical, but also has a metaphysical dimension. Henry repeatedly insists that, if his line is not "sinister," God will support his undertaking. This defines the line as an object of divine knowledge, apprehended by God in an atemporal manner, prior to its appearance as a sequence of sensibles [i.e., kings; for divine knowledge, see chapter 6]. Henry is not so naive as to claim that the "right" line in the mind of God automatically reproduces itself in appearances, for then all kings, including the Valois, ruled by "right." No, but God does help those who, being in the right, deliver their "puissance into" his "hand" [2.3.190], putting their own dextra "in a well-hallow'd cause" [1.2.293], :7 „A H C T o p m o aejiaeT t o t , k to pojnuicx nocjieaHHM“; in his poem “la zhil na pole Poltavskoi bitvy,” pt. 2.3, “Karl.” 212 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Hence Henry’s continual insistence that in capturing the French crown he serves as the instrument of the Lord. Among the many statements to that effect we may list his speech to the French ambassadors, with its appeal to will of God, “in whose name Tell you the Dauphin I am coming on, to venge me” [ / .2.289-92], The command for the French King to “lay apart The borrowed robes” also issues "in the name of God Almighty” [2.4.77-84], Henry claims that his "weak and sickly guard" marches with "God before" it [3.4.164]. The English pray that “God’s arm strike with us!” [4.3.4] and shout “God for Harry, England, and Saint George!” [3.1.34]. This, to be sure, is just lip-flapping. The real coup de grace comes with Agincourt. It is on Agincourt that Henry's metaphysical fortunes —and the play's argument (in the literary sense of the term) - are staked. Henry is at pains to ascribe the victory “to God, and not our strength” [4.7.89]; to ensure that everyone, both his present and future subjects, interpret it as an act of God attesting his superior right to the French crown. Here Agincourt provides the "ocular proof' [Oth. 3.3.366] of Henry's pedigree, its truth becoming all the more evident as the victory was improbable and miraculous, the “odds” so “fearful” [4.3.4], and the losses so uneven. After driving the latter point home by the protracted reckoning of the captured and the dead [4.8.78-108], Henry exclaims: HENRY: O God, thy arm was here; And not to us, but to thy arm alone, Ascribe we all! When, without stratagem,28 But in plain shock and even play of battle, Was ever known so great and little loss On one part and on th’other? Take it, God, For it is none but thine! EXETER: Tis wonderful! HENRY: Come, go we in procession to the village. And be it death proclaimed through our host To boast of this or take the praise from God Which is his only. FLUELLEN: Is it not lawful, an please your Majesty, to tell how many is killed? HENRY: Yes, captain; but with this acknowledgement. That God fought for us [4.8.108-22]. All of this is to imply that Agincourt is not your everyday "bloody constraint." If Henry V were a Hollywood movie, the against-all-odds victory of the "weak and sickly guard" would illustrate the triumph of the underdog, with the moral lesson reinforcing primarily economic pieties: you can make it big if you persevere and so forth. But it is a Renaissance portrayal of the fifteenth century, and the employed ideology is religious. The sorry state of the English army, the pre-battle scenes of uncertain English footsoldiers :* For elided stratagem, "at this time first invented," see Holinshed, "Henrie the Fift," 34-7. 213 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. contrasted against the flower of French chivalry glorying avcmt la lettre ["my way shall be paved with English faces," etc., 3.7.83-4], Henry's post-battle desire to ascribe the victory to God - all of these feed into the humility triumphant trope.29 As we have seen, Christian humility implicates the problem of appearances. The "humble" Henry frames the sensible battle as the "ocular proof' of his supposed prior right. The intangible superiority of his claim, as it were, produces the appearances that are the battle, eventually causing another appearance: that of his son becoming King of both France and England. These appearances "attest" [pr.l6\ to his nonsensual "right by line." In claiming merely to deliver his "puissance into the hand of God" [2.3.190], Henry allows God to ensure, by means of history, that the prior signified ("right by line") manifests itself as the posterior signifier (Agincourt). If Agincourt “proves” Henry’s right to the French crown, the arrest of Cambridge et al. “proves” his right to the English. Once again Henry ascribes the leading role to God, who “so graciously hath brought to light This dangerous treason” [2.3.185]. Modeling the trial on the Last Judgment and himself on Christ as God's instrument, the King fills his speech with biblical echoes, from the “fall of man” to “Get you therefore hence, Poor miserable wretches, to your death” [2.3.178-9, comp, with Mat:. 25:41-, see also Henry's apocalyptic contemplation of mercy vs. justice, 166-81]. The Last Judgment is supposed to "reveal" and make "known" everything that has been "covered" and "hid" [Luke 12:2]: in the great drama that is history, it is the Recognition Scene, the Last Show and Tell, the culminating display of all that has so far escaped sight. Each of the accused - i.e. everybody - reveals his darker purpose, makes his virtues and vices visible to all. In becoming entirely public, the sinner accuses himself, affirming the justice of his punishment [for self-accusation of the body at the Last Judgment, see Ostashevsky 221 - 263; English abstract 287-288]. The idea of the Last Judgment as the manifestation of invisible prior truths by appearances, generates the biblical resonances in the trial of Cambridge, Scroop and Grey. As they will at the final reckoning, the judged recognize both their sins and the justice of the punishment; each of them ascribes the discovery to God (“Our purposes God justly has discovered [...] God be thanked for prevention”) and requests his own execution [2.3.151- 65]. Even the “traitors” appear to accede to the implicit argument that God discovered the treason because the Lancaster claim is better to the Mortimer: thanks to divine intervention, "the truth of it stands off as gross As black and white" [2.2.103-4; Henry on the guilt of Scroop; note motif of writing, i.e. "attesting"]. :9 Note the christological motif in the French pre-battle method for dividing the spoils: "The confident and over-lusty French Do the low-rated English play at dice" [4.chor. 18-9\. 214 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If Henry’s account of his successes is that they offer ocular proof of his prior right, the play’s Epilogue contradicts him flat out with “Fortune made his sword” [6]. So Henry’s successes are due not to God, not to the rightness of his claim, but to the “furious Fickle wheel” of “giddy Fortune [...] That goddess blind, That stands upon the rolling restless stone,” as Pistol most tragico-comically declaims upon the imminent hanging of Bardolph [J.4.27-9]. Do we believe the Epilogue’s dismissal, so brief and cavalier as to be almost unnoticeable? It seems we must, for the Epilogue also refers to the pitiful plight of Henry IV, whose “right by line” to both thrones is of course no worse than his father’s. And so the explanation describes another O: the Ferris wheel of Fortune, the law that what goes up must comes down. In the sequence of appearances where Henry's real genealogy in all its ambivalence is followed by Agincourt, the "little place" determines the value of the signs coming before it, and conveys this nonsensual value to have been the prior cause of the posterior signifying sequence. Once again does "a crooked figure [...] Attest in little place a million." 5. Crooked Figures and Treacherous Crowns When we read or watch the Prologue, we do not immediately comprehend that the words "crooked figure" mean zero. The expression is anthropomorphic, and materializes in our imagination as a hunched human being. Only from the following lines ("Attest in little place a million," etc.) do we realize its denomination. A sufficiently thorough dictionary will tell us that the word "figure" may mean Hindu-Arabic numeral. But what of "crooked"? In this context it obviously intends zero's shape, but why choose that word and not some other synonym of "circular"? Wouldn't the more precise "round" fit the metrical bill? This section will argue that the employment of “crooked” in the Prologue erects a direct bridge between Hindu-Arabic notation and the improprieties of genealogical sequencing in the history plays. The same link implicates the "crooked figure" in stealing; the fact that the object of the theft is "crowns" - a pun uniting symbols of kingship and cold hard cash - will enable us to question the confused motives behind Henry's French Wars. For starters, "crooked" as curved or oblique is the antonym of "right" as straight. In the Bible, “crooked” and “straight” are coupled as geometrical metaphors signifying unrighteous and righteous respectively. So the Ecclesiastes, in a line that to my ear reverberates throughout the genealogical “right by line” problems in the history plays, sighs, “That which is crooked cannot be made straight” [1:15; the Vulgate continues with et 215 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stultorum infinitus est numerus]?0 Since "right" is also the antonym of “left,” “crooked” suffers a certain semantic transference onto the leftward —or "sinister" —movement along the line. The only other use of “crooked” in the play occurs at the end of Canterbury’s Salic Law speech, as applying to the Valois’ "crooked titles Usurped from you [Henry] and your progenitors" [1:2:99-100], In Shakespeare from the Margins, Patricia Parker relates these lines to the "indirect crook'd ways" by which Henry IV had acquired his crown [2 Henry 4, 4:5:184; Parker 44], In his prayer before Agincourt, Henry V refers to these "crook'd ways" as “the fault My father made in compassing the crown” [4.2.299-300; my italics], with “compassing” legible literally as a circling gesture, and therefore opposed to the straight, the right, the linear. The connection between “crooked” and genealogical "righting" by elision of elder stock is strengthened by the crooked figure of another usurper - Richard HI, "the crook-back prodigy" of 3 Henry VI [1:4:75-76; see ibid. 2:2:96, 5:5:30; connection made by Parker 44; see ibid. 24, 43 for “sinister”]. As such “crooked figures,” Hugh Capet “Convey’d himself as heir to the Lady Lingare” [H5 1.2.74] and Henry IV and Pepin declared themselves the rightful heirs to Richard H and Childeric respectively. Canterbury's "convey'd," with the sinister, antedating movement it denotes, is a many-sided pun, also implicating speaking and stealing. Let us focus on the latter. In the Merry Wives o f Windsor, Pistol expounds “convey” to be the euphemism for “steal” [1.3.27-28\ noted in Walter’s ed. of H5; OED, "convey," 6b]. In Richard II, when the “unrightful” king [5.1.63] Henry IV commands the deposed Richard to be “conveyed” to the Tower, Richard puns, “’Convey?’ Conveyers [i.e. thieves] are you all That rise thus nimbly by a true king’s fall” [4.1.317-9; noted by Parker 10 and ch. 5]. The "crooked figure" steals the crown. "Crown" is one of the two most common puns in Henry V (the other is "fault," and we shall get to it soon). Henry himself sanctions our regarding the word as simultaneously referring to symbols of kingship and monetary units, when he quips that “it is no English treason to cut French crowns, and to-morrow the king himself will be a clipper” [4.1.233-5]. We note how Henry's clever "equivocation" [see Ham. 5.1.138] connects the word "crown" to dishonest dealings and theft, with the "king himself' in the role of the agent. 30 So also the “crooked generation” [Deut 32:5] of the unrighteous moves in paths as “crooked” [Isa 598, also Ps 125:5, Prov 2.15] as those made by the “crooked serpent” [Job 26:13, Isa 27:1]. They cannot be corrected until Doomsday, when, Jesus ominously quotes Isaiah, “the crooked shall be made straight” [Luke 3:5, Isa 40:4; 45:2]. While, as I showed in my previous chapter, for Greek philosophers divine motion is circular and human linear, here the valorization is reversed. Crookedness, circling —and with them the zero - bear negative and fallen connotations. 216 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Both types of crowns appear in the Chorus of Act 2. While “all the youth of England are on fire [...] and honour’s thought Reigns solely in the breast of every man,” this lovely and exclusive piece of idealism finds itself the bedfellow of expectation of titles to be won: “crowns imperial, crowns and coronets, Promised to Harry and his followers” [ / - // ] . In the same speech, France has “found out A nest of hollow bosoms, which he fills With treacherous crowns,” i.e. money [20-27]. Henry later makes sure to claim the “treason” of Cambridge, Grey and the "bedfellow” Scroop, was motivated by “a few light crowns” of France (and not, by implication, the English crown for Mortimer), a claim Cambridge himself denies or rather qualifies [2.2.89, 155-7; "bedfellow" in 2.2.8]. Almost everywhere in the play are crowns as monetary units somehow associated with treason: for instance, Henry fills with crowns the glove employed in a rather heavy-handed jest against Williams [4.8.59', see also “crowns for convoy” for deserters in 4.3.37]. While Henry proclaims that in the French wars he covets only honor and not gold [4.3.24-9], the Chorus' punning mention of "crowns imperial [...] Promised to Henry” backpedals on the pronouncement. If the desire for the French crown lies at the root of the war, so does the desire for monetary crowns, for Canterbury’s legal justification of the war by exposing the "crooked" nature of French “titles” [1.2.94] is itself not unsinister. As Act 1, Scene 1 makes clear, the French Wars offer the English Church an opportunity to escape the confiscation of lands donated to the Church; the Salic Law speech follows Canterbury’s attempt to strike a deal with Henry by offering him a one-time grant, which, after the speech, becomes explicitly connected with the war effort [1.2.132-5]?1 The "severals and unhidden passages Of [Henry’s] true titles" happily permit Canterbury to exit into the summer sunlight of fiscal independence, which the Chorus might darkly refer to as "gilt" [see 1.1.72-81; pun on "gilt" and "guilt" in 2.chor.26], Although Henry does not go France to pillage. Pistol’s threat to cut the throat of Monsier le Fer at Agincourt unless given “crowns, brave crowns” [4.4.38] may be taken as a punning caricature of Henry’s enterprise in the same vein as Bardolph’s stealing the pax [3.6.40-6, 4.5.75; see section 8]. Both the royal and the fiscal crowns are circular. That, and their association with treason - that of Cambridge, Pepin, Hugh Capet, Henry IV, Williams and of the hypothetical coward of the Crispin’s Day speech - make both types of crowns apt to be referred to as "crooked figures." In an earlier section, I argued that the king is he who 31 And so when the king - who is certainly aware o f Canterbury’s conflict of interest —warns the Archbishop not to “nicely charge” his “understanding soul” [1.2.15], we might remember that he himself is not unbiased regarding the possible war in France, if only for his father’s deathbed advice that the crown met by “indirect crook’d ways” is best retained by busying “giddy minds With foreign quarrels” [2H4 4.5.184, 213-4; connection between advice and French Wars made by Parker, 164]. 217 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. appears like a king. Part of that appearance is wearing the crown. Once again the sign turns out to prior to, and in a "sinister" relationship with, the ostensible essence. But how is it possible that both the object of theft (the crown) and the thief (usurper) be "crooked figures"? Have we missed some key distinction? Are we reading the play through circular spectacles? 6. The C o u n t While describing the numerical and geometrical underpinnings of the concept of the patrilineal genealogical line, I illustrated its apparent rightward movement thus: king[----------->king2---------- >king3----------->king4----------->king5 Each member of the sequence can be compared to a whole number; each is, in this sense, a one ["numbering" in Dee *jv]. The process of naming kings is essentially the same process as counting them :32 I >1----- > 1------- >1------ >1 1 2 3 4 5 I now propose that we envision the genealogical line not as a sequence of kings, but as a sequence of things: of what Richard //calls "hollow" crowns [3.2.160]: crown j----------->crown2---------- >crown3----------->crown4----------->crown5 or O— >O >O— >O >O 1 2 3 4 5 Here, the object of count is not an I but an O. One of the aims of this and the following sections is to justify and explain the diagram above. Doing so shall entail reinscribing women into the genealogies of the history plays, as well as analyzing the way Henry V treats the distinction between male and female bodies. Let us start by considering two of the equivocations in Katherine’s English lesson after the taking of Harfleur, which insert the issues of sex and gender into that of the 3: See "reckon," section 8. Donne's Essays in Divinity contain an extensive treatment of genealogical naming as numbering in the Bible. Donne attributes the title of the The Book o f Numbers to God’s command for the Jews "to be numbred, and to be numbred by name" [Donne 53 referring to Num. 1:2], whence the book's interminable lists of who begot whom. He also quotes Jerome to the effect that Numbers contains "torius Arithmeticae mysteriae," presumably in its genealogical lists. For Donne, an avid reader of Pico della Mirandola and his school, the fact that numbering and naming in the Bible are essentially the same is solidified by gematria, yet the fact that Hebrew letters have numerical values manifests, rather than establishes, the equivalency (comp, with other meanings of the word "cipher" in my chapter 8). 218 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sequence. The lesson of course is entirely made of puns, but the weightiest ones arrive at the end: KATHERINE: Comment appelez-vous le pied et la robe? ALICE: Le foot, madame; et le count. KATHERINE: Le foot, et le count? O Seigneur Dieu! Ds sont les mots de son mauvais, corruptible, gros, et impudique, et non pour les dames d’honneur d’user [3.4.49-55], The “translations” are: “pied” = foutre, and “robe” = “cunt.” The first might be understood as a visual pun on the English “pied” in the sense of maculate, and by extension, adulterated, impure. What is pied, adulterated? Copulation, foutre. For the second, “robe” is an editorial correction. The Folio text says “roba,” which is generally understood as a pun on “loose woman” [Walter], whereas “count” is taken to be Alice’s pronunciation of “gown” [Mowat and Werstine’s Folger ed., Bevington’s Complete Works], Regardless of whether Shakespeare wrote “robe” or “roba,” it is as difficult not to hear a nod in the direction of “rob” (med. F. rober) as it is not hear a similar nod, in “cunt” being spelled “count” [see same spelling in Harington's "Of Lelia" in his Ajax 59, n. 28], in the direction of counting. The importance of this multiple pun to the play cannot be overestimated. At the risk of seeming improper - unfortunately, we shall run that risk throughout most of this section - I should like to employ the reader's imaginary forces in the following show-and-tell [see "show" in Ham.3.2.140-2], This spectacle is addressed to the male gaze, but Renaissance spectacles generally are, and we can do nothing about it. Lift the robe, and there it is: the "count." The former belongs to the outsides, the surfaces, the appearances; the latter underlies them. Beneath appearances, we discover counting. Lifting, so to speak, the robe of verbal justifications for the French wars, demonstrations of "right," Henry's claims to covet only honor, we come upon the "count" of crowns both royal and fiscal. Language robes numbers, rhetoric calculation. Backtracking, we realize that the appearance "robe" already holds the key to that which it hides, in its resemblance to the medieval French rober (= Italian rubare), to steal.33 To count is to steal. The pun accuses Henry’s actions in France in the same manner as does Bardolph's theft o f the pax. But the link between "count" and "crowns" goes much deeper than that. In an age with no DNA testing but plenty of anxiety about bastardy, physical resemblance bears the surest “ocular proof’ of paternity. The idea of child as the "image" of the father feeds into 33 Comp, with the juxtaposition of "robe” and robbery in rondel 83 of the historical Charles d’Orleans, where the "robe" of a character is "fait ablative" [464], 219 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the idea of conception as the copying of a male form by impressing it upon female matter [e.g. MND 1.1.49-51, Son. 9:5-8; but see child as image of mother in Son. 3.9-10]. It is therefore envisioned as printing, whether of text or of seal [e.g. MND, loc. cit., WT 5.1.124-7, 34 Son. 11:13:14], or as the impression of a die upon metal, coining [MM 2.4.45-6 speaks of those who “coin heaven’s image In stamps that are forbid”; comp, with "coining" in The Revenger's Tragedy, 2.2.144], Coining seems to be the metaphor particularly fitted to royal procreation, since the minting of coins was the king's prerogative [see Lr. 4.6.83-4]. The queen reproduces “the King’s reall,” while the royal mint reproduces “his stamped face” [Donne “Canonization” 1. 7]. Both are places where crowns multiply.35 Kate's "count" implicates all these themes, for the language lesson immediately follows the taking of Harfleur: she learns the English names of body parts in expectation of marriage to her conqueror, marriage that eventually shall produce an heir to the crowns of both England and France. While the idea of “right by line” is patrilineal, the males who succeed one another are, needless to say, men of women bom [comp, with Macbeth 4.1.80, 5.8.15-6]. The multiplication of “hollow” crowns is performed upon the female body. In an ideal genealogical line the woman is elided, invisible; the line appears, as it were, parthenogenic. When it forks, however, the woman becomes very visible indeed [see "forks" in Lr. 4.6.117; "forkhouse" in Partridge 116]. Coining entails the multiplication of identical copies. The woman perpetuates the line; but she does not stop there, at one. Her ability to stand at the source of several lines defines her as the origin of threats to the one line from other, the origin of theft and treason. As we have repeatedly seen, the “cause” of every genealogical problem in the history plays, the “cause” by which every crooked figure legitimates himself, and to which he appeals, is a female ancestor. Woman's ability to pass on succession lies at the origin of Carolingian, Capetian, Mortimer and York claims to the crown; Henry’s wars in France and the 34 MND: "To you your father should be as a god [...] one To whom you are but as a form in wax By him imprinted." WT: “Your mother was most true to wedlock, prince. For she did print your royal father off. Conceiving you [...], Your father's image is so hit in you." 35 The words “count," “cunt,” and “coin” might look and sound similar but they do not have the same etymology: Fr. corner, to count, comes from the L. computare, whose puta has absolutely nothing to do with the L. cunnus, Fr. con, denominating the same thing as a puta uses in her usury [TA 2.2.101] and which we more chastely describe with the L. word for scabbard [see e.g. Aeneid 4.579-80, “dixit, vaginaque eripit ensem Fulmineurn”]. Nor does cunnus have anything to do with the L. cuneus, meaning wedge, comer [“Britannia,” says Tacitus, “in cuneum tenuatur,” is shaped like a wedge, in Lewis and Short, "cuneus"; comp, with "nook-shotten (i.e. running into comers, indented - Walter) isle of Albion," H5, 3.5.14], whence, through F. coin, wedge, die for stamping money, comes our “coin.” However, Shakespeare does corral many of these senses in, for instance, Othello’s “keep a comer in a thing I love. For others’ uses’ [3.3.276-7]. 220 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mortimer threat in England occur “in right and title of the female” [Henry 5 7.2.59]; even Henry’s marriage to Katherine is meant to strengthen Henry’s claim to France in an anterior manner by compounding a “boy, half French, half English” [5.2.216-7], both Lancaster and Valois.36 At this point we should consider how Henry V treats the way male and female bodies are understood in the Renaissance. In his Making Sex, Thomas Laqueur argues that the Renaissance subscribed to the Galenic view of men and women as, in Laqueur's words, "hierarchically, vertically, ordered versions of one sex" [10]. Men are essentially the same as women: the cause of sexual differentiation is than men have and women lack perfection: women, in other words, are imperfect men. Perfection manifests itself in heat and is responsible for the difference in male and female genitals. These are regarded as isomorphic, with the penis and the scrotum being the vagina and the uterus turned inside out and projected outside the body by the greater heat of the male. The Galenic view was so strong that genital isomorphism was affirmed, rather than denied, by Renaissance anatomists performing dissections [Laqueur 1-149]. Male and female behavior were envisioned in the formal terms of male and female genitals and body-image in general. The male body is closed, the female open; the male contained and complete, the female uncontainable and incomplete.37 The Revenger’s Tragedy, for instance, is typical in superimposing the two images of the body upon the behavioral oppositions of hiding-showing, containing-leaking, keeping silent-telling: “why are men made close,” exhales Vindice in the ear of Count Lussorioso, “But to keep thoughts in best? I grant you this, Tell but some woman a secret over night, Your doctor may find it in the urinal i’ the morning” [1.3.82-5]. And in a later scene he pronounces, “That woman is all male, whom none can enter” [2.1.103]. Men here are conceived in terms of literal and metaphorical integrity, where virtue - from L. vir, man - consists of unity and closure. This notion of masculinity enables us to treat the concept of the patrilineal “right by line” in Henry V numerically. Critics of the play, after all, are always opposing the united and virile English to the scattered and effeminate French. English propaganda in the play stresses unanimity and singularity of purpose: so, for example, the Chorus’ insistence that 36 The marriage of course suppresses the Dauphin. There is a great deal of irony in Henry’s subsequent words that the boy “shall go to Constantinople and take the Turk by the beard,” not only in that they apply to Henry Vr, but also that Eastern wars are the preferred solution in both Lancaster and Capetian families for keeping the subjects’ minds off contested pedigrees. (Louis DC, who appears in the Salic Law speech as the lineally insecure Lewis X, was historically what Henry IV dreamt of being - a crusader.) 37 For ungendered use of open and closed bodies, see Bakhtin's Rabelais and his World-, for gendered, see Paster. 221 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “all the youth of England are on fire,” or his description of England as a “little body with a mighty heart” [2.ch.l and 16; my italics]. Appeal to “manhood” is very prominent in Henry’s St. Crispian speech, meant to unite the “few” the “happy few,” the “band for brothers” in the desire to have “not one man more” [4.3.66, 60, 23; echoing historical Henry's motto une sanz plus, Walter]. English propaganda represents manhood as unity and vice versa. In the preceding chapter, we have shown the synonymity between perfection and oneness; the Galenic thesis that perfection is the essential difference between men and women lends additional support to what follows. Closed and unified, the male is conceptualized as a one, a coherent and perfect self, a monad. We may represent him by the letter-numeral I. Not only the male body, but also the male self is complete and identical to itself; the male "I" is always present as whole. The key pun for us here (it is not in the play) is “integrity,” connected as it is to “integer,” from L. for untouched, entire, righteous (“Integer vitae scelerisque purus," says Horace in Odes 1.22). The patrilineal sequence Henry IV -> Henry V -> Henry VI may be imagined as integers 4, 5, 6 on a number line. Canterbury’s gendering insistence on unity and closure in the “order” of “a peopled kingdom”[/.2./S 9]38 is strange given its context, for it answers the possibility of rear action on the part of Scotland. Henry is concerned lest, while he is busy ransacking France, “the Scot on his unfumish’d kingdom Came pouring, like the tide into a breach, With ample and brim fullness of his force, Galling the gleaned land with hot assays” [7.2.148-151]. To put it roundly, Henry fears that by extending himself to France he will become the “bending author” [5.c/i2.2] of his own undoing. To these possible references to the preposterous practice of sodomy, we might add the King’s “bedfellow” Scroop [2.2.8-9, 9<5-9]39 and Exeter’s description of the “espous’d” death of York and Suffolk [4.6.7-27]. I am not interested in gossiping about the sexual lives of characters, who anyway are fictional, and dead. I am interested in words, in how affirmations of unity and closure go hand-in-hand with insinuations of penetration from behind. So in the case of Canterbury’s order speech; so also in the case of Scroop, whose disclosure follows another of Henry’s assertions of unanimity: “We carry not a heart with us from hence, That grows not in a fair consent with ours; Nor leave not one behind that does not wish Success and 38 According to Canterbury, each social stratum fulfills its own function, which all, “As many lines close in the dial’s centre [...], End in one purpose.” This is a common theory of the state as unity of parts “Congreeing in a full and natural close. Like music” [1.2.204-12, 182-3]. The metaphor map of the theory is the Neoplatonic equation between numbers in the center and along the circumference [see chapter 6], as well as the less specific idea of number as unity of plurality, or, on the level of music, of harmony. 39 Compare with the ascription of the following opinion to Marlowe in the so-called "Baines libel”: "That Saint John the Evangelist was bed fellow to Christ and leaves always in his bosome, that he vsed him as the sinners of Sodome" [Shirley 182], 222 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conquest to attend on us” [2.2.21-4]. These juxtapositions are ironic and their effect is to give the lie to the notions of unity and closure, particularly as these concern maleness. In Henry V, to curtail Donne, no man is an I. Of prime importance here are Shakespeare’s pregnant usages of “fault” or “breach,” words as equivocal and as frequent as “crown.” Both the possible attack of the Scots and especially the “treason” of Cambridge, Scroop and Grey are talked of in these very terms. The Scots, as I already quoted, might come “pouring, like the tide into a breach” [1.2.149; my italics], “Fault” is applied to the treason by the Chorus (the “fault” which France “fills With treacherous crowns,” 2.ch. 16-22), by Henry (‘T heir faults are open,” 2.2.142) and the traitors, each of whom confesses his “fault,” begging pardon for it but not for the body [2.2.76, 152, 165], Faults and breaches undermine the unity, the integrity of the body, making it susceptible to penetration [comp, with discussion of mines in 3.2.58-68 and eating the leek in 5.7]. At the same time, “fault” must also be felt in its other sense, as a sin. So Henry asks God to forgive “the fault My father made in compassing the crown” [4.2.299-300]. The King is constantly framing his speeches so as to lay the responsibility for the war onto the shoulders of other persons: Canterbury [1.2.13-28, 97]; the Dauphin [282-4]; the French King (2.4.105; the problem of how more than one person can bear the blame for the entire war is worthy of Parmenides 131b-c]. The citizens of Harfleur will be at fault if the city is sacked [5.5.79]; the soldiers bear the responsibility for the sins committed in Henry's service [4.1.150-192]; Cambridge and Co. did not show mercy and so he is justified in not showing them any [2.2.39-59, 79-83; see also 174-7]. In all of these episodes, the royal desire to avoid being "at fault" must be thought of as desire to retain "integrity" in all senses of the word, including closure, perfection, oneness. As opening, “fault” may also be understood in a more general sense as the propensity to sin, the imperfection of human nature. Original sin is implicitly recalled when Henry presents “the fault” of the Scroop to be “another fall of man” [2.2.742]. Conversely, according to the Archbishop, Henry’s reformation “came [...] scouring faults”, when it “whipp’d th’offending Adam out of him, Leaving his body as a Paradise, T ’envelop and contain celestial spirits” [1.1.28-34]. Because the ideas of closure and unity come thus gendered, and because of Canterbury's linking them to Genesis, we may overhear in the word “fault” a double reference to the “woman’s fault” both in the sense of the fall as the fault of Eve and in as it concerns the female anatomy. The fault that undoes male unity and closure feminizes maleness [for effeminacy as instability, see Laqueur 123]. Whatever can be entered, to paraphrase The Revenger's Tragedy, is not male. 223 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The connections between the "fault," corporeality, femininity, and openness to penetration are evident in the attack on Harfleur. The literary-historical origin of the scene lies in medieval allegory, which portrays the female body by the trope of the fortified city — to be entered or even taken by storm, since, as Ovid said, militat omnis amans [Amores 1.9]. There are two instances when the female body can be regarded as relatively closed; one of them is virginity ["maiden cities" "girdled with maiden walls that war hath never entered," 5.2.340-4; for hortus conclusis trope, see Sazonova 164], Once penetrated, however, the body turns open, its dilated fault in need of control. The awakening of female appetite (whose increase grows by what it feeds on. Ham. 1.2.4-5) can be countermanded only by possession which, by closing the body for the second time, erects, to paraphrase Augustus, a castitas restituta.*0 So Castiglione’s Lord Gasper claims that, during intercourse, woman "receiveth of man perfection"[ qtd. in Laqueur 125]. Henry's storming of Harfleur, with its “Once more unto the breach, dear friends, once more. Or close the wall up with our English dead” [3.1.1-2], is an allegorical rape. Its motive, the motive for filling the breach and closing the wall, is to reimpose possession but by a different master: Harfleur belonged to the French King, now she belongs to the English. Female and so in want of "perfection," the city will acquiesce. This scene precedes Kate’s language lesson and then the French lords fuming over “Norman bastards” [3.4.10]. The forced or willing surrender of French women to “the lust of English youth” [3.3.21, 3.5.30, 4.5.16] imagined by the lords presents the entire war in the light of Harfleur, whose taking they of course won't recognize as legitimate. We can taste the irony in their indignation. If 350 years earlier the Norman French had found the fault of England and populated the land with “bastard warriors,” now their descendents have found the fault of France [5.5.37; also Kate as stand-in for France in 5.2, esp. 338- 47]. Both countries are here feminized, with the occupation of their fault by the male body of the conqueror resulting in a new line [comp, to Revenger’s Tragedy 1.2.203-4, “A bastard by nature should make cuckolds, Because he is the son of a cuckold-maker'']. Finally, the female fault - as the "weak spot," the point admitting penetration and possession by another master - appears not only on the imperialist, but also on the genealogical level, in Canterbury’s “unhidden passages” that let Henry lay a matrilineal claim to France [1.1.86], and in the "fault" that Henry IV opened, by "compassing the crown" [4.2.299-300], for the rival and again matrilineal claims of the Mortimers. 40 The extension of the closed body image from the male first to the virgin and then to any “chaste” woman capitalizes on the notions of honesty as oneness (paronomasia reinforced by the antonymic dishonesty = duplicity) and constancy as a) faithfulness and b) immutability. Hence the dual gendering of England, “set in the silver sea. Which serves it in the office of a wall Or as a moat defensive to a house,” in its description by John of Gaunt [R2, 2.1.40-51] as both “sceptered isle” and “teeming womb." 224 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We shall now take the bull by the horns. Famously, the fault that Lear calls the "organs of increase” [1.4.271], Hamlet refers to as the “nothing” that lies “between maids’ legs” [3.2114-9], “Nothing” here appears as a slang term for female genitals in a system that conceived of male and female sexual organs in terms of the opposition presence - absence.41 As Patricia Parker has shown, the “nothing” that lies “between maids’ legs” is also likened in the anatomical discourse of the day - by a character with the apt name of Helkiah Crooke - to “the letter, o, small and wondrous narrow.” Crooke also refers to his “o” as the “Fissure” [Parker ch. 7, esp. 236-237]. Crooke's use of the letter “o” comes from slang. It is obviously implicated in another Shakespearean language lesson. In The Merry Wives o f Windsor 4.1. the Welshman Sir Hugh Evans teaches Latin declensions to a schoolboy with the strange name of Will Page. “What is the focative case, William?” asks Sir Hugh in his accented English. “O —vocativo, O. ” answers William. “Remember, William, focative is caret,” enjoins Sir Hugh [47-49].iZ The female genitalia as the small “o” or the dilated “O” show other notable connotations. The 1612 edition of Thomas Dekker’s English Villainies Discovered by Lantern and Candlelight, a study on thieves’ cant published under several different titles, bears a chapter entitled “O perse - O.” It contains addenda supposedly sent in by a former High Constable to the Bellman of London; the Constable claims to have acquired his knowledge of thieves’ cant from a beggar he denominates as O perse —O. In thieves’ cant the phrase refers to all things crooked [see song 289], but especially to the female role in the sexual act (the Constable translates the phrase wag fo r a win as ”...0 per se - O... for a penny,” and is of the opinion that many are drawn to beggary because of its practice of free - or in any case affordable - love, 286). The pun here is submerged, but obvious to anyone with a modicum of French: O perse sounds the same as O perce. The Constable feminizes his informant for divulging the secret speech of his trade, and making public that For "thing" vs. "nothing" as male and female organ, see Willbem 245 and Partidge "thing" 2, p. 203 quoting the thing "you cannot show me" in H8 1.4.48-9; "thing" female in Oth. 3.3.306 and IH 4 3.3.90-3. Most comically, another slang term for penis in the early modem period is “yard." Had binary notation been invented a century earlier, it certainly would have found employment here. Its inventor Leibniz saw in it a "Symbolum re rum omnium exnihilo creatarum per unitatem," with the unity being God, zero nothing, and all things as numbers that are combinations thereof. So he sent it to his Jesuit friends in China as an aid to convert the Emperor, because the Chinese thought creatio exnihilo a silly idea. The Emperor refused to grant an audience. See letter to Grimaldi dated February 1697. to Bouvet dated February 1701, and others in Leibniz korrespondiert mil China. *' For “case” as another slang term for vagina, see Partridge 85; connected to "declension" in Parker 118, 323 n. 76. Charles d’Orleans in the already mentioned rondel 83 has a character who “failli a en son cas genitif' [464]. The academic intendment of Shakespeare's joke is that the vocative, or as Sir Hugh says, the “focative," case in Latin may be demarcated in English by an O: Brute = O Brutus! This, however, is only the case for most types of masculine nouns of the second declension; in all other declensions and genders the vocative caret, is lacking, the endings that visually distinguish it from the nominative forms. 225 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which ought to be h id /3 Like Vindice of The Revenger’s Tragedy 1.3.82-5, he presents stereotypicalIy female linguistic incontinence in terms of the visual pun on mouth and vagina. What is the provenance of the phrase O perse - O? According to Dekker’s commentator E. D. Pendry, it comes from hom-books, wherein “in reading aloud, the pupil indicated a syllabic letter or character by ‘per se’; hence, O per se - O: O on its own” [351, “horns”]. Despite the lovely juxtaposition of "horns" and O perce, we can establishes a much more direct between O per se —O and the “nothing” that lies “between maids’ legs” via arithmetic. After all, “O” signifies nothing as numeral and not as letter. If then O per se - O may also be a line from an arithmetic lesson, in what context would it be appear? In my first chapter, I quoted Italian and Latin arithmetics where.per se is used in indicating a zero written with no numerals to the left of it: “zero [...] perse sola nulla significa,” “Nota autem circularis .o. per se sumpta nihil usus habef' [Calandri and Tzwivel in Smith and Karpinskii 59 n. 5 and 61 n. 1]. We have also seen that zero was the numeral of choice for explaining the dependence of value upon place in Hindu-Arabic notation because, unlike other figures, it “doth signyfie nothing” by itself [Recorde, Grounde Bvv], Thus the phrase O per se —O is instrumental to the most evident paradox of that notation: while zero “of himselfe signifieth not” [Digges 1], “she holdyng that place givethe others for to signyfie,” “dat signare sequentf' [Sacrobosco and Villa Dei in Steele 34, 5].44 7. O P er Se — O Let us consider the O per se /perce in light of the "memorable lines" of the history plays. There is a strange but evident analogy between the way the arithmetical O perse, which "signifieth not" [Digges 1] enables the signification of "signifiyng figures" [Recorde, Grounde Bvv], that follow/precede i t / 5 and the way the female O perce, while 43 The name may also refer to the crier’s call “O yes!” a bastardized version of the F. Oyez. hear ye, see TC 4.5.144. 44 Occupation of place is a concept extremely important to Othello, the very name of whose “arithmetician" [1.1.19] lieu-tenant may come from the early modem “cass,” from L. cassus, empty, void. Patricia Parker devotes considerable attention to the concept of place in the play, without, however, considering it mathematically. Any such attempt would, in my opinion, have to search Othello for terminology from the Rule of Coss, i.e. sixteenth-century algebra, connecting it to the terminology of the courtroom. The English “coss” comes from the Italian cosa, L. res, which denoted the unknown in early equations and means “thing” - as in “common thing” [3.3.306]. In the judicial “case” (L. causa, as in “It is the cause" [5.2.1] or res, as in rent [ram, 1.1.88] agere [ago]) of Desdemona (mono is Venetian for O perce), the thing is the "ocular proof' of the handkerchief, whose meaning is prejudged and misconstrued. 45 Follow according to the order of place, precede according to the order of rank; compare with difficulty in medieval manuals over "what follows," as seen in Villa Dei’s claim that the zero ”dat signare sequenti." 226 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not permitted to rule, nonetheless "augmenteth" the "places” that follow/precede it, by making possible the genealogical claims of their occupants. Hence of this crooked figure too we might say: while O by itself - perse - might be O, in combination (L. conjunctio, Picinelli “O”), it “maketh the other figures wherewith it is ioned, to increase more in value by their place” [Baret in Henry V, ed. Walter 6 n.I5-6]. The phrase per se means “by itself, alone” but in philosophical Latin it is used in opposition to per accidens, and means “with respect to essence.” Therefore, when applied to numerals, per se denotes not only the fact that the numeral stands alone, but also what it signifies when it does so, i.e. what Recorde calls its “certayne” value. For us, this distinction is null, for we equate numeral standing alone alone with its certain value, yet sixteenth-century arithmetics did not. Let us recall the explanation of Hindu-Arabic numerals in Leonard and Thomas Digges' Stratioticos: All numbers may be expressed by these characters following. I 2 3 4 5 6 7 8 9 0 whose simple ualue by themselues considered, you may heere-under behold. i ij iij iiij v vi vij viij ix O The Ciphra O augmenteth places, but of himselfe signifieth not [ 1]. As we saw in my first chapter, this passage tries to relate Hindu-Arabic numerals to Roman numerals I-IX which themselves stand for numbers as positively existent entities, multitudes of units. Leonard Digges' attempt to reconcile the new notation with referentiaiism is impeded by the fact that the numeral O cannot be taken to indicate any Roman numeral or, in fact, any "thing" at all. Therefore if 1 per se refers to i, 2 per se refers to ij, 3 per se refers to iij, and so forth, O per se refers only to itself, to O. O per se - O. All early arithmetics differentiate between O and numerals I - 9 on the grounds that they "signify" and it doesn't [Digges 1, Recorde Grounde Bvv, Villa Dei in Steele, 5, etc.]. In earlier chapters, we defined "signify" in this context as point to something that exists outside of oneself. This permitted us to predicate the classical number concept upon the Platonic understanding of signs as pointing to prior signifieds. Hence the zero plays a visible role in the restructuring of the general concept of signification, a restructuring that stresses syntax and reassigns priority to signifiers. On the level of numbers, this task was performed in 1585 by Simon Stevin, for whom zero became the "principe et commencement du nombre" [498; see my chapter 5]. If we interpret Shakespearean genealogies numerically, we must agree that for him too the zero and not the one lies at the beginning of the number line: 227 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O p e rc e ------->king[----------->king2----------- >king3---------- >king4 O per se --------> 1 -------------- > 2 > 3 -------------- > 4 While in Stevin zero usurps the role of the one as the origin of number and the arithmetical analogue of the point, in Shakespeare zero usurps the role of origin and the means whereby the one produces numbers: through division resulting in multiplication of parts [see Republic 525e and my chapter I]. Stevin disregards ontology, in effect equating the being of numbers with the being of numerals; in generating numbers by division, Shakespeare retains a key ontologizing element. This permits us to regard what follows as the peculiar transformation - following Dermot Moran, one might say "meontologization" —o f Platonist metaphysics. As the synecdochal female, O perce / "nothing" / "organs of increase" exhibits the characteristics of matter [comp, with Hamlet’s pun on “mother” - “matter” in 3.2.323, 3.4.9; see Parker 254-5: for "matter" as female, see Timaeus 50d and Metaphysics 986a24-8\. Matter is also the unlimited seat of plurality, identified with mutability as well as the first even number, the two [for matter as dyad, see e.g. Bovelles, Duodecim Numeris 149r_v]. In Shakespeare, the ascription of mutability to women may be seen in, for instance, Othello (“Sir, she can turn, and turn, and yet go on. And turn again,” 4.1.249-50); the number two surfaces in the related idea of female duplicity (e.g. “dishonest manners” of “German women,” i.e. their proclivity to breeding bastards, in H5 1.2.48-9', for “german” as double see Parker 127-133). 46 As we saw in my third chapter, Aristotle identifies the Indefinite Dyad, opposed to the One in Plato’s late ontology, with matter [Metaphys. 987b20] and that with me on or nothing [Physics 192a6-l0; in its capacity as no-thing, Wolfson 360]. All of these are vital for what follows.47 J6 The same concept lies at the root of harangues against make-up, such as those of Hamlet [3.1.143-6; 5.1.192-4], or of Vindice’s grim joke while pointing to the skull o f his beloved: “See, Ladies, with false forms You deceive men, but cannot deceive worms” [The Revenger's Tragedy 3.5.96-7, see also Crashaw's “Death's Lecture” 11. 23-4], J~ As we saw in earlier chapters, Neoplatonic ideas of number were still very much present in all sorts of Renaissance texts, consciously or unconsciously, with or without attribution. It is absolutely unnecessary for Shakespeare to have studies Plato or Aristotle for their concepts to have filtered down and become transformed in his texts. We may thus propose the following. Historically, Shakespeare's scenario of the birth of numbers incorporates and reinterprets the Neoplatonic thesis of the emergence of numbers from the monad that divides while remaining itself [see chapter 6]. In Shakespeare, however, the unity at the origin of all numbers is reinterpreted in a manner so radical as to replace the concept of unity with its opposite, i.e. difference. The O is not difference in the sense that it is something already differentiated; rather it is difference above differentiation in the same way as pure unity is said in the Sophist 245a to be “altogether devoid of parts.’’ If Plotinian monism interpreted late Plato to the effect that the Indefinite Dyad became entirely dependent on the One, here it is the Dyad that, as the ontological foreshadowing of the O, can be regarded as the "first principle:" the Dyad that, as I pointed out in the Plato section of the previous chapter. 228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. What does it mean to claim numbers come from the division of nothing? It is to claim that, in the same manner as in the classical number concept all numbers signify - in the sense given to this word by early arithmetics - the one itself, here they signify - in the same sense of the word - nothing. In the same manner as in the classical number concept all numbers are ones, here they "are" nothings. In the same manner as in the classical number concept all numbers are pluralities of monads joined by their being-one, here, being images [eikones, Proclus 4; Dee *v] of nothing that exists, they lack being [for classical numbers and the one, see my chapters 2, 3 and 6]. Applying the referentialist grammar of inquiry, we receive answers that referentialism cannot allow. For Shakespeare, the "nothing" that is the origin of numbers is characterized by instability, lack of identity - and, paradoxically, multiplicity. On the examples of maleness and of the uneasy fusion of arithmetical and notational increase in the Prologue, we have seen how the one in Shakespeare paradoxically equals the O. Therefore, when we generate numbers from the O by dividing it as Plotinus, Eriugena and Nicholas of Cusa divided the unity [see chapter 6], the next number is the two. At the same time, in the O that is the fault, division is always already present: the O is the two, the first number in the classical number concept, and therefore the symbol of multiplicity [see chapter 2]. Thus numbers, formerly multiplicities of ones ["Number is as much to say, as a multitude," Baker I; also Digges I, Baker vv, Dee *jr, etc.], lose the "of ones"; as a result, they are no longer individual and positively existent entities but embodiments of multiplicity as such. The replacement of one as unity by O as multiplicity in the role of the origin of number necessitates that any number intend and be all numbers on the "ontological" level.48 Therefore it is not entirely correct to call the O the "origin" of numbers. It is the origin of numbers because numbers come to be through its division; it is not the origin because it is never bereft of fault and is always present as a multiplicity. There is never an ontological staring-point that is integral, without the totality of numbers.49 What then permits the being of particular numbers I, 2, 3, etc.? The fact that Shakespeare reinterprets the general concept of signification in terms of the posteriority of the signified. The "being" of any number is zero in the sense that any number is nothing in itself, something in combination (conjunctio). In other words, what makes any number a particular number and not all numbers is its enchainment in what our last section shall call is always other to itself and therefore unknowable. 4!We would say 'on the existential level,' were it not for Shakespeare's retention of division as the mode of the coming to be of many 49 In the same way the zero is not the “commencement” o f Hindu-Arabic notation: it is a numeral just like any other, which we manipulate with other numerals in order to get numbers. There can be is no 0 without I. 2. 3, 4, 5. 6, 7. 8, 9. 229 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "account." The measurements of things do not signify numbers as they do in Plato; rather, things provide relationships that can be expressed numerically by the observer. Yet the observer Shakespeare insists on focusing on number as it "is," on its "essence" as the paradoxical jointure of nothing and multiplicity. Hence the deeply unsettling tone of his treatment of number, with its simultaneous themes of want and unbridled proliferation. For when, retaining the metaphysical grammar of inquiry, we examine number as it "is," we see that it isn't. Hence his dual employment of Hindu-Arabic notation as the prime emblem of the preposterous workings of accounts, as well as of the fact that that each individal unit of account "signifies" and "is" nothing. In sixteenth-century English, as David Willbem points out. "nothing" was pronounced noting [249; puns on "nothing" as noting in MAAN 2.3.53-6].50 The very word "notation" - which in Renaissance arithmetics bears the meaning of writing numbers with Hindu-Arabic, and not Roman, numerals —indicates, as it were, the essential vacuity of the signifier. A sequence of figures written in Hindu-Arabic notation offers the image of nothing producing nothing, with the these nothings appearing as something - but, alas, only appearing. Let us now examine the genealogical line as it "really is": O p e rc e >crown,---------- >crown2----------->crown3 ->crown4 O ------------ > O j ------------->o2------------ >o3 >O' 4a O------------------ > 1 ------------------- > 2 ------------------ >3- --------------->4 8. Accom pts and Reckonings The Prologue asks the audience to "let us, ciphers to this great accompt On your imaginary forces work." The word "accompt" - spelling for "account" somewhat more usually associated with its numerical meanings —is also implicated in the "robe" = "count" pun of Kate's English lesson. Its synonym on many levels is "reckoning," and we shall discuss the two words as if they were interchangeable. Both "account" and "reckoning" implicate words and numbers [comp, with section 6], The first is the Romance and the second the Germanic form for counting, enumerating serially, and then calculating: thus Henry asks God to take away his soldiers’ “sense of reckoning, if th’ opposed numbers Pluck their hearts from them” [4.2.297-8]. From this "account" and "reckoning" acquire their meanings in, well, accounting. Early business 50 Willbem's article is devoted in part to the generative powers of Shakespeare’s nothing, relating it to zero but in an isolated matter, without considering the properties of Hindu-Arabic notation. 230 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ledgers entered their debits and credits in a mixture of words and numbers [see Crosby 199-223, esp. examples on 209-10, 221-3]. It was probably bookkeeping that produced the verbal sense of the terms, as in "to give an account / reckoning of something," most often, but not exclusively, of one's actions. Yet the original numerical content was never far behind: thus the usage of "account" and "reckoning" in the context of the Last Judgment ("the king himself hath a heavy reckoning to make [...] at the latter day,” Henry 5 4.1.135- 43) generally takes the shape of bookkeeping metaphors, as for instance in Everyman, 11. 19-10, 70, 104-7, etc.51 We can think of an account (or reckoning) as a sequence of numbers, words, appearances that signify in the manner of Hindu-Arabic notation, with the signifiers pointing to a posterior signified. We perceive the priority of the signifier when we are confronted by several accounts of the same "thing," like Roman vs. Hindu-Arabic numerals, and the accounts do not match (e.g. one has sign for nothing and the other does not). When, however, we are enveloped in a single, totalizing account, the illusion of what Brian Rotman calls "the anteriority of things to signs" is restored. The account claims faithfulness to reality: that it indicates and represents prior signifiers. Once we use only Hindu-Arabic notation, for instance, we begin to think that it actually reflects the pre­ existent nature of number. Henry V proliferates in accounts which sometimes ignore each other and sometimes conflict. We have already described Henry’s account of his "right by line" framing Agincourt as "ocular proof’ of truths written in excelsis; it is contradicted by the Chorus’s "Fortune made his sword" [5.ch2.6\. We have also mentioned Henry’s reckonings of who is to bear the blame for the war - Canterbury, Dauphin, the French King —which contradict each other by over-determination. At one point we witness the manufacture of a reckoning that we simply know to be false. Henry orders the massacre of prisoners because the French are counter-attacking [4.6.36-7], whereas Gower later attributes it to revenge for the effect of the counter-attack: the slaughter of boys guarding the luggage [4.7.1-11; curiously, J. H. Walter buys into Gower’s account, see intro, xxx]. Part of what is at stake here is the vast distance between history, as a sequence of real events, and historiography, as a sequence of "mock’ries." The propensity of historiography to err appears especially strongly in two cases. Canterbury’s account of Salic Law, an account crooked by the virtue of the financial stake the Church has in the outcome, has not only historical (Lewis the Tenth instead of Louis IX) but also arithmetical errors. The Archbishop reckons that 421 years pass between the death of Pharamond in 51 For secular example, see Ballade 120 of Charles d’Orleans, in which his heart takes him into its office and together they examine the ledgers of past loves [366-70]. 231 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 426 AD and the capture of the Salic land in 805 .AD. It is, of course, 379. The miscalculation is Holinshed’s ["Henrie the Fift,” 10]. Does Shakespeare repeat it automatically? If he did, someone would surely have noticed it during rehearsals or performances. In a play so concerned with counting, sequencing, and representation, to repeat the false "reckoning" of a historiographical source in a passage that demonstrates the falsity of a text claiming historical legitimacy, seems to be yet another way of ironicizing the signifying claims of any account, of any histoire. History / story is also implicated in the case of Fluellen, whose comparison of Henry to Alexander inadvertently parodies humanist employment of ancient models for encomiastic and propaganda purposes. In this scene, we witness an account being bom from similarities - except that here they are exceedingly stretched: "There is a river in Macedon, and there is also moreover a river at Monmouth [...], and there is salmons in them both."52 Fluellen sees the objects of comparison as “one, [...] alike as my fingers is to my fingers” [4.7.26-32]. His words raise the much more general problem of oneness or identity on which any "alike" is based: Alexander and Henry become the model of the resemblance between any two things, thus standing at the source of any classification and judgment. An account arguing that Henry and Alexander are "all one" will map them onto each other. A host of problems follows. The analogy transmogrifies into a proportion - according to what sixteenth-century algebra called the Rule of Coss - of Alexander: Cleitus as Henry : Falstaff. Fluellen's tongue brought him where he never intended to go: to implicitly accuse Henry of treason and murder [4.7.35-43]. The proportion may also backfire upon Alexander: if the play shows us Henry as he is in all his ambivalence prior to the accounts of historians, it may lead us to doubt their accounts of Alexander.53 Fluellen's 53 I find it particularly ironic that the manufacturer of this account if Fluellen. With his accented, proliferating in synonyms, and syntactically faulty, i.e. mis-joined, English, the Welsh captain fights to further the conquests of his conqueror, the situation of his “nation” [comp, with Macmorris’s “What ish my nation?” 3.2.124-7] being most adequately expressed in another “error” made by his countryman in another play: “Melodious birds sing madrigals - Whenas I sat in Pabylon" [MWW 3.1.22-3, first line quoting Marlowe's "Shepherd." second Psalm 137], Henry glosses over the division of nations within his empire (illustrated by 3.2), and stages himself as of Fluellen's blood (“For I am Welsh, you know, good countryman” 4.7.109; see also same claim to Pistol, 4.1.51). Yes, he is Harry of Monmouth, and Monmouth is in Wales, but there may be more to the story. Perhaps the play here points towards Geoffrey of Monmouth and the Trojan refugee Brutus who, according to Geoffrey's account, founded and ruled Britain; sixteenth-century genealogists attempted to demonstrate a line of descent from this Brutus to the English kings [Williams 417; Sherman passim]. Whether or not Henry is, as he seems to claim, descended from Brutus, in its association of the Welsh with another group of defeated exiles the play again finds but tragedy, with Pistol taunting Fluellen as “base Trojan” [5.1.20; comp, also to the Hostess malapropically sending Falstaff to “Arthur’s bosom,” 2.3.9-10]. 53 Once we start thinking of classical models, the “excellent services committed at the bridge” [3.6.3-4; for “service” as theft, see 3.2.47] by Pistol remind us not, as Fluellen says, of the “valiant” Mark Anthony (the 232 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. comparison also leads us to pause before the Chorus’s comparison of Henry's return to the projected return of Essex [5.ch.31-5]. Fluellen himself attempts to explain how to work with resemblances. When Gower corrects his referring to Alexander as the "Pig" [i.e. the Big] to "the Great,” he objects: ‘‘the pig, or the great, or the mighty, or the huge, or the magnanimous, are all one reckonings. save the phrase is a little variations” [4.7.16-9; my italics; for Alexander the Pig, see Quint]. Although he wishes to say his synonyms have the same denotation, his lack of what the grammarians call “agreement,” i.e. correspondence in number, gender or case, complicates matters. The denoted things is one; it is also many. The way it appears in diverse accounts, in diverse conjunctions, is "a little variations.” What then constitutes identity? If the world is composed of "a little variations," how is judgment possible? The play endlessly poses this question, which also occupies such seminal Renaissance texts as Sanches's Quod Nihil Scitur and Montaigne's "Apologie de Raymond Sebond." We encounter it in Shakespeare's employment of montage: can we take the juxtaposition of two scenes, as two accounts, and erect a meta-account over them? For instance, Falstaff is dying in 2:1 and already dead in 2:3; the intermediate scene, however, shows us the trial of Cambridge, Scroop and Grey. Nym and Pistol [2.1.121-4] attribute FalstafPs death to the king’s breaking his heart with the Petrine “I know thee not, old man” [2H4 5.5.48]. Should the juxtaposition be read as affirming their claim, and, moreover, as intimating that he who judges traitors is himself a traitor? Or how do we interpret Bardolph’s stealing the pax and getting hanged for it [3.4.4111 As noted by commentators, the episode is historical; according to Holinshed ["Henrie the Fift," 30], the theft was o f the pyx, the vessel wherein the host is kept, and not of the pax, an image of the crucifixion used, in the locution of the OED, as an osculatory. Should we judge the pun to be another accusation leveled at Henry? Is it supposed to indicate that Bardolph gets hanged for “pax of little price” [3.6.45], whereas the stealers of a greater pax (L. “peace”) go unpunished? Why should the “treason” of Williams —a manipulation on Henry's part - be spoken of in the same terms as that of Cambridge? “Here is,” exclaims Fluellen; “praised be God for it! a most contagious treason come to light, look you, as you shall desire in a hero of Actium) but of Publius Horatius Cocles and his stand on the bridge against the troops of Porcenna. Horace is reported to have fought so many of Porcenna’s soldiers at once that the bridge collapsed under their weight; still, he survived. Fluellen does say he “did see” Pistol perform a “gallant service,” but whether he actually saw Pistol fight is unclear [Walter's H5, 76 n. 14-16]. If bravery be a virtue, ‘tis none of Pistol’s. Gower calls him one of those who go “to the wars to grace himself at his return into London under the form of a soldier," learning all the details of battles to give accounts with themselves in the starring role [3.6.68-83]. Fluellen does not argue and we know that this is exactly what Pistol is going to do when he returns [5.1.92-3]. Whether this casts any light on Horace is for the reader to decide. 233 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. summer's day” [4.8.22; comp with God bringing the first treason ‘‘to light” in 2.3.185; also with Canterbury’s “clear as is the summer’s sun,” 1.2.86]. Why, for that matter, should Henry’s jest repeat, albeit in a scrambled fashion, a number of motifs associated with Henry’s account of himself as God’s instrument, such as the empty glove [comp, with “God’s arm” 4.3.4, 4.8.108-9], the reward of crowns, and also Fluellen's imitation of Henry as Henry imitates God in the trials of Williams and Cambridge [65; Paul: “Imitatores mei estote sicut et ego Christi, ” 1 Cor 11:1]? Immediately after Henry gives the signal to attack, we behold Pistol’s threatening the already captured Monsieur le Fer with death unless given “crowns, brave crowns” [4.4.38]. Pistol’s rapaciousness may, as I already mentioned, be a caricature of the whole English enterprise. However, in a play whose hero imitates Christ as iudex venturus, the biblical echoes of the Pistol - le Fer scene may be significant as well. In this moment of comedy, the Frenchman's pathetic “Est-il impossible d'echapper la force de ton bras?" places the "God's arm" motif into an extremely apocalyptic light. It is also possible that the scene contains references to Nebuchadnezzar’s dream in the second chapter of Daniel, regarded as concordant with Revelations.5* The problem of judgment in a world of "a little variations" underlies the play's references to what was sometimes called the Day of Reckoning, such as the extensive debate between "Harry Le Roy" and Williams in 4.1.135-94.55 What is most at stake, of course, is the reckoning of the King himself. Henry's insistence on his integrity, his striving to lay the blame on the shoulders of others, must be understood not only as propaganda or psychology, but also - perhaps even primarily - in the prospect of "the latter day." His prayer on the night of Agincourt is weirdly arithmetical: I Richard's body have interred new; And on it have bestow'd more contrite tears 34 Nebuchadnezzar dreamt of a “great image” whose “head was of fine gold, his breasts and his arms of silver, his belly and his thighs of brass, his legs of iron, his feet part of iron and part of clay” [Daniel 2:32- 33], The idol, whose four parts also represent four kingdoms, crumbles when its feet are smashed by a stone, i.e. the kingdom of God [■44-45]. Strangely enough. Pistol’s bringing Monsieur le Fer to his knees [55] involves all or most of the above elements, some explicitly [crowns = gold, brass 20, iron = Fr. le fer], some implicitly [flesh = clay, moys = silver?]. Pistol's title. Ancient, is reminiscent of "Ancient of days," the name of God found only in Daniel 7. 55 If Henry's "cause be not good," says Williams, "the king himself hath a heavy reckoning to make; when all those legs and arms and heads, chopped off in a battle, shall join together at the latter day, and cry all, "We died at such a place”; some swearing, some crying for a surgeon, some upon their wives left poor behind them, some upon the debts they owe, some upon their children rawly left. I am afeard there are few die well that die in a battle; for how can they charitably dispose of any thing when blood is their argument?" Henry's rebuttal, whose ideational content is all the more important since its quibbles take up a good deal of stage time, does not even broach the subject of "the cause.” Rather, Henry absolves himself again, by proving that one's sins are one's own problem. 234 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Than from it issued forced drops of blood: Five hundred poor I have in yearly pay, Who twice a-day their wither’d hands hold up Toward heaven, to pardon blood; and I have built Two chantries, where the sad and solemn priests Sing still for Richard's soul. More will I do; Yet all that I can do is nothing worth, Since that my penitence comes after all. Imploring pardon [4.2.301-1 1]. The problem has two levels, the religious and the numerical. As concerning the former, Henry's atonement is faulty since he is “still possessed Of those effects for which” his father committed his crimes [Ham. 3.3.53-4, said of Claudius]. As concerning the latter, his penitential multiplication of numbers cannot adequately account for the deed that came before it. The lack of a positive exchange rate between names, numbers, things, deeds, etc., means that all of the above appearances are translatable only inasmuch as they are as nothings - notings - texts illegible with respect to prior signified.56 The sole denominator of all accounts is their most basic existential ground as conjunctions of zeroes [see "cipher" in chapter 8]. Reminiscences of the Last Judgment come naturally to a play about a battle, for battles end with reckonings of the dead. The herald gives Henry “the number of the slaughter’d French” [4.8.76]. What follows is a long catalogue of names and numbers; same for “the number of our English dead” [104], The "score" is 126 French princes and lords + 8,400 “knights, esquires, and gallant gentlemen” + 1,600 mercenaries to 4 Englishmen “of name” and 25 commoners. In this apocalyptic ledger entry, arithmetic is again inextricable from language: the “reckoning” handed Henry both counts and lists. "He is dead," jokes Lysander in A Midsummer Night's Dream, "he is nothing" [5.1.305-6], If God knows the eternal "number of our own name " [Dee *ijr] and shall publish it on the "latter day” [4.1.137], we do not. All we know of our life is that it returns to the same, "ad idem") that it ends "where it begunne," that "Of nothing he made us, and we strive too. To bring our selves to nothing backe." Once we bracket God's prior knowledge of "our selves," all that each of us, as seen by ourselves, amounts to in our circuit is, as Crashaw's Death declares, a "hyperbolized nothing."57 56 Since the idea of naming as numbering comes from Hebrew where the two systems are translatable, it is curious to think of English as retaining the fact of equivalency while losing the actual correspondences. In the case of English, then, letters then refer not to particular numbers but to the "being” of number as such: not(h)ing. 57 Lucarini in Picinelli, "circinus," referring to Gen. 3:19; Donne "Valediction Forbidding Mouming,"I. 36, and "First Anniversarie" 11. 155-6; Crashaw, "Death's Lecture," 1. 11, respectively. 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. On that level, the "reckonings" given to Henry by Exeter and the herald "actually" read: oooooooooooooooooooooooo. 9. Judgm ent "Les langues des homines," says Kate, "sont pleines de tromperies” [5.2.115]. The fault in Shakespeare is, in addition the everything else, that which is misinterpreted, from Quickly’s malapropism in ‘T hat was not her fault. She does so take on with her men; they mistook their erection” [MWW 3.5.36-8] to “GOWER; Gentlemen both, you will mistake each other. JAJME: A! that’s a foul fault” [H5, 3.2.137-8]. It is therefore connected to the polysemy of language, the fact that one word may branch out into many meanings, which permits it to be mis-taken - “equivocation will undo us” [Ham. 5.1.138]. Polysemy is here regarded as the “fault” of language, the “another fall” of Babel as the result of which “de tongue of de mans is be full of deceits."58 Polysemy is one of the phenomena clearly giving the lie to referentialism. A single linguistic form that has many meanings cannot possibly point at a pre-existent signified. I once had a student who wrote a paper on irony. She thought the noun had some strange relation to the adjective having to do with iron, the metal. “Irony” the noun and “irony” the adjective are of course not the same word, since their etymologies have nothing to do with each other. Still, should I ask the reader to define “irony,” how will the reader know what word to define? Only by context; “irony” per se, so to speak, is only “irony.” The same goes for the far more common instance when we are not dealing with two homonyms, but a single word. Given the word “chair,” how is one to know whether it refers to what we sit on, or to the head of an academic department? What pre-existent signified does the word “chair” refer to? Meanings are posterior to signifiers; to reckon what any one word means at any particular here and now we need to have several words joined by syntax. The signifier itself is an O per se - O', it is only a fork, a jointure between several possibilities. It acquires meaning only when locked into an account. Since any instance of conjunctio constitutes an account, accounts are infinite in number. 58 Comp. Kate's “tromperie” to letting out air in Othello's “wind instrument” whereby “hangs a tale” C3.1.6-10; for farting as parody of the trumpets of angels, see Inferno 21.139, where the leader o f demons “avea del cul fatto trombetta”). The duplicitous polysemy of language is thereby tied to the Great Equivocator himself [see Cassio on wine as “invisible [...] devil” that makes one “discourse fustian with one’s own shadow” 2.3.270-5], whose cloven hoof Othello searches for on the body of Iago [5.2.587]. For "honest" Iago as devil in the sense of origin of division, see esp. his “I am not what I am” [1.1.65], glaring parody of G od's Ego sum qui sum in Ex. 3:14. 236 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The signifier itself is only a form signifying nothing. In accounts, however, it multiplies: “Est-il impossible d'echapper la force de ton bras?” “Brass, cur!” [4.4.16-8]. What we see or hear as one is in fact two: two disparate accounts and the imprecise duplication of the form of a single "thing'' in them, the duplication that is, in fact, the doubling of the thing. Let us return to the I. In this chapter, sometimes explicitly, sometimes implicitly, I played with the fact that the I, a Renaissance figure for the one, is also the first-person singular pronoun. Shakespeare's treatment of numbers may also be regarded as a theory of the self. We have seen how Henry V gave the lie to the concept of the male as a perfect, enclosed and self-identical self. Rather, the self is, to recall a formula applied to zero, nothing in itself, something in combination. What does this mean? The principle at hand is what we have already encountered in Lear's "change places and handy-dandy, which is the justice, which is the thief?" [4.6.147-55]. It is "place makes the man." The self exists only as part of sequence and is determined by its other elements. What I "am" depends on the things I am conjoined with - for instance: my occupation, my speech, my actions. It is therefore posterior to them. There is no pre-existent, integral self: there is only a set of selves in accounts, i.e. in chains of appearances, chains of events.59 This means that what I am is my deeds; my deeds, since they depend on conjunctions with things that are not me, intend the world. What I am is an account composed of the intersections of infinities of other accounts. The origin of all of this in Shakespeare lies in his contradicting the Platonic model of signification whose signifiers point to autonomous, prior, and positively existent 59 The moment when we face two conflicting accounts, with the result that our account's illusion of naturalism breaks, we glimpse our self to be no I, but the O, the dyad, the origin o f multiplicity: “My lord is not my lord," [Oth., 3.4.121]; “Why, this is not Lear. Does Lear walk thus, speak thus? Where are his eyes? [...] Who is it that can tell me who I am?” [1.4.217-22]; “Leonato, stand I here? Is this the Prince? Is this the Prince’s brother? Is this face Hero’s? Are our eyes our own?" [MAAN 4.1.69-71]. No, our I’s - "if there be such a thing as I" - are not our own [Donne, “To Sir H.W. at his going Ambassador to Venice,” 1. 33). Remarkable here is the answer Othello provides to Emilia's whodunnit: “Nobody - 1 m yself’ [5.2.125]. Renaissance paradoxes such as Ulrich von Hutten’s Nemo [see Colie 296-9; text in Lateinische] quite explicitly relate Odysseus’s celebrated pun to the working of language in the exemplary case of what Bovelles calls the affirmative and the negative meanings of the word “nothing,” but, in the case of Shakespeare's captain declaring himself Nemo, the paradox goes much further. “Nobody” intends the situation - Iago. Cassio. Emilia, Aleppo, the handkerchief; it intends Othello as the cipher locked within the situation and controlled by it; it intends the fact that our eyes are not our own, and therefore somebody else is guilty. At the same time there is no somebody else - Othello might very well say, “je est un au tre” for what is Othello but the situation, Iago, Cassio, Emilia. Aleppo, the handkerchief? “’Twas I that killed her" [5.2.130]. Not that I think Shakespeare denies freedom of the will, but for him it forms a kind of circle, which this is neither the time nor the place to expound. In Othello’s case, however, the is precious little free will. He is passive and subordinate to his name [Gk. 8f[Xus = female]. 237 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. signifieds, replacing it with a model stressing syntax, and where the signifler has the priority. One example of the latter model is Hindu-Arabic numerals. Another is theater. If one may talk of the essence of theater, that essence is anti- essentialist on the level of ontology, and anti-referentialist on the level of signs. The poor player, when he struts and frets his hour upon the stage, represents someone he is not. His speech, his clothing, his actions - in short, all of his appearances - refer not to his “true” self, but to another self. He “writes” not his own, but his brother’s character. This latter signified self is not prior but posterior to the signifiers borne by the player. It is posterior in two senses: 1) of the signified being the effect of the signifier, and not, as in referentialism, the cause; 2) of the “integral” signified forming only at the end of the play, when the last in its sum total of signifiers becomes evident. 238 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 SKOVORODA, CARTESIAN PLATONIST Desired, the snow falls upward, the perfect future, a text of wheels. xoay B m c c t o in a n K H H a O h H a n e ji cKOBopony.1 0. Introduction Every dissertation is supposed to have a conclusion. Since I don't feel as if I had composed a dissertation, but rather something in its stead, a work whose qualities, as those of Abie Hadjitarkhani's Alice of the Fine Parts, "are so many and so different, that they do not make a whole" [84], it's more fitting that I write not a conclusion, but rather something in its stead: a chapter with a new protagonist. Luckily, the work of our new protagonist, the eighteenth-century Ukrainian philosopher Hryhorii Skovoroda, is of such a nature as to bring together almost all the strands we have dealt with up to now. Skovoroda is in many senses a Platonist, yet his philosophy foregoes any discussion of Platonic Ideas.2 This is a pretty striking omission. If the job of Ideas in Platonism is, recalling another Alice, to differentiate between a raven and a writing desk, Skovoroda does not make any attempt to explain the fact that some things are more equal that others, i.e. that they come in genera and species. I shall argue that this refusal to differentiate is a necessary consequence of the system Skovoroda proposes, a system focusing on existence rather than being, reified and separated from appearances. At the end of this chapter I shall also try to account for the omission of Ideas historically, by relating it to certain trends in seventeenth-century philosophy, notably Cartesianism. My basic claim here is: no Descartes, no Skovoroda. 1Michael Palmer, "Letters to Zanzotto: Letter 5"; Samuil Marshak, "Vot kakoi rasseiannyi;" 1.113. : One of his mentions of Plato's Ideas designates them as unknowable objects of faith, handy but not obligatory [Dialog, sig. 4; 1.190]; another cavalierly identifies them with what we shall call the geometical figures of things [Potop, sig. 7; 1.539]; „MoaceTb 6 b r r b , h n p a B a a , * rro T b ic j tt a e a H H O o 6 p a 3 H b tx n e n a T e H 3aK JuoH aeTC H b o a h o h T a tto it ace, a o a H a p a c x o a H T C H b T b tc a i a y , h e aH H C K yaejtbH btH M O je j ib co K p b tJicH b a e c jiT H TbicH H ax c o c y a o B . [...] E c j i h He n o H H M a e u ib , o o y a a a f i p a 3 y M c b o h b nocayuiaHie.“ „ B c t T pw M bipbi [macrocosm, microcosm, Bible] c o c t o h t H3 a B o x e a H H o cocTaBjifltouzHX ecTecTB, H a3biB aeM bix Marepia h <popMa. CiH $ o p M b i y f la a T O H a H aau B ato T C Ji HMSH, CHpiiHb BHjrkHHX, BHMbl, o6pa3bt.“ 239 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since I am devoting an entire chapter to somebody unknown to most of my readers, let us start with a biographical overview. Skovoroda, bom in 1722 into a Cossack family, in 1734 enrolls in the Mohyla Academy in Kiiv, from where he is summoned to St. Petersburg in 1741, to sing in the Imperial Choir. After three years at Court, he returns to Mohyla to finish his course-work at the penultimate level, that of philosophy. At that point, the philosophy taught at Mohyla is exclusively scholastic; from the summaries of lectures attended by Skovoroda we see him exposed to scholastic logic, metaphysics and physics, including treatments of universals, body, number, and the unlimited [Stratii, Litvinov, Andrushko 289-302]. In 1745 Skovoroda accompanies Major-General Vishnevskii to Tokay, in order to ensure a steady supply of Hungarian wine for Her Imperial Majesty, who was much dependent on it.3 He makes the most of his sojourn abroad, traveling to Vienna, Budapest, Bratsilava, and perhaps elsewhere, in an attempt to "acquaint himself with people famous for learning and knowledge," according to the testimony of his friend and first biographer Mikhail Kovalinskii [Skovoroda 2:489]. Back in Russia in 1750, he receives a post in poetics at Pereiaslav College, and gets promptly thrown out for teaching poetry according to a new and improved method, and also for impertinence. Again at Mohyla from 1751 to 1753, Skovoroda studies theology with professor Georgii Koniskii [Makhnovets 73-5]. Off and on from 1759 to 1768 he teaches various subjects at Kharkiv College, until he is again fired, this time for unorthodox lectures on the Catechism.4 From then onwards he lives at the estates of his admirers, engaging them in philosophical conversation, taking long walks, and composing his treatises and dialogues. He dies in 1794. I do not pretend to cover all of his philosophy. I also admit that every now and then he makes a statement I cannot reconcile with the rest. However, such incongruities are few, and primarily concern matters subject not to reason but revelation.5 1. Figurative Speech Skovoroda is a hermeticist. The hermeticism I have in mind is not alchemical or astrological but theoretical. Its ingredients are many and include the trope of the book of nature, Egyptian hieroglyphs as understood in the Renaissance, the view of Platonism stressing signification, the so-called Corpus Hermeticum, the theory of prisca theologia, 3 "I ju s t c a n 't do w ithout it," 0 6 0 h t h t o k 6e3 o H a ro He M o ry ,“ w rote E lizabeth; qtd. in B alabash 92. 4 A rev ised sum m ary o f w hich is now available as Nachal'naia D ver'ko Khristiankomu Blogonraviu; S k o v o ro d a 1.14-26. 5 1 do not understand how , given S kovoroda's portrait o f G od, he can still say th a t G od "gets horribly furious" „y*acH O »pHTCJi“ [Narkiss sig. 31; 1.57]. Perhaps he is ju s t being figurative. 240 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the Holy Scriptures. It fuses all these into a coherent whole by interpreting things as signs and signs as things, with signs pointing to what I called the prior signified. I shall now give a brief overview of the hermetic tradition, and clarify the constitutive role it accords to Plato’s understanding of signification.6 Perhaps this shall impress upon the reader a sense of wonder before Skovoroda, the hermeticist who finds a way to ignore possibly the most important tenet of the system he inherits. 1.1 The Hermetic Hypothesis Sir Thomas Browne provides a succinct formulation of the parallel between the Bible and the book of nature: There are two bookes from whence I collect my Divinity; besides that written one of God, another of his servant Nature, that universall and publik Manuscript, that lies expans’d unto the eyes of all; those that never saw him in the one, have discovered him in the other; [...] surely the Heathens knew better how to joyne and reade these mysticall letters, that wee Christians, who cast a more carelesse eye on these common Hieroglyphicks, and disdain to suck Divinity from the flowers of nature [78-9]. What Browne means is that the Creator has encoded natural forms with religious meaning; thus, for instance, "those strange and mysticall transmigrations that" Browne has "observed in Silkwormes" direct his mind to the Resurrection [110]. "Nature [,] the art of God," [81] is as full of allegory as the Scripture; every object presents an image of a higher truth; Browne, above, refers to objects as "Hieroglyphicks." The appellation is standard. It comes from the false decipherment of Egyptian hieroglyphs which took the Renaissance by the storm, giving rise to, among other things, the art of the emblem [Iversen 73-5], The Florentine discovery, in 1419, of the Hieroglyphica of Horapollo reinforced and transformed the medieval version of the trope of the book of nature [for medieval versions, see Curtius 319-326]. Writers on Egyptian hieroglyphs during the Roman period, one of whom appears under the name of Horapollo, no longer knew how to read these "sacred carvings," but interpreted them symbolically [Iversen ch. 2, Boas in Horapollo 3-40, Zhmud' 169 n. 15]. The Egyptians, so the story went, used pictures of objects instead of letters. These pictures indicated neither sounds nor words, but concepts: "as in by the Bee they signified a King ruling his Commons with great moderation and gentleness, by the Gos-Hauk, they meant speedy performance of their affairs" [Polydore Vergil 3.8]. The "mute rhetoric" [Ripa, Iconologie eijv] of the land of the pyramids was 6 Reader seeking more details should turn to the work of Iversen, Heninger, and Gombrich, as cited in my bibliography. 241 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. alleged to be universal and eternal, open to decipherment by anyone familiar with its principle and with the properties of portrayed objects. The Hieroglyphica was discovered at the same time as the Corpus Hermeticum, a collection of Platonist-influenced texts dating from approximately the same period but attributed to the most unfortunately fictitious Hermes Trismegistus, whom the Renaissance regarded as the founder of Egyptian civilization [Yates 43, 48-9]. Sir Thomas Browne characterizes "the Philosophy of Hermes" as that this visible world is but a picture of the invisible, wherein as in a pourtraict, things are not truely, but in equivocall shapes; and as they counterfeit some more reall substance in that invisible fabric [74]. According to their discoverers and interpreters, the Corpus Hermeticum and other writings of Hermes7 are passionately dualist, regarding the world as composed of the less real sensibles and the real and immutable non-sensuals, existing in the Platonist signifier- signified relationship. The principle of "equivocall shapes" counterfeiting something invisible and "more reall" was also thought to be at work in hieroglyphs, their invention or discovery being attributed to the same sage.8 Giving credence to Horapollo and monstrously antedating the Corpus Hermeticum, Renaissance hieroglyphers reversed the true line of intellectual descent, and placed Hermes at the source of Platonism. A late Greek tradition sent Pythagoras and Plato to Egypt to study with the local priests [Riginos 64-65, n. 15-16; Zhmud 63-4; Ficino 1.764], What the two studied was Egyptian hieroglyphs. It was in imitation of Egyptian hieroglyphs that Pythagoras developed his so-called symbola, allegorical injunctions such as "Abstain from beans,"9 whereas Plato simply "tira de ces Figures Hyeroglifiques la meilleure partie de sa doctrine" [Ripa, Iconologie eiijv]. Plato's ontology therefore reported on the signifying nature of the physical world as disclosed by Hermes. Hermes was also thought to have influenced the Jews. God's own secretary, the future writer of the Pentateuch was, of course, brought up at the court of the Pharaoh, 7 Other writings already known to the Renaissance and ascribed to Hermes include The Smaragdine Tablet, which most accords with Browne's summary, the Liber XXIV Philosophorum, which defines God as "a sphere whose center is everywhere and circumference nowhere.” and a number of alchemical treatises. 11Some Renaissance hieroglyphers identified Hermes with Theuth, the Egyptian inventor of writing in Plato's Phaedrus [274c-5b]; others, elaborating upon the Jewish Antiquities o f Josephus [1.70-2], ascribed the invention of hieroglyphs to Adam or his close descendants, with Hermes assigned the role of post- deluvian decipherer and popularizer. For the latter version, see Spafarii 127; Giarda in Gombrich 150; also Kircher; comp, with Manetho in Copenhauer's Hermetica xv. 9 See Plutarch's "Of Isis and Osiris," Moralia 354 e-/and other references in Heninger 246-7, 254 n. 35. Skovoroda's interpretation of this symbol [Kol'tso sig. 22; 1.274] repeats that given by Plutarch in other essays; see Heninger 257, 284 n. 87. 242 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where he acquired perfect knowledge of the local "philosophy conveyed in symbols."10 Thus Egyptian hieroglyphs inspired the stylistics of the Bible, and the Jewish mode of sacred speech in general, as encountered in the language of the Hebrew prophets. Finally, not only the prophets but Christ himself, participating in the tradition, "cacha sous des Paraboles la plus-part de ses diuins secrets."11 Children mimic the speech patters of their parents. Since God the Father wrote the book of nature, whose functioning inspired hieroglyphs, it is only natural that "the Word which came forth from the Light” as the "son of God" [Corpus Hermeticum 1 p. 6] should also be in the habit of saying one thing, and meaning another. As the dualist ontology of Hermes and Plato had its rhetorical counterpart in the figurative speech of Pythagoras and hieroglyphs, so the figurative speech of the Bible had its "ontological” counterpart in Paul's dictum that "the invisible things of [God] from the creation of the world are clearly seen, being understood by the things that are made" [Rm. 1.20; Vulgate: "a creatura mundi"]. We can see that this whole syncretist network of preposterous identifications rested on the concept of signifying by figures, whether in the sense of rhetorical tropes, or of physical objects, or of their depictions. Our overview shall end by distinguishing between the ontological and the religious signifieds of these figures. The popularity of Egyptian hieroglyphs is in large part due to the fact that they offered Platonists of all stripes a pristine model of signification as such. Lacking in syntax and apprehended at once, hieroglyphs partook in "the non-discursivess of the intelligible world" [Plotinus 5.8.6.7] as human language did not. Moreover, their signs were general: the hieroglyphic cat portrayed cat pure and simple and no cat in particular. Already Plotinus regards Egyptian hieroglyphs as the most adequate way to designate the Plato's Ideas [5.8.6.1-10]. Ficino goes further, explaining the general aspect of Egyptian signs by the character of divine thought [1768; see Gombrich, 157-9, 232, who quotes both passages, for discussion and other examples]. The fact that the relationship between words and things on the one hand, and Ideas on the other, is far more complex, privileged rather than imperiled the hieroglyphic model. However, neither Plotinus nor Ficino intimates that hieroglyphs represent Ideas only. What hieroglyphs as well as objects in the book of nature express is religious and moral truths. They do so by means of Ideas, but mediating them through metaphor: thus the hieroglyphic vulture stands for "mother," because, as everybody knows, all vultures are female [Horapollo 1.11], and therefore the obligatory gender of vulture must be 10 Philo De Vita Mosis 1.23; see Heninger 247; Yates 11, 26; Scott 33 for this idea in the Renaissance. 11 Ripa, Iconologie eiij"; same comparison in Valeriano in Iversen 72-3; Quarles in Lewalski 185-6, Iversen 83. 243 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. inscribed in its Platonic Idea. Although the book of nature speaks in particulars rather than universals, the "common Hieroglyphicks" of God signify according to the same general framework. 1.2 Biblical Figures in Skovoroda It is with Egyptian hieroglyphs that we shall start our study of the concept of figure in Skovoroda. Skovoroda is acquainted with the intellectual genealogies spun by Renaissance hermeticists, and subscribes to them. As far as the putative influence of Egyptians on the Greeks is concerned, he alleges it to extend not only to philosophy, but also mythology. He regards the Greek myths of Narcissus and of the Sphinx as garbled versions of the symbols used by Egyptians in their moral teachings. The optimism of his "reconstruction” o f these myths is truly hair-raising [Narkiss sig. 1-2; 1.27-9; Kol'tso sig. 18; 1.267-9], Although he passionately affirms the general thesis of prisca theologia [St'on 293], Skovoroda shows himself of two minds in evaluating how closely the religion of ancient Egyptians resembled that of the Bible. At one point he refers to what is evidently Egyptian "theology" as "the mother of the Hebrew,"12 whereas at other times he regards the Egyptians as having the right idea, but erring in details. The right idea consists in the principle of figuration. The Egyptians invented the "figures" which were subsequently "borrowed" by Moses [Silenus ch. 17; 1.400]. Elsewhere, Skovoroda expands on this claim; "all heavenly and earthly figures employed by the ancients, and particularly by the philosophers of Egypt, were dispossessed by Israel from signifying their mysteries, and dedicated to God."13 Thus the figurative speech of the Bible is nothing else but the discursive analogue to Egyptian hieroglyphs: "the prophets [...] sing in figures;" "you’ve heard the figures o f prophetic speeches. Figure, image, parable, tale - it's all one.”14 Skovoroda's understanding of the word "figure" in this context is dual. First of all, "figure" means metaphor, the representation of one thing by another. The Bible "borrows words from you that designate your base superficies, for example: feet, hands, eyes, ears, head, clothing, bread, vessels, house, soils, cattle, earth, water, air, fire;" but employs them in a strictly spiritual manner.15 Thus, the word "foot" may mean human desire, 12 „ o 6 B e T u ia J n .u i o o r o c j io B iH enmeTCKiji, xxe e c r b M a T e p e B p e f i c i c ij i ," [N in tr o .; 1.26]. 13 „ H e T O JibK o 3 e M JH o c e s r u io z ta M H p o a c a e H is M H , ho Bcfe H e 6 e c H b iit h 3 e M H b is c f m r y p b t, a o M a u iH iH apeBHHM, H a n n a n e b E n u r r t jn o 6 o M y ix p a a M , o t h jib o t o 3 H a n e H iji tbhhctb hx, n e p e H e c H 3 p a m ib b n o cB H m eH ie r o c n o a e B H " [Kol'tso s i g . 47; 1.307; s e e a l s o s i g . 45; 1.304]. 14 „noK>T b < j ) n r y p a x , “ [Dvoe s ig . 7; 1.168]; s e c o n d q u o te f r o m Sion 270. 15 „ O H a 3 a H M C T B y e T o t T e 6 e caoBa, no/myio t b o k j oK O JD tH H O CTb 3HanauiiH, H a n p H M tp : Hont, p y K H , o h h , yum, r o j i o B y , o a e * a y , x jif c 6 , c o c y n u , h o m , rpyirra, c k o t , 3 e M J n o , Boay, B 0 3 n y x , o r o H b ,“ [Askh. s ig . 37; 1.135; e x a m p le s sig . 38; 1.135-6]. 244 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. whereas "tree" may mean man. If we recall the patristic opposition between the Alexandrian (Clement, Origen) and the Antioch (Chrysostom) schools of exegesis, Skovoroda is clearly on the side of the former. He focuses on the allegorical, and disdains the literal sense - quite vehemently, as we shall see. The second meaning of "figure" in this context differs from the first only superficially. Whereas the first treats "feet, hands, eyes," etc., as words, the second regards them as verbal descriptions of things, and attends to the latter. Here Skovoroda draws upon the tendency among some seventeenth-century exegetes to speak of Biblical objects like the burning bush, Samson’s lion, Joseph's coat of many colors, as analogous to Egyptian hieroglyphs [Joseph Hall qtd. in Lewalski 185]. Not some, however, but all objects in the Holy Scriptures are, for Skovoroda, hieroglyphic. The discussion of "the figures" used by the ancients "to secretly signify eternity," in Kol'tso is immediately followed by the claim that the Bible itself was created by God from sacred and mysterious images: the heavens, the moon, the sun, stars, evening, morning, cloud, rainbow, paradise, birds, beasts, man, and so forth. All these are images of [...] divine wisdom shown to Moses upon the mountain; all these and all creatures are shadow signifying eternity.16 Clearly, Egyptian schooling prepared Moses for collaboration with him, who had used the same rhetorical principle in composing the book of nature. In a lovely burst of vernacular, Skovoroda imagines the two brainstorming their literary endeavor. God: Hey Moses! Let sunlight be my figure! It will point to my truth, incomprehensible to mortals, which shines in your perishable nature.17 In other words, Skovoroda sees the second, invisible level of the Bible as, so to speak, one extremely elongated cartouche: a sequence of things as discrete signifiers, "divine menagerie, where people, cattle, beasts and birds are figures."18 16 „ M H e K aaceT ca, h t o h c a M a 6 h 6 ju x e c r b 6 o r o M c o 3 a a H a a c b h iu c h h o —TaHHCTBeHHbix o 6 p a 3 0 B . H e 6 o , n y H a , c o j i m i e , 3 B t3 n b i, B e n e p , y r p o , o 6 jia ic , a y r a , p a ir , rr rH n b i, 3 B t p a , q ejio sfeic, h n p c m j t . Bee c ie c y T b o 6 p a 3 b i B b ic o T b i, H eSecHO H n p e M y a p o c r H , n o ic a 3 a H H O u M o a c e i o H a ro p fe ; Bee c ie h b c h T B a p b e c r b c r e H b , o o p a 3 y i o m a « B tH H o c r b “ [ Kol'tso s ig . 2 3 ; 1 .2 7 5 ; s e e a ls o Kol'tso sig 4 5 ; 1 .303- 4; Silenus c h . 10: 1 .3 9 2 -3 ; c h . 17 1 .4 0 0 ], 17 „ H p e n e 6 o r : — Cjiymafi MoHceti! IlyiiiaH 6yaeT c o j i h c h h u h catT (JjHrypoio Moeio! OHa CTaHeT noKa3biBaTb nepcroM Hcnmy mok>, cuuomyio b miHHofi Barnefi Harypfc, HesfcpoaTHoio C M epT H biM ," [Silenus ch. 6 ; 1 .3 8 6 ], 18 „ 6 H 6 jiia e c r b Mip CHM6ojniHHwfi h aaepHHea 6o*ifi, a moae, c k o t u h rrrHUbi cyrb <t>Hrypbi h xepyB H M b i, ciiprfeHb B03KH, Be3yuiie BkiHOCTH coKpoBHiue," [Silenus ch. 10; 1 .3 9 2 ]. 245 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since all that the Bible stands to lose if it were translated into ancient Egyptian is the flotsam and jetsam of language, those parts of speech that serve to connect, not picture, Skovoroda calls it the "Hebrew Sphinx" [Zhena ch. 4; 1.408]. Maiden-headed and lion­ bodied, the de-winged monster here associated with the other Thebes killed all those who could not decipher her riddle [Kol'tso sig. 18; 1.268; Sion 270, 291]. Such too is the Bible - "think not that of anything else [...] it is written 'like a roaring lion ravening his prey'" [Zhena ch. 4; 1.408 quoting Eze 22.25]. Sometimes, when Skovoroda speaks of this "Hebrew Sphinx" - well, we can just hear the sudden choking of his listeners: The Bible is a lie [...]. This seven-headed dragon (the Bible), vomiting cataracts of acrid waters, has covered the whole earth with superstition [...]. The Bible is a rather bad and uncomplicated reed-pipe [...]; thorn-happy brambles, foolishness, to say it with Paul, of God; or, to speak vulgarly, dirt, garbage, merds, human pus [...]. She is that lion [lev], circumambulating the Universe, roaring and ravening, pouncing on the poor reader from the left [levoi] side. How he perishes in her infernal jaws! [...] Isn't she also that whore of Solomon? [...] She is that whore! "Words like floodwaters, flattering is the tongue of that whore." She causes universal flood. 19 What on earth is going on? Skovoroda is putting his foot down on what he designates as "idolatry." With respect to biblical exegesis, Skovoroda’s "idolatry" equals what we call fundamentalism, and may be defined as privileging the sensus litteralis sive historicus, concentrating on literal meaning at the expense of the allegorical. "The letter killeth," says Paul, "but the spirit giveth life" [2 Cor 3.6]. If we read the "Let there be light" of Genesis as referring to a physical stage of Creation, we are already idolaters. The phrase may refer to it or it may not (Skovoroda does not dwell on either possibility), but that's simply beside the point. The point is spiritual. Yet words do have literal meanings. "Let their be light" does appear to describe Creation, thereby misleading its naive readers, making them focus on the physical: "What a great multitude reads the Bible! Yet without profit is this house of God locked and sealed" 19 „Ei6aia e c T b a o x b “ ; „Ceit c e a M iir a a B b iH a p a ic o H ( 6 i 6 m x ) , eon ropK H X x j u 6 u H 3 6 a e B a a , Becb cboh map 3eM HbiH n o K p b u i c y e B tp ie M “ [Silenus intro.; 1.375 and 1.373]. „ 6 h 6 j u j i B e cb M a e c T b nypHoio h H ecaojK H O io a y a o t o , e c T jm ee o 6 p a iu a e M k u a m H M i u i o t c k h m a fc a a M , 6 o a y u iH H TepHOBHHK, r o p b K a a h HeBfcycHaa B o a a , a y p a a e c T B o , e c r a H c f l a B a o M c ic a 3 a T b , 6 o * H e , a a n cicaacy aaiiHo, M O T b iaa, a p x H b , r p r n b , i h o h H eao B tH ecK iH 1* [Kol'tso sig. 16; 1.265]. Bible a s lion and w h o re of Solomon from Sion 292; alluding to Pr. 6.24?. 246 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for them.20 It is in reference to this illusionism of language that Skovoroda speaks of the "historical or moral hypocrisy" of the Scriptures.21 In calling the Bible the "Hebrew Sphinx," Skovoroda in part draws attention to its duality of aspect.22 For him, biblical speech resembles a wall, one of whose faces is turned towards darkness and the other, the inner, towards the Orient: gilded is that second face with the light of the Almighty.23 The visible, "dark" face of the wall presents a series of Figures, "uneven, degenerate, curled, strange; to speak straightly - steep, circumlocutory, knotted up, unusual; to put it briefly - women's tales, vain eccentricities, lies out of the mouths of babes" [Sion 270]. But when these figures meet their decipherer, their Oedipus, then does the Bible turn into "a pharmacy acquired by divine wisdom for the healing of the world of the soul," "a testament, with the world of God sealed within, [...] a fenced-in paradise of amusement, a treasure encased," a shell enclosing a precious pearl.24 2. Figure in the Book o f Nature 2.1 Parallel Scripture and Positionality The hermeticist tenet of the book of nature as parallel Scripture has been illustrated for us by Sir Thomas Browne. Skovoroda subscribes to it fully. In the same way as "the Bible is a lie, [yet] the foolishness of God does not consist in teaching us this lie, but only in imprinting upon it the vestiges and tracks that lead the crawling mind up to the altitudinous truth," so "all creation is a lie mendacious and inconstant, and all creation is the same field of divine vestiges."25 The word I translate as "creation" is tvar', which like the Latin creatura can stand for any creature as well as their totality, but not the act. Like the "figures" of Moses, all creation is a "shadow signifying eternity," for "God made all tvar' 20 „ K o J ib M H o ro e m h o h c c c tb o HHTaeT 6 h 6 j u i o ! H o 6 e 3 n o J ib 3 b i c e fi a o .v t 6 o a d i t 3 a n e p T h 3 a n e H a T a H ,” [Kol'tso s ig . 14; 1 .2 6 3 ]. 21 „ C e a e c T b n p u p o u H b iH u m u i b 6 h 6 j u h ! H c r o p i a j r a o f o h j i h M o p a j m o f o J u r a tM e p H o c T b fo T a x c o rm e c T b ({ m ry p b i h c h m b o j ib i , h t o H H oe H a J u r a t , a H H oe H a c e p a i r t , “ [Silenus c h . 7; 1 .3 8 8 ]. “ H is S p h in x , b e in g E g y p tia n , h a s n o w in g s ; s e e e .g . Kol'tso s ig . 18; 1 .3 7 2 . 23 „Ci« c r t H a HMteT Te.viH yio c T o p o H y , r y , K O T o p a » c m o t p h t k o Tvrfe. H o C T p a H a e » , k BOCTOKy o o p a u ie H H a H , e c r b B H yrp eH H su t h b c h cB tT O M B b n u H x ro 6 o r a n o 3 a a iu e H H a * ,“ (TV sig . 4 2 ; 1 .6 8 ]. :4 „ o H a e c r b a ir r e K a , o o a c ie io n p e M y a p o c r i e i o n p i o 6 p t T e H H a » , a r m y B p a n e B a m ji a y u ie B H a r o M u p a , h h oaH H M 3eM H biM a e x a p c T B O M H e H 3 irtJ U ie M a ro ,“ [Kol'tso s i g . 17 ; 1 .2 6 6 ]. „ E h 6 j u b HaM o t npeaK O B H auiH X 3 aB tT O M o c ra B J ie H a , a a h c a w a o H a e c T b 3 a B tT , 3 a n e H a T a e B m a ji B H yrpH c e o e m h p o o a c iit, KaK o rp a a c a e H H b m p a n y B e c e a H T e a b H b iH , Kaic 3aK jno<ieH H biH k h b o t c o K p o B iim e , KaK n e p a o B a M a T b , a p a r o i r t H H t i t u i e e n e p a o BHyrpb c o 6 j n o a a B i u a j i , “ [Kol'tso s ig . 1 9 ; 1 .2 7 0], 15 ,E i 6 a i j i ecTb a o * b , h 6 y « C T B O 6 o a c ie He b t o m , « it o 6 j c k h H a c H a y n a a a , h o t o j i b k o b o JDict H a n e n a T a e a a c j r t a b i h c re 3 H , n o j n y u i H i i yM B 0 3 B o aK iu iJi k n p e B b ic n p e H H tw H C T H irt [...] . Bc« * e T B a p b e c r b a o x b H enocTO H H H a h o 6 M a m H B a , h b c s T B a p b e c r b t o n o a e c a ta o B 6 o * i a x , “ [Silenus intro.; 1.375]. 247 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. be the figures of his glory".26 Skovoroda is quite indiscriminate in whether to explain his basic concepts by examples culled from the world or from the Scripture. Admittedly in one place he does claim that "the symbolic, secretly signifying world of Moses is a book [and] does not concern the inhabited world at all."27 Yet if the Bible is entirely self-enclosed and non-referential, one could not speak about it and the world interchangeably, nor think the former has any effect on the latter, as Skovoroda constantly speaks and thinks. And, if the Bible did not concern us, who happen to be part of the world, what reason should we have for reading it? Rather, what this sentence argues is that the Bible has nothing to do with the world regarded as an end of itself. Skovoroda’s definition of idolatry intends not only fundamentalists, but all those who shower appearances with undue attention: like pagans worshipping visible things [Nach. Dver' ch. 1; 1.17; Narkiss sig. 17; 1.45; Silenus ch. 4; 1.383], or even scientists. For Skovoroda, the arts and sciences, including arithmetic and geometry, lose their value as soon as they are employed for profane needs, i.e. those divorced from figuration [Kol'tso sig. 48; 1.307]. Investigations of the infinity of "Copemican worlds," the fact that "the earth is a planet," that "there is a moon near Saturn, and perhaps even more than one"28 are misdirected methodologically. As with the historicity of the Bible, whether or not Skovoroda believes in the factual accuracy of science is of no importance. One should behave in the world like at the opera: make do with what comes before one's eyes, and not go peering backstage [Nach. Dver' ch. 4; 1.20]. Skovoroda’s system offers a position and not absolute and universal knowledge. This position consists of Skovoroda's unique version of the book of nature. The reason we read the Scriptures is that they provide us with a marker: the more obvious fact that natural objects in the Bible stand for something else suggests that the same objects in the world - men, beasts, birds, heavenly spheres - are also figurative.29 ;s „c T e H b , o6pa3ytoiuajc B tH H o c r b ” [Kol'tso s ig . 2 3 ; 1 .2 7 5 ]. „ 6 o r , b c io T B a p b 3 a ± a a B cnaBbi C Boea (])HrypaMH,'‘ [Silenus c h . 9 ; 1 .3 9 0 ]. 27 „M oH cefiC K H H ace, CHM OoaHHeciciH (T a H H O o o p a a H b rii) M ip e c r b K H u ra . O H a h h b n e w He T p o r a e T o o H T e jib H o ro M ip a ,“ [Silenus c h . 6 ; 1 .3 8 4 ] :s „K onepH H K aH C K iH M b ip b i,“ [Narkiss s i g . 14; 1 .4 1 ].„ 3 e M Jix e c r b o n a H e T o io [ . . . ] o k o j i o C a T y p H a e c r b J ly H a , a M oaceT 6 b m > h He o a H a “ [Kol'tso s ig . 5 ; 2 5 2 ]. :9 For m a r k e r, s e e , e .g ., „ K t o 6 m M o r a o r a a a T b c a , h t o H o e B a a y r a o 6 p a 3 e c r b CBnmeHHMX o H o n iH , e c r jiH 6 b i He c m h C H p a x o B , n o x B a j u x 6 o a c ec T B e H H y io n p e .v iy a p o c r b , c ic a 3 a n : „ O i a B a b m c o tm , T B e p a b HHCTOTbi... JlyHa B ceM ... v rfecau n o HMeHH C B oevty e c r b . . . . J J o 6 p o T a H e 6 e c e , c a a B a 3B C 3a... Bnacab a y ry h o a a r o c a o B H c o T B o p m a r o k>, a t a o n p e x p a c H a cix H ieM c b o h m “ ( m . 4 3 ) “ [Kol'tso sig . 2 3 ; 1 .2 7 5 ]. „ E c jih 6 m Henofffcic n p o 3 M B a a c a k c 3 h o m , a h h k t o T o r o He 3 H aeT , b t o BpeMX h HM eHOBaTb e r o k o 3 H o m e c r b a f e a o n y c r o e , n o T O M y h t o c n y u ia T e n e B a Mbicab H a K 0 3 J ii o c ra H O B H T c a , He a o ii u io B a o n e a o B t K a ,“ [Askhan’ sig. 2 3 ; 1 .1 1 5 ]. 248 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 Figure as G eom etrical A spect of Things Skovoroda also employs the term "figure" or its equivalents in talking about the world. Instead of definitions, he provides illustrations. One of them opposes the paint of a fresco to its line drawing (risunok). He dismisses paint as "dust and emptiness"; the "power" of the picture, that which endows it with mimetic abilities, is the "drawing, or proportion and distribution of paints."30 While Skovoroda's opposition of paint to drawing is unambiguous, the opposition of paint to its "proportion and distribution" is a good deal less so. Do "proportion and distribution" refer to the juxtaposition of colors, the relations of hues, or simply to where the paints are located by the line drawing? If we take the next example and its use of the word "proportion," it seems as if the latter is the proper alternative. To be sure, this scandalizes our aesthetics. Yet all we know of Skovoroda’s aesthetics points to him conceiving of paintings as emblems - the fresco in question, for instance, bears a lemma - and in the art form of emblematics the line is the dominant, or rather exclusive, vehicle of signification. The next example deals with architecture. If, looking at a church, you see only its bricks and lime, do you see the church? "By no means! In this way I see only its outermost and final appearance, which even cattle can see, but as long as I do not comprehend the symmetry [of the church], its proportion and size," I cannot perceive the church itself.31 In both of these passages, therefore, figure may be described as the geometrical shape or structure of things. This "figure" differs from the geometrical figures of the triangle, the cube, and so forth, only in complexity and not in kind. Like them it is subject to measure and consequently, as we shall see, to expression by means of numbers. In the final section of this chapter, I shall argue it originates in Descartes. What is the relationship between geometrical and rhetorical figures? Well, it's rather simple and, at the same time, astounding. In Skovoroda's fresco, the drawing identifies the image by telling us what the latter depicts. In this case, the depiction is that of a man bruising the head of a serpent. When it comes to objects not depicted but really existing in the book of nature, geometrical figure allows the thing to be a figure in the sense of trope or hieroglyph. The figure defines the thing, and by knowing what thing is before us, we know what it is a figure of, what it stands for. To have figure is to be figurative. 30 ..Kpacxa He H H oe h t o , xax n o p o x h nycTouia; pwcyHOK, m m n p o n o p u i a h p a c n o n o a c e H ie KpacoK — t o c m i a , “ [ N s ig . 11; 1.38]. 31 „H HK aic! TaK H M o 6 p a 3 0 M , o n H y TOJiiK O fc p a m u o io h n o c jif c /m io io H a p y a tH o c r b BHacy b h c h , KOTOpyiO H CKOT BHUHT, a CHMMTpiH eH , HJtH npO tlO piU H II p a 3 M t p a , KOTOpblil BCeMy CBJttb H ro /iO B a M a T e p ia jiy , n o Hence b Hew He pa3y\rfeK >, hjw T o r o h e » He BHacy, He B tia a e » r o n o B b i," [N sig. 12; 1.39]. 249 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The tie between geometry and signification is why, in illustrating what he means by things' figures, Skovoroda also turns to the example of reading. Here he distinguishes between "the paint on words" and "the figures of letters": if all you see is the paints but not their shapes, you cannot read the word, thus ignoring what the inscription refers to. "The figure in letters is the same," says Skovoroda, "as the drawing in a painting, and the plan in an edifice."'’2 This is an exceedingly strange comparison. First of all, if a painted man resembles a living man by virtue of similarity in their geometrical figures, the word "man" looks nothing like a living man. Secondly, Skovoroda identifies recognizing graphemes in words with knowing what the words themselves mean. Let us take the word "dog": is it not one thing for me to put together the letters d, o, and g, and quite another to know what this combination of letters refers to? Skovoroda makes no allowance for the distinction. Is this an accidental oversight? Had we gallivanted into the dialogue and pointed out the inconsistency, would he have altered his example? No. Let us return to the fresco. The fresco is allegorical. It obviously refers to God's promise that the seed of Eve "shall bruise thy [i.e. the serpent's] head” [Gen. 3.15], which for Christian exegetes foreshadows redemption. The man, in other words, is Christ. If that half of the allegory is unremarkable, the half dealing with the serpent is not. The serpent is one of Skovoroda's many images of the Bible; when its connotations are negative, it stands for the Bible understood literally.33 The fresco urges us to pursue allegorical and not literal meaning; the lemma "The wise man's eyes are in his head; but the fool walketh in darkness" aims at the same point [Eccles. 2.14]. Thus Skovoroda assumes that simply by isolating the figure of the fresco we know what it depicts not only literally but also allegorically. The two signifieds are one, and they are both equally immanent in the geometry of the signifier. 3: „ E c j i h k t o K pacK y H a c jio B a x b h u h t , a rm cM eH n p o H e c r b He M oaceT , x a x T e 6 t KaaceTCJt? B h j i h t j t h TaiciH nH CbM eHa? [...] H t o b x p a c ic a x pHcyHOK, t o ace c a M o e ecrb c |)H ry p o io b m ic b M e H a x , a b C T poeH iH n jiaH O M ,“ [ N sig . 12; 1.39; s e e a ls o Kol'tso s ig . 12; 1.261]. 33 Fresco prefaced by discussion of Christ bruising head of serpent in N, sig. 8-9; 1.35. The metaphor of the serpent is ambivalent, and encapsulates both modes under which the philosopher regards the Holy Scriptures: „Bh6ju}i ecrb TOHHbiH 3Miit“ [Kol'tso sig. 39; 1.297]. Although "accursed among beasts" [Gen 3.15 in Kol'tso sig. 10; 1.258], the serpent also rolls himself up into a circle, for which reason Egyptians used the image of ouroboros, i.e. serpent forming a circle and biting its own tail, to figure eternity [Kol'tso sig. 39; 1.297; hieroglyph borrowed by Moses Silenus ch. 17; 1.400], On another level the fresco signifes figure triumphing over matter [for Christ as figure, see Nach. Dver' ch. 4; 1.20-21; and iVsig. 46; 72-3]. 250 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 Figure as T h o u g h t This immanence indicates the fact that the nature of Skovoroda’s geometrical aspect of things is far from exclusively geometrical. Going over his examples of painting, writing, and architecture, he maintains that "as the drawing, so the figure [of letters], and the plan, the symmetry, the size [of the building], are nothing else but thoughts." Elsewhere, talking o f Biblical Figures, he says, "every figure [...] is a thought; for it preaches something to u s . " 34 The identity of figure and thought is affirmed by Skovoroda's version of the injunction nosce teipsum, "know thyself," which, he thinks, forms the intellectual kernel of the Narcissus myth [Narkiss sig. 1; 1.27]. Staring at his reflection in the water, Narcissus realizes that he is a bearer of figure: "On the canvas of my transient flesh I perceived my image, not made by human hands, image that is 'the brightness of [the Father's] glory'."35 As in the examples of the fresco, the church, and letters, the figure of Narcissus is the real Narcissus: not the material of his body, i.e. flesh, but the organization of the material. At the same time, the real Narcissus is also the thought of Narcissus, his cogitating self. "Mens cujusque, is est quisque. T he mind of a person is the person,'" Skovoroda quotes Cicero, and comments: "hence the Germans call man Mensch, that is mens, i.e. thought, mind; whereas the Greeks refer to man as phos, that is light, i.e. mind." OK, proclaims one of the dimmer characters in the dialogue, "from now on I am going to call myself thought. "3<s From Skovoroda's treatment of "thought" and "mind" as synonyms, we see that he is talking not about this or that thought, but thinking as such, which is equated with the (cogitating) self.37 The concept of mind must also be strictly divorced from the brain; nowhere in Skovoroda is the mind said to exist in one portion of flesh more than in any other. Rather, if what I am is my figure, and what I am is my thought, then my figure is my thought, or my mind is none other that the geometrical aspect of my body.38 34 ,,K aK pucyH O K , T a x ( { w r y p a , h r u i a n , h c H M M eT p ia, h p a 3 M t p He H H oe h t o e c r b , x a x m m cjih," [N sig. 12; 1.39]. „Bcsica cjiHrypa [...] e c T b t o Mbicjib. H tn T O 6 o HaM n p o n o B t i i y e T , " [Askhan’ sig. 2; 1.84]. 35 „ y 3 p e ji a Ha n o a o T irfe n p o T e x a i o m i a M o e a i u i o t h H e p y x o T B o p e H H b iH o o p a 3 , „ H * e e c r b cHXHHe c n a B b i OTHia,“ “ [TVs ig . 3; 1.30; r e f e r r in g to Heb. 1.3 s p e a k i n g o f C hrist], 36 „OTCK>ny y TeBTOHOB H ejio B ex H a 3 b iB a e T c a Memii, c n p e n b m e n s , t o e c r b MbicJTb, yvi; y eJUUiHOB ace H a p n u a e T c a M y * tpoc, c n p a b c a t r , t o e c r b y M .“ R ecourse to su ch ety m o lo g ies com es easily to a p e r s o n who id e n tifie s th e le tte r s d, o, g w ith th e c o n c e p t " d o g ." ..o T c e n fe c r a H y c e 6 e Mbicmto 3 B a n > ,“ [Narkiss s ig . 7n; 1.34]. 37 A sim ilare identification to thought w ith the thing that thinks m ay be found in D escartes' Principles o f Philosophy, 1.63: 'T h o u g h t [...] can be regarded as constituting the n ature [...] o f intelligent substance [...; it] must then be co nsidered as nothing as nothing else but thinking su b sta n ce itse lf [...] - that is. as mind." 38 Comp, to A ristotle's d efin itio n o f soul as the form o f body, relating to it as the shape does to the wax bearing it [De Anima, 4I2a20-412b-10\. 251 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Such is the epiphany of Narcissus and, of course, also of Oedipus, the decipherer of the Sphinxian riddle whose answer is man [Kol'tso sig. 18; 1.268],39 Skovoroda prescribes the transformative experience encoded in these "Egyptian” myths to his audience as the fulcrum from which all other conclusions regarding God and the figurative nature of the universe may be deduced. The fulcrum is personal, a thought experiment each one of us must perform for him or herself, but its results are the same for everyone.40 2.4 Second Nature All things possess geometrical figure. Inasmuch as any one thing is held together by the "nerves, and arteries, and veins"41 of geometry, so the contiguity of things forming the world may be regarded as having one composite structure, as being a figure composed of smaller figures.42 "All matter," says Skovoroda, "is painterly dirt and dirty paint [...], whereas the blessed nature is the beginning itself, that is the beginningless invention [...], and most wise delineation, that beareth all visible paint."43 If here Skovoroda presents the world's structure in the familiar terms of line drawing, elsewhere he again conceives of its figure as thought: The whole world is composed of two natures: one visible, another invisible. Visible nature is called creation [tvar'], and invisible God. This invisible nature, or God, penetrates and holds together all creation; he always was everywhere, is, and will be. For example, the human body is visible, but the mind, which penetrates it and holds it together, is invisible. For this reasons the ancients called God the universal mind.44 39 It is n o t c l e a r w h e th e r O e d ip u s 's a n s w e r h a s a n y th in g to d o w ith h is e a r l i e r tr ip to D e lp h i, w h o s e te m p le b o re th e in s c r ip tio n " k n o w th y s e lf." C o m p a r e S k o v o ro d a 's ’’r e c o n s tr u c tio n " o f th e O e d ip u s m y th to B ro w n e 's "e v e ry m a n s o w n r e a s o n is h is b e s t Oedipus " [6 6 ], a n d to A th a n a s iu s K ir c h e r ’s c a llin g h is stu d ie s o f h ie r o g ly p h s Sphynx Mystagoga a n d Oedipus Aegyptiacus. In m y o p in io n . Leads " l e a r n e d T h e b a n " [3.4.153] in te n d s O e d ip u s a s h e w h o k n e w w h a t a th in g is th is p o o r, b a re , f o r k e d a n im a l, w ith th e " g o o d A th e n ia n " b e in g S o c r a te s , w h o k n e w h e k n e w n o th in g [174, c o m p . " W h a t is th e c a u s e o f th u n d e r ? " w ith s a m e in th e Clouds o f A r is to p h a n e s ] . 40 S a m e r e c o m m e n d a tio n fo r th e cogito e x p e r im e n t in D e s c a rte s ' Principia 1.1. 41 A n d re w M a r v e ll, " D ia lo g u e b e tw e e n th e S o u l a n d B o d y ," 1. 8. 42 „.w ip c e it [ . . . ] o y a b T O B tH e ii H3 b ^ h o h k o b , h ji h M a u i h h ti m e , H3 M auiHH OK c o c ra B Jie H H b iH ," [Silenus c h . 3 ; 1 .3 8 1 ]. F o r w o rld as m a c h in e in D e s c a rte s , s e e Principia 4.188 a n d 4.203. 43 „ B c e B e iu e c T B o e c T b tc p a c n a a r p s 3 b a r p a a H a a ic p a c ic a h acHBO im cHbiH n o p o x , a 6 jia a c e H H a a H a T y p a e c r b ca.M a H a n a iio M , t o e c r b 6 e 3 H a tta jiH o to H H B e tm ie to , h j i h H 3 o 6 p tT e H ie M , h n p e M y u p rfc u iu e fo Iie JiH H e a iiie io , b c k j BHiiH M yto 4 > a 6 p y H o c a m e i o ," [Razg. Piati s ig . 5 1 ; 1 .2 4 5 ], 44 „Becb M ip c o c t o h t H3 U B yx H a T y p : o j u t a — BHUHMaa, u p y r a a — H eBU O H M aa. BmiHMaa H a T y p a H a 3 b iB a e T ca T B a p b , a HeBHUHMaa — 6 o r . C i a H eB H ju tM aa H a T y p a , h j i h 6 o r , b c s T B a p b n p o H H u a e T h c o n e p a c H T ; Beazrfe 6 b iJ i, e c r b h 6 y .n eT . H a n p H M t p , T fcno a e J i o B ta e c K o e b h u h o , h o n p o H H iia io u iiH h c o jie p a c a tn iH o n o e yM He B H aeM . Flo c e ii npiiH H H e y upeB H H X 6 o r H a 3 b iB a jic a y \ t bccmhphuh , “ [Nach. D ver'ch. 1; 1.16]. 252 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Several pages later, "the universal mind" is compared to seal impression, to "architectural symmetry," to order in a peopled kingdom, and so forth [Nach. D ver'4; 1.18]. "The word of God, his counsels and thoughts: this it is the plan impalpably spread throughout the material of the Universe, containing and filling all."45 These passages have many ramifications; let us consider those that are most striking. First of all, if God is the figure of all things, their structure, then each thing's figure is also God: 'Ts he not the being of all? He is the true tree in the tree, the grass in the grass, the music in the music, the house in the house, in the clay of our body the new body."46 He is the body in the body because he is the figure defining the clay; the grass in the grass because he is the figure defining the grass, the house in the house because he is the figure of the house, and so forth. The being of a thing lies in, and is the same as, its figure. Therefore the notions of God as ens entium and forma formarum undergo an equally radical transformation. God is, so to speak, the presence of mathematics in the world, and the presence of mathematics in the world is God. (This last thesis generalizes the earlier identification of figure and thought.) Also known as true nature ["nature" as name of God in e.g. Razg. Piati sigs. 9-10; 1.213-4], God can be beheld only in the mental act of abstracting structure from matter, and "seeing" the former. The Bible calls for such separation in the verse "And God [...] divided the waters [...] from the waters" [Gen. 1.7 in Askhan' sig. 2; 1.83-4; for other verses, see Kol'tso sig. 53; 1.313], Thus mentally sundered, each one thing becomes a two, a visible thing of matter and the invisible thing of figure: the arm you see hides another arm; there are two men in you; man and man, tongue and tongue.47 Yet Skovoroda is no dualist. Let us return to his understanding of signification. He explains: "The spirit [of God] divides his ambivalent word into two: the signifier and the secret signified," literally: that which images and that which is imaged.48 The relationship between signifier and signified may be seen in the already cited metaphor of biblical speech as a wall, one of whose sides is dark, and the other illumined [N sig. 42; 1.68]. "Let us not ..CjTOBo o o a d e , c o B tT b i h Mbicsin e r o — c e ft e c T b t u i a n , n o B ceM y M a T e p ia Jiy b o [...] B ceneH H ofi He HyBCTBHTejibHO n p o c T e p u iiH c s , Bee c o a e p a c a m i i i h H c n o JiH H io u iiH ,“ [N s ig . 14; 1 .42]. „He o h m i obiTH eM B ceM y? O h b a e p e B e HCTHHHbiM a e p e B O M , b T p a B e T p a B o io , b M y3biKe M y B b iK O io , b a o M e a o M O M ,“ [Narkiss s ig . 13; 1 .4 0 ]. 17 „ B e z ib T b i H H T aji, h t o „eaH H O K ) m a r o j i a 6 o r “ , a T a M p a 3 y \rfc e T C ji a s o e — n e a o B t i c h H ejioafeic, H3biK h H 3biK ,“ [Narkiss s i g . 5 3 , 1 .7 9 ; s e e a l s o s i g s . 10, 1 7 -1 9 ; 1 .3 7 , 4 5 - 7 .] „ C m o t p h ac, Kaic IT a B e ji p a 3 a e j w e T n e ji o B t K a . Kan CKOTHHy H a a B o e : „ e c T b T t a o a y m e B H o e , h e c r b T e n o a y x o B H o e “ [Askhan' s ig . 2 0 ; 1.1 1 0 in r e f e re n c e to corpus animate and corpus spirituale of I Cor. 15.44 ]. B u t in fact th e tw o a r e o n e : " B 3 rm n rre H a ztHBHoe a fc jio o o a c e e : H3 a B y x H ejio B tico B co cT aB JieH o aH H " [Razg. Piati.. s ig . 7 ; 1 .2 1 1 ]. u „M ayx aBofiHoe c B o e c j io b o Ha j i b o e p a 3 y M ie T : H a o 6 p a 3 y i o u j e e h n a T a H H O o 6 p a 3 y e M o e ,“ [Kol'tso s i g . 2 4 ; 1 .2 7 7 ], 253 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. think that the word of God is dual in fact," and that it possesses two autonomous essences; rather, "the word of God appears as one, and is heard as two [...]. The face is single, but of a double aspect: hollow and divine, visible and hidden, signifying and signified."49 The visible portion of a thing, i.e. its appearance, signifies the presence of God in its figure and by means of its figure. Skovoroda unites the signifier and signified into a single entity, diune like the Sphinx. The process of signification here requires mentally sundering the entity into signifier and signified in order to make out the latter. Thus, whether signification is to occur at all depends not on the entity itself, but on how it is confronted by that poor, bare, forked animal, the beholder. "Whoever can’t count up to two is really dumb,” exclaims Skovoroda, and adds: "Every figure is a mirror" [Sion 281, 270]. 2.5 Figure as Unity and as Point To define figure as a thing's geometrical structure is to tell half the story. That the concept of thing's figure extends past geometry may be seen in the already quoted claim that God "is the music in the music." Another of Skovoroda’s illustrations of figure makes his meaning quite clear, for it opposes sheer sound, which any ear may hear, to concordance or harmony [Kol'tso sig. 15; 1.265; sig. 12; 1.261]. The concept of harmony here is clearly Pythagorean and intends parts considered with respect to each other and the whole, i.e. unity [Heninger 93-104]. Does harmony have a geometrical analogue? Yes. In the third section of Narkiss, Skovoroda re-evaluates the architectural illustration of figure that was set forth the previous day. "If you measured the length and breadth of the church with a [ruler] or a rope, do you think you would have discovered its measure?" "I don’t think so," answers another of the colloquitors. "I would have discovered only the space of its materials, whereas its exact measure, which contains the materials, I discover only when I comprehend its plan."50 In this passage Skovoroda differentiates between figure as extended structure and the same structure apprehended in its unity. Like Pythagorean harmony, the unity of the figure appears to consist of proportions (the word appears in the previous day's example of the church, N sig. 12; 1.39). In this respect, the curious difference between music and geometry is that whereas musical proportions are among whole numbers only (Pythagorean 49 „C jiobo 6oacie sbjm ctcji oztHHaicoBoe, ho cjiuiiihtcji ztBOHHoe [...] JIh u o o a h o , ho a Be n o p o a b i : nycraa h 6oaci*, HBHaa h Tail Has, o6pa3yioma* h o6pa3yeM aa,“ [Kol'tso sig. 46; 1.305]. 50 „E cjih 6bi Tbi a o a ro T y h iim poTy uepKBH H3\rfepHJi caacHeM hjih BepeBKoio, Kaic T eo t KaaceTca, y3Haji jm Tbi vrfepy e«? [...] 51 6bi y3HaJi o a h o to jib k o npocTpaHCTBO MaTepiajioB ea; a TOiHyio ea .vrfepy, conepacam yio MaTepiaJibi, b t o BpeMy y3Haio, K oraa noHHMaio HJiaH e a ," [Narkiss sig. 14, 1.41], 254 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. theory of proportions in Euclid book 7), proportions in geometry apply to incommensurable as well commensurable magnitudes (Eudoxus' theory of proportions in Euclid book 5). While the Greeks did not think of magnitudes as numbers, Skovoroda, as we shall see from his statement that "measure contains number,” probably does. Up to now, we have been identifying God with the world's figure. However, since the latter is extended, its lines must be infinitely divisible. But "God is indivisible; rather, he is the unity spread through all ages, places, and creatures."51 God appears to relate to the world as unity of figure relates to figure; and yet Skovoroda was certainly not misrepresenting his views earlier, when he identified the entire (divisible) figure with God. How can both be possible? Skovoroda cites the Platonist comparison of God to geometer or mathematician, "for he endlessly exercises in proportions or scales, molding sundry figures, for example: grass, trees, animals, and so forth."52 Here God is an agent acting upon figures. Skovoroda himself prefers to describe the world as a clock, employing the key image of the seventeenth-century mechanistic world-picture [see Burtt 113; Funkenstein 317-24]. However, if Descartes, Boyle, Locke, Leibniz, and Newton regard God as the craftsman who manufactured the clock, and therefore as separate an agent as Plutarch's geometer, Skovoroda's God is the ghost in the machine. "The most blessed nature, or spirit, keeps the whole world in motion, like the mechanical subtlety [that maintains the motion of] a clock machine on a tower."53 Skovoroda’s altering the traditional form of the clock simile indicates his refusal of the craftsman-geometer type of agency; indeed, "rational antiquity" represented the "mechanical subtlety” that is God in the anthropomorphic form of a geometer because, one presumes, clocks had not been invented yet. Several pages later, Skovoroda compares the presence of God in the world-clock to "the tempo in the clock's motion,"54 i.e. to its ticking, with each tick being a one. He also compares God to a point. An extremely common image in Skovoroda, the point often occurs in the context of machines. Taking up Christ's words that "the kingdom of God is within you" [Luke 17.21], he describes his Narcissus as "neither chiromancer 51 „b 6 o 3 t pa3nejieHia Hicrb, ho oh ec rb n p o crap aio iu eecji no BcfeM BtfcaM, M tcraM h TBapsM enHHCTBO," [Dialog sig. 6; 1.193]; Hanajio [...] He nacrb h He c o c t o h t H3 HacreH,“ [Silenus ch. I; 1.379]. 53 „pa3y.MHaa upeBHOCTb cpaBHHBaiia ero c MaTeMaTHKOM hjih reoMeTpoM, noTOMy h to HenpecraHHo b n p enopniax hjih pa3M tpax ynpaaam eT cs, BburknjntBaji no pa3HbiM tfmrypaM, HanpHM ip: TpaBbi, aepeBa, 3Btpeii h Bee npoHee,“ [Nach. Dver' ch.3, 1.17; simile attributed to Plato in Plutarch’s Table-talk, a.k.a. Symposiacs or Quaestiones convivialium, 8.2]. 53 „CiH-To ojiaaceHHiimiaji HaTypa, hjih ayx, Becb vtip, 6 y jrro MaiUHHHcroBa xHTpocrb nacoByio Ha o au iH t MauiHHy, b ABHacemH coAepacHT,” [Nach. D ver' 3, 1.17; see also Narkiss sig. 54; 1.80]. 54 „b ABHaceHiH nacoBOH MauiHHbi TeMno,“ [Nach. Dver' ch. 4; 1.19]. 255 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nor anatomist, but one who perceived within himself the main point of the machine - the kingdom of God.”55 This main or center point of one’s self, i.e. one's very soul, contains "peace and happiness" within it.56 When we open a clock, we see a configuration of wheels. Each wheel rotates around its particular point. Skovoroda conceives of the point as the microcosm and origin of a circle: "as in the circle, so in the point: the first and last point is the same, and where it began, there it ended," he says, explaining eternity.57 We have seen the interplay of all these concepts - God, eternity, the Greek definition of circle - in our sixth chapter, the analogy between the point and the one has appeared in other chapters as well. Skovoroda's clock, whose center points and different-sized circles all work towards one single tempo, recalls Plotinus describing the Transcendent One as the center point of an intelligible circle composed of points each of which is the center of the revolution of one soul [6.9.8.1-16]. Since each center point already contains within it all points of the circle, and since an equal circle may be erected around any of these points, the Transcendent One and the universe may both be regarded as present within each soul in their essential entirety. Plotinus's image prefigures the formula for divine omnipresence, attributed to Hermes Trismegistus and first found in the twelfth-century Liber XXIVphilo sophorum. that God is a sphere (or circle) whose center is everywhere and circumference nowhere. This formula is often quoted by Skovoroda [e.g. Kol'tso sig. 37; 1.294]. As description of divine ubiquity, which becomes reinforced once we conceive of real space in geometrical terms (God is in every point in the continuum), it bears extremely important ramifications for the concept of geometrical figure, and allows us to reconcile the problem of its divisibility with his indivisibility. Any point on any figure is the locus of God himself. In any point on any figure at which I deign to direct my finger, God and all other figures dwell in one indivisible whole, being that point's foundation and presupposition. The center of the world is everywhere, and from any point may the whole of it be re­ created. The circle (writes Skovoroda) is the original figure, the father of squares, triangles and countless others. But even its circumference depends on the center underlying it. 55 .He npeicpacHbiH H a p u b ic c , n e XHpoMaHTHK h He aHaTOMHK, h o yBHirtBiuiH B H yTpb c e 6 e rnaBHbiH M auiH H bi nyHKT - u a p c T B ie o o a c ie - cew y3Haji c e 6 e , H a u ie n b MepTBOM acH Boe, b o T\rfc c B tT , s a x aJiM a 3 b rp x 3 H ,‘‘ [Alfavit s ig .7 ; 1 .3 1 9 ]. C o m p , w ith " T h in e e v e r m o r e , m o s t d e a r la d y , w h ils t th is m a c h in e is to h im ,” in Ham. 2.2.122-3 a n d w ith cogito a s A rc h im e d ia n p o in t in D e s c a rte s ' Second Meditation p. 24. 56 „Mnp h m a c r rie b c a M o ii c p e a H k im ie it Tonici H a u ie it a y u iH ,“ „cepne«tH O H H a u ie it TOHbKH,“ „ b TOHKy M e n u , b c a M y io a y m y m o k j , “ [ K ol’tso s ig s . 1 3 . 9 , 19; 1. 2 6 2 , 2 5 6 , 2 6 9 ]. S1 „B Heft T a x , x a x b x o j i u t : n e p B a x h nocjrfeaH X X T o x ic a e c r b T a ace, h m e n a n a J io c b , T aM h K O H 'iH Jiocb,” [Silenus c h . 1; 1 .3 7 9 ], 256 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. And it is noteworthy that the center point of a circle is the tiniest figure, smaller than a poppy-seed or a grain of sand, [and yet is] the progenitor of dense and corpulent figures with their huge machines. And if out of such a minuscule, void of appearance and almost nonentitous, point were bom ships with city walls, towers, pyramids, colossi, labyrinths, then why could it not have given birth, from the secret abyss of its eternal womb, to the innumerable number of the machines of the world? That invisible, unbegun beginning which underlies all points and whose center is everywhere, circumference nowhere, by the continuous birth of creatures even here attests its yearless sojoum.S8 Because every point harbors the All, to know oneself by perceiving one's figure is to know one is God, for "for the true man and God are one and the same.” It also implies knowing oneself to be the microcosm containing all creation: "You yourself are the earth, and the beast, and the cattle, and the tree, and the grape, and the house."59 All creatures share their being God, and, inasmuch as they are God, any creature is the same as any other. If God is each point in each figure, he is also "the tempo in clock's motion," that is each now in time. Significantly, Skovoroda interprets God as Being in terms of his presence in the duration, i.e. existence, of the world. Although the absolute name of God cannot be known, the best of his knowable names is that which he himself discloses to Moses: "I AM THAT I AM [...]. Thus shalt thou say to the children of Israel, I AM hath sent me unto you" [Ex. 3.14], Skovoroda renders God's "I am" in Church Slavic as Az esm ' syi, I am he who exists, I am he who is in being (syi is present participle of "to be")."This name of mine," God informs Moses, "is identical with my essence":60 i.e. God 58 .U H p fcy jib e c n . H a n a jib H a x ( J jn r y p a , o r e u K B aap aT O B , T p e y ro jn > H iu co B h a p y r n x o c m H C J ie m ib ix . Ho o a H a K o /K h c a M a a o fc p y acH o cT b 3aBHCHT o t c B o e r o u e i r r p a , n p e n B a p a e .M a a o h u m . H n p H M tn a H ia a o c r o H H o e , h t o unpicyjT bH biH nyH K T c a M a a K p o in e H H a a tjm r y p ic a , y M a jie H H ie M aK O B aro 3 e p H a h ntcH H HK H, - p o n H T e jie M e c r b i u i o t h m x h a e o e n b i x 4 m r y p c h x M aiuHHa.vtH o rp o M H b iM H . H K o r a a o t c t o j u . M ajn o ceH b K O H , B ita a He y M y u ie w h i i o h t h h h h t o j k h o h t o h k h n o p o m i j i H c a K o p ao JiH c ro p o x tcK H M H c rtH a M H , x o p o M a M H , MOCTaMH, 6 a u iH a .M H , rm paM H ua.M H , KOJiocca.M H, aaoH pH H Ta.M H , t o j u i x H e ro He m o m o H3 TaHHOH 6e3n.H U afeHHbix H e a p c b o h x n o p o a H T b Bee 6e3HHCJieHHoe m h p c k h x M aniH H h h c ji o ? O H o e B c a x in nyH K T n p e a B a p a i o m e e 6 e 3 H a H a jib H o e HeBHOHMoe H a n a n o u e H T p c b o h Be3zrfe, oKpyacHOCTH H H ra e He H M y u ie e , h 3 n t c b H enpepbiB H bi.M T B apew p o a c n e H ie M C B m rfeT ejibC T B yiom ee o 6e3JieTH O M C BoeM n p e 6 b iB a H iH ,“ [ Kol'tso s ig . 4 1 ; 2 9 9 -3 0 0 ] , I f G o d is a p o in t, h o w c a n h e a c tiv e ly m o ld f ig u r e s ? S k o v o r o d a d o e s n o t p ro v id e an a n s w e r ; m y g u e s s re s ts o n th e id e n tif ic a tio n o f th e p o in t w ith u n ity . T h e p r e s e n c e , in a fig u re , o f th e u n ity o f th a t fig u re m u s t b e lo c a te d in e v e r y p o in t u p o n a n y o f th e lin e s th a t c o n s t it u te th e fig u re . T h e p re s e n c e o f u n ity in th e p o in ts o f th e lin e s p r e s u m a b ly fo rc e s th e la tte r to c o a le s c e in to t h a t p a r t ic u l a r fig u re in th e fir s t p la c e : th is is h o w G o d m o ld s f ig u r e s . S in c e e a c h fig u re is o n ly a p a rt in th e e n s e m b le o f fig u re s to g e t h e r c o n s titu tin g th e w o rld , e a c h p o in t e n f o r c e s th e u n ity o f th e th in g a n d o f th e w o rld . 59 „ HCTHHHbiH HeJiofffeK h 6 o r e c T b t o * e , “ [Narkiss s ig . 20 ; 1.47 ]; „Tbi h 3eM JW , h 3 B ip b , h c k o t , h n e p e B O , h B H H o rp a a , h a o M , “ [Askhan' s ig . 27 ; 1. 120]. 60 „Cia e jH H im a BceM y m a B a , a c a M a 6 e 3 H a H a jib H a a h h B pevteH eM , h h vrfecTOM, h h h o ji o m He o rp aH H H e H H ajt, h h HMeHeM. C i« - T O m a t h h o r e u OTBtnaeT M o n c e i o , h t o e fi h m c h h HtT. K r o - a e iiu ie T M o e r o h m c h h , t o t He b h j i h t e c r e c r B a M oero. H m b Moe h e c rec T B O e c r b t o ace: „A3 ecM b 257 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as the "thing" that exists is ontologically the same as his act of existing. Skovoroda then identifies God's existence-essence with the intelligible "sound” emitted by the world-clock, which is also present in us as tick, our own being, i.e. our being God. We can hear him within us, thundering, "I am, I am syi" [see Razg. Piati sig. 9, 22; 1.213, 223]. God himself is present in his entirety in every now, as the now.61 We can interpret this scenario in light of signification by remembering that figures are not only geometrical, but also rhetorical. When I speak, my words have many levels of meaning - literal, figurative, etc., - the most fundamental of which attests to my existence. The proclamation "I am" is equally present in any utterance, whatever its source and whatever else it may be saying. In the same way, God (the speaker) is present in the world (the utterance) inasmuch as he is its most fundamental aspect, which can be described as follows. Rather than being some particular signified or some particular thing, God is the general fact of signification in signification and the general fact of being in being. In contrast to Platonism, where things signify Ideas and then the One, Skovoroda's God is not the result of signification. Rather, he occupies a different plane, one where all signs are equal in their being signs, the plane on which they exist as signs pure and simple. On the plane where each sign is (a sign), it is God. Also in contrast to Platonism, he is not Being in the autonomous and reified sense of the term, but rather that which is universal in the particular beings of all things that are. This universality is not a particular shared characteristic, like a member common to all sets which include other, non-shared members, but the general characteristic of being a member in a set. As with signs, each thing is God on the plane where it is simply a thing, and not this or that thing. We can also regard this plane as a mode under which a thing (or sign) is perceived by the beholder: in the mode of particulars, each thing (or sign) appears as its particular self; in the universal mode, it appears as God. cbiH“ . 9i to t , hto e c M b Be3.ofc, B c e ra a b o Bce.M, h He b h u h o M eH e, a n p o T H e e B ee b h u h o , h irfcT T o r o HHHero," [Alfavit sig. 7; 1.318-9;]. 61 The clock of Descartes, unlike that of Leibniz, requires God's constant attendance, the reason being that Descartes regards time as not inherently continuous: "the nature of time is such that its parts are not mutually dependent." In order for any change to occur, therefore, there must be something ensuring the succession of nows, something "which continually reproduces us, as it were, that is to say, which keeps us in our existence." So writes Descartes, arguing that "the fact that our existence has duration is sufficient to demonstrate the existence of God" [Principia l.21\ see also 2.42]. The now points to the being of God directly and bears, so to speak, the imprint of his hand. This is why Skovoroda's world-clock with its discrete nows, each identified with God as keeping all things in being, resembles the clock of Descartes much more than that of subsequent thinkers. 258 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. The Book of Number 3.1 Figure and Number "Measure and number always pertain to God," argues Skovoroda.62 Like many exegetes before him, he finds numbers to be an especially significant part of the Bible. Naturally, he is very fond of the number two, and sees even such passages as Judith's having to strike Holofemes twice [.Judith 13.8] as referring to his doctrine of dual nature. He also indulges in the same type of calculations as employed in gematria, rather blithely sticking Hindu-Arabic numerals in the place of Hebrew letters, and thereby obtaining incorrect results [Kol'tso sig. 52; 1.311]. Overall, however, his approach differs from standard procedure in that it focuses much less on interpreting particular numbers than on the general fact that "the multitude [of figures] in the Bible often becomes delimited by some known number."63 Thus, the number of persons accompanying Abraham (318) or Gideon (330) matters primarily inasmuch as it is a number, for all numbers already point to one single truth: how things came to be and how they subsist in being. Several chapters earlier, I mentioned the Pythagorean-Platonist opinion that number came to be from the interaction between the principles of limit and unlimited: "For number, beginning with unity, is capable of indefinite increase, yet any number you choose is finite; magnitudes likewise are divisible without end, yet the magnitudes distinguished from one another are all bounded, and the actual parts of the whole are limited" [Proclus p. 6]. Skovoroda clearly thinks of figures as limited in some such way, for he occasionally calls them "termini, limits' in Slavic, encasing the power of God within,"64 and speaks of termini while referring to Job 38.4-5: "Where wast thou when I laid the foundations of the earth? [...] Who hath laid the measures thereof, if thou knowest?" [Kol'tso sig. 53; 1.3 13].65 God's "laying the measures" of things is conceived in terms of their becoming limited "by some known number": "this formless dirt, returning to its beginning, assumes 6: „ M tp a a hhcjio B c e r a a e c T b 6 o a c ie ,“ [Kol'tso sig. 52; 1.311] *3 „ h t o c e il M HoroHHCJieHHRH p o a [ o o p a a o B n p u c H o c y m if l o o a c ia ] H a trro b 6 h 6 j u h m r tc T H b iM HHCjioM o rp aH H H H B aeT ca, n a n p . A B p a a M c o H T e c b o h a o M o n a a i ib i 300 h 18. r e a e o H B 3«n c c o d o io 330 M yaceii h n p o n . Bee c ie c y a a ace c n y a c i r r h a t a a e T c a h j ih no T O M y , h t o o h h Bcfc c y T b icaic o a n a a r e n a , p a a c a a i o u i a a o a H o T B e p a o e h H 3B ecT H oe H a n a a o , K o r a a 6 e c H H c a e H H o ro n p a x a n e p e n e c T b Heab3»,“ [Kol'tso sig. 29; 1.283-4], w „CiH ac 3iiaMeHiH H a 3 b iB a io T ca T epM H H aM ii: caaB eH C K ii - n p e a fc a a M H , c u n y 6 o a a n o BHyrpb ce6e 3 a K a K iH a io iU H M H ,“ [Kol'tso sig. 33; 1.288]. 65 In Job, God continues with "or who hath stretched the line upon it?" but Skovoroda switches from measure to weight, probably recalling Wisdom o f Solomon 11.7. 259 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from it the well-tempered stamp of species and measure, and measure contains number."66 What does the last clause, "measure contains number," mean? In other words, how does Skovoroda conceive number? For starters, nowhere does he offer any discussion of number as pluralities, monads, etc., nor dees he ever speak of number as existing autonomously. This silence cannot only be due to his omission of Platonic Ideas, for neither does he seem to observe the Aristotelian distinction that "a plurality is a kind of quantity if it is numerable, a magnitude is a kind of quantity if it is measurable" [Metaphys. l020a7-9]. Rather, it is measure that is said to "contain number,” and since the word "measure" appears here with a geometrical meaning, as in the earlier-quoted usage of Narkiss [sig. 14, 1.41], one may propose that Skovoroda, unlike the Greeks, regards magnitudes as numerable. If that be the case, his concept of number ought to be of such nature as to include whole numbers, fractions, and irrationals (limited to radicals, one would presume).67 At the same time, his concept of limit must differ from that of Proclus, for whom magnitudes are limited with respect to themselves by virtue of being divisible into an aliquot number of parts, whereas their infinite divisibility is explained by their partaking in the unlimited. Since Skovoroda identifies the unlimited with God [see ogranichennost' in, e.g., Nach. Dver' intro; 1.14] and speaks of figure, also identified with God, as terminus or limit, his concept of limit must intend only the fact that the lines that compose figures are, as Proclus says, "bounded."68 Each line is then a particular limiting of the unlimited.69 Since magnitudes are numerable, the length of each line can also be expressed by Hindu-Arabic numerals. To have magnitudes, i.e. to be expressible by numerals, is 66 „cix 6 e 3 o 6 p a 3 H a j j r p » 3 b , B 0 3 B p a m a a c b k C B o e M y H a n a j i y , n p ie M J ie T o t e r o n e n a T J i i e M o e H a c e 6 e 6 n a r o o 6 p a 3 i e BH.ua h v rfep b i, a . \ r t p a h h h c j i o b c e 6 e a a ic ju o H a e T [ . . . ] . K o r n a c i x HeBUH HM a h HeycrpoeHHa i u io T b H3 h h t t o j k h o c t h C B o e u B b ix o iiH T b T O H H o e C B oe H a n a j i o , T o r n a c o 3 H a a e T C J i 113 HHHTOJKHOCTH B HfeHTO H Iip e C T a e T 6 b IT b HHHTO, T .e . lU IO T ilO H T bM O K ), B TO BpeM H , K O H ia 6 o r , H 3H H T aaH M HO»cecTBO 3 B t3 x i h H a p H tia a H e c y m a s , s k o c y u i a a , p e n c T : „ J 3 a o y a e T c B e T ,“ [Kol'tso s i g . 29-30; 1.284], 07 For brief history of transcendental numbers, see 85-6. 6a In this he conceives of limit and unlimited in a manner diametrically opposed to that of the Platonists. As we have seen in my third chapter [section 6], the limit and the unlimited may be identified with Plato's One and the Indefinite Dyad respectively [sec Metaphys. 987b25-7], In their attempt to make the system monist. Christian Platonists identified the One with God, locating both limit and the unlimited in the One's transcendent nature. Skovoroda never thinks of God as separate from the world; in the world, however, he is the unlimited that exists only under the self-imposition of some limit. w Georges Poulet, in his Les Metamorphoses du Cercle, relates certain usages of the circle metaphor to the incarnation perceived as the nebulously mathematical paradox of inscribing the infinite into finite [38-9]. Skovoroda must have something like this in mind when he very strangely but insistently identifies figure with Christ [Nach. Dver' ch. 4, 1.20-21, and N sig. 46; 72-3]. Yet if figure is Christ, it is also, as we have seen, the Father, in the sfense that he is its unity. 260 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tantamount to being this or that thing. How then does one express the general and universal fact of being, i.e. God? 3.2 C ircles "All creation is,” exclaims Skovoroda, and embarks on a long string of interchangeable invectives which can be rendered very approximately as, "junk, mess, morass, moss, trash, ruin, gutter, balderdash, crush, and flesh, and flashes." "Everything visible," he says, "is an idol." "Every appearance is an image, and every image is flesh, shadow, idol and nothing." "'An idol,"' he quotes Paul, "’is nothing.' Idol, figure,70 image are the same thing and nothing" [/ Cor. 8.4]. "Flesh is nothing."71 You're not a man, the man who does not know himself is informed, "you’re [...] nothing, you're umbra, you're shadow" [Sion 175]. "You'll wind up telling me I got no ears or eyes," the same type of personage protests. "Yea, I already said long ago that there's none of you at all." "The idolater," exclaims Skovoroda, "honors that which is empty." "You see that in yourself which is nothing, and you see nothing." "Your empty eye beholds emptiness everywhere."72 And so forth. In designating the emptiness, hollowness, nothingness of appearances, Skovoroda elaborates upon that line of Baroque, especially Tridentine, imagery, whereby each thing is a "bulla... quae vitreis turgida fertur aquis," a swollen bubble carried on glassy waters (Tavorskii in Maslov xxiii; see Poulet ch. 2 for other examples and discussion]. His images of the world tend to feature conglomerations of circles: the clock is one; "the base fleece," with its curls, another; water, with its bubbles, the third.73 All of them present the world as a set of objects, each one of which is a circle and a nothing. Among Skovoroda's many circular figures are "wheel," "compass," "ring," "eye." The Latin word for ring, annulus, reminds us that time is composed of circles, for the 70 In s e v e r a l in s ta n c e s , th e w o rd " fig u re " m e a n s s im p ly a p p e a r a n c e : „ K a a c a o H T B apw ( j w r y p a e c r b H enecTH B aH n y c r o i u b , e c jm B o n jio m e u ie M H B.vrfeuieHieM c b o h m He o c B u m a e T eaH H c b h t , " [Silenus ch. 4; 1.3831. 71 „ B c h T B a p b e c r b p y x n u a b , c.vrfccb, C B o a o n b , ji o m , x p y u i b , c r e n b , B a a o p , c n a o H b , h tu x o T b , h njieTKH.** [Nach. Dver' in tro .; 1 .1 5 ]. ,B c e t o a a o a , h t o B H a H M o e ," [Narkiss c h . 3 ; 1 .4 5 ]. „ B c ju c a n B H ru tM o cT b e c r b o 6 p a 3 , a K aacabiH o 6 p a a e c r b a a o T b , cfeHb, H a o a h h h h t o , “ [Kol'tso s ig . 3 0 ; 1 .2 8 5 ]. „ „ M a o a h h h t o ace e c r b . " H aoa, tj w r y p a , o 6 p a 3 e c r b t o ace h h h h t o a c e ." [Silenus c h . 4 ; 1 .3 8 3 ], „ r iJ iO T b h h h t o a c e ," [Alfavit sig . 7; 1 .3 1 9 ], 72 „JT yica: Tbi H a r o B o p m u b , h t o y M e a e h h y in e i i, h h o n e i! H frr. H p y r : H a a , a yace aaB H O c ic a 3 a a , h t o T e o e B c e r o trfcT." [Narkiss s ig . 6 ; 1.33], „ H a o a o n o i c a o H e i i n y c r o e h t c t " [Silenus c h . 4 ; 1 .3 8 3 ] „BHaHUJ B c e o t TO, HTO HHHTO, H HHHeTO He BHaHIUb." ..Ily C T O e TBOe OKO CMOTpHT BO BCeM Ha ny cro u n o ," [Narkiss sigs. 6, 9; 1.32, 36]. [Kol'tso s ig . 3 8 ; 1 .2 9 5 ]; 73 „p y H O c ie n o a a o e , " f o r w a te r , s e e Kol’tso s ig . 5 0 ; 1 .3 0 9 , a n d passim. 261 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Romans called "the circle of year" annus; the circle also stands for eternity.74 The wheel for Skovoroda points to Ezekiel's vision of the Merkabah, The appearance of the wheels and their work was like unto the color beryl: and they four had one likeness; and their appearance and their work was it were a wheel in the middle of a wheel. When they went, they went upon their four sides: and they turned not when they went. As for their rings, they were so high that they were dreadful; and their rings were full of eyes around them four [Ezek. 1.16-8]. Skovoroda’s interpretation of the Merkabah is unusual in that for him it represents not, in the words of Ovid, the "Palatia caeli" of medieval Jewish mystics [Metamorph. 1.176; Scholem ch. 2], but the world itself.75 Skovoroda must see Ezekiel’s sweet chariot, with its wheels and wheels within wheels, as the biblical image of the world-clock, for he immediately quotes David's "the voice of thy thunder in the wheel" [Pr 77.18; King James has "was in the firmament"] in obvious reference to the "I am" of God's tempo. Each thing is a wheel because it is nothing, and also because it is God. "Why is God depicted by the wheel?" asks Skovoroda, and answers: because he is the unbegun beginning of everything; and also the endless end; and such, as we have seen, is the nature of the circle.76 God can also be envisaged as the center point of the circle that stands for the 74 „H to e c r b n e p c r e H b ? E c r b iib ip K y n , K O Jieco, B tH H o c T b ,“ [Askhan', s ig . 5 3 ; 1 .1 5 5 ]; „ K p y r h K O Jibuo - to ace [ ...] B c n o M m u i x K O Jieca 6 e3K O H eH H aa h a o p a a y v rf e jic ji, hto c i a ( J ja r y p a H an o M H H a eT o 6e3JTtTHOM B tn b H a r o n p H C H O cy u iiH . K p y r j r t r a phmckh - a n n u s , a K O jn .n o - a n n u iu s ," [Kol'tso. s ig . 3 1 ; 1 .2 8 6 ]. C o m p , w ith c a M o e B pcM x K o r a a o rp a H H H e B a e T c a , T o r a a p a 3 y v f t e T c a o o a c ie h B tH H o e ,“ [Kol'tso s ig . 2 9 ; 1 .2 8 3 ], 75 „ k t o He cK aaceT , h t o n e p c r b h j i h r p « 3 b o t K o n e c a 3H anH T T JitH H y io n p n p o a y ? C i e e c r b M e c r o T . v it . K t o ace o n a T b He b h j i h t , h t o K O Jieco n p io c fe H a e T B tH H a ro B tn H o c r b ? C i a e c r b 3eM JW , B K JU O H aioinaa b c e 6 t c a k r BtnHOCTH. 3 H a.M eH y eM a» K O JiecoM , 6 y a r o KOJieco b K O Jiecn 3aKJTK3HHJioca: b 3eMHOM H e 6 e c H o e , b T jrtH H O M H eTJifcHHoe, KaK ro B o p H T Ie3eK iH Jn», B H aeB U iin K O Jieca: „H a t r i o h x o a u i e , h k o a m e 6 u 6 b u i o k o j i o b KOJiecH.“ Ho c h h K O Jieca He n p o c r b i H O bL ra: „ H B H ueH ie KOJiec h c o T B o p e m e h x , h k o BHZteHie 4 > a p c H c a “ ( p o a a p a r o n t H H a r o k b m h ji) . T o b o p h t h J X „ r j i a c r p o M a T B o e r o b K O Jie c a ...“ . B h u h t c , tcy n b i ciw K O Jieca a o K a T H JiH c a . T e n e p b , K aa ceT ca, bhjiho , h t o K O Jieco e c r b o o p a 3 , 3 a K p b iB a io u a H B H y rp b c e o e oe3K O H eH H oe K O Jieco o o a c ia BtnH OCTH, h e c r b o y a T o n e p c r b , H pH JibH yB U iax k Hew: „ Z ly x acH3HH o a u i e b K O Jiecax “ . JXy x acH3HH H B tH H O C T b - OZIHO TO ace. [ ...] C aM b lB eTO H CTbipe aCHBOTHbtX, HTO TaCKatOT 3 a COOOH BHirfeHHblJl eM y K O Jieca, K a a c e rc a , t o ace H an ep H H B aio T o o p a a K o e H - jr a 6 o T B a p u , B K jn o H aio iu iH b c e 6 e o jiH c r a T e jib H b iii BHa BtnHOCTH; e c r b o y a T o 6 b i B e3 y u iiH c o K p o B H ia e o o a c ie B 030K , eB p eacK H n y T b j i h He x ep y B H M “ [Kol'tso sig . 2 5 ; 1 .2 7 8 ]. 76 „ a j w n e r o 6 o r H 3 o 6 p a a c a e T c a K o a e c o M ? [ . . . ] O h naH H H aeT Bee - He H aH H H aeT ca, T tM e c r b H a n a a o ; He M oaceT o b tT b H a n a jio M h h h t o , ecT JiH n p e a c a e T o r o 6 b i a o h t o - j i h 6 o . To o a n o e c r b ncT H H H o e H a n a jio , h t o Bee n p e a B a p a e T h c a M o HHHtM He n p e a B a p jie M o . O a H H TOJibK o 6 o r e c r b p o a H o e H a n a jio , h t o B ee n p e a B a p a e r . O h B ee n p e a B a p a e T h n o c j r t B c e ro o c r a e T c a , n e r o h h o ne.M a p y r o M C K a3aT b He M oacHO,“ [Kol'tso s ig . 3 9 ; 1 .2 9 6 ] . „ H a n a J i o 6e3KOHeHHoe, h caM O B ceM y KOHeii, h B ceM y rH 6 jn o u ie M y KaK H anaTO K , TaK h o c r a T O K ,“ [Askhan' sig . 10; 1 .9 6 ]. „6e3B H H O B H oe aanajio, 6 e 3 H a n a jib H a a Buna [ ...] Cia B b ic o H a n u ia a BHHa B c e o 6 u iH M HMeHe.M H M eH yeT ca 6 o r , C B O H C T B eH H aro HM eHH e it H t T , “ [Kol’tso sig. I I ; 1 . 2 5 9 ] . ,, y K o a b u a H tT K O H n a, n o n o c n o B H i i t , h y TBoeii p tH H .“ [Kol'tso sig. 3 3 ; 1 .2 8 9 ], 262 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. world: "the celestial is in the terrestrial, [and] the immortal is in the mortal," as the axis is in the wheel and as the pupil is in the eye.77 In one sentence in the Merkabah passage that reminds us of the state of roads in the Russian Empire, the wheel is an image of eternity at the same time as it is "like unto the dirt clinging to it." For Skovoroda, the circle alternately symbolizes mutable and immutable nature, matter and figure, creation and God, nothing and everything; it is as ambivalent as the Sphinx, the Bible, the world, the sign. Skovoroda proposes a new method for reading the Bible that is as brilliant as it is utterly insane. All that really matters in the text is references to circular figures: "when a compass-like, flat-rounded, or spherical figure starts - such as a ring, breadloaf, coin, and so forth, or grapes and fruits with branches and seeds - pay attention!"78 This commandment is enlarged in the greatest possible detail in a dialogue appropriately titled Kol'tso, The Ring. Its characters take turns expounding the nature of circularity in ecstatic, meandering, serpentine79 circumlocutions; the idolatrous skeptic among them dismisses their speech pattern as okoliosnaia,30 which may be roughly rendered as "wheeling about," and therefore ironically points to the Merkabah. "Inasmuch as I can see," humph-humbugs the skeptic. Into the count of your wheels you will soon pile up sieves, dishes, loaves of breads, wafers, bliny with plates, with eggs, with spoons and nuts and other junk [...]. Add peas with beans and with raindrops - 1 think all of this is in the Bible. Don't forget the fruits from Solomon's gardens and Jonah's gourd and other watermelons. Finally, Zachariah’s menorah with its glasses and with oil burning in its nut-like cups; for the cup has more of a sphere than the plate does, and the plate more than your wheel rim.81 77 Kol'tso s ig . 2 5 ; 1 .2 7 8 , q td . a b o v e ; a ls o Kol'tso s i g . 4 1 ; 1 .3 0 0 ; „ O ko e c r b n p H p o a H b m im p f c y a b , u e t r r p — e r o 3 tH H ita . [ ...] H to kojio b kojkch , hto 3 eH H iia b oicy [ ...] , t . e . 6 o r , b H eO ecH bix a 3 e M H b ix T M a x 3H aM eH iH .“ As I m e n tio n e d e a r l ie r , S k o v o r o d a e m p lo y s th e tr a d itio n a l P la to n is t v ie w o f c e n t e r p o i n t a s m ic ro c o s m o f c ir c le . 78 „T aK /K e, K o r n a H anH H aeTCjt ( j w r y p a tib ip tc y a b H a a , n a o c i c o K p y r a a a , u ia p o B t ia H a a — K ajcoB aa e c r b n e p c r r e H b , xjrfeo, M o n e T a h n p o T H ., m m B H H o rp a n H b ie h c a a o B b ie r u i o a b i c atT B a.M H h cfeM eHaM H h n p o T H . C m o t p h 6 o a p o ! , “ [Silenus c h . 7 ; 1 .3 8 8 ], '9 For Bible as the "serpentine" river Meander, see Askh. sig. 37; 1.134-5. 80 „Kyaa B a .v t, o p a T U b i, n o H p a B H Jia c b o ic o a e c H a a ! OattH y c rr a ji, apyrofi H a H a a ,“ [Kol'tso sig . 34; 1 .2 9 1 ], 81 „ B o t h T p e T iu n p H H a jic a r a y r b rfe ace a y n i c JiyK ouuca.M H h o 6 p y n b a M H . C ico ab K O B tt a tT b MoacHo, Bbi c K o p o H a ic a a a e T e b rneT B a u m x ic o a e c p b u ie T a , o j n o a a , x a fe o b i, o n p tc H O K H , 6 a H H b i c T ap ejiK aM H , c a itu a M H , c aoacicaM H h o p t x a M H h n p o a y i o p y x a a a b . . . I lp H a a H T e r o p o x c 6 o 6 a M H h c a o ac a e B b iM H x a n a a M H - b 6h6juh, a y M a t o , B ee c ie e c r b . H e 3 a 6 y a b T e n a o a o B H3 C o a o M o n o B b ix c a a o B c I o h h h o i o T biK B oto h a p o y 3 a M H . H a x o H e a , h 3 a x a p iH H ceaM H C BeuiH H K c KpyaceHKaMH h c ro p a u iH M b o p b x o a H a B b ix n a m x a x e a b e M ; a H a u ix a e c r b 6 o a b u i a a a a c r b u i a p a , Heaceaw T a p e a x a , a r a p e a x a - H e a c e a n B a m o 6 o a , “ [Kol'tso sig. 3 7 ; 1 .2 9 4 ], 263 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. He hits the nail on the head in spite of himself. Each of the hieroglyphical figures of men, beasts, heavenly bodies in the cartouche that is the Bible is, on yet more fundamental level, a circle. Cracking open the Scriptures, the "natural circle" of the eye encounters this: OOCKDOOOOOOOOOO.82 The circles that are figures —Skovoroda sometimes refers to them as "knots," although the pun is absent in Russian - have a kind of syntax. In Kol’tso, they come in pairs: The sea and the whale, the whale and Jonah, Jonah and the gourd. In the eyes of the whale, the sea is nothing, and the whale in eyes of Jonah is emptiness. Jonah, in the eyes of the gourd that cools his head, and the gourd in the eyes of the glory of the Eternal One are nothing.83 In any pair OO, the meaning of each O depends the meaning of the other O: if O, is nothing, then O, is the All, and the other way around. Neither of the two possibilities is more present in the sign: if an O is nothing, then "it yeveth power in signification to other" [OED "cipher," n. 1], but it too may shortly receive the same power. Each pair thus stands for the uniformly ambivalent nature of being: look at any appearance as a non-signifier, and it is nothing, look at it as a signifier, and it is God.84 8: Natural circle: „O ko ecrb n p a p o iiH b iH im p K y jib ," [Kol'tso sig. 41; 1.300]. If m y reader needs any further evidence for the Baroque nature of Skovoroda's thought, here is a lovely lyric by Tristan L'Hermite: Amarille en se regardant Pour se conseiller de sa grace Met aujourd'hui des feux dans cette glace Et d'un cristal commun fair un miroir ardent. Ainsi touchd d’un soin pareil Tous les matins I'astre du monde Lorsqu'il se leve en se mirant dans I'onde Pense tout etonn£ voir un autre soleil. First two stanzas of "Pour un excellente Beaut£ qui se mirait," in Mathieu-Castellani, 405. Compare with Lear's "there was never yet fair woman but she made mouths in a glass" [3.2.35-6], The juxtaposition of two zeroes is not, unfortunately, the origin of our infinity sign; nor is Ex. 33.23; see Cajori, Notations 196. 83 „ C .v to T p H , K ax BbeTCH 3M iit c e it h H rp a e T ! M o p e h kht , kht h I o H a , Io H a h T biK Ba. M o p e b paccy -acn eH iH K irra e c r b hhhto ; a kht b p a c c y a m e H iH I ohm - n y c r o u i . Io H a B a n r a a o M TbiKBbi rojioBy e r o n p o x ji a a c u a io m ij t, a TbiKBa B 3 o p o M c jra B b i fffcH bH aro hhhto ace ecTb,“ [Kol'tso sig. 51; 1.310], 84 „Cin y 3 J ib i, 3 mih , BbtoTCJt h n e p e o n e T b iB a io T C J t M e a u iy co6 ok >, H e n a a H H O T a M hbjuw r o j i o B y , rz rh 6 b i J i H e n a B iio xboct , h H a n p o T H B . H K a x n a f e n p H p o n b t : r n a B H a a h H H 3 U ia a , a fe H b H a a h T jr iH H a a , B e e c o c r a B J W io T , T a x h n B a o 6 p a 3 a , c o c r a B J U i i o t u i e chmboji, n o B c eM y c B s iu e H H O M y n o j n o hbjWiotcji, M a c r o n e p e M tH H B M t c r o c B e n tm w , n a 3 e M JU o B c e j u n o m a r o c x c a f c r a , H H a n p o T H B ,“ [Kol'tso sig. 49; 1.309], 264 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In his claim that each figure is an O, Skovoroda expresses his refusal of assigning particular essences to things and his determination to give the constructive role to the beholder. Each figure is a mirror [Sion 270], reflecting the natural O of the eye. One wonders to what extent his reading method and its concept o f circular object equivalent to the reader’s circular self is an inside joke on his own name. Skovoroda, meaning "frying pan. ..35 3.3 F ig u re A rithm etical Skovoroda never uses the word zero, but then again he never makes references to any sources other than Plutarch, Cicero, and the Bible, although his definitions of God, some of his "Egyptology," and quite a few of his concepts are obviously taken from writings more recent. Whatever we may think of such reluctance to show one's cards, a circular figure signifying nothing by itself and something in combination, I think, provides a foot fitting the shoe at hand. The circle appears in Skovoroda as the sign for both nothing and the All: if we regard appearances as all there is, then God is nothing; if we regard God as all there is, appearances are nothing. Either way, whatever we regard as all there is, is nothing at the same time: God because he is the unlimited, invisible and impalpable no thing, resisting reification; appearances because once we disregard their figures, they too become unlimited no-things, unformed matter. Zero as a sign for nothing, fine; but how can zero be the sign for the All? I suggest that zero for Skovoroda attests to the general presence of number, not this or that number but numbers any and all. It functions therefore as the symbolic equivalent of the concept "number" as separated from, and applying equally to, all particular numbers: which, if we trust our speculation concerning Skovoroda's usage of the word "measure," include whole numbers, fractions, and radicals. I argued that for Skovoroda numbers express the lengths of the lines of his figures. A particular line is therefore a particular number n, with rt > 0. The being of particular figures, therefore, consists of many such "numbers" in geometrical relationships with one another. However, the sign for the general presence of line or number cannot be a line or *s According to Dal"s Russian dictionary, skovorodka, the diminutive of skovoroda, also means egg sunny side up, which in today's Russian is called glazun'ia, from glaz, eye. Unfortunately, as with “knot," the obvious English pun on "eye" and the less obvious English-Greek pun on "pan," must be dismissed as sheer coincidences. Also coincidentally, some of Skovoroda's key terms — kolesa, wheels, telega, cart — have since acquired drug-related meanings that are not entirely foreign to the ecstatic uses he had put them to, with kolesa being certain psychodelics, and telega, roughly, what you say when you're on them. Still. nomina omina, as Ralph Cramden used to say: and it certainly does make Russian undergraduates particularly open to lectures on Skovoroda's philosophy. 265 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. particular number. As for Stevin, zero for Skovoroda is not one of the numbers but their origin and source: also as in Stevin it is considered the arithmetical analogue of the point. Since zero stands for the general existential being of numbers, all numbers at their most fundamental level "are” zeroes; we can also say that they are "made up" of zeroes (and lines of points) in the same way as all loaves are made up of bread. Or, in the words of Donne's "Second Anniversary," though [...] quantities Are made of lines, and lines from Points arise, None can these lines or quantities unjoynt. And say this is a line, or this a point (11. 131 -5]. As with numbers, so with things whose figures numbers express. Here the circle that is zero symbolizes existence: the type of being that is common to all things and yet itself is no thing. The name of this type of being is God, or true nature. In the same way as we convey our thoughts by "the figures of letters or pulsations of air," in order to show himself the unlimited God must encipher himself.86 And so he authors the book of nature, appearing in it under the guise of figures: figures hieroglyphical, geometrical, arithmetical. The reason that the well-meaning reader should focus only on circular figures is that all figures are, in essence, ciphers. 4. Skovoroda and Seventeenth-Century Philosophy I picked Skovoroda's philosophy as the topic of my conclusion not only because it collects and rounds off most of the strands running through my dissertation, but also because I see the manner in which it envisages their total as marked by those intellectual developments of the seventeenth century which the rest of my dissertation leads up to and abuts against. Skovoroda's hermeticism - if you can call it that - appears to have been devised in a world understood mechanistically. To be sure, Skovoroda offers a spiritual antidote; but his antidote incorporates a bit of the poison: not only some of the metaphors (clock, machine), but also some of the assumptions of mathematical physics. The initial philosophical theory' behind such physics is the work of Descartes, despite the fact that his project to unify mathematics and physics proved unrealizable for his 86 „A(()aHaciH: Hy, jw x Hero 6 o r H 3 o 6 p a * a e T c a o 6 p a 3 a M H ? £ kob: A tm juw n e r o tboh mmcjih H3o6pa>Kaeuib (|)H ry p a M H 6yK B hjih y a a p e H i e M B 0 3 a y x a ? A < t> aH acm : Jinx T o r o , « r r o mmcjih moh He BaaHbi. .Hkob : A 6 o r bo cto t u o ih p a 3 c o K p o B e H fffe e tbohx M b ic J ie ii, H e a c e jm tboh B 0 3 a y iU H a ji Mbicjib b HapyjKHOM T B o eM 6 o j iB a H e „ “ [Kol'tso sig. 48; 1.308]. 266 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. own physics, which, as a whole, relies on non-mathematical explanations.87 Reading Skovoroda does make one (or at least me) think of Descartes. This is not too great an anachronism: ironically, the French academic establishment embraced Descartes only at the time when, towards the middle of the eighteenth century, the workable mathematical physics of Newton had been acquiring acceptance in more serious quarters [Jolley in Cottingham, 418-9]. I would like to finish this chapter by relating certain features of Skovoroda's system to their parallels, and, I think, origins, in Cartesian thought. Skovoroda never mentions Descartes but that, given his avoidance of naming modem sources, means nothing. When in his life might an exposure to Cartesian thought have occurred? The more obvious guess is during his five-year sojourn abroad, where his biographer reports him to have sought out unnamed local intellectual luminaries.88 Since Kovalinskii's claim is so nebulous, however, we might do well to consider another possibility. In my introduction to this chapter, I said that philosophy at Mohyla was exclusively scholastic. This is true only until Skovoroda's third return to his Academy. The lecture notes of Mohyla professors published in 1982 include a philosophy course delivered in 175189 by professor Georgii Scherbatskii, which is Cartesian with a vengeance [Stratii, Litvinov, Andrushko 289-302]. Exceptionally well-read, Scherbatskii peppers his critical analysis of Descartes’ Meditations and Principles o f Philosophy with references to Spinoza, Gassendi, Galileo, as well as Lucretius, Epicurus, and Augustine. He claims that "mathematically [...], the world is studied exclusively from the point of view of extension," equates extension and body, regards all matter as homogenous and composed of spherical corpuscules while circulate in vortices, and so forth. He also devotes considerable time to analyzing the formula cogito ergo sum, which he regards as lifted from Augustine. Admittedly, by 1751 Skovoroda had already completed his philosophy credits. Still, Scherbatskii's course is so explosively unlike anything previously taught at Mohyla, that it must have inspired tremendous buzz. It seems likely that, had Skovoroda not exposed himself to Descartes through the learned men he ostensibly sought out in Vienna, 87 Its failure is due both to the narrow range of what may be subject to mathematical treatment [Bum 106- 111 ] as well as to the fact that the Cartesian identification of mathematics and physics is such as to not leave sufficient space for experiment [Gaukroger 133-5], 88 Pierre Larousse's Grand Dictionnaire Universel du XIXe siecle claims that Skovoroda, in his years abroad, studied in Halle with the prominent Leibnizian Christian Wolff. To the best of my knowledge, there is no other evidence to that effect, and since every single other sentence in the entry contains errors, the claim ought to be taken with more than a grain of salt. (Larousse’s entry is lent credence by Maknovets [57-8] and, more enthusiastically, by Dan B. Chopyk [33, with entry quoted on 34].) 89 Same year as the Sorbonne approves the Cartesian doctrine of innate ideas [Jolley in Cottingham, 418]. 267 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Budapest, Bratislava, etc., he would have done so here. The world of Mohyla was small and tight; Kovalinskii insists on Skovoroda's intellectual curiosity as a young man [Skovoroda 2.489]; and he was turning thirty, with three years at Court and five years in Europe under his belt - the kind of student whose interests are never defined by his homework, nor friendships limited to his much younger and far less experienced classmates.90 All of this is, of course, speculation. Turning to Skovoroda's writings, let us start by taking his omission of the prior signifieds of Ideas as conjoined with the fact that, for him, a thing's geometrical figure (figura) determines the particular being of the thing. This, of course, conjoined with a parallel omission (of Aristotelian substantial forms), is the role figure (figura) also performs in Descartes. For Descartes, to be a corporeal thing (i.e. a body) is to be extended in length, width and breadth; differences between bodies consist solely in what he terms primary characteristics. These include figure [for figure as "limit of extended thing," see Regulae 12 p. 418-9], "size [...], motion, position, divisibility of component parts, and the like” [Principia 1.48]. Qualities such as color, smell, taste, etc. (the so-called secondary characteristics), have no being in the things themselves, and appear in our thought as the result of the interaction between our senses and primary characteristics. For instance, Descartes suggests that what we see as distinctions between colors are actually distinctions between geometrical patterns, i.e. figures [Regulae 12 p. 412-3; Principia 4.196-8]. If in Platonism the difference between two things like, say, a raven and a writing desk is the effect of their pointing to, or participating in, Ideas, for Descartes a raven is a raven not because it possesses some kind of "ravenness," but because its figure is such as may be labeled with the term "raven." Thus in both Descartes and Skovoroda, figure, as the geometrical aspect of a thing, determines the nature of the thing. The scenario does not account for the fact that some things are more like others, as neither does Skovoroda. Why should figures be divisible into discrete classes, like one class of ravens and another of writing desks? Why is there no shading off among them, and how would such discreteness come into being in the first place? These questions are 90 Judging by figures in Makarii Bulgakov, in 1751 Mohyla contained roughly 1000 students in all, with scarcely a 100 in philosophy, and in theology even less [108, 147]. Biographers are unanimous in describing Skovoroda as one of the best among them, and his interest in and command of classical Greek and Hebrew did set him at the top o f the pyramid [see Makarii 154-6]. In 1753, Timofei Scherbatskii, the Archbishop of Kiiv, whose relation to Georgii Scherbatskii I have been unable to ascertain, recommended Skovoroda for a job, although the word "best" in our evidence may be a hyperbole [see Makhnovets 67]. Finally, I should note that Stratii refers to Skovoroda's theology professor as another admirer of Descartes [Stratii 27], although the philosophy course taught by Konisski in 1749 is not Cartesian by any means [Stratii, Litvinov, Andrushko, 303-308]. 268 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not considered. As a result, objects in Cartesian physics must be judged ontologically indeterminate.91 Such indeterminacy is a necessary component of the system. Burtt argues that "the real criterion" for classification of characteristics as primary is "the possibility of mathematical handling" [117]. Objects, in other words, are considered upon one plane only: the plane that contains all of their aspects that may be expressed mathematically. This plane represents all of them as homogenous, as having one and the same essence. We do not treat them as ravens and writing desks: the only thing that matters is that mode of being in which they are bodies. Let us try to characterize this mode of being. What we isolate as objects of "mathematical handling" are aspects of outside appearances. Instead of being interested in the raven as a raven, we are interested in, say, its wingspan; instead of being interested in the writing desk as a writing desk, we are interested in, say, its height. Thus, the being in which all objects become homogenous in the Cartesian sense is the opposite of pure Being defined platonically, for it concerns appearances, not essence; its nature is existential. Skovoroda's omission of Platonic Ideas occurs on similar grounds. As we have seen, he asks us to perceive God not in the prior or posterior signified, but in the fact of signification common to all. This fact of signification, or fact of being, is apprehended from the identity (or homogeneity) of things: things are identical inasmuch as they all, in equal measure, are their figures, whether in the geometrical (Cartesian) or the rhetorical sense of the word. All things are alike in their figures being figures, in their very fact of appearing in an ontologically homogenizing plane. They may or may not have prior signifieds, but on this plane it does not matter. The relationship between figure and number in Descartes also recalls what may be trawled concerning the same topic from the writings of Skovoroda. While Greek geometry regards the multiplication of two lines as yielding a plane, and the multiplication of three lines as yielding a solid, Descartes' Geometrie (1637) enables its algebraic treatment of the subject by making operations with lines yield other lines. Whereas in the Geometrie figures are expressed by algebraic letter signs, which incidentally Descartes calls chiffres [ed. 91 Descartes is forced to confess that the world itself might be entirely different from how it is known through his physics:"Just as the same craftsman could make two clocks which tell the time equally well and look completely alike from the outside but have completely different assemblies of wheels inside, so the supreme craftsman of the real world could have produced all that we see in several different ways" [Principia 4.204]. He does think the possibility to be so very remote [Principia 4.205] and all but rejects it due to methodological considerations: "If a cause allows all the phenomena to be clearly deduced from it. then it is virtually impossible that it should not be true" [Principia 3.45]. Hooking onto the possibility that things are otherwise, his more conservative critics refuse to see Cartesian physics as abolishing that of his predecessors, and charge him with multiplying explanations [Stratii, Litvinov, Andrushko 269]. 269 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. princ. 299], the lines that constitute figures of things in the world have determinate length, and therefore can be expressed by numbers. Since Descartes regards number as not "separate from the things which are [...] numbered, but [... as one of the] modes under which we consider the thing in question" [Principia 1.55], the "difference between quantity and the extended substance" is said to be "merely a conceptual one, like that between number and things numbered" [Principia 2.8]. In permitting number to express magnitude, Descartes, like Stevin, overcomes the Aristotelian distinction that "a plurality is a kind of quantity if it is numerable, a magnitude is a kind of quantity if it is measurable" [Metaphys. 1020a7-9]. The fact that number is deprived of ontological autonomy92 is once again coupled with recognizing radicals and fractions as numbers in the same way as integers. The lines constituting Skovoroda's figures should be conceived as res numeratae in the Cartesian fashion, as not really different from quantities to be expressed by numbers. Here Stevin's lack of distinction between the being of number and the writing of number in Hindu-Arabic numerals justifies Skovoroda's identification of zero with that ontological plane on which things are identical. Zero symbolizes the presence of figure because it is the origin of number, but no number. Conceived in such a way and with all numbers being positive entities, zero proclaims "there is some number here,” but refuses to state what. It says, "something will be counted," but nothing is as yet counted; "there will be magnitude," but there is as yet no magnitude. Thus it, and not the one or some other "signifiyng fygure" [Recorde, Grounde Bvv], may be identified with the being of number as number, and with the being of figure as figure: the kind of being that is considered independently of its particulars. Regarded only inasmuch as they manifest this kind of being, i.e. only inasmuch as they exist, all numbers become zeroes, all magnitudes become points, ail things become God. Despite the fact that Cartesian physics, dedicated as it is to the study of appearances without regard to their allegorical meanings, must be judged by Skovoroda as idolatrous, it seems to be responsible for several of the basic features of his system. Even his definition of self as thought (Cicero's "mens cujusque, is est quisque"), is redolent the cogito, especially since Skovoroda, like Descartes, presents it as the necessary personal realization from which the rest of the system may be drawn [similar recommendation by Descartes in Principia 1.1]. Whatever his differences from Descartes may be —these are, of course, 9: "Number, when it is considered simply in the abstract or in general, and not in any created things, is merely a mode of thinking" [Principia 1.58]. For symbolic nature of the Cartesian concepts of number and figure, see Klein 197-211. 270 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. innumerable, and start with his identification of figure with thought93 —Skovoroda ought not be regarded as dissociated from the modem philosophical tradition. 93 This act, however, is not as out of the blue as it might seem: the location of res cogitans, which is unextended, in the body (= extension) is an insoluble problem for Cartesian philosophy. Descartes' general claim that the soul has its principal seat in the brain, although it also informs the entire body [Principia 4.189], treats that which is indivisible as if it were composed of parts. O f the soul’s "informing" the body he writes that "the soul is really joined to the whole body, and [...] we cannot properly say that it exists in any one part of the body to the exclusion of others. For the body is a unity which is in a sense indivisible because of the arrangement of its organs, these being so related to one another that the removal of any one of them renders the whole body defective. And the soul is of such a nature that it has no relation to extension, or to the dimensions or other properties of the matter of which the body is composed: it is related solely to the whole assemblage of the body's organs. This is obvious from our inability to conceive of a half or a third of a soul, or of the extension which a soul occupies. Nor does the soul become any smaller if we cut off some part of the body, but it becomes completely separate from the body when we break up the assemblage of the body’s organs" [Passions o f the Soul 30]. The thesis that the soul literally informs the body, that, rather than having "no relation to extension," it is identical to the body's figure, is one mincing step away from this passage. (Res cogitans and res extensa are joined by Spinoza as a single substance regarded under different modes. Taking Descartes up on his concession that God is the only autonomous substance [Principia 1.51], Spinoza defines God or Nature as the solely existing and infinite substance, of which all finite things are merely contractions or modifications; in this way, God is identical with his creation. A work on the similarities between Spinoza and Skovoroda should prove extremely interesting.) 271 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B IB LIO G R A PH Y Titles of Shakespeare plays and books of the Bible in references are abbreviated according to the fifth edition of the MLA Handbook. Other abbreviations will, I hope, be obvious (e.g. Metaphys. = Metaphysics, Principia = Principles o f Philosophy). I tried to be as ecumenical as possible in locating passages: the volumes, chapters, and sometimes the page numbers of the editio princeps are given in italics, whereas regular typeface indicates those of the edition in my bibliography. Since I could not get a hold of the latest Collected Works of Skovoroda, I provided both the signature numbers of the manuscripts used in the 1961 edition, as well as the page numbers of the same edition. Arden was used for Shakespeare citations, the King James and the Vulgate for the Bible, Hippocrates G. Apostle for Aristotle’s Metaphysics, and Loeb for those of Plato’s dialogues which have a separate entry below. English translations of foreign texts are taken from the English-language versions in the bibliography. When no English edition is cited, the translation is mine and the original is invariably in the footnotes. I omit the originals only for Skovoroda's Sion, whose Russian transliteration periodically meddles with his grammatical forms, and is therefore unquotable. Agrippa, Henry Cornelius. Three Books o f Occult Philosophy. Trans. J. F. 1651. London: Chthonios, 1987. Aristotle. Aristotle's Metaphysics. Trans, and ed. Hippocrates G. Apostle. Bloomington, IN: Indiana UP, 1966. Aristotle. The Basic Works o f Aristotle. Trans, and ed. Richard McKeon. New York: Random House, 1941. Augustine. De Libero Arbitrio (Libri Tres). The Free Choice o f the Will (Three Books). Latin text with English trans. by Francis E. Tourscher. Philadelphia: Peter Reilly, 1937. Augustine. On Free Choice o f the Will. Trans. Anna S. Benjamin and L. H. Hackenstaff. NY: Bobbs-Merrill, 1964. Augustine. The Trinity. Trans. Stephen McKenna. Fathers of the Church Ser. Washington, DC: Catholic U of America P, 1963. Baker, Humphrey. The Well-Spring o f Sciences. 1562. London, 1655. Bakhtin, M. M. Literatumo-kriticheskie stat'i. Moskva: Khudozhestvennaia literatura, 1986. 272 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Barabash, Iu. "Znaiu cheloveka": Grigorii Skovoroda: poeziia, filosofiia. zhizn'. Moskva: Khudozhestvennaia literatura, 1989. Boeio, Guiseppe. Dizionario del dialetto veneziano. Venezia: Santini, 1829. Boethius. Boethian Number Theory. A Translation o f the De Institutione Arithmetica. Trans, and intr. by Michael Masi. Amsterdam: Rodopi, 1983. Boethius. De Institutione Arithmetica. De Institutione Musica. Ed. Godofredus Friedlein. Lipsiae: B. G. Teubner, 1868. Boss, Valentin. Newton and Russia: The Early Influence, 1698 - 1796. Cambridge, MA: Harvard UP, 1972. Bovelles, Charles de / Carolus Bovillus. Liber de intellectu. Liber de sensu. Liber de nichilo. Ars oppositorum. Liber de generatione. Liber de sapiente. Liber de duodecim numeris. Epistole complures. Paris, 1510. Bovelles, Charles de / Charles de Bovelles. Le livre du neant / Liber de nichilo. Trans, and ed. Pierre Magnard. Paris: Vrin, 1983. Browne, Sir Thomas. The Major Works. Ed. C. A. Patrides. New York: Penguin, 1977. Brunschvicg, Leon. Descartes et Pascal, lecteurs de Montaigne. New York: Brentano, 1944. Bulgakov, Makarii. Istoriia Kievskoi Akademii. Sankt-Petersburg: tip. Zhemakova, 1843. Bumyeat, M. F. "Plato on Why Mathematics Is Good for the Soul." MS. to be publ. in British Academy Symposium on Mathematics and Philosophy in the History o f Philosophy. Ed. Timothy Smiley. Burton, Robert. The Anatomy o f Melancholy. Ed. Lawrence Babb. S.u., MI: Michigan State UP, 1965. Burtt, E. A. The Metaphysical Foundations o f M odem Science. Rev. ed 1952. Atlantic Highlands, NJ: Humanities P, 1992. Cajori, Florian. A History o f Mathematical Notations. Vol. 1. Chicago: Open Court, 1928. Cajori, Florian. A History o f Mathematics. NY: Macmillan 1919. Charles d'Orleans. Ballades et rondeaux. Ed. Jean-Claude Miihlethaler. Paris: Livre de Poche, 1992. Chopyk, Dan B. "G. S. Skovoroda - The Fable Writer. His Life and Times.” Fables and Aphorisms. By Gregory S. Skovoroda. Trans. Dan B. Chopyk. New York: Peter Lang, 1990. 1-70. Chyzhevs'kyi, Dmitro. Fil'osofiia H. S. Skovorodi. Warszawa: Shevchenko, 1934. Clulee, Nicholas H. John Dee’s Natural Philosophy : Between Science and Religion. London: Routledge, 1988. 273 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Colie, Rosalie L. Paradoxia Epidemica: The Renaissance Tradition o f Paradox Princeton: Princeton UP, 1966. Copenhauer, Brian P., trans. and ed. Hermetica. Cambridge, Eng.: Cambridge UP, 1992. Copleston, Frederick. A History o f Philosophy. Vol. 4. M odem Philosophy: Descartes to Leibniz. Garden City, NY: Image - Doubleday, 1963. Cotgrave, Randle. A Dictionarie o f the French and English Tongues. London, 1611. Columbia, SC: U of South Carolina P, 1950. Cottingham, John, ed. The Cambridge Companion to Descartes. Cambridge, Eng.: Cambridge UP, 1992. Crashaw, Richard. The Complete Poetry o f Richard Crashaw. Ed. George Walton Williams. New York: Norton, 1974. Crosby, Alfred W. The Measure o f Reality: Quantification and Western Society, 1250- 1600. Cambridge, Eng.: Cambridge UP, 1997. Cunnar, Eugene R. "Donne's "Valediction Forbidding Mourning" and the Golden Compasses of Alchemical Creation." Literature and the Occult. Ed. Luanne Frank. Arlington, TX: U of Texas P, 1977. 72-110. Curry, Anne. The Hundred Years War. London: Macmillan, 1933. Curtius, Ernst Robert. European Literature and the Latin Middle Ages. Trans. Willard R. Trask. Bollingen Ser. 36. New York: Harper and Row, 1963. Dante Alighieri. Divina Commedia. With commentary by Giovanni Fallani and Silvio Zennaro. Roma: Newton, 1994. Danzig, Tobias. Number: The Language o f Science. 1930. 4th ed. New York: Free P, 1954. Dee, John. The Autobiographical Tracts o f Dr. John Dee, Warden to the College o f Manchester. Ed. James Crossley. S.I.: Chetham Society, 1851. Dee, John. The Mathematicall Praeface to the Elements of Geometrie o f Euclid o f Megara. 1570. With introduction by Allen G. Debus. New York: Science History, 1975. Dekker, Thomas. The Wonderful Year. The Gull's Horn-Book. Penny-Wise, Pound- Foolish. English Villainies Discovered by Lantern and Candlelight. Selected Writitngs. Ed. E. D. Pendry. Cambridge, MA: Harvard UP, 1968. Descartes, Rene. Discours de la Methode. Paris: Gamier - Flammarion, 1966. Descartes, Rene. Oeuvres de Descartes. Vol. 8.1. Principia Philosophiae. Ed. Charles Adam and Paul Tannery. Paris: Vrin, 1973. Descartes, Rene. The Geometry o f Rene Descartes With a Facsimile o f the First Edition. Trans. David Eugene Smith and Marcia L. Latham. 1925. New York: Dover, 1954. 274 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Descartes, Rene. The Philosophical Writings o f Descartes. Trans. John Cottingham, Robert Stoothoff and Dugald Murdoch. 2 vols. Cambridge, Eng.: Cambridge UP, 1984-5. Dictionnaire de Theologie Catholique. Ed. A. Vacant, E. Mangenot and E. Amann. Vols. 3 and 9. Paris: Letouzey, 1927, 1939. Digges, Leonard and Thomas. An Arithmetical Vvarlike Treatise Named Stratioticos. 1572. London, 1590. Dionisii Areopagit. O bozhestvennykh imenakh. O misticheskom bogoslovii. With commentary by Maximus the Confessor. Greek ed. and trans. G. M. Prokhorov. Sankt-Peterburg: Glagol, 1994. Dionysius, pseudo-. The Complete Works. Trans. Colm Luibheid. Classics of Western Spirituality Ser. London: SPCK, 1987. Donne, John. Essays in Divinity. Ed. Evelyn M. Simpson. Oxford: Oxford UP, 1952. Donne, John. The Complete English Poems o f John Donne. Ed. C. A. Patrides. Rev. ed. London: Everyman - Dent, 1994. Donne, John. The Sermons o f John Donne. 10 vols. Ed. Evelyn M. Simpson and George R. Potter. Berkeley, CA: U of California P, 1958. Du Cange, Charles Du Fresne. Glossarium mediae et infimae latinitatis. 1688. Expanded ed. 10 vols. Niort: Favre, 1883-7. Eriugena, Johannes Scottus. Periphyseon. 3 vols. Ed. and trans. I. P. Sheldon-Williams with Ludwig Bieler. Scriptores Latini Hibemiae, vols. 9-11. Dublin: Dublin Institute for Advanced Studies, 1968-81. Eros baroque : anthologie de la poesie amoureuse baroque 1570-1620. Ed. Gisele Mathieu-Castellani. Paris: Nizet, 1986. Euclid. The Thirteen Books o f Euclid's Elements. Trans, and ed. Thomas L. Heath. 2nd ed. 1925. 3 vols. New York: Dover, 1956. Evans, Gillian R. "From abacus to algorism: Theory and practice in medieval arithmetic." British Journal fo r the History o f Science, 10.35(1977): 114-131. Everyman and Medieval Mystery Plays. Ed. A. C. Cawley. London: Everyman —Dent, 1962. Ficino, Marsilio. Opera omnia. 1576. 2 vols. Torino: Bottega d'Erasmo, 1959. Fineman, Joel. "The Sound of O in Othello: The Real of the Tragedy of Desire." Critical Essays on Shakespeare's Othello. Ed. Anthony Gerard Barthelmy. New York: G. K. Hall, 1994. Freccero, John. "Donne’s 'Valediction forbidding mourning'." ELH, 3.4 (Dec. 1963). Freeman, Rosemary. English emblem books. London: Chatto and Windus, 1948. 275 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fridugis / Fredegisus. "Epistola de substantia nihili et de tenebris an sint." Monimenta Germaniae Historica. Epistolae Karolini Aevi. Ed. Ernest Diimmler. Vol. 2. Berlin: Weidmann, 1895. Fridugis. "Letter on nothing and darkness." Trans. Hermigild Dressier. Medieval Philosophy: from St. Augustine to Nicholas ofCusa. Ed. John F. Wippel and Allan B. Walter. New York: Free P, 1969. 103-108, 468. Funkenstein, Amos. Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century. Princeton: Princeton UP, 1986. Galileo. The Discoveries and Opinions o f Galileo. Ed. Stillman Drake. Garden City, NY: Anchor - Doubleday, 1957. Gaukroger, Stephen, ed. Descartes: Philosophy, Mathematics and Physics. Totowa, NJ: Barnes & Noble, 1980. Gerson, Lloyd P. The Cambridge Companion to Plotinus. Cambridge, Eng.: Cambridge UP, 1996. Gilson, Etienne. Le Thomisme: Introduction a la philosophie de Saint Thomas d'Aquin. Paris: Vrin, 1948. Grant, Edward. Much Ado About Nothing: Theories o f Space and Vacuum from the Middle Ages to the Scientific Revolution. Cambridge UP: 1981. Gullberg, Jan. Mathematics: From the Birth o f Numbers. New York: Norton, 1997. Hadjitarkhani, Abie. "Alice of the Fine Parts." 6,500, 2 (Spring 2000): 84-6. Halliwell-Phillipps, James Orchard, ed. Rara Mathematica. 2nd ed. London: Maynard, 1841. Harington, Sir John. A New Discourse o f a Stale Subject, Called The Metamorphosis o f Ajax. Ed. Elizabeth Story Dunno. New York: Columbia UP, 1962. Heninger, S. K. Touches o f Sweet Harmony: Pythagorean Cosmology and Renaissance Poetics. San Marino, CA: Huntington Library, 1974. Higgins, Dick. Pattern Poetry: Guide to an Unknown Literature. Albany, NY: SUNY P, 1987. Holinshed, [Raphael]. Holinshed's Chronicles: Richard II 1 398-1400, Henry TV and Henry V. Oxford: Oxford UP, 1923. Horace. The Works o f Horace. Ed. Thomas Chase. Philadelphia, PA: Eldridge, 1892. Howard, Jean E. and Phyllis Rackin. Engendering a Nation: A Feminist Account o f Shakespeare's English Histories. London: Routledge, 1997. Isidore of Seville. Etymologiarum sive Originum Libri XX. Ed. W. M. Lindsay. Vol. 1. Oxford: Oxford, 1911. 276 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Iversen, Erik. The Myth o f Egypt and its Hieroglyphs in European Tradition. Princeton: Princeton UP, 1993. Jaffe, Michele Sharon. The Story o f O: Prostitutes and Other Good-for-Nothings in the Renaissance. Harvard Studies in Comparative Literature, 45. Cambridge MA: Harvard UP, 1999. Johnson, Francis R. Astronomical Thought in Renaissance England: A Study o f the English Scientific Writings from 1500 to 1645. Baltimore: Johns Hopkins, 1937. Jones, Roger Miller. "The Ideas as the Thoughts of God." Classical Philology, 21 (January - October 1926): 317-26. Jonson, Ben. Ben Jonson's Conversations with Drummond o f Hawthomden. Ed. R. F. Patterson. London: Blackie, 1924. Jordan, Leo. "Materialen zur Geschichte der arabischen Zahlzeichen in Frankreich." Archiv fu r Kultur-Geschichte, 3.3 (1905): 135-195. Karpinski, Louis Charles. The History o f Arithmetic. NY: Russel & Russel, 1965. Kharms, Daniil. "O vremeni, o prostranstve, o suschestvovanii," etc. Logos 4 (1993): 102-24. Khlebnikov, Velimir. Tvoreniia. Moskva: Sovetskii Pisatel’, 1986. Kircher, Athanasius. Oedipus Aegyptiacus, hoc est Vniversalis hieroglyphicae veterum doctrinae temporum iniuria abolitae instauratio. Vol. I. Roma, 1652. Klein, Jacob. Greek Mathematical Thought and the Origin o f Algebra. Trans. Eva Brann. 1968. New York: Dover, 1992. Klibansky, Raymond, Erwin Panofsky and Fritz Zaxl. Saturn and Melancholy: Studies in the History o f Natural Philosophy, Religion and Art. London: Thomas Nelson, 1964. Kuhn, David. The Codebreakers. Abr. ed. London: Sphere Books, 1973. Lachtin, Mikhail. Meditsina i vrachi v Moskovskom gosudarstve. M.: Univers. tip., 1906. Laqueur, Thomas. Making Sex: Body and Gender from the Greeks to Freud. Cambridge, MA: Harvard UP, 1990. Lateinische Gedichte deutscher Humanisten. Ed. Harry C. Schnur. Stuttgart: Reclam, 1967. Leibniz, Gottfried Willhelm. Leibniz korrespondiert mit China: der Briefwechsel mit den Jesuitmissionaren (1689-1714). Ed. Rita Widmeier. Frankfurt am Main: Klostermann, 1990. Lewalski, Barbara Kiefer. Protestant Poetics and the Seventeenth-Century Religious Lyric. Princeton, N.J.: Princeton University Press, cl979. Lewis, Charleton T., and Charles Short. A Latin Dictionary. Oxford: Oxford UP, 1966. Losev, A. F. Mif, Chislo, Suschnost'. Moskva: Mysl', 1994. 277 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Makhnovets’, Leonid. Hryhorii Skovoroda. Kiiv: Naukova Dumka, 1972. Male, Emile. The Gothic Image: Religious Art in France o f the Thirteenth Century. 3rd ed. Trans. Dora Nussey. New York: Harper and Row, 1972. Mamardashvili, Merab. Kartezianskiie razmyshleniia. Moskva: Progress - Kul'tura, 1993. Marshak, Samuil. Sobranie sochinenii v chetyrekh tomakh. Vol. 1. Moskva: Khudozhestvennaia literature, 1957. Marvell, Andrew. The Complete Poems. Ed. Elizabeth Story Donno. NY: Penguin, 1985. Maslov, S. I. Biblioteka Stefana Iavorskago. Kiev: Meinander, 1914. Matter, E. Ann. The Voice O f My Beloved: The Song of Songs In Western Medieval Tradition. Philadelphia, PA: U Penn P, 1990. Menninger, Karl. Number Words and Number Symbols. Trans. Paul Broneer. 1969. NY: Dover, 1992. Merriam-Webster New Book o f Word Histories. Springfield, MA: Merriam-Webstar, 1991. Montaigne, Michel Eyquem de. Essais. 2 vols. Paris: Gamier, 1962. Moran, Dermot. The Philosophy o f John Scottus Eriugena: A Study o f Idealism in the Middle Ages. Cambridge, Eng.: Cambridge UP, 1989. Mullachius, Guilelmus Augustus, ed. Fragmenta philosophorum graecorum. Vol. 2. Paris: Didot, 1867. Murray, David. Chapters in the History o f Bookkeeping, Accountancy and Commercial Arithmetic. 1930. New York : Amo Press, 1978. Nicholas of Cusa. On Learned Ignorance. A Translation and an Appraisal o f De Docta Ignorantia. Jasper Hopkins, ed. and trans. 2nd ed. Minneapolis: Arthur J. Banning, 1990. Nicolas de Cusa. Opera omnia. Iussu et Auctoritate Academiae Litterarum Heidelbergensis. Vols. 1, 3, 5. Lipsiae: Felix Meiner, 1932, 1972, 1983. Nicomachus of Gerasa. Introduction to Arithmetic. Trans. Martin Luther D'Ooge. 1926. New York: Johnson Reprint, 1972. Nostradamus, Michel. Interpretation des Hieroglyphes de Horapollo. Ed. Pierre Rollet. Barcelona: Ramoun Berenguie, 1968. Oleinikov, Nikolai. "Stikhi." Avrora 1989, no. 6: 123. Ostashevsky, Eugene. "Strashnyi Sud, ili Kak tainoe stanovitsia iavnym." Vidimost' nezrimogo / The Visuality o f the Unseen. Ed. Dmitry Golynko-Volfson. Sankt- Peterburg: Borey, 1996. 221-262. Ovid. Amores. Medicamina faciei feminae. Ars amatoria. Remedia amoris. Ed. E. J. Kenney. Oxford: Oxford UP, 1961. 278 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oxford English Dictionary. Compact ed. New York: Oxford UP, 1971. Palmer, Michael. A t Passages. New York: New Directions, 1995. Panchenko, A. M. Russkaia stikhotvomaia kul'tura X VII veka. Leningrad: Nauka, 1973. Parker, Patricia. Shakespeare from the Margins: Language, Culture, Context. Chicago: U of Chicago P, 1996. Parschikov, Aleksei. Figury intuitsii. Moscow: Moskovskii rabochii, 1989. Parshall, Karen Hunger. "The art of algebra from Al-Khwarizmi to Viete: A study in the natural selection of ideas." History o f Science, 26:72 (June 1988). 129 - 164. Rev. ver. <http-//viva.lib.virginia.edu/sc>. Accessed 10 December 1998. Partridge, Eric. Shakespeare’s Bawdy. New York: Dutton, 1960. Pascal, Blaise. Pensees. Ed. Leon Brunschvicg. Paris: Gamier - Flammarion, 1976. Paster, Gail Kem. The Body Embarassed: Drama and the Disciplines o f Shame in Early M odem England. Ithaca, NY: Cornell UP, 1993. Pelikan, Jaroslav. The Christian Tradition: A History o f the Development o f Doctrine. Vol. 1. The Emergence o f Catholic Tradition (100-600). U of Chicago P, 1971. Petit Robert 2: Dictionnaire universel des noms propres. Ed. Alain Rey. New ed. Paris: Le Robert, 1990. Philo, of Alexandria. "Life o f Moses." Works. Trans. F.H. Colson and G.H. Whitaker. 10 vols. Cambridge, MA: Loeb - Harvard UP, 1956. Picinelli, Filippo / Philippus Picinellus. Mundus Symbolicus. Trans, and ed. Augustus Erath. 1694. 2 vols. New York: Garland, 1976. Pico della Mirandola. Giovanni. Conclusiones nongentae : le novecento tesi dell’anno I486. Ed. Albano Biondi. Firenze : Leo S. Olschki, 1995. Plato. Cratylus. Parmenides. Greater Hippias. Lesser Hippias. Trans. Harold North Fowler. Rev. ed. Cambridge, MA: Loeb - Harvard UP, 1953. Plato. The Collected Dialogues o f Plato, Including The Letters. Ed. Edith Hamilton and Huntington Cairns. Princeton: Princeton UP, 1961. Plato. Theatetes. Sophist. Trans. Harold North Fowler. Cambridge, MA: Loeb - Harvard UP, 1921. Plotinus. Enneads. Trans. A. H. Armstrong. Vols. 5-7. Cambridge, MA: Loeb - Harvard UP, 1984-8. Plutarch. "Isis and Osiris." Moralia. Vol. 5. Trans. Frank Cole Babbitt. Cambridge, MA: Loeb - Harvard UP, 1957. 3 - 191. Plutarque. Oeuvres Morales. Vol. 9.3. Propos de table [Questiones convivialium], Livres 7 - 9. Trans. Fran^oise Frazier and Jean Sirinelli. Paris: Belles Lettres, 1996. 279 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Poetry gruppy "0BER1U." Ed. M. B. Meilakh. Biblioteka poeta, bol'shaia ser. Sankt- Peterburg: Sovetskii pisatel', 1994. Poulet, Georges. Les Metamorphoses du Cercle. 1961. Paris: Flammarion, 1979. Pound, Ezra. Personae: The Shorter Poems o f Ezra Pound. Rev. ed. Ed. Lea Baechler and A. Walton Litz. New York: New Directions, 1990. Praz, Mario. Studies in Seventeenth-Century Imagery. 2 vols. London: Warburg, 1939, 1947. Princeton Encyclopedia o f Poetry and Poetics. Ed. Alex Preminger. Enlarged ed. Princeton: Princeton UP, 1974. Prioison, Al'f. Pro kozlenka, kotoryi umel schitat' do desiati. Trans. V. Ostrovskii. Moskva: Rosmen, 1998. Proclus. A Commentary on the First Book o f Euclid’s Elements. Trans. Glenn R. Morrow. Princeton: Princeton UP, 1970. Prokl. Kommentarii k pervoi knige "Nachal" Evklida. Vvednie. Trans, and ed. Iu. A. Shichalin. Moskva: Greko-latinskii kabinet, 1994. Puttenham, George. The Arte o f English Poesie. Facs. of 1589 ed. Amsterdam: Theatrum Orbis Terrarum, 1971 Pye, Christopher. The Regal Phantasm: Shakespeare and the Politics o f Spectacle. London: Routledge, 1990. Quint, David. "'Alexander the Pig’: Shakespeare on History and Poetry." Boundary 2:10 (1982). Rabanus Maurus. B. Rabani Mauri [...] Opera Omnia. Vol. 2. Ed. J-P Migne. Patrilogia Latina, 111. Tumhout, Belgium: Brepols, s. a. Rabelais, Francois. Oeuvres Completes. Ed. Pierre Jourda. Vol. 1. Paris: Gamier, 1962. Ramus. Petrus. The Art o f Arithmeticke in Whole numbers and Fractions. Trans. William Kempe. London, 1592. Raspe, Rudolph, et al. The Singular Adventures o f Baron Munchausen. Ed. John Carswell. New York: Heritage, 1952. Record[e], Robert. The Grounde o f Artes. 1542. Amsterdam: Theatrum Orbis Terrarum, 1969. Record[e], Robert. The pathway to knowledg, containing the first principles o f geometrie. London: R. Wolfe, 1551. Record[e], Robert. The whetstone o f witte, which is the seconde parte o f Arithmetike. London, 1557. Rich, Audrey N. M. "The Platonic Ideas as the Thoughts of God." Mnemosyne. 4th ser., 7.2 (1954): 123-133. Richter, Vil'gelm. Istoriia meditsiny v Rossii. Vol. 2. Moskva: Universitetskaia tip., 1820. 280 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Riginos, Alice Swift. Platonica: The Anecdotes Concerning the Life and Writings o f Plato. Leiden: E. J. Brill, 1976. Ripa, Cesare. Iconologia. Milano: TEA Arte, 1992. Ripa, Cesare. Iconologie. Paris, 1644. Rotman, Brian. Signifying Nothing: The Semiotics o f Zero. London: Macmillan, 1987. Russell, Bertrand. Introduction to Mathematical Philosophy. 2nd ed. 1920. New York: Dover, 1993. Sanches, Francisco / Franciscus Sanchez. That Nothing is K now n/ Quod Nihil Scitur. English ed. Elaine Limbrick. Latin ed. and trans. Douglas F. S. Thompson. Cambridge, Eng.: Cambridge UP, 1988. Sanford, Vera. "Robert Recorde's Whetstone o f witte, 1557." From five fingers to infinity. Ed. Frank J. Swetz. Chicago: Open Court, 1994. Sartori, Fausto. L'Arte dell'acqua di vita: Nascita e fine di una corporazione di mestiere veneziana (1618-1806). Venezia: Fondazione Scientifica, 1996. Sayre, Kenneth M. Plato's Late Ontology: A Riddle Resolved. Princeton UP, 1983. Sazonova, L.I. Poeziia russkogo barokko (Vtoraia polovina XVII —nachalo XVIII v.). Moskva: Nauka, 1991. Schoenbaum, S. William Shakespeare: A Compact Documentary Life. 1977. New York: New American Library, 1986. Scholem, Gershom G. Major Trends in Jewish Mysticism. New York: Schocken, 1973. Scott, Walter, ed. and trans. Hermetica. Vol. 1. Introduction, Texts, and Translation. Boston: Shambala, 1985. Shakespeare, William. Hamlet. Ed. Harold Jenkins. New York: Arden - Methuen, 1982. Shakespeare, William. King Henry V. Ed. J. H. Walter. New York: Arden —Methuen, 1954. Shakespeare, William. King Lear. Ed. R. A. Foakes. London: Arden - Nelson, 1997. Shakespeare, William. Measure fo r Measure. Ed. J. W. Lever. 1965. New York: Arden - Routledge, 1988. Shakespeare, William. Othello. Ed. M. R. Ridley. 1958. New York: Arden —Routledge, 1989. Shakespeare, William. The Complete Works o f Shakespeare. Ed. David Bevington. 4th ed. New York: Longman, 1997. Shakespeare, William. The First Part o f King Henry the Fourth: Texts and Contexts. Ed. Barbara Hodgdon. New York: Bedford, 1997. Shakespeare, William. The Life o f Henry V. Ed. Barbara A. Mowat and Paul Werstine. New York: Folger - Washington Square, 1995. Shakespeare, William. The Merchant o f Venice. Ed. John Russell Brown. 1959. New York: Arden - Routledge, 1988. 281 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Shakespeare, William. The Tempest. Ed. Stephen Orgel. New York: Oxford UP, 1988. Sherman, William H. John Dee: The Politics o f Reading and Writing in the English Renaissance. Amherst, MA: U of Massachusetts P, 1995. Shirley, John W. Thomas Harriot: A Biography. Oxford: Oxford UP, 1983. S i m o n o v , R. A. Matematicheskaia mysl' Drevnei Rusi. Moskva: Nauka, 1977. Simson, Otto von. The Gothic Cathedral: Origins o f Gothic Architecture and the Medieval Concept o f Order. 2nd ed. Bollingen Ser. 48. Princeton: Princeton UP, 1974. Sinfield, Alan. Faultlines: Cultural Materialism and the Politics o f Dissident Reading. Berkeley: U of California P, 1992. Skovoroda, Grigorii. "Sion.” Sochineniia. Trans. I. V. Ivan'o and M.V. Kashuba. Vol. 1. Moskva: Akademiia Nauk SSSR, 1973. 266-294. Skovoroda, Hryhorii. Tvori. 2 vols. Kiiv: Akademiia Nauk URSR, 1961. Smet, Antoine de. "John Dee et sa place dans 1'histoire de la cartographie." My Head is a Map: Essays and Memoirs in Honour o f R. V. Tooley. Ed. Helen Wallis and Sarah Tyacke. London: Francis Edwards and Carta, 1973. 107-116. Smith, David Eugene and Louis Charles Karpinski. The Hindu-Arabic Numerals. Boston: Ginn, 1911. Soboleva, T. A. Tainopis' v istorii Rossii (istoriia kriptograficheskoi sluzhby Rossii XVIII - nachala XX veka). Moskva: Mezhdunardnye otnosheniia, 1994 Spafarii, Nikolai. Esteticheskie Traktaty. Leningrad: Nauka, 1978. Steele, Robert, ed. The Earliest Arithmetics in English. Early English Text Society 118. Oxford: Oxford UP, 1922. Stevin, Simon. Principal Works o f Simon Stevin. Vol. II a-b: Mathematics. Ed. by D. J. Struik. Amsterdam: Swets & Zeitlinger, 1958. Stratii, la. M, V.D Litvinov and V. A. Andrushko. Opisanie kursov filosofii i ritoriki professorov Kievo-Mogilianskoy Akademii. Kiev: Naukova Dumka, 1982. Stratii, la. M. "Spetsyfika rozrobky naturfilosofskoi problematiki u Kyievo-Mohylians'kii akademii." Ed. V. M. Rusanivs'kii. Rol' Kyievo-Mohylians'kii akademii v kul'tumomu iednanni slov'ians'kikh narodiv. Kiiv: Naukova Dumka, 1988. 26-30. Swetz, Frank. Capitalism and Arithmetic: The New Math o f the 15th Century, Including the Full Text o f the Treviso Arithmetic o f 1478 Translated by David Eugene Smith. La Salle, IL: Open Court, 1987. Taylor, E. G. R. Tudor Geography, 1485-1583. London: Methuen, 1930. Thomas a Kempis. The Imitation o f Christ. Mod. ver. of Richard Whitford's 1530 trans. Ed. Harold C. Gardiner. New York: Image - Doubleday, 1955. 282 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tourneur, Cyril. The Revenger's Tragedy. Three Jacobean Tragedies. Ed. Gamini Salgado. Rev. ed. New York: Penguin, 1969. Ushkalov, Leonid. Hryhorii Skovoroda i antychna kul'tura. Khar'kiv: Znannia, 1997. Vergil, Polydore. De rerum inventoribus. Trans. John [i.e. Thomas] Langley. New York: Agathynian Club, 1868. Victor, Joseph M. Charles de Bovelles, 1479-1553: An Intellectual Biography. Travaux d’Humanisme et Renaissance 161. Geneva: Librarie Droz, 1978. Vigenere, Blaise de. Traicte des Chiffres ou Secretes Manieres d'Escrire. Paris, 1586. Virgil. The Aeneid o f Vergil. Ed. Charles Knapp. Rev. ed. Glenview, IL: Scott, Forseman 1951. Waerden, B. L. van der. A history o f algebra from Al-Kwarizmi to Emmy Noether. Berlin: Springer-Verlag, 1985. Willbem, David. "Shakespeare’s Nothing.” Representing Shakespeare. Ed. Coppelia Kahn and Murray M. Schwartz. Baltimore, MD: The Johns Hopkins UP, 1980. Williams, Penry. The Late Tudors: England 1547-1603. Oxford: Oxford UP, 1995. Wolfson, H. A. "The meaning of ex nihilo in the Church Fathers, Arabic and Hebrew Philosophy, and St. Thomas.” Mediaeval Studies in Honor o f Jeremiah Denis Matthias Ford. Ed. Urban T. Holmes, Jr. and Alex. J. Denomy. Cambridge, MA: Harvard UP. 1948. 355-70. Yates, Frances. Giordano Bruno and the Hermetic Tradition. Chicago: U of Chicago P, 1964. Zhmud', L. Ia. Nauka, filosofiia i religiia v rannem pifagoreizme. Sankt-Petersburg: Aleteiia - VGK, 1994. 283 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.