# Talk:Topology

The term 'topology* *

The representation on a map of the physical features of a landscape.

In psychoanalysis the term is used to describe the differentiation of the mind or psyche into subsystems with specific functions and characteristics.

Freud's topographies of the psyche owe much to nineteenth-century theories of cerebral localization, which ascribe different mental functions to different areas of the brain.

It was the study of dreams that led Freud to the conclusion that unconscious activities such as dreaming are quite divorced from the conscious mind and literally take place on *ein anderer Schauplatz* (another stage or theatre).

Freud evolved two distinct topographies.

The first, elaborated between 1900 and 1915, describes an apparatus comrpising unconscious, preconscious and conscious systems, with mechanism of censorship to prevent ideas from moving between them.

Considerations of representability and other mechanisms of the dream-work filter or censor the content of dreams and fantasies before allowing them to enter the conscious mind, usually because their sexual content is unacceptable to conscious thought-processes.

The second or 'structural' topography, elaborated from 1920 onwards, describes a structure of three agencies known respectively as the id, the ego and the super-ego.

Topology refers primarily to the branch of mathematics that rigorously treats questions of neighborhoods, limits, and continuity.

Psychoanalysts have applied it to the study of unconscious structures.

Topology (*topologie*) is a branch of mathematics which deals with the properties of figures in space which are preserved under all continuous deformations.

These properties are those of continuity, contiguity and delimitation.

The notion of space in topology is one of topological space, which is not limited to Euclidean (two- and three-dimensional space), nor even to spaces which can be said to have a dimension at all.

Topological space thus dispenses with all references to distance, size, area and angle, and is based only on a concept of closeness or neighbourhood.

## Contents

## Freud

In what have been called his two "topographies" (the first dating from 1900 and the second from 1923), Freud resorted to schemas to represent the various parts of the psychic apparatus and their interrelations.

These schemas implicitly posited an equivalence between psychic space and Euclidean space.

Freud used spatial metaphors to describe the psyche in *The Interpretation of Dreams*, where he cites G. T. Fechner's idea that the scene of action of dreams is different from that of waking ideational life and proposes the concept of 'psychical locality'.

Freud is careful to explain that this concept is a purely topographical one, and must not be confused with physical locality in any anatomical fashion.^{[1]}

His 'first topography' divided the psyche into three systems: the conscious (Cs), the preconscious (Pcs) and the unconscious (Ucs).

The 'second topography' divided the psyche into the three agencies of the ego, the superego and the id.

## Lacan

Early on, Jacques Lacan noted that the limitations of such a naive topology had restricted Freudian theory, not only in the description of the psychic apparatus (a description that in the end required an appeal to the economic point of view), but also in the specificity of clinical structures.

The hypothesis that the unconscious is structured like a language, that is, in two dimensions, led Lacan to the topology of surfaces.

The concept of 'foreclosure', for example, which he constructed on the basis of this topology, confirmed the heuristic value of his approach.

In his 1961-1962 seminar "Identification", Lacan unveiled a collection of topological objects — such as the torus, the Möbius strip, and the cross-cap — that served pedagogical aims.

But already he saw them as more than just models.

With the Borromean knot, introduced in 1973, he took the position that these objects were a real presentation of the subject and not just a representation.

Below are several of Lacan's topological objects.

## The Cut and the Signifier

Far from being given a priori, every space is organized on the basis of cuts and can actually be considered as a cut in the space of a higher dimension.

We are familiar with the subjective impact of this: The events of our lives only become history through the castration complex, which organizes our reality at the price of an imaginary cutting off of the penis.

According to Freud, by introjecting a single trait of another, the subject identifies with the other (at the price of losing this person as a love object).

In the single trait Lacan found the very structure of the signifier: A cut allows the lost object to fall away. He called this cut the "unary trait."

The linguist Ferdinand de Saussure insisted on the fundamentally negative, purely differential character of the signifier.

Lacan formalized this property in the double loop, or "interior eight," in which the gap created by the cut is closed after a second trip around a fictional axis.

The difference of the signifier from itself is indicated by the difference between the two trips around the loop (Figure 1).

## The Möbius Strip and Interpretation

If a signifier represents the subject for another signifier, then the subject would be supported by a surface whose edge would be a signifying cut.

Note that the plane—the usual screen for the subject's images, figures, and dreams, that is, plans—is a surface that does not meet these conditions.

The double loop cannot be drawn on a plane without showing a cut.

The same is true of a sphere, a simple representation of the universe.

The Möbius strip, on the other hand, can represent this cut and symbolize the subject of the unconscious.

Since a Möbius strip only has one surface, it is possible to pass from one side to the other without crossing over any edge—an apt representation of the return of the repressed.

The Möbius strip also has certain other peculiarities.

A cut that runs one-third from the edge and parallel to the edge divides the strip into a two-sided strip linked to what remains of the original Möbius strip.

But if this cut is made in the center, it does not divide the Möbius strip in two.

Instead, the entire strip is transformed into a strip with two sides.

This characteristic illustrates the equivalence between the Möbius strip (the subject) and the medial cut that transforms it, and also provides a model of how interpretation functions.

Interpretation does not abolish the unconscious.

On the contrary, it makes the unconscious real for the subject by its transformed appearance as another (an Other) surface (figure 2).

## The Torus

Lacan made different uses of the torus.

By drawing Venn diagrams, traditionally used to illustrate basic logical operations, on the surface of the torus, he demonstrated the extent to which our thinking depends upon the plane surface, and he also provided another possible basis for the logic of the unconscious (Figure 3).

By inscribing the same circles on the surface of the torus, Lacan revealed the logic of the unconscious discovered by Freud (Figure 4).

On the torus, only symmetrical difference is consistent.

Thus we have a demonstration of how the signifier can be different from all other signifiers and also from itself.

Lacan also used the torus to represent the subject as the subject of demand.

In this sense, the torus can be conceived as the surface created by the iteration of the trajectory of the subject's demand.

This trajectory turns around two different empty spaces, one that is "internal," D, the lack created in the real by speech, and one that is "central," d, corresponding to the place of the elusive object of desire that the drive goes around before completing the loop (Figure 5).

For every torus, there is a complementary torus, and the empty spaces of the two are the inverse of each other.

Lacan made this structure of complementary toruses the support of the neurotic illusion that makes the demand of the Other the object of subject's desire and, conversely, makes the desire of the Other the object of subject's demand.

This structure also arises from the fact that on a torus, the signifying cut (the double loop) does not detach any fragment.

Neurotic subjects, insofar as they give in to neurosis, insofar as they are "in the torus," are not organized around their own castration, but instead excuse themselves by substituting the Other's demand for the object of their fantasy (figure 6).

## The Cross-Cap

The cross-cap, or more precisely, the projective plane, can represent the subject of desire in relation to the lost object.

A double loop drawn on its surface in effect divides this single-sided surface into two heterogeneous parts: a Möbius strip representing the subject and a disk representing object a, the cause of desire.

The disk is centered on a point that is related to the irreducible singularity of this surface, which Lacan identified with the phallus.

Unlike the representation of the subject produced on the torus, here a single cut, which symbolizes castration, produces both the subject and the object in its divisions (figure 7).

## The Borromean Knot

Introduced by Lacan in 1973, the Borromean knot is the solution to a problem perceivable only in Lacanian theory but having extremely practical clinical applications.

The problem is: How are the three registers posited as making up subjectivity—the real (R), the symbolic (S), and the imaginary (I)—held together?

Indeed, the symbolic (the signifier) and the imaginary (meaning) seem to have hardly anything in common—a fact demonstrated by the abundance and heterogeneity of languages.

Moreover, the real, by definition, escapes the symbolic and the imaginary, since its resistance to them is precisely what makes it real.

This is why Lacan identified the real with the impossible.)

In psychoanalysis, the real resists, and thus is distinct from, the imaginary defenses that the ego uses specifically to misrecognize the impossible and its consequences.

If each of the three registers R, S, and I that make up the Borromean knot is recognized to be toric in structure and the knot is constructed in three-dimensional space, it constitutes the perfect answer to the problem above, because it realizes a three-way joining of all three toruses, while none of them is actually linked to any other: If any one of them is cut, the other two are set free. Reciprocally, any knot that meets these conditions is called Borromean. Note that the subject is now defined by such a knot and not merely, as with the cross-cap, as the effect of a cut (figure 8).

Unfortunately, this ideal solution, which could be considered normal (without symptoms), seems to lead to paranoia. Lacan considered this to be the result of failure to distinguish among the three registers, as if they were continuous, which indeed occurs in clinical work. Being identical, R, S, and I are only differentiated by means of a "complication," a fourth ring that Lacan called the "sinthome." By making a ring with the three others, the sinthome (symptom) differentiates the three others by assuring their knotting (figure 9).

In this arrangement, the sinthome has the function of determining one of the rings. If it is attached to the symbolic, it plays the role of the paternal metaphor and its corollary, a neurotic symptom.

Lacan also drew upon non-Borromean knots, generated by "slips," or mistakes, in tying the knots. These allowed him to represent the status of subjects who are unattached to the imaginary or the real and who compensate for this with supplements (Lacan, 2001). In such cases the sinthome is maintained.

By using knots, Lacan was able to reveal his ongoing research without hiding its uncertainties. The value of the knots, which resist imaginary representation, is that they advance research that is not mere speculation and that they can grasp—at the cost of abandoning a grand synthesis—a few "bits of the real" (Lacan, 1976-1977, session of March 16, 1976). Even though he knew something about topology as practiced by mathematicians, Lacan advised his students "to use it stupidly" (Lacan, 1974-1975, session of December 17, 1974) as a remedy for our imaginary simplemindedness. He also recommended manually working with the knots by cutting surfaces and tying knots. Finally, for Lacan, topology had not only heuristic value but also valuable implications for psychoanalytic practice.

## More

Lacan criticises these models for not being topological enough.
He argues that the diagram with which Freud had illustrated his second topology in *The Ego and the Id* (1923b) led the majority of Freud's readers to forget the analysis on which it was based because of the intuitive power of the image.^{[2]}
Lacan's interest in topology arises, then, because he sees it as providing a non-intuitive, purely intellectual means of expressing the concept of structure that is so important to his focus on the symbolic order.
It is thus the task of Lacan's topological models "to forbid imaginary capture."^{[3]}
Unlike intuitive images, in which "perception eclipses structure", in Lacan's topology "there is no occultation of the symbolic."^{[4]}

Lacan argues that topology is not simply a metaphorical way of expressing the concept of structure; it is structure itself.^{[5]}
He emphasises that topology privileges the function of the cut (*coupure*), since the cut is what distinguishes a discontinuous transformation from a continuous one.

Both kinds of transformation play a role in psychoanalytic treatment. As an example of a continuous transformation, Lacan refers to the moebius strip; just as one passes from one side to the other by following the strip round continuously, so the subject can traverse the fantasy without making a mythical leap from inside to outside. As an example of a discontinous transformation, Lacan also refers to the moebius strip, which when cut down the middle is transformed into a single loop with very different topological properties; it now has two sides instead of one. Just as the cut operates a discontinuous transformation in the moebius strip, so an effective interpretation proferred by the analyst modifies the structure of the analysand's discourse in a radical way.

While [[schema L] and the other schemata which are produced in the 1950s can be seen as Lacan's first incursion into topology, topological forms only come into prominence when, in the 1960s, he turns his attention to the figures of the torus, the moebius strip, Klein's bottle, and the cross-cap.^{[6]}
Later on, in the 1970s, Lacan turns his attention to the more complex area of knot theory, especially the borromean knot.^{[7]}

## Miscellaneous

The topology of the subject is the trace that there were psychoanalysis and thus symptom. (Tyings)

Topology: according to the dictionary the Robert “Study of the invariant properties in the geometrical deformation of the objects and the continuous transformations applied to beings mathematics-Structure where these properties in a unit intervene. Topology initially was called geometry of situation or analyzes situs.
Topology general (where sets), combinative (or algebraic). “
General topology corresponds to the language of the categories, of mathematics.

Traditional canonical logic corresponds to the language of the predicates, the proposals, it is consistent, supplements, totalitarian. It is difficult to distinguish logical and mathematical. Physical topology corresponds to the geometry Topology treats transformations, the equality between two object is defined there like the possible way of a presentation with another, by a deformation continues one with the other, out metric, proportion. Topology, a geometry of rubber!

topology lacanienne: But what is this thus? It is the topology of the subject!

Topology studies the relations of the place with the speech, it is a rhetoric (space of castration), not a code; with the crossroads of the statement (of the littoral structure of the letter) and gesture (drawing) metaphorical, it is an esthetics; it is the structure even. It wants to transmit what there is like reason of the desire, even irrational desire him, which produces the disorder in topology. The topology of the subject is the setting of the phallic function outstanding. The psychoanalysis of Freud and Lacan is the last phase of the social psychosis (for Freud it is in bond with the rejection of homosexuality; for Lacan in bond with xenophobia and the anti-semitism) of which we are prone and topology is pulsatory closing of the diagram of Freud. (Psychic Apparatus, diagram L, letter 52). The exclusion of sexuality and unconscious melt the social pact which generates, legitimates and maintains violence. [Recall: the psychoanalysis of Freud and Lacan (and “not-analyzes it”) is the plague even and not pastoral, (even on Internet!) and the policy of malignant agitating the red rag will not solve anything, quite to the contrary, reinforcing only the problem, tightening only the node!.)] The psychoanalysis builds the object has and is completed with object A. (1) The psychoanalysis is rational, false and irrefutable; it is resolution; holds of the goods and symptom; is a business of reason not inevitably scientific. It puts in act structure of involution meaning (copula which links the identical one with the different one), is to pass from unilateral to the bilatère because of an edge which does not function very well in the psychic apparatus. Topology makes it possible to account for the psychoanalysis outside the psychoanalysis. Topology presents waste of the scientific and mathematical theory. It refers to the structure of the language and not to measurement (reserved to the technocrats), it littéralise a geometrical problem without resorting to the number, If the subject is the effect of meaning, if the Other is the preliminary site of the pure subject of meaning, the circuit of the word and its combinative on the graph of the desire of Lacan, is a place of inscription, where the great Other rests.

The subject is questioned by topology because it has to answer problems of logic that Traditional Canonical Logic (LCC) forsakes. Indeed how to solve the problems of the relation of the subject to the other or the Other; with its context; with the concept of interior, outside; of intrinsic, extrinsic; of room and total; differentiation and identification. The LCC is founded on the principle of noncontradiction and the excluded third. Modified logic (moebienne), while preserving a rigour of writing, accounts for the paradoxes rather than to be unaware of them.

Topology requires the concept of dimension like topological and nonmetric invariant. We must consider the object in its obstruction and that a dimension consists of other dimensions. It is the cut which disjoins an object in two parts. We perceive the world by our retina on a surface in two dimensions (and not in three). The concept of dimension allows the passage of intrinsic extrinsic, distinction which makes it possible on a subject (in intrinsic space) to take its own body like object (in extrinsic space), which qualifies narcissism; where allows to show the identity of two distinct situations of appearance.

Topology of surfaces (in connection with the act of the analyst, the meeting, the effects on the psychical reality of the subject, with the effects of the word, the words); topology of the nodes (relates to the series of acts which constitute the direction of the cure, the relation of transfer, its duration, its evolution, the articulation between the various elements of the structure; topology of the cut (how the cut is manufactured), topology of the structure (how articulates the cuts), topology of the transfer!. The operations of continuous transformation, cut, redoubling, reversal on surfaces create butting together, the substitution of the node. To withdraw the node of the psychoanalysis and it is is delirious it, the node is the achievement of a cut. The movements of the speech they is the nodes. The node is a simple machine, almost not tied, is not a writing, it is in bond with the impulses. there is introduced in psychoanalysis by the 3rd dimension, it is no node in dimension two. The node consists between various extrinsic positions and is erased in certain intrinsic situations. This disappearance where pulsation is what constitutes the function known as paternal. Chains and nodes propose like a practice covering the whole of the spectrum of the writing, of the mathème to the poem.

Plunging is a setting flat while respecting dimension (tops below), representation of the nodes. The immersion is a setting in the plan and product of the graphs.

## See Also

- Knot
- Schema L
- Schema R
- Seminars
- Thalassa. A Theory of Genitality
- Unary trait
- torus
- borromean knot
- extimacy
- subject
- cross-cap
- knot

## References

- Lacan, Jacques. (1975). La troisième, intervention de J. Lacan, le 31 octobre 1974. Lettres de l 'École Freudienne, 16, 178-203.
- Lacan, Jacques. (1974-1975). Le séminaire, livre XXII, R.S.I. Ornicar? 2-5.
- Lacan, Jacques. (1976-1977). Le séminaire XXIII, 1975-76: Le sinthome. Ornicar? 6-11.
- Lacan, Jacques. (2001). Joyce: Le symptôme. In his Autres écrits. Paris: Seuil.