by Daniel Smith
Modern mathematics also leaves us in a state of antinomy, since the strict finite interpretation that it gives of the calculus nevertheless presupposes an axiom of infinity in the set theoretical foundation, even though this axiom finds no illustration in the calculus. What is still missing is the extra-propositional and sub-representative element expressed in the Idea by the differential, precisely in the form of a problem. (DR 178)
There are several reasons why Deleuze would refuse Badiou's identification of ontology with axiomatized set theory and maintain the ontological irreducibility of problematics. Most obviously, Badiou's ontology presumes the eventual reduction of physics (and the other sciences) to mathematics, which at present is itself no less a matter of faith than the eighteenth-century belief in the ghosts of infinitesimals. Freeman Dyson, to give one example among many, has strongly questioned this reductionistic presumption, predicting that “the notion of a final statement of the laws of physics [in a finite set of mathematical equations] will prove as illusory as the notion of a final decision process for all of mathematics.”41 More importantly, within mathematics itself, there are notions that remain outside the grasp of the discretization program—most notably the geometric continuum itself, the non-discrete “continuous continuum,” which still maintains its problematic status. “According to this intuitive concept,” mused Gödel, “summing up all the points, we still do not get the line; rather the points form some kind of scaffold on the line.”42 Or as Hermann Weyl put it,
in spite of Dedekind, Cantor, and Weierstrass, the great task which has been facing us since the Pythagorean discovery of the irrationals remains today as unfinished as ever; that is, the continuity given to us immediately by intuition (in the flow of time and in motion) has yet to be grasped mathematically.43
(The term “continuum” is still used to denote both types of continuity—the continuous geometric continuum and the discrete arithmetic continuum—even though the two notions differ in kind.) In a seminar, Deleuze noted that “the idea that there is a quantitative becoming, the idea of the limit of this becoming, the idea that an infinity of small quantities tends toward the limit—all these were considered as absolutely impure notions, as non-axiomatic or non-axiomatizable” (29 Apr 1980). One of the aims of Deleuze's own theory of multiplicities is to assess the status of such notions as problematic.
A more recent example can help serve to illustrate the ongoing tension between problematics and axiomatics within contemporary mathematics. Even after Weierstrass's work, mathematicians using the calculus continued to obtain accurate results and make new discoveries by using infinitesimals in their reasoning, their mathematical conscience assuaged by the (often unchecked) supposition that infinitesimals could be replaced by Weierstrassian methods. Despite its supposed “elimination” as an impure and muddled metaphysical concept, the ghostly concept of infinitesimals continued to play a positive role in mathematics as a problematic concept, reliably producing correct solutions.
Even now [wrote Abraham Robinson in 1966], there are many classical results in differential geometry which have never been established in any other way [than through the use of infinitesimals], the assumption being that somehow the rigorous but less intuitive ε, δ method would lead to the same result.44
In response to this situation, Robinson developed his non-standard analysis, which proposed an axiomatization of infinitesimals themselves, at last granting mathematicians the “right” to use them in proofs. Using the theory of formal languages, he added to the ordinary theory of numbers a new symbol (which we can call i for infinitesimal), and posited axioms saying that i was smaller than any finite number 1/n and yet not zero; he then showed that this enriched theory of numbers is consistent, assuming the consistency of the ordinary theory of numbers. The resulting mathematical model is described as “non-standard” in that it contains, in addition to the “standard” finite and transfinite numbers, non-standard numbers such as hyperreals and infinitesimals.45 In the non-standard model, there is a cluster of infinitesimals around every real number r, which Robinson, in a nod to Leibniz, termed a “monad” (the monad is the “infinitesimal neighborhood” of r). Transfinites and infinitesimals are two types of infinite number, which characterize degrees of infinity in different fashions. In effect, this means that contemporary mathematics has “two distinct rigorous formulations of the calculus”: that of Weierstrass and Cantor, who eliminated infinitesimals, and that of Robinson, who rehabilitated and legitimized them.46 Both these endeavors, however, had their genesis in the imposition of the notion of infinitesimals as a problematic concept, which in turn gave rise to differing but related axiomatizations. Deleuze's claim is that the ontology of mathematics is poorly understood if it does not take into account the specificity and irreducibility of problematics.
With these examples in hand, we can make several summary points concerning the relation between the problematic and axiomatic poles of mathematics, or more broadly, the relation between minor and major science. First, according to Deleuze, mathematics is constantly producing notions that have an objectively problematic status; the role of axiomatics (or its precursors) is to codify and solidify these problematic notions, providing them with a theorematic ground or rigorous foundation. Axiomaticians, one might say, are the “law and order” types in mathematics: “Hilbert and de Broglie were as much politicians as scientists: they reestablished order” (TP 144). As Albert Lautman noted, “irrational numbers, the infinitely small, continuous functions without derivatives, the transcendence of e and of π, the transfinite had all been accepted by an incomprehensible necessity of fact before there was a deductive theory of them.”47 In this sense, axiomatics is a foundational but secondary enterprise in mathematics, dependent for its very existence on problematics. As Jean Dieudonné suggests,
In periods of expansion, when new notions are introduced, it is often very difficult to exactly delimit the conditions of their deployment, and one must admit that one can only reasonably do so once one has acquired a rather long practice in these notions, which necessitates a more or less extended period of cultivation [défrichement], during which incertitude and controversy dominates. Once the heroic age of pioneers passes, the following generation can then codify their work, getting rid of the superfluous, solidifying the bases—in short, putting the house in order. At this moment, the axiomatic method reigns anew, until the next overturning [bouleversement] that brings a new idea.48
Nicholas Bourbaki puts the point even more strongly, noting that “the axiomatic method is nothing but the ‘Taylor System’—the ‘scientific management’—of mathematics.”49 Deleuze adopts a similar historical thesis, noting that the push toward axiomatics at the end of the nineteenth century arose at the same time that Taylorism arose in capitalism: axiomatics does for mathematics what Taylorism does for “work.”50
Second, problematic concepts often (though not always) have their source in what Deleuze terms the “ambulatory” sciences, which includes sciences such as metallurgy, surveying, stonecutting, and perspective. (One need only think of the mathematical problems encountered by Archimedes in his work on military installations, Desargues on the techniques of perspective, Monge on the transportation of earth, and so on.) The nature of such domains, however, is that they do not allow science to assume an autonomous power. The reason, according to Deleuze, is that the ambulatory sciences
subordinate all their operations to the sensible conditions of intuition and construction—following the flow of matter, drawing and linking up smooth space. Everything is situated in the objective zone of fluctuation that is coextensive with reality itself. However refined or rigorous, “approximate knowledge” is still dependent upon sensitive and sensible evaluations that pose more problems than they solve: problematics is still its only mode. (TP 373)
Such sciences are linked to notions—such as heterogeneity, dynamism, continuous variation, flows, etc.—that are “barred” or banned from the requirements of axiomatics, and conseq
uently they tend to appear in history as that which was superseded or left behind. By contrast, what is proper to royal science, to its theorematic or axiomatic power, is “to isolate all operations from the conditions of intuition, making them true intrinsic concepts, or ‘categories’ … Without this categorical, apodictic apparatus, the differential operations would be constrained to follow the evolution of a phenomenon” (TP 373–4). In the ontological field of interaction between minor and major science, in other words,
the ambulant sciences confine themselves to inventing problems whose solution is tied to a whole set of collective, nonscientific activities but whose scientific solution depends, on the contrary, on royal science and the way it has transformed the problem by introducing it into its theorematic apparatus and its organization of work. This is somewhat like intuition and intelligence in Bergson, where only intelligence has the scientific means to solve formally the problems posed by intuition. (TP 374)
Third, what is crucial in the interaction between the two poles is thus the processes of translation that take place between them—for instance, in Descartes and Fermat, an algebraic translation of the geometrical; in Weierstrass, a static translation of the dynamic; in Dedekind, a discrete translation of the continuous. The “richness and necessity of translations,” writes Deleuze, “include as many opportunities for openings as risks of closure or stoppage” (TP 486). In general, Deleuze's work in mathematical “epistemology” tends to focus on the reduction of the problematic to the axiomatic, the intensive to the extensive, the continuous to the discrete, the non-metric to the metric, the non-denumerable to the denumerable, the rhizomatic to the arborescent, the smooth to the striated. Not all these reductions, to be sure, are equivalent, and Deleuze analyzes each on its own account. Deleuze himself highlights two of them. The first is “the complexity of the means by which one translates intensities into extensive quantities, or more generally, multiplicities of distance into systems of magnitudes that measure and striate them (the role of logarithms in this connection)”; the second, “the delicacy and complexity of the means by which Riemannian patches of smooth space receive a Euclidean conjunction (the role of the parallelism of vectors in striating the infinitesimal)” (TP 486). At times, Deleuze suggests, axiomatics can possess a deliberate will to halt problematics. “State science retains of nomad science only what it can appropriate; it turns the rest into a set of strictly limited formulas without any real scientific status, or else simply represses and bans it.”51 But despite its best efforts, axiomatics can never have done with problematics, which maintains its own ontological status and rigor.
Minor science is continually enriching major science, communicating its intuitions to it, its way of proceeding, its itinerancy, its sense of and taste for matter, singularity, variation, intuitionist geometry and the numbering number … Major science has a perpetual need for the inspiration of the minor; but the minor would be nothing if it did not confront and conform to the highest scientific requirements. (TP 485–6)
In Deleuzian terms, one might say that while “progress” can be made at the level of theorematics and axiomatics, all “becoming” occurs at the level of problematics.
Fourth, this means that axiomatics, no less than problematics, is itself an inventive and creative activity. One might be tempted to follow Poincaré in identifying problematics as a “method of discovery” (Riemann) and axiomatics as a “method of demonstration” (Weierstrass).52 But just as problematics has its own modes of formalization and deduction, so axiomatics has its own modes of intuition and discovery (axioms are not chosen arbitrarily, for instance, but in accordance with specific problems and intuitions).53
In science an axiomatic is not at all a transcendent, autonomous, and decision-making power opposed to experimentation and intuition. On the one hand, it has its own gropings in the dark, experimentations, modes of intuition. Axioms being independent of each other, can they be added, and up to what point (a saturated system)? Can they be withdrawn (a “weakened” system)? On the other hand, it is of the nature of axiomatics to come up against so-called undecidable propositions, to confront necessarily higher powers that it cannot master. Finally, axiomatics does not constitute the cutting edge of science; it is much more a stopping point, a reordering that prevents decoded flows in physics and mathematics [= problematics] from escaping in all directions. The great axiomaticians are the men of State within science, who seal off the lines of flight that are so frequent in mathematics, who would impose a new nexum, if only a temporary one, and who lay down the official policies of science. They are the heirs of the theorematic conception of geometry. (TP 461)
For all these reasons, problematics is, by its very nature, “a kind of science, or treatment of science, that seems very difficult to classify, whose history is even difficult to follow.”54 None the less, according to Deleuze, the recognition of the irreducibility of problems and their genetic role has become “one of the most original characteristics of modern epistemology,” as exemplified in the otherwise diverse work of thinkers such as Canguilhem, Bouligand, Vuillemin, and Lautman.55 Beyond its significance in the interpretation of mathematics, problematics plays a significant role in Deleuze's theory of Ideas as well as his ontology (“Being” necessarily presents itself under a problematic form, and problems themselves are ontological). In all these domains, Deleuze's theory of problematics is extended in a theory of multiplicities, and it is to the nature of such multiplicities that we now turn.
DELEUZE'S THEORY OF MULTIPLICITIES
One of Badiou's most insistent claims is that Deleuze's theory of multiplicities is drawn from a “vitalist” paradigm, and not a mathematical one. The primary point I would like to establish in what follows is that, contra Badiou, Deleuze's theory is in fact drawn exclusively from mathematics—but from its problematic pole. Badiou at least admits that Deleuze's conception of multiplicities is derived in part from the differential calculus, but he concedes this point only to complain that Deleuze's “experimental construction of multiplicities is anachronistic because it is pre-Cantorian.”56 Cantor's set theory, however, represents the crowning moment of the tendency toward “discretization” in mathematics (the conception of sets as purely extensional), whereas Deleuze's project, as we have seen, is to formalize the conception of multiplicities that corresponds to the problematic pole of mathematics. In other words, problematics, no less than axiomatics, is the object of pure mathematics. Abel, Galois, Riemann, and Poincaré are among the great names in the history of problematics, just as Weierstrass, Dedekind, and Cantor are the great names in the discretization program, and Hilbert, Zermelo-Fraenkel, Gödel, and Cohen the great names in the movement toward formalization and axiomatization. Deleuze is fully aware of the apparent “anachronism” involved in delving into the pre-Weierstrassian theories of the calculus (Maimon, Bordas-Demoulin, Wronski, Lagrange, Carnot …). “A great deal of truly philosophical naiveté is needed to take the symbol dx seriously,” he admits, while none the less maintaining that “there is a treasure buried in the old so-called barbaric or prescientific interpretations of the differential calculus, which must be separated from its infinitesimal matrix” (DR 170). The reason Deleuze focuses on role of the differential (dx), however, is twofold. On the one hand, in the calculus, the differential is by nature problematic; it constitutes “the internal character of the problem as such,” which is precisely why it must disappear in the result or solution.57 On the other hand, whereas Plato used geometry as a model for his conception of transcendent “Ideas” because he saw the latter as unchanging theorematic forms, Deleuze uses the calculus as a model for his conception of immanent Ideas because the differential provides him with a mathematical symbolism of the problematic form of pure change (Bergson had already spoken of the differential or “fluxion” as a mean of capturing, via mathematics, a vision of the élan vital).58 Deleuze will thus make a strong distinction between “differential relations” and “axiomatic relations” (29 April 1980). Even in Diff
erence and Repetition, however, the calculus is only one of several mathematical domains that Deleuze utilizes in formulating his theory of multiplicities: “We cannot suppose that differential calculus is the only mathematical expression of problems as such … More recently, other procedures have fulfilled this role better.”59 What is at issue, in other words, is neither the empirical or intuitive origin of mathematical problems (e.g., in the ambulatory sciences) nor the historical moment of their mathematical formalization (pre- or post-Cantorian). “While it is true that the [continuous] continuum must be related to Ideas and to their problematic use,” Deleuze writes, “this is on condition that it no longer be defined by characteristics borrowed from sensible or even geometrical intuition” (DR 171). What Deleuze finds in pure mathematics is a rigorous conception of the constitution of problems as such, divorced not only from the conditions of intuition, but also from the conditions of their solvability. It is on the basis of this formalization that Deleuze, in turn, will be able to assign a precise status to mathematical notions such as continuous variation and becoming—which can only be comprehended under the mode of problematics. Space precludes a more detailed analysis of Deleuze's theory of multiplicities here; for our purposes, I would simply like to highlight three mathematical domains that have formalized the theory of the problem, and which Deleuze utilizes in formulating his own conception of multiplicities as problematic.60
1. The first domain is the theory of groups, which initially arose from questions concerning the solvability of certain algebraic (rather than differential) equations. There are two kinds of solutions to algebraic equations: particular and general. Whereas a particular solution is given by numerical values (x2 + 3x – 4 = 0 has as its solution x = 1), a general solution provides the global pattern of all particular solutions to an algebraic equation (the above equation, generalized as x2 + ax – b = 0, has the solution x = √a2/2 + b – a/2). But such solutions, writes Deleuze, “whether general or particular, find their sense only in the subjacent problem which inspires them” (DR 162). By the sixteenth century, it had been proved (Tataglia-Cardan) that general solvability was possible with squared, cubic, and quartic equations. But equations raised to the fifth power and higher refused to yield to the previous method (via radicals), and the puzzle of the “quintic” remained unresolved for more than two centuries, until the work of Lagrange, Abel, and Galois in the nineteenth century. In 1824, Abel proved the startling result that the quintic was in fact unsolvable, but the method he used was as important as the result. Abel recognized that there was a pattern to the solutions of the first four cases, and that it was this pattern that held the key to understanding the recalcitrance of the fifth. Abel showed that the question of “solvability” had to be determined internally by the intrinsic conditions of the problem itself, which then progressively specifies its own “fields” of solvability.
