The following article was rejected for publication by the Buffalo Center of Psychoanalysis as being “incomprehensible” to the audience of the journal Umbra (which we were informed addresses a type of understanding peculiar to students of critique and literature). Be that as it may, others less hampered by such strictures will find below, for the first time in the English language, an introduction to Lacan’s generalization and extension of the psychoanalytic cure. R.G.
If Jacques Lacan had been interrogating the limits of Freudian psychoanalysis from the beginning, it was not until 1956 with D’une question préliminaire a’ tout traitement possible de la psychose ( Of A Preliminary Question To Any Possible Treatment of Psychosis) that he showed the necessity of reformulating the problem in a topological presentation that went beyond an “abstract theory of the faculties of the subject” 1 .
Although Lacan continued to purify what he called the “ideology of psychoanalysis” through a topology of surfaces, it was not until February 9th 1972, in his seminar Ou Pire, that he changes to a theory of knots and announces his discovery of the Borromean Rings. What remains invariant in this transformation is the insistence that the use of such topological structures consists not in illustrating the theory of psychoanalysis, but of initiating a practice of psychoanalysis itself:
For is not structuralism what permits us to pose our experience as the field where it speaks? If yes, “the distance to experience” of structure disappears since it [the structure] operates not as a theoretical model, but as the original machine which puts in scene the subject. (Remarques Sur La Rapport De Daniel Lagache, Lacan, 1958-60)
By the time of R.S.I., Lacan corresponds the three closed chords of the Borromean Lock to the Real, Symbolic, and the Imaginary, while noting this tertiary grouping of rings was only a minimum and required a fourth ring ∑ (in black) that was only implicitly indicated in Freud’s use of the term psychic reality 2. Further still, by explicitly equating this fourth ring with the Nom-du-Pere and the Oedipus complex Lacan isolates it as what analysis comes to operate on: “To be knotted otherwise, this is what is essential to the Oedipus Complex and it is precisely on what analysis operates.” (R.S.I., January 14th 1975)
Four Ring Borromean Lock
Previously, Lacan had noted that the ethics of psychoanalysis would depend on introducing something ‘useful’ precisely at the point not only of the failures of his Ecole, but analysis itself (Preliminary to the Seminar of R.S.I., November 9th 1974). By doing so, Lacan began not only to render account of the ideals of psychoanalysis (its possibilities and psychotherapies) and its closures (its applications and techniques in the Heideggerian sense), but to open up a place for the treatment of the aporias of its cure.
Statement of the Problem: From Aporia to Structure
In the contemporary scene of Lacanian psychoanalysis, we are faced with a double obstacle of a too contextual understanding of psychoanalysis and a completely de-contextualized understanding of its relation to topology.
On one hand, we have the infinite psychoanalytic histories describing the nature of its aporias – the incomplete nature of Freud’s analysis of neurosis, the impossibility of analyzing psychosis and perversion, negative therapeutic reaction, etc.- and the repetitive epistemological investigations into the nature of its impostures, immaturity, and psuedo-scientificity. Our introduction here is less ambitious in its scope, yet has the advantage of allowing us to situate what is crucial: the structural and operatory modes that a discontinuity of such natural themes refer to. For this reason our entry is only superficially historical or epistemological, for if it were, our analysis would remain at the level of a succession of themes natural to psychoanalysis. Our introduction is different in so far as it focusses on the manner in which not only Lacan’s topology, but a theory of psychoanalysis resists such historical and epistemological accounts. It is this resistence that is necessary to listen to and translate in the presentation of a theory and practice of analysis.
On the otherhand, we have become use to those who bravely state that a Borromean falls apart — or unlocks — when anyone of its three components is taken away or cut, while never explaining why any of this should be significant to the aims and goals of the analytic cure in the first place. For if it were not only with a bit less haste, surely they could find the time to report why Lacan never tired of explaining the real problem was to be found in how the fourth ring of the Sinthome – Psychic Reality,Oedipus Complex, or Name of the Father — comes to bear in this problem of unlocking.
My short intervention cannot hope here to resolve this two sided figure of understanding too much too quickly. Rather it suffices that we isolate the problem and decipher the reasons for such complacency, while laying out a few landmarks so that the reader may begin to orient themselves differently.
One should begin, for instance, with the December 16th 1975 seminar of Lacan’s Sinthome in which he had already formulated a topological movement that would not only undo a knot, but also unlock the four ring Borromean lock by allowing the fourth ring to slip off 3. In so doing, Lacan isolates a separation from the Sinthome that was only avoided by the traditional attempts to describe the aporias of the psychoanalytic cure negatively and not as a positive moment of dé-nouement. For surely the Gordian problem is to show that to un-tie a knot – or Sinthome - is not to non-tie or cut a knot, but more positively to tie one by adding its per-verse . Without these precautions, the psychoanalytic in-curable looses its structure and effectivity, while trivializing into the morbid consciousness of the non-curable.
What is difficult for some to admit, in the way I have just spoken, is that a psychoanalytic theory would be able to disengage itself from the themes that have become so natural and intuitive to its practice. It would be wrong, however, to allow oneself to be intimidated by such a separation, as it can serve here as a guide since it bears witness to a moment of psychoanalytic history that has become incomprehensible to scholars and psychoanalysts alike. Far from being an obstacle, it is an indication of how one can learn from a psychoanalytic theory to read its history: a moment when the Lacanian introduction of topology, before being develloped in and of itself, was conceived as a certain ‘instrument’ that not only resolved certain problems of analytic theory, but created others. For instance, Lacan’s theory of the mirror stage, still embroiled in the mechanics of representation, was at the origin destined to rigorously elaborate what Freud had only discursively isolated as a problem of perversion 4. Inneutralizing the aporias that such a case presentation signals to the analyst, Lacan introduced the term perversion in structural terms: as a type of inversion associated to the relation of an object with its symmetric image 5. It was precisely at this moment that it became possible for Lacan to ask: at what point would have Freud needed his fable of primordial masochism and the infant, if he had had an adequate topology ? Inversely, it also becomes necessary to ask: at what point does such a structural account itself pose an obstacle to an understanding that had previously been natural to the practice of psychoanalytic theory (neurosis, the talking cure, woody allen, etc.) ? Or again, as if it were a question of recuperation: in what respect does such a structural account, in all its detours and generality, still permit the carrying out of the ‘initial’ practice of psychoanalysis ? Are the words ‘ego’, ‘super-ego’, ‘id’ etc. a literary way for old psychoanalysts to be able to continue to think their relation to a theory and practice of psychoanalysis ?
If today it has become common place to narrate the aporias of psychoanalysis – those historical moments that show themselves in the construction and deconstruction of its natural themes of expression – it is only in the isolation of a pure material of psychoanalysis – its knot or structure - that such aporias can be shown to not onlygeneralize its theory, but extend its field and practice.
Topological Presentation of the Incurable: The Pere-versely Oriented
It is not in the rupture of Symbolic, Imaginary, and the Real that defines
perversion, it is that they are already distinct and that it is necessary to
pose a fourth which is the Sinthome in occasion [...] that perversion is
nothing other than the version ver le pere, and that, in sum, the father
is a symptom or Sinthome, if you want.
J.Lacan, Le Sinthome, 1975-76
The remarks of our last section, as brief as they may have been, begin to justify the diagrams below as they present a psychoanalytic place not as a description of what occurs on the analytic couch, but as an inscription of what occurs in the structure of an analytic practice. In such a movement we are no longer asking the trivial topological question of whether an analysand should sit or lie down on a couch, or where the analyst should be in the session, rather we are interrogating the act and topos of psychoanalysis itself, without trivializing this encounter to the professional status of the doctor/patient relation. Here then, reformulating our questions at theplace of the Subject and the Other, we must ask: What is the interaction of two Borromeans: analyst and analysand, are they one or two ?
In figure 2 at the top of this article the structure of the subject is presented with two Borromeans (Bos) that have been brought together and embedded in the plane. The rings of the Bo are colored red, green, and blue with the lock on the left having a fourth black ring ∑ interwoven amoung the three. We have joined the Bos by putting them in correspondence by bands – double strands – joining rings of the same color. This putting into correspondence by bands assures that the rings remain closed curves after joining (a closed curve joined to a closed curve by a band remains a closed curve). Such a correspondence between two Bos was first proposed in the literature by Sourry and Lacan as a way to ‘un-do’ a lock, not by cutting, but by adding another perversely oriented lock. This act of dénouement being nothing other than, psychoanalytically speaking, what occurs in the interpretation of the Sinthome.
Dénouement of the Sinthome
Here, then, in fig.3 we have shown only the end result of the process of interpretation: a veritable dénouement where the various arcs of the rings of the Bos in the plane are deformed so that the Sinthome falls away (the dimensions of the problem permitting no cutting, re-drawing, tearing, etc. of the figure).
To conclude, I hope the reader will find the time and enjoyment to construct this clinical problem for themselves by actually transforming figure  into figure , thereby filling in the missing diagrams. For it is by addressing this separation from the Sinthome that contemporary Lacanian theory passes from a practice in intensionto a practice in extension:that is to say, operates an involution from the place of a psychoanalytic practice to a practice of a psychoanalytic place.
1/ Appearing initially in la Psychanalyse, then republished in his Ecrits, “Of A Preliminary Question…” p.531-583, Lacan only adds the topological presentation of the Mobius band and the corresponding footnote in 1966. This much said, the importance of “Of a Preliminary…” becomes apparent, not only in so far as this article is the only Ecrits (Writings) with a workable topological presentation, but when the article is juxtaposed with his earlier articles and still illustrative use of graphs in such articles as the Subversion of the Subject and the Dialectic of Desire, p.793-827 Ecrits, (1960).
2/ The over and under weaving is indicated in the following way: over = solid line, under = broken line.
3/ The Borromean is neither properly speaking a knot nor a chain, but a lock. Or at least, the necessity for this triadic classification of spatial connection was first put forward in P.G. Tait’s On Knots(1876) Later, with the work of Milnor - Link Groups (1954) - this triadic classification would trivialize into a binary relation between knots and homotopy chains.
4/ The psychoanalytic scholar will perhaps be perplexed over the use of the term ‘perversion’ in reference to Lacan’s preliminary article, “The Mirror Stage as the Formative Function of the I” (1949), which is purportedly about paranoia and the “paranoiac alientation that dates back to time at which the specular I turns into the social I“. Yet, later, by sharpening his concept of paranoia beyond the representational metaphors, Lacan would drop the illustrative use of mirrors to introduce a more precise topological definition of perversion.
5/ There are two delicate questions here that have been presented in the seminars and will be presented in future articles: firstly, the case where the image is an inversion of its object, must be distinguished from the case where it is a perversion. The use of this language goes back at least as far as Listing’s Vorstudien zur Topologie (1847) and is today standard in the traditional textbooks of optics. Secondly, the question of at what point the relation of the image to the object must be problematized as a certain nonrelation - as not a 1-1 correspondance – is most often brought out by distinguishing two different theories in which the figure is posed: the image as a mathematical entity and the image as a psychological entity (the figure ‘seen’). The former being a 1-1 correspondance, the latter not, thus, introducing considerations of what became known after Locke as ‘secondary qualities’: color, error, luminosity, orientation, etc. This difference goes back at least as far as Kepler’s distinction between ‘pictures’ and ‘images of things’; the former being what allows the formation of a science of optics, the latter being what would open up to, at first, a theory of psycho-physiological vision: from Bouguer’s Essai d’optique sur la graduation de la lumiere (1729) to Weber’s (1831) and Fechner’s study of differential perception and the formulation of a constant: the logarithm of the psycho-physical relation. If modern research has critiqued their results by showing that this constant is only the case in what concerns a statistical average of a zone of excitation conforming to the law of Gauss, there is still an open question on the structural problem of continuity presupposed, but not explicity brought out in the natural themes of psycho-physiology. See Poncaire’s topological critique of the Weber/Fechner conjectures.