The Logic and Topology of the Other Jouissance

Paper presented at English Speaking Seminar

International Forum of the Lacanian Field

 June 2009

published in Analysis 15, 2009

by the Australian Center for Psychoanalysis

ISBN 972-0-646-52715-4

By: Yehuda Israely, Ph.D.

 Tel-Aviv Forum

Introduction

Topology is the mathematics of discontinuity (Barr, 1964, 2). Topological structures differ from one another in the way they embody this discontinuity. Their surfaces have a variety of holes, edges, loops or nodes. Since the subject is an effect of a cut by the signifier, Lacan uses topology to structure the complex forms of this cut. 

The subject exists within a logical structure. Lacan identifies the subject in Descartes' cogito not through the proof of thinking but through the logical structure of the cogito, if – then (Lacan, 14.12.1966). Just like geometry is a graphic form of algebra, topology is a graphic form of logic. It is the same subject that exists in topology and logic. 

Binary logic of oppositions can be represented in simple topological structures that differentiate between inside and outside. Binary logic and topology can be used to structure the subject in its phallic aspects, as having or not having the phallus, as subjected to desire as a limit to Jouissance. When Lacan explores feminine Jouissance beyond Freud, phallic logic and phallic topology are not enough. To speak of the Other Jouissance he relies on multidimensional topological bodies, and paradoxical logic that transcend cause and effect.

My goal in this paper is to trace the development of the logic and topology of the Other Jouissance throughout Lacan's seminars. I will begin with Seminar Twenty as a pivot, with the topological quality of "compactness" – infinity within limits. 

From this middle point I go back and trace the origin of these ideas in Lacan's topological structures: the Graph of Desire, the Moebius Strip, the Torus and the Sphere equipped with a Cross-cap. In the third part I will show how from Seminar Twenty to the later seminars, the ideas continue to develop, culminating in the subversion of Freud's Psychic Reality in the Borromean Knott.  

Paradox

"The logic of the Other Jouissance" sounds like a paradox. On one hand, classical logic requires binary or phallic opposition between true and false, and on the other hand, the Other Jouissance is precisely "not phallic." To understand the paradox of non-phallic logic, let us first try to understand what is a paradox for Lacan.

Lacan's approach to solving a paradox is to question its underling premise. In Russell's paradox, there is a catalog of all the catalogs that do not contain themselves as members. Does it contain itself or not? It must contain itself because its one of the catalogs that do not contain themselves, but as soon as it becomes a member of itself as a set, it ruins the validity of the set as containing only members that do not contain themselves. 

The underlying premise that Lacan exposes is the assumption that the catalog as a containing set and the catalog as a contained member are the same catalog (Lacan, 24.1.1962). The Other Jouissance is a paradox. How can we speak about what is structurally defined as unspeakable? What kind of writing and logic can support what cannot stop not being written? 

As in the case of the catalog, Lacan's solution is the annulment of premises. The basic premise of phallic logic is that a signifier is identical to itself, and different from other signifiers. A equals A and A doesn't equal B. ( A=A and A≠ B.) This is necessary for signifiers to produce meaning. Since the Other Jouissance goes beyond meaning, within the Other Jouissance a signifier is not identical to itself A≠ A.    The name of the catalog on the cover of the book and the name of the catalog as a bibliographic note are not the same catalog.

Phallic Jouissance is based on A=A. In the clinic it will present itself as the premise of the subject that Jouissance can be reclaimed through repetition. Neurosis is the search for the missing signifier of primary repression, of ultimate Jouissance. It is the phantasm that the next signifier will satisfy.

This is destined to fail. The failure of the Phallic Jouissance requires that there would be another, an Other Jouissance, that is linked to that constantly missing signifier. In Seminar Fourteen "The Logic of the Phantasm", Lacan ( 23.11.1966) uses the device of a statement written on a board: "Write the smallest whole number that is not written on the board". As soon as it is written it is no longer the smallest number that is not written. The number can never be written and be a satisfaction of the demand. The number that is always yet to be written is the unary trait, the S1 not linked to S2. It is the Signifier of the Lack of the Other that remains outside the parenthesis of the barred Other, outside of the signifying chain. 

The Other Jouissance is rooted in the primordial signifier, in S1, a "stand-alone" signifier that does not participate in the metaphor-metonymy dispositive. The movement that represents Jouissance does not flow as in phallic Jouissance from one signifier to another but within the signifier itself. It does not progress with a phantasm of culmination toward the ultimate signifier of Jouissance. There is no Other to the Other Jouissance. The infinite Jouissance is not the final destination but the movement itself. The Topological quality that Lacan uses to structure this movement that doesn't reach S2, is Compactness (Lacan, 1972-1973, 9.) 

Compactness

In Seminar Twenty Lacan takes the topological quality of compactness as a structure for the Other Jouissance. Compactness means that you can continue to fill a closed set and never finish. You can fill a box with sand and there will always be room in the box for smaller grains. 

Figure 1

As in the Zeno's paradox of Achilles and the Tortoise, we can proceed toward a goal, shortening the distance by half in every step, and never reach the goal. With no S2 as a reachable goal, with no S2 as a caption point that momentarily suspends the flow and replaces Jouissance with meaning, the endless movement of desire is the object itself. 

Compactness is demonstrated in the mystic's yearning toward God, which is identified by Lacan in Seminar Twenty as one of the forms of the Other Jouissance (Lacan, 1998 [1972-73], 76.) 

In seminar nine "Identification", he takes the example of the Sufi ethics of perpetual desire as a Jouissance (20.6.1962). The Sufi mystic as other mystics, takes his yearning to god not as a means to an end, desire toward Jouissance, but as desire as Jouissance itself. In Topological terms, the Sufi mystic in compact space is shortening the distance between him and God without the phantasm of grasping him. The mystic takes his own yearning, his own desire, as Jouissance. The analyst's suspended attention is rooted in this desire as Jouissance, and Lacan states clearly that this is what we have in common with mystics.

Compactness in the imaginary appeared in Lacan's teaching in a latent form as early as the Mirror Stage. The pieces of the fragmented body do not add up to the whole imaginary form. The Jouissance that escapes the form is represented by it's lack, the phallic signifier. 

Compactness in the symbolic appeared through the phallus as signifier; Signifier of the ratio between Jouissance obtained and the ultimate incestual Jouissance that is promised.  In Seminar fourteen, the Logic of Phantasy (8.3.1967)  he shows how phi plus phi squared plus phi to the third and so on, never add up to one; never fill up S1 and progress to S2.

Graph of Desire

The Graph of desire (Lacan, 1999 [1960], 692) provides the topology for the phallic ratio (Lacan, 1999 [1958], 575). It demonstrates how Jouissance progresses from S1 to S2, precisely because it does not fill up S1. The metonymy is a response to S1 not satisfying. It is as if Lacan's interest in topology could not have been sparked without a notion of the Other Jouissance and the promise of going beyond Freud on questions of femininity. It is as if Seminar Twenty had an après-coup influence on the Graph of Desire.

Figure 2

If we take S1 as a starting point and S2 as an end, the graph of desire shows the suspension of Jouissance between the two. 

In its basic form, the Graph of Desire shows the Après-Coup nature of meaning formation in the signifier chain. S1 cannot fill up before S2 because S1 is determined retroactively by S2.

In Seminar Five, "The Formations of the Unconscious" Lacan brings from Freud's "Wit and its Relation to the Unconscious" the example of "famillionaire" to demonstrate this point (20.11.1957)  The two signifiers that comprise the neologism, "familiar" and "millionaire" are taken from the treasury of signifiers by the subject who is traveling along the synchronic vector. 

The signifiers are combined at the next junction that is designated as "Signifier of the Other", but they don't have meaning yet. The meaning crystallizes only after another movement along the diachronic vector from S1 back to S2. 

There is a moment, after the words were combined but before this combination is designated as a new word. This is the moment between the telling of the joke and the laughter of understanding. What exists at that moment? What is this thing that is made of language but is not a word yet? What Other Jouissance exists with S1 as foundation but without S2 as limit? 

This solitary S1 is not outside of language, since it is already a combination of two signifiers, but it is not language in the sense that it is not a word in the dictionary yet. One of the forms of this S1 is Lalangue – Utterances saturated with Jouissance, not yet tamed by meaning (Lacan, 1998 [1972-73], 44.) In analysis it could be the utterance, eh, emm, which is what the analysand is saying when he is lacking words. This utterance functions as a unary trait, a signifier that takes the place of the lack of words. 

At the top left corner of the full graph is the signifier of the Lack in the Other.

Figure 3

It is a development of the basic cell; another way to show the movement between S1 and S2 in terms of movement from Jouissance to castration. The Signifier of the Lack in the Other holds the place evacuated within language to designate that there is no universe of discourse. It is placed on the left side of the vector from Jouissance to castration. The appropriation of the neologism "Famillionaire" as meaningful by the code on the right hand side of the basic cell is repeated at the top level. On the castration end of the vector, Jouissance is captured by the demand of the other and is phallicized as drive. The appropriation of the Real body by language divides it into erogenous zones. 

Figure 4

The topology of graphs does not eliminate oppositions. It relies on them as nodes, as poles of a tight rope than enables suspension. To circumvent this limitation Lacan turns to a strange shape that has three dimensions but only one surface and one edge, the Moebius Strip.

Moebius Strip

Classical logic requires oppositions. Inside is defined in opposition to outside. Freud's insight, that there is no negation in the unconscious (Freud, 1905 [1901], 57 n. 2), is Lacan's premise in formulating logic of continuity between opposites. At first glance the Moebius strip has two sides that connect at the point of the twist, bridging opposites. This view gives shape to the sexual opposites, man and woman. The point of the twist is the mythical point of sexual relation, where opposites meet. 

Figure 5

At a second glance, the strip would be seen as one surface with no opposites at all, with one surface and one edge. The place of the twist would no longer be a localized point on the strip but the nature of the strip itself. (Lacan 1981 [1963-64], 156)

The Moebius strip would be, but is not, seen as one sided, because as subjects determined by language we cannot grasp it in one glance. We have to break the Moebius Strip to manageable pieces in order to reassemble it in our mind. The subject that would transcend the opposites that language imposes is the subject of the Other Jouissance. 

The existence of the Moebius Strip is the proof of the possibility of existence for such a subject. It is the topological shape of the logic that structures such a subject. 

Torus

The Graph maps the suspension between S1 and S2, Jouissance and castration. The Moebius Strip requires us as subjects to be suspended there in order to grasp it's non-duality beyond Euclidean space. The Torus provides a topology for understanding the logic of Phallic Jouissance as equivalent to the One, and the Other Jouissance as equivalent to the infinite. The topology of the One and the infinite on the Torus is a step in Lacan's thought toward the Idea of compactness, infinity within the One, in Seminar Twenty.

The Torus in its skeletal form is a ring. It is the same ring that forms the Borromean Knot (Lacan 19.11.1974) . The ring is formed by taking an infinite line, 

Figure 6

Bending it into a loop, 

Figure 7

Tying a knot, and cutting the excess infinite lines. 

Figure 8

This cut is the price paid in Jouissance, in the flesh of the body, for the One of the signifier.

The knot that holds the curved line as a Torus is the unary trait. It is the device that transforms the infinite into a one, is in between them, and is the seat of the point of identification as an edge on the circle (Lacan, 14.12.1966) . The unary trait is both an element of language, a letter that ties the mythical subject into a one, as well as tail of the Real, a trace of the infinite, the Jouissance of the body from which it came. The masculine side would be represented as "the one"" of the ring.  Since the infinite cannot be written, what replaces the infinite is its tail, the unary trait or the letter. The Unary Trait is the placeholder of the woman side as "the infinite" of the trimmed endless line. Kierkegaard exemplifies the Other Jouissance in his disavowal of the One in favor of the endless. (Lacan, 1988 [1972-73], 77). 

Again, this knot is not localized, for the simple reason that we could dissolve the One by cutting anywhere on the ring. The point of the knot will be established retroactively after the cut.

The full Torus is made of two rings or circles, one is the skeletal torus itself, the empty circle, and the other is the circumference of the bent tube, the full circle (Lacan, 1.3.1961). 

Figure 9

The full Torus shows the impossibility of the Other Jouissance. The circle around the hole designates desire and the circle around the tube designates demand. The lost petit a' is at the center of the Torus, at the empty circle of desire, while the neurotic repetitively searches for it in the spirals of the tube, the defiles of the signifier, the full circle of demand (Lacan, 6.6.1962).

When two Tori link as subject and Other, the impossibility of the Other Jouissance is employed as cause of desire. The two linked Tori show the Demand of the subject as the desire of the Other in hysteria, or the desire of the subject as the demand of the Other in obsessional neurosis lesson (Lacan, 30.5.1962). 

Figure 10

But the Other Jouissance exists. The Other Jouissance, as the mystic's desire as Jouissance, or Spinoza's intellectual love of God, or a passionate love with no price attached, ex-sists, precisely because it is impossible. It is the incestual root of any phantasm. It can be impossible to the point of putting the subject in a precarious position in mad love or passage a l'act. 

The logic of the Other Jouissance is such that its impossibility is its cause. As a writing for the impossibility of the Other Jouissance as its cause, Lacan employed the impossible topological shape, the Sphere Equipped with a cross-cap.

Cross-cap

The cross- cap integrates previous topological forms: the graph, the Moebius, and the Torus. It is a Torus that underwent a Moebian twist, and its hole reduced to zero. Its importance for mapping the topology of the Other Jouissance is found in the act of cutting it in two. (Lacan, 13.6.1962)

Figure 11

If we walked on its surface we would trace a Moebian path. If someone walked on the other side of the torus and reached the twist together with us, the Moebian paths would have to intersect. This is impossible because the planes cannot cross each other. This impossibility engendered a fourth dimension of space for it to exist. 

The Moebian path forms a loop called interior eight that can be traced or cut on the surface of the cross-cap. The cut intersects itself twice like the après-coup from the graph of desire (Nasio, 2004). The cut creates the surface on which it is cut. The signifier creates the reality of which it is a product.

Figure 12

The interior eight divides the cross-cap into two parts: a Moebius strip that designates the subject and a disc that designates the object. This is the topology of Phantasm. The cut is made on the surface, but which surface? The surface before or after the cut? The line that creates its surface is a topological form of Russell's paradox. This shows the limits of classical logic, which does not account for this après-coup movement of a cut made on a surface that the cut made. The masculine Jouissance is after the cut. It is the phantasm of subject – cut – object. It is phantasmatic in the sense that it takes the surface to be ontologically given while ignoring the fact that the surface is a result of the cut. The Other Jouissance is the ungraspable riddle, the impossibility of a cut creating its surface.

In Seminar Ten on Anxiety (Lacan, 30.1.1963), Lacan uses the point in the center of the cross-cap to designate the anxiety of the lack of the lack. The paradox of its impossibility is felt in the horrifying image of looking at one's own eye balls. This point is the point of convergence, of subject and object, subject and Other, and desire and Jouissance. This convergence threatens the subject but it is also the entry to the Other Jouissance.

A cut that converges opposites but leaves a unary trait for the identification of the subject, would be a cut that leaves a disc that is infinitely small. Minimizing the object in this way leaves the object and subject at the same point. 

Figure 13

The Jouissance of the mystic is in relation to an object so minimal, that the subject and object, desire and Jouissance collapse into each other. Lack as object is desire as Jouissance. 

The Other Jouissance as "desire as Jouissance", appears in Seminar Eight "Transference" (1.2.1961).  In Lacan's interpretation to Plato's symposium, the agalma that is the object of Alchibiades's attraction to Socrates is Socrates' desire. 

Socrates shows Alchibiades that his love is phallic, that it is based on the mutual exclusion of the lover and the loved. In topological terms it is the line that cuts and separates subject and object. 

The feminine position is represented by Socrates as he is quoting Diotima right before Alchibiades enters the scene. The ultimate Eros according to Diotima is toward eternity through beauty. It would be a kind of beauty that links man to the Real of the Gods, not caught up in the phallic erogenous zones and their metonymical  goods. In Diotima's version of Eros there is no difference any longer between the subject as lover, the other as loved (Lacan, 25.1.1961 ). The collapse of these distinctions is what characterizes the Other Jouissance 

Diotima's version of Eros is neither a man nor a god. It’s the link, a daemon; In between. It is In the place of the unary trait, in between the Symbolic and the Real. In Seminar Seven on the Ethics of Psychoanalysis, Lacan says that beauty is in between the Symbolic and the Real, the signifier and Das-Ding. If not for the link to the Symbolic, if not for the taming of the gaze by art, the Real would show itself as horror. 

Lacan's topological development went through paradox, suspension of meaning, Après-coup logic, Moebian non-locality, The infinity of the line in relation to the one of the ring, and finally, the convergence of desire and Jouissance in the horizon of the fourth dimension. These have set the stage for the topology of compactness, the perpetual motion toward the infinite horizon, as a structure for the Other Jouissance. 

Borromean Knot

The logic and topology of the Other Jouissance serves a stepping stone on the way to formulating a topology of psychic reality. The irreducible lack as the link between desire and Jouissance is taken by Lacan in Seminar RSI as the Borromean quality. The rings hold together without being linked with each other. What links them is the Borromean structure they form. 

The same non-phallic logic enabled Lacan to transcend the Freudian Rock of Castration in analysis, enabled him to formulate a non-phallic Jouissance, and finally in RSI, to transcend the Freudian Oedipal myth and formulate a non-phallic reality (Lacan, 14.1.1975).

The structural irreducibility of the gap, the impossibility of the sexual relation, is the foundation of the Other Jouissance and psychic reality. As a name, it is a signifier that enables Joyce to ex-sist. He experiences the epiphany of the Other Jouissance with this device that frees him from his phallic lineage, which supplements a unary trait as sinthome to ward off psychosis. 

We cannot grasp the Borromean knot in its Real tangible string form. In order to let it speak Lacan flattens it. The flat diagram of RSI is the Imaginary form, the Borromean Knott in it's speakable dimension.

Figure 14

Within the flat dimension of speaking, the intersection of the Symbolic and the Imaginary is marked as Sense. The Other Jouissance is the adjacent area of the Real, the part that is in ex-sistence to speakable sense. The flat diagram localizes the Other Jouissance as the Jouissance of no-sense (Lacan, 14.1.1975

In the triad: Real ex-sistence, imaginary consistence and symbolic hole; ex-sistence is the contribution of the Real to reality. Ex-sistence is the essence of non phallic logic. It requires absence to fill a paradoxical role, not as an alternative of existence but as its foundation.  Finally, The non phallic logic and topology of the Other Jouissance as ex-sistence, is a formal structure that serves us analysts. It allows us to trace with the analysand the path from the symbolic envelope of the symptom as phallic, to the kernel of Real Jouissance in sexuality, to the lack at the core of being.

References

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Lacan, J. (1957-1958). Seminar V Formations of the Unconscious. Unpublished.

Lacan, J. (1999) [1958] The Signification of the Phallus. in Ecrits. New-York, Norton.

Lacan, J. (1999) [1960] The Subversion of the Subject and the Dialectic of Desire in the Freudian Unconscious in Ecrits. New-York, Norton.

Lacan, J. (1960-1961) Seminar VIII  Transference. Unpublished.

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