SEXUATED TOPOLOGY AND THE
SUSPENSION OF MEANING

A NON-HERMENEUTICAL PHENOMENOLOGICAL
APPROACH TO TEXTUAL ANALYSIS

WILLIAM J. URBAN

CHAPTER 7

SETTING THE FORMULAE OF SEXUATION IN MOTION

Well then, you are going to take up L'étourdit again and off you go, give us what will be, not your reading, but the way in which you are going to follow it.651

As the derivation, explication and relevance to textual theory of the Lacanian logical square are the primary aims of the previous chapter, the thesis of Part II that this logical square can be used to think the suspension of meaning has not yet been satisfactorily demonstrated. Nevertheless, by establishing the spatial asymmetry of the logical square important groundwork has been laid and the present chapter seeks to expand on this understanding by turning to L'étourdit.652 For in this text Lacan takes up topology in a manner suggestive of the further spatial manipulations to be had with his sexuated formulae. Indeed topology permits thought to aim beyond the set theory at stake in the logical square as presented above. This is a crucial step towards demonstrating our thesis since thinking in terms of set formation is what largely restricted the previous discussion of the logical square to its hermeneutical horizon. As was seen there, in the circumstance where well-defined sets encounter elements of the same order existing in domains refusing to similarly collectivize into meaningful entities, one could at best speculate that these entities might be suspended. That is, approached from the perspective of set formation alone the precise mechanism of such a suspension remains elusive. However, by merging the four discourses with the four sexuated formulae, the Lacanian logical square effectively begins to revolve and the spatiality it inscribes becomes transformed. Following these spatial twists and turns makes it apparent that the hermeneutical circle rotates in a field which is itself suspended from a singular point. In Part I this suspension point was spoken of alternately as a sublime object or a nonsensical element. But in sexuation theory it is identified with the Not-all. The wager of the present chapter is that topology leads thought to this suspension point by effectively equating the Not-all with the subversive structure of the cross-cap. The recent major commentary on L'étourdit by Fierens proves invaluable and is closely considered.

Section 7.1 introduces a twofold path or circuit through the Lacanian logical square leading up to the suspension of meaning. This introduction is first made using the logic of negation and second by making use of the Kantian table of nothings. Comprised of an initial trajectory with a subsequent reversal, the two legs of this circuit are expanded upon in the following sections. Section 7.2 establishes the hermeneutical circle in the right deixis by integrating the four discourses into the logical square in order to demonstrate the breakdown of the meaning-relation. Section 7.3 explicates Lacan's use of topology and then integrates it into the logical square to demonstrate the suspension of meaning.

7.1 Counting Nothings in the Land of Nyania

If the Lacanian logical square is to be used to demonstrate the suspension of meaning, it must ultimately be placed into motion. Hence a major effort of the present chapter is to establish how each of its four propositions should not be thought of as isolated to its respective quadrant but rather as inscribing elements and spatial relations that overflow one from the other. Certainly the previous chapter has already taken a significant step in this direction by establishing their interrelation. But as each proposition was explicated in turn through the others, what may have been lost is how these relations are not chaotic but do have a certain order or ranking beyond what has already been said regarding the status of the exception and the a priori nature of the left deixis. This chapter suggests one way to conceive this ordered movement and the present section provides an introduction to it, sketching out the suggested twofold path to follow so that the suspension point of meaning can be unveiled. Laying all our cards on the table, this path is none other than the movement through quadrants → → → of the logical square, with a return trajectory that underscores how the Not-all of plays a part in the constitution of each of the other three while itself being constructed by them. These two trajectories are taken up in more detail in Sections 7.2 and 7.3. While this section is largely introductory, by tracing out the entire circuit once via the logic of negation and once more via Kant's table of nothings, it nevertheless does prove valuable in underscoring the importance of both prohibition (i.e., the prohibitive 'No!' of negation) and spatiality (i.e., inscribed nothings) to the Lacanian logical square.

One way to understand the movement → → → is by recognizing how it proceeds as a series of successive negations. Simply stated, the universal affirmative is negated by the particular affirmative , which in turn is negated by the universal negative , followed by a further negation by the particular negative . This succession of three negations (necessitating the use of the bar of negation four times) is indispensible for entering the space of the Not-all. In more detail, the movement begins with the 'No!' said by Epimenides to the phallic function . This effectively divides the right deixis into two halves and what is at stake in this division can be reduced to the logic of sets. That is, where quadrant holds the set of the phallic, in quadrant there exists the set of that which is not phallic. Yet this neat division is upset as soon as a move is made into quadrant . For the proposition found there, identical to the one proceeding it but with a negating bar over the existential quantifier, affirms that 'There is no that is not phallic.' This additional negation calls the bluff of the One and thus effectively empties the Epimenidean set. But this does not leave the set of the phallic intact, for the further movement into quadrant finds there a proposition which negates the universal as well. Thus the phallic set turns out to be Not-all there is.653 So at the end of this trajectory what exactly is left? The present chapter endeavors to show that here lies the sought-after sublime object, a contingent piece of the real capable of suspending the structured rotation of the hermeneutical circle whose meaning subsists in the imaginary register articulated in the deixis to the right. This can already be understood by more consequentially following the third arrow of the above succession. In this case directly negates and this can fruitfully be conceived as the impossible ontologization of the void inscribed in the latter quadrant.654 As seen below, this ontologization becomes a crucial factor when it comes to the question of the suspension of meaning.

For the present moment, the suspicion that the last two negations in the above succession have effected a departure from set theory is well confirmed by Lacan himself in L'étourdit. In introducing the two propositions from the left deixis, he notes how

'[t]heir inscription is not usual in mathematics. To deny, as the bar over the quantifier marks it, to deny that there exists one is not done, and still less that forall is fornotalled (pourpastoute). It is here nevertheless that there is revealed the sense of the saying, from the fact that, combining there the nyania (thereisnotonewasdenied)... it supplies for the fact that between them, there was no relationship (de rapport nyait pas).'655

What Lacan draws attention to is the fact that the writing of and departs from ordinary mathematical usage to arrive elsewhere. In terms of the present discussion this is a shift from the theory of simple set formation (in play in the previous chapter) towards a dimension best served by topology.656 This is also a shift from the fact of enunciating (i.e., that of the Epimenidean order) to the fact of saying. Either way it is a shift from the right deixis of sets to the specific topology inscribed in the left, a shift caused by the nyania which inscribes the fact that there is no relationship between the two sides. This term nyania is a neologism by Lacan and provides a sense of the extraordinary difficulty of following L'étourdit. But as is generally the case, even such mysterious signifiers can be understood provided one is prepared to work through the Lacanian text.657 As it turns out this nyania is rather simple to unwind. It is composed of two negations. The nia [was denied] refers to the exception in the right deixis, the One who enunciates a 'No!' and thereby exempts itself from the set of the phallic. Epimenides, the man who enunciates how 'All men are mortal' and the Freudian primordial father all fit this bill. The second negation, n'y a [is not], is the negation of the existence of this exception which creates an exit out of the logic of sets and an entrance into the specific topological domain of the left deixis. With nia equivalent to and nya to , the combined nyania is thus none other than . According to Figure 6.2, this proposition is equivalent to , while the Not-all is equivalent to the existence of the exception so that a set might exist part phallic and part not phallic. Distance from such an understanding was taken soon after the Lacanian logical square was formally presented in Chapter 6, but the path suggested by L'étourdit allows one to truly bury the notion of such equivalencies. For traversing the logical square as → → → thoroughly breaks with any and all symmetry between the two sides. The crucial moment arrives with the movement into the left deixis beginning with the nyania which clears the space to allow 'everything' to be said of the element 'even if it proceeds without reason.' Yet this 'everything' does not return us to the All, for 'it is an all (tout) outside universe, which is read right away from the second quantifier as notall.'658 As the text proceeds it becomes clear that 'reason' concerns the way the exception limits the universal affirmative, providing it with a boundary that effectively makes of the right deixis the place of the One. But what happens when reason fails, that is, when the exception is denied? In this case it is not so much that the All is suppressed; rather, it immediately begins to overflow the confines which rendered it a universal One, an overflow which can be conceived as occurring in the space cleared by the nyania. This all that is not All is the Not-all. The Not-all is what exceeds the All, being 'outside' or 'greater than' the universe and the universal All.

But there is a surprise awaiting the reader on the following page. Contrary to what one might expect from studying Figure 6.2, the fact that the Not-all negates the universal nevertheless does not make it a particular. Lacan instead characterizes the Not-all as 'the singular of a "confine" ("confin")' which he immediately links with jouissance. Now, in English as well as in French, the term confine suggests a bounded region and if this term stood alone it would certainly fail to distinguish the Not-all from the All, the latter of which must precisely be conceived as such a region kept within the limits set by the exception.659 The Not-all must of course be conceived differently and Lacan achieves this conception quite precisely with the expression just cited. Understanding this requires a brief etymological digression. In both French and Old French the noun of this expression is always said in the plural, as confins, which additionally translates as a 'border' or 'boundary.' Its Latin roots are found with confinium [boundary, limit] derived from confinis [bordering on] which in turn is derived from com- [with] and finis [an end, limit]. By unconventionally dropping the final 's,' the endeavor here is to return to a sense better captured by its etymological roots and by further stating how we are to take the singular of this term, Lacan doubly marks this fact to ensure it will not be missed. His reading of 'confine' in the singular thus frustrates the conception that the Not-all qua confine is a region restricted to set limits. Instead we are effectively asked to recognize in the Not-all the very limit as such. Now, by embodying the very limit itself the Not-all completely subverts the logic of the universal and of the particular. For it directly negates the former but cannot be equated with the latter either, having attained itself by passing through the nyania which negates the exception. Since the singular is usually conceived in the sense of 'the one and only' and since the exception can legitimately be read as affirming the existence of just that, this cannot be what is captured by the Not-all. If it did the Not-all would reduce to the exception. But the Not-all only comes to be through the denial of the exception. Thus the Not-all must be radically reconceived: the singular of a confine is that which paradoxically carries within it the universal, the exception which unifies and limits, and the negation of this exception. That is, all of the propositions in the Lacanian logical square are effectively comprehended in the space of the Not-all. As Fierens says, what is at stake with the Not-all 'is a new way of thinking, which no longer classifies in boxes or logical places, but whose logic, since it is a journey, is constituted from changes of logic.'660 So initially the path proceeds quite linearly through the quadrants of the logical square, but upon reaching the Not-all the path suddenly takes a paradoxical twist. Here a return trajectory commences, underscoring how the sequence → → → had taken place within the space of the Not-all of all along.

Neither universal nor particular, the Not-all is singular. The importance of having arrived at this result cannot be overemphasized for the project of suspending the hermeneutical circle of meaning. This singular element, suspected to exist in certain aesthetic theories in Section 4.2 and directly theorized by Freudian-Lacanian thinkers throughout Chapter 5, is now established as occupying quadrant of the Lacanian logical square. What should be recognized is how the Not-all of is an element doubly-inscribed, operating both as part of and the space within which the sequence → → → proceeds. The Not-all thus schematizes an element which overlaps content with form, an element found in the sequence yet paradoxically embodying the very sequence which disclosed it. To identify with this element would thus be to identify with a suspension point to the sequence. This has direct implications for textual analysis. Although it has not been directly shown how the classical hermeneutical circle is confined to the right deixis, already the fact that the Not-all breaks with the particular makes it highly likely that this is the case. For where else could the dynamic between the universal and particular play out if not between the first two quadrants? The third quadrant is thoroughly empty and the fourth contains a singular element. If this is the case, it might be conceived how the universal (whole) and particular (part) turn in an empty space, generating a singular element that simultaneously embodies the very form of their reciprocal exchange. Extracting out and identifying with this element would then suspend the circular turn. In more concrete terms the circle begins with the provisional possibility of the universal notion (perhaps an interpretive thesis like 'Gadamer is Heideggerian') and the necessity of its particular constitutive exception ('At least one text of Gadamer is not Heideggerian'), continues through the recognition of the impossibility of any such exception ('There is not one text of Gadamer that is not Heideggerian'), only to conclude with the final contingency of the interpretive act, the act which underscores each of these moments and forever ensures the inaccessibility of meaning and the indetermination of interpretation. The point here would be that the very approach of the interpreter to the text must be paradoxically embodied in an element of that text. Identification with this textual element would then necessarily suspend the interpretive process and the associated hermeneutical pursuit of textual meaning. Before turning to a more detailed discussion, the path and return trajectory through the quadrants of the logical square can be repeated in another (Kantian) way.

Lacan recalls at the bottom of the first page of L'étourdit how 'it is from logic that [analytic] discourse touches on the real by encountering it as impossible, which is why it is this discourse that raises it to its final power: science, I have said, of the real.'661 This investigation of the real qua impossible is exactly what one should expect of the Lacan of the 1970s and Fierens intriguingly suggests linking it with Kant's own investigation into the impossible which comes on the final pages of the Transcendental Analytic of his first Critique.662 The 'Table of Nothings' found there is something to which Lacan himself made references to a decade earlier. The link is established through the identification of each of the four Kantian nothings with one of the four propositions of the Lacanian logical square.663 Accordingly, the nothing that Kant calls an ens rationis [Latin, a being of reason] is an 'empty notion without an object' and is to be associated with the pure essence of the universal of quadrant . It can be grasped by considering the ontological status of the hypothetical proposition 'If p then q ' from Section 6.3 above when no particular p is affirmed to exist. What results is a Gedankending [German, a thing of thought] which is the status Kant assigns to the noumenon – that which cannot be known but can be thought. It is a nothing which no positive feature can fill out. In Fregean terms it is a function without an argument and in Lacanian algebra it is S2 when S1 is barred to it, or . The classic Kantian examples include the soul, God and the universe as a whole. The point here is that these are easily imagined but are never actually encountered since they transcend the limits of our experience. Slightly less graspable is a nihil privativum [Latin, a privative nothing], an 'empty object of a notion' and this is to be situated at the level of quadrant which harbors a particular element that says 'No!' Here the corresponding matheme would be the exceptional S1 secretly supported by the void of $, or . The term legitimately brings to mind the passage from Summa Theologiae (1256–72) where St Thomas Aquinas cites St Augustine of Hippo's characterization of evil as the privation of good664 and this is itself a good starting point since the nothing in question does concern a positive notion with a positive object that just happens to be missing, like the holes in a sponge. Yet a better approximation of this nothing can be had by considering it as a real opposition, as the result of the conflict between two positive forces. This nothing is not the absence or lack of a positive force, but the 0 which results when two positive forces of equal strength cancel each other out, like the rope which remains stationary in a tug-of-war match when two groups of men of equal strength pull it in opposite directions. However, the problem with these examples is that they are too determinate, readily providing images which 'fill in' the nothing and thus what is lost is how the nihil privativum which emerges as the outcome of the mutual privation of two opposing forces has an ontological status somewhat 'less than' that of the ens rationis.

The purest evacuation of existence, however, is the nothing Kant calls an ens imaginarium [Latin, an imaginary being]. This 'empty intuition without an object' is of the order of quadrant since both are clearly concerned with the absence of all substance. Hence in Lacanian terms it would be written as the void of $ without objet a, or . Less than either of the previous two nothings, this absence nevertheless functions as the form of intuition without itself being an intuitable object. The classic examples are pure space and pure time and Kant opens his first Critique with a discussion of each of these in turn since they are the very condition of possibility of all representation. But by reaching this point the journey through nothings has not yet ended. There is still one more to be considered, one which is somehow 'less than nothing,' if 'nothing' is most consequentially defined as the pure absence of the empty set ∅ named by the ens imaginarium. The nothing in question is a nihil negativum [Latin, a negative nothing], an 'empty object without a notion' which correlates to quadrant . The corresponding matheme is the empty objet a devoid of its notional S2, or . This occurs when the very notion of an object contradicts itself and thereby cancels itself. The object which results is a nothing of this type, what Kant calls an Unding [German, a non-thing]. Such self-contradicting notions not only fail on the notional level but cannot be grasped through our imagination either; that is, in no way can we imagine what a 'square circle' looks like. So it cannot be approached either through the logic of the universal or of the particular but is rather to be experienced as the impossible ontologization of the empty space within which the universal and particular subsist. In previous terms it is singular, a punctuated point paradoxically found within the very field it itself discloses. The most elusive nothing of the four, one can say nothing of it except by keeping silent.

In terms of ontological ranking, the four nothings thus follow the same sequential path through the logical square as that forged by the logic of negation. Starting with that which harbors the highest being, it proceeds as: (ens rationis) → (nihil privativum) → (ens imaginarium) → (nihil negativum). The 'naturalness' of initially proceeding along this path can be confirmed by attempting to grasp these nothings through the imagination, the hermeneutical phenomenological faculty par excellence. For instance, the hypothetical 'If p then q ' occupying the Platonic heavens is as easily imagined as the souls of the departed and God Himself in the Christian heaven, but less imaginable are the holes without the delimiting sponge material or the nothing of a motionless rope. Still less can pure space and pure time be represented by an image since by definition these are the a priori of representation itself. And although Lacan tells us that objet a is not without its imaginary dimension, whatever image does come to attach itself to this real object is more fleeting than any other, as is witnessed whenever one attempts to imagine what the object of a self-contradictory notion like a square circle might look like. In following this sequence, what should not to be overlooked is how the very movement of encountering the impossible ends up in actually constructing this final paradoxical object or alternatively, how the other three moments are but different forms of this object. To sense this construction a theoretical application of the fruitful scientific technique of 'reverse engineering' might be made. Consider that by attempting to grasp the object of the negative nothing directly, one inevitably slips backward into an imaginary being. But whatever comes to represent the empty set (e.g., ∅) fails on that count alone, for the a priori of representation cannot be represented. Thus the slide into a privative nothing is equally inevitable since it seems that the object can only be approached at a glance, as if a minimal representation were necessary to delimit its contours (e.g., empty parentheticals { }). Yet again this image fails to satisfy for long and eventually gives way to the temptation to treat the object as a transcendent being of reason, as an object safely stored away in the noumenal beyond. At every stage the form of the object to be grasped is reduced to the preceding one. What this movement underscores is how the objet a of permeates all the others while simultaneously being constructed by them. Its double presence is why there is a persistent viscosity to the sequence, preventing a smooth run through the logical square.

Having traced out the path through the Lacanian logical square twice, once by tarrying with the negative and once more by pursuing a series of lesser and lesser nothings, a more detailed discussion can now be had to better account for the topological twisting of space that is the movement from one quadrant to the next which ends in the suspension of the hermeneutical field of meaning. The following section integrates the theory of the four discourses with the logical square to establish how this square concerns the production of meaning, the hermeneutical circle and the potential suspension of this circle, while simultaneously showing the passage between its quadrants in the revolving light of discourse theory. The final section then provides an account of the path through the logical square by making use of the topological figures discussed by Lacan in L'étourdit. It is held that topology functions heuristically, teaching the interpreter who engages in its study that the hermeneutical pursuit of textual meaning is not all there is to interpretation. Topology, much better than (post)structuralism, aesthetic theory or set theory alone, accesses a dimension which can answer the obvious question of 'What now?' that arises after the proper initiatory interpretive stance has been struck, a stance well-captured by Fierens' excitable introductory exclamation: 'Interpretation is in no way to be reduced to explaining the meaning of the text!'665 Here it must be stressed how the insights of L'étourdit will be lost to anyone not willing to clear his workspace of the usual tools of interpretation and instead take up scissors, tape and paper and follow Lacan in making the requisite cuts and sutures to produce his sexuated topological figures – this said fully in the spirit of Freud who often admonished his readers that his own discoveries would not be properly recognized if they neglected the analysis of their own dreams.

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