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What Lacan distinctly opposes is the classical, even “moralistic” dimension of the Hegelian infinity: the recurring circle completely closed in upon itself, the repetitive enfolding of the infinity in the One—the point, in short, at which the infinite ceases to be the other of the finite as One. Lacan will have little of Hegel’s unity of the one and the multiple. For it does not suffice to say that the recurrence of the One—its ability to become “its own other” by becoming another One (which is nothing other than the ability of the One to sublate infinity)—exhausts the function of the Other. And so it would seem that Lacan would be quite at home with other criticisms of Hegel in his efforts to uphold the Other against this sublation of infinity.

If I may be forgiven for stating the obvious, Lacan makes it clear that the repetition of the One cannot exhaust the other without generating a new other in turn. Is this not what Freud teaches us in Beyond the Pleasure Principle?

For the sake of clarity, let’s assume that the entirety of my conscious life is governed by the pleasure principle. Every attempt I make to recover an earlier state—every time I “fill in” what I am missing through the sequential recurrence of signifying elements—demands that I repeat myself. In repeating myself, I am pushed forward, towards somewhere far away from the earlier state I incessantly attempt to regain. Repetition replaces the first object (the lack I fill in with various names and numbers) with a second object, the void I circumscribe when I leap from the future (from which I am guided by repetition) towards the past (in which I am guaranteed the possibility of repeating again). There are, of course, many ways in which I can apprehend objet a, but few are ever so tangible as this. The fact that there can be no substantial “beyond” to the pleasure principle (the fact that this beyond can only ever be supposed outside the pleasure my ego confines me to) can be attributed to the bad timing inherent in the pleasure principle. To go backwards towards an earlier state of affairs, I must make a step forwards. I repeat by necessity, creating my object anew.

Consequently, what Lacan surely means when he upholds the “function of the other” in a repetitive system is this very inclusion of a heteronomous element (the “interval,” if you will) which any system aiming at continuity must invariably include. But this object does not disrupt the consistency of a perfectly closed system; by including a heteronomous element within pleasure, it is what provides that system with consistency itself. We can witness the distinctiveness of Lacan’s reproach to Hegel. As Jacques-Alain Miller repeatedly states, the objet a is not simply a product of otherness. It is a logical object, that which sustains a system in the absence of the Other. It replaces the once full presence of the Other (the place from which meaning can be guaranteed) Hegel Unsutured with the Other’s function—that which repetition strives towards.2 And it is towards this that Lacan gestures when differing from Hegel.

If it were all this simple, there would be no point to the present discussion.

There are two purposes for writing this paper. One, obviously, is to sort out, at a very elementary stage, certain differences between Lacan and Hegel.

This is no easy task given the variety of viewpoints on the matter. Some will say that Lacan is “bad philosophy”—period; there is no point in discussing him alongside Hegel. Others will read Lacan in line with contemporary, “post-structuralist,” critiques of Hegel, critiques which, as I have alluded to above, for the most part, undercut the unity of Hegel’s absolute through the intervention of otherness, or difference, into Hegel’s system. Finally, there are those, most notably Slavoj Žižek, who will attempt to “rescue” Hegel from his critics by proclaiming him a Lacanian. If no one reading seems sufficient (although I admit a partiality to the final interpretation), it is most likely the case that any of the above agendas (saving Hegel, saving Lacan, critiquing both) overrides the difficulty—one could even say impossibility—of taking either Hegel or Lacan at their word.

Quite simply, it seems that distinctions need to be made, and if it is my intent to do so here, it will be for the purposes of delimiting the above example

**of Hegelian infinity that Lacan takes issue with. Hence, my second purpose:**

what I propose is not solely a reading of Hegel avec Lacan, but to explicate Alain Badiou’s (Lacanian) critique of Hegel. Badiou’s is not a simple thesis— but it does, to be sure, disclose Lacanian principles. What Badiou objects to in Hegel is the rejection of the mathematical in favor of the essential finitude of self-consciousness. Rather than viewing the mathematical as an independent foundation of truth from which various other discourses are derived (as in Plato, Descartes, or Leibniz), Hegel views the philosopher’s task as being one in which the mathematical (the infinite) is placed in a subordinate relation to subjective reflection. Well, it seems clear enough where a Lacanian could differ; when acknowledging Lacan’s use of cybernetics in the fifties, it is obvious that the unconscious process of counting always exceeds what the conscious subject can think at any one point. A symbolic or mathematical foundation of existence cannot be sufficiently absorbed by the essential finitude of subjective self-reflection. But to effect such an absorption seems to be Hegel’s intent.

This, crudely put, would be a starting point for understanding Badiou: for the latter, the mathematical imposes a structure which cannot be globally enveloped by a conscious subject.

This is only a very preliminary reading of Badiou, a reading which will be far from exhaustive. Let it suffice to say that whereas Hegel (in his efforts to subordinate the infinite to the status of the repetitive One) seeks to establish the subject as a global site of truth, Badiou’s subject is always a local part of a logical structure in which truth is present. This subject is an indispensable part of this system, and, to be sure, there is no philosophy without a subject.

20 Penumbra But this subject is always only a finite subject. There are four axioms, derived

**from Badiou,3 which can be briefly given:**

a. Any finite formula expresses a subject. The subject is not a transcendental agency or perceiving consciousness, but a point expressed by a finite number or signifier.

b. The Subject is the local status of truth. The subject is a point in a chain of knowledge (in Lacanian terms: S2 … S3 … S4 … ) which is located somewhere between an event that has been presupposed (the “supernumerary name” which inaugurates the procession of signifying elements: S1...) and the point towards which that chain is directed (“signification”). The subject is caught in the chain at any one of these points. It is a part of the situation that the supernumerary name of the event constitutes.

c. Truth is constituted by a hole in knowledge. Truth is not qualified through an intelligible intuition. A truth is indiscernible within knowledge; it is the unnamed towards which the signifying elements which comprise knowledge as such are directed, but never reach.

a. The subject is not this void. The void is inhuman and a-subjective. Truth is realized through the multiplicity of elements that the void generates of which the subject is a part. The subject is, in effect, a finite part which is caught between an event and its truth. It is the local status of this situation as truth. Ultimately, saying that the subject is a local status of truth is very different from defining the subject as the hole in knowledge which is truth.

b. This final point may come as a surprise—do we not usually conceive the subject as the void which is represented by a signifier? Is this not how a subject is “sutured” into a symbolic; that is, as a void that is named? This is usually how suture is understood: the element which is sutured is the void of the subject. Badiou suggests something different, something, in fact, which comes much closer to the actual definition of suture in psychoanalysis. What is sutured, strictly speaking, is not the subject to the discursive chain, but the relation between the Symbolic as knowledge (or, to use Badiou’s terms, situation) and being (the Real). It would hardly seem necessary to review the entirety of the original theory of suture that Jacques-Alain Miller wrote thirty years ago if his thesis had been sufficiently understood.4 Since the case is otherwise, an exegesis will prove necessary. To expound both Badiou and his reading of Hegel requires that the reader devote his or her attention to the original relation between the One and the multiple.

Hegel Unsutured Ordinarily, suture is read as an Imaginary process through which a subject is included in a given system while disavowing, or annulling, Symbolic difference. But in Miller’s argument, the point is this: for a symbolic system to become a closed economy, it must account for the element it excludes (the subject). The agent of suturing is that which puts the Symbolic in communication with the Real, it installs “something” in the place where the subject is absent. And were it not for the inclusion of the “something” (an absence which is not nothing) within a given set (or symbolic system), distinctions between its elements could not even be drawn, since these distinctions cannot be empirically determined. This was a primary necessity for Frege’s mathematics: the exclusion of an empirical thing (its substitution by number) was necessary to sustain a logical system. Yet this substitution could not occur without marking the fact that the subject has already been excluded. But if distinctions are no longer drawn between actually existing things, then there must, in that system, be some other means of differentiating its elements.

The answer appears to be easy enough: what is sutured is the lacking subject to its signifier or representative. We could imagine that a subject is sutured when it is named as an individual. Were this not possible, something would be missing from the set—there would simply be a series of empty numbers. If the reader takes further notice, however, he or she will realize that it would be contradictory to say that the subject is what completes the set, what provides for the missing element, since it is precisely Frege’s point that the subject be excluded. The goal is something other than a merely Symbolic rewriting of the subject; for Frege, it is the formal structure of the set that interests him. The missing element, in other words, must be logical, not subjective.

In any event, when turning to the original problem that Miller presents, it is admittedly true that one is dealing with the inclusion of the subject within a given set. For Miller or Frege, there are two relations formed between the subject and the set: there is the relation between the subject and its given concept (subsumption) and there is also the relation between the subject and the number which comes to represent it in the set (assignation). Given a hypothetical set consisting of the “members of F,” neither the concept (“member of F”) of the set nor the elements (counted terms) which comprise it, comes first.

The perfect logic of the system demands that the concept exist exclusively through the inclusion of the members which it subsumes. Yet these members, as objects, are only insofar as they fall under the given concept (that is, so long as they are no longer things). The paradox, or “performativity,” of the set necessitates that neither assignation nor subsumption is primary: a subject is subsumed at the same time that it is assigned a number. To be included, the subject must be counted. So it is clear that if a thing is counted as a number, it is no longer equal to itself but to the number which assigns its place in the set.

When counted, one does not emerge as a “member of F,” but as equal to the 22 Penumbra concept “member of F.” One is included through being equal to its representative, to the number which stands in for the self.

But a volatile loss of truth is invoked by the very principle of exclusion which founds a logical system. The subject’s emergence in a set means that it is counted as one, and this one (1) is what becomes repeatedly representative for all members in the set. We can see clearly where the potential loss of truth occurs: how is it that one thing can be distinguished from another if they are both counted as one, if they can no longer be empirically differentiated as things? How is counting even possible if the distinction between “one” and “two” is no longer evident? Let me begin again: to be truly distinct, any one element must be equal to itself. One is “one” insofar as it is equal to itself: it cannot be exchanged for “two” without a loss of truth. And in order for this to be true, the number needs a “substance” of sorts, it needs a self to be equal to.

But this self cannot be an empirical thing. This, in fact, is the very problem.

Something, Miller adds, must be added to the set in order to make counting possible, in order to close the set, to make each element equal to itself.

This “something” is the inclusion of that which is not equal to itself—conceptually, zero, the empty set. We arrive at the empty set when we conceive of a set having no members, that is, of a set whose members are not equal to themselves.

This follows (as we shall see with regards to Hegel) when we conceive of the possibility of an empty set: of a set which contains no elements, yet has a property nonetheless. Let ω represent an infinite set of which x is a member precisely

**when it is not equal to itself. The empty set can be written thus:**

Ø = {x ϵ ω: x ≠ x} Our first set (“members of F”) is “sutured” through the inclusion of this other set as its member. Given this reading, our first impulse would be to inscribe the empty set between the numbers in the set, as if it is that which emerges between 1 and 2 (e.g. 1 = 1, ( 1≠ 2), 2 = 2). We could, in this instance, call the empty set the interval which emerges between the successive counting of elements. This is not the point for Miller: the empty set is not generated by, or even between, repetitions—it is what allows for repetition itself.