by Daniel Smith
THE DIFFERENTIAL RELATION
If Deleuze appeals to the model of the differential relation, then, it is because it offers an initial mathematical response to this question. The differential relation was discovered in the seventeenth century by Leibniz and Newton, and can be distinguished from fractional relations, which had been known since antiquity, and algebraic relations, which had only received a rigorous status in Descartes's work. Already in fractions, there appears a kind of independence of the relation from its terms. The fraction 2⁄3, for example, is not a whole number, because there is no assignable number which, when multiplied by three, equals two. To be sure, we can decide, by convention, to treat fractions as numbers: that is, to subject them to the rules of addition, subtraction, and multiplication. Moreover, once we have the fractional symbolism at our disposal, we can treat numbers as if they were fractions: we can write 2 as 4⁄2 or 6⁄3. But in themselves, fractions are complexes of whole numbers, and as such, they provide a first approximation of a relation that is independent of its terms. None the less, this first approximation remains limited. In a fractional relation, the relation is still between two terms, and a determinate value must be assigned to the terms; that is, the terms must be given and specified (2 and 3). By contrast, in an algebraic relation, such as x2 + y2 – R2 = 0, a determinate value no longer needs to be assigned to the terms; the terms of the relation are variables. It is as if the algebraic relation acquires a higher degree of independence than the fractional relations, since the terms of the relation no longer need to be specified as such. However, even though I need not assign a determinate value to the terms, the variable must none the less have a determinable value. The variables x and y can have various singular values, but they must none the less have a value. In an algebraic relation, in other words, the relation is indeed independent of any particular value of the terms, but it is still not independent of the determinable value of the variable. The same is true in symbolic logic, in which the variable, though undetermined, must none the less sustain its identity throughout the arguments.
The differential relation constitutes the third step in this history of mathematical relations. In 1701, Leibniz wrote a short, three-page text entitled “Justification of the Infinitesimal Calculus” in which he illustrated the nature of the differential relation using an example from ordinary algebra.22 In the differential relation dx/dy, dy in relation to y is equal to zero, and dx in relation to x is equal to zero—they are infinitely small quantities. Thus it is possible to write, as was done frequently in the seventeenth century, that dx/dy = 0/0. Yet the relation 0/0 is not equal to zero; in the differential relation, the relation subsists even when the terms disappear. In this case, the terms between which the relation is established are neither determined, nor even determinable; the terms themselves have neither existence, nor value, nor signification (DI 176). The only thing that is determined is the reciprocal relation between the terms. The terms are reduced to vanishing terms, to vanishing quantities (or virtualities), yet the relation between these vanishing quantities is not equal to zero, but refers to a third term that has a finite value: dx/dy = z. Applied to a circle, for example, the differential relation dx/dy tells us something about a third thing, a trigonometric tangent. We can say that z is the limit of the differential relation, or that the differential relation tends toward a limit. When the terms of the relation disappear, the relation subsists because it tends toward a limit, z. When a relation is established between infinitely small terms, it is not annulled along with its terms, but tends toward a limit. This is the basis of the differential calculus as it was interpreted in the seventeenth century—an interpretation that was identical to the comprehension of an actual infinity. Weierstrass and Russell would eventually give the calculus a static and ordinal interpretation, which liberated the calculus from any reference to infinitesimals, and integrated it into a pure logic of relations.
This, then, is the importance Deleuze ascribes to the differential relation. It is a pure relation that provides an example of what Deleuze will call the concept of “difference-in-itself.” Difference is a relation, and normally—that is to say, empirically, it is a relation between two things that have a prior identity (“x is different from y”). Relations require prior relata, and differences require prior identities. With the notion of the differential relation, however, Deleuze takes the notion of difference to a properly transcendental level: that is, to a domain where relations no longer depend on their terms. This is what distinguishes the model of the calculus from axiomatic set theory: the latter establishes relations between non-specified elements (axiom of extensionality), whereas in the former the elements are reciprocally determined by the relations themselves. Not only is the differential relation external to its terms, and not only does the relation persist even when its terms have disappeared, but one could also say that the relation determines its terms. Difference here becomes constitutive of identity—that is, it becomes productive and genetic, thus fulfilling Maimon's demand: a genetic philosophy finding its ground in a principle of difference.
SINGULARITIES AND MULTIPLICITIES
In a certain sense, one could say that this principle of difference is the starting point of Deleuze's philosophy, from which he will deduce a number of related concepts that constitute the conditions of real experience.23 When a differential relation reciprocally determines two (or more) virtual elements, it produces what is called a singularity, a singular point. This is the first concept Deleuze deduces from the differential relation. In mathematics, the singular is not opposed to the university (as in logic), but rather to the ordinary or the regular; a singular point (or singularity) is distinguished from ordinary or regular points, particularly when speaking about points on a determinate figure. A square, for instance, has four singular points, its four corners or extrema, and an infinity of ordinary points that compose each side of the square (the calculus of extrema). Simple curves, like the arc of a circle, are determined by singularities that are no longer extrema, but maximum or minimum points (the calculus of maxima and minima). The singularities of complex curves are far more intricate: they constitute those points in the neighborhood of which the differential relation changes sign, and the curve bifurcates, and either increases or decreases (the differential calculus).
Such an assemblage of ordinary and singular points constitutes what Deleuze calls a multiplicity—a third concept. One could say of any determination in general—that is, of any individual—that it is a combination of the singular and the ordinary, of the remarkable and the regular. The singularities are precisely those points where something “happens” within the multiplicity (an event), or in relation to another multiplicity, causing it to change nature and produce something new. For instance, to take the example of a physical system, the water in my kettle is a multiplicity, and a singularity in the system occurs when the water boils or freezes, thereby changing the nature of the physical multiplicity (its phase space). Similarly, the point where a person breaks down in tears, or boils over in anger, is a singular point in their psychic multiplicity, surrounded by a swarm of ordinary points. Every determinate thing is a combination of the singular and the ordinary, a multiplicity that is constantly changing, in perpetual flux.
One can see that Deleuze is here breaking with a long tradition that defined things in terms of an essence or a substance—that is, in terms of an identity. Deleuze's philosophy of difference replaces the traditional concept of substance with the concept of multiplicity, and replaces the concept of essence with the concept of the event.24 The nature of a thing cannot be determined simply by the Socratic question “What is …?” (the question of essence), but only through such questions such as How?, Where?, When?, How many?, From what viewpoint?, and so on (questions concerning events).25 The question “What is singular and what is ordinary?” is one of the fundamental questions posed in Deleuze's ontology, since, in a general sense, one could say that “Everything is ordinary!” as much as one can say t
hat “Everything is singular!” (FLB 91). A new-found friend might unexpectedly erupt in anger at me, and I might wonder what I could have done to provoke such a singularity in his psychic multiplicity; but then someone might lean over to me and say, “Don't worry, he does this all the time. It's nothing singular, it has nothing to do with you, it's the most ordinary thing in the world. We're all used to it.” Assessing what is singular and what is ordinary in any given multiplicity is a complex task, which is why Nietzsche could characterize the philosopher as a kind of physician who assesses phenomena as if they were symptoms that reflected a deeper state of relations within the multiplicity at hand.
TWO EXAMPLES: LEIBNIZ AND SPINOZA
Curiously, although Leibniz was one of the great partisans of the thesis that relations are internal to their terms, his actual analyses—particularly in the New Essays—often tended to illustrate the opposite thesis. Like most great thinkers, he had one foot in the past (his theology) and one foot in the future (everything else). This is particularly true in Leibniz's theory of perception, which Deleuze often appeals to as an illustration of these notions. Leibniz had argued that the genesis of our conscious perceptions must be found in the minute and unconscious perceptions of which they are composed, and which my conscious perception “integrates.” For instance, I can apprehend the noise of the surf at a beach, or the buzz of a group of people at a party, but not necessarily the sound of each wave or the voice of each person that they include. A conscious perception is produced when at least two of these minute and virtual perceptions—two waves or two voices—enter into a differential relation (dx/dy) that determines a singularity, which “excels” over the others and becomes conscious. Every conscious perception constitutes a constantly shifting threshold; the multiplicity of minute and virtual perceptions are like the obscure dust of the world, its background noise, what Maimon liked to call the “differentials of consciousness,” and the differential relation is the psychic mechanism that extracts from this multiplicity my finite zone of clarity on the world. Leibniz thus divides the Cartesian appeal to the “clear and distinct” into two irreducible domains (DR 146, 213). My conscious perception of the noise of the sea, for example, is clear but confused (not distinct), since the minute perceptions of which it is composed remain indistinct. Conversely, the unconscious perceptions are themselves distinct but obscure (not clear): distinct, in so far as all the drops of water remain distinct as the genetic elements of perception, with their differential relations, the variations of these relations, and the singular points that they determine; but obscure, in so far as they are not yet “distinguished” or actualized in a conscious perception, and can only be apprehended by thought, or at best, in fleeting states close to those of drowsiness, or vertigo, or dizzy spells.
This is why Deleuze can say that the elements of a genetic philosophy demanded by Maimon were already present in Leibniz. Leibniz determines the conditions of real experience by starting with the obscure and the virtual (a multiplicity); a clear perception is actualized from the obscure by a genetic process (the differential mechanism). These minute perceptions do not indicate the presence of an infinite understanding in us—which was Kant's own criticism of Maimon (DR 192–3; FLB 89)—but rather the presence of an unconscious within finite thought—a differential unconscious that is quite different from the oppositional unconscious developed in Freud. As Deleuze writes,
Salomon Maimon—the first post-Kantian to return to Leibniz—drew all the consequences from this kind of psychic automatism of perception. Far from having perception presuppose an object capable of affecting us, and conditions under which we would be affectable [Kant], it is the reciprocal determination of differentials (dy/dx) that entails both the complete determination of the object as perception, and the determinability of space-time as a condition … The physical object and mathematical space both refer to a transcendental psychology (differential and genetic) of perception. Space-time ceases to be a pure given in order to become the totality or nexus of differential relations in the subject, and the object itself ceases to be an empirical given in order to become the product of these relations in conscious perception. (FLB 89)
This is what Deleuze means when he says that the conditions of real experience must be determined at the same time as what they are conditioning; space and time here are not the pre-given conditions of perception, but are themselves constituted in a plurality of spaces and times along with perception.
As a point of comparison, we might note that Deleuze also makes use of these same notions in a completely different seventeenth-century context: no longer Leibniz's theory of perception, but Spinoza's theory of individuation.26 Spinoza held that an individual is composed of an infinity of parts, which he called “simple bodies.” But what exactly counts as a simple body? Deleuze's thesis is that, in Spinoza, simple bodies are actually infinite. The actual infinite—which is one of the richest notions of the seventeenth century, at once metaphysical, mathematical, and physical—must be distinguished from both the finite and the indefinite. The formula of the finite says that, in any analysis, one reaches a term where the analysis ends—a term such as the “atom.” The formula of the indefinite says that, no matter how far one pushes the analysis, whatever term one arrives at can always be divided or analyzed further, indefinitely, ad infinitum—there is never a final or ultimate term. The formula of the actually infinite, however, is neither finite nor indefinite. On the one hand, it says that there are indeed ultimate or final terms that can no longer be divided—thus it is against the indefinite; but on the other hand, it says that these ultimate terms go to infinity—thus they are not atoms but rather terms that are “infinitely small,” or as Newton would say “vanishing terms.” The notion of actual infinity thus implies a double battle against finitism and against the indefinite.
Three problems posed by this distinction are indicative for our purposes. First, since the terms of an actual infinity are smaller than any given quantity, they can never be treated one by one; that is, they cannot be treated numerically. It would be nonsensical to speak of an infinitely small term that can be considered singularly. Rather, infinitely small terms can only exist in infinite collections. Spinoza's simple bodies, in other words, are in fact multiplicities: the simplest of bodies exists as infinite sets of infinitely small terms, which means that they exist collectively and not distributively.27 These types of multiplicities, however, have no parts: that is, they are intensive. Cantor's set theory, of course, would later rediscover the notion of actual infinity on a completely different basis, as extensive, and this raises an essential point for Deleuze. The theory of multiplicities, he argues, always entails two types of multiplicity (intensive and extensive, continuous and discrete, non-metric and metric, and so on), since what is important is precisely what takes place between the two: their constant transformations and becomings (WP 152). Second, even in Spinoza, these infinitely small terms have no interiority; they have strictly extrinsic relations of exteriority with each other. They form, in Spinoza's own language, a “modal” matter of pure exteriority. This points to a second feature of Deleuze's approach: although he constantly appeals to the “empiricist” dictum of the exteriority of relations, Deleuze winds up exploring this dictum most thoroughly in the “rationalist” thought of Spinoza and Leibniz (less because of their so-called rationalism, however, than because of their introduction of the problem of the infinite). Finally, Spinoza's claim that individuals are composed of an infinity of simple bodies, which are themselves infinite multiplicities, raises the question, How can we distinguish the simple bodies that belong to one individual from those that belong to another? Put differently, how can we distinguish one infinite collection from another infinite collection (as they cannot be distinguished by their parts, since infinite sets exceed any assignable number)? Here again, extensive and intensive multiplicities generated different responses: Cantor's theory of transfinite numbers implied an infinity of infinite extensive sets (aleph numbers),
since the set of subsets of a given set is necessarily larger than the original set (WP 120); whereas in the seventeenth century, infinite multiplicities can be distinguished by their differential relation (of a greater or lesser power), since the infinitely small, or a vanishing quantity, cannot be defined independently of a differential relation (which Spinoza called a “relation of movement and rest”). But the question itself points to a third aspect of Deleuze's thought: the search for the conditions of real experience implies a turn away from the “constitutive finitude” of Kant's transcendental subject in favor of the question that preoccupied both Spinoza and Leibniz: Under what relation can the infinite belong to a finite individuality?
THE PROBLEMATIC, THE VIRTUAL, AND THE INTENSIVE
With these Leibnizian and Spinozistic examples in mind, we can return, in our deduction of Deleuzian concepts, to three final notions: the problematic, the virtual, and the intensive. The first two of these concepts correspond to the question: What is the status of the multiplicities constituted by these minute and unconscious perceptions? Deleuze will say that they are objects of Ideas, in a modified Kantian sense, because even though they are not given directly in phenomenal experience, they can none the less be thought as its conditions. They are, as it were, the noumenon closest to the phenomenon (DR 222). To move from conditions to the conditioned is to move from a problem to its solution or, what amounts to the same thing, from the virtual to the actual. It remains for us to examine the parallel structure of these two remaining concepts.
