(PDF) The Surd

The Surd Those who have exposed themselves to mathematics in only small doses may have the impression that math is a done deal. In grade school, a math problem is only ever a contrivance, as the solution is already known and indeed precedes the problem: given 2+5 one must produce 7, and even when the so-called problems become much more sophisticated, they don’t really get more problematic; derived from the solution, the problem is only a means of testing the pupil’s ability to return to it. This is the image of mathematics that most of us are left with, where solutions are already contained within their problems, waiting to be teased out by appropriate reductive methods; the answers are there already, even if they are as yet hidden from us problem-solvers. But there is another side of mathematics, where the solution is not a foregone conclusion and the problem is worthy of its name. Answers do not follow immediately from questions, but must be secured through resolute and patient labor on the part of the mathematician. This minor mathematics — exemplified for instance in the numerous mathematical examples throughout the oeuvre of Gilles Deleuze — contrasts with the static and formalized mathematics that we might call State or royal mathematics (by analogy with Deleuze’ and Guattari’s notion of State science). Whereas State mathematics comprises formal rules and an established set of symbols that confine problems to the leading edges of research, minor mathematics apprehends math at its most problematic, at its precarious or uncertain moments, when contingency overwhelms the security of the solutions, and only the urgent intervention of the mathematician can rescue a suitable result from the calculation. Such active nurturing attends every advance in mathematical research, but certain examples stand out most clearly, when the mathematics ties its fate to that of the mathematician and can be constructed and maintained only through her activity. In such moments, it is as though the mathematics will not solve itself, as though the solution is not yet there in the problem, as though the math had reached out of the abstract into the empirical, and so could not be taken for granted in its progress. Historically, such moments of minor mathematics are called constructivist. Constructivism underlies each advance in the history of mathematics, but the history books relegate it to a small and somewhat forgotten niche. Like minor science, minor mathematics is overwritten, claimed and formalized by the standard theorems and techniques of State mathematics. If constructivism elevates the problem to the highest position, then its formalization by major mathematics strips it of its problematic potential and asserts once again the priority of the solution. Of course, State mathematics dominates the history books, so we should not be surprised to find constructivism under-represented there. For example, Deleuze must reach into the obscure history of mathematics at the outset of Chapter Four of Difference and Repetition, to recover mathematical techniques of differential analysis that are ontologically substantive, generative and not merely reductive. Borrowing from a usually forgotten history of the calculus, he presents the differential, dx , in a process of differentiation, a process that generates the world and the -2- ideas that make sense of it. In the modern calculus, the differential does not perturb the formulas in which it appears; it is largely a placeholder to indicate which variable is to be integrated over or which derivative to take. But for earlier mathematicians — Deleuze draws upon the researches of mathematicians Carnot, Lagrange, and Wronski — the differential had to be willfully and actively manipulated in its equations; it was an extra term, left over after the rest of the equation had been reduced, and the methods for dealing with it could not be decided in advance. Constructive methods, such as successive approximation, ensured an active role for the mathematician, who reduced the equation bit by bit, sculpted it into a simpler form. One chapter later, Deleuze offers a prototypical example when he proposes that the universe is the remainder in God’s calculations. The remainder mandates a construction, for it is an element left over after the calculations have been performed, a problematic surplus that does not cancel out of the equation and requires contingent methods, techniques developed “on the ground.”1 Though constructivism propels even State mathematics, it disagrees with the formal essence of mathematics as defined by the State. Math is traditionally marked by its universality, its wholesale abstraction, its divorce from contingency, personality, and the empirical. Constructivism, on the 1 A Thousand Plateaus perpetuates such constructivist examples, from Riemann spaces that allow no overarching perspective and must be navigated locally and singularly, to fractals that are generally apprehended in the process of construction and not as completed figures, to the numbering number, which could be a general term for the constructive impulse in mathematics, for it is the intensive number, determined always in relation to its context, and insisting upon a uniqueness, atypical of number, that ensures that it can only be understood and manipulated in context. -3- contrary, clings to contingency, and develops its methods in a particular context, outside of which they may well lose their relevance or their force of solution. Math claims no politics, but constructivism politicizes mathematics, as it is no longer a matter of determinate formalities. In response, royal mathematics elides or cancels out minor mathematics, claiming its results while neutralizing or sterilizing its activism. The State restores to the mathematics a pure formality, universalism, and objectivity. In the one-and-a-half examples to follow, this essay attempts to distill the constructivist moment and generalize it, moving from mathematical history to intellectual history: constructivism, and in particular the surd, as the motor of becoming. The surd is the anomalous element, the unassimilable, that disrupts the linear progress of history, fractures an established discipline to open the way to new methods and ideas. At its initial appearance, the surd erupts into a constructivist heterodoxy, techniques to be determined in the moment, contextual methods that must be worked out on the spot. Before long, however, constructivism cedes its hold on these methods, as they are formalized and claimed by the State, the revolutionary potential of the surd neutralized. While the mathematical narrative here is historical, the language of history is ultimately provisional, like the constructivist methods it refers to. The claims in this paper are not primarily historical but genealogical; that is, they account for events in terms of their motive forces, but the actions of these forces may not be historically distinct from their reclaiming (and official elision) by royal sciences. Thus, as we will see, the same forces that employ the surd toward new disruptive methods are the same forces that attempt to -4- corral the surd and restore to mathematics a legitimacy and formality that the surd resists. Intuitionism and The Surd In mathematics, a surd is an irrational number, a real number that cannot be expressed as the ratio of two integers. It also is the name for the “root” sign, . These two meanings are not incidentally related: one easy way to generate irrational numbers (surds) is to take roots of rational ones. The square root of two, for example, is a surd. Typically, traditional (royal) mathematics has little trouble with the surd. In grade school, we just write out the first few digits after the decimal point, and then write an ellipsis to indicate the continuation of the sequence. “The square root of two is 1.41421….” The question — practically ignored by State mathematics — is the meaning of this ellipsis. In a rational number, such as one-third, the ellipsis in the decimal representation seems unambiguous; in 0.333… it means “just keep writing 3s.” But in the square root of two, the ellipsis feels somehow inadequate, for it refers to no pattern; the decimal expansion of the square root of two (or of just about any irrational number2) is a seemingly haphazard sequence of digits that is fully determined (by the definition of square root in this case), but in a strong sense unpredictable. The ellipsis in the representation of an irrational does not succeed in specifying any particular number, so that one must know beforehand what number is being referred to in order to interpret those dots 2 One can construct irrational numbers that have a pattern, and hence are exceptions to the rule. E.g., 0.030030003000030000030000003…. But this is a special, contrived case, and does not typify irrational numbers. -5- correctly. If the ellipsis means continue in this manner, an irrational number does not suggest just which manner is intended. Luitzen Egbertus Jan Brouwer found this ellipsis problematic; even more problematic for Brouwer was the tacit willingness of traditional mathematics to leave the ellipsis unexamined. In response, he founded intuitionism as a branch of mathematics that proceeds first of all from an epistemological (rather than a mathematical) commitment. Amidst the intellectual rubble of post-World War I Europe, Brouwer drew together some strands of nineteenth-century constructivism to forge a mathematics that restores truth and certainty where none had been guaranteed. The principle tenet of intuitionism is this: no mathematical claim is acceptable that cannot be experienced as certainly true by the mathematician; allow only what can be grasped in the mathematician’s immediate intuition.3 (We shall see what this means….) In intuitionism, the mathematician is no longer the one who works on the mathematics; the mathematician is its vessel and locus. For Brouwer mathematics is mental activity, independent of language and independent of the notation used to remember and convey it. Only intuitions are mathematics. Thus, the symbols of math and other written or verbal communications about it are not the genuine item, but only a means of reminding oneself or others how to “have” the same mathematics again. 3 The flattening of ontology onto epistemology, the sense of reconstruction on a firm foundation, the reliance on intuition, and the notion of a critique from within all attest to Kant’s parentage in Brouwer’s work. -6- Intuitionism insists on the active role of the mathematician in doing the mathematics, a constructivism in principle. But what difference does it make in practice to demand that the truth of the mathematics be experienced? Was there something uncertain about prior mathematics? Were mathematicians allowing claims into the mathematics that could not be immediately intuited as true? Brouwer argues that, indeed, many claims were being routinely but illegitimately accepted into the calculus. While claims about finite quantities are, in principle at least, experience-able as true in an immediate intuition, claims about the infinite are another story.4 In particular, it is the surd that fractures mathematics in the crucible of post-World War I math and logic. The ellipsis in a surd, say the intuitionists, cannot stand for an unwritten infinity of digits, since no such completed infinity can be directly experienced in the mind of the mathematician. Instead, Brouwer offers a definition of number, including irrational number, that is purely finitary, a definition in terms of the construction of the number that does not posit some finally constructed infinity of digits. In Brouwer’s words from his early work in 1913, a number is a “law for the construction of an elementary series of digits after the decimal point, built up by means of a finite number of operations” (85). In other words, a number is some procedure by which we can unfailingly specify further digits 4 In the finite realm, intuitionist mathematics is pretty much functionally identical to traditional math. The breaking point lies between countable and uncountable infinities. (Roughly speaking, an infinite number of discrete objects is countable, while a continuum is an uncountable infinity.) For the intuitionists, uncountable infinities were nearly incomprehensible, while countable infinities could be dealt with one discrete element at a time. Discrete objects can be intuited, while a true continuum is beyond intuition. -7- to arbitrary precision. For instance, start with 0.74, then continue to add digits, alternating between 7 and 4. This would yield a rational number. Another example: start with 1.41421 (the first few digits of the square root of two), then continue to add successive digits, choosing each next digit to be the highest digit such that the square of the resulting rational number is less than 2.5 This would be the intuitionist definition of the square root of two. By defining number in terms of a procedure, intuitionism demands that number be not a fixed, eternal entity, existing in some ideal space of number-hood. Rather, an intuitionist number is a rule, a method of construction. Even in this simple definition, we can see already the surd at work, its irrationality calling upon constructive procedures, an engagement by the mathematician. But the surd does not stop at this call for mathematical commitment; its perturbations are more dramatic, as Brouwer illustrates. He defines a number r, called the pendulum number, as the limit of a sequence of elements, c1 ,c2 ,c3 ,…. The sequence is 1 1 1 1 , , , ,… . 2 4 8 16 5 In other words, pick a next digit, say, 4, and add it to the end of the digits you have so far: 1.414214. Multiply this number by itself, 1.414214 2 = 2.0000012378 . This is slightly greater than 2 so replace the terminal 4 with a 3 and repeat: 1.4142132 = 1.99999840937 . 3 is thus the largest digit that can be appended to make the square of the result less than 2. Continue in like manner to generate the next digit and each successive digit ad infinitum. -8- It’s the geometric sequence defined by the inverse powers of two, where the sign switches back and forth with each element. The general formula is 1 n c n = . 2 This sequence bounces back and forth around 0, getting closer to 0 the farther you get in the sequence (Figure 1). It’s limit is in fact 0, and there is not yet any disagreement between intuitionist and traditional mathematics. But Brouwer adds a twist. He alters the definition of the sequence by freezing it beginning at the mth term of the sequence, so that starting at c m , all the rest of the terms will just be equal to c m . Now, under this altered definition, the limit of the sequence is no longer 0, but c m . The question of course is, What is m? Figure 1: The sequence of cs, bouncing back and forth as it approaches 0. In an original stroke of genius, Brouwer defines m so as to render the limit of the sequence, r, a rather strange number. m is defined as the first place in the decimal expansion of where the digits 0123456789 occur: Start to write out , counting its digits. As soon as you get to a place where 0123456789 occurs, note how far along you are in your count. This is m, as is illustrated in Figure 2. (Just to avoid confusion, note that has no prior relationship to r; Brouwer could have chosen any irrational number, and -9- picked arbitrarily because it is a universally recognized irrational number. The essential criterion is that is a surd, whose digits can be successively calculated to arbitrary precision, but whose yet-to-calculated digits cannot be predicted or patterned in advance of their calculation.6) Figure 2: The sequence of digits of , and the (hypothetical) mth position, where 0123456789 occurs. The thing is, we don’t know where such a sequence of digits occurs in the decimal expansion of . We don’t even know if such a sequence ever occurs in the decimal expansion of . If such a sequence occurs at an even-numbered place in the decimal expansion of , for instance, at the 1000th place, then c m will have an even exponent, it will be equal to 1 1000 c1000 = , 2 which is a positive (but small) number. The sequence of cs will “freeze” at that point, and r, the limit of the sequence, will be equal to this small positive number. If, on the other hand, the sequence 0123456789 occurs at an odd 6 As such, it may be significant that is not only an irrational number, but also a transcendental one, i.e., a real, irrational number that cannot be expressed as the root of a polynomial. This intensifies the “surdity” of , inasmuch as it has a singular relationship to the universe and does not just express a relationship among numbers. It is as though reaches beyond mathematics, generating its identity in the empirical domain (in the relationship of a circle’s area to its diameter). Other transcendental numbers, such as e, share this property of reaching out of mathematics and into nature. Perhaps this is also true of certain algebraic (non-transcendental) numbers, such as the golden mean, but the relationship between the golden mean and natural phenomena is more approximate, not exact like . -10- place in , then c m will have an odd exponent and r will be a negative small number. But if (on the third hand!) that sequence never occurs in the decimal expansion of , then there is no m, and the sequence of cs will just keep getting closer to 0, with a limit, r, of 0. A traditional mathematician, say a platonist, would not object to this definition of the pendulum number. She would allow that r is a well-defined real number whose value we don’t at present know, since we don’t know where 0123456789 occurs in (nor whether it occurs at all). In spite of our ignorance about r, though, she would insist that as a well-defined real number, r is either positive, negative, or equal to zero. This is where the surd fractures mathematics, splitting intuitionist from traditional mathematics. Brouwer too allows that r is a well-defined, real number: it meets the definition of an intuitionist real number in that we can specify it to arbitrary precision, simply by writing out the digits of . However, we cannot say of r that it is either positive, negative, or equal to 0. We cannot say this because we cannot immediately experience the truth of any of these three possibilities, and therefore none of them is true. It is not true that r is greater than 0, nor is it true that r is less than 0, nor is it true that r is equal to 0. None of the three usual ordering relations holds between r and 0. (In effect, Brouwer’s pendulum number instantiates a new logical value: it is not true that r > 0, but neither is it false.7) 7 Different versions of intuitionism treat this other truth-value differently. Brouwer adhered to the belief that there are only two truth-values, true and false, so that this value of not-true was not a formal element of the calculus but only a step in thinking about it. Other intuitionists codified the value of not-true, making it a third term alongside true and false. -11- This counterintuitive conclusion of intuitionist analysis deserves reemphasis. It is not a deferral in regards to r, a refusal to answer questions about r due to insufficient data. Rather, r, in all its indefiniteness, enters the mathematics, carrying its ambiguity along with it. It thereby perturbs traditional, even sacred theorems that had held sway for hundreds of years. In this case, r defies the fundamental ordering of the continuum — the property that every real number is either greater than, less than, or equal to any other real number. Here, uncertainty (about the digits of ) is not an extramathematical concern that leaves a gap in the math waiting to be filled in, but is incorporated directly into the calculus. Epistemology intrudes upon ontology, to tie the truth of the mathematics to the concrete mental processes of the mathematician. No longer an eternal posit, an ideal relation among ideal objects, mathematical truth is now a contingent event, something that happens to the mathematician and to the mathematics. If at some future moment, the sequence 0123456789 is discovered in the decimal expansion of , or if it is someday proved that no such sequence can ever occur in the decimal expansion of , then r will take on a decimal value, and it will obtain one of the three ordered relationships to 0. The future-historical event will change the truth of the mathematics. By virtue of the surd (in this case, ), mathematics bleeds over from the realm of the pure abstract into the empirical. It becomes dependent on human history and human progress. And it insists, above all, on the participation of the mathematician as an essential element of the mathematics. With the entry of the pendulum number into the mathematics, Brouwer replaces the ordered continuum of real numbers with a disunified -12- set of incommensurables, scattered surds that cannot be adequately compared to other numbers. By definition, the surd is incomparable, singular, and it lends this quality to intuitionist number in general. The surd makes numbers into immeasurable ordinal quantities whose nature can be determined only by pursuing a process that may never yield a certain result. r and 0 cannot be adequately compared to each other, for each is a singular experience, constructed in such a manner that they diverge, r becoming skew to 0. The pendulum number depends for its definition upon what Brouwer terms an opaque fleeing property.8 This is a mathematical property that describes a (hypothetical) mathematical object, such that no object is known that holds the property, and it is also not known that no such object exists. Moreover, there is no finite process that will surely determine whether or not there is such an object. Thus, an opaque fleeing property is a property such as “the first place in the decimal expansion of where the digits 0123456789 occur.” This opaque fleeing property locates and rarefies the surdity of , its uncanny simultaneous contingency and necessity. is contingent for wholly 8 A fleeing property is any property such that (1) it can be determined for each natural number n that ( n) either holds or is absurd, (2) no method is known for calculating a number with the property , and (3) the assumption that some number exists with the property is not known to be absurd. The stipulation of opaqueness adds the further condition that (4) the assumption that some number exists with the property is also not known to be non-contradictory. Verification for “0123456789 in ”: (1) For each number, n, we can check whether it is the first place in the decimal expansion of where the series 0123456789 occurs (or not), simply by expanding out to the n+9th place. (2) We have no way of calculating the first place where that series occurs, except by calculating the successive digits of . (3) We have no reason to believe that the series never occurs in . Thus, this property is fleeing. (4) We have no reason to believe that the series does occur. Thus, this property is also opaque. -13- singular, without genus, and governed by no law but its own; nevertheless it is woven into the fabric of the universe, the singular relationship between circumference and diameter of a circle. Its specificity defies its abstraction. It is effectively split between the empirical and the abstract, a pure abstraction with the infinite, irreducible detail that generally accrues only to the concrete. thus reaches out of the eternal mathematical world and into the empirical human one.9 It is a strange and disconcerting move, therefore, when Brouwer, consistent with traditional mathematical practice, abstracts from this particular opaque fleeing property to speak of opaque fleeing properties in general. (For instance, “as long as a fleeing property exists such that […]” Or, “for each fleeing property, ƒ, […]” [1981 42].) That is, he ceases to specify a particular fleeing property, and instead reifies the concept of opaque fleeing property, using it as a mathematical effect in the analysis: “Consider a number defined by some fleeing property, .” This is tantamount to taking the form of an opaque fleeing property and voiding it of its content. In this formalization, the surd, formerly crucial to the fleeing property, is neutralized, drained of its disturbing singularity so that it can serve as a general term.10 Contingency is no longer the contingency of , but becomes a 9 In one sense, Brouwer’s use of ties intuitionist mathematics to time and space. The math becomes spatial inasmuch as is a geometric quantity, a relationship among (abstract or ideal) spatial phenomena. More significantly, though, the math becomes temporal to the extent that is treated in intuitionism as an object to be discovered, a process for generating digits but a process that necessarily takes its time. 10 Brouwer seems mildly concerned about formalizing the opaque fleeing property, as though he suspects that he may be losing the surd so crucial to the radical consequences of his calculus. On occasion, he laments the fact that his proofs (such as those involving the pendulum number) depend on the existence of unsolved mathematical problems (such -14- purely formal element of the calculus, with rules to govern its manipulations and effects. When fleeing properties are formalized, the surd is not so much evinced as insisted upon, spontaneously generated in its form but without content. However, a surd without content is impotent. For a surd is precisely an excess of content, an intensity that cannot be contained by a form. Though its results persist in the calculus, they no longer require the engaged attendance of the mathematician. As a formality, intuitionism loses its tie to the human, and becomes just another branch of mathematics, a universal and agreeable formalism. Historically, formalization of the surdity of intuitionism made it acceptable to other mathematicians. Troelstra for instance attempts as his chief project to formalize intuitionist mathematics to the point where we no longer need extra-mathematical conceptual apparatus, but can regard intuitionism as a set of rules for manipulating symbols. The triumph of royal mathematics. Brouwer, on the other hand, wishes to maintain the surdity of his mathematics, for it is just this surdity that attaches the math firmly to the mental activity of the mathematician. Having lost the opaque fleeing property to formalization, he needs to conjure another surd, to trump the opaque fleeing property with a more insistent concept. The opaque fleeing property was something of a parlor trick anyway, a game to demonstrate as the existence of 0123456789 in ). In general, Brouwer is somewhat torn between his commitment to a constructivist mathematics, in which the math is empirical, and an adherence to the traditional epistemology of mathematics, in which math is universal and atemporal. He doesn’t desire that the conclusions of intuitionism actually change over time, but neither does he want to allow a universalizability. -15- some of the easy and dramatic implications of intuitionist analysis, but not a serious contribution to mathematics.11 To up the ante, Brouwer introduces the notion of a free choice sequence, and a new concept of number to go with it, more radical still than the pendulum number. To generate a free choice sequence, epistemology must be further twisted, the very notion of choice remade as a mathematical concept. One of the problems for traditional mathematics was to explain the continuum. How is it that points can somehow aggregate to form the continuum, or number line? How do points of dimension 0 fully occupy a line segment, which has a dimension of 1? How can discrete points, that have no length, form a continuous line? Traditional mathematicians posited an axiom of the continuum; they simply asserted as a basic assumption of the mathematics that when you take all of the (infinity of) rational numbers and all of the (even bigger infinity of) irrational numbers, you have covered the continuum to form a continuous line.12 (As the continuum is most easily conceptualized in geometrical terms, I will henceforth refer interchangeably to number or point. The same thing is intended by each term.) This claim, say the intuitionists, is not immediately verifiable as true in the mind of the mathematician. In fact, it’s altogether counterintuitive. 11 Brouwer tends to use the opaque fleeing property in discussions aimed at more general audiences and not so much in his formal papers. (Compare the conference address, Brouwer 1923b, which, while still mathematically rigorous, is not so laden with formalities as some of his other writings.) On the other hand, as an intuitionist, Brouwer did not distinguish sharply between formal and “everyday” modes of mathematics, and has been criticized for using a plain language style even in his formal presentations. 12 Technically, there are not just an infinite number of irrationals, but a non-denumerable or uncountable infinity of them, more irrationals than integers. A higher level infinity of points is required to create a sufficient density to constitute the continuum. -16- Intuitively, no matter how many points you place side-by-side, there is no way that they should ever be able to cover a positive linear area, no way that they can ever form a continuous line. To compensate for this incorrectness in the calculus — and it is an important claim that underlies a great deal of higher mathematics — the intuitionists offer a further definition of number, one that is intended to overcome this problem by building into number the continuity that number is expected to cover. This new definition of number is based on the idea of a free choice sequence, and it once again relies on the surd, which imparts to number a power of the continuum. The early intuitionist definition, above, makes of number a process for determining with arbitrary precision successive digits of a decimal representation. The new definition introduces the innovation of the free choice sequence, which allows a certain freedom or latitude in the process. Now, the rule that determines successive digits need not be wholly determining; rather, the rule for choosing successive digits can allow the mathematician a choice of next digit, and this still defines a particular point (number) in the intuitionist calculus. For example, here (Figure 3) is an intuitionist point: start with 0.31, then choose successive digits so that each next digit is either a 4 or a 2. This definition simply fails to specify a number that a traditional mathematician would recognize as well-defined. For a traditional mathematician, this is not a number, but a method for generating different numbers. And while a traditionalist would agree that any number generated according to this method will have certain properties (such as the property of being greater than 0.31 and less than 0.32), the existence of these properties does not make the number itself well-defined. -17- But the intuitionist understands this definition differently. For Brouwer, this rule defines a point. Though two different mathematicians might make different choices, before any choices have been made or as long as all of the choices made so far coincide, both mathematicians are working Figure 3: An illustration of a single number, as defined by a free choice sequence. with the same point. As soon as they make different choices, they are then no longer working with the same point. A given point, therefore, consists of choices already made plus the possibility of making various choices in the future. As a process of choice, a point includes the potential for any number of points, so that each point contains an internal difference, equal to itself and different from itself. Unlike a point in traditional mathematics, an intuitionist point, defined in relation to a free choice sequence, is no longer an unimaginable idealization of dimension 0. On the contrary, using the definition of free choice sequence, a point is a progressive narrowing of intervals, a process of honing in that keeps open an interval and never finally closes it down to 0. Points thereby become fuzzy, each equal to itself and to its neighbors (but -18- less so). Indeed, in Brouwer’s formal treatment of free choice sequences, he does not refer to successive digits of a decimal number; rather posits a sequence of nested intervals, and the choice is a matter of how each interval is to be fit within its enclosing interval. The first few “choices” of intervals for a point are illustrated in Figure 4. Figure 4: The first few “choices” of nested intervals that define a point. Now the problem of the continuum is solved almost by default: the continuum is just the totality of all choices, or the freedom to make choices however one will. Each point, as a process of continual refinement, covers a vanishing but positive linear area. The power of the continuum is built into the points that constitute it, requiring no extra axiom to justify its continuity. The points do not aggregate or bunch together side-by-side to create the continuum. Rather, as choices, the points are in motion, smearing themselves across intervals, blurring their own boundaries to leave the continuum in their wake. Intuitionist mathematics does not exactly promote the line over the point, but, as Deleuze and Guattari recommend, defines both line and point in terms of a motion that generates each of them. The result of this definition of number in terms of free choice sequences is to impart a maximum of continuity into relations among numbers. But it is a continuity that derives from the fact that the elements, the numbers themselves, incorporate difference into their identity. Because each number has difference within itself — a part still to be determined — -19- the differences between numbers are less dramatic, generating a maximum of continuity among numbers. Every number leaves part of itself not yet determined, and this indeterminacy is a difference that lives within number and forms part of its nature. Figure 5: A continuous function and a discontinuous function, at least according to the definition of continuity in traditional mathematics. -20- The continuity of intuitionist number shows up in the calculus in the intuitionist theorem that every function of the unit continuum is continuous. This departs dramatically from the claims of traditional mathematics. Whereas the pendulum number is something of a contrivance, a rather forced example of the oddball cases allowed by the early intuitionist definition of number, this revision of the notion of continuity radically alters the landscape of mathematical analysis and demonstrates the vast distance, opened by the surd, between intuitionist and traditional conclusions. In traditional mathematics, the continuity of a function is a question of its smoothness. A function is continuous if it has no breaks or gaps, no jumps in it. For instance, the function in Figure 5A, though quite a rollercoaster, is nevertheless continuous: there are no jumps or gaps in the graph of the function; it is “filled in” from one end to the other. Placing a fingertip at one end of the function, one could trace its entire path without lifting one’s finger. The second example, Figure 5B, is a discontinuous function; this is called a jump discontinuity, since the function jumps suddenly at a point from one value to another. Brouwer’s radical claim is that, in spite of this jump, the function in 5B is still continuous. 5B represents a continuous intuitionistic function because points (or numbers) are themselves defined so as to be smudges or vanishing intervals rather than finished points of dimension 0. In traditional mathematics, the function in 5B is discontinuous because we can specify a point, l, at which the function suddenly jumps from one value to another. On the left side of this point, the function equals 1, and on the right side, the function equals 2. But for the intuitionists, one cannot specify such a point because there are no -21- absolute points. There are only neighborhoods, intervals surrounding l, honing in on the classical point l, but never actually reaching it. And any neighborhood around the classical point l will include space to either side of l. However closely one hones in on l, one still keeps open the possibility of making further choices so as to determine a value of 1 for the function, or of making other choices so as to determine a value of 2. Thus, putting it rather too bluntly, near l the function evaluates to both 1 and 2. Instead of a point at which the function jumps, the intuitionists see a neighborhood that effectively connects the two sides of the function by tying their ends together in a single point, a point that includes difference within itself. The intuitionist point l functions as a kind of worm hole, a black hole or singularity that causes two otherwise distant points in space to overlap each other. The surd appears in this case in the guise of the concept of choice. The intuitionist definition of a point includes within its formalisms an openness, an indeterminacy that, not yet filled in, might be satisfied in any number of ways. As other intuitionists have argued (compare Troelstra 12), choice here is not so much about the actual fact of choosing as about the form of choice, the fact that one can define a point and work with it without having narrowed down all of the intervals that would limit that point to a dimension of 0. Choice is thus simultaneously made and suspended; proofs are carried out as though only so many choices have been made, but the results are valid no matter how many have actually been made. There are always more choices to be made, so that the interval remains open, with a dimension of 1 but ever shrinking toward 0. Thus, in Brouwer’s proof of the continuity of -22- every function of the unit continuum, he relies crucially on the fact that, as we hone in on l, selecting increasingly narrow intervals surrounding l, it will always be possible to imagine for the “next” choice that we are honing in on a point to the left of l or that we are honing in on one to the right of l. Once the next choice, say the qth choice, is actually made, it will undoubtedly cut off certain possibilities that existed prior to choice q, but it will still be possible to make the next choice so as to aim toward a point to the right or one to the left of l. The formal procedure of choice keeps the point from collapsing to zero, and calls once again upon the participation of the mathematician, this time an ideal mathematician, endowed with a power of abstract choice. Choice is the surrogate of the surd. Though this definition of number is mathematically cumbersome, it is nevertheless intuitively appealing, even “natural.” No longer an idealization of dimension 0, number is now a process of narrowing, mirroring the physical or mental process we might go through in determining a point, and incorporating the necessarily inexact and provisional endpoint at which we inevitably curtail our process. While empirical pointing happens over a specific time period, intuitionist points only refer to an abstract time, once again drawing upon form (of time) without reference to a specific content.13 13 Though editor and mathematician Charles Parsons refers to the free choice sequence as “a process in time” (notes to Brouwer 1927 446), this is merely heuristic, as they function atemporally. Free choice sequences imply the formal structure of time, for there is always a part of the sequence that has already been chosen, a part that has not yet been chosen, and the immediate choice to be made. However, proofs involving free choice sequences demonstrate that each of these three dimensions of time (past, future, present) exists all at once at different stages. That is, the proof treats a free choice sequence as having at the same time many different (but related) pasts, many different (but related) futures, etc. It is as though the free choice sequence establishes a notion of past per se, future per se, -23- Corresponding to the notion of an idealized temporality is the posit of an ideal mathematician (or “creative subject” in some of the literature). Brouwer did not want for free choice sequences to subject the mathematics to the caprice of the mathematician, as though the results of the calculus depend on just which choices the mathematician chooses to make. Rather, choice in intuitionism has a Nietzschean resonance, in that every choice gets made. Proofs involving free choice sequences do not take for granted any particular choice; they assume only that some choice gets made, and that some are still to be made. (Again, it is a matter of idealization: choice as a form without content.) Choice is affirmed as a principle so that it is not a matter of any particular choice. “But what is relevant from a mathematical point of view is not any individual choice sequence as such, but the ‘mathematical’ fact that there exist many perfectly well-defined (lawlike) operations on sequences which can be carried out without assuming the arguments to be determined by a law” (Troelstra 12). The principle of choice defines number as a process, with a maximum of continuity between numbers and a maximum of difference within each number. Every choice is effectively made, every choice is affirmed, and this generates the universe of number (the continuum). Formalizing or idealizing the concept of choice freezes a dynamic process into a static element of the mathematics. Whereas traditional mathematics idealizes number, ignoring its processual aspect, intuitionism without having a wholly specific past or future. Phenomenologist van Atten argues that Brouwer is committed philosophically to the identification of a choice sequence with its specific moment of origin, but though possibly true in principle, this temporal localization seems to be nearly irrelevant in practice. -24- formalizes the process as process, capturing this motion of numbering in vivo, formalizing not number but the genesis of number. Intuitionism discovers the essence of number in numbering, naming that essence and giving it a place amongst the symbols of the calculus. Brouwer thus preempts the process of numbering, seizes numbering with its virtuality intact, before it has cancelled that virtuality with the actuality of a fixed and determinate value. Notably, the power of the free choice sequence is a power of determination, a finite determination that can always be further determined, but is never finally determined. Intuitionism knows not to explicate too far, as Deleuze puts it. Brouwer thus suspends the mathematics in between virtual and actual, refusing complete determination to promote the process that gives rise to the determinable. This hold on the vital essence of number could not endure. Much of the language of choice, and the epistemological and ontological commitments associated with Brouwer’s intuitionism, were at odds with traditional mathematical standards. Later intuitionists attempted to preserve the results of the intuitionist calculus while discarding the philosophical underpinnings. These intuitionist reformers rejected the “unmathematical” ideological and epistemological commitments of intuitionism, but wished to retain its formal results; after all, its alterations to traditional numerical analysis and to logic are at least interesting and possibly even useful. Troelstra and Heyting each formalize significant parts of intuitionism, so that it becomes only another system of symbolic manipulation, a formal alternative to traditional calculus stripped of the surd’s original revolutionary power. A term like free choice sequence is replaced with the less -25- provocative infinitely proceeding sequence, or just a sequence that is lawless or even non-lawlike. The human operator is thereby eliminated from the mathematics, and sterile if productive research can continue without epistemological threats to its universality and objectivity. Such is the history of the surd, in math as elsewhere. Its introduction disrupts standard practices, opens a break in the linear progress of a field, inviting markedly new understandings, concepts, and techniques. Initially, these novel elements do not settle comfortably amidst their established cousins, and so demand a real labor and lend the whole radical enterprise an empirical or contingent character. The arrival of the surd involves a struggle that spills over the edges of the discipline, mathematics becoming politics, philosophy, aesthetics, and other concrete and value-laden productions. Eventually — sometimes it takes centuries while other times it is coincident with the appearance of the surd — these results are claimed in the name of State technique, and the surd loses its revolutionary force. The discipline seals off the openings that connect it to contingency and assumes once again an air of self-sufficiency and even self-evidence. Its surdity formalized, intuitionism remains of interest primarily to historians and philosophers of mathematics. The hopes that Brouwer held for practical applications went unrealized, and sometime in the middle of the twentieth century, mathematicians mostly stopped worrying about epistemology, preferring to go about their business as undeclared formalists. -26- Sound and the Surd With mathematics as a privileged model, my hypothesis is to elevate the surd to a general term: the spur of becoming, the juncture where ideas diverge. The surd inaugurates new ideas, not just in math but across disciplines — arts, sciences, studies of the spirit. History, or rather genealogy, shocked ahead by the surd.14 Well short of an adequate demonstration of this hypothesis, this essay will content itself with only one further example. Aside from its mathematical usage, English has retained another specialized meaning of the term surd. Etymologically, surd is a Latinate translation of the Greek alogos, the irrational or rootless. But alogos also refers to what is outside of speech, what cannot be spoken. Linguistics preserves this connotation: in linguistics, the surd is an unvoiced sound or phoneme, that which is not spoken but is nevertheless carried in the speech. (French retains an oddly converse meaning in sourd, the word for deaf.) Linguists fail to appreciate the broad scope of this phenomenon and its essential role in making meaning out of speech. The surd is the sonic analog of the textual supplément, the excess of meaning that hides in the pauses 14 As this essay offers primarily two examples, generalizations about the surd are relegated here to a footnote. In general, the surd can be recognized by the following phenomena: (1) a lack of official sanction or recognition, (2) a pressure to formalize one’s results, (3) the conflation of theory and practice (or of epistemology and practical knowledge), (4) an emphasis on the contingent and contextual as opposed to the general case (nothing can be taken for granted, each result must be thought each time), (5) an insistence on thought as an activity and not just an arrival or end, (6) labor and genius side-by-side, (7) interdisciplinarity, (8) rediscovery of the simplest ideas as now problematic and complex again, (9) a suspicion of the abstract and a faith in the immediacy of experience, (10) activity or motion in the objects under consideration, (11) a willingness to let objects have their way, to work with them rather than to dominate them, (12) a sense of precariousness, a real risk that things might not work out, that they are still up in the air or in development, and (13) maximum libidinal investment. -27- between words, the implicit commas, the white noise around sounds that surrounds spoken language and lends itself to the progressive and concrete generation of new meaning. A sentence always says more than its words: the surd, that which cannot finally be treated in words. Deleuze notes in The Logic of Sense that a word can never say its meaning, so that there must always be another word to name the meaning of the first. However, he acknowledges one exception to this rule: the nonsense word, the absurd, the only word that says its own meaning by saying that it doesn’t say. To speak, to sound off, is to draw upon an active if unconscious not-saying, to deploy the surd as the very possibility of initiating meaning in language. To detect this linguistic phenomenon that generates meaning, examine the edges of spoken language, the thresholds that divide an utterance from the silence that surrounds it. The surd marks that point of fracture, where sound develops from silence and where silence overtakes reverberant sound. Every sound irrupts from silence, beginning with a noisy nonsense, and fades eventually back into silence via a senseless, irregular chaos. Even the formal symbols and numerical indexes of acoustics cannot tame sound’s ecstatic origins, which rend a hole in the rigid fabric of physics. Two measurable, empirical phenomena evince the effect of the surd in sound: the uncertainty principle and Gibbs phenomenon. In both cases, it is a matter of suddenness, of stopping or starting, of sharp edges, of singular moments. Both phenomena relate to the dual nature of oscillating signals such as sounds. A sound (or other signal) can be represented in two complementary manners. Typically, a sound is represented (in a graph) as an amplitude varying over time. Sound is the oscillation of air pressure, and by charting -28- this change in air pressure over time, one represents the sound in all its details. (For comparison, think of a seismogram that is similar but represents the motion of the earth instead of the change in air pressure, or a barometer which also measures changes in air pressure but over a coarser scale of time.) However, instead of showing a sound as a change of pressure over time, one can also represent it as the composite of perfectly regular oscillations. Every sound, no matter how complex, can be constructed by adding together simple sounds (sine waves), and one can therefore identify a sound by noting which sine waves (frequencies) it comprises. In fact, while it is clearly significant when a sound happens — there is little point in yelling “Look out!” after your friend has already been crushed by the falling piano — it is perhaps more significant which frequencies contribute to the sound, as its characteristic frequencies determine what it actually sounds like, high or low, harsh or soothing, aaaah or ooooo, bell-like, string-like, or percussive. The surd at the starts and stops of sound places a limitation on the complementarity of these dual representations. There is an uncertainty principle of acoustics (due to Gabor) — strictly analogous to Heisenberg’s uncertainty principle for quantum mechanics — which holds that a sound cannot be fully determinate with respect to both frequency and time. The more precisely located is a sound in time, the less precise we can be about its frequency content. And the more precisely we describe its frequencies, the less precise we can be about when the sound occurs. “A signal can be represented either as a function of time or as a function of frequency (i.e. its spectrum) and as it is compressed in one representation so it expands in the other” (Stuart 62). Only a sound with no beginning or ending has an exact -29- frequency; every sound with a duration, every sound that starts and stops must include physically inexact frequencies, patches of noise describable by Gaussian distribution functions (bell curves), wherein pitch is defined statistically over a fuzzy range instead of discretely at a specific note. Thus, a singular sound, one that occurs at a particular time, in a particular context, must always begin and end with noise, indeterminacy, the surd. The most sudden events — transients, as they are called by engineers and audiophiles — these sudden transitions are inevitably marked by noise that obscures and even distorts them. A sound located at a specific moment loses its definition, becomes a smudge of energy across the frequency spectrum, a pure noise whose meaning is only its temporal singularity but not its (atemporal) timbral characteristics. Conversely, sounds with an exact frequency or set of frequencies can’t be placed in time at all; they are idealizations, omnitemporal sounds that can never begin or end. If the meaning of a sound is a matter of its location in time and its frequency spectrum, then the surd guarantees that meaning is irreducible, beyond acoustic analysis. Physics cannot account precisely for both components of sound’s meaning. The surd is that indeterminate excess of meaning that slurs the edges of sounds and blurs the contributing frequencies, disallowing any absolute distinctions, insisting that every sound must already have begun, since its point of origin is precisely non-localized. Frequency analysis is the technique by which engineers analyze sound (and many other signals). And the chief method for this analysis is the Fourier analysis or Fourier transform. This is a mathematical technique for taking a signal, represented as a changing amplitude over time, and -30- generating its complementary representation, as a spectrum of frequencies. The claim of the uncertainty principle, manifesting the surd, is that sharp or sudden events in either representation will correlate with broad and smooth events in the other representation. But there is a further wrinkle…. The sharpest or most singular events, those that are most closely anchored to a moment in time, manifest a more severe distortion, a behavior Figure 6: A square wave and an illustration of the Gibbs phenomenon or overshoot that is an artifact of its Fourier transform. known as the Gibbs phenomenon. Since Fourier analysis yields a representation of a sound in terms of frequency, one can take this representation as a kind of recipe, such that by recombining these frequencies one can recreate the original sound.15 But, for signals that have a 15 Fourier analysis includes a large family of mathematical techniques. One primary distinction is between the Fourier transform, in which a signal is represented as a continuous function of its component frequencies, and a Fourier series, in which a signal -31- discontinuity (of the sort discussed in relation to intuitionism above), the Fourier analysis does not yield a strictly accurate representation of the frequencies of the sound. Instead, the recreated signal overshoots the original signal at the point of discontinuity, and this inaccuracy persists, as the recreated signal rises above the original signal then drops below it in a perpetual oscillation. This oscillatory deviation from the original signal near a point of discontinuity is the Gibbs phenomenon (Figure 6). The singular point of discontinuity evades capture by the usual means of analysis, and engineers are forced to alter their methods, tailor their analysis to suit the specific and exceptional case at hand.16 There are ways to compensate for this deviation. By modifying the Fourier analysis using a multiplicative factor called the Lanczos sigma, one can eliminate the overshoot of the Gibbs phenomenon. Of course, this alteration has its own consequences, as the surd does not simply step aside. For one thing, the introduction of the Lanczos sigma factors causes the entire recreated signal to fall short of the original signal by a small percentage. A second result is an increase in the rise time of the recreated signal: eliminating the overshoot causes the recreated signal to take longer to approach the level of the original signal (Figure 7). The surd — in this case, a is represented as the sum of (an infinite number of) discrete sinusoid components. While the Gibbs phenomenon does apply to the Fourier transform, the discussion here refers chiefly to the Fourier series. 16 This is not just a hypothetical example. Engineers routinely deal with discontinuities in signals, as the square wave, whose edges are discontinuities, is a frequent basis for signal construction. Other discontinuities occur when, for example, a signal is “brick wall” filtered, or when a noise gate is applied that suddenly shuts off the signal when it falls below a certain level of amplitude. The Gibbs phenomenon is a genuine hurdle for engineers, whose audio and stereo component designs include attempts to combat it. -32- discontinuity that represents the specificity, the unique moment of the original signal — the surd ensures that no wholly accurate recreation is possible, that no analysis can do justice to the original signal. The Lanczos sigma factors, responding to the surd inherent in the discontinuity in the original function, do not succeed in purging the surd from the analysis. Indeed, they reintroduce the surd in another form. They Figure 7: A square wave and its Fourier approximation including compensation with Lanczos sigma factors. Though the overshoot of the Gibbs phenomenon has been eliminated, note the slow rise time and the overall low amplitude of the approximation relative to the original square wave. sin x represent the sinc function, generally defined as , which is effectively a x smoothing function; it concentrates its energy at its center, but gently spreads out from that center so as not to have any sharp or sudden events (see Figure 8). Thus, the Lanczos sigma factors make localized, contextual alterations to a function, alterations that soften the function without changing its basic form. They are a means of piecemeal or spot-correction, a kluge as it’s called in engineering. The Lanczos sigma smoothes the sudden -33- jump at a discontinuity, but otherwise doesn’t alter the overall form of the signal. This smoothing, which is tantamount to spreading the burst of energy at the discontinuity over a larger area, is the reason for the slower rise time (since the suddenness is smoothed into a diagonal rise) as well as the overall sin x Figure 8: The sinc function, , upon x which the Lanczos sigma factors are based. The horizontal line is the x-axis, which helps to show how most of the energy of the wave is concentrated at its center. shortfall of the function (since the energy required to reach the original amplitude has been spread out slightly). Which is to say, the distortion of the Gibbs phenomenon (which tends to sound like ringing in acoustic signals) can be eliminated only by constructive methods that are tailored specifically to the situation at hand. The Lanczos sigma factor is the ultimate local intervention; it is a tool, a magic wand to wave over particular trouble spots, but its effects are mostly local and are designed to tame an otherwise unruly situation. Confronted -34- with the surd in the form of a discontinuity, engineers apply the Lanczos sigma factor, another surd to combat the effects of the first. These are the phenomena that occur at the birth of meaning in sound, the jumps where sound arises out of silence. ******* Clearly, this theory of the surd needs much further testing. Even within the domains of mathematics and speech, the surd is more promise than result, and there may be better theories of progress, as well as exceptional cases that would call into question the very notion of a general theory of progress within and without these domains. Moreover, the drastic difference in scope between these two examples raises the possibility that the commonality of the term surd in each discipline is an accident of history with no further implications. Still, the research thus far is compelling if not decisive, and my initial investigation of other fields, from “primitive” ethnography to digital technology, discovers the surd there as well at the decisive moments of progress. At least in the digital, the pattern holds true: the digital encounters events or objects that it cannot accommodate, and it must reshape itself in order to make room for these new ideas, but eventually settles back into a placid or rigid formula, neutralizing the novelty that challenged it to develop. -35- BIBLIOGRAPHY Brouwer, Luitzen Egbertus Jan. “Intuitionism and Formalism” (1913), in Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press 1983. ———. “On the Significance of the Principle of Excluded Middle in Mathematics, Especially in Function Theory” (1923b), in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press 1967. ———. “On the Domains of Definitions of Functions” (1927), in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press 1967. ———. “Intuitionistic Reflections on Formalism” (1927a), in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press 1967. ———. “Consciousness, Philosophy, and Mathematics” (1949), in Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press 1983. ———. “Addenda and Corrigenda” (1954), in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press 1967. ———. “Further Addenda and Corrigenda” (1954a), in J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge: Harvard University Press 1967. ———. Brouwer’s Cambridge Lectures on Intuitionism, ed. D. van Dalen. Cambridge: Cambridge University Press 1981. Deleuze, Gilles. Difference and Repetition, trans. Paul Patton. New York City: Columbia University Press 1994. ———. The Logic of Sense, trans. Mark Lester. Ed. Constantin Boundas. New York: Columbia University Press 1990. Deleuze, Gilles, and Félix Guattari. A Thousand Plateaus, trans. Brian Massumi. Minneapolis: University of Minnesota Press 1987 [1980]. Gabor, Dennis. “Theory of Communication.” Journal of the Institution of Electrical Engineers 93, no. 3: 429–457. 1946. ———. “Acoustical Quanta and the Theory of Hearing.” Nature 159, no. 4044: 591. London 1947. -36- Heyting, Arend. “The Intuitionist Foundations of Mathematics” (1931), in Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press 1983. ———. Intuitionism: An Introduction. Amsterdam: North-Holland Publishing Co. 1956. ———. “Disputation” (1956), in Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press 1983. ———. “Axiomatic Method and Intuitionism” (1961), in Y. Bar-Hillel et al. (eds.), Jerusalem: The Magnes Press 1966. Stuart, R. D. An Introduction to Fourier Analysis. London: Methuen & Co. Ltd. 1966. Troelstra, A. S. Choice Sequences: A Chapter of Intuitionistic Mathematics. Oxford: Clarendon Press 1977. Van Atten, M. S. P. R. Phenomenology of Choice Sequences. Utrecht: Zeno 1999. -37-

Virtual Mathematics " ~~ '! i · the logic of difference ,.,.i ! ·I Edited by Simon Duffy 'I :.1 'CLINAMEN PRESS Contents Notes on contributors vii Acknowledgements X 1 Simon Duffy: Deleuze and mathematics 1 2 Alain Badiou: Mathematics and philosophy 12 3 Gilles Chatelet: Interlacing the singularity, the diagram and the metaphor 31 4 Jean-Michel Salanskis: Mathematics, metaphysics, Copyright @ Clinamen Press Ltd 2006 philosophy 46 Copyright@ Simon Duffy 2006 Individual essays @ contributors 5 Charles Alunni: Continental genealogies. Mathematical confrontations in Albert Lautman and Gaston Bachelard 65 The publishers specifically waive copyright on any format o!her than hard copy (book) for this and any subsequent 6 David Webb: Cavailles and the historical a priori in editions of Virtual Mathematics: the logic of difference Foucault 100 wi!h respect to the texts contributed by Charles Alunni and Gilles Ch!ltelet. 7 Simon Duffy: The mathematics of Deleuze's differential logic and metaphysics 118 First edition published Manchester 2006 8 Daniel W. Smith: Axiomatics and problematics as two Published by Clinarnen Press Ltd modes of formalisation: Deleuze's epistemology of 80 Bromwich Street mathematics 145 Bolton BL21JE 9 Robin Durie: Problems in the relation between maths and philosophy 169 www.clinamen.co .uk 10 Arkady Plotnitsky: Manifolds: on the concept of space All rights reserved. No part of this edition may be reproduced, in Riemann and Deleuze 187 stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, 11 Aden Evens: The surd 209 recording or otherwise) without the written permission of !he publishers 12 Manuel DeLanda: Deleuze in phase space 235 A catalogu~ record for this book is available from The British Library Bibliography 248 ISBN 1 903083 311 260 Index Typeset in Times by Northern Phototypesetting Co. Ltd., Bolton Printed in Great Britain by Antony Rowe Ltd, Chippenham SPACE IN RIEMANN AND DELEUZE Dreifaltigkeit. The concept of manifold carries a sense of a "'L""JO'l1(;uv points, neighbourhoods, mappings, connections, etc., but is richer. 5 The best access to the English translation, originally published in is found at http://www.m1tth1; .tcd.ic;Vp:ub/Hi!;tfvJ[attlfP•eo~~le/Rit~mlmn/( WKCGeom.html, which I shall cite throughout this article. 6 Cf., Laugwitz's discussion, to which I an indebted here but which more conventional view of Riemann's conceptual mathematics 303-7). . 7 Riemann actually considered the so-called differential or smooth (in mathematical, rather than Deleuze and Guattari's sense) manifolds, means that one can define differential calculus on such objects. 8 It is at this juncture that they note that, 'the confrontation between and Einstein on the topic of Relativity is incomprehensible if one place it in the context of the basic theory of Riemannian manifolds, as who have exposed themselves to mathematics only in small doses fied by Bergson' (1987, 484). have the impression that maths is a done deal. In grade school, a 9 Deleuze and Guattari see Benoit Mandelbrot's fractals as moving problem is only ever a contrivance, as the solution is already very general definition of smooth space' (1987, 486). I am not altogc~thc~r and indeed precedes the problem: given 2+5 one must produce 7, tain why fractals (so called because they can, as topological spaces, be . even when the so-called problems become much more sophisticated, fractional, rather than whole, spatial dimensions) are singled out here, ' · don't really become more problematic; derived from the solution, cially given that Riemannian spaces (their dimensions are not problem is only a means of testing the pupil's ability to return to it. offered as a/the primary mathematical model of smooth space a bit , . This is the image of mathematics with which most of us are left, There may be, however, aspects of fractal spaces, which indicate that a where solutions are already contained within their problems, waiting to tal space 'builds itself' more naturally or more structurally as a smooth (p. 486-88). This particular point does not appear to me, however, to be teased out by appropriate reductive methods; the answers are there ·· ·· even ifthey are as yet hidden from us p~oblem-solvers. But there the overall argument of this essay. 10 The resulting smooth spaces are not without interest here, including in is another side of mathematics, where the solution is not a foregone con- context of the question of temporality in Bergson and then Deleuze and the problem is worthy of its name. Answers do not follow as indicated earlier, especially in The Logic of Sense (1990). JllLlH"'''""''"~J from questions, but must be secured through resolute and 11 On these issues I permit myself to refer to Arkady Plotnitsky, The Knowable. labour on the part of the mathematician. This minor mathematics and the Unknowable: Modern Science, Nonclassical Thought, and the 'Two _:'exemplified for instance in the numerous mathematical exa~ples Cultures' (2002, 1-107). · throughout the oeuvre of Gilles Deleuze contrasts with the static a?d 12 See Paul de Man's reading of Kant in Aesthetic Ideology (1987). The ques·~ · formalised mathematics that we might call State or royal mathemattcs tion of materiality and the body in phenomenology of Husser!, Bergson and · (by analogy with Deleuze and Guattari's notion of State science (1987, Maurice Merleau-Ponty, is also an important reference here, which, however, 367-74)). Whereas State mathematics comprises formal rules and an would require a separate analysis. established set of symbols that confine problems to the leading edges of . ~esearch, minor mathematics apprehends maths at its most problematic, ·., at its precarious or uncertain moments, when contingency overwhelms .the security of the solutions, and only the urgent intervention of the math- ematician can rescue a suitable result from the calculation. Such active nurturing attends every advance in mathematical research, but certain examples stand out most clearly, when the mathe- .• matics ties its fate to that of the mathematician and can be constructed and maintained only thro~gh her activity. Iri such moments, it is as 208 ADEN EVENS THE SURD though the mathematics will not solve itself, as though the solution is its divorce from contingency, personality, and the empirical. ,., ..... ctrnrt•v•"m on the contrary, clings to contingency, and develops its I',, yet there in the problem, as though the maths had reached out abstract into the empirical, and so could not be taken for granted metuul'" in a particular context, outside which they may well lose their or their force of solution. Maths claims no politics, but progress. Historically, such moments of minor mathematics are · constructivist. ~n111struc1tivl politicises mathematics, as it is no longer a matter of I Constructivism underlies each advance in the history of det<~mun:are formalities. In response, royal mathematics elides or cancels matics, but the history books relegate it to a small and somewhat minor mathematics, claiming its results while neutralising or sterilis- ten niche. Like minor science, minor mathematics is activism. The State restores to the mathematics a pure formality, claimed and formalised by the standard theorems and <onh~•m · and objectivity. State mathematics. If constructivism elevates the problem to the In the one-and-a-half examples to follow, this essay attempts to position, then its formalisation by major mathematics strips it of the constructivist moment and generalise it, moving from mathe- lematic potential and asserts once again the priority of the soluhorr:· history to intellectual history: constructivism, and in particular course, State mathematics dominates the history books, so we surd, as the motor of becoming. The surd is the anomalous element, be surprised to find constructivism under-represented there. unassimilable, that disrupts the linear progress of history, fractures an For example, Deleuze must reach into the obscure t~tu.u"''"'u discipline to open the way to new methods and ideas. At its mathematics at the outset of Chapter Four of Difference and appearance, the surd erupts into a constructivist heterodoxy, tech- (1994), to recover mathematical techniques of differential to be determined in the moment, contextual methods that must be are ontologically substantive, generative and not merely reductive: out on the spot. Before long, however, constructivism cedes its rowing from a usually forgotten history of the calculus, he · on these methods, as they are formalised and claimed by the State, differential, dx, in a process of dif.fere~tiation, a process that .....~,.~"'•~ •revolutionary potential of the surd neutralised. the world and the ideas that make sense of it. In the modem calculus . While the mathematical narrative here is historical, the language of differential does not perturb the formulae in which it appears; it is is ultimately provisional, like the constructivist methods to which a placeholder to indicate which variable is to be integrated over or · ·.refers. The claims in this paper are not primarily historical but derivative to take. But for earlier mathematicians - Deleuze draws ''"'"'a•.vJ:.•"'~"·· that is, they account for events in terms of their motive the researches of mathematicians Camot, Lagrange, and w,·nn.c1r1 , but the actions of these forces may not be historically distinct differential had to be wilfully and actively manipulated in its their reclaiming (and official elision) by royal sciences. Thus, as we it was an extra term, left over after the rest of the equation had' see, the forces that employ the surd toward new disruptive methods reduced, and the methods for dealing with it could not be ,.l,.,,;,.l,.;:s::l~ the same forces that attempt to corral the surd and restore to mathe- advance. Constructive methods, such as successive a legitimacy and formality that the surd resists. ensured an active role for the mathematician, who reduced the bit by bit, sculpted it into a simpler form. . Intuitionism and the surd One chapter later, Deleuze offers a prototypical example whe~· proposes that the universe is the remainder in God's calculations: · :mathematics, a surd is an irrational number, a real number that cannot remainder mandates a construction, for it is an element left over afte!' ~ei<~xo1re~;sed as the ratio of two integers. It is also the name for the 'root' calculations have been performed, a problematic surplus that dobs y. These two meanings are not incidentally related: one easy way cancel out of the equation and requires contingent methods, ,,.,,h,,m·•"' 'generate irrational numbers (surds) is to take roots of rational ones. developed 'on the ground' .1 root of two, for example, is a surd. Though constructivism propels even State mathematics, it . ' , Typically, traditional (royal) mathematics has little trouble with the agrees with the formal essence of mathematics as defined by In grade school, we just write out the first few digits after the deci- State. Maths is traditionally marked by its universality, its ': point, and then write an ellipsis to indicate the continuation of the 210 211 THE SURD ADEN EVENS sequence. 'The square root of two is 1.41421. .. .'The question into the calculus. While claims about finite quantities are, in tically ignored by State mathematics - is the meaning of this vUJliJI>IS#I at least, experience-able as true in an immediate intuition, a rational number, such as one-third, the ellipsis in the decimal about the infinite are another story.4 In particular, it is the surd that tation seems unambiguous; in 0.333 ... it means 'just keep mathematics in the crucible of post-World War I maths and But in the square root of two, the ellipsis feels somehow ,.,,.,~r... -•c . The ellipsis in a surd, say the intuitionists, cannot stand for an it refers to no pattern; the decimal expansion of the square root of two :ini);IOtlten infinity of digits, since no such completed infinity can be 2 of just about any irrational number) is a seemingly haphazard experienced in the mind of the mathematician. Instead, Brouwer of digits that is fully determined (by the definition of square root hti a definition of number, including irrational number, that is purely case), but in a strong sense unpredictable. The ellipsis in the reiJre:;enta a definition in terms of the construction of the number that does tion of an irrational does not succeed in specifying any particular posit some finally constructed infinity of digits. so that one must know beforehand what number is being ,.~.-.~--·" · In Brouwer's words from his early work in 1913, a number is a order to interpret those dots correctly. If the ellipsis means Cotltmue:itr< for the construction of an elementary series of digits after the deci- this manner, an irrational number does not suggest just which point, built up by means of a finite number of operations' (1983a, intended. In other words, a number is some. procedure by which we can unfail- · Luitzen Egbertus Jan Brouwer (1967, 198:3) found this specify further digits to arbitrary precision. For instance, start with problematic; even more problematic for Brouwer was the tacit then continue to add digits, alternating between 7 and 4. This would ness of traditional mathematics to leave the ellipsis un•eXEtmlnecL a rational number. Another example: start with 1.41421 (the first response, he founded intuitionism as a branch of mathematics digits of the square root of two), then continue to add successive ceeds first of all from an epistemological (rather than a ma:the:ma:ti<ia choosing each next digit to be the highest digit such that the square commitment. Amidst the intellectual rubble of post-World War I ·the resulting rational number is less than 2.5 This would be the Brouwer drew together some strands of nineteenth-century •muJnl,lJlH"L definition of the square root of two. tivism to forge a mathematics that restores truth and certainty vvu"''"' '"" By defining number in terms of a procedure, intuitionism demands had been guaranteed. The principle tenet of intuitionism is this: no number be not a fixed, eternal entity, existing in some ideal' space of ematical claim is acceptable that cannot be experienced as certainly hP.r-hooa. Rather, an intuitionist number is a rnle, a method of con- by the mathematician; allow only what can be grasped in the m2tth~:m~lti asttu<;Ucln. Even in this simple definition, we can already see the surd at dan's immediate intuition.3 (We shall see what this means ...) its irrationality calling upon constructive procedures, an engage- In intuitionism, the mathematician is no longer the one who by the mathematician. But the surd does not stop at this call for on the mathematics; the mathematician is its vessel and locus. ~n1attterr1atical commitment; its perturbations are more dramatic, as Brouwer mathematics is mental activity, independent of language,aJ!d ... ~'""''""'illustrates. independent of the notation used to remember and convey it. Only · · He defines a number r, called the pendulum number, as the limit of itions are mathematics. Thus, the symbols of maths and other written or verbal communications about it are not the genuine item, but only a means of reminding oneself or others how to 'have' the same mathemat~ ics again. Intuitionism insists on the active role of the mathematician ·m ' 'fhe sequence is doing the mathematics, a constructivism in principle. But what difference does it make in practice to demand that .the 1 1 1 1 truth of the mathematics be experienced? Was there something uncertain - 2'4'-8'16 ,... about prior mathematics? Were mathematicians allowing claims into " mathematics that could not be immediately intuited as true? Brouwer argues that, indeed, many claims were being routinely but illegitimately 212 213 THE SURD ADEN EVENS It is the geometric sequence defined by the inverse powers of two, The thing is, we don't know where such a sequence of digits occurs the sign switches back and forth with each element. The general decimal expansion of :n:. We don't even know if such a sequence is occurs in the decimal expansion of :n:. If such a sequence occurs at even-numbered place in the decimal expansion of :n:, for instance, at c.=HJ 1000'h place, then c., will have an even exponent, it will be equal to This sequence bounces back and forth around 0, getting closer the farther you proceed in the sequence (Figure 1). Its limit is in and there is not yet any disagreement between intuitionist and rrwJmon~lll mathematics. 1t = 3.14159265358979323 ... 0123456789 ... ca 4th i digit m th td.lgl't CJ Cs c.., (may not exist) c c. c. Figure 2: The sequence of digits of :n:, and the (hypothetical) m•h position, .· where 0123456789 occurs. -0.5 0 0.25 which is a positive (but small) number. The sequence of c's will 'freeze' Figure 1: The sequence of c's, bouncing back and forth as it approaches 0. at that point, and r, the limit of the sequence, will be equal to this small positive number. If, on the other hand, the sequence 0123456789 occurs at an odd place in :n:, then c'" will have an odd exponent and r will be a But Brouwer adds a twist. He alters the definition of the sequence small negative number. But if (on the third hand!) that sequence never by freezing it beginning at the m'" term of the sequence, so that ···-.,·..=..~.occurs in the decimal expansion of 1t, then there is nom, and the sequence at c,, all the rest of the terms will just be equal to c.,. Now, under this of c's will just keep getting closer to 0, with a limit, r, ofO. altered definition, the limit of the sequence is no longer 0, but em. The A traditional mathematician, say a Platonist, would not object to question of course is, What is m? this definition of the pendulum number. She would allow that r is a well- In an original stroke of genius, Brouwer defines m so as to render defined real number whose value we don't at present know, since we the limit of the sequence, r, a rather strange number. m is defined as the don't know where 0123456789 occurs in :n: (nor whether it occurs at all). first place in the decimal expansion of :n: where the digits 0123456789 In spite of our ignorance about r, though, she would insist that as a well- occur: Start to write out :rt, counting its digits. As soon as you reach a ·defined real number, r is either positive, negative, or equal to zero. place where 0123456789 occurs, note how far along you are in your 1 This is where the surd fractures mathematics, splitting intuitionist count. This ism, as illustrated in Figure 2. (Just to avoid confusion, note from traditional mathematics. Brouwer too allows that r is a well- that :n: has no prior relationship tor; Brouwer could have chosen any irra" · defined, real number: it meets the definition of an intuitionist real number tiona! number, and picked 1t arbitrarily because it is a universally-recog- in that we can specify it to arbitrary precision, simply by writing out the nised irrational number. The essential criterion is that 1t is a surd, whose. digits of :n:. However, we cannot say of r that it is either positive, nega- digits can be successively calculated to arbitrary precision, but whose · tive, or equal to 0. We cannot say this because we cannot immediately yet-to-be-calculated digits cannot be predicted or patterned in advance of experience the truth of any of these three possibilities, and therefore none their calculation.)6 of them is true.It is not true that r is greater than 0, nor is it true that ris 214 215 THE SURD ADEN EVENS less than 0, nor is it true that r is equal to 0. None of the three usual that describes a (hypothetical) mathematical object, such that no ing relations holds between r and 0. (In effect, Brouwer's nn.~-'--•·""' is known that holds the property, and it is also not known that no number instantiates a new logical value: it is not true that r >'; object exists. Moreover, there is no finite process that will surely neither is it false.) 7 whether or not there is such an object. Thus, an opaque fleeing This counterintuitive conclusion of intuitionist analysis is a property such as 'the first place in the decimal expansion of reemphasis. It is not a deferral in regard tor, a refusal to answe the digits 0123456789 occur'. This opaque fleeing property tions about r due to insufficient data. Rather, r, in all its'-'"''-"'"--'· · and rarefies the surdity of ;n;, its uncanny simultaneous contin- enters the mathematics, carrying its ambiguity along with it. It and necessity. ;n; is contingent for wholly singular, without genus, perturbs traditional, even sacred theorems that had held sway for: · governed by no law but its own; nevertheless it is woven into the dred~ of years. In this case, r defies the fundamental ordering of. of the universe, the singular relationship between circumference contmuum - the property that every real number is either greater diameter of a circle. Its specificity defies its abstraction. It is effec- less than, or equal to any other real number. Here, uncertainty (about split between the empirical and the abstract, a pure abstraction with digits of~). is not an extra-mathematical concern that leaves a gap irreducible detail that generally accrues only to the concrete. maths wrutmg to be filled, but is incorporated directly into the thus reaches out of the eternal mathematical world and into the Epistemology intrudes upon ontology, to tie the truth of the mathematii human one.9 to the concrete mental processes of the mathematician. No 10 n1 11:er:,.~ It is a strange and disconcerting move, therefore, when Brouwer, eternal posit, an ideal relation among ideal objects, mathematical with traditional mathematical practice, abstracts from this par- now a contingent event, something that happens to the IH<LLu<,1muJcia opaque fleeing property to speak of opaque fleeing properties in and to the mathematics. If at some future moment, the general. (For instance, 'as long as a fleeing property exists such that .. .' 0123456789 is discovered in the decimal expansion of;n;,orifit is 'for each fleeing property, f, .. .' (1981, 42).) That is, he ceases to day proved that no such sequence can ever occur in the decimal .-specify a particular fleeing property, and instead reifies the concept of sion of ;n;, then r will take on a decimal value, and it will obtain one fleeing property, using it as a mathematical effect in the analysis: three ordered relationships to 0. The future-historical event will 'Consider a number defined by some fleeing property, a•. This is tanta- the t~th of the mathematics. By virtue of the surd (in this case, ;n;), . mount to taking the form of an opaque fleeing property and voiding it of ematlcs bleeds over from the realm of the pure abstract into the erntviri;;·.!'~~its content. In this formalisation, the surd, formerly crucial to the fleeing ~al: It becomes dependent on human history and human progress. And property, is neutralised, drained of its disturbing singularity so that it can mststs, above all, on the participation of the mathematician as an essel1- serve as a general term. 1°Contingency is no longer the contingency of ;n;, tial element of the mathematics. but becomes a purely formal element of the calculus, with rules to govern With the entry of the pendulum number into the mathematics' its manipulations and effects. ' Brouwer replaces the ordered continuum of real numbers with a disuni~ When fleeing properties are formalised, the surd is not so much fied set of incommensurables, scattered surds that cannot be adequately evinced as insisted upon, spontaneously generated in its form but with- compared to other numbers. By definition, the surd is incomparable, siilc out content. However, a surd without content is impotent. For a surd is gular, and it lends this quality to intuitionist number in general. The surd precisely an excess of content, an intensity that cannot be contained by a makes numbers into immeasurable ordinal quantities whose nature can form. Though its results persist in the calculus, they no longer require the be determined only by pursuing a process that may never yield a certain, engaged attendance of the mathematician. As a formality, intuitionism r~sult. rand 0 cannot be compared adequately to each other, for each is a· loses its tie to the human, and becomes just another branch of mathemat- smgul~ experience, constructed in such a manner that they diverge, 'f ics, a universal and agreeable formalism. Historically, formalisation of becommg skew to 0. ." the surdity of intuitionism made it acceptable to other mathematicians. The pendulum number depends for its definition upon what Troelstra (1977) for instance attempts as his chief project to formalise Brouwer terms an opaque fleeing property. 8 This is a mathematical intuitionist mathematics to the point where we no longer need extra- 216 217 THE SURD ADEN EVENS mathematical conceptual apparatus, but can regard intuitionism as ~o:re:;•cu<au•uu. The new definition introduces the innovation of the free of rules for manipulating symbols. The triumph of royal ...........:,ut~lUCl~,.; sequence, which allows a certain freedom or latitude in the Brouwer, on the other hand, wishes to maintain the surdity. of fnrcJcess. Now, the rule that determines successive digits need not be mathematics, for it is just this surdity that attaches the maths firmly. determining; rather, the rule for choosing successive digits can mental activity of the mathematician. Having lost the opaque the mathematician a choice of next digit, and this still defines a par- property to formalisation, he needs to conjure another surd, to point (number) in the intuitionist calculus. For example, here the opaque fleeing property with a more insistent concept. The 3) is an intuitionist point: start with 0.31, then choose successive fleeing property was something of a parlour trick anyway, a so that each next digit is either a 4 or a 2. This definition simply demonstrate some of the easy and dramatic implications of ..· "u'"•umRr to specify a number that a traditional mathematician would recog- analysis, but not a serious contribution to mathematics.U To up the· well-defined. For a traditional mathematician, this is not a Brouwer introduces the notion of a free choice sequence, and a but a method for generating different numbers. And while a cept of number to go with it, more radical still than the peJildtOur iiiitl'aa.1uuoHau"' would agree that any number generated according to this number. To generate a free choice sequence, epistemology must be will have certain properties (such as the property of being greater ther twisted, the very notion of choice remade as a mathematical 0.31 and less than 0.32), the existence of these properties does not One of the problems for traditional mathematics was to ~~ 1r-...., .. ,,11c the number itself well-defined. continuum. How is it that points can somehow aggregate to form the tinuum, or number line? How do points of dimension 0 fully 4 line segment, which has a dimension of 1? How can discrete points, have no length, form a continuous line? Traditional mathematicians· 4 /~ posited an axiom of the continuum; they simply asserted as a basic or 4 assumption of the mathematics that when you take all of the (infinity of) /2-or rational numbers and all of the (even bigger infinity of) irrational num- 4 2 bers, you have covered the continuum to form a continuous line.tZ (As the 0.31 -or etc. continuum is most easily conceptualised in geometrical terms, I will 2 4 henceforth refer interchangeably to number or point. The same L l l l l l l " " " '\ 4- or or 2 intended by each term.) 1 , This claim, say the intuitionists, is not immediately verifiable as 2\:r true in the mind of the mathematician. In fact, it is altogether counterin~ 2 tuitive. Intuitively, no matter how many points you place side-by-side, there is no way that they should ever be able to cover a positive linear . Figure 3: An illustration of a single number, as defined by a free choice area, no way that they can ever form a continuous line. To compensate sequence. for this incorrectness in the calculus and it is an important claim that underlies a great deal of higher mathematics- the intuitionists offer a fur" But the intuitionist understands this definition differently. For ther definition of number, one that is intended to overcome this problem Brouwer, this rule defines a point. Though two different mathematicians by building into number the continuity that number is expected to cover,. might make different choices, before any choices have been made or as This new definition of number is based on the idea of a free choice long as all of the choices made so far coincide, both mathematicians are sequence, and it once again relies on the surd, which imparts to number working with the same point. As soon as they make different choices, a power of the continuum. they are then no longer working with the same point. A given point, The early intuitionist definition, above, makes of number a process therefore, consists of choices already made plus the possibility of making for determining with arbitrary precision successive digits of a decimal various choices in the future. As a process of choice, a point includes 218 219 THE SURD ADEN EVENS the potential for any number of points, so that each point ~.;uJua1ms., not yet determined, and this indeterminacy is a difference that lives internal difference, equal to itself and different from itself. number and forms part of iis nature. Unlike a point in traditional mathematics, an intuitionist The continuity of intuitionist number shows up in the calculus defined in relation to a free choice sequence, is no longer an the intuitionist theorem that every junction of the unit continuum is able idealisation of dimension 0. On the cohtrary, using the This departs dramatically from the claims of traditional a free choice sequence, a point is a progressive narrowing of tuu•"'u""''""· Whereas the pendulum number is something of a con- process of honing in that keeps open an interval and never finally. a rather forced example of the oddball cases allowed by the it down to 0. Points thereby become fuzzy, each equal to itself intuitionist definition of number, this revision of the notion of con- neighbours (but less so). Indeed, in Brouwer's formal treatment radically alters the landscape of mathematical analysis and choice sequences, he does not refer to successive digits of a rdemonst:rates the vast distance, opened by the surd, between intuitionist number; rather he posits a sequence of nested intervals, and the traditional conclusions. a matter of how each interval is to be fit within its enclosing intervaL . In traditional mathematics, the continuity of a function is a ques- first few 'choices' of intervals for a point are illustrated in Figure 4/ tion of its smoothness. A function is continuous if it has no breaks or gaps, no jumps in it. For instance, the function in Figure 5A, though quite arollercoaster, is nevertheless continuous: there are no jumps or gaps in the graph of the function; it is 'filled in' from one end to the other. [ [ [E ffJ]~ ] a fingertip at one end of the function, one could trace its entire path without lifting one's finger. The second example, Figure 5B, is a discontinuous function; this is called a jump discontinuity, since the Figure 4: The first few 'choices' of nested intervals that define a point. function jumps suddenly at a point from one value to another. Brouwer's radical claim is that, in spite of this jump, the function in 5B is still Now the problem of the continuum is solved almost by default: continuum is just the totality of all choices, or the freedom to 5B represents a continuous intuitionistic function because points choices however one will. Each point, as a process of continual .. (or numbers) are themselves defined so as to be smudges or vanishing ment, covers a vanishing but positive linear area. The power of: rather than finished points of dimension 0. In traditional math- continuum is built into the points that constitute it, requiring no , the function in 5B is discontinuous because we can specify a axiom to justify its continuity. The points do not aggregate or point, l, at which the function suddenly jumps from one value to another. together side-by-side to create the continuum. Rather, as '"'"'u''"''"'"• the left side of this point, the function equals 1, and on the right side, points are in motion, smearing themselves across intervals, blurring the function equals 2. But for the intuitionists, one cannot specify such a own boundaries to leave the continuum in their wake. Intuitionist point because there are no absolute points. There are only neighbour- ematics does not exactly promote the line over the point, but, as hoods, intervals surrounding l, honing in on the classical point l, but and Guattari recommend, defines both line and point in terms of a never actually reaching it. And any neighbourhood around the classical that generates each of them. point l will include space to either side of 'I. However closely one hones The result of this definition of number in terms of free in on l, one still keeps open the possibility of making further choices so sequences is to impart a maximum of continuity into relations .as·to determine a value of 1 for the function, or of making other choices numbers. But it is a continuity that derives from the fact that the so as to determine a value of 2. Thus, putting it rather too bluntly, near l ments, the numbers themselves, incorporate difference into their the function evaluates to both 1 and 2. Instead of a point at which the Because each number has difference within itself- a part still to be function jumps, the intuitionists see a neighbourhood that effectively mined- the differences between numbers are less dramatic, generating co'nnects the two sides of the function by tying their ends together in a maximum of continuity among numbers. Every number leaves part: single point, a point that includes difference within itself. The intuition- 220 221 THE SURD ADEN EVENS ist point l functions as a kind of wormhole, a black hole or singularity interval remains open, with a dimension of 1 /but ever shrinking towards causes two points otherwise distant in space to overlap each other. ;... o. Thus, in Brouwer's proof of the continuity of every function ofthe unit continuum, he relies crucially on the fact that, as we hone in on l, select- ing increasingly narrow intervals surrounding l, it will always be possible to imagine for the 'next' choice that we are honing in on a point to the left of lor that we are honing in on one to the right of l. Once the next choice, say the q'h choice, is actually made, it will undoubtedly cut Df! certain possibilities that existed prior to choice q, but it will still be A possible to make the next choice so as to aim toward a point to the right or to the left of l. The formal procedure of choice keeps the point from collapsing to zero, and calls once again upon the participation of the mathematician, this time an ideal mathematician, endowed with a power of abstract choice. Choice is the surrogate of the surd. · Though this definition of number is mathematically cumbersome, 2.5 it is nevertheless intuitively appealing, even 'natural'. No longer an idealisation of dimension 0, number is now a process of narrowing, mirroring the physical or mental process we might go through in deter- mining a point, and incorporating the necessarily inexact and provisional 1.5 endpoint at which we inevitably curtail our process. While empirical B pointing happens over a specific time period, intuitionist points only refer to an abstract time, once again drawing upon form (of time) without reference to a specific content.'l 0.5 Corresponding· to the notion of an idealised temporality is the positing of an ideal mathematician (or 'creative subject' in some of the literature). Brouwer did not want free choice sequences to subject the mathematics to the caprice of the mathematician, as though the results of Figure 5: A continuous function and a discontinuous. function, at least ·the calculus depend on just which choices the mathematician chooses to according to the definition of continuity in traditional mathematics. make. Rather, choice in intuitionism has a Nietzschean resonance, in that every choice gets made. Proofs involving free choice sequences do not The surd appears in this case in the guise of the concept of choice. take for granted any particular choice; they assume only that some choice The intuitionist definition of a point includes within its formalisms an gets made, and that some are still to be made. (Again, it is a matter of openness, an indeterminacy that, not yet filled in, might be satisfied in. idealisation: choice as a form without content.) Choice is affirmed as a any number of ways. As other intuitionists have argued (see Troelstra . principle so that it is not a matter of any particular choice. 'But what is 1977, 12), choice here is not so much about the actual fact of choosing as · relevant from a mathematical point of view is not any individual choice about the form of choice, the fact that one can define a point and work sequence as such, but the 'mathematical' fact that there exist many per- with it without having narrowed down all of the intervals that would limit' fectly well-defined (lawlike) operations on sequences which can be that point to a dimension of 0. Choice is thus simultaneously made and carried out without assuming the arguments to be determined by a law' suspended; proofs are carried out as though only so many choices have (Troelstra 1977, 12). The principle of choice defines number as a process, been made, but the results are valid no matter how many have actually . with a maximum of continuity between numbers and a maximum of been made. There are always more choices to be made, so that the difference within each number. Every choice is effectively made, every 222 223 THE SURD ADEN EVENS choice is affirmed, and this generates the universe ~esll1uu"''"'" cousins, and so demand a real labour and lend the whole continuum). radical enterprise an empirical or contingent character. The arrival of the Formalising or idealising the concept of choice freezes a involves a struggle that spills over the edges of the discipline, math- process into a static element of the mathematics. Whereas ematics becoming politics, philosophy, aesthetics, and other concrete and mathematics idealises number, ignoring its processual aspect, value-laden productions. Eventually sometimes it takes centuries while ism formalises the process as process, capturing this motioni, other times it is coincident with the appearance of the surd - these results numbering in vivo, forma1ising not number but the genesis of · are claimed in the name of State technique, and the surd loses its revolu- Intuitionism discovers the essence of number in numbering, naming· tionary force. The discipline seals off the openings that connect it to essence and giving it a place amongst the symbols of the ""''""'"" contingency and assumes once again an air of self-sufficiency and even Brouwer thus pre-empts the process of numbering, seizes nmnb 1~rin ~elf-evidence. Its surdity formalised, intuitionism remains of interest with its virtuality intact, before it has cancelled that virtuality ""'h'.'" primarily to historians and philosophers of mathematics. The hopes that actuality of a fixed and determinate value. Notably, the power of the · Brouwer held for practical applications went unrealised, and sometime in choice sequence is a power of determination, a finite deter;mi1nation the middle of the twentieth century, mathematicians mostly stopped that can always be fuLther determined, but is never finally de1tennir1ed. worrying about epistemology, preferring to go about their business as Intuitionism knows not to explicate too far, as Deleuze puts it. Kr11nu'"''". undeclared formalists. thus suspends the mathematics in between virtual and actual, refusing complete determination to promote the process that gives rise to the determinable. · Sound and the surd This hold on the vital essence of number could not endure. Mu~~­ . With mathematics as a privileged model, my hypothesis is to elevate the of the language of choice, and the epistemological and ontological . surd to a general term: the spur of becoming, the juncture where ideas commitments associated with Brouwer's intuitionism, were at odds with diverge. The surd inaugurates new ideas, not just in maths but across dis- traditional mathematical standards. Later intuitionists attempted ,t0 ' ciplines- arts, sciences, studies of the spirit. History, or rather genealogy, preserve the results of the intuitionist calculus while discarding the shocked ahead by the surd. 14 Well short of an adequate demonstration· philosophical underpinnings. These intuitionist reformers rejected the of this hypothesis, this essay will content itself with only one further 'unmathematical' ideological and epistemological commitments of· . example. itionism, but wished to retain its formal results; after all, its alterations to · Aside from its mathematical usage, English has retained another traditional numerical analysis and to logic are at least interesting and pos" specialised meaning of the term surd. Etymologically, surd is a Latinate sibly even useful. Troelstra (1977) and Heyting (1956, 1966, 1983) each · translation of the Greek alogos, the irrational or rootless. But a logos also formalise significant parts of intuitionism, so that it becomes only .· refers to what is outside of speech, what cannot be spoken. Linguistics another system of symbolic manipulation, a formal alternative to tradi" preserves this connotation: in linguistics, the surd is an unvoiced sound tiona! calculus stripped of the surd's original revolutionary power. A or phoneme, that which is not spoken but is nevertheless carried in the term like free choice sequence is replaced with the less provocative speech. (French retains an oddly converse meaning in sourd, the word for infinitely proceeding sequence, or just a sequence that is lawless or deaf.) Linguists fail to appreciate the broad scope of this phenomenon even non-law-like. The human operator is thereby eliminated from the • and its essential role in making meaning out of speech. The surd is the mathematics, and sterile if productive research can continue without · sonic analogue of the textual supplement, the excess of meaning that epistemological threats to its universality and objectivity. hides in the pauses between words, the implicit commas, the white noise Such is the history of the surd, in maths as elsewhere. Its introduc- around sounds that surrounds spoken language and lends itself to the pro- tion disrupts standard practices, opens a break in the linear progress of a gressive and concrete generation of new meaning. A sentence always field, inviting markedly new understandings, concepts, and techniques. says more than its words: the surd, that which cannot finally be treated in Initially; these novel elements do not settle comfortably amidst their words. Deleuze notes in The Logic of Sense (1990) that a word can never 224 225 THE SURD ADEN EVENS say its meaning, so that there must always be another word to name the Heisenberg's uncertainty principle for quantum mechanics which holds meaning of the first. However, he acknowledges one exception to this that a sound cannot be fully determinate with respect to both frequency rule: the nonsense word, the absurd, the only word that says its own and time. The more precisely located is a sound in time, the less precise meaning by saying that it does not say. To speak, to sound off, is to draw we can be about its frequency content. And the more precisely we upon an active if unconscious not-saying, to deploy the surd as the very describe its frequencies, the less precise we can be about when the sound possibility of initiating meaning in language. occurs. 'A signal can be represented either as a function of time or as a To detect this linguistic phenomenon that generates meaning, function of frequency (i.e. its spectrum) and as it is compressed in one examine the edges of spoken language, the thresholds that divide an representation so it expands in the other' (Stuart 1966, 62). Only a sound utterance from the silence that surrounds it. The surd marks that point of with no beginning or ending has an exact frequency; every sound with a fracture, where sound develops from silence and where silence overtakes duration, every sound that starts and stops must include physically inex- reverberant sound. Every sound irrupts from silence, beginning with a . act frequencies, patches of noise describable by Gaussian distribution noisy nonsense, and fades eventually back into silence via a senseless; functions (bell curves), wherein pitch is defined statistically over a fuzzy irregular chaos. Even the formal symbols and numerical indices of · range instead of discretely at a specific note. Thus, a singular sound, one acoustics cannot tame sound's ecstatic origins, which rend a hole in the that occurs at a particular time, in a particular context, must always begin rigid fabric of physics. Two measurable, empirical phenomena evince and end with noise, indeterminacy, the surd. The most sudden events - the effect of the surd in sound: the uncertainty principle and the Gibbs transients, as they are called by engineers and audiophiles these sudden phenomenon. In both cases, it is a matter of suddenness, of stopping or transitions are inevitably marked by noise that obscures and even distorts starting, of sharp edges, of singular moments. them. A sound located at a specific moment loses its definition, becomes Both phenomena relate to the dual nature of oscillating signals a smudge of energy across the frequency spectrum, a pure noise whose such as sounds. A sound (or other signal) can be represented in two com- meaning is only its temporal singularity but not its (atemporal) timbral plementary manners. Typically, a sound is represented (on a graph) as an characteristics. Conversely, sounds with an exact frequency or set of amplitude varying over time. Sound is the oscillation of air pressure, and frequencies cannot be placed in time at all; they are idealisations, by charting this change in air pressure over time, one represents the omnitemporal sounds that can never begin or end. sound in all its details. (For comparison, think of a seismogram that is If the meaning of a sound is a matter of its location in time and its similar but represents the motion of the earth instead of the change in air frequency spectrum, then the surd guarantees that meaning is irreducible, pressure, or a barometer which also measures changes in air pressure but beyond acoustic analysis. Physics cannot account precisely for both com- over a coarser scale of time.) However, instead of showing a sound as a ponents of sound's meaning. The surd is that indeterminate excess of change of pressure over time, one can also represent it as the composite meaning that slurs the edges of sounds and blurs the contributing of perfectly regular oscillations. Every sound, no matter how complex, frequencies, disallowing any absolute distinctions, insisting that every can be constructed by adding together simple sounds (sine waves), and sound must already have begun, since its point of origin is precisely one can therefore identify a sound by noting of which sine waves non-localised. (frequencies) it is comprised. In fact, while it is clearly significant when Frequency analysis is the technique by which engineers analyse a sound happens there is little point in yelling 'Look out!' after your sound (and many other signals). And the chief method for this analysis is friend has already been crushed by the falling piano - it is perhaps Fourier analysis or the Fourier transform. This is a mathematical tech- more significant which frequencies contribute to the sound, as its char- nique for taking a signal, represented as a changing amplitude over time, acteristic frequencies determine what it actually sounds like: high or low, and generating its complementary representation, as a spectrum of harsh or soothing, aaaah or ooooo, bell-like, string-like, or percussive. frequencies. The claim of the uncertainty principle, manifesting the surd, The surd at the starts and stops of sound places a limitation on the is that sharp or sudden events in either representation will correlate with complementarity of these dual representations. There is an uncertainty broad and smooth events in the other representation. But there is a principle of acoustics (due to Gabor 1946, 1947) strictly analogous to further wrinkle .... 226 227 ADEN EVENS THE SURD · eliminating the overshoot causes the recreated signal to take The sharpest or most singular events, those that are most longer to approach the level of the original signal (Figure 7). anchored to a moment in time, manifest a more severe rh•tn....e:~c. behaviour known as the Gibbs phenomenon. Since Fourier yields a representation of a sound in terms of frequency, one can representation as a kind of recipe, such that by recombining frequencies one can recreate the original sound. 15 But, for signals have a discontinuity (of the sort discussed in relation to ,u.,cuuumsm' above), the Fourier analysis does not yield a strictly accurate rHTlr''"o" ....: .. tion of the frequencies of the sound. Instead, the recreated signal shoots the original signal at the point of discontinuity, and this 1D2Lcc:uraov· persists, as the recreated signal rises above the original signal then below it in a perpetual oscillation. This oscillatory deviation from ulc··""''"'' original signal near a point of discontinuity is the Gibbs phenomenon ·. (Figure 6). · Figure 7: A square wave and its Fourier approximation including compensation with Lanczos sigma factors. Though the overshoot of the Gibbs . phenomenon has been eliminated, note the slow rise time and the overall low amplitude of the approximation relative to the original square wave. The surd in this case, a discontinuity that represents the speci- . ficity, the unique moment of the original signal- ensures that no wholly . accurate recreation is possible, that no analysis can do justice to the orig- inal signal. I, The Lanczos sigma factors, responding to the surd inherent in the Figure 6: A square wave and an illustration of the Gibbs phenomenon or discontinuity in the original function, do not succeed in purging the surd overshoot that is an artefact of its Fourier transform. from the analysis. Indeed, they reintroduce the surd in another form. The singular point of discontinuity evades capture by the usual · · 1 fi d sin x h. h · They represent the smc function, general y de ne as -x-· w lC 1s means of analysis, and engineers are forced to alter their methods, tailor 16 effectively a smoothing function; it concentrates its energy at its centre, their analysis to suit the specific and exceptional case at hand. but gently spreads out from that centre so as not to have any sharp or There are ways to compensate for this deviation. By modifying the · Fourier analysis using a multiplicative factor called the Lanczos sigma, sudden events (see Figure 8). Thus, the Lanczos sigma factors make localised, contextual alter- one can eliminate the overshoot of the Gibbs phenomenon. Of course, · ations to a function, alterations that soften the f'\lnction without changing this alteration has its own consequences, as the surd does not simply step its basic form. They are a means of piecemeal or spot-correction, a kluge aside. For one thing, the introduction ofthe Lanczos sigma factors causes as it is called in engineering. The Lanczos sigma smoothes the sudden the entire recreated signal to fall short of the original signal by a small jump at a discontinuity, but otherwise does not alter the overall form of percentage. A second result is an increase in the rise time of the recreated . 229 228 THE SURD ADEN EVENS the signal. This smoothing, which is tantamount to spreading the burst commonality of the term surd in each discipline is an accident of history energy at the discontinuity over a larger area, is the reason for the .··with no further implications. Still, the research thus far is compelling if rise time (since the suddenness is smoothed into a diagonal rise) as not decisive, and my initial investigation of other fields, from 'primitive' as the overall shortfall of the function (since the energy required to ethnography to digital technology, discovers the surd there as well at the the original amplitude has been spread out slightly). decisive moments of progress. At least in the digital, the pattern holds tfue: the digital encounters events or objects that it cannot accommodate, and it must reshape itself in order to make room for these new ideas, but eventually settles back into a placid or rigid formula, neutralising the that challenged it to develop. Notes , 1 A Thousand Plateaus perpetuates such constructivist examples, from 1. Riemann spaces that allow no overarching perspective and must be navigated locally and singularly, to fractals that are generally apprehended in the process of construction and not as completed figures, to the numbering number, which could be a general term for the constructive impulse in math- ematics, for it is the intensive number, determined always in relation to its Figure 8: The sine function, si~ x, upon which the Lanczos sigma factors are context, and insisting upon a uniqueness, atypical of number, that ensures that based. The horizontal line is the x-axis, which helps to show how most of the it can only be understood and manipulated in context. energy of the wave is concentrated at its centre. ·: 2 One can construct irrational numbers that have a pattern, and hence are exceptions to the rule. E.g., 0.030030003000030000030000003 .... But this is a special, contrived case, and does not typify irrational numbers. Which is to say, the distortion of the Gibbs phenomenon (which 3 The flattening of ontology onto epistemology, the sense of reconstruction on tends to sound like ringing in acoustic signals) can be eliminated only by a firm foundation, the reliance on intuition, and the notion of a critique from constructive methods that are tailored specifically to the situation at within all attest to Kant's parentage in Brouwer's work. hand. The Lanczos sigma factor is the ultimate local intervention; it is a ·. , 4 In the finite realm, intuitionist mathematics is pretty much functionally tool, a magic wand to wave over particular trouble spots, but its effects identical to traditional math. The breaking point lies between countable and are mostly local and are designed to tame an otherwise unruly situation. uncountable infinities. (Roughly speaking, an infinite number of discrete Confronted with the surd in the form of a discontinuity, engineers apply objects is countable, while a continuum is an uncountable infinity.) For the the Lanczos sigma factor, another surd to combat the effects of the first. intuitionists, uncountable infinities were nearly incomprehensible, while These are the phenomena that occur at the birth of meaning in sound, the countable infinities could be dealt with one discrete element at a time. jumps where sound arises out of silence. Discrete objects can be intuited, while a true continuum is beyond intuition. 5 In other words, pick a next digit, say 4, and add it to the end of the digits you have so far: 1.414214. Multiply this number by itself, 1.4142142 ******* 2.0000012378. This is slightly greater than 2 so replace the terminal4 with a Clearly, this theory of the surd needs much further testing. Even within 3 and repeat: 1.4142132 1.99999840937. 3 is thus the largest digit that can the domains of mathematics and speech, the surd is more promise than be appended to make the square of the result less than 2. Continue in like result, and there may be better theories of progress, as well as exceptional manner to generate the next digit and each successive digit ad infinitum. cases that would call into question the very notion of a general theory of 6 As such, it may be significant that :n; is not only an irrational number, but also progress within and without these domains. Moreover, the drastic differ- a transcendentlil one, i.e., a real, irrational number that cannot be expressed ence in scope between these two examples raises the possibility that the as the root of a polynomial. This intensifies the 'surdity' of :n:, inasmuch as it 230 231 THE SURD ADEN EVENS has a singular relationship to the universe and does not just express a reia,, . , 11 Brouwer tends to use the opaque fleeing property in discussions aimed at tionship among numbers. It is as though .n: reaches beyond mathematics. more general audiences and not so much in his formal papers. (Compare with generating its identity in the empirical domain (in the relationship of a circl~·~ the conference address 1967a which, while still mathematically rigorous, is area to its diameter). Other transcendental numbers, such as e, share this not so laden with formalities as some of his other writings.) On the other property of reaching out of mathematics and into nature. Perhaps this is iuso hand, as an intuitionist, Brouwer did not distinguish sharply between formal true of certain algebraic (non-transcendental) numbers, such as the golden and 'everyday' modes of mathematics, and has been criticised for using a mean, but the relationship between the golden mean and natural phenomen'a plain language style even in his formal presentations. is more approximate, not exact like .n:. 12 Technically, there are not just an infinite number of irrationals, but a non- 7 Different versions of intuitionism treat this other truth-value differently: denumerable or uncountable infinity of them, more irrationals than integers. Brouwer adhered to the belief that there are only two truth-values, true and A higher level infinity of points is required to create a sufficient density to false, so that this value of not-true was not a formal element of the calculus constitute the continuum. but only a step in thinking about it. Other intuitionists codified the value of 13 Though Charles Parsons, editor of Brouwer (1967c) and himself a mathe- not-true, making it a third term alongside true and false. matician, refers to the free choice sequence as 'a process in time' (notes to 8 A fleeing property is any property such that (1) it can be determined for each p. 446), this is merely heuristic, as they function atemporally. Free choice natural number n that either holds or is absurd, (2) no method is known for sequences imply the formal structure of time, for there is always a part of the calculating a number with the property , and (3) the assumption that some sequence that has already been chosen, a part that has not yet been chosen, number exists with the property is not known to be absurd. The stipulation of and the immediate choice to be made. However, proofs involving free choice opaqueness adds the further condition that (4) the assumption that some sequences demonstrate that each of these three dimensions of time (past, number exists with the property is also not known to be non-contradictory. future, present) exists all at once at different stages. That is, the proof treats a Verification for '0123456789 in :rt': (1) For each number, n, we can check free choice sequence as having at the same time many different (but related) whether it is the first place in the decimal expansion of where the series pasts, many different (but related) futures, etc. It is as though the free choice 0123456789 occurs (or not), simply by expanding out to the n+9th place. (2) sequence establishes a notion of past per se, future per se, without having a We have no way of calculating the first place where that series occurs, except wholly specific past or future. Phenomenologist van Atten (1999) argues that by calculating the successive digits of .n:. (3) We have no reason.to believe that Brouwer is committed philosophically to the identification of a choice the series never occurs in .n:. Thus, this property is fleeing. (4) We have no sequence with its specific moment of origin, but though possibly true in prin- reason to believe that the series does occur. Thus, this property is also opaque.: ciple, this temporal localisation seems to be nearly irrelevant in practice. 9 In one sense, Brouwer's use of .n: ties intuitionist mathematics to time and 14 As this essay offers primarily two examples, generalisations about the surd space. The math becomes spatial inasmuch as ,n; is a geometric quantity, a are relegated here to a footnote. In general, the surd can be recognised by the relationship among (abstract or ideal) spatial phenomena. More significantly, following phenomena: (1) a lack of official sanction or recognition, (2) a though, the math becomes temporal to the extent that .1t is treated in intu- pressure to formalise one's results, (3) the conflation of theory and practice itionism as an object to be discovered, a process for generating digits but a (or of epistemology and practical knowledge), (4) an emphasis on the con- process that necessarily takes its time. tingent and contextual as opposed to the general case (nothing can be taken lO Brouwer seems mildly concerned about formalising the opaque fleeing for granted, each result must be thought each time), (5) an insistence on property, as though he suspects that he may be losing the surd so crucial to thought as an activity and not just an arrival or end, (6) labour and genius the radical consequences of his calculus. On occasion, he laments the fact that side-by-side, (7) interdisciplinarity, (8) rediscovery of the simplest ideas as his proofs (such as those involving the pendulum number) depend on the now problematic and complex again, (9) a suspicion ofthe abstract and a faith existence of unsolved mathematical problems (such as the existence of . in the immediacy of experience, (10) activity or motion in the objects under 0123456789 in .n:). In general, Brouwer is somewhat torn between his consideration, (11) a willingness to let objects have their way, to work with commitment to a constructivist mathematics, in which the math is empirical, them rather than to dominate them, (12) a sense of precariousness, areal risk and an adherence to the traditional epistemology of mathematics, in which that things might not work out, that they are still up in the air or in develop- math is universal and atemporal. He doesn't desire that the conclusions of ment, and (13) maximum libidinal investment. , intuitionism actually change over time, but neither does he want to allow a 15 Fourier analysis includes a large family of mathematical techniques. One universalisability. primary distinction is between the Fourier transform, in which a signal is 232 233 THE SURD represented as a continuous funct ion of its component frequencies , and a Fourier series , in which a signal is represented as the sum of (an infinite 12 number of) discrete sinusoidal components. While the Gibbs phenomenon Oeleuze in phase space does apply to the Fourier transform, the discussion here refers chiefly to the Fourier series . 16 This is not just a hypothetical example. Engineers routinely deal with dis- continuities in signals, as the square wave, whose edges are discontinuities, Manu el Delanda is a frequent basis for signal construction. Other discontinuities occur when, for example, a signal is ' brick wall' filtered , or when a noise gate is applied that suddenly shuts off the signal when it falls below a certain level of ampli- tude. The Gibbs phenomenon is a gen uine hurdle for engineers , whose audio and stereo component designs include attempts to combat it. The semantic view of theories makes langu age largely irrelevant to the subject. Of course, to present a theory, we must present it in and by language. That is a trivial point. ... In addition, both because of our own history - the history of philosophy of science which became intensely language-oriented during the first half of [the last] century - and because of its intrinsic importance, we cannot ignore the language of sc ience. But in discussion of the structure of theories it can largely be ignored. (Van Fraassen 1989, 222) Van Fraassen is perhaps the most important representative of the empiri- cist tradition in contemporary analytical philosophy. But why use a quotation from an analytical philosopher, however famous , to begin a discussion of the work of an author who many regard as a member of the rival continental school of philosophy? The answer is that Gilles Deleuze does not belong to that school, at least if the latter is defined not geo- graphically but in terms of its dominant traditions (Kantian and Hegelian). As is well known, Deleuze himself arg ued for the superiority, in some respects, of Anglo-American, or empiricist, philosophy (Deleuze and Parnet 2002, ch. 2) . In addition, Deleuze's work was in large part a sustained critique of language (or more generally, of repre- sentation) as the master key to philosophical thought and, as the opening quotation attests, van Fraassen is also a leader of the emerging faction of phjlosophers of science disillusioned with the linguistic approach. There are, then, several points of convergence between the two authors but also rnany divergences. This essay will explore both. Let us first of all clarify van Fraas sen's position. What does it mean to say that in discussing the structure of scientific theories the language in which they are expressed is irrelevant? Or to put it differently, in what approach towards the nature of scientific theories is language itself 234