by Daniel Smith
Third, to be a condition of real experience, the condition can be no broader than what it conditions—otherwise it would not be a condition of real experience, capable of accounting for the genesis of the real. It is for this reason that there can be no categories (at least in the Aristotelian or Kantian sense) in Deleuze's philosophy, since, as Deleuze puts it, the categories cast a net so wide that they let all the fish (the real) swim through it. But this requirement—that conditions not be broader than the conditioned—means that the conditions must be determined along with what they condition, and thus must change as the conditioned changes. In other words, the conditions themselves must be plastic and mobile, “no less capable of dissolving and destroying individuals than of constituting them temporarily.”9
Fourth, in order to remain faithful to these exigencies, Deleuze continues, “we must have something unconditioned” that would be capable of “determining both the condition and the conditioned” (LS 123, 122), and which alone would be capable of ensuring a real genesis.10 It is the nature of this unconditioned element that lies at the basis of Deleuze's dispute with the general movement of the post-Kantian tradition: is this unconditioned the “totality” (Kant, Hegel), which necessarily appeals to a principle of identity (the subject), or is it “differential” (which is Deleuze's position, modifying a position hinted at by Leibniz)? Indeed, Deleuze's appeal to the interrelated concepts of the foundation [fondation], the ground [fond, fondement], and the ungrounded [sans-fond] reflects his complex relation to the traditions of pre- and post-Kantianism. Both Spinoza and Leibniz, in their shared anti-Cartesian reaction, complained that Descartes had not gone far enough in his attempt to secure a foundation for knowledge; erecting a foundation is a futile enterprise if the ground itself is not firm and secure. Before laying the foundation, in other words, one must prepare the ground—that is, one must inquire into the sufficient reason of the foundation.11 Deleuze describes Difference and Repetition in its entirety as an inquiry into sufficient reason, but with this additional caveat: in following the path of sufficient reason, Deleuze argues, one always reaches a “bend” or “twist” in sufficient reason, which “relates what it grounds to that which is truly groundless,” the unconditioned (DR 154). It is like a catastrophe or earthquake that fundamentally alters the ground, and destroys the foundations that are set in it. All three of these aspects—foundation, ground, and the ungrounded—are essential to Deleuze's project.
Sufficient reason or the ground [he writes] is strangely bent: on the one hand, it leans towards what it grounds, towards the forms of representation; on the other hand, it turns and plunges into a groundlessness beyond the ground which resists all forms and cannot be represented. (DR 274–5)
In Deleuze's theory of repetition (temporal synthesis), for instance, the present plays the role of the foundation, the pure past is the ground, but the future the ungrounded or unconditioned: that is, the condition of the new.
Finally, the nature of the genesis that is at play here must therefore be understood as what Deleuze calls a static genesis (a genesis that takes place between the virtual and its actualization), and not a dynamic genesis (a historical or developmental genesis that takes place between actual terms, moving from one actual term to another).
These five themes recur in almost all of Deleuze's early writings as elaborations of the two post-Kantian demands that Deleuze appropriates from Maimon: the search for the genetic elements of real experience and the positing of a principle of difference as the fulfillment of this demand.
THE MODEL OF THE CALCULUS
However, it is one thing to lay out a general project like this; it is another thing to find a “method” (to use a term Deleuze disliked) capable of providing a way of thinking the conditions of the real that would fulfill these requirements. If logical principles determine the conditions of the possible, and the categories determine the conditions of possible experience, where can one go to search for the conditions of real experience (that is, the conditions for novelty itself)? Deleuze in fact appeals to several non-philosophical models in his work. One of them is artistic creation, and in a sense Deleuze's transcendental empiricism can be read, in large part, as a reworking of Kant's transcendental aesthetic.12 Another model is molecular biology, which defines individuals in terms of a genetic structure that constitutes the real conditions of its external and visible properties, and thus constitutes a profound break with the traditional approach of “natural history” (DR 214–21). But the model I would like to focus on here is the mathematical model of the differential calculus. Many of the concepts that Deleuze develops in Difference and Repetition to define the conditions of the real—the differential relation, singularities, multiplicities or manifolds, the virtual, the problematic, and so on—are derived from the history of the calculus.
There are a number of reasons why Deleuze would turn to the model of calculus. Philosophy, of course, has always had a complex relationship with mathematics, but the particular branch of mathematics privileged by philosophers often says much about the nature of their philosophy. Since the late nineteenth and early twentieth centuries, for instance, philosophers have tended to focus on axiomatic set theory, since they were preoccupied with the question of the foundations of mathematics, with its twin programs of formalization and discretization. Plato, by contrast, famously appealed to Euclidean geometry as a model for Ideas because it defined forms or essences that were static, unchanging, and self-identical. Deleuze could be said to appeal to the calculus for the exact opposite reason: it is the calculus that provides him with a mathematical model of a principle of difference. The calculus is the primary mathematical tool we have at our disposal to explore the nature of reality, the nature of the real—the conditions of the real. When physicists want to examine the nature of a physical system, or engineers want to analyze the pressure on a weight-bearing load, they model the system using the symbolism of the calculus. What spawned the “scientific revolution” of the last three centuries was what Ian Stewart has called the differential equation paradigm: “the way to understand Nature is through differential equations.”13 Hermann Weyl wrote that “a law of nature is necessarily a differential equation” (FLB 47), and Bertrand Russell, perhaps even more strongly, claimed that “scientific laws can only be expressed in differential equations.”14 In this sense, one might say that the calculus is existentialism in mathematics, “a kind of union of mathematics and the existent.”15
This is why Leibniz—who invented the calculus, along with Newton—remains an important figure for Deleuze. In the history of philosophy, Deleuze suggests, there were two great attempts to elucidate the conditions of the real, albeit in two different directions: Hegel (the infinitely large) and Leibniz (the infinitely small).16 Deleuze's strategy, with regard to the history of philosophy, seems to have been to take up Maimon's critiques of Kant and to resolve them, not in the manner of the post-Kantians, such as Fichte and Hegel, but rather by following Maimon's own suggestions and returning to the pre-Kantian thought of Hume, Spinoza, and Leibniz. Of these three, it is Leibniz who plays a decisive role—at least with regard to the question of the real that concerns us here—since, in Deleuze's reading, he already had an implicit response to the two post-Kantian demands formulated by Maimon. “All the elements to create a genesis as demanded by the post-Kantians,” Deleuze noted in one of his seminars, “are virtually present in Leibniz” (20 May 1980). The calculus, to be sure, takes us into a complex and heavily mined territory, with its own intricate history. Moreover, the calculus is not the only mathematical domain to which Deleuze appeals: group theory, topology, and non-Euclidean geometry, among others, also make frequent appearances throughout Deleuze's texts. It is not that Deleuze is setting out to develop a philosophy of mathematics, nor even to construct a metaphysics of the calculus. Deleuze appeals to the calculus primarily to develop a philosophical concept of difference, to propose a concept of difference-in-itself. “We tried to constitute a philosophical concept from
the mathematical function of differentiation,” Deleuze writes in the preface to Difference and Repetition. “We are well aware, unfortunately, that we have spoken about science in a manner which was not scientific” (DR xvi, xxi). Starting with the differential relation, we can follow, in a rather schematic manner, a “deduction” of the concepts that Deleuze extracts from the calculus for his philosophical purposes. This analysis would constitute a segment of a broader consideration of Deleuze's philosophy of difference.17
THE LOGIC OF RELATIONS
Let me turn first to the nature of the differential relation. To understand the importance of this concept for Deleuze—and the way in which this type of relation differs from logical relations, or even from real and imaginary relations in mathematics—we can perhaps make a brief foray into the philosophical problem of relations in general. The problem of relations has tortured philosophy since its inception, and since the Greeks, the question of relations has been linked to the problem of judgment. The simplest form of judgment is the judgment of attribution, A is B (e.g., the sky is blue), although it was recognized early on that every judgment of attribution is a kind of offense against the principle of identity (14 Dec 1982). It is easy to understand the relation of identity, A is A (the sky is the sky, a thing is identical to itself), but how is it possible to say that A is B? Philosophers explained it by saying that, in a judgment of attribution, A and B are not the same thing: a judgment of attribution attributes a property (blue) to a subject (sky), or an attribute to a substance, and thus could be said to lie at the origin of every metaphysics of substance. A more complicated form of judgment, however, is a judgment of relation, such as A is smaller than B (Peter is smaller than Paul). In a judgment of relation, we can no longer say that we are attributing a property to a subject, because if I say that “being smaller than” is a property of A, I would also have to say, at the same time, that “being taller than” is also a property of A, since there is also a C that is smaller than A (“Peter is smaller than Paul, but taller than Mary”). Plato had already pointed out that this would entail attributing contradictory properties to the same subject (“being smaller than,” “being taller than”), which would seem to be an offense against the principle of non-contradiction, just as the judgment of attribution seems to be an offense against the principle of identity. (One could object by saying that the properties being predicated are not simply “smaller than” and “taller than,” but rather “smaller than Paul” and “taller than Mary,” but this does not solve the problem. Paul and Mary are themselves real beings, and while the concept of Peter may contain properties, it is not possible for the concept of Peter—a real being—to contain other real beings, and not simply properties.) A judgment of relation (“Peter is smaller than Paul”), in short, cannot be reduced to a judgment of attribution (“Peter has blue eyes”). When I say, “Peter is smaller than Paul,” this relation is neither a property of Peter, nor a property of Paul; rather, it is something between the two. But what is this “between the two”? “The foundation [fondement] of the relation cannot be found in either of the terms that it unites: the mystery seems unfathomable” (1 Mar 1983). Where, then, is it to be found? Philosophy has offered at least three responses to this question.
First, Plato's ingenious response was to say that relations are pure Ideas that go beyond the sensible world: there is an Idea of the Small, and an Idea of the Large. When we say that “A is smaller than B, and greater than C,” we are simply saying that A participates in the Idea of the Small in relation to term B, and that it participates in the Idea of the Large in relation to term C. But this simply restates the problem: relations are irreducible to the attributes or properties of a thing. Indeed, once one has discovered the world of relations, one might ask if, in the end, every judgment is not a judgment of relation—that is, if there are no properties at all, but only relations. Leibniz, second, took a quite different approach: he tried to show, at all costs, that every judgment of relation is reducible to a judgment of attribution, and he was willing to draw the necessary conclusion: every concept that designates a real being (Peter, Paul, Adam, Caesar) must contain the totality of all other concepts. Why? Because Peter is related to Paul, and more distantly to Caesar, and even more distantly to Adam, which amounts to saying that the concept of every real being necessarily expresses the totality of the world. Leibniz tried to save the judgment of attribution by claiming that relations are internal to their terms: if the concept of Peter contains all other terms, then all imaginable relations can be reduced to attributions—that is, to properties of the concept.18 The greatness of Hume, finally, is to have come along and argued the exact opposite: namely, that relations are external to their terms. “For me,” Deleuze said, “this proposition was like a clap of thunder in philosophy” (14 Dec 1982). Rather than invoking Ideas, or undertaking operations as complex as those of Leibniz, Hume's admonition is to accept exteriority. Prior to Hume, philosophers tended to support interiority; to comprehend something was to internalize it, whether in a concept or in one's head. Some sort of interiority was necessary. Hume arrives on the scene and says, “You are not seeing the world in which you live, which is a world of exteriority, an infinite patchwork of bits and pieces, irreducibly external to each other.”19 While properties are internal to the terms to which they are attributed, relations are exteriorities; one cannot give an account of relations by relating them to their terms. As Deleuze argued in his first book, on Hume, the fundamental thesis of empiricism is less that knowledge derives from sensible experience but, more profoundly, that relations are external to their terms (ES 98–9). Empiricism was thus faced with the formidable task of inventing a new logic, a logic of relations that breaks with the logic of attribution, which would begin with Hume's theory of probabilities and take on its definitive form in Bertrand Russell (PI 37–8). If the empiricists discovered the sensible world, what they discovered in it was an absolutely new formal logic of relations.
Deleuze will adopt Hume's empiricist stand on relations, but in developing his own transcendental empiricism, he will modify and develop it in several directions. One the one hand, he will link this analysis of relations with the concept of becoming (which does not assume its full importance as a concept in Deleuze's thought until A Thousand Plateaus).20 If relations are external to their terms, and do not depend on them, then the relations cannot change without one (or both) of the terms changing. A resembles B, Peter resembles Paul: this relation is external to its terms, it is contained neither in the concept of Peter nor in the concept of Paul. If A ceases to resemble B, the relation has changed, but this means that the concept of A (or B) has changed as well. If properties belong to something solid, relations are far more fragile, and are inseparable from a perpetual becoming. Hitchcock's entire cinema, to give a non-philosophical example, was modeled on the evolution of relations (the innocent taken to be guilty) and a varied play of conjunctions (because … although … since … if … even if …). In Mr. and Mrs. Smith, a minor comedy, Hitchcock asks, What happens to a couple who suddenly learn that their marriage was not legal, and thus that they have never been married (MI x, 202)? When a relation changes, what happens to its terms?
One cannot think relation [Deleuze noted in a seminar] independent of a becoming that is at least virtual, whatever the relation might be. In my opinion, the theorists of relation, however strong they might have been, have not seen this … A relation is not only external to its terms, but it is essentially transitive, in the sense of “transitory.” (14 Dec 1982)
On the other hand, Deleuze will also push the thesis of the exteriority of relations to its radical limit. If relations are external to their terms and do not depend on them, if relation is the domain of becoming (if every relation envelops or implies change), then one might say that, at the limit, or at a deeper level, there are not even terms, but only packets of variable relations. What we call a term in itself is only a packet of relations. In this sense, one could say that Deleuze is in the pr
ocess of dissolving the notion of substance into that of multiplicity. “In a multiplicity, what counts are not the terms or the elements, but what is ‘between’ them, the in-between, a set of relations that are inseparable from each other” (D viii). It is not unlike Virginia Woolf's dictum in literature: not to see things, but to see between things.21 But what, then, does it mean to speak of a “pure relation”: that is, a relation that is not only external to its terms, but a relation that persists even when its terms have disappeared?