Lately, I’ve been reading up a bit on psycho-analysis, tried to get through Grothendieck’s La clef des songes (the key to dreams) and I’m in the process of writing a series of blogposts on how to construct a topos of the unconscious.
Stella Maris is set in 1972, when the math-prodigy Alicia Western, suffering from hallucinations, admits herself to a psychiatric hospital, carrying a plastic bag containing forty thousand dollars. The book consists entirely of dialogues, the transcripts of seven sessions with her psychiatrist Dr. Cohen (nomen est omen).
Alicia is a doctoral candidate at the University Of Chicago who got a scholarship to visit the IHES to work with Grothendieck on toposes.
During the psychiatric sessions, they talk on a wide variety of topics, including the nature of mathematics, quantum mechanics, music theory, dreams, and the unconscious (and its role in doing mathematics).
The core question is not how you do math but how does the unconscious do it. How it is that it’s demonstrably better at it than you are? You work on a problem and then you put it away for a while. But it doesnt go away. It reappears at lunch. Or while you’re taking a shower. It says: Take a look at this. What do you think? Then you wonder why the shower is cold. Or the soup. Is this doing math? I’m afraid it is. How is it doing it? We dont know. How does the unconscious do math? (page 99)
Before going to the IHES she had to send Grothendieck a paper (‘It was an explication of topos theory that I thought he probably hadn’t considered.’ page 136, and ‘while it proved three problems in topos theory it then set about dismantling the mechanism of the proofs.’ page 151). At the IHES ‘I met three men that I could talk to: Grothendieck, Deligne, and Oscar Zariski.’ (page 136).
I don’t know whether Zariski visited the IHES in the early 70ties, and while most historical allusions (to Grothendieck’s life, his role in Bourbaki etc.) are correct, Alicia mentions the ‘Langlands project’ (page 66) which may very well have been the talk of town at the IHES in 1972, but the mention of Witten ‘Grothendieck writes everything down. Witten nothing.’ (page 100) raised an eyebrow.
The book also contains these two nice attempts to capture some of the essence of topos theory:
When you get to topos theory you are at the edge of another universe.
You have found a place to stand where you can look back at the world from nowhere. It’s not just some gestalt. It’s fundamental. (page 13)
You asked me about Grothendieck. The topos theory he came up with is a witches’ brew of topology and algebra and mathematical logic.
It doesnt even have a clear identity. The power of the theory is still speculative. But it’s there.
You have a sense that it is waiting quietly with answers to questions that nobody has asked yet. (page 68)
I did read ‘The passenger’ first, which is probably better as then you’d know already some of the ghosts haunting Alicia, but it’s not a must if you are only interested in their discussions about the nature of mathematics. Be warned that it is a pretty dark book, better not read when you’re already feeling low, and it should come with a link to a suicide prevention line.
1. The unconscious is structured as a topos (Jacques Lacan argued it was structured as a language), because we need a framework allowing logic without the law of the excluded middle for Lacan’s formulas of sexuation to make some sense at all.
2. This topos may differs from person to person, so we do not all share the same rules of logic (as observed in real life).
3. Consciousness is related to the points of the topos (they are not precise on this, neither in the talk, nor the book).
4. All these individual toposes are ruled by a classifying topos, and they see Lacan’s work as the very first steps towards trying to describe the unconscious by a geometrical theory (though his formulas are not first order).
Surely these are intriguing ideas, if only we would know how to construct the topos of someone’s unconscious.
Let’s go looking for clues.
At the same meeting, there was a talk by Daniel Sibony: “Mathématiques et inconscient”
Sibony started out as mathematician, then turned to psychiatry in the early 70ties. He was acquainted with both Grothendieck and Lacan, and even brought them together once, over lunch, some day in 1973. He makes a one-line appearance in Grothendieck’s Récoltes et Semailles, when G discribes his friends in ‘Survivre et Vivre’:
“Daniel Sibony (who stayed away from this group, while pursuing its evolution out of the corner of a semi-disdainful, smirking eye)”
In his talk, Sibony said he had a similar idea, 50 years before Connes and Gauthier-Lafaye (3.04 into the clip):
“At the same time (early 70ties) I did a seminar in Vincennes, where I was a math professor, on the topology of dreams. At the time I didn’t have categories at my disposal, but I used fibered spaces instead. I showed how we could interpret dreams with a fibered space. This is consistent with the Freudian idea, except that Freud says we should take the list of words from the story of the dream and look for associations. For me, these associations were in the fibers, and these thoughts on fibers and sheaves have always followed me. And now, after 50 years I find this pretty book by Alain Connes and Patrick Gauthier-Lafaye on toposes, and see that my thoughts on dreams as sheaves and fibered spaces are but a special case of theirs.”
This looks interesting. After all, Freud called dream interpretation the ‘royal road’ to the unconscious. “It is the ‘King’s highway’ along which everyone can travel to discover the truth of unconscious processes for themselves.”
“The dream brings blocks of words, of “compacted” meanings, and we question, according to the good old method, each of these blocks, each of these points and which we associate around (we “unblock” around…), we let each point unfold according to the “fiber” which is its own.
I introduced this notion of the dream as fibered space in an article in the review Scilicet in 1972, and in a seminar that I gave at the University of Vincennes in 1973 under the title “Topologie et interpretation des rêves”, to which Jacques Lacan and his close retinue attended throughout the year.
The idea is that the dream is a sheaf, a bundle of fibers, each of which is associated with a “word” of the dream; interpretation makes the fibers appear, and one can pick an element from each, which is of course “displaced” in relation to the word that “produced” the fiber, and these elements are articulated with other elements taken in other fibers, to finally create a message which, once again, does not necessarily say the meaning of the dream because a dream has as many meanings as recipients to whom it is told, but which produces a strong statement, a relevant statement, which can restart the work.”
Key images in the dream (the ‘points’ of the base-space) can stand for entirely different situations in someone’s life (the points in the ‘fiber’ over an image). The therapist’s job is to find a suitable ‘section’ in this ‘sheaf’ to further the theraphy.
It’s a bit like translating a sentence from one language to another. Every word (point of the base-space) can have several possible translations with subtle differences (the points in the fiber over the word). It’s the translator’s job to find the best ‘section’ in this sheaf of possibilities.
This translation-analogy is used by Daniel Sibony in his paper Traduire la passe:
“It therefore operates just like the dream through articulated choices, from one fiber to another, in a bundle of speaking fibers; it articulates them by seeking the optimal section. In fact, the translation takes place between two fiber bundles, each in a language, but in the starting bundle the choice seems fixed by the initial text. However, more or less consciously, the translator “bursts” each word into a larger fiber, he therefore has a bundle of fibers where the given text seems after the fact a singular choice, which will produce another choice in the bundle of the other language.”
This paper also contains a pre-ChatGPT story (we’re in 1998), in which the language model fails because it has far too few alternatives in its fibers:
I felt it during a “humor festival” where I was approached by someone (who seemed to have some humor) and who was a robot. We had a brief conversation, very acceptable, beyond the conventional witticisms and knowing sighs he uttered from time to time to complain about the lack of atmosphere, repeating that after all we are not robots.
I thought at first that it must be a walking walkie-talkie and that in fact I was talking to a guy who was remote control from his cabin. But the object was programmed; the unforeseen effects of meaning were all the more striking. To my question: “Who created you?” he answered with a strange word, a kind of technical god.
I went on to ask him who he thought created me; his answer was immediate: “Oedipus”. (He knew, having questioned me, that I was a psychoanalyst.) The piquancy of his answer pleased me (without Oedipus, at least on a first level, no analyst). These bursts of meaning that we know in children, psychotics, to whom we attribute divinatory gifts — when they only exist, save their skin, questioning us about our being to defend theirs — , these random strokes of meaning shed light on the classic aftermaths where when a tile arrives, we hook it up to other tiles from the past, it ties up the pain by chaining the meaning.
Anyway, the conversation continuing, the robot asked me to psychoanalyse him; I asked him what he was suffering from. His answer was immediate: “Oedipus”.
Disappointing and enlightening: it shows that with each “word” of the interlocutor, the robot makes correspond a signifying constellation, a fiber of elements; choosing a word in each fiber, he then articulates the whole with obvious sequence constraints: a bit of readability and a certain phrasal push that leaves open the game of exchange. And now, in the fiber concerning the “psy” field, chance or constraint had fixed him on the same word, “Oedipus”, which, by repeating itself, closed the scene heavily.
Okay, we have a first potential approximation to Connes and Gauthier-Lafaye’s elusive topos, a sheaf of possible interpretation of base-words in a language.
But, the base-space is still rather discrete, or at best linearly ordered. And also in the fibers, and among the sections, there’s not much of a topology at work.
Perhaps, we should have a look at applications of topology and/or topos theory in large language models?
Items recently added to the online Bourbaki Archive give us new information on time and place of the birth of the concept of schemes.
From May 30th till June 2nd 1955 the ‘second caucus des Illinois’ Bourbaki-congress was held in ‘le grand salon d’Eckhart Hall’ at the University of Chicago (Weil’s place at that time).
Only six of the Bourbaki members were present:
Jean Dieudonne (then 49), the scribe of the Bourbaki-gang.
Claude Chevalley (then 46), who wanted a better, more workable version of algebraic geometry. He was just nominated professor at the Sorbonne, and was prepping for his seminar on algebraic geometry (with Cartan) in the fall.
Armand Borel (then 32), a Swiss mathematician who was in Paris from 1949 and obtained his Ph.D. under Jean Leray before moving on to the IAS in 1957. He was present at 9 of the Bourbaki congresses between 1955 and 1960.
Serge Lang (then 28), a French-American mathematician who got his Ph.D. in 1951 from Princeton under Emil Artin. In 1955, he just got a position at the University of Chicago, which he held until 1971. He attended 7 Bourbaki congresses between 1955 and 1960.
The issue of La Tribu of the Eckhart-Hall congress is entirely devoted to algebraic geometry, and starts off with a bang:
“The Caucus did not judge the plan of La Ciotat above all reproaches, and proposed a completely different plan.
I – Schemes
II – Theory of multiplicities for schemes
III – Varieties
IV – Calculation of cycles
V – Divisors
VI – Projective geometry
etc.”
In the La Ciotat-Tribu,nr. 35 there are also a great number of pages (page 14 – 25) used to explain a general plan to deal with algebraic geometry. Their summary (page 3-4):
“Algebraic Geometry : She has a very nice face.
Chap I : Algebraic varieties
Chap II : The rest of Chap. I
Chap III : Divisors
Chap IV : Intersections”
There’s much more to say comparing these two plans, but that’ll be for another day.
We’ve just read the word ‘schemes’ for the first (?) time. That unnumbered La Tribu continues on page 3 with “where one explains what a scheme is”:
So, what was their first idea of a scheme?
Well, you had your favourite Dedekind domain $D$, and you considered all rings of finite type over $D$. Sorry, not all rings, just all domains because such a ring $R$ had to have a field of fractions $K$ which was of finite type over $k$ the field of fractions of your Dedekind domain $D$.
They say that Dedekind domains are the algebraic geometrical equivalent of fields. Yeah well, as they only consider $D$-rings the geometric object associated to $D$ is the terminal object, much like a point if $D$ is an algebraically closed field.
But then, what is this geometric object associated to a domain $R$?
In this stage, still under the influence of Weil’s focus on valuations and their specialisations, they (Chevalley?) take as the geometric object $\mathbf{Spec}(R)$, the set of all ‘spots’ (taches), that is, local rings in $K$ which are the localisations of $R$ at prime ideals. So, instead of taking the set of all prime ideals, they prefer to take the set of all stalks of the (coming) structure sheaf.
But then, speaking about sheaves is rather futile as there is no trace of any topology on this set, then. Also, they make a big fuss about not wanting to define a general schema by gluing together these ‘affine’ schemes, but then they introduce a notion of ‘apparentement’ of spots which basically means the same thing.
It is still very early days, and there’s a lot more to say on this, but if no further documents come to light, I’d say that the birthplace of ‘schemes’, that is , the place where the first time there was a documented consensus on the notion, is Eckhart Hall in Chicago.
As it is probably better to run years behind than to stand eternally still, I’ll try out how much of a blogger I am in 2004. If you want to read more about blogging, Rebecca’s pocket is a good place to start. She has written an essay on the early days of
the weblog, an article on weblog ethics and a couple of (pretty obvious) tips for a better
weblog.
But let us not talk about ‘better’ or ‘ethical’ at this moment, I’m just starting out. Give me a couple of weeks/months to develop my own style and topics and I’ll change the layout accordingly.
For now, I’ve taken the free blog-tool pMachine which uses only PHP and MySQL on the server-side so I should be able to get the layout suited to my own mood shortly. A major advantage of a weblog over a homepage is that you can feed it to programs called news aggregators by subscribing.
The program can then be tuned so that it ventures out on the net at regular intervals checking whether any of the blogs it is subscribed to has new material and reports back with the running title and opening lines.
If more people would turn their homepages into weblogs, the frustrating job of checking (usually in vain) whether their pages has been updated could be left to the aggregator. For now, I’m using Rancheros’s NetNewsWire as my Mac OSX news
aggregator.
Okay, now it is time to make some final preparations for endyear, but tomorrow I’ll wake up to become a blogger…
