(PDF) Math Anxiety

ANGELAKI journa l of the the oretical humani tie s v olum e 5 n umber 3 december 2000 H ere is a math problem, a puzzle. You remove the plastic wrap, and open the lid. Pieces swim before you: amorphous blobs that overlap and stick to each other. As you begin to fit uncooperative pieces together, contours form, edges rigidify into straight lines or complex curves, knots appear and twist the goo of a piece into an unmanageable tangle, or a perfect sphere, or a Julia set. Pieces break apart, generate spon- taneously, cover and eat each other. As pieces are brought into proximity, they take on not only contours or shapes, but also colors and patterns on their surfaces. Sometimes pieces “match,” sometimes they clash, sometimes you can’t tell. This makes the puzzle significantly more complex (!), and the pieces even harder to place; an exact fit is out of the question. When one solu- aden evens tion loses its promise, you shuffle pieces and try others. In every solution, some pieces fit better than others, and some don’t fit at all. It needn’t be emphasized that this puzzle has no correct MATH ANXIETY solution; it determines itself only as it is being fit together. this kind of puzzle; they demand to be solved, Such a puzzle might sound rather boring; no but one cannot say in advance what pieces may right answers, no logic, no sense; just pieces be brought to bear in response to this demand or doing as they will. But look closer. Wherever you how the problem will develop as one responds to look, the pieces tangle, meld, and shift in relation it. Confronted with a problem, one draws on bits to each other. Not just shape, but everything of history, geography, biology, physics, psychol- about a piece is in flux, from color to weight, ogy, rhetoric, aesthetics, etc. Not just anything strength to poverty, what it ate for breakfast to will make sense in relation to a problem, but one how it deals with gravity. This flux is not a chaos, cannot foretell what might end up making sense. far from it, for in chaos there can be no issue of When your boyfriend threatens to call your wife color or weight, strength or poverty. Chaos does and tell her “everything,” you’ve got a problem, not admit of any sense, whereas this puzzle makes and who knows what constellations this situation lots of sense, it generates sense as its pieces act will lead to? When you have an hour to kill and and react in relation to each other. Color and aren’t sure what to do, this is a problem; prob- weight, affect and defect are sensible, and the lems may even be welcome, but pleasant or way that these sensible notions fit together is how unpleasant, whatever the outcome, the problem sense is made. calls for something, demands something not-yet- How would you work on this puzzle? You determined of you and of the world. I mean to already are working on it. I mean, you have prob- emphasize here the complex and open nature lems. Doesn’t everyone? And problems are just of problems, the way they do not break down ISSN 0969-725X print/ISSN 1469-2899 online/00/030105-11 © 2000 Taylor & Francis Ltd and the Editors of Angelaki DOI: 10.1080/09697250020034788 105 math anxiety cooperatively, dissolve into clear and distinct that give them meaning in the world. Sense is elements. They drag otherwise unrelated people, thus produced after the fact, and ontogenesis places, and objects into the mix. Use your shoe describes the generation of objects but not their for a pillow. Freeze your compact discs. Hire a relations. This standard metaphysics often philosopher to reorganize your company. Maybe explains genesis as a spontaneous modal flip dogs are good at sexing chicks. Your problem (from possible to actual), and sense as an extra- happens as solutions are attempted and neous imposition of a human perspective on a discarded, feeding into the problem to alter its fully formed pre-existent object. By contrast, parameters; new obstacles and new paths may Deleuze’s account holds that genesis is not a crop up at any time, any place. Think laterally, sudden change of modes, but a process of deter- act locally. mination, beginning not from a previous deter- When rain drips onto a leaf, the paths it takes mined state (the possible), but from the across that uneven surface are solutions to a undetermined. This process is a puzzle being fit complex problem at the juncture of differential together, an insistent demand or pressure. To topology, fluid dynamics, evolutionary biology, understand why things are the way they are, we and metaphysics. If this problem is a puzzle, look at what problems they respond to, what then who is the puzzler?; the pieces – water, leaf, forces create just these tensions, conflicts, and time, gravity, inertia, happenstance, etc. – are congruencies. Things make sense when the only agents. As such, the problem becomes connected to their problematic origins. clear only in the course of its solution; one The problem unifies its cases of solution, not recognizes the problem and its boundaries after because each element of the solution is inter- the fact, or as it is working itself out. The prob- changeable, not because each part of the solution lem makes sense of its solutions, and these solu- responds to the problem in the same way, but tions make sense only when one grasps their because the entire solution responds to the prob- unity in the problem. lem together. The parts of a machine do not each I am still discussing puzzles, but these puzzle do the same thing, but they are unified by virtue pieces do not come shrink-wrapped in a box. of the machine they constitute. The problem is a Instead, you find them all around you, wherever machine, a function or field, and this field orders there is a tension, wherever something must be and skews the objects to which it gives rise. The done, wherever things are created, destroyed, problem thus creates a kind of continuity, a altered, augmented, combined, and juxtaposed. continuity that connects its cases of solution in Anywhere that sense is made, it is made by a sense and sensibility, but the various parts of the problem working itself through, a puzzle that solution are heterogeneous, each playing its own demands to be worked and re-worked, and whose role in relation to the problem and to the other pieces cannot but respond to that demand. This elements of the solution. Even to talk of solutions puzzle is ontogenesis, the way things become here is misleading, for the most problematic what they become. At the beginning of Chapter problems do not disappear in their solutions. Four of Difference and Repetition,1 Gilles Rather, the so-called solution is only that Deleuze offers an account of ontogenesis as a moment when the problem reaches its greatest problematic puzzle, pieces that pose and respond degree of clarity, when the forces that constitute to problems, acting and interacting to determine the problem ally themselves clearly on one side the relations that make sense of them or make or another. If we consider only this ultimatum, them sensible, give them character. The problem, this static snapshot of the problem, the problem in working itself out, determines both the disappears. If instead we focus on the process by elements that figure in its solutions and the which the problem comes to be, then we survey (sensible) qualities of those elements. To become not only the elements of the problem, but the is to become sensible. The dominant history of determination of their relations, the generation of Western metaphysics holds that things first of all their sense in what Deleuze refers to as the prob- come to be, only later to assume the relationships lematic Idea. 106 evens Why does he call this puzzle – that ties together meaningfully as a means of working out together its elements into the continuity of a a problem. One cannot say in advance what will sensible solution – an “Idea”? Platonic Ideas take shape, which forces will be significant, gather a number of objects under a common which events will stand out, who will play a term, such as “beautiful” or “virtuous.” The pivotal role. Guests arrive at a cocktail party with participants in an Idea share that Idea in all sorts of history, tendencies, characters, but common, though there is nothing else in particu- with no set plan. The situation itself is a problem. lar that they need share. That is, participation in What is going to happen? Whom will Ed talk to? the Idea is irreducible. Still, participation in an Is it going to get political again? Are there Idea is not haphazard, for it makes sense: the set enough hors d’oeuvres? And if the host poses of participants in an Idea forms a sensible contin- these problems to herself, she may be mostly uum. For example, the Idea of human phenotype ineffectual as the problem of the party will is irreducible: there are no necessary and suffi- undoubtedly find its own solution, agreeable or cient elements that make a body human. But otherwise. Over the course of the evening, the there are plenty of characteristics of human guests permute their arrangements according to bodies, and these characteristics vary continu- the problematic dynamic determined by their ously–height, density, skin color, eye color, shape singular personalities, their prior relationships, of head, etc. The Idea of the body yields a conti- the availability of alcohol, the style of music and nuity, a mass of individual bodies that range locations of the loudspeakers, and who-knows- continuously in a variety of ways. what else. The result is a complex continuity of Kant, as well, uses Ideas to make sense of movement that constitutes the only sense of the things, to determine continuities, but unlike cocktail party and is unified only by the problem Plato, he does not begin with ready-made things that inheres in the color of each tie, the strength which must be parceled out, sorted into the of each drink, and the wit in every quip. correct Ideas. Kant makes a great leap when he The problematic Idea, says Deleuze, provides focuses on the process by which continuities a “systematic unity” (169) to its cases of solution; form. The understanding makes sense of the but the unity that the puzzle gives to its pieces is world by fitting it into concepts and their rela- never final nor is the fit ever exact. Instead of a tions. Concepts alone are inadequate to make single big picture on its surface, the puzzle is sense of anything; they form isolated cases that always broken by unaligned edges and sudden offer no reason for their existence. Only by relat- changes of color from piece to piece. That is why ing concepts in the problematic Idea do they bear the usual methods of investigating continuity are relations that make sense of them, for they flawed. One attempts to start with the completed become not isolated instances but cases of solu- puzzle and work backwards, taking it apart into tion, responses to a problem that determines its pieces to determine what it’s made of and how their histories and gives them a sufficient reason. it’s put together. But this method is doomed to Whereas concepts are determinate things, fail, for breaking the puzzle into pieces yields comprising specific qualities, the manifold of only smaller puzzles but hides the process of intuition that we must make sense of is wholly determination by which those pieces take shape. indeterminate, a vague jumble with no particular The philosophical investigation of sense generally characteristics, nothing accessible to the under- starts with one determined state of affairs and standing. The process by which the world is attempts to trace it back to another determined divided into concepts is thus a process of deter- state of affairs whence it came, searching for the mination, from the undetermined to the determi- reasons for things among those very things. nate or from manifold to concept. Ontogenesis is thought to proceed from determi- In Kant, the Idea still gives rise to continuity, nation to determination, thus ignoring the but it is no longer, as in Plato, the continuity of process by which things become determined. If a common quality. The complex continuity we start with the continuous number line made generated by the problematic Idea brings things of points, we can break that line down into 107 s7 s5 s3 s1 P s8 s6 s4 s2 s0 Math Anxiety -1- Aden Evens math anxiety Fig. 1. A sequence of points, s0, s1, etc., that approaches the point P. Note that no matter how small an interval we consider around P, at some point in the sequence of sn , all further points in the sequence will be contained in that interval. smaller and smaller continuous pieces, but we close to the center of the interval, since no matter will never discover what makes the line or those how small we make the interval, the tail of the pieces continuous, no matter how small the parts line is still contained in it. The fundamental get. Working backwards from continuity to its concept of calculus, the concept of the limit, is cause will not succeed. Instead, we must discover defined in just such finitist terms. A sequence of an ideal cause of continuity, and examine how terms is said to approach a limit when it can be that cause can generate the continuity alongside shown that no matter how small an interval you the continuous things. We must acknowledge choose around that limit, there is some point in that things and their senses come to be simulta- the sequence after which every term is contained neously and interdependently. As that ideal cause within that interval (Figure 1). So, in proving of continuity, the problematic Idea that draws something about limit points, we only ever things together into sense, Deleuze proposes the consider intervals of finitely small size, for one mathematical differential, dx. can of course always find a smaller interval. I did mention at the outset that this was a Likewise, in proofs using the differential, we math problem. A math problem, even a calculus never actually involve the infinite; rather, we use problem, seems at first to be a rather poor exam- variables, which are manipulated as arbitrarily ple of a problem. The math problems one finds small (but still finitely small) numbers. The at the end of the chapter in math textbooks just important point here is that as long as our sit there, seemingly indifferent to whether or not symbolism reduces the limit, and by reference they are solved. (No doubt, because they have the differential as well, to finite quantities and already been solved.) Indeed, most of us feel variables that represent such quantities, the limit little compulsion to attempt a solution. They are and the differential become just more quantities, not very problematic problems. Fermat’s Last a finite value that is special only in that it is Theorem, on the other hand, certainly was a unspecified or unknown. Without the power of problem: it begged to be solved.2 Deleuze discov- problem, the differential loses its unique ers in archaic interpretations of the differential a dynamic, and we lose sight of its unique powers. problematic power, the power to problematize. The limit is Lesson One in a modern calculus How does the differential, dx, cause problems? class. This lesson teaches the mathematical In the modern calculus, the differential is ill- formalisms that correspond to the notion of suited to be the ideal cause of continuity, because “arbitrarily small.” So, one learns which symbols it is understood on a finitist model, or, at best, as to use and what rules govern the manipulation of an infinitesimal. In other words, the modern those symbols, or what constitutes legitimate interpretation of the differential makes it out to proof in the domain of the arbitrarily small. be an arbitrarily small quantity. In modern calcu- Lesson Two makes use of Lesson One to show lus, when we use the differential (say, to prove how to take a derivative. Though we are intro- something or to take a derivative), what we do is duced to it by means of the limit or differential to demonstrate that something-or-other is true at its core, the derivative is no sooner learned for an arbitrarily small quantity. Usually, it is a than the differential is discarded, and even its question of intervals: a line “approaches” a point symbol, dx, is often omitted. Calculating deriva- when, no matter how tiny an interval around the tives becomes a matter of the rote manipulation point you take, it still encloses the end of the of symbols, and the intimate relationship line. That line, we reason, must get arbitrarily between the derivative and the differential is 108 evens forgotten, a distant murky memory that won’t be on the exam. If the differential still appears in certain formulas, it is only an annoying vestige of the higher mathematics that birthed it. So, let me remind you of the relationship between limits and derivatives. Derivatives are first introduced by posing this problem: given a curve that represents some function, called the primitive function, and a point on that curve, what is the slope of the tangent to the curve at that point? (See Figure 2.) Well, why do we care? First, we must define some terms (and these are paraphrases with attendant inaccuracies). The tangent at a given point is the line that touches the curve at that point without crossing the curve. The slope of the tangent is how steep it is, mathematically a matter of the relation between the change in the vertical direction and the change in the horizontal, or the increase in y divided by the increase in x, or “the rise over the run” (see Figure 3). What is the significance of the slope of the tangent to the primitive function at a given point? It tells us the slope of the curve at that point. And the slope of a curve at a point is how fast it is changing there, how quickly the curve is “moving” up or down. A steep tangent means a Fig. 2. A curve and the line tangent to it at the point (x,y). steep curve at that point, one that is rising or falling quickly. A flat tangent means a curve that is not moving up or down at all at that point. If the curve represents distance over time, then the slope of the tangent at a point tells us the veloc- ity at that time. If the curve represents popula- tion over time, then the slope tells us the growth rate. And if the curve represents the elevation of land not over time but over some area, then the slope of the tangent repre- sents the slope of the land at any given point. In all, it is quite a useful calcu- Fig. 3. Four line segments, with their slopes indicated. lation to make. 109 math anxiety But finding the slope of the line tangent to a which simplifies to curve turns out not to be such a simple matter. Since the tangent touches the curve without cross- lim f(x+D x)- f(x) D xÞ0 D x ing it (nearby), it only touches it in one point (locally). But it takes two points to define a line provides the general formula for the derivative of and determine its characteristics such as slope. f(x) (see Figure 5). Given two points, it is easy to find the slope of the Here, D x represents a finitely small quantity, a line that runs through them: it is just the differ- finite and measurable distance, that is allowed to ence between the y values divided by the differ- approach zero. So it comes as no surprise that the ence between the x values. If the two points are differential, dx, which is defined as D x “on its (x1,y1) and (x2,y2), then the slope of the line they way to 0,” should also be regarded in modern define is (y1–y2)/(x1–x2) (see Figure 4). calculus in finite terms. In school, we begin with discrete whole numbers: 0, 1, 2, 3, and so on. Eventually, we fill in the gaps between these numbers using ratios of them to produce ratio- nal numbers (fractions). Then, we fill gaps be- tween rationals by taking Fig. 4. The slope of this line is (y1–y2 )/(x1–x2). sequences of rational numbers such that the numbers in the sequence Since we know the function, and can calculate get closer and closer together – in other words, it at any point, we know at least one point on the line in question, the point of tangency; but how do we calculate the slope with only one point specified? The method is to choose a second point on the curve some distance away, and calculate the slope of the line that runs through both of those points. This yields an approxima- tion of the slope of the tangent. The closer the second point is to the point of tangency, the closer will be the approximation of the slope of the tangent.3 To find the slope of the tangent at a point, we must take the limit – as the “other” point approaches the point of tangency – of the slope of the line between the two points. Instead of x1 and x2, we can represent the two x coordi- nates as x (the x-value of the point of tangency) and x+D x (the x-value of a point D x away from x), and allow D x to approach 0. The two points are (x, f(x)) and (x+ D x, f(x+ D x)). The slope is the difference between the y values ( f(x) and Fig. 5. Finding the slope of the tangent: a line drawn between two points on a curve. As D x gets smaller, the f(x+ D x)) divided by the difference between the x lower point gets closer to the upper point, and the line values (x and x+ D x). Taking the limit, gets closer to being tangent. The formula for the deriv- ative is the slope of this line as D x approaches 0: lim f(x)- f(x+D x) lim f(x+D x)- f(x). D xÞ0 x- (x+D x) D xÞ0 D x 110 evens we take a limit – and this produces irrational precedes numbers and intervals of finitely small numbers, completely filling out the number line. size, but the differential relation, dy/dx, precedes Finally, this concept of an arbitrarily small inter- the “primitive” function whose slope it is said to val, or limit, is applied not just to the numbers represent. In calculus class we are presented with but to their relations, functions, where it a function and told to differentiate it, to take the becomes the differential. Instead of producing derivative or produce the differential relation. In the differential from the number line in this Deleuze’s rereading of the calculus, the primitive manner by an operation on arbitrarily small function does not precede the differential rela- intervals, Deleuze reverses the usual priority to tion, but is only the ultimate result or byproduct start with the differential and to generate number of the progressive determination of that relation. from it. That is, he places the differential at the The differential is a problem, and its solution origin of number, as the power of difference that leads to the primitive function. deviates from itself to generate the entire number We have thus far been considering just one line and eventually the points that populate it. point at a time, though it could have been any The differential is not a finite distance, not even point. But inasmuch as it generates the primitive an ever smaller one. If the differential is an infin- function, the differential relation characterizes itesimal, it is a substantive and positive infinites- not only one point but a whole range of points, imal; not a finite approach toward zero as modern an entire function. It is not a relation of variables, calculus would have it, but a movement of zero where one is considered independent (x), and the away from itself. The differential, dx, is a torsion other dependent (y). The differentials, dx and dy, that inflects the point, x. It is the instantaneous do not take on single values nor do they vary over velocity of x, x being only the apparent point of a range of values; they are neither particular nor departure for a movement that precedes any general, but universal, inasmuch as their relation departure. establishes an irreducible character that is not Though we can make some sense of x by constant, but incorporates multiplicity. itself, as a variable that represents any of certain In other words, the differential relation is not specifiable values, we can make almost no sense a formula that relates x to y over some range of of dx by itself. Inasmuch as it is a movement, values for x, though this is how we are taught to it precedes those coordinates (x and y) that interpret it: in school, the differential relation, or eventually measure it, an arrow set against the derivative, is just another formula, another func- void. As Deleuze says, on its own it is “strictly tion akin to the primitive function. Rather, the nothing” (171). It obtains a value only in differential relation relates x to y not in breadth, relation. dx, though undetermined, points toward over a range of values, but in depth; it operates its own determination in relation to another in each point on the function, condensing the differential, dy. The above formula for the deriv- quality, the character of the entire function into ative necessarily involves a comparison, change every point. If you are given the value of some in y versus change in x. Differential calculus is differentiable function at a point, then that’s keenly interested in this relation, dy/dx, the almost all you know about the function. You differential rise over the differential run. The have little idea how the function behaves at any terms of this relation are neither constants nor other point; you know the function only at that variables, for dx and dy mean nothing outside of one point. However, the existence of a derivative their relation. They determine their relation reci- means that you can take a limit at that point, and procally, determining each other in relation as in order to take a limit the function must be rela- they determine the primitive function that relates tively smooth, not broken up or wildly chaotic. x to y. The function does not jump around too much in Deleuze’s reversal of the priority of the differ- the vicinity of that point. Which means that near ential and the finite numbers that normally the point whose value you know, the function define it thus carries over from Lesson One to stays relatively close to that value. If you are told Lesson Two. It is not just that the limit itself that the differentiable function passes through 111 math anxiety the point (3,5), then you know that as long as x function is doing at each point. It’s not that the is close to 3, y will be close to 5.4 You can differential relation represents the slope of the approximate the function near that one point: function at each point; it’s that by representing since it stays close to 5 near 3, it must look some- the slope of the function at each point, the differ- thing like a horizontal line passing through (3,5). ential relation presents or characterizes the whole However, if you also are given the slope of the function in each of its points. tangent to the function at that point, then you The differential relation captures the univer- know how quickly and in what direction the func- sal character of the whole function, and this tion is moving near that point. Let’s say the slope overall character or shape of the function is a at the point (3,5) is steep, maybe 100. You can matter of how many times it changes direction, now take a better guess at the values of nearby how many bumps it has and how regularly they points. Specifically, you can assume that, at least occur, how often it becomes infinite, and how near that one point, the function looks like a often it crosses the x-axis. These criteria divide steep line passing through (3,5) with a slope of the function into pieces, whose endpoints are 100. If you also know how fast the slope is chang- defined by the critical points that mark the ing at that point – the slope of the slope or the changes in direction, the local maxima and value of the second derivative – then you can minima, the zero-crossings, the approach to make an even better estimate of the behavior of infinity (Figure 6). Deleuze refers to these criti- the function near that point. The function may cal or distinctive points as singular points, for be steep, but is it getting steeper, or more shal- their number and relations uniquely determine low (at that point)? The second derivative the singular character of the function itself. They answers this question, and gives you an even are points of articulation where the function better idea of the shape of the function near the alters its behavior: the function gets steeper to one point you know. Third derivative: the func- the left of a singular point and shallower to the tion is steep and getting steeper, but how quickly right; or it’s positive on one side and negative on is it getting steeper? Etc. the other; or it’s curvy and then suddenly flat. Note that your increasingly refined estimates These singular points break the function into of the behavior of the function do not rely on the parts and so determine the type or species of formula for the derivative nor on any formula, function in question. We can judge the kind of but only on the actual values of the derivatives. function according to the number and arrange- You are given one point on a function and a ment of its distinctive points. Trigonometric sequence of numbers representing the values of functions tend to have an infinite number of the derivatives of the function at that point, and distinctive points with a certain regular recur- from these numbers, you can reconstruct an rence. (A sine wave has an infinite number of approximation of the whole function not just at peaks and valleys, an infinity of maxima and that one point, but also in an area around that minima.) Polynomial functions have a finite point. Each successive (value of the) derivative at number of distinctive points, one fewer than the that point provides a more accurate sense of the degree of the polynomial. (Think of the shape of the whole function. This process of parabola, a binomial, with its single minimum or successive approximation is not merely heuristic; maximum.) Capturing the character of the prim- in fact, if you know the values of all the deriva- itive function, the differential also holds the key tives of a function at a given point, you can to the singular points, for where the primitive construct a polynomial, another function, that is function has a peak, its tangent will be horizon- equivalent to the primitive function near that tal, so its slope will be zero; and where it point. This is the sense in which the differential approaches infinity, its slope too will become is a universal: the differential packs into each infinite (see Figure 6). By examining the number point the nature of the entire function, for the and distribution of zeros and infinities in the differential relation generates not the value of the differential relation, we can determine the function, but its behavior, its character, what the species of primitive function. 112 evens tions that take place in this instant. Problems determine themselves incrementally and always in relation; the pieces of the problem don’t fall into place until they are already solved, hesitating, negotiating, calculating in a game of strategy and diplomacy with goals to be determined as the game is played. Problems do not come out of nowhere, though it may sometimes seem so, as they collapse, in an instant, the gap between the vague dread of recip- rocal determination and the abject terror of complete determination. Fig. 6. Each of the circles surrounds a singular or distinctive point, where the curve has a local maximum Consider the increas- or minimum. Note that the tangent at each of the ingly successful attempts to harness the differen- circled points is flat, so its slope equals 0. tial power of the problem in quantum computers. Classical computers deal only with pseudo-prob- The differential relation, dy/dx, applies to each lems, problems already solved, since the condi- point and also to the universal character of the tions of the problem and the number of possible entire function. Thus, the reciprocal determina- solutions are strictly determined in advance, and tion of dx and dy in each point is simultaneously always boil down to a choice among two alterna- a reciprocal determination of the character of the tives. There is no problem in the problems tack- whole function. Reciprocal determination occurs led by classical computers; we can’t even pose a at once point by point and part by part, deter- real problem to such a computer, much less mining the shape of the whole function as it deter- arrive at a solution. Quantum computers, on the mines the points that constitute it. The function other hand, survey not one value but a range at thus takes shape gradually, progressively, as the a time, allowing each “variable” not only to vary singular points shift and glide relative to each but to statistically inhabit multiple and infinites- other, tense and relax to alter their configuration. imally distinct values at once. A physical quan- A problem forms like a soap bubble stretched tum bit can be 0, or 1, or in between, or across the wire outline of an abstract geometric something else altogether, since the bit is in fact figure. How to connect the vertices most effi- a wave, and the differences between one value ciently, how to find the correct degree of curva- and another are not immaculate, but shifting and ture, how to distribute density so as to bend subtle. Of course, the equations that describe the without breaking? And when a weakness is value of the quantum bit (qubit) are filled with stretched beyond its breaking point, the bubble differentials, and these do not disappear until the snaps into a new shape, determining new criteria, wave collapses into its final determination. new boundary conditions, posing and solving a Quantum bits do not collapse into specific values new problem in a flash, so that one would never until they have solved the problem, until they suspect the whole network of differential calcula- find a mutually acceptable solution that deter- 113 math anxiety mines not only the final outcome but also the nature of the problem itself. Qubits can be entangled, such that they only operate conjointly, posing a differential and statisti- cal problem of reciprocal determination. Quantum computers are thus unlike their classical cousins, which begin and end with already determined possi- bilities for individual bits. They pose and solve prob- Fig. 7. The curve is a graph of part of the function, lems more like artists, who must generate the 1/(1+x). The dashed line shows the domain of conver- right stroke or choose the right materials or gence of a power series for this function centered at 0. Outside of this domain, the power series is undefined, invent the right sound that will advance the so we would require more than one power series to composition. (What artist would not be tortured? approximate the entire function. They deal exclusively with problems, problems of determination.) the series of infinite order polynomials they I discussed earlier how the values of the deriv- represent. But there is a problem with power atives for a function at a given point allow the series: they only represent the primitive function construction of a polynomial that is equivalent to in some interval, possibly a very small one, that function near that point. Specifically, these around their center point. They are only guaran- values are used to determine the coefficients of a teed to represent part of a function. Far from its polynomial called a power series or Taylor center, a power series may no longer be equal to series.5 Incidentally, we are probably now up to the primitive function, and it may even become Lesson Fifteen or Twenty in first-year calculus. undefined. Singular points divide a function into Given a differentiable function, by taking deriva- parts, and each part can have a different power tives of that function at a particular point, much series to represent it. Deleuze thus raises this as I described earlier, we can produce a power question: for any particular function, can it be series representation of the function. The point at constructed from just one power series over its which the derivatives are taken is called the entire domain, or must it be built in pieces from center of the power series. For example, the multiple power series (see Figure 7)? power series for sin x centered at 0 is Much rides on this question of the number of power series. If only one is required, then the x3 x5 x7 ... n x 2n+1 sin x= x- + - + +(- 1) + ... entire function is built according to one overrid- 3! 5! 7! (2n+1)! ing principle, and there is a smooth and unbro- Again, Deleuze reverses the usual priority. ken path from any point on the function to any Whereas we normally begin with the primitive other point. A function can be regarded as a rule function and generate the power series, he for transformation, and a single universal power regards the power series as a way of constructing series guarantees that this transformation will the primitive function. Since the power series is carry any point on the function to any other with itself derived from the differential relation, it is no complications, no convolutions. But if any the differential which is powerful, and exercises point can be smoothly transformed into any its power to generate the primitive function, the other, then we have eliminated radical difference. sensible solution. Sine and cosine, logarithm and Deleuze opposes himself here to Leibniz, who exponent become just a convenient shorthand for effectively builds his differential calculus and his 114 evens monadic philosophy one out of the other. Leibniz that must be constantly pieced together again, insists on smooth transformation, for only then that is never the same thing twice or for two do the differences among monads boil down to different puzzlers. And this makes sense, for mere shifts of perspective, reconcilable by a sense is not a complacent and comfortable fit, but continuous movement without a hitch. Every a problematic puzzle that never quits asking. The argument is resolvable, every point of view best puzzles, the ones that make the most sense, comprehensible in terms of every other, though are also the hardest problems vast differences might be crossed before debating and the most compelling. To parties see eye to eye. The entire function, the solve a problem is to follow the entire universe is summarized, captured in a differential through its twisted single, overarching power series; and this one calculus. And you thought you power series, that covers the whole of existence didn’t like math. and encompasses every point of view, is God himself, God as the ultimate mathematical entity. notes For Leibniz, God and the universe include only reconcilable difference, only solvable problems – 1 Gilles Deleuze, Difference and Repetition, trans. Paul Patton (New York: Columbia UP, 1994) such is the best of all possible worlds. 168ff. All further page references are to this text. But what is so great about a world where disagreements are only apparent, where difference 2 We might conclude that mathematicians work is always superficial? On the contrary, Deleuze on real problems, while math students deal only wishes to allow for disagreement that cuts to the with neutralized problems. Really though, math students, from infants to grade schoolers to grad- bottomless bottom of the ontology itself, a differ- uate students, are already working on real prob- ence that cannot heal and does not want to. lems. Learning always proceeds by confronting a Numerous power series are required, uncountably problem, so that the transition from math student many, which can make for some ugly and to mathematician feels relatively smooth; one con- contorted functions, bent and broken, but it also tinues to work in the familiar manner. means that problems can form in spite of, or even 3 This formulation is not strictly true. But you can because of, radical difference. The lesson never find an interval around the point of tangency in taught in calculus, for the teachers do not know which it is true. it, is that sense is problematic, and the problems that give sense to things are not limited to the 4 “Close to” is a very relative term here. In fact, the function can vary by an arbitrarily large solvable or the reconcilable. If we look into the amount over an arbitrarily small interval near the process of determination, if we look at more than point. Strictly speaking, the smoothness (or conti- the actual states of things, but follow the differ- nuity) of the function at a given point means that no ential lines that connect solutions to the problems matter how small a variance you want, you can that make sense of them, then the world becomes find some neighborhood around that point in a Pirandello play, or a garden of forking paths. which the variance is that small. Only a problem makes sense, and problematic 5 The general formula for a power series is sense does not tie things into a neat bundle, does a0 +a1x +a 2x 2 +a3 x 3+ ... +a n x n + ... . The a values are not wrap up all the loose ends, but always points calculated using the values of the derivatives of the in new directions, demands to be worked on, primitive function. shaped into another problem with another sense. So your puzzle really was a math problem, for it was a differential puzzle, a puzzle of determi- nation. The puzzle never fits together, and there Aden Evens would be no sense in its doing so, or only a dead 76 Brantwood Road sense, a totalizing sense that would homogenize Arlington, MA 02476-8004 the puzzle, flatten its surface, and make each USA piece uniform. Instead, let the world be a puzzle E-mail: