The text below is a constantly evolving summary bringing together psychoanalysis and topology.

Why Topology and Psychoanalysis?

 “Topology is the discipline that comes to bear on the traits at the origins of civilization
                                                                                                                                                                              Maurice Fréchet

The difficulty that post-Freudian psychoanalysis faces today is that it asks important questions without having an effective means to respond. By effective we mean the means by which to account for the necessity of a psychoanalytic discourse beyond the appeal to speech and the inertia of descriptive commentary.  Why is this effectivity important?  Clinically, anyone who addresses a symptom poses it. Ethically, there is no real cause to work in analysis if it can not be assumed.  Of course, it is always possible to begin an analysis just because someone mentions a theory and practice or a cure.  But this ‘possible’ analysis will always remain at the level of a rumour or transfer, if it does not engage the problem of its use or necessity.

Though not alone, Lacan was the first to render account of an effective analysis by extending it to the research of linguistics, logic, and mathematics.  What these domains of rationality have in common is a focalization on the material and formal procedures of inscription beyond the contents of description. In avoiding such domains, psychoanalysis has historically been assimilated to an instruction in the humanities; or worse yet, sought to prop itself up on the contents of scientific disciplines – neuroscience, biology, genetics, etc. – having little or no correlation to an intrinsic necessity. In spite of the extrinsic of the humanities and science,  it is important to show how such an intrinsic necessity may be achieved in the construction of the structure of a theory- practice.  This structural achievement was first brought out by Lacan as a problem of topology. Though we provide a descriptive overview below, the reader is invited to check the references on this website for a more extensive presentation in mathematics and logic.

Why Study Topology?

To date the results in psychoanalytic theory and practice have been underwhelming.  Despite the claims of progress, contemporary psychoanalysis, be it post-freudian or post-lacanian, seems to be no better or worse than what came before. When it has become possible to say and publish anything in the culturally accepted jargon of analysis, one must ask at what point future analysts will look back upon the current scene as nothing more than literary fantasy and ‘folk psychology’. Without denying the folklore, it would be a progress to introduce a purely analytic clinic that proceeds on the basis of its proper discourse without being appropriated by the missions of, on one hand, psychology, psychotherapy, and psychiatry, on the other, philo-lit-crit theory. The interest of our topological work is to make room for this more precise development.

To get a handle on what is at stake, look at a similar situation: ask an architect what his or her profession would be like if you took away the geometry and logic of a construction. Within no time at all such an occupation would be reduced to the skill set of a building code inspector.  Though geometry may appear without a direct utility or appear as a mere conceptual basis of the theory, with a second look it is also what allows the architect to disengage a practice from the power relations of the building code inspector. Without this basis, the architecture of modern psychoanalysis consistently falls back into the mission of a psycho-inspector: someone – a nurse, social worker, psychologist, literary professor, psychiatrist, life coach, spiritual guide, etc. – who may arrive to monitor and describe someone in  a coded language of psychoanalysis without ever being able to assume the consequences of the analytic act.  Call this all too normal state of affairs with Lacan the ideology of analysis:

What the ideology of contemporary psychoanalysis suffers from today is the lack of an adequate topology

J. Lacan, Seminar XVI, D’un autre à l’Autre

This is not to suggest that psychoanalysis is based on topology in the same way that architecture is based on geometry, but that any theory that does not effectively construct the site and logic of its practice is a participation in ignorance. In short, it is the propagation of a technical ‘know-how’ having little to do with the place of the analyst. Today, much of the stagnation of psychoanalysis occurs precisely at this place of ignorance: the well-intentioned asking important questions that, in the lack of an effective and constructive response, are co-opted into the ideological mission of a psycho-inspector qua technician. Of course, it is always possible to deny the place of analysis itself and take up a philosophical position with regard to its theory-practice. But we find it not only ineffective but worrisome that psychoanalysis not only in the U.S. but elsewhere – Europe and South America – often finds itself split into this divide. As a counter-measure, what is wanted is a practice of a theory that is open to anyone, yet does not fall back into the false innocence of the technician or the anarchy of the philosopher; what is required is a place to determine the ethics and effectivity of a theory and practice in a way that does not devolve into a professional institute or a utopia.


Twenty-five years of intensive work have had as a consequence of assigning to psychoanalytic technique goals immediately different from those of the beginning. At the beginning, all the ambition of the medical analyst was to conjecture what was hiding in the unconscious of the sick person, and to reunite these elements in a whole and communicate them when it was proper. Psychoanalysis was above all an art of interpretation. But the psychotherapeutic task was not however resolved by this.  A new approach has come to light that consists in obtaining from the ill person a confirmation of a construction […]”

(S. Freud, Beyond the Pleasure Principle, 1920)

“Psychoanalysis is not an art of interpretation, it is a construction.  Interpretation comes to bear on a material element ( missed acts, lapsus, etc.).  Construction, on the contrary, comes to bear on the entire course of an existence, most notably on the initial and determining phases.”

(S. Freud, Constructions in Analysis, 1932

Concrete Topology

Despite the recognition of the debt Lacanian analysis owes to topology, many today have not arrived to incorporate topology into their practice beyond a mere ornamental use.  Indeed, in combining topology with psychoanalysis one encounters two theses with regard to the formation of its place.

A) The abstract position that views topology as a formal discipline that is only concerned with conceptualization and concerns of theory, while the clinic is merely a place where the hypotheses of the theory are to be verified or not.

B) The concrete position that views topology as a doctrine of structure concerned not simply with theory, but with the foundation of the clinic.

The former, (A), views topology as a complement to psychoanalysis that is inessential for its presentation to the public or the clinic.  Here, the clinic is only approximative and an inertia of speculative ideas about patients and clinical cases, while the theory, more precisely ‘philosophy’, works absolutely or it does not work at all.

The latter, (B), takes topology as a supplement to psychoanalysis that is, however, necessary both to its presentation and its clinic.  In this respect, the clinic is not approximative; nor does it have anything to do with verifying theoretical hypotheses on the backs of patients in order to construct a ‘clinical case’. Here, the theory proceeds not as the whole truth or as a philosophical system, but with regard to an experimental framework.

The former, (A), as may be expected does little with the topology except through metaphor and historical commentary, i.e., the ‘late’ Lacan, etc.  Remaining at this level, the results have been largely disappointing as ‘Lacanian topology‘ digresses into an academic and esoteric reference to Borromean Rings, Sinthomes, and Mobius bands that function as icons. At best, such topological icons function as mere conceptual landmarks of an argument that needs to be developed (but usually never is); at worst, they become ways of imagining a clinic as a way to brand people with a psychoanalytic terminology of barred subjects, psychosis neurosis, and perversion. In either case, it is not a question of denying anyone a delirious use of topology or psychoanalysis, but simply a question of how not to remain there.

Though it is less well-known and its participants fewer in number, (B) founds a purely analytic clinic through an experimental framework and topological presentation whose diagrams are not only functional, but calculable according to a matheme and deducible according to a psychoanalytic logic.  Such an approach opens up the clinic in a new and more serious way, while cutting across disciplinary boundaries. Yet, there is no need to call such a structural approach ‘interdisciplinary’ since it is not a question of transferring contents from one discipline to another, but of determining the formal and material means of a discourse. There is no need to fetishize a knot or mobius band since what is at stake is the logic and mathematics that such intuitive constructions provide for the analytic act.

Since there is a lot of work in the field to come*, our first aim is to furnish the necessary references to those seeking to develop a more precise practice of this topology.  These references do not simply consist in following the indications left by Lacan, but in constructing the object that he discovered – the little object a – beyond a literary, philosophical,  political, or coded reference.

* Because of the relative newness of the field, there are currently no books in the English language that serve as a rigorous introduction to Topology and Psychoanalysis. At this time PLACE is, however, producing a manual that will be made available by the fall semester of 2017.

The Place of the Subject and the Object 

Freud and his first generation inheritors focalized on the voice as speech, puns, metaphor, and commentary, while the gaze was only considered passively and indirectly – as the image of a dream or a scene of fantasy.  In each case, the gaze was never directly constructed in practice, but left as a problem of imagination, then constructed in theoretical models of the neuron and optics.

By the time of Lacan the gaze is constructed directly in practice and theory with a special concern for its interpretation in a topology both in the mathematical sense of the term and in the more ancient sense of a topics or place of discourse. A practice of topology in this respect consists in separating both the voice from mere speech and the gaze from imagination while drawing the implications this has for a reading and writing of the subject. One way to get a hold on what is involved in this correlation of voice and gaze is to notice in the history of geometry and topology there have been many blind researchers, from Dr. Moyneaux to B.Morin , but there have never been any without a voice.  Far from being a mere instrument of speech, the voice can be shown to have an effective relation to the gaze and constructions in space, if the conditions are established for this to be brought out. The term ‘subject’ has been introduced in Lacanian psychoanalysis to account both for the birth of modern science and what Freud first postulated as the id: an agency of a drive incongruous to the ego and its self-representations. If psychoanalysis has arrived to disengage such a subject from the ego through its effects of impediment, break, and slips of speech, it often does so without ever being able to give the subject a place beyond a field of passion and human error. Left at this theatrical level, the subject only finds a place in a passionate development; that is to say,  in the masquerade of psychotherapy or in reference to the arts – film, literature,etc. – and social critique.  It is not surprising, therefore, that the most current and readable applications of psychoanalysis to the social scene – film, literature, politics, etc. – also produce some of the most retrograde and stereotypical conceptions of actual psychoanalytic practice.

The source of this avoidance consists in confusing desire with passion which in turn revolves around a reduction of the voice to an instrument of speech and the gaze to vision. Since Lacan the practice of psychoanalysis consists in making room for a place of the subject whose reading and writing would not be systematically effaced with a reference to the place of passions – whether it be reference to the dramas of the therapeutic ‘talking cure’ or applied psychoanalysis. The familiar Woody Allen version of psychoanalytic therapy is pre-Lacanian to the degree that the voice is reduced a place for passionate speech (commenting on one’s dreams, life, experience, etc.) and its relation to the gaze remains un-constructed, i.e., reduced to imagination or a psychological conception of visual experience. What is wanted is a place for a psychoanalytic discourse: a topos that would not reduce the singularity of the relation of the voice to the gaze to a psychology of the faculties or an aesthetics. Without this topology of the subject the object of desire is avoided and assimilated to a passion through a set of historical fossils –psychiatry, psychology, and psychotherapy – and ritual. Both the experimental and clinical framework of Lacanian analysis founds its basis on this seperation of the voice and the gaze in the construction of a topology.  In a more naive and colorful way, such a topological construction is nothing other than a practice of narcissism.


Example of a past 2014 Summer Immersion

Summer Immersion 2014:  Algebra, Logic, & Topology

Tuition: $300.

First Meeting: 11:00-1:30 Saturday July 12, 2014; then three more Saturdays till Aug. 2.

Requirements: Open to the Public. The immersion is for all levels and requires no previous background in logic or mathematics.  Class limited to first six applications (reception of tuition required to confirm registration).

Descriptive: The immersion will concentrate on developing a translation between Algebra, Logic, and Topology. It proceeds in the manner of an atelier in the sense that the participant will need paper, color pencils-pen, scissors to construct topological objects and images that conform to the writing of Logic and Algebra.  The immersion is not a lecture, but is a collaborative effort in presenting constructions that responds to a problem posed during the course.  The immersion is developed in the context of the recent ‘Constructing Oedipus’ course, especially the problem of symmetries and dualities in space. This much said, the immersion will not develop the translation into the psychoanalytic vocabulary, but will concentrate solely on the Algebra, Logic, and Topology.

Text: The immersion will use as its reference Groups and Their Graphs by Israel Grossman and WIlhem Magnus; American Mathematical Association, 1964.

Pre-immersion assignment: it is advised to read Groups and Their Graphs chpts. 1–4 with a focus on writing a presentation of a triangle in space (2 and 3 dimensions).

Plan of Immersion

I – July 12: Call for presentations and constructions of the Group of a triangle in space. Call for the presentation of the multiplication table of the Group for the triangle. Call for the presentation of the axioms of a Group.  Discussion of problems/results: symmetries, rotations, identities, etc.

Assignment for July 19: Read Chpts. 5, 7,10. Focus: writing the presentation of a tetrahedron in space

II – July 19: Call for presentations of the graph of the group of a triangle. Call for presentations and constructions of the Group of a tetrahedron in space (3 dimensions and 4). Call for the graph of the group of a tetrahedron.

Assignment for July 26: Read 11,12, 13.

III – July 26: Call for presentations and constructions of the Normal Sub-group, Galois Groups, and Quarternions.

Assignment for August 2: Read Chpt. 14 on the Group of Links and handout of immersion on Groups of Knots and Logic. Focus on the Group of the Knot, Link, and Lock – with the translation in to Logic.

IV – Aug. 2: Call for constructions and/or presentations of the Group of the Knot, Link, or Lock.  Call for a translation of the Algebraic (Group) notation into a Logic of the Knot.  Concluding summaries from the participants (if they have one) and proposals for future work.



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