the topos of unconsciousness

Since wednesday, as mentioned last time, the book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient” is available in the better bookshops.



There’s no need to introduce Alain Connes on this blog. Patrick Gauthier-Lafaye is a French psychiatrist and psycho-analyst, working in Strassbourg.

The book is a lengthy dialogue in which the authors try to find a use for topos theory in Jaques Lacan’s psycho-analytical view of the unconscious.

If you are a complete Lacanian virgin, it may be helpful to browse through “Lacan, a beginners guide” (by Lionel Bailly) first.



If this left you bewildered, for example by Lacan’s strange (ab)use of mathematics, rest assured, you’re not alone.

It is no coincidence that Lacan’s works are the first case-study in the book “Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science” by Alan Sokal (the one of the hoax) and Jean Bricmont. You can download the book from this link.



If now you feel that Sokal and Bricmont are way too harsh on Lacan, I urge you to have a go at the book “Writing the structures of the subject, Lacan and topology” by Will Greenshields.



If you don’t have the time or energy for this, let me give you one illustrative example: the topological explanation of Lacan’s formula of fantasy:

\[
\$~\diamond~a \]

Loosely speaking this formula says “the barred subject stands within a circular relationship to the objet petit a (the object of desire), one part of which is determined by alienation, the other by separation”.

Lacan was obsessed with the immersion of the projective plane $\mathbb{P}^2(\mathbb{R})$ into $\mathbb{R}^3$ as the cross-cap. Here’s an image of it from his 1966-67 seminar on ‘Logique du fantasme’ (213 pages).



This image includes the position of the objet petit $a$ as the end point of the self-intersection curve, which itself is referred to as the ‘castration’, or the ‘phallus’, or whatever.

Brace yourself for the ‘explanation’ of $\$~\diamond~a$: if you walk twice around $a$ this divides the cross-cap into a disk and a Mobius-strip!

The mathematics is correct but I fail to see how this helps the psycho-analyst in her therapy. But hey, everyone will tell you I have absolutely no therapeutic talent.

Let’s return to the brand new book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient”.



It was to be expected that they would defend Lacan’s exploitation of (surface) topology by saying that he was just unfortunate not to have the more general notion of toposes available, as well as their much subtler logic. Perhaps someone should write a fictional parody on Greenshields book: “Lacan and the topos”…

Connes’ first attempt to construct the topos of unconsciousness was also not much of a surprise. According to Lacan the unconscious is ‘structured like a language’.

So, a natural approach might be to start with a ‘dictionary’-category (words and relations between them) or any other known use of a category in linguistics. A good starting point to read up on this is the blog post A new application of category theory in linguistics.

Eventually they settled for a much more ambitious project. To Connes and Gauthier-Lafaye every individual has her own topos and corresponding logic.

They don’t specify how to construct these individual toposes, but postulate that they are all connected to a classifying topos, which is their incarnation of the world of ‘myths’ and ‘fantasies’.

Surely an idea Lacan would have liked. Underlying the unconscious must be, according to Connes and Gauthier-Lafaye, a geometric theory! That is, it can be fully described by first order sentences.

Lacan himself used already some first order sequences in his teachings, such as in his logic of sexuation:

\[
\forall x~(\Phi~x)~\quad \text{but also} \quad \exists x~\neg~(\Phi~x) \]

where $\Phi~x$ is the phallic function. Quoting from Greenshield’s book:

“While all (the sons) are subject to ($\forall x$) the law of castration ($\Phi~x$), we also learn that this law nevertheless resides upon an exception: there exists a subject ($\exists x$) that is not subject to this law ($\neg \Phi~x$). This exception is embodied by the despotic father who, not being subject to the phallic function, experiences an impossible mode of totalised jouissance (he enjoys all the women). He is, quite simply, the exception that proves the law a necessary beyond that enables the law’s geometric bounds to be defined.”

It will be quite hard (but probably great fun for psycho-analysts) to turn the whole of Lacanian theory on the unconscious into a coherent geometric theory, construct its classifying topos, and apply the Joyal-Reyes theorem to get at the individual cases/toposes.

I’m sure there are much deeper insights to be gained from Connes’ and Gauthier-Lafaye’s book, but this is what i got from a first, fast, cursory reading of it.

Grothendieck meets Lacan

Next month, a weekend-meeting is organised in Paris on Lacan et Grothendieck, l’impossible rencontre?.



Photo from Remembering my father, Jacques Lacan

Jacques Lacan was a French psychoanalyst and psychiatrist who has been called “the most controversial psycho-analyst since Freud”.

What’s the connection between Lacan and Grothendieck? Here’s Stephane Dugowson‘s take (G-translated):

“As we know, Lacan was passionate about certain mathematics, notably temporal logic and the theory of knots, where he thought he found material for advancing the theory of psychoanalysis. For his part, Grothendieck testifies in his non-strictly mathematical writings to his passion for the psyche, as shown by many pages of his Récoltes et Semailles just published by Gallimard (in January 2022), or even, among the tens of thousands of pages discovered at his death and of which we know almost nothing, the 3700 pages of mathematics grouped under the title ‘Structure of the Psyche’.

One might therefore be surprised that the two geniuses never met. In fact, a lunch did take place in the early 1970s organized by the mathematician and psychoanalyst Daniel Sibony. But a lunch does not necessarily make a meeting, and it seems that this one unfortunately did not happen.”

As it is ‘bon ton’ these days in Parisian circles to utter the word ‘topos’, several titles of the talks given at the meeting contain that word.

There’s Stephane Dugowson‘s talk on “Logique du topos borroméen et autres logiques à trois points”.

Lacan used the Borromean link to illustrate his concepts of the Real, Symbolic, and Imaginary (RSI). For more on this, please read chapter 6 of Lionel Baily’s excellent introduction to Lacan’s work Lacan, A Beginner’s Guide.

The Borromean topos is an example of Dugowson’s toposes associated to his ‘connectivity spaces’. From his paper Définition du topos d’un espace connectif I gather that the objects in the Borromean topos consist of a triple of set-maps from a set $A$ (the global sections) to sets $A_x,A_y$ and $A_z$ (the restrictions to three disconnected ‘opens’).

\[
\xymatrix{& A \ar[rd] \ar[d] \ar[ld] & \\ A_x & A_y & A_z} \]

This seems to be a topos with a Boolean logic, but perhaps there are other 3-point connectivity spaces with a non-Boolean Heyting subobject classifier.

There’s Daniel Sibony‘s talk on “Mathématiques et inconscient”. Sibony is a French mathematician, turned philosopher and psychoanalyst, l’inconscient is an important concept in Lacan’s work.

Here’s a nice conversation between Daniel Sibony and Alain Connes on the notions of ‘time’ and ‘truth’.

In the second part (starting around 57.30) Connes brings up toposes whose underlying logic is much subtler than brute ‘true’ or ‘false’ statements. He discusses the presheaf topos on the additive monoid $\mathbb{N}_+$ which leads to statements which are ‘one step from the truth’, ‘two steps from the truth’ and so on. It is also the example Connes used in his talk Un topo sur les topos.

Alain Connes himself will also give a talk at the meeting, together with Patrick Gauthier-Lafaye, on “Un topos sur l’inconscient”.

It appears that Connes and Gauthier-Lafaye have written a book on the subject, A l’ombre de Grothendieck et de Lacan : un topos sur l’inconscient. Here’s the summary (G-translated):

“The authors present the relevance of the mathematical concept of topos, introduced by A. Grothendieck at the end of the 1950s, in the exploration of the structure of the unconscious.”

The book will be released on May 11th.

Chevalley’s circle of friends

Last week, Danielle Couty ArXiVed her paper Friendly views on Claude Chevalley (in French).

From the abstract: “We propose to follow the itinerary of Claude Chevalley during the last twenty years of his life, through the words of Jacques Roubaud, Denis Guedj and Alexander Grothendieck. Our perspective is that of their testimonies filled with friendship.”

Claude Chevalley was one of the founding fathers of Bourbaki. Two of the four pre-WW2 Bourbaki-congresses were held in “La Massoterie”, the Chevalley family domain in Chancay (see this post, update: later I learned from Liliane Beaulieu that the original house was destroyed by fire).

In 1938 he left for Princeton and stayed there during the war, making it impossible to return to a position in France for a very long time. Only in 1957 he could return to Paris where he led a seminar which proved to be essential for the development of algebraic groups and algebraic geometry.



Picture from N. Bourbaki, an interview with C. Chevalley

The Couty paper focusses on the post-1968 period in which Chevalley distanced himself from Bourbaki (some of its members, he thought, had become ‘mandarins’ and ‘reactionaires’), became involved with the ecological movement ‘Survivre et vivre’ and started up the maths department of a new university at Vincennes.

The paper is based on the recollections of three of his friends.

1. Jacques Roubaud is a French poet, writer and mathematician.

On this blog you may have run into Roubaud as the inventor of Bourbaki’s death announcement, and the writer of the book with title $\in$.

He’s also a member of Oulipo, a loose gathering of (mainly) French-speaking writers and mathematicians. Famous writers such as Georges Perec and Italo Calvino were also Oulipo-members (see also Ouilpo’s use of the Tohoku paper).

Chevalley introduced Roubaud (and others) to the game of Go. From Couty’s paper this quote from Roubaud (G-translated):

“. . . it turns out that he had learned to play go in Japan and then, in Paris, he could not find a player […] I played go with him […] and then at a certain moment , we thought, Pierre Lusson and myself, it would still be good to create circumstances such that Chevalley could have players. And so, we had a lot of ambition, we said to ourselves: “We’re going to write a treatise on go, and then lots of people will start playing go”. »

The resulting Go-book is A short treatise inviting the reader to discover the subtle art of Go. Here’s Georges Perec (left) and Jacques Roubaud playing a game.



Picture from Petit traite invitant a la decouverte de l’art subtil du Go

2. Denis Guedj was a French novelist, mathematician and historian of science professor, perhaps best known for his book The Parrot’s Theorem.

In May 1968, Guedj was a PhD-student of Jean-Paul Benzecri (the one defining God as the Alexandroff compactification of the univers), working in the building where ‘Le Comité de Grève’ installed itself. Here he met Chevalley. A Guedj-quote from Couty’s paper (G-translated):

“Claude Chevalley was one of the three professors of the Faculty of Science to commit himself totally to the adventure until the end, occupying the premises with the students on the Quai Saint-Bernard […] and sleeping there frequently . That’s where I met him.

We had taken possession of this universe which until then had only been a place of study and knowledge, and which, in the mildness of this month of May, had become a place of life, of a life wonderfully exhilarating. The college was ours. At night we walked down the aisles yet? lined with tall trees, entered the empty lecture halls, slept under the stars. Needless to say that at the beginning of the school year, in the fall of 1968, it was impossible for us to find our place in these undressed spaces from which the magic had withdrawn. »



Picture from Décès de l’écrivain et universitaire Denis Guedj

In June 2008, Guedj was one of the guests at the special edition of France Culture on the occasion of Grothendieck’s 80th birthday, Autour d’Alexandre Grothendieck.

3. Alexander Grothendieck, mathematician and misogynist, deified by some of today’s ‘mandarins’.

The paper by Danielle Couty may shed additional light on Grothendieck’s withdrawal from Bourbaki and mathematics as a whole. A G-translated Grothendieck quote from the paper:

“It was Chevalley who was one of the first, with Denis Guedj whom I also met through Survivre, to draw my attention to this ideology (they called it “meritocracy” or a name like that), and what there was in her of violence, of contempt. It was because of that, Chevalley told me […] that he could no longer bear the atmosphere in Bourbaki and had stopped setting foot there. »

Claude Chevalley stayed at Vincennes until his retirement in 1978, he died on June 28th 1984.

Grothendieck stuff

January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books.



Here’s a G-translation of Gallimard’s blurb:

“Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to his own words. The mythical typescript, which opens with a sharp criticism of the ethics of mathematicians, will take the reader into the intimate territories of a spiritual experience after having initiated him into radical ecology.

In this literary braid, several stories intertwine, “a journey to discover a past; a meditation on existence; a picture of the mores of a milieu and an era (or the picture of the insidious and implacable shift from one era to another…); an investigation (almost police at times, and at others bordering on the swashbuckling novel in the depths of the mathematical megapolis…); a vast mathematical digression (which will sow more than one…); […] a diary ; a psychology of discovery and creation; an indictment (ruthless, as it should be…), even a settling of accounts in “the beautiful mathematical world” (and without giving gifts…)”.”

All literary events, great or small, are cause for the French to fill a radio show.

January 21st, ‘Le grand entretien’ on France Inter invited Cedric Villani and Jean-Pierre Bourguignon to talk about Grothendieck’s influence on mathematics (h/t Isar Stubbe).

The embedded YouTube above starts at 12:06, when Bourguignon describes Grothendieck’s main achievements.

Clearly, he starts off with the notion of schemes which, he says, proved to be decisive in the further development of algebraic geometry. Five years ago, I guess he would have continued mentioning FLT and other striking results, impossible to prove without scheme theory.

Now, he goes on saying that Grothendieck laid the basis of topos theory (“to define it, I would need not one minute and a half but a year and a half”), which is only now showing its first applications.

Grothendieck, Bourguignon goes on, was the first to envision the true potential of this theory, which we should take very seriously according to people like Lafforgue and Connes, and which will have applications in fields far from algebraic geometry.

Topos20 is spreading rapidly among French mathematicians. We’ll have to await further results before Topos20 will become a pandemic.

Another interesting fragment starts at 16:19 and concerns Grothendieck’s gribouillis, the 50.000 pages of scribblings found in Lasserre after his death.

Bourguignon had the opportunity to see them some time ago, and when asked to describe them he tells they are in ‘caisses’ stacked in a ‘libraire’.

Here’s a picture of these crates taken by Leila Schneps in Lasserre around the time of Grothendieck’s funeral.



If you want to know what’s in these notes, and how they ended up at that place in Paris, you might want to read this and that post.

If Bourguignon had to consult these notes at the Librairie Alain Brieux, it seems that there is no progress in the negotiations with Grothendieck’s children to make them public, or at least accessible.

Huawei and topos theory

Apart from the initiatives I mentioned last time, Huawei set up a long term collaboration with the IHES, the Huawei Young Talents Program.

“Every year, the Huawei Young Talents Program will fund on average 7 postdoctoral fellowships that will be awarded by the Institute’s Scientific Council, only on the basis of scientific excellence. The fellows will collaborate with the Institute’s permanent professors and work on topics of their interest.”

Over the next ten years, Huawei will invest 5 million euros in this program, and an additional 1 million euros goes into the creation of the ‘Huawei Chair in Algebraic Geometry’. It comes as no particular surprise that the first chairholder is Laurent Lafforgue.

At the launch of this Young Talents Program in November 2020, Lafforgue gave a talk on The creative power of categories: History and some new perspectives.

The latter part of the talk (starting at 47:50) clarifies somewhat Huawei’s interest in topos theory, and what Lafforgue (and others) hope to get out of their collaboration with the telecom company.

Clearly, Huawei is interested in deep neural networks, and if you can convince them your expertise is useful in that area, perhaps they’ll trow some money at you.

Jean-Claude Belfiore, another mathematician turned Huaweian, is convinced topos theory is the correct tool to study DNNs. Here’s his Huawei-clip from which it is clear he was originally hired to improve Huawei’s polar code.

At the 2018 IHES-Topos conference he gave the talk Toposes for Wireless Networks: An idea whose time has come, and recently he arXived the paper Topos and Stacks of Deep Neural Networks, written jointly with Daniel Bennequin. Probably, I’ll come back to this paper another time, for now, the nForum has this page on it.

Towards the end of his talk, Lafforgue suggests the idea of creating an institute devoted to toposes and their applications, endorsed by IHES and supported by Huawei. Surely he knows that the Topos Institute already exists.

And, if you wonder why Huawei trows money at IHES rather than your university, I leave you with Lafforgue’s parting words:

“IHES professors are able to think and evaluate for themselves, whereas most mathematicians just follow ‘group thinking'”

Ouch!