Amalgamating Lacan’s Formulae of Sexuation, Discourse Theory and Topology
WILLIAM J. URBAN
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[Start of second talk]
At the end of my previous talk I had indicated that the two sides of Lacan's logical square in no way enter into a relationship. This thoroughly frustrates reading the square all at once, for any such 'global' understanding would place it into a meaningful space governed by a universal notion. But it was seen how such a notion is attacked by the propositions on both sides, revealing that the notion as such is inherently split. I'd like to now turn to the similar split which traverses meaningful space itself and I'm bolstered by the fact that Lacan directly characterizes meaning as sexual shortly after placing his formulae on the blackboard at the start of his lecture on March 13, 1973. This raises the intriguing possibility that these formulae might be used to demonstrate the suspension of meaning.
One strategy to do this would be to establish how a certain path can be forged through the quadrants. Doing so would permit conceiving the square as rotating, which alone would truly bury the notion that the two sides are equivalent. More importantly it would lead us to the suspension point of this path. Critical to this undertaking is the amalgamation of the square first with the four discourses and second with topology. But before turning to this, I'd like to briefly introduce this sequential path in two different ways.
The first way might be called 'The Path of Negation' and is directly discernible from the propositions themselves where Lacan indicates negation by the horizontal bar he places over the function, the argument, both, or neither. The path starts out in quadrant where 'all say yes,' but proceeds to quadrant where 'one says no' and then on to quadrant where this exception is denied: 'no one says no.' The path thus seems to terminate in quadrant but since 'not-all say yes' amounts to negating all the previous contradictions, the Not-all delimits a space within which the entire sequence must be said to have taken place. Lacan tells us as much in his 1972 écrit entitled L'étourdit where he asks us to read the Not-all not as a particular, but rather as a singular element which paradoxically encompasses the universal All. In other words, the entire sequence → → → is effectively subverted by the singularity of quadrant .
The second way, 'The Path of Nothings,' is more involved, but recommends itself for a couple of reasons. By directly inscribing the nothings of Kant's Table of Nothings into the square, it allows us to imagine how space itself is transformed as we proceed from quadrant to quadrant. This is most productively done with classic examples of these nothings in mind. So God who inheres in quadrant is more easily envisioned than the nothing of evil defined as the privation of good in quadrant ; but while the purest evacuation of existence is found in quadrant , as in the case of pure space and pure time which function as the very form of our intuition according to Kant, the nothing of a 'square circle' in quadrant is even less than nothing, again qualifying this quadrant as thoroughly paradoxical, somehow permeating all the other nothings while simultaneously being constructed by them. The other advantage to aligning our thought to these nothings is that it allows us to begin assigning mathemes from discourse theory to each of these quadrants as per Kant's definitions. It's thus a reassuring guide when it comes time to assign a unique Lacanian discourse to each of the four quadrants.
Perhaps a greater reassurance is had with Žižek who recently mapped out a proposal for a 'unified theory' of the formulae of sexuation and the formulae of the four discourses. In hindsight it's all rather obvious. Given the four mathemes of discourse theory, can there be any better matheme than S2 to mark quadrant ? Here is the matheme denoting the capital 'O' Other, defined as a battery of signifiers which inscribes meaning within its signifying chains and thus nicely corresponds to the universal All. By extension, this makes S1 a natural fit for quadrant where an escapes the universal function, for this matheme is likewise an exception, the one signifier uniquely sticking out from all the others. Turning to quadrant where such exceptions are denied so that a spatial void is thereby inscribed, there is no better matheme to find there than the split subject [$], the empty lack between signifiers devoid of any substantive meaning. And finally, by the process of elimination, the paradoxical objet a is placed in quadrant which we can conceive as the impossible ontologization of the void. This is fitting as this matheme does in fact act as the impossible objectal correlate to the emptiness of an alienated subject faced with a constitutive split.
There are many benefits to having identified each quadrant with a single matheme. For instance, as S1 and S2 articulate the field of meaning, and as these have come to mark the first two quadrants, this establishes the hermeneutical circle as isolated to the right side of the logical square. This is a crucial step towards demonstrating how meaningful space is suspended by an element found in the left side. Additionally, this analysis allows us to treat the entire logical square as one giant rotating discourse so long as each quadrant is taken as one of the four places of discourse theory. Thus, with a 90° counterclockwise rotation the Md emerges and suddenly the analysis of meaning with respect to this discourse is readily applicable to the logical square. (As an aside, this also requires the two quadrants on the right to be interchanged. Doing so would realign the square with Lacan's own presentation of the sexuated formulae which doesn't strictly respect the original Aristotelian arrangement. In contrast I present them so as to allow for easy comparison with Aristotle. And while we're on the topic of presentation, you've no doubt also noticed that I've interchanged the two sides of both Aristotle's and Lacan's squares. This was done for a variety of reasons, but I assure you it's strictly a cosmetic change).
We can also treat each matheme in each quadrant as the agent of its own discourse. This permits assigning a unique discourse to each quadrant. Accordingly, the Ud is found in quadrant , the Md in quadrant , the Hd in quadrant and the Ad in quadrant . It now becomes obvious that amalgamating the square with the discourses places it into motion. This is easily seen by the arrows of discourse theory whose clockwise turn effectively trace out a sequential path through the quadrants of the square. You can follow this path in many ways. For instance, you can do so with the elemental meaning-relation in mind, a relation that is defined as the relation between S1 and S2. Now, as this pair travel through the square together, they provide a stabilizing force for each new discourse they come to occupy and thus allow it to maintain minimal cohesion in the face of those impossibilities and impotencies which continually threaten to tear them apart. But such stabilization-through-meaning is only the case for the first three discourses. For in the Ad, there is no longer an arrow to be had between S1 and S2. Instead, two parallel lines intervene between them, indicating that the meaning-relation has thoroughly broken down. In other words, meaning cannot stabilize the Ad, it cannot momentarily cover over impossibility and impotency as it does in the other discourses. Why? Because here the meaning-relation is itself rendered impotent. Again, this does not simply take place at the termination of the sequential path, for the Ad of quadrant must be seen as the real engine of the discursive turns which constitute it. So once again, where classical logicians have done their best to avoid logical impossibilities and impotencies, Lacan instead attempts to inscribe such aporia directly by making the very breakdown of the meaning-relation a resource. That is, the Ad nourishes itself with the breakdown of meaning. This should not to be overlooked. Yes, it's true, the Ad is the very science of the rotation of the discourses. It can thus easily aim its interpretation at the circulation of meaning in the other discourses. But it must be stressed that it does so keenly aware that what ultimately drives discursive rotations is the failure of the meaning-relation to definitively establish itself. We can thus look upon the Ad as offering respite from the pursuit of meaning. This is directly observable in the matheme for this discourse, where we see objet a in the place of agency. And as I've said many times already, this is a nonsensical point from which meaning can be suspended.
One way to arrive at this point is by following the meaningless twists, cuts and sutures of space that is the mathematical discipline of topology.
Lacan discusses the surface of these objects – called topological spaces – in the aforementioned L'étourdit which, by the way, might be translated as 'the-turns-said' provided we not lose sight of the fact that those who complete such a circuit of 'saids' are blunderers and scatterbrains. This particular écrit is unusually difficult even by Lacanian standards and I'd recommend anyone wishing to follow it to do so with the book length commentary provided by Christian Fierens. There you'll find ample illustrations of many figures discussed in L'étourdit which might provide you a nice guide as you battle through it. But I want to warn you of something, which I feel qualified to do given the strips of paper and other assorted objects in front of me: simply reading Lacan or Fierens will not be enough. Recall how Descartes defended his use of analysis in the Meditations by equating perfect understanding with self-discovery. And how Freud often lamented across the pages of his dream book that his discoveries were not widely accepted because so few possess the self-discipline needed to take up the hard work of analyzing their own dreams. Such statements have always terrorized me for their truth, a truth which I've come to learn somehow links the failure to understand with a reluctance to supplement the Word with the difficult practice of locating one's own subjectivity in the matter at hand. I feel this has never been more true than with Lacan's texts on topology which stand little chance of being understood until you clear your workspace of the usual tools of interpretation and take up a pair of scissors, a spool of tape and some sheets of paper to rediscover what Lacan initially discovered by cutting and suturing topological space. How else to understand Lacan's repeated claim that the möbius strip is nothing but its own absence and, moreover, is to be equated with the subject? Or else his more general claim that topology is not a metaphor for the universe but rather that which thoroughly captures the universe?
I feel it's time to get to work whenever such claims no longer appear obvious. And the work plan I've been using is the one before you. Here is an amalgamation of Lacan's logical square with topology. It not only demonstrates how the propositions have inscribed the self-split of meaningful space itself, but it will also lead you to the Ad where such space is suspended, provided you follow its sequential path. This path of course begins with the sphere of quadrant , whose perfection has made it a longtime favorite stand-in not only for the universe itself but also for the original harmony said to exist between man and woman. But with a pinch and a suture, the sphere can be transformed into the torus of quadrant , provided at least one point of the suture is cut away to the space of the exception. But you only truly leave the realm of the spherical when you cross to the left side where you are plunged into the topology of aspherical surfaces. Lacan does this by means of his 'conjuring trick' where the torus is deflated with a twisting pinch, then cut, then shuffled, then sutured back together. What emerges is the möbius strip of quadrant whose single surface is nonorientable unlike the sphere and torus where meaningful vectors can always be plotted. Moreover, whereas spherical surfaces have no boundary yet nevertheless bound space, the boundary of the möbius strip fails to do so. But it only achieves its escape from meaningful space through its pure absence. A more radical escape is had when the single edge of the möbius strip is sutured to the single edge of a spherical disk to produce the cross-cap of quadrant . This is a surface which has no boundary, yet unlike spherical surfaces the cross-cap cannot bound space. It accomplishes this by means of the spherical disk itself which is just as it sounds: a single disk cut away from a surface like a sphere or a torus. The spherical disk is thus quite curious in that it exercises a 'presence' on spherical surfaces, yet nevertheless is somehow able to stand outside meaningful space. Lacan calls this disk the 'out-of-line' point. But we know it by its more common name: the objet a.
Here is the suspension point of meaning.
Since the cross-cap embodies the entire path which led up to it, a cut to its surface will not transform it into a new topological space. If it did, it would be legitimate to ask what follows the cross-cap. Nothing follows. But this nothing is curiously ontologized. That is, upon reaching the cross-cap, we come upon the paradoxical cornerstone of the logical square whose own fragmentation coincides with the operation that lays it securely into place. The a-sphericity of the cross-cap thus provides a radical opportunity. For a cut to its surface will void a space of spherical meaning and kick the subject outside of meaningful space. Yet at the same time this releases objet a which ontologizes this newly opened domain of non-meaning. Here is a singular point such that if occupied by the subject, is capable of suspending the hermeneutical circle.
One final word of caution. It's tempting to conceive this suspension as an object which appears within the subject's visual horizon. True, this would testify to the subject's insight into how the ultimate impotency of the hermeneutical circle to deliver meaning nevertheless does not warrant its wholesale dismissal which might otherwise place the subject into an absurd realm. But the problem with this experience is that it registers itself too much on the subject side of the subject-object dichotomy, perhaps as in the case of the predominately academic Lacanian who anxiously finds himself in close proximity to the practicing Lacanian psychoanalyst. Of course anxiety is more 'objective' than any other subjective experience, as Lacan convincingly demonstrates in his tenth seminar from the early 1960s. But from the perspective of his work ten years later, it's clear that the experience of anxiety is not an automatic pass to the analyst's chair. For the analyst needs additional insight into how nonsensical encounters are only made possible because the hermeneutical circle is not up to its task to deliver a consistent field of meaning; here it's understood that non-hermeneutical phenomena wholly take place within the very space opened up by the failure of meaning. Saying it this way places the crucial accent on the object side, acknowledging how the subject only ever achieves an empty distance from the suspended field. This occurs when the subject 'fully' assumes the surplus object of meaning whose sudden appearance thoroughly coincides with the suspension. Simply said, objet a 'is' the suspended hermeneutical circle of meaning.
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