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Amalgamating Lacan’s Formulae of Sexuation, Discourse Theory and Topology


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[This talk was first given at the Toronto Psychoanalytic Society in the Summer of 2014.]

title slide reading Amalgamating Lacan's Formulae of Sexuation, Discourse Theory and Topology

I'd like to start off by thanking _____ and _____ for inviting me here to speak about some things I've been thinking about for the last few years. It's a pleasure to be here. Thank you both.

And I also thank all of you for coming.

Let me introduce myself. My name is William Urban and the title of my talk today is "Amalgamating Lacan’s Formulae of Sexuation, Discourse Theory & Topology." As you can see I have a script before me [wave it] and I intend to follow it, not simply because I'll get lost without it but because I'm convinced you can do alot worse than proceeding methodically, at least as a first approximation. I'll talk about my plan of attack in a moment.

But since I intend to eventually speak to you about that last term in the title – topology – it occurs to me that it might be appropriate to draw your attention to the fact that we're all sitting together in this room so as to occupy a common space. Now certainly this space has its general purpose and thus a meaning we all share, yet it no doubt also holds a personal significance for each of us. The question I ask is, which of these 2 levels – the universal or the particular – has the upper hand? For my part I'm tempted to vote in favor of the particular, at least today, given that I think of this room very much as the place where I got my official start as a student of Lacan and now I'm presenting as if I've achieved some sort of expertise. I assure you that this causes me no small amount of anxiety, especially given the fact that there is at least one practicing Lacanian psychoanalyst that I know of present in the room today. So I feel compelled to stress upfront that while I also self-identify as a Lacanian, that should be taken mostly in the sense of scholarship as I'm decidedly not a clinician. I've been on the couch reciting dreams but never at its head writing them down. But I can attest to the fact that my own analysis has proven invaluable to my understanding of Lacan, and I whole-heartedly recommend analysis to anyone wishing to further their understanding of what this Frenchman was trying to achieve, especially in the last two decades of his life which are so often glossed over in the university.

Now I've been advised that about an hour for a formal presentation is appropriate, so that's what I'm going to take. But I'm not going to talk continuously for that hour – I think that would be rather tiring for everyone.

slide giving outline of talk (1) Lacan on Meaning and Sexuation (40 mins); (2) Sexuation, Discourse Theory and Topology (20 mins)

Instead I've broken things up into 2 unequal chunks, the first being 40 minutes in length and the second 20 minutes which I'll deliver on either side of the break with a discussion period following each one.

So back to the title: that first term "Amalgamating" is very interesting. I came across it when I was initially asked to signify this talk back in May. This talk, by the way, is entirely based on my Dissertation which I wrote last year and defended 2 months ago at York University to earn a PhD in the Humanities. More specifically, it ideally targets the last 2 chapters where I amalgamate the three elements of sexuation, discourse theory and topology as they stood with Lacan in the early 1970's. But I didn't actually use this term there. In fact I rarely spoke of the amalgamation as such and when I did, I used perfunctory terms like "combining" and "integrating." In retrospect this is because I had not consciously set out to make an original contribution to the literature on Lacan but rather to the literature on textual interpretation which I thought would better satisfy the requirements of my degree. Accordingly, I set out to explore historical conceptions of textual meaning and ended up offering a radical alternative – namely, the suspension of meaning – which was done precisely by amalgamating these three Lacanian elements. The amalgamation of Lacan thus functions as a vehicle for the official aim of the Dissertation which concerns itself above all with meaning. But now that the amalgamation has become consciously centered in my mind, it's become an end in itself, and what continuously surprises me is how this end – which itself has no meaning – came about only because of a concern for meaning. So its entirely appropriate that it initially functioned as a means to another end, and also that I stumbled upon the term amalgamation whose root term amalgam cannot but bring to mind Mercury, the Roman messenger of the gods. Here is the original interpreter whose task was to render the strange pronouncements of the gods into meaningful understanding for us mortals. What Mercury thereby marks for us is the fact that already at the dawn of history there is a struggle with meaning and one that has been ongoing to this very day despite Lacan having revealed to us its suspension point. After the break I'll discuss my own efforts to rigorously and systematically demonstrate this point, but I'd like to first contextualize Lacan's thoughts on meaning so we can better appreciate his historically unprecedented accomplishment.

slide plotting historical timeline of Lacan's and Heidegger's careers; Reformation and Romanticism also plotted; other text includes: meaning qua object, meaning qua medium, hermeneutical circle, part-whole, hermeneutical phenonmenology, 1933 to mid-1950s herm phen, mid-1950s-60s post-structuralism, 1960s-1981 non-herm phen, 16th c., 19th c., 20th c.

Throughout the medieval period there was really only one text to speak of – Scripture – and struggles with its meaning were exclusively managed through the discourse of a single master, the Roman Catholic Church. This all changed in the early 16th century when the Protestant Reformers broke with this authority to decide on questions of interpretation themselves. Very quickly biblical exegesis fragmented into a multiplicity of interpretive techniques, eventually coalescing into hermeneutics, a field which borrows its name from Hermes, the Greek counterpart to the Roman god Mercury. Now in the hands of the early 19th century Romantic thinkers, the scope of hermeneutics was expanded to encompass any text, not just Scripture. It's here articulations of the famous hermeneutical circle first appear, which simply says that the part derives its meaning from the whole, while the meaning of the whole must be sought at the level of the part – a seemingly endless, inescapable circle. At this time the status of meaning remained as it always was, one which still largely informs our spontaneous understanding of it. That is, here is a focus on the object-like quality of meaning, the treatment of meaning like an object external to us whereby meaning is said to be 'over there' or 'in the book' but in a way that is often hidden from us so as to require the employment of appropriate interpretive techniques to unearth it. My research has lead me to conclude that this particular conception dominated historical theories of meaning up to the third decade of the last century when an irreversible event took place which fatally challenged this conception. This event was the 1927 publication of Martin Heidegger's Being and Time which effectively announced to the world that hermeneutical phenomenology had arrived. What Heidegger's questioning into the "meaning of being" revealed is a meaningful dimension logically prior to the subject's treatment of meaning as an external object. That is, according to Heidegger meaning should no longer be thought of as an object but rather as a substance or medium which encompasses any subject-object dichotomy. By all accounts this newer treatment of meaning as a medium has proved highly persuasive, dominating the theorization of textual approaches ever since and indeed, must be said to characterize Lacan's initial approach from just after his Dissertation in 1932 up to the mid-1950s. This first period of Lacan's career culminates with his Rome Discourse of 1953 – an écrit which undeniably bears the distinctive mark of Heidegger.

But given Lacan's subsequent turn to structuralism, we can assume he wasn't completely satisfied with the hermeneutical phenomenological approach he inherited from Heidegger. While this second period of his career is relatively brief, it nevertheless does produce a series of écrits most favored amongst his academic followers in the university. Such a reputation is well-earned as Lacan elegantly reiterates many basic structuralist insights into how meaning is defined by differential signifying systems such that, for instance, any one signifier can only mean what the others do not. Yet Lacan can never be labeled a simple structuralist or even a (post)structuralist despite the frequent comparisons to Jacques Derrida. For in these years he takes pains to stress the difference between the meaningless structural mechanisms of signifiers on the one hand, and their imaginary meaning-effects on the other. That is, meaning is demoted to the imaginary register, as that which covers over its own constitution in the symbolic register of signifiers. Accordingly, we are to set aside the meaning-effects of signifiers to instead turn our analytic gaze to decipher the movement of these signifiers in an effort to uncover the truth of unconscious formations. This is to better delimit the causal forces he believes to be at work in the field of meaning. Indeed Lacan parts company from thinkers like Derrida by his instance that there is a decentered cause of meaning which he equates with the signifying structure itself.

slide depicting the Retroactive trajecetory of meaning of the Signifying chain which proceeds from left to right through Lacan's signifiers S1 and S2; text below this figure reads as It was light... (a) outside (b) so he could carry it (c) reading; phallus qua signifier vs. phallus qua signified

It's during this time he introduces the notion of the button tie which momentarily anchors down meaning, thereby solving the subject's dilemma of having been caught between a meaning which retroactively posits itself on past elements and one which has a future orientation, anticipating the meaning to come. To illustrate, if I begin but don't complete a sentence with the words "It was light..." at best these words anticipate a limited array of possible meanings. But it's only by actually completing the sentence that the meaning is buttoned-down. It's also during this time Lacan makes a distinction between the meaning of the phallus (the subjective experience of pulsating between a joyful grasp of meaning and the painful inability to fully articulate it) and the phallic signifier (a pure signifier which means meaning as such). In the early 1970s, the phallus will additionally be conceived in its capacity as a function within the context of inscribing sexual difference into logical terms.

If Lacan's second period is characterized as taking a negative distance from hermeneutical phenomenology so as to land on (post)structuralism (for lack of a better term), his final 2 decades offer a more direct confrontation with Heideggerian thought. Starting in the 1960s, Lacan recognizes his previous work with signifying systems as ultimately involving a real dimension which only ever registers itself as a disturbance in those systems. The real – once simply the sensory-laden background of a symbolic subject – is now that which acts as the absent cause of the symbolic itself and accordingly the results of any exploration into the ultimate Cause of meaning would not land one in the symbolic but rather in the real. Now, if this real Cause is said to possess a being, this definitely is not a Heideggerian being. Lacan effectively addresses such concerns towards the end of Seminar XI when he discusses the vel of alienation.

slide depicting 3 venn diagrams from Lacan's Seminar XI (1964): The Vel of Alienation and Separation between money and life, Being and Meaning, and subject and Other; Lacanian mathemes/algebra $, S1, S2, a, S1-S2; non-meaning and nonsense

Just like how the choice of life is rather forced upon you when a robber tells you to choose between it and your money, the same with the subject with respect to its choice between its being and its meaning. For Lacan, meaning is compensatory damages for a loss of being, but unlike Heidegger who calls for a deep hermeneutic to reestablish a connection to this being, Lacan paradoxically calls for a further separation of the subject from this entire scenario. This lands the subject on the famous objet a which momentarily halts the meaningful vacillation of the alienated subject between its being and its meaning; so separated, the subject gains an 'empty' distance towards this meaningful existence precisely by identifying with this object which embodies the jouissance that supports the experience of alienation. The possibility of such a separation is what I wanted to demonstrate in a rigorous and systematic fashion using Lacan's early 1970s work in sexuation, discourse theory and topology. The reason I turned to this period of time is that in the 1960s, the real was still largely considered only in its traumatic impact on the subject which somewhat hinders its full theoretical articulation. Indeed the Venn diagram of Separation originates with Jacques-Alain Miller; it's actually found nowhere in Seminar XI nor anywhere else in Lacan's work. I felt that this theory of separation could be better demonstrated by turning to the 1970s where the real is conceived not only as traumatic but also as impossible. Accordingly, Lacan's efforts in these years moves in a more 'scientific' direction. He shifts his focus to logic and mathematics and what results is a great development of his mathemes which allow for a more 'objective' articulation of the subjective experience of separation.

These efforts begin with Seminar XVII where Lacan introduces his Theory of the Four Discourses.

slide depicting the Theory of the Four Discourses from Lacan's Seminar XVII (1969-70) using the mathemes $, S1, S2, a; the four places of discourse are marked by agent, truth, other, product/loss

This theory is designed to address the question of how the signifier and jouissance are related and thus clarifies the subject's relation to the system of signifiers. In terms of mathemes, this is a question of establishing a link between objet a and the signifying chain. You can see this in the Master's Discourse (Md) where the arrow used to represent the retroactive trajectory of meaning has been superimposed. As the signifying chain proceeds from left to right (from S1 to S2), it produces objet a, but as immediately lost to a subject which also emerges from this very chain. Yet the mathemes indicate that the emerging subject does so as split. Now, the simple question I've always had ever since first hearing of Lacan's split subject is the following: "Between what two or more elements is the subject split?" The Venn diagrams Lacan uses in Seminar XI seem readymade to provide the answer: of course the subject is split between its being and its meaning. This is essentially correct, but again it must be understood how this being is quite unlike the being pursued by the hermeneutical phenomenologist. First of all, it's so devoid of meaning that it lacks even a modicum of sense. Rather, it must be said to be a bit of nonsense. And second, it appears as utterly impossible. For if meaning is a forced choice, there is really no other option before the subject other than assuming itself as a meaningful subject. So again I ask: why call the subject split? The mystery is cleared up at once when it's realized how the split is, in a way, doubled. That is, the subject is split between itself in its meaningful dimension and... the split itself. This latter element is of course nothing. But it's a curious nothing in that it nevertheless has doubled itself into an 'something' which carries a certain material weight in the subject's meaningful universe. Here is the other choice before the subject, an impossible choice – the objet a – which, once assumed, can suspend the hermeneutical questioning into the meaning of being. In other words, when the subject identifies with this object, meaning is effectively suspended.

But in no way can the subject grasp such an impossibility from within the Md. In his usual cryptic style, Lacan explains this on March 11, 1970. I quote:

'I am a little analyst, a rejected stone initially, even if in my analyses I become the cornerstone. As soon as I get up off my chair I have the right to go for a walk. That is reversed, the rejected stone which becomes a cornerstone. It may also be, inversely, that the cornerstone goes for a walk. It's even like that that I will perhaps have some chance that things will change. If the cornerstone left, the entire edifice would collapse. There are some who are tempted by this.'

Lacan here speaks of objet a in two ways. While it's in the position of loss in the Md, he speaks of it as a rejected stone. But when it assumes the position of agency in the Analyst's Discourse (Ad), it becomes a cornerstone. Now, in as much as we read objet a as embodying the very limits of the subject's meaningful universe, the possibility is thus raised that when the subject fully assumes this cornerstone by acting in the capacity of an analyst, the interminable slide of signifiers is halted and the subject's life-world and all the meaning it entails is suspended.

These two discourses are thus uniquely linked and are, in fact, inversions of each other. The Md is the essential discourse retroactively producing meaning without restraint and the Ad reverses this process, putting an end to meaning production through identification with a nonsensical object. The Md thus produces the object of its own demise provided the subject fully appropriate it and it's in this sense that this 'other side of psychoanalysis' is worthy of an entire seminar being devoted to it.

Now, what strikes me about this seminar is how the formulization of discourse in general and the exploration of the Md in particular is what originally allowed Lacan to uncover a notion for which he is infamous, namely, the fact that 'there is no such thing as a sexual relation.' He explicitly tells us as much at the end of his lecture that particular day in March of 1970. Given this, it's not surprising that in the following seminar he turns to directly address this notion, introducing in March of 1971 four formulae which attempt to inscribe this notion in logical terms. However, this is only its first writing, for these four formulae undergo a series of revisions over the course of another year to only reach their finalized state in early March of 1972. We have Guy LeGaufey to thank not only for having traced through these reformulations, but especially for his explanation of the finalized version of the formulae themselves which he uses to critique existing clinical practices. His work has greatly informed my own attempt to adapt the formulae for textual purposes with the ultimate aim that they can be used to demonstrate the suspension of meaning once amalgamated with discourse theory and topology. As I said before, I'll directly address this after the break, but in the remaining minutes I'd like to provide a sense of what these formulae attempt to capture by way of their original derivation from the Aristotelian logical system which Lacan supplements with modern logic. LeGaufey's account of this derivation is excellent and I'd recommended his work on this basis alone.

slide depicting the Classical Aristotelian Logical Square with 4 propositions: All S are P (universal affirmative), Some S are P (particular affirmative), All S are not P (universal negative), Some S are not P (particular negative), with arrows expressing the logical relationships as contradictories, contraries, subcontraries, subalterns; includes traditional A, E, I, O shorthand

Soon after Aristotle's death, logicians began to arrange his work systematically and eventually produced what is known as the Classical Aristotelian Logical Square. Such a square is something you might take up in a first year course in logic and to the non-professional logician, it can certainly be daunting. And as you'll soon see, this only gets uglier after Lacan gets done with it, as the conceptual demands he makes of us has discouraged even the professionals. Yet I believe Lacan's transformation of the Aristotelian square into his own square – one that is effectively equivalent to his sexuated formulae – is something that is readily graspable, at least in a general sense.

To understand this, consider how Aristotle's original square appears highly structured: like any square it has four quadrants and in each of these quadrants is written a proposition in generic form, where S stands for Subject and P for Predicate. The two universal propositions in the top two quadrants affirm and deny that a predicate belongs to all of the subject, while the two particular propositions along the bottom do the same but only for some of the subject. As well, each of the propositions are logically related to each of the others. The result is a square which appears well balanced, symmetrical and highly defined. Once you enter it, it appears you can only move from one quadrant to the next in a pre-determined logical path, eventually ending up back at your starting point.

The circular aspect to the square might remind you of the one likewise forged by the hermeneutical circle of meaning, doubly so when you consider the analogy to be had between the universal and particular levels of Aristotle's square and the whole and the part of the hermeneutical circle. Such an analogy raises the possibility that we could rigorously extract out truthful meaning from a text. To illustrate, suppose you want to make a study of Lacan but know nothing of him. But you get it on good authority that Lacan is Heideggerian (no doubt from a first-period Lacanian). Now if you read some passages from his Rome Discourse, it's likely to confirm for you the truth of his Heideggerianism. But upon encountering his 'Instance of the Letter' written during his second-period, this universal thesis would need to be rejected. Yet given your reading of the Rome Discourse, this new evidence would not permit now calling Lacan Derridean. Rather, an entirely new thesis would need to be formulated. And this thesis would remain intact until additional textual evidence is encountered requiring yet another reformulation. It's obvious this process can continue ad infinitum, ever fine-tuning the universal thesis on Lacan through which nuanced meaning is continually unearthed from his particular texts.

This might all seem like common sense, but that's exactly the point. The example is convincing because it's logical. While I didn't identify the logical relations in play here, the example follows the rigorous contours of the Aristotelian square. However, what Lacan realizes is that such rigor is in appearance only. He reaches this conclusion after reading a 1969 article by Jacques Brunschwig who uncovers an irreducible equivocation harboring in the square. Now, given that logicians have attempted to improve on Aristotle for centuries, we can conclude that they have also either explicitly known of this problem, or at least had their suspicions. But far from aligning himself with their attempts to construct an unequivocal logical system, Lacan does something quite different: he centers the equivocation, he builds with this equivocation, he tarries with it, he blows it up. In short, what his formulae of sexuation do is inscribe that very equivocation itself into its set of logical formulae.

slide depicting the The Lacanian Logical Square (The Formulae of Sexuation) just below The Classical Aristotelian Logical Square with 4 propositions written with Lacan's mathemes of sexuation with accompanying text: All x are submitted to the phallic function (universal affirmative), There is (at least) one x which is not submitted to the phallic function (particular affirmative), There is no x which is not submitted to the phallic function (universal negative), Not-all x are submitted to the phallic function (particular negative); with arrows expressing the logical relationships as contradictories, equivalences; numbers 1, 2, 3, 4 mark the four quadrants of the logical square

What Lacan accomplishes here is no mean feat. But it's crucial to recognize how he in no way removes the equivocation. In a way he makes it much worse. If, in Aristotle's square it was somewhat localized, in Lacan's square it's rather spread out everywhere, deeply problematizing the propositions through specified relations that defy common sense. To be sure this has made the understanding of the formulae of sexuation notoriously difficult; the dearth of commentary on them in the secondary literature is no doubt proof of this.

So where is the equivocation? Well, it has to do with the particular proposition. Here's an example. If I say that 'Some of Lacan's texts are Heideggerian,' few of us would conclude that this is just a particular instantiation of the truth that 'All of Lacan's texts are Heideggerian.' But that's what a classical logician would do. In contrast, most of us would naturally hear this statement as excluding such a universal proclamation about Lacan, even if we know nothing of the man. Already with this example we go a long way towards understanding why Lacan chooses to place the universal and the particular levels into contradictory relation whereas Aristotle largely worked with them as implying each another.

But the way Lacan paradoxically conceives compatibility through contradiction and alternatively, contradiction in equivalency, cannot be fully appreciated without attending to the most obvious change he makes to the Aristotelian square discernible at the level of the writing of each proposition. For Lacan clearly rewrites Aristotle by way of Frege's invention of quantifier notation which 'mathematically' analyzes sentences of ordinary language. What Frege does is re-conceive the subject as an argument and the predicate as a function. Thus, when the argument 'Lacan' is placed into the function '( ) is Heideggerian,' this significantly changes the way we conceive the declaration that 'Lacan is Heideggerian.' Each of the four sexuated formulae are composed with these two parts, namely, a specific argument which is or is not submitted to a function – the phallic function, of course. This function in itself adds an enormous amount of complexity, but I suggest to you to initially treat it as any old function in mathematics. What this encourages is thinking with the sexuated formulae in realms of thought that are not inundated with pre-conceived notions of sexual difference. Indeed, it's often overlooked how these formulae readily lend themselves to other fields of thought precisely because they're written as logical propositions which, you'll notice, come complete with the algebraic symbol algebraic symbol x, a symbol traditionally operating as a variable ready to be filled with a variety of terms.

Strictly attending to the logical level will also confront you with how these formulae inscribe the very notion of notion, with the notion as such. To better bring this out Lacan uses the existential quantifier for two of the arguments. This is most readily understood in the propositions to the right where the notion that all algebraic symbol x are submitted to the function algebraic symbol x is confronted by the exceptional existence of at least one algebraic symbol x which is not so submitted. A blatant contradiction, but one that is nevertheless resolvable since the notion does not imply the existence of any algebraic symbol x and, moreover, the algebraic symbol x's that do exist can only aspire to the notion by arrogantly drawing a line over their own existence. So despite the contradiction, since these propositions inscribe two radically different levels, the truth of one does not necessarily interfere with the truth of the other. Hence it's always possible to declare Lacan to be Heideggerian or Derridean despite the lack of conclusive textual evidence; instead what is actually found are texts which, at one extreme, fall far short of these men and at the other extreme, parody their work to ridiculous excess. Here is the basis of the endless hermeneutical circle operating in the secondary literature on Lacan whereby one scholar will critique another by pointing out the poor choice of supporting textual material, to then give its 'correct' reading or else propose another thesis altogether. Yet as the two propositions to the right make clear, this is really a foregone conclusion, for any use of existing textual material cannot but undermine the interpretive thesis it's meant to illustrate.

And that's the point. The sexuated formulae effectively inscribe Lacan's concerted effort to undermine the universal notion once and for all, but in a way which leaves Heidegger's own efforts to do the same far behind. Indeed hermeneutical phenomenology also takes issue with the notion, echoing as it does the standard Romantic complaint of how abstract thought fails to capture the wealth of life itself. This is certainly a good start and one which of course defines Lacan's first period. The problem is that it's incapable of offering up a termination point that might suspend its endless investigation into life for an existence deemed essential to unveiling the being of the world. This limitation of hermeneutical phenomenology can be articulated with Lacan's square as it rather confines itself to the right side. But Lacan overcomes this limitation. That is, he supplements the right-side attack on the universal notion by a subversive attack launched from the left. Compared to the right which has a more classical orientation, the left side uses modern conceptual tools like Russell's paradox to better delimit the equivocation originally found to haunt the Aristotelian square. In this left side the setting aside of the notional All by the existence of its particular is affirmed in the Not-all of quadrant symbol for quadrant four of Lacanian logical square, a numerical 4 inside a box. But those algebraic symbol x's which do not submit and the Not-all which do submit don't magically recombine back to the whole of the universal. To prevent this from happening, Lacan supplements the Not-all with the proposition written in quadrant symbol for quadrant three of Lacanian logical square, a numerical 3 inside a box which states that there is no algebraic symbol x which does not submit. Thus, in the hands of Lacan, the notional All takes a double hit, first from the right where an exception trumps the notion's universal reign by its very existence, and then fatally so from the left insofar as there is no exception and what does exist cannot form into an All. In its most consequential reading the notion must be said to be thoroughly split and what ultimately does this splitting is the Not-all. This Not-all indexes a singular existence entirely without essence – a prospect unimaginable to Heidegger.

But the conclusion that Lacan attempts to inscribe the inherent split of the notion into logical notation is not the only way to characterize his logical square. Another way proceeds by attending to the spatiality that is likewise inscribed by the same notation. Already it might be suspected that Lacan's square fractures the unity and cohesiveness of the meaningful space traversed by the Aristotelian square whose propositions on either side stand in well-defined logical relations. In contrast, equivocation so pervades the Lacanian square that it cannot be said to traverse a unified space which might allow its two sides to enter into a relationship. For his square is thoroughly asymmetrical, inviting us to think the one and same space as thoroughly self-undermining, or how meaningful space is isolated to its right side with a suspension point to the left, which amounts to the same thing. After the break I'll discuss my own efforts to isolate this suspension point to quadrant symbol for quadrant four of Lacanian logical square, a numerical 4 inside a box by amalgamating the square with discourse theory and topology. Not surprisingly, this suspension point turns out to be objet a. But now I'd like to open it up for discussion.

[End of first talk]

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