SEXUATED TOPOLOGY AND THE
SUSPENSION OF MEANING

A NON-HERMENEUTICAL PHENOMENOLOGICAL
APPROACH TO TEXTUAL ANALYSIS

WILLIAM J. URBAN

7.3 The Suspension of Meaning

About midway through L'étourdit Lacan announces that it is time 'for a little topology.'674 In the ensuing pages the reader is guided through a series of operations to be performed on a few standard topological figures which transform them one into the next. This digression is nothing new. By this time Lacan has already devoted over a decade of study to this branch of pure mathematics, as any glance of his seminars from the 1960s will confirm. This conforms to the methodological approach in play since his 'structuralist' period and Lacan only increases his recourse to mathematical inscription and logical forms as he enters his final decade. But although this move to topology can be seen as a natural progression, here it is more important than ever to emphasize the real aspect of these methodological endeavors. The problem is that topological figures too readily lend themselves to the imagination, and falling into this imagery is a temptation difficult to resist. The danger here is to treat topology as a metaphor which would miss the n'espace [nospace] into which mathematical discourse can lead us, as Lacan writes a couple of pages later. Much safer is to approach this n'espace through pure literal algebra, a mathematical writing which today seems to increasingly take second seat to the figural presentation centering itself on the page. In the past the topological figure appeared with rarity and the reader's only recourse was to work through its algebra. Indeed a survey would likely show that the vast majority of topology textbooks from decades past employ only algebraic expressions within their pages, refraining from illustrative material. No doubt due to prohibitive publishing costs of the past, from the Lacanian perspective the technological advances made in recent years in graphical design allowing for more illustrations in publications today is not a welcomed development. The lesson here – that topology ultimately depends on metaphor as little as do the mathemes of sexuation and discourse theory – should be kept ever in mind in this final discussion which does make use of topological images to illustrate the suspension of meaning.

Before turning to topology as it is found in L'étourdit, an additional caution should be made, which additionally serves to underscore the critical relation Lacan strikes with mathematics and logic. Undeniably his belief that mathematics can serve as a model for psychoanalysis places him squarely in the Enlightenment tradition. In this respect he echoes Descartes' own project as per his unfinished treatise Rules for the Direction of the Mind (1619–28),675 which seeks to adapt the methodology of mathematics to the pursuit of philosophical knowledge. Within the Regulae (the Rules, as this treatise is commonly called) the successful advancement of arithmetic and geometry is traced back to the fact that mathematics deals with pure and uncomplicated objects which unproblematically lead to indubitable knowledge since the senses offer no resistance in grasping them. However, the senses do prove prejudicial in our grasp of the corresponding objects of philosophy. This problem is addressed in his Meditations on First Philosophy (1641).676 More specifically, his technique of hyperbolic doubt in the first meditation intends to purify the mind of sensory prejudice so that philosophical first principles can be purely grasped. In an analogous fashion Lacan champions mathematical discourse as that which is founded on something other than meaning. He thus sees in it a model for a psychoanalysis taken as a non-hermeneutical phenomenology. But unlike the Descartes of the Regulae this is not a blind embrace of mathematics. Again it must be emphasized that Lacan is well aware of the impasses reached by classical logic whenever it is taken to the 'meta' level, just as he is aware of the limits of reason whenever it is employed in a similar fashion. That is to say, the writing of his mathemes of sexuation and discourse theory bear witness to his having read both Russell and Kant, inscribing as they do the very aporia of logic and reason that others before him merely treated as exceptional cases.677 And his development of non-metaphorical topology performs precisely the same task, albeit in a different venue. Here one can already appreciate how topology might serve as a model for a psychoanalysis which goes beyond the perspective of meaning. For if Lacan speaks of making cuts into the surface of topological figures, this can certainly be read as equivalent to the cut preventing a universally consistent logic or even the cut which traverses reason as it aims for the unconditional, but it can just as well be read as the cut severing the meaning-relation S2 // S1.

So what exactly is topology? It is a branch of pure mathematics based on set theory. For its objects of study, called topological spaces, are effectively sets with extra structure. More precisely, topological space is a set each of whose points is enclosed in a collection of subsets called open sets. Even in the most introductory of textbooks this rough conception is formalized via mathematical script in a foundational definition that begets a myriad of other definitions, theorems, lemmas, corollaries and formal propositions. Within the space of a few chapters a highly specialized algebraic writing quickly establishes itself and to the non-mathematician any illustration provided is a welcomed relief. As said above such writing inscribes the real, but in order to keep within manageable limits the discussion which follows refrains from topological algebra to instead highlight pertinent attributes of a few select topological spaces.

A more informal discussion of topology might thus begin by noting its etymological roots in the Greek word topos [place] or else cite from a dictionary of the American language that might define topology as 'the study of geometric properties and spatial relations rendered unaffected by the continuous change of the shape or the size of the figures.' In common parlance topology has been dubbed 'rubber-sheet geometry' in a phrase which nicely captures the intuition that geometrically quite distinct shapes like the coffee mug and the donut are topologically equivalent (or homeomorphic) since each could be stretched, twisted or otherwise deformed into the other without being ripped apart. Standard examples of topological spaces include the line, the circle, the plane, the sphere, the torus, the möbius strip and the cross-cap. The last four are of present concern and can be identified with one of the four quadrants of the Lacanian logical square. This is shown in Figure 7.2.

Free PDF of Lacan and Meaning
(a book based on this dissertation)

Figure 7.2 simply adds to Figure 7.1 a standard topological space. Each of these topological spaces or figures is discussed in turn below. By way of introduction the most natural way to proceed is to begin with what is in many ways the simplest topological figure of the four, viz., the sphere. Now, by performing specific operations on the sphere it can be transformed into the torus, which can in turn be cut and sutured into the möbius strip, whose supplementation with additional topological features can then produce the cross-cap. This path thus travels through the quadrants of the Lacanian logical square as per the familiar sequence → → → . However, it must be stressed how for Lacan these transformative operations are not to be conceived as external interventions into the field of topology but rather as part and parcel of this very field itself. More exactly, the cuts and sutures are not operations to be performed on these figures so much as they are effectively equivalent to the figures themselves. Here again the privilege will go to the left deixis whose topological figures stand logically prior to those of the right. So although Lacan largely proceeds along this sequential trajectory in order to make himself initially understood, there is also discernible in his work an effort to have us recognize this trajectory as taking place entirely within the space of the cross-cap of quadrant . Again this quadrant is the cornerstone of the logical square in the sense that it harbors a paradoxical point such that if extracted, the entire sequence is suspended. Furthermore, there is both a theoretical and a practical advantage to now express the conclusions already drawn from the above analysis of the logical square in topological terms. The contention is that engaging with Lacan's topological discussion in L'étourdit, one arrives at a purity of thought on these matters, but a thought which paradoxically only emerges through the initial mistaken immersion into the topological image. The image is thus not so much to be forgotten and set aside; rather, it is to be recognized as that which spits out its own suspension point. On the practical front the obvious advantage in turning to topology is that its imagery provides for nice visual presentations of different aspects surrounding the phenomenological exploration of meaning, inclusive of its structural form as well as its breakdown and ultimate suspension. In general such visuals often help to stabilize that university discourse known as academic writing, from the simple essay to the PhD dissertation.

As suggested above, forging a path through the logical square might begin with the sphere. The naturalness of this approach receives Lacan's tacit blessing when he writes how '[n]othing is more of a nature to take itself to be spherical.'678 Yet Lacan does not begin his topological lesson with the sphere but rather with the torus. One explanation for this might be had by considering Lacan's claim 'that the sphere is what does without topology.'679 This is a curious statement since the sphere is by all accounts a spatial figure well defined in topology. A clue as to what motivates this claim is found in the sentence which immediately precedes it: 'Naturally there are saids that form the object of predicative logic and whose universalizing supposition belongs simply to the sphere, I say: the, I say: sphere, in other words: that precisely structure finds in it only a supplement which is that of the fiction of the true.' This seems to suggest that the sphere is to be associated with the imaginary covering over of the real or in terms of the foregoing analysis, of the fact that the logical square does not occupy uniform space. In more traditional sexuated terms the sphere would be equated with the attempt to compensate in the phantasmatic dimension for the traumatic fact that 'there is no such thing as a sexual relation.' Articulated in this way immediately brings to mind Lacan's analysis of Aristophanes' myth from Plato's Symposium (c. 384 BC)680 which he takes up around the time of his Seminar XI.681 Briefly, Aristophanes held the view that humans were originally spherical beings lacking nothing but unfortunately were split into two by a jealous Zeus; ever since then the human qua divided being has striven to find its compliment in the hope that it might provide complete satisfaction by returning it to the One. Historically it is notable that the notion of the harmonious perfection of the sphere was not confined to the level of individual fantasy. For instance, a similar notion informed the scientific understanding of the universe for centuries in the sense that the topological space of the sphere provided the standard by which the erratic movements of astronomical objects were to be reconciled. This does not imply that after Johannes Kepler the hope for a final reckoning with the ultimate meaning of the universe was abandoned. The existence of today's New Age philosophies and the ecological movement itself bear witness to the contrary insofar as they seek wholesome communion with the universe and/or nature. But such mythology continues to motivate the hard sciences as well, as in the case of present day physics in its effort to formulate a grand and unified Theory of Everything. The point here is that operating behind these projects is a 'universalizing supposition' to be associated with the topology of the sphere, whose universal quality Lacan highlights in the above citation by isolating out the definite article 'the' (la); that is, one should always speak of the sphere as the sphere. But whether these projects take the guise of (pseudo-)science or embrace the original platonic fantasy of the sphericity of primordial Man matters little, for Lacan would have us dismiss them all as so many efforts to harmonize the sexual divide.

That the sphere is to be associated with the universal682 and that any aspiration for universality is ultimately phantasmatic is likely the reason why Lacan begins with the torus and not the sphere. Nevertheless its necessary and illusory quality makes the sphere the implicit starting point of any discussion. So for the sake of completion we should begin one step prior to Lacan's own discussion, one which offers a topological description of the sphere and the operation which would transform it into a torus and thus place it in line with the operations Lacan himself performs on the other three topological figures which move them one to the next as per the sequence → → → .

One way to take up the sphere is to consider it a surface or, as the topologist calls it, a manifold. This is true for all four of the topological spaces now occupying the logical square.683 To begin thinking in terms of such surfaces, it is helpful to contrast them with surfaces that are strictly flat. These are the prototypical surfaces in topology and are called Euclidean planes. While the plane is customarily understood to be a flat surface extending infinitely far in all directions, Euclid himself conceived it to be of finite (though arbitrarily large) extent. Euclid's own conception is closer to our spontaneous understanding of the disk as a plane circle with its interior. Extending finitely, the disk obviously has an edge or boundary. Now, what makes the sphere an apt topological space for the notion of the universality of meaning is the fact that it is an example of a surface completely lacking boundaries. A small insect traveling on the surface of a sphere will not fall off the sphere since it lacks an edge. It is also possible for it to travel in any direction and return to its starting point. Such a surface is said to be closed. This makes it appropriate to deem meaning to be spherical: any traveler finding himself on its surface is fully immersed in its substance as there are no boundaries to cross which could provide a minimal distance to make of meaning an external object. In a word, the spherical surface rules out the subject-object dichotomy and can only offer the subject meaningful subjectivization. The Heideggerian-Gadamerian brand of hermeneutical phenomenology dwells within the topological space of spherical meaning and so makes its home in the quadrant marked by the matheme S2. In a happy coincidence, topologists use the similar notation of S2 to refer to the sphere (also written as 2-sphere) to mark its difference from the circular disk (a 1-sphere). But while the sphere is not homeomorphic to the disk, a cut can be made to its surface to make it so. Simply defined, the operation of cutting (or ripping or puncturing) introduces a discontinuity in the original surface and thus converts its space into a topologically nonequivalent one (while stretching or shrinking surfaces preserves continuity and thus topological equivalency). In the present case, by cutting the sphere in half two hemispheres result each of which is homeomorphic to the disk,684 or else the sphere can be flattened into a polygon by a cut which stops short of splitting the sphere in two. These cuttings resulting in spherical disks become important below. For now the task is to enquire into the operation which transforms the sphere of quadrant into the torus of quadrant .

Two methods to accomplish this are suggested. The first begins by recognizing how the removal of at least one point from the sphere results in a space that is homeomorphic to the plane. Refraining from an algebraic proof, this might be grasped by imagining how all spheres, like an inflated beach ball, have the feature of dividing space into a bounded region and an unbounded region so that it is impossible to travel within space from one region to the other without passing through the sphere. But if the beach ball is punctured (i.e., a single point is removed) it ceases to divide space into two regions, just like a Euclidean plane. This becomes clearer if a large section of the beach ball is cut away so that the remaining surface can be easily stretched flat. Now, if this puncturing or cutting procedure is performed on the beach ball in not one but two places on its surface, what results are the homeomorphic spaces of the annulus or a disc with one hole or a cylinder (since a sphere with two holes is simply an inflated version of a cylinder which additionally flattens into an annulus or a disc with one hole). The final step to transform the sphere with two holes into the torus requires suturing (or gluing or taping). Generally speaking the operation of suturing is the reverse of cutting. If the edges of a cut are sutured back to the way they were joined previous to the cut, then the original (or an equivalent) topological space is recovered;685 but if different edges not originally together before the cut are instead sutured, then a new topological space may result. In the present case suturing the two ends of the cylinder together (or suturing together the outer and inner rims of an annulus or the outer and inner edges of a disc with one hole) results in the torus. This doughnut-shaped topological space is another example of a manifold with no boundary. As in the case of the sphere, the torus is a closed surface and thus a small insect will similarly encounter no edges as it travels its surface.686 However, because what separates the two topological spaces is a suture which does not perfectly reverse the cut, the torus is clearly not homeomorphic to the sphere.

There is another method of transforming the sphere into the torus. This method reverses the order of the two basic procedures operative in the previous method so that here it is suturing which precedes cutting. But there is a preliminary operation to be performed. This involves what we might call 'pinching' the sphere between any two nonadjacent points, perhaps between and the antipode of (the point that is opposite through the origin). The two most northern and southern points occupying the polar regions of the sphere fit this bill, yet any two nonadjacent points will do. Deforming the sphere through pinching results in a surface which is homeomorphic to the non-pinched sphere, but this equivalence vanishes with the next two operations. The first step involves suturing. First, the and antipode of are sutured together; as well, all the points which immediately encircle are sutured to those which immediately encircle the antipode of . Second, the sutured and antipode of (now one point) is cut away. What results is the topological space of the torus. As with the previous method, this procedure becomes clearer if a larger section of points is pinched together, sutured and cut away; so as long as those points around the two edges of the removed section are also sutured together a torus is produced, although in this case the result is more in line with the typical visualization of this topological space as a donut with a 'big' hole.

This second method is more pertinent for present purposes. At first glance it appears that it makes no difference which method is used, for both produce the torus. As was said, the torus is a manifold with no boundaries and so it is a closed surface just like the sphere. Because of this similarity it is tempting to consider the topology of meaning to be toric to the same extent that it is spherical. Yet this would be going too far. Indeed meaning has something to do with the torus and the great advantage of taking up these topological surfaces within the context of the logical square is that such 'somethings' become more manifest. In general much of the analysis undertaken of the logical square with respect to meaning since Chapter 6 can now be rearticulated in topological terms. In the case of the sphere and the torus it is a matter of considering their relation as the relation between the two quadrants of the right deixis, a relation which was articulated above using various elements and terminology. Considering the sphere and the torus with these in mind gives us a better picture of the topology of meaning. For instance, despite the greater clarity which ensues by removing a large section of sutured points from the pinched sphere so that a big-holed donut visually presents itself, it must be stressed how the successful transformation of the spheric into the toric surface simply necessitates at least one (so sutured) to be cut away. This italicized phrase cannot but bring to mind the particular affirmative proposition of quadrant which of course is the quadrant of the torus. This cut away is the One, the exceptional Epimenides who refuses to submit to that function which collectivizes All into a universal and spherical set. As cut away it thus confirms the truth of the Md which has it that there is no such exception (much like how the center of gravity of the torus is 'missing,' positioned as it is at the heart of the empty space the torus encircles). Yet this nevertheless does not prevent it from forming a limit to the univeralizing pretentions of the sphere of meaning (even though missing the toric center of gravity still makes its 'presence' felt and thus carries a certain 'material weight' with respect to this particular topological space). That the existential particular affirmative operates as a limit to the universal was graphically presented above when a hyperbolic function was examined. Specifically, the asymptote = 0 was found to form a limit to = , although an ambiguity persisted as to whether this escaped should be counted amongst the All of this function. In topology as well there is the concept of a limit point whose relation to the topological surface is often just as ambiguous. In the case of the torus this limit is its missing towards which all the points on its surface converge. Again this might be visualized by imagining a small insect whose path of endlessly revolving in a spiral fashion around the core of the torus simultaneously also encircles (yet never encounters) its central axis point of rotation.687

In general an effort must be made to conceive the torus in contradictory relation to the sphere, as that topological space which constitutes universality in its very stance as exception and limitation. This contradictory relation may be envisaged rather crudely by way of personifying these topological spaces. Accordingly, the universal sphere might be said to aspire in its efforts to encompass the particular existential torus. Yet it finds that the best it can do is to fit itself into the latter's empty center. It is obvious that any stability derived from such an amalgamation would be short lived. On the one hand the universal cannot abide being demoted to an exception, which is precisely the status it would derive from its occupation as the toric center. On the other hand the torus cannot tolerate having been so 'completed;' indeed this is impossible since the very constitution of universal completeness is predicated on the existence of the exception. The two topological spaces of the right deixis of the logical square are thus forever at odds: the universal aspiration of meaning to encompass all aspects of existence is juxtaposed against the stance of the particular which attempts to extract itself from its universal grip to make instead a manageable object of meaning. What we have here are topological descriptions of that hermeneutical circle manifest in the right deixis but whose rotation is secretly driven by the spatial manifolds in the left which nevertheless do not refrain from offering up a suspension point to the rotation. To appreciate this, the torus of quadrant must first be transformed into the möbius strip of quadrant . It is with this transformation that Lacan begins his lesson in topology.

Lacan asks us to take a torus and deflate or empty it. This manipulation is a preparatory step in a series of cutting and suturing operations to come which will tear the torus out of the realm of the spherical (characteristic of the topological spaces in the right deixis) and plunge it into the topology of the aspherical surface (characteristic of those in the left deixis). However, this will not be accomplished by respecting the topological structure of the torus. A simple emptying which reduces the volume of the torus to nil and so results in the flat tire shape of the annulus will not do. For flattening the torus in this manner produces two separate folds along the two circumferences of the annulus and what is needed instead is a single fold which likewise completes two circuits around the central axis of rotation, but one which simultaneously completes a single circuit around its core.688 As per Lacan's instructions, this can be brought about by running the length of the torus between our pinched fingers, but doing so in such a way that the finger on top at the beginning is at the bottom by the end of a single complete turn around its axis. This procedure places a one-half twist (180°) into the deflated torus and the resulting figure looks something like a möbius strip. However, this is in appearance only. The surface of a deflated torus still has both an inner and outer face even if its inner face has been caved-in on itself as a result of the pinching operation. This is a characteristic shared by the torus and the sphere: as spherical surfaces, whether inflated or deflated, they both retain an inner and outer face.

Now, an actual möbius strip is obtained by this operation if the caved-in inner face is considered fused together. But Lacan follows another route to demonstrate this transformation 'in a less crude fashion' which leads up to what he calls his 'conjuring trick.'689 Beginning with a cut that follows the single fold of the deflated torus, we appear to end up with two pieces or laminas of the toric surface. Yet when this is stretched out the actual result is a single continuous strip. This strip, what Lacan calls the 'bipartite möbius strip' (or more simply the 'bipartite strip') has two sides and two edges and so should not to be confused with the 'true möbius strip.'690 Obviously, if the edges of the cut just performed are sutured back together with points along either side of the cut matching as they did before the cut, the flattened and one-half twisted torus is restored. But if one of the two laminas is slid out from under the other in either direction so that one edge lines up not with the other edge but with itself and this is then sutured together, what is produced is the one-sided, one-edged true möbius strip of quadrant .

What follows this conjuring trick is a series of cryptic sentences all of which may be viewed as leading up to and thus validating the paradoxical Lacanian claim that the möbius strip is nothing but its own absence. If this is indeed the case we can certainly appreciate Lacan's warning 'that it is not from the ideal cross-section, around which a strip is twisted in a half-turn, that the Moebius strip is to be imagined.'691 This was precisely the procedure used to construct the möbius strip as per the commentary surrounding Figure 5.5 above. But while certain nuances of Lacan's discussion are lost by proceeding down this forbidden path which ends up placing a very present möbius strip into our hands, nevertheless certain operations can be performed on this construction so that the möbius strip qua absence is experienced. More specifically, such an experience (which is not without a fleeting image of this absence) is had by comparing two different types of cuts of the möbius strip. To see this, take a möbius strip constructed out of a strip of paper with a one-half twist and cut it lengthwise a third of the way from the edge (any distance will do so long as it is not equidistant from the 'two' edges). Such a cut will not meet up with its starting point until it completes two circuits of the möbius strip. At the end of these two circuits three apparent laminas are produced but when stretched out, there are in fact only two strips. One strip is twice the length and a third the width of the original strip. This is the bipartite strip with its two sides and two edges, but what is remarkable is that the other strip produced is linked to it. Moreover, this second strip is identical to the original möbius strip before the cut and is in fact its central third. Thus a trisection of the möbius strip produces a narrower möbius strip linked to the longer bipartite strip.692 So far the only absence to speak of concerns the partial disappearance of the original möbius strip, transformed as it is into a bipartite strip due to the off-center cutting; but as its central portion is retained the möbius strip must be said to maintain a continual presence from start to finish. Now, to conceive the möbius strip as absent requires making a different type of cut while simultaneously thinking the results of the off-center cutting operation. First, take another constructed möbius strip and make a cut down its center lengthwise. It becomes clear that instead of two circuits this median cut is complete in only one and that this results in a double-length bipartite strip with no other strip linked to it. However, if we treat this median cutting operation as an off-center cutting operation so as to require an additional circuit to be completed (that is, we are to effectively double the cut where none is needed by pretending to cut through the empty space opened up by the initial median cut), we can imagine producing the möbius strip linked to the bipartite strip. Of course there is no such linked strip. But that is Lacan's point: the möbius strip in its essence is absence and this absence is as much produced by the single median cut which imaginarily produces the möbius strip as it is actually produced by the doubled and off-center cut. This is why Lacan can claim 'that the Moebius strip is nothing other than this very [single median] cut, the one by which it disappears from its surface.'693 The ramifications of equating the möbius strip to the cut run deep. For despite (or more accurately, because of) its absence from the spheric and toric topological spaces of the two quadrants of the right deixis of the logical square, it nevertheless makes its 'presence' felt in them whenever they undergo transformations since these transformation necessitate the cutting operation. Recalling from Figure 7.2 how the matheme associated with the möbius strip is $, this should not be surprising, for cutting is nothing more than the topological expression of the active intervention of the subject. Even better: the cutting operation is the subject. This is yet another way to understand the logical precedence the left deixis enjoys over the right despite only coming at the tail end of the sequential path through the quadrants of the logical square. Expressed in terms of textual analysis, the very act of interpretation is a cut into the topology of the text, and by tarrying with these cuts the hermeneutical pursuit of meaning can be suspended.

But the nature of the möbius strip is not simply to be of the cut. It is of the suture as well. For a single median cut is equivalent to the suturing operation which transforms a bipartite strip into a möbius strip, an operation that works by making along the whole length of the bipartite strip only one of its front and back sides so that '[t[here is not one of these points where the one and the other are not united.'694 If Lacan's use of the double negative, which effectively states how there is no exceptional point on the single surface of the möbius strip, is reminiscent of the universal negative proposition, this is quite appropriate given the quadrant the möbius strip occupies. Furthermore, the lack of exceptional points also implies that no exceptional line of points exists. This fact is lost when privileging a conception of the möbius strip as imaginarily produced by the median cut that follows a line of points thought to be somehow exceptional. For when it comes to the real möbius strip any single-turn cut following any one of its infinity of lines will produce this paradoxical unilateral surface. This makes Lacan's claim that such cuts are 'lines without points' understandable, for if the exception does not exist these lines cannot be conceived as composed of a set of points. Thus the möbius strip qua act of cutting and suturing thoroughly departs from those collectivized sets of points known as the torus and the sphere of the right deixis. Without exceptional points there is no way to orient oneself on the surface of the möbius strip, which is a problem not encountered on the latter two surfaces. Indeed orientability is a concept used by topologists to distinguish between topological surfaces. This concept can be grasped by imagining an arrowhead placed on a circle that indicates its rotation. If this oriented circle itself moves about a surface in an arbitrary manner and manages to preserve its orientation, the surface is said to be orientable. Orientability is a defining feature of the torus and the sphere of the right deixis. However, in the case of moving along any path on the möbius strip, by the time the rotating circle returns to its starting point its rotation will have reversed.695 The möbius strip is in fact the prototypical nonorientable space and if a surface can be said to contain an embedded möbius strip, this is enough to qualify that surface as nonorientable. As the surface of the cross-cap satisfies this condition, the line between the two sides of the Lacanian logical square can be newly expressed as that which divides the orientable spherical surfaces of the right from the more paradoxical nonorientable aspherical surfaces of the left.

This topologically expresses the fact that 'there is no such thing as a sexual relation,' or what Lacan often speaks of in L'étourdit as 'the ab-sens of the sexual relationship.'696 A French neologism translating as 'lack-of-sense' or 'ab-sense,' this term must not be confused with the absence that is the möbius strip. But how exactly is ab-sense to be distinguished from absence? For Lacan clearly holds these to be different, as when he writes 'that the ab-sense that results from the single cut, brings about the absence of the Moebius strip.'697 By combining the basic thesis of Chapter 3 on (post)structuralism with the discussion of discourse theory from Section 7.2, a relatively straightforward answer to this question presents itself. As long as one maintains the elemental distinction between meaning and sense whereby meaning is that which is 'caught between' signifiers and sense is what adheres to the formal structural framework of signifiers, then the familiar matheme of the meaning-relation can again be utilized. Accordingly, the structural level inscribed by writing S1 → S2 concerns sense, which implies that when this relation breaks down what results is ab-sense. Ab-sense is then to be equated with the matheme written S2 // S1 in distinction to the absence that is $. Redefining ab-sense in terms of the (breakdown of the) meaning-relation makes it legitimate to speak of the line dividing the right from the left deixis as one which also concerns 'the ab-sens of the meaning relationship.'698 But while the transformation of the torus into the möbius strip takes us across this line, this does not imply that the discourse first encountered there makes a practice of ab-sense. As already seen, the meaning-relation is still operative in the discourse associated with the möbius strip. It is only with a further turn that meaning is put into parenthesis in a discourse which straight away goes beyond the meaning-relation. Indeed the very posing of the question of this beyond is to refer back to this discourse which at once proceeds from this beyond. This discourse is of course the Ad: which, as evident in its writing, is a discourse founded on, operating from and aiming at the ab-sense of the meaning-relation. That such ab-sense is utilized as a resource in addressing the other qua absence helps keep these two distinct, but recognizing how the occupant in the place of agency is objet a further distinguishes both of these from nonsense. Each of these mathemes – $ (absence), S2 // S1 (ab-sense) and objet a (nonsense) – only appear together in the Ad. As repeatedly emphasized, this discourse takes logical precedence over all others. The quote immediately above states one way this is true, claiming as it does that ab-sense is what brings about the absence of the möbius strip. For present purposes it remains to be seen how objet a, repeatedly identified with the paradoxical suspension point of meaning throughout this study, becomes isolated so that it can act in this capacity. To anticipate, this is accomplished by making a cut on the cross-cap of quadrant which at the same time releases those elements constitutive of the topological spaces of the other three quadrants. But before turning to this operation and its results, the transformation of the möbius strip into the topological space of the cross-cap must first be briefly related.

In order to prepare for this transformation and to better appreciate the magnitude of what results, recall how the sphere and the torus of the right deixis are both manifolds with no boundary. Further, they are both capable of bounding the nothingness of space within their closed surfaces. But by turning to the möbius strip of the left deixis this trait is reversed. Instead of a surface with no boundary bounding space, the one-sided and one-edged möbius strip is a bounded surface thoroughly incapable of bounding space. It is simply nothing or as previously stated, it is nothing but its own absence. Now, it stands to reason that if this single edge was sutured somehow so that it no longer had a boundary, a closed manifold would result which would again bound space. However, this reasoning is only half correct: a closed manifold does indeed result, but one which nonetheless is incapable of bounding space. To use terminology from Section 7.1 above, what results is not so much nothing as it is less than nothing. Far from being a defect feature, this is instead its undeniable strength and the 'secret' to its dominant influence in the logical square – provided, of course, that it is closed with the proper operation. So exactly how is the one-edged möbius strip to be closed? Actually, there are a number of ways in which this can be done and Lacan mentions one699 before moving on to the operation that produces the topological space which most interests him. To arrive at this space, recall how a sphere can be cut into spherical disks. Whether a full hemisphere or portion thereof, such a cutting is a disk with a single circular edge which can be sutured to the single edge of the möbius strip. Performing this suturing operation 'caps off' the boundary of the möbius strip. The result is what is called the cross-cap,700 a surface which has no true inside or outside. It looks rather like a dented, brimless hat. Unlike its companion surface in the left deixis, the surface of the cross-cap has no boundary and thus is a closed manifold, a feature it shares with the sphere and the torus. Yet it differs from those surfaces of the right deixis in not being capable of bounding space. What is important to note, however, is that this 'failure' to bound space is successfully accomplished by the crosscap in an even more radical way than the möbius strip which only does so through its absence. In Lacan's words, the suturing of a spherical disk onto the edge of the möbius strip is an operation 'reduced to the point: out-of-line point which, in supplementing the line without points, happens to compose what in topology is designated as cross-cap.'701 It is thus not so much absence as rather a curious 'presence' that is at stake in the cross-cap and this singularity, what Lacan here calls the out-of-line point, is what allows the cross-cap to suspend that which obligates other closed manifolds to bound space. Now, if the bounded space of the right deixis is the space traversed by the hermeneutical circle, then the singular out-of-line point harboring within the cross-cap is unique in its ability to stand outside meaningful space. Here is the topological expression for the objet a, the sublime object capable of acting as the suspension point of meaning once it is cut away from the cross-cap.

What is suggested here is that there are two operations to consider when taking up the cross-cap, this fourth and last topological surface of the logical square. On the one hand, there is the suture which reveals the surface and on the other, there is the cut which breaks with all continuity. Yet these are not two separate operations, as if suturing first constitutes the cross-cap and then cutting subsequently tears it apart; if this were the case it would be legitimate to ask what topological surface follows the cross-cap. Nothing follows. But this nothing is a nothing curiously ontologized. For the foundational stone of the logical square has indeed been laid upon reaching the cross-cap – provided of course that this stone is understood as the very embodiment of those cutting and suturing operations which have led to it. In the final analysis then the specific task will be to conceive the suture and the cut as simultaneous operations.702 Yet despite being of the same gesture, these two operations can nevertheless be considered separately to appreciate how the cross-cap is privileged in the logical square in the sense of underscoring all the other topological spaces. Generally speaking, the suturing operation is what establishes the bounded topological space of meaning in the right deixis while the inverse operation cuts away a singularity capable of suspending meaningful trajectories through this bounded space.

More specifically, with respect to the suturing operation it is clear that what forms the single out-ofline point is the spherical disk which infinitely curves in upon itself due to its suture to the one-half twisted möbius strip. Recognizing how the out-of-line point of the spherical disk is objet a while the möbius strip is $, this suturing operation can be read as their elemental combing as per the fundamental fantasy that Lacan formally writes as $ ◊ a. Roughly equivalent to the Kantian transcendental schemata, this matheme is to be read as the split subject in relation to objet a. This relation is what grants to the subject a meaningful grasp of reality and through it the subject achieves a phantasmatic sense of wholeness, completeness and fulfillment. In a word, this is the basic formula for the meaningful subjectivization of the subject. In addition, the constitution of the cross-cap bears on the object side of the subject-object divide by establishing the universal field of meaning in which the subject can either immerse or oppose himself. One might say that out from the singularity of the out-of-line point 'grows' spherical surfaces of meaning703 and in terms of discourse theory this concerns the putting into relation the two mathemes associated with the spherical surfaces of the torus and the sphere. The relation in question is of course the meaning-relation S1 → S2. The suturing operation thus assures a smooth and continuous surface bounding a universal space of meaning for the subject of the hermeneutical circle.

The cutting operation can be expressed in various ways on the cross-cap and its significance likewise extends across the larger logical square. On the subject side it will be recalled from Section 5.3 that the top half of the Ad: inscribes a halt to the incessant slide of meaning. But positioned as it is in this discourse the matheme a → $ also writes the impossible relation between the two elements which compose the fundamental fantasy. This would then mark the appearance, in discourse theory, of what Lacan calls 'the traversing of fantasy.'704 Here the temporally paradoxical move made by the subject 'at the end of analysis' is to inhabit the very cause of his split subjectivity and to thus depart from an existence requiring compensation for the split by way of meaning. In other words, traversing the fantasy is an active assumption of the very place of the cause of meaning and this opportunity arrives at the bequest of the cut – thereby demonstrating that the 'impossible' nevertheless does happen. Also previously expressed as a separation from the forced choice of meaning which attends alienated subjectivity, more simply said the cut is that which allows a movement away from subjectivization to subjectivity proper. On the object side the cut is most discernible as the shift away from meaning to the domain of non-meaning, something understood in Part I as the shift from hermeneutical phenomenology to (post)structuralism. This cut breaks the continuity of the spherical surface and thus disallows it from bounding a universal space of meaning. With both the sphere and the torus severed, their relation also becomes one of discontinuity. Such severing is at once a severing of the hermeneutical circle or what has alternately been called the breakdown of meaning or ab-sense. As seen in the Ad this is concisely written as S2 // S1.

What is crucial to recognize is how this cut simultaneously offers up a 'stable' suspension point outside the purview of the hermeneutical circle it severs. Occupying this point would transform any failure to acquire meaning into ontological success. To explain this, consider how the cross-cap is produced by suturing the spherical disk to the möbius strip.705 Yet it will be noticed how this spherical disk only comes to be by having been first cut out of the spherical surface that is the bounded limit of meaning and as just discussed, this cut is the breakdown of meaning whose matheme S2 // S1 clearly pertains to the discourse associated with the cross-cap itself. This implies that the cut to the spherical space of meaning of the right deixis is at the same time the operative cut which constitutes the cross-cap of the left. Being at the tail end of the trajectory of transformations leading away from the meaningful realm of the right deixis, it should not be surprising that the cross-cap is constituted by an operation conceivable as either a cut or a suture. For performing an additional cut and suture to the cross-cap does not transform it into yet another topological space but instead fragments it into those spaces already discussed which, through a series of cutting and suturing operations, we have seen form a trajectory toward it. Here a moving away is a leading back, for the cross-cap is entirely singular in participating in its own constitution. The result is a topological space whose tautological aspect might be characterized as the ontologization of the fact that the möbius strip is nothing but its own absence. Yet despite being constituted by the tautological operation of a cut/suture, this ontologized nothing is nevertheless best understood by way of the cut. Recall how the aspherical surface of the cross-cap is nonorientable because it contains an embedded möbius strip. It then follows that by cutting away the interior of a disk from the cross-cap a möbius strip results. A cut thus splits the cross-cap into a spherical disk and a möbius strip. This result is well-established in the field of topology and can readily be seen in the Ad: where such a cut is doubly-inscribed, first on the lower level in the form of ab-sense which captures the severing of the hermeneutical circle and second on the upper level where the absence that is $ marks the cut tout court.706 Together these 'two' cuts release the a-spherical surface of objet a, the very embodiment of the nothing of this double-inscription: in a first move ab-sense voids a space of spherical meaning and kicks the subject outside the hermeneutical circle, while in a second move this newly opened domain of non- meaning is ontologized into a singular point such that if occupied by the subject, is capable of suspending the hermeneutical circle of meaning. Figure 7.3 provides a representation of this suspension.

In this figure the entire Lacanian logical square has been rotated clockwise a quarter turn from the way it was presented in its three previous incarnations. Since the matheme associated with each quadrant remains unchanged as well as the sequential order of the quadrants, this modification is entirely cosmetic. It has solely been made so that what it seeks to convey best strikes the eye. As can be seen, the hermeneutical circle in play between S1 and S2 of quadrants and is now at the bottom of the square with objet a of quadrant now on top, which allows for the empty distance between them to be marked by $ of quadrant . This arrangement visually confirms the existence of a 'suspension' point to the hermeneutical circle of meaning, a term which is to be taken as per the original emphasis given by its Latin roots.707 In the present context, however, the noun form of the Latin suspensio (derived from suspensus) is insufficient, as the temptation here is to read suspension as 'suspense' which thus confers upon it the sense of 'uncertainty' or 'indeterminacy.' We have seen how such states are not at all foreign to the subject who finds himself in the hermeneutical circle where the final retroactive production of meaning is often anxiously awaited. Far better than simply resigning oneself to objective meanings is to recognize in their mode of futur antérieur a subjective stamp. This can be gathered by looking instead to the original verb form where the Middle English suspenden and Old French suspendre are found to both derive from the Latin suspendere – a verb often defined as 'to hang up.' Here, however, it becomes important to consider suspendere in its full compounded sense and resist further breaking it down into its two components, sus- [for sub-, under] and pendere [to hang]. For if its prefix is neglected and only its base word is considered, suspension is once again taken as the situation where the final production of an objective meaning pends for a subject who can, for that very reason, easily wash his hands of the entire affair. This is not to say that to suspend meaning it simply suffices to recognize in it a relativity since its point of origin can be traced back to the particular perspective of the subject. For treating meaning as a strict subjective phenomenon is as deficient as its strict objective treatment. Rather, a curious mix of both the subjective and the objective is needed to reach suspension proper which in no way concerns a middle ground between the two: if the subject of the hermeneutical circle finds himself at the mercy of meanings retroactively produced by an objective mechanism beyond his control, this phenomenal experience in toto must of course be recognized as itself bearing his subjective mark; but the crucial point is that this recognition must be accomplished in an objective way.708 In terms of Part I this objective experience moves beyond the objectivity of the non-meaningful mechanisms of (post)structuralism to condense instead into a non-hermeneutical phenomenological singularity. Here is the ontological point from which the subject 'hangs up' the hermeneutical circle within whose turn meaning at any given moment hangs in abeyance.

This ontological point is of course objet a, which suspends the endless circular relation of S1 and S2. Figure 7.3 denotes this suspension by a short vertical line which 'links' objet a to the hermeneutical circle, but it will be noted that this line is dotted and that the void of $ is positioned 'between' them. This figure can be read in two ways. In a first approach, the non-hermeneutical phenomenological experience in question is one in which the hermeneutical field of meaning suddenly pulls away from the subject it just enveloped. Yet it does not disappear altogether but very much hangs nearby in the air, as if tethered just above the subject's head from some anchoring point still higher up. The nature of this anchoring point could be expressed in terms of the (barred) knowledge of the analyst709 in a twofold manner: his insight into the ultimate impotency of the hermeneutical circle is certainly what keeps it suspended, yet his further insight that this nevertheless does not warrant a wholesale dismissal of meaning is what keeps it within his horizon. In other words, although the Ad leaves the hermeneutical circle thoroughly incapacitated so that it is unable to exercise its influence on the subject's experience of reality, this does not imply that it thereby dissolves into the void so that what remains of reality is the utterly absurd. Remaining close enough to ensure this does not happen, what is at stake instead with the Not-all of meaning is not the absurd but the nonsensical. So while the inability of the hermeneutical circle to intervene in the phenomenal field makes it tempting to deride the impotency of meaning – to point a finger at it and smirk as it dangles and kicks its legs helplessly above our heads – derision is not the proper attitude to strike.710 For while derision successfully points out the inconsistency and failure of the hermeneutical circle to deliver meaning, it nevertheless overlooks how such a subjective stance is only made possible because the hermeneutical circle is not up to its task.

In broader terms, it is not that the domain of the nonsensical harbors some essential Thing which pre-exists meaning and thus causes it to stumble and fail; rather, nonsensical encounters happen due to the very inconsistency of the field of meaning. Indeed those non-hermeneutical phenomena experienced in the analyst's chair wholly take place within the very space opened up by the failure of meaning. Saying it this way places the crucial accent on the surplus objectivity of any situation and thus allows for a more consequential reading of Figure 7.3. Accordingly, where the first approach sees two distinct elements (what is hung up is done so through support from above), the second approach sees only one: what is to be fully acknowledged is the fact that the subject only ever achieves an empty distance – denoted by $ – from the hermeneutical circle so that its suspension is viewed as accomplished without attachment.711 Here the suspension of the hermeneutical circle is to be read as coinciding with the sudden appearance of objet a, what might be called the surplus object of meaning.712 Of course one need not be an analyst to experience its effects; the wavering, say, between a too much and a too little meaning is equally had by the analyst and the non-analyst alike. Yet such experiences will not halt the desire to acquire meaning until the subject begins forging a 'link' between the breakdown of meaning and its object-cause. Thus it might be understood that what prompts such wavering is the parasitic-like quality of this surplus object, its ability to attach itself to any textual or aesthetic work which, for that very reason, incites both an infinite production of its meaning and prevents its final hermeneutical reckoning. But this link must be forged in such a way that only a non-relation persists, for at its most basic level objet a is a point where a cause is immediately its own effect. What is then needed is not so much a link between the breakdown of meaning and its object-cause but rather the ontologization of the subjective experience of this forged link. This would move a mere subjective experience of this breakdown, which at best only ever 'negatively' confirms the existence of a persisting non-meaningful Thing, to its 'positive' registration in the analyst's discourse as the nonsense of objet a. Far from reaching some beyond of meaning, the move accomplished here is much more modest.713 Absent from the hermeneutical field as a meaningful element, objet a is nevertheless present there as a nonsensical surplus object of meaning. And whenever this object makes its presence 'fully' known the hermeneutical circle of meaning is at once suspended.

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