As a part of the project to collect essays representing the contributions of Olga Cox Cameron, Dan Collins, and other participants in the Dublin Colloquium on Seminar IX: Identification, these essays, edited by Carol Owens and Sarah Meehan O’Callaghan began as a summary expansion of “the graphics gap” — the considerable difference between the French texts available through Staferla and the translations of Cormac Gallagher. In contrast to the professionally drawn diagrams (often in color) and high-definition reproductions of works of art, the texts available at LacanInIreland present hand-drawn and sometimes inaccurate reproductions, possibly because the manuscripts available to Gallagher were difficult to decipher.
Collins and Cox Cameron presented written essays, from which their chapters will be developed. My presentations were primarily commentaries on the possible uses and meanings of restored graphics; there was no development of themes or critical analyses. Each of the five essays proposed for the collection was begun afresh but based on themes assigned by the editors, drawn from presentations.
This page archives draft editions of the essays, with introductions outlining the claims and findings. Generally, the revised studies quickly moved from a project of restoration to one of re-interpretation. Lacan’s Seminar IX is widely regarded as marking the inception of his interest in topology. Lacan himself cites IX as the beginning of his ‘serious interest in topology’ (Rome Discourse, p. 85*). This claim depends on a narrow view of what constitutes topology. If limited to specific references to form of projective geometry (the torus, Möbius band, cross-cap, etc.), these are not mentioned before Seminar IX (1961). If, however, the operations, elements, and relations key to topology are considered (cuts, holes, doubling, Euler circles, lacing, etc.) Lacan’s interest in ‘spatial’ operations begins with his first seminar (1953) or possibly before, with his interest in the Three Prisoners’ Dilemma (1937).
This understanding of Lacan’s investment in topology requires a different theory of reading. For this I drew on Freud’s recommendations to the analyst, that to hear the free-association of the analysand properly, it was necessary that the ear of the analyst ‘evenly distribute’ its attention (Gleichswebende Aufmerksamkeit). This soft-focus approach strives to equalize the appearance of terms referring to the spatiality of the subject, signifier, and unconscious, so that any incident has a ‘fractal’ value within the whole of Lacan’s development.
Evenly distributed attention attunes itself to a completely ignored but extremely important fact. Lacan’s topology is (as, I argue, all projective topology) based on inversion. There are over 276 references to some kind of inversion in Lacan’s writings before 1961. The comparatively fewer number after that date (60 mentions in the principal works) suggests that, as Lacan realized how projective topology figures took up the cause of inversion, he was less obliged to mention inversion specifically.
What is the significance of this? Apart from offering proof that Lacan’s ‘spatiality’ began very early, inversive geometry is a formal field of mathematics centered on operations governing the equalization of forms inside the ‘inversion circle’ as they are inverted to the outside of the circle. This is identical to the way Lacan introduces and develops his ‘original’ idea of extimité! Although Lacan uses this word only twice in Seminar VII, The Ethics of Psychoanalysis, 1959–1960, Jacques-Alain Miller argues that the idea of extimity is everywhere in Lacan’s thinking. Unfortunately, Miller develops this thesis without any apparent knowledge of inversion geometry, the obvious and clear mathematical link to Lacan’s idea of extimité.
Inversiveness explains the distribution of topology-like terms throughout Lacan’s career. It explains how his early interest in topological forms connects to his later interest in knots and braids. It grounds the logic connecting operations on the torus and cross-cap to conditions of symmetrical difference, the ‘union without intersection’ of two overlapping Euler circles, Lacan’s idea of the unary trait as iterative and self-intersection, and Lacan’s demonstration of the relation of the unary trait to the objet a in his “slide-rule” analogy in Seminar XIV, The Logic of Phantasy, about which virtually nothing has been written.
In fact, Lacans students and followers are almost totally silent on the questions of inversive geometry, extimity, and the iterative structure of the unary trait that connects it to the Golden Ratio and, thence, number theory. The fact that I am not a mathematician would suggest that I am not qualified to go beyond the claim that there are large zones in Lacanian theory that have yet to be theorized. Therefore, I do not go in this direction. In fact, I recommend that Lacanians do not go further in mathematics than Lacan himself was able to go. This would venture into territory that lacks any means of verification or corroboration. Speculation, so necessary for any theorizing, would be untethered, un-relatable. It would be Lacanian psychoanalysis without Lacan, an entirely unadvisable project in my view.
Instead, I would recommend that theory go where inversion itself goes — namely to the ways in which cultures of all periods and places have grounded their own ‘operations’ on inversion. Much of this has been explored by archaeologists, anthropologists, and critical theorist who have found, in the idea of liminal space and the operations of ‘rites of passage’, inversion in the form of rituals, folktales, architecture, art production, cuisine, performances, myths … in other words, all cultural operations emphasizing transformation and non-orientation: visits to Hades (katabasis), relations and motifs of life and death (isomerics), marriage customs, eschatology, and foundation rites — in other words, actions involving inversive uses of the spaces of perception (parallax) and building.
This “ethno-topological” approach is in fact what Lacan began with his visits to the Musée des Antiquités Nationales de Saint-Germain-en-Laye, where he found marks on deer bones that gave him the idea of the unary trait as a “one of 1.” Lacan extended his ethnological interests to, as I myself recommend, the works of art of all ages, kinds, and degrees of sophistication. Where only a few Lacanians can claim the mathematical expertise to pursue the option of extending topology beyond Lacan’s own expertise (which I do not recommend!), there are no Lacanians unable to take a more detailed interest in the cultures they inhabit, or to develop the ability to look at other cultures with a more sophisticated theoretical eye.
In my view, there are only a few ethnological guides who seem pre-disposed to Lacanian studies. Certainly, the works of Arnold van Gennep and Victor Turner would be fruitful starting places. But, nothing compares to the comprehensive although difficult work of Giambattista Vico. Vico seems to have read Lacan before starting work on his New Science of 1725/1744. With a similar focus on the subject’s relation to the signifier, Vico used inversion geometry at all levels to explain how cultures in different places, evolving at different times, would follow similar developmental patterns.
The argument for a Vico-Lacan connection cannot be made here, and anyway it is not essential to the study of Seminar IX that I am presenting in these five essays. Besides, the case has already been made more eloquently and forcefully than I will ever be able to make, by the French scholar Baldine Saint Girons. Until I am able to continue this thesis, I advise the reader to engage with her work on the connection of Lacan to this 18c. Neapolitan philosopher of culture.
In the meantime, I invite colleagues to review the essays for Owens and Meehan’s collection here, as they can be made available. A title and brief description will suffice to present what I hope will be a case for opening a new field of study with the present domain of Lacanian scholarship, fusing an ‘ethnotopology’ with inversive geometry.
ESSAY 1 / Why Lacan Should Be Read with Inattention (Gleichswebende Aufmerksamkeit)
Here I argue that Lacan’s early interest in the spatiality of psychoanalysis is founded on inversiveness, a precursor of his formal interest in projective geometry. By restoring the graphics of the seminars for English readers, it is possible to see broad spatial themes and procedures. PDF. MS-Word.
Essay 2 / The Interior-8 as a Model of Lacan’s Ethno-Topology
The simple interior-8 is not simplistic. It is the geometro-logical basis of the torus and, thence, the Klein bottle, cross-cap, and other figures involving the (katagraphic) double-edged Möbius cut. Restoring the Staferla graphics to the English translations is critical to restoring Lacan’s extimité to its central position, as inversion geometry, to Lacan’s ‘evenly-distributed topology’ (Freud’s Gleichswebende Aufmerksamkeit; lacan’s mi dire), key to a revised reading of his œuvre with an inattention geared to its fractalized ideas. PDF. MS-Word.
Essay 3 / Lacan’s and Inversive Geometry
Lacan’s mantra supporting topology should be extended. To the sequence Real > Structure > Topology, we should add > Inversion. This is critical for the alternative presented in Seminar IX, namely to connect psychoanalysis to the logic and works of culture. The example of Chaplin’s City Lights shows that it is easier to see the schema of the double torus in terms of the après coup of demonstration when its alternative, presentation, is literally canceled by the role played by blindness. PDF. MS-Word.
Essay 4 / Lacan, Pataphysics, and the Polish Signifier
In his early days, Lacan hobnobbed with famous Surrealists who propelled the arts and culture of Europe. Aragon, Breton, Jarry, and Dalí were personal friends. Picasso was one of his analysands. So, when Lacan quotes a joke from Alfred Jarry’s play, Ubu Roi, we must take seriously the connection he makes with the highly incompetent King of Poland with his evolving idea of the Other and master signifier. From surrealism’s “pataphysics” to inversion geometry, this chapter aims to look beneath the hood of Seminar IX’s topology. At the rock bottom lies “induction puzzle,” a strategy of rejecting simplistic evidence of appearance in favor of a structural solution. This opens the door to ethnology even further. PDF. MS-Word.
Essay / Every Frame is a Double Frame
The idea of the double frame is critical for the transfer of ideas, from the clinic of psychoanalysis to the culture of the everyday as well as the arts, religion, folklore, popular culture, etc. This is not the ad hoc spoof that Žižek claims to have made to astonish naïve academics at a boring conference but, rather, a consequence of parallax experience that can be modeled by projective topology, especially as the role of inversion is revealed. Lacan’s foundation story of the sardine can that refused to look at him (Seminar XI) has a deeper meaning if we consider the double frame in relation to a spherical model of visibility. The inversive, inductive double frame is nothing less than the unconscious, which can be modeled through the analogy to the electrical circuit function of capacitance. Insulation and the fourth wall idea are key to the clinic-to-culture transfer, which can be corroborated using the idea of induction and inversion as precursors to projective topology). PDF. MS-Word.
*The English-language version of this is Anthony Wilden [Translator], The Language of the Self: The Function of Language in Psychoanalysis, by Jacques Lacan (Baltimore: The Johns Hopkins Press, 1968).