by Daniel Smith
Building on Abel's work, Évariste Galois developed a way to approach the study of this pattern, using the technique now known as group theory. Put simply, Galois “showed that equations that can be solved by a formula must have groups of a particular type, and that the quintic had the wrong sort of group.”61 The “group” of an equation captures the conditions of the problem; on the basis of certain substitutions within the group, solutions can be shown to be indistinguishable in so far as the validity of the equation is concerned.62 In particular, Deleuze emphasizes the fundamental procedure of adjunction in Galois:
Starting from a basic “field” R, successive adjunctions to this field (R1, R2, R3…) allow a progressively more precise distinction of the roots of an equation, by the progressive limitation of possible substitutions. There is thus a succession of “partial resolvants” or an embedding of “groups” which make the solution follow from the very conditions of the problem. (DR 180)
In other words, the group of an equation does not tell us what we know about its roots, but rather, as Georges Verriest remarks, “the objectivity of what we do not know about them.”63 As Galois himself wrote, “in these two memoirs, and especially in the second, one often finds the formula, I don't know ….”64 This non-knowledge is not a negative or an insufficiency, but rather a rule or something to be learned that corresponds to an objective dimension of the problem. What Deleuze finds in Abel and Galois, following the exemplary analyses of Jules Vuillemin in his Philosophy of Algebra, is “a radical reversal of the problem–solution relation, a more considerable revolution than the Copernican.”65 In a sense, one could say that “unsolvability” plays a role in problematics similar to that played by “undecidability” in axiomatics.
2. The second domain Deleuze utilizes is the calculus itself, and on this score Deleuze's analyses are based to a large extent on the interpretation proposed by Albert Lautman in his Essay on the Notions of Structure and Existence in Mathematics.66 Lautman's work is based on the idea of a fundamental difference in kind between a problem and its solution, a distinction that is attested to by the existence of problems without solution. Leibniz, Deleuze notes, “had already shown that the calculus … expressed problems that could not hitherto be solved, or indeed, even posed” (DR 177). In turn Lautman establishes a link between the theory of differential equations and the theory of singularities, since it was the latter that provided the key to understanding the nature of non-linear differential equations, which could not be solved because their series diverged. As determined by the equation, singular points are distinguished from the ordinary points of a curve: the singularities mark the points where the curve changes direction (inflections, cusps, etc.), and thus can be used to distinguish between different types of curves. In the late 1800s, Henri Poincaré, using a simple non-linear equation, was able to identify four types of singular points that corresponded to the equation (foci, saddle points, knots, and centers) and to demonstrate the topological behavior of the solutions in the neighborhood of such points (the integral curves).67 On the basis of Poincaré’s work, Lautman was able to specify the nature of the difference in kind between problems and solutions. The conditions of the problem posed by the equation are determined by the existence and distribution of singular points in a differentiated topological field (a field of vectors), where each singularity is inseparable from a zone of objective indetermination (the ordinary points that surround it). In turn, the solution to the equation will only appear with the integral curves that are constituted the neighborhood of these singularities, which mark the beginnings of the differenciation (or actualization) of the problematic field. In this way, the ontological status of the problem as such is detached from its solutions; in itself, the problem is a multiplicity of singularities, a nested field of directional vectors which define the “virtual” trajectories of the curves in the solution, not all of which can be actualized. Non-linear equations can thus be used to model objectively problematic (or indeterminate) physical systems, such as the weather (Lorenz); the equations can define the virtual “attractors” of the system (the intrinsic singularities toward which the trajectories will tend in the long term), but they cannot say in advance which trajectory will be actualized (the equation cannot be solved), making accurate prediction impossible. A problem, in other words, has an objectively determined structure (virtuality), apart from its solutions (actuality).68
3. But “there is no revolution” in the problem–solution reversal, continues Deleuze,
as long as we remain tied to Euclidean geometry: we must move to a geometry of sufficient reason, a Riemannian-type differential geometry which tends to give rise to discontinuity on the basis of continuity, or to ground solutions in the conditions of the problems. (DR 162)
This leads to Deleuze's third mathematical resource, the differential geometry of Gauss and Riemann. Gauss had realized that the utilization of the differential calculus allowed for the study of curves and surfaces in a purely intrinsic and “local” manner: that is, without any reference to a “global” embedding space (such as the Cartesian coordinates of analytic geometry).69 Riemann's achievement, in turn, was to have used Gauss's differential geometry to launch a reconsideration of the entire approach to the study of space by analyzing the general problem of n-dimensional curved surfaces. He developed a non-Euclidean geometry (showing that Euclid's axioms were not self-evident truths) of a multi-dimensional, non-metric, and nonintuitable “any-space-whatever,” which he termed a pure “multiplicity” or “manifold” [Mannigfaltigkeit]. He began by defining the distance between two points whose corresponding coordinates differ only by infinitesimal amounts, and defined the curvature of the multiplicity in terms of the accumulation of neighborhoods, which alone determine its connections.70 For our purposes, the two important features of a Riemannian manifold are its variable number of dimensions (its n-dimensionality), and the absence of any supplementary dimension which would impose on it extrinsically defined coordinates or unity.71 As Deleuze writes, a Riemannian multiplicity is
an n-dimensional, continuous, defined multiplicity … By dimensions, we mean the variables or coordinates upon which a phenomenon depends; by continuity, we mean the set of [differential] relations between changes in these variables—for example, a quadratic form of the differentials of the co-ordinates; by definition, we mean the elements reciprocally determined by these relations, elements which cannot change unless the multiplicity changes its order and its metric. (DR 182)
In his critique of Deleuze, Badiou suggests not only that a Riemannian manifold entails “a neutralization of difference” (whereas Riemannian space is defined differentially) and a “preliminary figure of the One” (whereas Riemannian space has no preliminary unity), but also that it finds the “subjacent ontology of its invention” in set theory (whereas its invention is tied to problematics and the use of infinitesimals). What Badiou's comments reflect, rather, is the inevitable effort of “major” science to translate an intrinsic manifold into the discrete terms of an extensive set (though, as Abraham Robinson noted, it is by no means clear that results obtained in differential geometry using infinitesimals are automatically obtainable using Weierstrassian methods).72
In Difference and Repetition, Deleuze draws upon all these resources to develop his general theory of problematic or differential multiplicities, whose formalizable conditions can be briefly summarized as follows.
1. The elements of the multiplicity are merely “determinable”; their nature is not determined in advance by either a defining property or an axiom (e.g., extensionality). Rather, they are pure virtualities that have neither identity, nor sensible form, nor conceptual signification, nor assignable function (principle of determinability).
2. They are none the less determined reciprocally as singularities in the differential relation, a “non-localizable ideal connection” that provides a purely intrinsic definition of the multiplicity as “problematic”; the differential relation is not only external to its terms
, but constitutive of its terms (principle of reciprocal determination).
3. The values of these relations define the complete determination of the problem: that is, “the existence, the number, and the distribution of the determinant points that precisely provide its conditions” as a problem (principle of complete determination).73
These three aspects of sufficient reason, finally, find their unity in the temporal principle of progressive determination, through which, as we have seen in the work of Abel and Galois, the problem is resolved (adjunction, etc.) (DR 210). The strength of Deleuze's project, with regard to problematics, is that, in a certain sense, it parallels the movement toward “rigor” that was made in axiomatics; it presents a formalization of the theory of problems, freed from the conditions of geometric intuition and solvability, and existing only in pure thought (even though Deleuze presents his theory in a purely philosophical manner, and explicitly refuses to assign a scientific status to his conclusions).74 In undertaking this project, he had few philosophical precursors (Lautman, Vuillemin), and the degree to which he succeeded in the effort no doubt remains an open question. Manuel De Landa, in a recent work, has proposed several refinements in Deleuze's formalization, drawn from contemporary science: certain types of singularities are now recognizable as “strange attractors”; the resolution of a problematic field (the movement from the virtual to the actual) can now be described in terms of a series of spatio-temporal “symmetry-breaking cascades,” and so on.75 But as De Landa insists, despite his own modifications to Deleuze's theory, Deleuze himself “should get the credit for having adequately posed the problem” of problematics.76
DELEUZE AND BADIOU
Equipped now with a more adequate understanding of Deleuze's conception of problematics, we can now return to Badiou's critique and see why neither of his two main theses concerning Deleuze articulates the real nature of their fundamental differences. Badiou's thesis that Deleuze is a philosopher of the One is the least persuasive, for several reasons. First, Badiou derives this thesis from Deleuze's concept of univocity, proposing the equation “univocity = the One.” But already in Scotus, the doctrine of the “univocity of Being” was strictly incompatible with (and in part directed against) a Neo-Platonic “philosophy of the One.” Moreover, Deleuze's explicit (and repeated) thesis in Difference and Repetition is that the only condition under which the term “Being” can be said in a single and univocal sense is if Being is said univocally of difference as such (i.e., “Being is univocal” = “Being is difference”).77 To argue, as Badiou does, that Deleuze's work operates “on the basis of an ontological precomprehension of Being as One” is in effect to argue that Deleuze rejects the doctrine of univocity.78 In other words, “Being is univocal” and “Being is One” are strictly incompatible theses, and Badiou's conflation of the two, as has been noted by several commentators, betrays a fundamental misunderstanding of the theory of univocity.79 Second, while it is none the less true that Deleuze proposed a concept of the One compatible with univocity (e.g., the “One-All” of the plane of immanence as a secant plane cut out of chaos; see WP 35, 202–3), Badiou seems unable to articulate it in part because of the inconsistency of his own conception of the One, which is variously assimilated to the Neo-Platonic One, the Christian God, Spinoza's Substance, Leibniz's Continuity, Kant's unconditioned Whole, Nietzsche's Eternal Return, Bergson's élan vital, a generalized conception of Unity, and Deleuze's Virtual, to name a few.80 The reason for this conceptual fluidity seems clear: since the task of modern philosophy, for Badiou, is “the renunciation of the One,” and since for him only a set theoretical ontology is capable of fulfilling this task, the concept of the “One” effectively becomes little more than a marker in Badiou's writings for any non-set-theoretical ontology. But the fact that Augustine—to use a famous example—became a Christian (believer in God) by renouncing his Neo-Platonism (adherence to the One) is enough to show that these terms are not easily interchangeable, and that renouncing the One does not even entail a renunciation of God. Moreover, Kant had already showed that the idea of the “World” is a transcendent illusion. One can only speak of the “whole” of Being (“the totality of what is”) from the viewpoint of transcendence; it is precisely the “immanence” of the concept of Being (univocity) that prevents any conception of Being as a totality. Third, and most important, the notion of the One does not articulate the difference between Badiou and Deleuze even on the question of “an immanent conception of the multiple.” Extensive multiplicities (sets) and differential multiplicities (e.g., Riemannian manifolds) are both defined in a purely intrinsic or immanent manner, without any recourse to the One or the Whole or a Unity. The real differend must be located in the difference between axiomatics and problematics, major and minor science.
Badiou's thesis concerning Deleuze's “vitalism,” by contrast, comes closer to articulating a real difference. (Badiou recognizes, to be sure, that Deleuze uses this biological term in a somewhat provocative manner, divorced from its traditional reference to a semi-mystical life-force.) Although Deleuze's formal theory of multiplicities is drawn from mathematical models, it is true that he appeals to numerous non-mathematical domains in describing the intensive processes of individuation through which multiplicities are actualized (biology, but also physics and geology). “Vitalism” enters the picture, in other words, at the level of individuation—hence the distinction, in Difference and Repetition, between the fourth chapter on “The Ideal Synthesis of Difference” (the theory of multiplicities, which appeals to mathematics) and the fifth chapter on “The Asymmetrical Synthesis of the Sensible” (the theory of individuation, which appeals to biology). But this distinction is neither exclusive nor disciplinary. Even in mathematics, the movement from a problem to its solutions constitutes a process of actualization; though formally distinct, there is no ontological separation between these two instances (the complex Deleuzian notion of “differen t/c ation”). As Deleuze explains,
we tried to constitute a philosophical concept from the mathematical function of differentiation and the biological function of differenciation, in asking whether there was not a statable relation between these two concepts which could not appear at the level of their respective objects … Mathematics and biology appear here only in the guise of technical models which allow the exposition of the virtual [problematic multiplicities] and the process of actualization [biological individuation]. (DR xvi, 220–1)
Deleuze thus rejects Badiou's reduction of ontology to mathematics, and would no doubt have been sympathetic to Ernst Mayr's suggestion that biology might itself be seen as the highest science, capable of encompassing and synthesizing diverse developments in mathematics, physics, and chemistry.81
Badiou's resistance to this “vitalism” can be accounted for by his restricted conception of ontology. For Badiou, the term ontology refers uniquely to the discourse of “Being-as-being” (axiomatic set theory), which is indifferent to the question of existence. For Deleuze, by contrast, ontology encompasses Being, beings, and their ontological difference (using Heideggerian language), and the determinations of “Being-as-such” must therefore be immediately related to beings in their existence. This is why the calculus functions as a powerful test case in comparing Deleuze and Badiou. The calculus has been rightly described as the most powerful instrument ever invented for the mathematical exploration of the physical universe. In its initial formulations, however, as we have seen, the calculus mobilized notions that were unjustified from the viewpoint of classical algebra or arithmetic; it was a fiction, as Leibniz said, irreducible to mathematical reality. From these origins, however, one can trace the history of the calculus along two vectors, so to speak: toward the establishment of its foundations, or toward its use in an ever-deepening exploration of existence. The movement toward rigor in mathematics, by “royal” science, was motivated by the attempt to establish a foundation for the concepts of the calculus internal to mathematics. Badiou situates his work exclusively on this path
, characterizing axiomatic set theory as “rational ontology itself.”82 Deleuze, by contrast, while stressing the foundational necessity of axiomatics, equally emphasizes the role of the calculus in the comprehension of existence.
Differential calculus [he writes], is a kind of union of mathematics and the existent—specifically, it is the symbolic of the existent. It is because it is a well-founded fiction in relation to mathematical truth that it is consequently a basic and real means of exploration of the reality of existence.83