by Daniel Smith
LEIBNIZ AND THE PRINCIPLE OF IDENTITY
Scene one focuses on Leibniz, who would have been a perfect philosophical movie star, since he is a man of contradictions. He is somewhat reactionary, a defender of law and order, of the status quo, of “policing” in every sense of the term; he says malicious things about Spinoza; but at the same time he invents the calculus, and undertakes one of the most remarkable adventures of thought in the history of philosophy. The reason: Leibniz took the most basic principle of logic—the principle of identity—and attempted to make it penetrate existence in its entirety by formulating the reciprocal of the principle of identity: namely, the principle of sufficient reason. Scene one briefly shows how.3
The classical formulation of the principle of identity is “A is A.” In this formula, the principle of identity is an absolutely certain thought, but it is none the less an empty thought. Are we truly thinking when we say “A is A”? It's not clear. The popular formulation of the principle of identity would be: “A thing is what it is.”4 Though this formula seems trite, it actually goes further than the formula “A is A” because it points to the domain of Being governed by the principle of identity—namely, the domain of essences: the principle of identity asserts the identity between a thing and what the thing is (its essence). But Leibniz's originality lies in the logical formulation he provides of the principle of identity: “every analytic proposition is true.” It is this formula that lies at the basis of Leibniz's attempt to extend the principle of identity beyond the domain of essences and into the domain of existence. What is an analytic proposition? An analytic proposition is a proposition in which the subject and the predicate reciprocate with each other. The principle of identity is presented in the form of a reciprocal proposition: there is a subject, A; then the verb is; and then a predicate or attribute, A. The principle of identity states that, in the proposition “A is A,” there is a reciprocity between the subject and the predicate, even though the distinction between subject and predicate remains. What Leibniz calls analysis is the operation that discovers a predicate in a notion taken as a subject.
But how can this new formula allow us to think existing beings? The principle of identity posits the identity of the thing and what the thing is, even if the thing itself does not exist; existing things thus appear to lie outside the principle of identity. In order to think existing beings, Leibniz will derive a new principle from the principle of identity, which he will call the principle of sufficient reason. The popular formulation of the principle of reason would be “everything has a reason.” This is the fundamental battle cry of rationalism—everything has to have a reason, there must be a reason for everything that takes place—and the entire philosophical project of Leibniz, the greatest of the rationalists, is animated by his pursuit of the problem of sufficient reason. But how can the principle of sufficient reason allow us to think existing beings? This becomes clear in Leibniz's metaphysical formulation of sufficient reason: “all predication has a foundation in the nature of things” (FLB 42). What this means is that everything that is predicated of a thing must be included in the concept of the thing. What is said or predicated of a thing? If we say that what is predicated of a thing is its essence (what is is), then there is no difference between the principle of identity and the principle of sufficient reason. But what is said or predicated of a thing is not only the essence of the thing; it is also the totality of events that happen to the thing in its existence.5 For example: Caesar crossed the Rubicon. Since this is a true proposition, Leibniz will insist not only that the predicate “crossed the Rubicon” must be contained in the Caesar's notion or concept, but moreover that we should be able to demonstrate, through an analysis, that this is the case.
This is an astonishing philosophical move, which would make for dramatic cinema, if thought itself could be filmed. The principle of identity provides us with a model of truth that is certain, yet it does not make us think anything, so Leibniz simply reverses the formulation of the principle of identity using the principle of reciprocity. The principle of identity says that an analytic proposition is necessarily a true proposition, whereas the principle of sufficient reason says that a true proposition is necessarily an analytic proposition. The principle of sufficient reason, in other words, is the reciprocal of the principle of identity, but this reversal was only made possible through Leibniz's logical reformulation of the principle of identity. It is through this reversal that Leibniz will attempt to use the principle of identity to conquer the domain of existence.
There are two things that might be said about Leibniz's principle of sufficient reason. The first is that it seems absolutely crazy; it is hard to see how anyone could take it seriously. Ian Hacking once wrote that “Leibniz's claim that in every true proposition the predicate is contained in the subject is the most absurd theory of truth that has ever been advanced.”6 It is easy to see why: Leibniz is claiming that, just as we can demonstrate that the predicate “three sides” is included in the subject triangle, we should be able to demonstrate that the predicate “crossing the Rubicon” is contained in the concept of Caesar. One can hardly imagine the conditions under which such a thing would be possible, unless we were God himself, with his infinite understanding. But the second point is this: Leibniz's posing of the problem of sufficient reason would mean nothing if he had not had the means to create the philosophical concepts that were necessary to explore the conditions of this problem (This is Deleuze's definition of philosophy: the creation of concepts in response to shifting problematics.) Here, we introduce a shot of Leibniz standing on a precipice, about to plunge into the labyrinth of the continuum, the maelstrom of the actual infinite. He is calm, tranquil, and confident, however, because for every problem posed by his search for sufficient reason he will create a concept adequate to it, even as he is falling into the abyss. Here are a few of those concepts—just enough to feel the power of Leibniz's thought.
First, if everything we attribute with truth to a subject is contained in the notion of the subject, then we must also include the totality of the world in the subject. Why? Because the principle of sufficient reason is not the same thing as the principle of causality. To say that “everything has a cause” means that A is caused by B, B is caused by C, and so on—a series of causes and effects extending infinitely in all directions. To say that “everything has a reason,” however, means that we must give a reason for these causal relations; it means that the causal relation between A and B must itself be included in the notion of A. This is how the principle of sufficient reason provides the ground for the principle of causality: the principle of causality states the necessary cause of a thing but not its sufficient reason.7 But opening up of this ground is precisely the abyss into which Leibniz is plunging. Once he says that the predicate “crossing the Rubicon” is included in the notion of Caesar, he cannot stop himself; he is forced to include the totality of the universe in Caesar's concept. The event “crossing the Rubicon” has multiple causes and effects that stretch to infinity, such as the establishment of the Roman Empire and the crucifixion of Jesus, and Leibniz cannot say that “crossing the Rubicon” is included in the notion of Caesar without saying that all the causes and effects of this event are also included in the notion of Caesar. This is the first hallucinatory concept Leibniz creates: the concept expression. Each of us, in our concept, expresses or contains the entirety of the universe.
But Leibniz immediately confronts another danger. If the concept of each subject expresses the totality of the world, would this not mean that there is only a single subject (like Spinoza's substance), and that individuals are merely the appearances or modifications of this universal subject? But Leibniz's philosophy is fixed on the individual, and he cannot admit such a claim without renouncing the principle of sufficient reason.8 No problem, Leibniz responds. To avoid this danger, he will simply create another concept: each individual notion may indeed include the totality of the world, but it does so from a certain point of view.
This marks the beginning of “perspectivist” philosophy, which Leibniz derives from the theory of conic sections.9 Leibniz's claim is not that each individual expresses the totality of the world from its own point of view, as if everything were “relative” to the viewpoint of the subject, since in fact the exact opposite is the case: it is the subject that is constituted by the point of view. Point of view, in other words, is the sufficient reason of the subject; the individual notion is the point of view through which the individual expresses the totality of the world.10
But this propels Leibniz into a third problem: What determines this point of view? How does one distinguish between points of view? Once again, Leibniz plunges on: each of us expresses the totality of the world from a certain point of view, he tells us, but we necessarily express most of the world in an obscure and confused manner, as if it were a mere background noise. The totality of the world is really included in the individual notion, but primarily in the form of infinitely small perceptions—another concept. None the less, if there is finite neighborhood of the world that I express clearly and distinctly, it is precisely that neighborhood of the world affects my body; Leibniz here deduces the necessity of the body as that which occupies the point of view. I myself do not express clearly and distinctly the crossing of the Rubicon, for example, since that concerns Caesar's body. Each individual substance occupies a different point of view on the world because each of them has a different zone of clear and distinct expression on the world as a function of its body.
But Leibniz still cannot stop, for he now confronts a final danger: How do we know that each of these individuals is expressing the same world? This is a problem for the following reason. The principle of identity allows us to determine what is contradictory: that is, what is impossible. A square circle is a circle that is not a circle; it is impossible because it contravenes the principle of identity. But at the level of sufficient reason, Caesar not crossing the Rubicon or Adam not sinning in the Garden of Eden is neither contradictory nor impossible; Caesar could have not crossed the Rubicon, and Adam could have not sinned, whereas a circle cannot be square. The difficulty is: How can Leibniz hold that everything Adam did is contained for all time in his individual concept, and that Adam the non-sinner was none the less possible? This is where Leibniz invents perhaps his most notorious concept, which is in fact an entirely new logical relation: incompossibility. At the level of existing things, it is not enough to say that a thing is possible in order to exist; it is also necessary to know with what it is compossible. Adam the non-sinner was possible in itself, but it was incompossible with rest of the actualized world. Leibniz derived his most famous doctrine from the concept of imcompossibility; among the infinity of incompossible worlds God had in his mind at the moment of creation, God chose “Best” of all possible worlds to pass into existence, governed by a harmony that is pre-established by God. Thus, Leibniz says, when I want to demonstrate that the predicate “sinner” is contained in the concept of Adam—that is, when I perform an analysis of the concept “Adam”—I move backwards from Adam the sinner to Eve the temptress, and from Eve the temptress to the evil serpent, and from the evil serpent to the apple, and so on; moving forward, I show that there is continuity between Adam's sin and the Incarnation and Redemption by Christ. In other words, there are series that are going to begin to fit into each other across the differences of time and space, and the ultimate aim of Leibniz's great book Theodicy is to justify God's choice of this world as the “best” world, with all its interlocking series, and with all its suffering and cruelty.11 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 sufficient reason for the inclusion of the predicate “sinner” in the individual notion of Adam.
It is here that we end scene one, for we seem to have reached a blockage. It seems to go without saying 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. Infinite analysis is possible for God, to be sure, but this is hardly a satisfactory answer. We may be happy for God (close-up of God, smiling), but then we would wonder why Leibniz went to such trouble to present this whole story about sufficient reason if it remains inaccessible to us as finite beings. This apparent blockage in Leibniz's thought will return at the end of our scenario, but what we have seen in scene one is the “delirious” creation of concepts one finds in Leibniz. Expression, point of view, minute perceptions, incompossibility: all these are concepts that are generated in Leibniz—created by him—as a result of his positing the problem of sufficient reason. This is why Deleuze says is it useless to pose objections to a philosopher; the more important thing, at least initially, is to extract the “problematic” generating their thought, and to follow it as far as one can.
HEGEL AND THE PRINCIPLE OF NON-CONTRADICTION
But now scene two intervenes. It begins with a tracking shot that moves past a number of philosophical figures: Descartes, Leibniz (again, briefly, in a flashback), then Kant and Fichte, and finally Hegel. Hegel is the culminating point of the second scene, which charts out the trajectory through which philosophy attempted to conquer existence, no longer through the principle of identity, as in Leibniz, but through the principle of contradiction.
Scene two begins with Descartes, who is another good philosophical movie star; suave, debonair, long hair, goatee, he sleeps until noon every day, and likes working in bed, which I personally admire. Christina, the Queen of Sweden, the story goes, forces him to get up early; this gives him pneumonia, and he dies. Prior to dying, however, Descartes had attempted to think existence in his own manner, and his undertaking would have even greater repercussions in philosophy than Leibniz's. In the Meditations, Descartes claimed that, in order to doubt, I must be thinking; hence I am a thinking being. The question of doubt, it is true, does not bear on the existence of things, but rather on the knowledge I have of the existence of things. In so far as I doubt, there is a knowledge that I cannot doubt, which is the knowledge of myself as a thinking being. But in this manner, Descartes was the first thinker to introduce into philosophy a formula that would later be developed extensively in German philosophy: the “I = I” or the “Self = Self” (Ich = Ich). Now, although the “I = I” might appear to be simply a re-formulation of the principle of identity “A = A,” in fact it has a completely different status. The identity A = A is the identity of the thing thought, and as such it is a hypothetical judgment. Its complete formulation would be: if there is A, A is A; if A exists, then A = A. But perhaps A does not exist, perhaps there is nothing. (This is why the principle of identity corresponds to the question, “Why is there something rather than nothing?”) What Descartes showed was that the principle of identity is a purely hypothetical judgment; I can always doubt A, not only in its existence, but even in its concept. Thus, when Descartes says that there is one thing I cannot doubt, I = I, he did something radically new in philosophy: he discovered an identity that is no longer subject to this hypothetical condition; he discovered an unconditioned identity, or what came to be called a thetic or categorical judgment. This is the discovery of subjectivity: the position, or auto-position, of the subject, the I = I. Fichte would develop this thesis to its ultimate conclusion: one can only say “A is A” because A is thought, but what grounds the identity of what is thought is the reality of the thinking subject, the identity of the finite “I.” Thus, the principle of identity, “A is A,” founds its ground in the auto-position of the subject, the “I is I.” In Descartes, the principle of identity left the sphere of logic and took a first baby step into the real, or into existence.
A brief flashback to Leibniz, for this is precisely where his philosophy intervened. For, although the I = I allowed Descartes to conquer a small island of existence, the Cartesian cogito is, as it were, enclosed in a citadel. Affirming something other than the think
ing subject—such as the reality of something thought (mathematics) or the reality of something experienced (the sensible world)—will require an entire acrobatics, a series of complex reasonings on Descartes's part, all of which will appeal to the guarantee that God exists and is a truthful being. So, although Descartes had obtained his little island of existence—the cogito—what Leibniz sought to attain was the adequation of thought with existence in its entirety, the real in its totality. What Descartes did not see was that the I = I does not simply refer to the little island of the cogito, posited in the certitude of itself, but rather expresses or comprehends the totality of the world as the set of its own predicates. Such is the significance of the shift from Descartes's cogito to Leibniz's monad.
But now scene two jumps ahead to Kant, famous for the regularity of his daily walks, and the bizarre garters he made to hold up his socks. Kant and the post-Kantians would take up Leibniz's project, but in a new manner, taking it in a different direction. The reason: after Leibniz, no one could affirm that every true proposition is analytic. What had intervened was Kant's fundamental discovery of synthetic judgments. For Kant a judgment such as “The three angles of a triangle are equal to two right angles” is no longer an analytic judgment but a synthetic judgment, since its demonstration must pass through the concept of a square; the proposition is therefore a synthesis of two concepts. The results of discovery were profound. Although Descartes had located the ground of the principle of identity in the “I = I,” what Kant discovered was that the “I = I” is a synthetic identity, and no longer simply an analytic identity.
