Lacan opens with a discussion of what I have elsewhere named the property of “borromean dependence:” the borromean knot/chain is such that if one ring is detached the other two will also, by necessity, become detached, and the whole thing will fall apart.
The question that Lacan is pursuing seems to be the relationship of language and logic to topology. Indeed, this is what I tried to show in a recent paper published (kindly and with a great amount of risk, no doubt) by PsychoanalysisLacan, titled “A Portrait of Baudelaire as a ‘Man of Genius’: Ordinary Psychosis within the Age of Modernity” (here). In that article I tried to show that the logical formulations of the oedipal metaphor indicate a borromean knot already – they already point toward a topological understanding. Minimally, this is discovered by noticing the way in which the parentheses are nested within the formula.
In any case, Lacan says: “Should language be broached in its grammer? — in which case, this is certain, it hinges on a topology.”
Topology, as defined by Lacan, is mathematical but it does not deal with “units.” This is a curious statement given the focus on Frege in previous classes, and, indeed, Frege’s focus on “units” in his Foundations of Arithmetic. Thus, topology insists that “it can be pulled out of shape.”
And then Lacan claims that the tetrahedrons from the previous class were actually links, circles, rings even. “Each of them was a closed and flexible circle that only holds fast when linked to the others.” What does this mean? The pentagram from the previous class probably makes this very clear: these are two interlocking tetrahedrons.
But we can see the topology of the rings most clearly by imagining the ternary structure of the initial triangle, and each vector or vertex as an interlocking ring.
There are a few paragraphs on taking language “on the basis of signifying combinatorics” or “semantics,” which I can not understand whatsoever. And then there is something important that I am missing about the problem of being compelled to take very different types of action under the heading of “verbs.” I hope that somebody can make this comprehensible for me.
In any case, Lacan is putting forward something original and inventive: language having its origin in a topological structure. And he finds this only by pursuing the discovery of the impossible sexual relation.
And then we get those two modalities of the “real” define so easily by Fink in his book The Lacanian Subject: there is the “first order” real and the “second order” real. Lacan rather asks the question of a new real. For so long the real was that which limited the symbolic. Now, we have the following question: “does the speaking being speak because of this something that happened to sexuality, or did this something happen to sexuality because he is a speaking being?” It seems to me that the question might also be phrase in the following way: “does the speaking being speak because of the impossible real of sexual relations, or does the impossible real of sexual relations occur because he is a speaking being?”
On the one hand, there is a question of the real arises as a consequence of the symbolic (the imposition of the name-of-the-father, if you like). On the other hand, there is a question of the symbolic (signifier, name of the father, etc) arising as a consequence of the real itself. I believe that both are correct, but the latter is probably more correct because it accounts for the origin of the name of the father as signifier.
And then Lacan makes his famous statement about Simone de Beauviour’s work in The Second Sex. He says:
She called me by phone to tell me that she would certainly need my advice in order to enlighten her as to what the psychoanalytic contribution to her book would be. As I remarked to her that it would take at least a good five or six months for me to unravel the question for her — which is a minimum because I’ve been speaking about it now for twenty-years, and this is not by chance — she announced that of course it was out of the question for a book that was already being finalized to have to wait so long, the laws of literary production being such that it seemed to her that having more than three or four consultations with me was impossible. Following which, I declined the honour.
The fundament of what I’ve been coming out with for a while now, since last year, is very precisely that there is no second sex.
This final statement is fascinating: it is that the woman does not exist. He continues, returning to the function of language, to state that it is from the moment that language begins to function that the second sex ceases to exist.
Lacan introduces his infamous formulae of sexuation:
Ex! Phi(x) [whereby ! indicates a negation]
Lacan makes a clever statement about the biblical meaning of “know,” which, it seems to me, means “to have sexual intercourse with” (a verb, then, from genesis in the old testament).
This is an ontology, then, for Lacan, since he believes it states something fundamental about Being. He points to the Chinese who have “long called upon two fundamental essences that are respectively the feminine essence, which they call Yin, in opposition to the Yang, […] underneath.”
There is a sense in which the feminine and masculine essences have a “one to one” relationship which corresponds with number. It seems that there is a one to one form of coupling among the senses — the “soul to soul” encounter, which, Lacan derives from the animistic model.
Lacan goes on to call this model a “fantasy.” And it is this fantasy that repeats, and it repeats the unique encounter of the speaking being. And what does the fantasy do? It tries to conceal the function of language, which sexuates the beings: “the fantasy is there to say language does not exist.” The fantasy is also that there is a sexual relation, which, it seems, amounts to the same thing as saying that language does not exist.
Thus, that language does not exist implies also that there is a sexual relation. Yet, these are both fundamental to the fantasy.
Lacan claims that when it comes to the sexual relation … the Other is “an emptiness.” But he qualifies this: “by means of something extra.” Thus, the Other is an emptiness by means of something extra. This “something extra” relates to the homophone Lacan next produces: “the Hun,” which sounds like “the One” in French.
Lacan claims that Socrates is a hysteric! Lacan simply points to the reports of his “cataleptic manifestations.” Which is an embodied symptom. And then Socrates is also responsible for sustaining a discourse that gave way to the discourse of science. And how? It was by bringing “the subject to the place of semblance.” I am not sure what this means … I presume it means that the subject, instead of being located in the real, comes to the imaginary. Thus, he places the subject in the position of phallic signifier who stands in the way of enjoyment. This would explain Lacan’s point that “he had to adjure her kindly to withdraw so as to allow his death to take on its full political signification.”
It is the analytic discourse — not the hysteric’s discourse — that is able to claim, finally, that there is no such thing as sexual relation. This implies, does it not, that Lacan believes himself to be beyond Socrates (the hysteric) in some way. And perhaps he was! Thus, when my friend asks me: “what is socrates to the capitalist discourse?” My answer, here, might be: “a capitalist!, and what is the analyst within capitalist discourse?” Socrates seems no longer to be the hero that he may have once been for Lacan.
The analyst discourse observes that there is no sexual relation but also that there is a “phallic function,” phi(x,y). And it is the phallic function that spells out clearly the point of exception: there is “one” of the two terms that is not typified by sexual relation. What does this mean? He links this expression also to the homophonic “hun,” there is one, hun, that is not typified by sexual relation: and it is the woman. The hun is not Male or Yin/Yang. It is an organ — but only through its function.
Lacan makes a claim in passing that helps us to understand the discourse of science. First, as I wrote above, Socrates gave way to the discourse of science by bringing the subject to the place of semblance. Second, the discourse of science inserts “language onto the mathematical real.” Does this not imply, at some level, that there was no foreclosure? When foreclosure is absent then the symbolic — language — returns in the real. This, at least, was the position Lacan spelled out decades earlier in his psychosis seminar (sem. 3).
Although it is true that we continue, within language, to spell out the relation of the sexes through the materials of language — signifiers — as “man” and “woman,” in their “one-to-one” relation, this is a “mirage” of language. It does not accurately refer to the being of man and woman. Analytic discourse therefore rejects the ancient writing of man and woman. Lacan remarks that even he too is writing these down, in a way, but only, finally, with recourse to topology. And this is different! His writing moves from logic toward topology – and this is crucial. There is a difference, then, in logic and in topology. Topology gives us something that logic keeps only as mirage, fantasy.
It is topology that allows us to focus on the “empty place” and to see it as operative already within logic(al discourse). Moreover, topology permits us to see how the “empty place” determines the argument itself. It was, once again, the “empty place” of zero as the non-identity with 1 that determined the logic of succession for Frege.
It is not the phallic function that makes the sexes different, since both men and women are subject to the phallic function.
There are within propositional logic four fundamental relationships: negation, conjunction, disjunction, and implication. But Lacan points to a limitation in propositional logic: the relationship of “negation” can not be sustained: negation is not only falsehood, it is not only all that is not true. Why? Because truth is what is not written (which is different from that which is falsely written, or, rather written falsely). It is what is not written that challenges the supremacy of the phallic function.
In the relationship of man to woman, one always stands as an obstacle to the other — to the relationship as such.
A woman cannot be castrated, thus !Ax Phi(x) (Not All x are submitted to castration).
Lacan has gone through negation a bit, but turns to conjunction. Conjunction is grounded upon the bringing together of two true propositions. But this, claims Lacan, is not what is permitted by his formulae: there can be no conjunction of sexuation, of sexual relation. Or, perhaps, sexual relation is the mirage of conjunction.
Similarly, disjunction can not be operative in his formulae either. Disjunction occurs when it is impossible for two propositions are false at the same time. Disjunction can not allow for two propositions to be false at the same time. Lacan goes on to give proof that disjunction is not operative in his formulae – I have opted not to think too hard about this stuff.
We return for a moment to the relation of 1 to 0. It is the relation of the phallus to the not-all, and this is also the “at least one.” “There is at least one x, …” and “there does not exist an x” … is this correct?
There is feminine jouissance and phallic jouissance; feminine jouissance does not concern the phallic function, and this is how Lacan chooses, at this time, to define it. And Lacan claims that Tiresias disclosed feminine jouissance when he came back Theresa. Tiresias was a blind prophet of Apollo who transformed into a woman for seven years.
When !E(x) (top level of the feminine formulae of sexuation), there is the possibility of sexual relation. But here, when there is sexual relation the Other ceases to exist: the Other is absent. Why? Because the Other would be the obstacle to the sexual relation? Because the Other would be the barred signifier of the Phallus?
0 and 1 makes 2, but on the symbolic plane. And this is where existence takes root: in the symbol.
The speaking being is ungraspable — and all the more so because he requires the symbol to support himself: Phi.
Existence is not Being, then. This is the statement Lacan returns to …
-to exist depends upon the Other.
-but Being does not depend upon the Other.
What does this mean: “All of you here, from some angle or other, do indeed exist, but when it comes to your Being, you’re not so tranquil.” It means, I think, that existence does not guarantee Being.
Aristotle maintains that existence can in no way be established except outside the universal, Ax. So existence is particular, rather than universal. This particular, which is associated with existence, is related to the use of the word some.
“Some x …” implies “there exists some x…” and this is a particular proposition.
Lacan informs us that this is formalized as a quantifier. Apuleius argued that whenever Aristotle used all and some, we are dealing with quantifiers, with issues of quantity. But Lacan contends that all and some are not actually quantifiers. Rather, they are “two modes of […] the incarnation of the symbol.” Thus, you can incarnate the symbol all of the time, or you may incarnate the symbol some of the time. Or, rather: some symbols or all symbols.
Every language uses “alls” and “somes.” This is fascinating. It is for this reason that Lacan believes that all languages have a “common root.” I wonder if his is lalangue? It is possible since lalangue is the babbling of language. This makes all the more sense for me since Lacan says that the common root has to be “related to something that is not language.” The real seems to be the common root, in some way.
There is a gap between the 0 and the 1 within Frege’s system, and that is the non-relation, which is also, one might say, non-identity (as a principle). Lacan likens this to the ternary structure … but this time for mathematics.
On the “at least one,” it is the “name of the father.” But here Lacan makes a really nice move by saying that the name of the father is “supported by” the “at least one.” Thus, the name of the father does have the primacy of place that it used to have for Lacan.
AxPhix indicates that “every male is serf to the phallic function.” We see here a way of transposing the formulae of sexuation into Marxian analyses, or economic analysis. However, it is not so simple: it is not that only men are “serfs,” clearly. But women are “serfs” too, perhaps, but in their own way.