The following is a reworking of part of my talk before the Toronto Lacan Society on Lacan’s Universes of Discourse and the New Symptom.
The virtue of the Lacanian mathemes is two-fold. First, the mathemes allow us to recognize structural identities and similarities where we might not otherwise discern them. Take the classical formulation of the Oedipal structure. A father intervenes in the relationship between the mother and child, prohibiting her as an object of jouissance for the child. Here the Oedipus is formulated in terms of biologically sexed bodies: Fathers are male, mothers are female. When confronted with an analysand in the clinic that grew up under two women, an analyst that thinks in terms of images might jump the gun and conclude early on that this analysand will be psychotic because we conclude that the father is foreclosed in this family structure (or a pervert because the father is disavowed in this family structure) because no father was present.
In the clinic, the mathemes help us to avoid this hasty judgment. Rather than referring to fathers and mothers which invite thinking in terms of images, Lacan instead articulates the Oedipal structure in terms of a series of algebraic symbols, represented on the left-hand side of the diagram above under the heading “Homme”. The upper line “∃x~Φx” can be read as “there exists a being that is not castrated or alienated in language”. The second line “∀xΦx can be read as “all beings are castrated or alienated in language”. Taken together, the upper line articulates the father-function as a being that hasn’t sacrificed jouissance and that lays down the law, while the lower proposition articulates the subject that is subjected to the law or that has had to sacrifice jouissance to enter the symbolic order.
Why articulate things in this way, rather than simply talking about Oedipal drama in terms of mothers and fathers? Talk of mothers and fathers leads us to think of penises, vaginas, and breasts. This has diagnostic implications for what we assume about the analysands that enter the clinic. The advantage of Lacan’s highly abstract formalization is that it helps us to avoid these images, so as to attend to functions. Returning to the case of a family structure composed of two women, we can now see that one of these women can serve the paternal function of instituting the prohibition against incest, despite the fact that she isn’t male. In other words, the Oedipus– and by extension, patriarchy –isn’t about biological organs, but is a particular structure where biological men and women alike can serve certain functions. This has important consequences for feminism as well. For if it is true that patriarchy is the Oedipal structure, and if it is true that biological men and women can occupy the position of the paternal function, then it follows that overcoming patriarchy cannot be a matter of simply placing a biological female in a position men once occupied. It is the structure itself that is patriarchal, regardless of whether or not a male or female occupies the upper line of the equations, not the organ that the person possesses that makes something patriarchal.
In addition to the clinical value of these mathemes, they also have an analytic value. Initially, when talking about the left-hand of Lacan’s graph we might have thought we were just talking about the family Oedipal structure, or mommy, daddy, and child. However, insofar as we’re talking about a structure rather than empirical types of individuals, it becomes possible to think fractally and discern this structure in a variety of very diverse social relations that initially appear quite different from one another. We see this patriarchal structure articulated in Lacan’s two propositions in the myth of the primal father articulated by Freud in Totem and Taboo, nationalisms, structures of sovereignty characteristic of dictatorships and monarchies, certain forms of party politics, Laplace’s demon, the theologies of theistic religions such as Christianity, Islam, and Judaism, certain management structures in the workplace, and so on. While these social formations and belief systems have very different contents, they nonetheless have the same fractal or formal structures.
The second virtue of the mathemes is that they allow us to actually discover relations that we might not have anticipated. I suspect this is how Lacan discovered his four discourses. Lacan was initially looking for a way to represent his definition of the subject in a set of mathemes. In earlier formulations Lacan had articulated his definition of the subject with the aphorism “the signifier represents the subject for another signifier.” How to represent this aphorism in mathemes in a way that also takes into account the loss of jouissance that takes place upon the subject’s introduction to language (which is not indicated in the aphorism)? Lacan’s solution was the matheme to the left above. We have one signifier (S1) relating to another signifier (S2), representing a subject ($) for that signifier S2. This representation of the subject for another signifier produces a loss of jouissance, and Lacan represents that loss or surplus-jouissance with the matheme “a”.
Look at how nicely this all works! In a single formula, Lacan has managed to represent a very complex body of theory and everything seems to fall into place. For example, on the lower portion of the formula to the left above we see the relation $-a, or “barred subject related to objet a. This is the formula of fantasy that Lacan represents as $ <> a (read “barred subject punch a”). I’ll have more to say about fantasy in this connection in a moment.
This little formula allowed Lacan to discern three additional forms of social relation. If we keep the relations between the four terms– $, S1, S2, a –constant, we can permute the structure to arrive at three other social relations. For example, the next permutation is what Lacan calls “the discourse of the hysteric” (right). We’ll notice that in this discourse, that the barred subject ($) has moved from it’s position in the lower left-hand position to the upper left-hand position, S1 has moved from the upper left-hand position to the upper right-hand position, that S2 has moved from the upper right-hand position to the lower right-hand position, and that a has moved from the lower right-hand position to the lower left-hand position.
In other words, the three additional social relations are discovered by advancing each matheme forward in a clockwise motion while retaining the ordered relations between these terms. We thus find that if we keep the order of the terms constant, there are only four possible ways they can be combined (left). There’s no need to memorize the four discourses, for so long as you know the initial discourse or ordering– the discourse of the master –you can derive the other three by following the rule of keeping the relations between the terms constant and advancing each term one position forward. This is how Lacan discovered the four discourses. It wasn’t that he first believed that there’s a discourse of the master, university, hysteric, and analyst and then found a way of representing them with the mathemes. Rather, it’s that he formulated the formula for the subject and then wondered what structural permutations follow from it.
Structural analysis such as this– and this is what true structuralism is –have the effect of transforming psychoanalysis. This, at least, is what happened with Lacan’s own teaching. Over the course of his teaching he was perpetually revising and transforming both Freud based on bringing Freud’s discoveries into contact with the findings of linguistics, mathematics (especially set theory), symbolic logic, and ethnography. But Lacan’s own teaching changed significantly across time both as a result of his encounter with the matheme and what took place in the clinic. We can thus distinguish between two types of Lacanians: Oedipal or Talmudic Lacanians and post-mastery Lacanians that work on the premise that “there is no Other of the Other” and that “the big Other does not exist.” A Talmudic Lacanian is a Lacanian that restricts their discussion of Lacan and clinical practice to what Lacan taught, treating him as a master or Father who knows the truth, and endlessly interpreting that text in much the same way that the Talmudic scholar endlessly interprets Talmud without ever adding anything to it. The post-mastery Lacanian, by contrast, holds that Lacan showed us the way in terms of how he read and interpreted– for example, we get something entirely new in his way of approaching Freud, not a rote repetition of Freud –and in terms of how he worked with the mathemes. Recognizing that every Father or Master is castrated, that they’re all shams or imposters and semblances of mastery, the post-master Lacanian recognizes that Lacan said many valuable things, but that he didn’t say it all— indeed, Lacan constantly emphasizes that no one can say it all because “truth can only be “half-said” –and works with his teaching not as a closed system, but as a generative methodology for generating new insights that are remote from anything Lacan himself ever articulated.
Another way of expressing this difference would perhaps be as the difference between Dogmatic Lacanians and Dynamic Lacanians. The former are trapped in the endless interpretation of the letter of the seminar and internecine battles over who has interpreted Lacan correctly, while the latter are building on Lacan and developing his orientation (not thought), rather than simply repeating it. We see something of Dynamic Lacanianism in the Millerian school, where, in their work with the Borromean knot and “ordinary psychosis” (thank you Svitlana Matviyenko!), they’ve gone well beyond anything Lacan himself ever said or thought as a result of their encounters with the clinic and the strange new symptoms they’ve encountered that defy traditional diagnostic categories and structures. We see it also among Dublin Lacanians such as Rik Loose who have patiently attempted to develop a clinic of addiction that Lacan hardly ever touched or discussed. Indeed, as I argued last Saturday, we’re no longer even living in a universe of neurosis or the discourse of the master– though I didn’t realize I was arguing this, until Svitlana told me I was –but have entered a new form of social relation with new apparatuses of jouissance (Lacan calls the four discourses “apparatuses of jouissance) that require new formalizations of the social relation to be understood. I’ll save that for the book I someday write on Lacan. As an aside, if someone wants to organize a conference or edited collection on the enigmatic concept of jouissance, I’m game (though I won’t co-edit). After the symposium of Saturday– and it really was a symposium in the fullest and best sense of the word –I’m convinced that we’re now living in the age of the object, of the objet a or j-symptoms (jouissance-symptoms), rather than the age of s-symptoms or signifier-symptoms as were characteristic of the universe of the master. This has taken place for structural reasons. Lacanianism today requires a return to Lacan in the sense that Lacan returned to Freud rather than in the sense of endless Talmudic interpretation. Something similar could be said of the need for a return to Marx that would forget Marxism (though not entirely, of course) of the last 100 years. Perhaps the only thing that really needs to be remembered (rather than returned to) for us Lacanians today is the forgotten and repressed Bakunin and Kropotkin, whose anarchical politics is most resonant with the Lacanian teaching that the big Other does not exist and that there is no Other of the Other.
I digress. Let’s return to the question of fantasy. We saw, in the discourse of the master, that the formula for fantasy (for obsessional neurosis) appears in the lower portion of the graph ($<>a). I emphasize that this is the formula for obsessional fantasy, because, in Seminar 6, Desire and its Interpretation, Lacan distinguishes between obsessional structures of fantasy and hysteric structures of fantasy. Obsessional structures of fantasy– $ <> a –attempts to negate the Other through a detour in which the subject that would be directly related to the lost object of jouissance without the mediation of the Other. This is why the obsessional often seems so solipsistic. By contrast, in Seminar 6, Lacan argues that the fantasy of the hysteric has a very different structure represented by the formula “a <> -A-“; which can be read as “objet a punch barred or non-existent Other”. Both of these fantasies correlate quite well with Lacan’s aphorisms for obsession and hysteria. Of the obsessional, Lacan says that “the obsessional has a desire for an impossible desire.” While this formulation is much more complex than I’m here suggesting (and I’ve written quite a bit about it elsewhere on the blog), what could be more impossible than a $ or subject related to a without the traumatic and intermediary desire of the Other?
By contrast, of the hysteric, Lacan says that “the hysteric has the desire for an unsatisfied desire.” We have to be careful here as Lacan’s conception of subjectivity is always relational. Where does the hysteric wish there to be dissatisfaction? In himself or in the Other that is his partner? It can occur in either place. What terrifies the hysteric is the possibility of an uncastrated Master or S1. Why? Because if the Master to whom the hysteric’s desire were addressed were uncastrated (~$), then there would be no place for him to be. In other words, the hysteric wants to be the solution to the master’s lack or incompleteness. Therefore he uses his fantasy as a way maneuvering the Other to reveal a lack or incompleteness in the Other so that he might come to fill that lack. For example, we see the discourse of the hysteric in protest movements and the way that once they get once they want they remain dissatisfied and conclude their new master’s are shams or imposters because they need the master to be a sham so as to continue to exist as desiring subjects ($) and occupy the place of that which would fill or plug up the lack in the master. In the discourse of the hysteric we see this fantasy represented in the relation underneath the discourse as $-S2, where S2 is equivalent to ~A~ or the barred Other.
So we see that two of Lacan’s formulations of the fundamental fantasy appear in the discourse of the master (obsessional neurosis) and the discourse of the hysteric respectively. Here we see the power of the matheme to expand psychoanalysis, for the fact that we see these two fantasies appear in these two discourses now invites the question of whether there’s a new fantasy structure for the discourse of the analyst and the discourse of the university. Underneath the discourse of the analyst we see the relation S2 <> S1, while underneath the discourse of the university we see the relation S1 <> $. Are these new structures of fantasy that have hitherto gone unexplored in Lacanian theory?
Let’s start with the discourse of the analyst. There’s something scandalous in suggesting that there might be a fantasy underlying the analyst’s discourse as a well analyzed and trained analyst is someone who’s supposed to have traversed the fantasy, thereby becoming capable of setting their fantasy to the side so as to occupy the position of the remainder, loss, or surplus of objet a for their analysand. This is the ideal case. However, we could hypothesize that nonetheless there are perils of the analytic position that arise from the possibility of a residual fantasy structure at work in their way of relating to their analysand. In Seminar 10, Anxiety, Lacan teaches that the “punch”– <> –is to be read in terms of the logical functions of “and” (&, sometimes represented in symbolic logic as an A without the line) and “or” (represented in symbolic logic as v), as well as the greater than (>) and less than (<) signs in arithmetic. How does Lacan arrive at this thesis? Through a homology in the present that takes place when we break apart the four points of the losange, “punch”, or diamond that relates the two elements of each fantasy. This is why we potentially have 16 structures of fantasy or 4 for each of the 4 discourses, rather than just four fantasies simpliciter.
So for the analysts discourse we would have four potential fantasies that could disrupt practice: (S2 & S1), (S2 v S1), (S2 > S1) and (S2 < S1). Is there evidence for this? Let’s shift to an issue that initially seems unrelated. One debate that’s raged in Lacanian theory for years now is that of where to place perversion in the four discourses. Perversion, unlike psychosis, is a social link because the Other is not absent in the universe of the pervert. Lacan teaches that the pervert is the one that claims to have knowledge of jouissance who occupies the position of objet a rather than a position of desire (where desire is the absence of jouissance and an actual defense against jouissance). The story goes that the sadist treats himself as the agent of the Other’s jouissance (think of the torturer that says “I don’t like to do this, but it’s what God or the dictator commands”), while the masochist treats himself as the object of the Other’s jouissance (“Do with me what you will”). The best reading of sado-masochism, I think, is Deleuze’s Coldness and Cruelty (and Lacan concurs, in Seminar 16, as I recall). In each case, the subject presents not as a subject, but as an object determined by the will of the Other.
The controversy in psychoanalysis arises from the fact that the only place suitable for perversion in Lacan’s four discourses seems to lie in the discourse of the analyst, where objet a appears in the position of the agent. The problem is that the analyst recognizes that he is only a semblance of the objet a, whereas the pervert believes he really is the objet a. Fantasy theory allows us to distinguish the position of the analyst from the position of the pervert. The pervert is one who has the fantasy of S2 <> S1, who believes that they have knowledge of jouissance (consider Sade’s metaphysics that dictates the necessity of what he does based on beliefs about what nature wants), whereas the analyst recognizes that the relation between S2 and S1 has the form (S2 // S1) or that there’s always a gap between knowledge (S2) and our ability to totalize knowledge in a master-signifier (S1).
Fantasy theory nicely accounts for the position of the pervert and allows us to distinguish an analyst from a pervert. But we get three additional theories as well. In addition to the fantasy S2 <> S1, characteristic of perversion, we get the fantasies (S2 v S1), (S2 > S1), and (S2 < S1). The first of these fantasies, (S2 v S1), would be an exclusive disjunction that says “either S2 or S1″. This would seem to be a refusal to ever unite a symptom (S1) with its interpretation (S2). Perhaps, in the clinic, we find this fantasy at work in medically inclined psychotherapists, who hold to the thesis that symptoms never signify anything but that they’re just brute phenomena to be treated by pharmaceuticals (and I think some symptoms might have this form). In other words, this fantasy formation refuses interpretation or meaning. The formula of fantasy “S2 > S1” would be what Freud described as interminable analysis, where the analyst refuses to recognize a privileged, non-sensical, signifier as that which “names” the sinthome of the analysand, instead demanding more interpretation. In the final Lacan, the end of analysis is conceived as “making a name for yourself”– i.e., subtracting a non-sensical signifier from the order of the symbolic that would function as one’s source of jouissance in the Real –and distinguishes between “believing in the symptom” and “identifying with the symptom”. A person believes in their symptom when they think that there will be a final signifier– what Derrida called a “transcendental signifier” –that ties everything together and finally explains it all. Analysts can suffer from this fantasy with respect to their analysands. Lacan himself suffered from this fantasy when he wrote the Rome Discourse insofar as he argued that the symptom could be completely resolved in interpretation (i.e., he believed the big Other exists or forms a totality), and Freud suffered from it earlier in his teaching. The paradox is that this fantasy renders analysis infinite or interminable because rather than encircling an irreducible Real of jouissance as a “symptom-machine”, they believe that there’s a final signifier and so they keep interpreting these S1’s in terms of S2’s, never allowing the patient to leave analysis (“you’re not done yet!”). This is also a fantasy of mastery or the idea that S2 or the ~A~ can be mastered. Finally, we would have the fantasy structure S2 < S1. This would be a rather straightforward fantasy where the analysand treats some symptom (S1) as totalizing S2, telling their analysand to leave and that they’re done when there’s more work to be done.
The theory of the four fantasies for the analyst’s discourse could play a pivotal role in training analysis, helping the analyst to be to avoid perils of perversion, mastery, infinite semiosis, and premature conclusion. I’m sorry I haven’t been more clear in this post explaining the details of Lacanian discourse theory. I invite readers to fill out the other twelve fantasies corresponding to the other three discourses with their own clinical background and examples.