by Daniel Smith
THE PRINCIPLE OF THE IDENTITY OF INDISCERNIBLES
This sets us on the path of the third principle, the identity of indiscernibles. The principle of sufficient reason says: for every thing, there is a concept that includes everything that will happen to the thing. The identity of indiscernibles says: for every concept, there is one and only one thing. The principle of the identity of indiscernibles is thus the reciprocal of the principle of sufficient reason. Unlike Leibniz's first act of reciprocity, however, this reciprocation is absolutely necessary. (The move from the principle of identity to the principle of sufficient reason, by contrast, was Leibniz's coup de force as a philosopher; he could undertake it only because he created the philosophical means to do so.) Banally, this means that there are no two things that are absolutely identical; no two drops of water are identical, no two leaves of a tree are identical, no two people are identical. But more profoundly, it also means—and this is what interests Deleuze—that in the final analysis every difference is a conceptual difference. If you have two things, there must be two concepts; if not, there are not two things. In other words, if you assign a difference to two things, there is necessarily a difference in their concepts. The principle of indiscernibles consists in saying that we have knowledge only by means of concepts, and this can be said to correspond to a third reason, a third ratio: ratio cognoscendi, or reason as the reason of knowing.
This principle of indiscernibles has two important consequences. First, as we have seen, Leibniz is the first philosopher to say that concepts are proper names: that is, that concepts are individual notions. In classical logic concepts are generalities which, by their very nature, cannot comprehend the singularity of the individual. But can we not say that the concept “human,” for instance, is a generality that applies to all individual humans, including both Caesar and Adam? Of course you can say that, Leibniz retorts, but only if you have blocked the analysis of the concept at a certain point, at a finite moment. But if you push the analysis, if you push the analysis of the concept to infinity, there will be a point where the concepts of Caesar and Adam are no longer the same. According to Leibniz, this is why a mother sheep can recognize its little lamb: it knows its concept, which is individual. This is also why Leibniz cannot have recourse to a universal mind, for he remains fixed on the individual as such. This is Leibniz's great originality, the formula of his perpetual refrain: substance is individual.
Second, in positing the principle of indiscernibles (every difference is conceptual), Leibniz is asking us to accept an enormous consequence. For there are other types of difference, apart from conceptual difference, that might allow us to distinguish between individual things. For example, numerical difference: I can fix the concept of water and then distinguish the drops numerically, disregarding their individuality—one drop, two drops, three drops. A second type of difference: spatio-temporal difference. I have the concept of water, but I can distinguish between different drops by their spatio-temporal location (not this drop here but that drop over there). A third type: differences of extension and movement. I can retain the concept of water and distinguish between drops by their extension and figure (shape and size), or by their movement (fast or slow). These are all non-conceptual differences because they allow us to distinguish between two things that none the less have the same concept. Once again, however, Leibniz plunges on; he calmly tells us, no, these differences are pure appearances, provisional means of expressing a difference of another nature, and this difference is always conceptual. If there are two drops of water, they do not have the same concept. Non-conceptual differences only serve to translate, in an imperfect manner, a deeper difference that is always conceptual.
It is here that we reach the crux of the matter in Deleuze's reading of Leibniz. Although no one went further than Leibniz in the exploration of sufficient reason, Leibniz none the less subordinated sufficient reason to the requirements of “representation.” By reducing all differences to conceptual differences, Leibniz defined sufficient reason by the ability of differences to be represented or mediated in a concept.
According to the principle of sufficient reason, there is always one concept per particular thing. According to the reciprocal principle of the identity of indiscernibles, there is one and only one thing per concept. Together, these principles expound a theory of difference as conceptual difference, or develop the account of representation as mediation.11
In Aristotle, what “blocks” the specification of the concept beyond the smallest species is the individual itself. The concept provides us with a form for which the individual constitutes the matter; in Kant, it will be the forms of space and time that block the concept. Leibniz is able to reconcile the concept and the individual only because he gives the identity of the concept an infinite comprehension; every individual substance, or monad, envelops the infinity of predicates that constitutes the state of the world. Where the extension of the concept = 1, the comprehension of the concept = ∞. It is one and the same thing to say that the concept goes to infinity (sufficient reason) and that the concept is individual (indiscernibility). In pushing the concept to the level of the individual, however, Leibniz simply rendered representation (or the concept) infinite, while still maintaining the subordination of difference to the principle of identity in the concept.
For Deleuze, this subordination of difference to identity is illegitimate and ungrounded. We have seen that, in Leibniz, the principle of sufficient reason is the reciprocal of the principle of identity, and that the principle of indiscernibles is in turn the reciprocal of the principle of sufficient reason. But would not the reciprocal of the reciprocal simply lead us back to the principle of identity (6 May 1980)? The fact that it does not, even in Leibniz, points to the irreducibility of the principle of difference to the principle of identity. Deleuze's thesis is that, behind or beneath the functioning of the identical concept, there lies the movement of difference and multiplicity within an Idea. “What blocks the concept,” writes Deleuze in Difference and Repetition, “is always the excess of the Idea, which constitutes the superior positivity that arrests the concept or overturns the requirements of representation” (DR 289). Difference and Repetition in its entirety can be read as a search for the roots of sufficient reason, which is formulated in a theory of non-representational Ideas. But “the immediate, defined as the ‘sub-representative,’ is not attained by multiplying representations and points of view. On the contrary, each composing representation must be distorted, diverted, and torn from its centre”—in order to reveal, not the immediacy of the given, but rather the differential mechanisms of the Idea that themselves function as the genetic conditions of the given.12 Deleuze understands the term “Idea” largely in its Kantian sense, except that Kantian Ideas are totalizing, unifying and transcendent, whereas Deleuzian Ideas are differential, genetic, and immanent. It is on the basis of his post-Kantian return to Leibniz that Deleuze will develop his revised theory of Ideas in Difference and Repetition.
THE LAW OF CONTINUITY
These considerations bring us to the law of continuity. What is the difference between truths of essence (principle of identity) and truths of existence (principles of sufficient reason and indiscernibility)? With truths of essence, says Leibniz, the analysis is finite, such that inclusion of the predicate in the subject can be demonstrated by a finite series of determinate operations (such that one can say, “Q.E.D.”).13 The analysis of truths of existence, by contrast, is necessarily infinite: the domain of existences is the domain of infinite analysis. Why is this the case? Because if the predicate “sinner” is contained in the concept of Adam, and if we then follow the causes back and track down the effects, the entire world must be contained in the notion of Adam. When I perform the analysis, I pass from Adam the sinner to Eve the temptress, and from Eve the temptress to the evil serpent, and from the evil serpent to the forbidden fruit, and so on. Moving forward, I show that there is a direct connection between Adam's sin and the Incarnation and Re
demption by Christ. There are series that are going to begin to fit into each other across the differences of time and space. (This is the aim of Leibniz's Theodicy: to justify God's choice of this world, with all its interlocking series.) Such an analysis is infinite because it has to pass through the entire series of elements that constitute the world, which is actually infinite; and it is an analysis because it demonstrates the inclusion of the predicate “sinner” in the individual notion of Adam. “In the domain of existences, we cannot stop ourselves, because the series are prolongable and must be prolonged, because the inclusion is not localizable” (FLB 51). This is the Leibnizian move that matters to Deleuze: at the level of truths of existence, an infinite analysis that demonstrates the inclusion of the predicate (sinner) in the subject (Adam) does not proceed by the demonstration of an identity. What matters at the level of truths of existence is not the identity of the predicate and the subject, but rather, that one passes from one predicate to another, from the second to a third, from the third to a fourth, and so on. Put succinctly: if truths of essence are governed by identity, truths of existence are governed by continuity. What is a world? A world is defined by its continuity. What separates two incompossible worlds? The fact that there is a discontinuity between the two worlds. What defines the best of all possible worlds, the world that God will cause to pass into existence? The fact that it realizes the maximum of continuity for a maximum of difference.
Now the notion of infinite analysis is absolutely original with Leibniz; he invented it. It seems to go without saying, however, that we, as finite beings, are incapable of undertaking an infinite analysis. In order to situate ourselves in the domain of truths of existence, we have to wait for experience; we know through experience that Caesar crossed the Rubicon or that Adam sinned. An infinite analysis might be possible for God, whose divine understanding is without limits, but this is hardly a satisfactory response. While we may be happy for God, we might also ask ourselves why Leibniz went to such trouble to present this whole story about analytic truths and infinite analysis if it were only to say that such an analysis is inaccessible to us as finite beings. It is here, however, that we begin to approach the originality of Deleuze's interpretation of Leibniz. For Leibniz, says Deleuze, indeed attempted to provide us finite humans with an artifice that is capable of undertaking a well-founded approximation of what happens in God's understanding, and this artifice is precisely the technique of the infinitesimal calculus or differential analysis. As finite beings, we can none the less undertake an infinite analysis, thanks to the symbolism of the calculus. The calculus brings us into a complex domain, having to do not only with the relation of Leibniz to Newton, but also with the debates on the mathematical foundations of the calculus, which were not resolved until the development of the limit-concept by Cauchy and Weierstrass in the late nineteenth and early twentieth centuries.14 I would here like to focus on two aspects of Leibniz's work on the metaphysics of the calculus that come to the fore in Deleuze's own reading of Leibniz: the differential relation and the theory of singularities. These are two theories that allow us to think the presence of the infinite within the finite.
THE DIFFERENTIAL RELATION
Let us turn first to the differential relation. What is at stake in an infinite analysis is not so much the fact that there is an actually existing set of infinite elements in the world. The problem lies elsewhere. For if there are two elements—for example, Adam the sinner and Eve the temptress—then there is still a difference between these two elements. What, then, does it mean to say that there is a continuity between the seduction of Eve and Adam's sin (and not simply an identity)? It means that the relation between the two elements is an infinitely small relation, or rather, that the difference between the two is a difference that tends to disappear. This is the definition of the continuum: continuity is defined as the act of a difference in so far as the difference tends to disappear—continuity is a disappearing or vanishing difference. Between sinner and Adam I will never be able to demonstrate a logical identity, but I will be able to demonstrate (and here the word demonstration obviously changes meaning) a continuity—that is, one or more vanishing differences.
What is a vanishing difference? In 1701, Leibniz wrote a three-page text entitled “Justification of the Infinitesimal Calculus by That of Ordinary Algebra,” in which he tries to explain that, in a certain manner, the differential calculus was already functioning before it was discovered, even at the level of the most ordinary algebra.15 Leibniz presents us with a simple geometrical figure (Figure 3.1). Two right triangles—ZEF and ZHI—meet at their apex, point Z. Since the two triangles ZEF and ZHI are similar, it follows that the ratio y/x is equal to (Y – y)/X. Now if the straight line EI increasingly approaches point F, always preserving the same angle at the variable point Z, the length of the straight lines x and y will obviously diminish steadily, yet the ratio of x to y will remain constant. What happens when the straight line EI passes through F itself? It is obvious that the points Z and E will fall directly on F, and that the straight lines x and y will vanish; they will become equal to zero. And yet, even though x and y are equal to zero, they still maintain an algebraic relation to each other, which is expressed in the relation of X to Y. In other words, when the line EI passes through F, it is not the case that the triangle ZEF has “disappeared” in the common sense of that word. The triangle ZEF is still “there,” but it is only there “virtually,” since the relation x/y continues to exist even when the terms have vanished. Rather than saying the triangle ZEF has disappeared, Leibniz says, we should rather say that it has become unassignable even though it is perfectly determined, since in this case, although x = 0 and y = 0, the relation x/y is not equal to zero, since it is a perfectly determinable relation equal to X/Y. Unassignable, yet perfectly determined—this is what the term “vanishing difference” means; it is when the relation continues even when the terms of the relation have disappeared. The relation x/y continues when Z and E have disappeared. This is why the differential relation is such a great mathematical discovery; the miracle is that the differential relation dx/dy is not equal to zero, but rather has a perfectly expressible finite quantity, which is the differential derived from the relation of X to Y.
Figure 3.1
The differential relation is thus not only a relation that is external to its terms, but a relation that in a certain sense constitutes its terms. It provides Deleuze with a mathematical model for thinking “difference-in-itself” (the title of the second chapter of Difference and Repetition). The differential relation signifies nothing concrete in relation to what it is derived from—that is, in relation to x and y—but it signifies something else concrete—namely, a z—which is something new, and this is how it assures the passage to limits. Thus, to consider several famous examples, Leibniz can comprehend rest as an infinitely small movement, coincidence as an infinitely small distance, equality as the limit of inequalities, the circle as the limit of a polygon the sides of which increase to infinity. The “reason” of the law of continuity is thus the ratio fiendi, the reason of becoming. Things become through continuity: movement becomes rest; the polygon, by multiplying its sides, becomes a circle. This is the source of the popular formulation of the law of continuity in Leibniz: nature never makes leaps (there is no discontinuity in nature). What, then, is an infinite analysis? An infinite analysis fills the following condition: there is an infinite analysis, and a material for infinite analysis, when I find myself before a domain that is no longer directly ruled by identity, but a domain that is ruled by continuity and vanishing differences.
To understand what this theory of the differential relation means in concrete terms, consider the corresponding theory of perception that Leibniz develops in relation to it.16 Leibniz had observed that we often perceive things of which we are not consciously aware. We recall a familiar scene and become aware of a detail we did not notice at the time; the background noise of a dripping faucet suddenly enters our consciousness at ni
ght. Leibniz therefore drew a distinction between conscious perceptions (“apperceptions,” or molar perceptions) and unconscious perceptions (“minute” or molecular perceptions), and argued that our conscious perceptions must be related, not simply to recognizable objects in space and time, but rather to the minute and unconscious perceptions of which they are composed. I apprehend the noise of the sea or the murmur of a group of people, for instance, but not the sound of each wave or the voice of each person that compose them. These unconscious minute perceptions are related to conscious “molar” perceptions, not as parts to a whole, but as what is ordinary to what is noticeable or remarkable: a conscious perception is produced when at least two of these minute and “virtual” perceptions enter into a differential relation that determines a singularity: that is, a conscious perception. Consider the noise of the sea: at least two waves must be minutely perceived as nascent and “virtual” in order to enter into a differential relation capable of determining a third, which excels over the others and becomes conscious. Or consider the color green: yellow and blue can be perceived, but if the difference between them vanishes by approaching zero, then they enter into a differential relation (db/dy = G) that determines the color green; in turn, yellow or blue, each on its own account, may be determined by the differential relation of two colors we cannot detect (dy/dx = Y). The calculus thus functions in Leibniz as the psychic mechanism of perception, a kind of automatism that determines my finite zone of clarity on the world, my point of view. Every conscious perception constitutes a threshold, and the minute or virtual perceptions (infinitely small perceptions) constitute the obscure dust of the world, its background noise. They are not “parts” of conscious perception, but rather the “ideal genetic elements” of perception, or what Salomon Maimon called the “differentials of consciousness.” The virtual multiplicity of genetic elements, and the system of connections or differential relations that are established between them, is what Deleuze terms the “Idea” of sensibility. It is the differential relations between these infinitely small perceptions that draw them into clarity, that “actualize” a clear perception (such as green) out of certain obscure, evanescent perceptions (such as yellow and blue). “The Idea of the world or the Idea of the sea are systems of differential equations, of which each monad only actualizes a partial solution.”17
