## August 31, 2006

### Letter from Grothendieck

#### Posted by John Baez

Alexander Grothendieck was the most visionary and radical mathematician in the second half of the 20th century - at least before he left his home and disappeared one fine day in 1991.

For a quick tale of his life, try clicking on his name above. For a longer version, try this:
• Allyn Jackson, Comme apellé du néante - as if summoned from the void: the life of Alexandre Grothendieck. Part 1, Part 2.

This newly available document will be interesting to his fans, and also students of n-category theory:

Ronnie Brown was one of the first champions of higher-dimensional algebra, studying topology using first groupoids, and then n-groupoids and related structures. For more on his philosophy, try these: Grothendieck became interested in n-groupoids and n-categories in the 1970s, so he began corresponding with Brown an others on these topics. In 1983 he sent a 593-page letter about these topics to Daniel Quillen - a letter which has now become extremely influential:
In fact, it was released in installments to a number of people including Larry Breen, Ronnie Brown, and Tim Porter. For more on this story, try: In the newly released mail to Brown, Grothendieck wonders why Quillen didn’t reply to his letter! He also comments on how homotopy theorists seemed uninterested in higher-dimensional algebra:

It is all too evident that I am not an expert on homotopy theory, and the books I am writing now on foundational matters are very likely to be looked at as “rubbish” by most experts, unless I show up with π147(S123) as a byproduct (whereas it is for the least doubtful I will…).

His ideas were too far ahead of his time for easy acceptance. Only now are the times catching up!

According to Ronnie Brown, Pursuing Stacks will be published in Documents Mathématiques, with various correspondence as an appendix, edited by Georges Maltsiniotis. Maltsiniotis and Brown are now editing 69 letters exchanged between Grothendieck and Brown; this one is number 17.

I also hear that Colin McLarty is working on a biography of Grothendieck.

I thank Ronnie Brown for making this letter from Grothendieck available, and also for converting his 1987 paper “From groups to groupoids: a brief survey” into LaTeX.

Posted at August 31, 2006 2:34 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/912

### Re: Letter from Grothendieck

Posted by: Florifulgurator on September 1, 2006 3:00 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Just so everyone can know what the question was, let me say a bit more. Leila Schneps, an expert on Grothendieck’s theory of dessins d’enfants and a founding member of the Grothendieck Circle, was one of the last mathematicians to meet Grothendieck and correspond with him. Sam Leithe writes:

One of the last members of the mathematical establishment to come into contact with him was Leila Schneps. Through a series of coincidences, she and her future husband, Pierre Lochak, learned from a market trader in the town he left in 1991 that ‘the crazy mathematician’ had turned up in another town in the Pyrenees. Schneps and Lochak in due course staked out the marketplace of the town, carrying an out-of-date photograph of Grothendieck, and waited for the greatest mathematician of the 20th century to show up in search of beansprouts.

‘We spent all morning there in the market. And then there he was.’ Were they not worried he’d run away? ‘We were scared. We didn’t know what would happen. But he was really, really nice. He said he didn’t want to be found, but he was friendly. We told him that one of his conjectures had been proved. He had no idea. He’d stopped being interested in maths at that stage. He thought his unpublished work would all have been long forgotten.’

Grothendieck’s first disappearance, in a sense, came in 1970, when at the very height of his powers he abandoned a post that had been created for him at the Institute of High Scientific Studies (and which remains probably the most prestigious tenure in his field of mathematics) on the grounds that it was partly funded by the military industrial complex. In 1988, he was awarded the Crafoord Prize. He refused to accept it.

Grothendieck’s father was an anarchist who died in Auschwitz; Alexandre, along with his German mother Hanka, was interned in France during the war as an ‘undesirable’. These days, what we know of Grothendieck’s thinking suggests his guiding preoccupation is the problem of evil. He lives alone and works, for 12 hours a day, on a 50-volume manuscript which addresses, among other things, the physics of free will.

One story has it that Grothendieck is now convinced that the Devil is working to falsify the speed of light. Schneps ascribes his concerns with the speed of light to his anxiety about the methodological compromises physicists make. He talks constantly, however, about the Devil, semi-metaphorically, sitting behind good people and nudging them in the direction of compromise, of the fudge, of the move towards corruption. ‘Uncompromising’ is the expression Schneps favours.

In his correspondence with Leila Schneps, he told her he would be willing to share his research into physics with her if she could answer one question: ‘What is a metre?’

She and Lochak, baffled, took a month to write back — and did so at length. But before this letter arrived, Grothendieck dispatched three letters in quick succession. His first letter appeared to threaten suicide. His second was ‘the warmest, warmest thing … saying it’s just amazing anyone cares….’ The third addressed ‘Leila Schneps’ in bitterly sarcastic inverted commas. They found their subsequent letters returned unopened. ‘We went to see him and he slammed the door in our faces….’

Has Grothendieck — runs the obvious question — gone mad? Well, possibly. It all depends on what you mean by ‘mad’.

‘He lives alone and he writes on really deep ideas,’ says Schneps. ‘In the past, what about saints or prophets? Did people think they had gone mad? He cannot bear to live in the world we’re in…. He’s certainly abnormal. I could not possibly call him mad. People say there’s normal and there’s insane. These are not the only two categories….’

I can give some more clues about Grothendieck’s question if people want.

Posted by: John Baez on September 1, 2006 4:25 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

if people want.

Posted by: urs on September 1, 2006 4:39 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Try Connes and Navarro on the ‘meter question’:

Posted by: Quidam on September 1, 2006 6:30 PM | Permalink | Reply to this

### What is a meter?

What is a metre?

It’s 3.28 feet. And don’t tell me you don’t know what a foot is!

More seriously, I can imagine several perspectives from which this question is subtle and interesting. I am not sure which one of these subtleties we are supposed to be interested in.

For instance: are we allowed to assume that we know what a positron, and an electron are?

If so, there is a natural number $n$ - which I can tell you if desired - such that a meter is $n$ times the mean distance of the electron from the positron in the ground bound state of positronium.

(Actually, the official definition also involves a photon, and various quarks - namely the radiation emitted from caesium atoms. But that’s just a variation of the above theme.)

Of course, that explanation only works if we agree that we know what “distance” means in the first place.

So is the question maybe really: “What is distance?”

I must say I am slightly afraid that the real intention of the question is something like “What does it mean to observe something (like the ground state of positronium)?”

Hopefully David Corfield will be back from his vacation soon…

Posted by: urs on September 1, 2006 5:57 PM | Permalink | Reply to this

### Re: What is a meter?

I am not sure which one of these subtleties we are supposed to be interested in.

Based on my impression upon skimming the story, I can guess that it is intended to be a toned down version of the similar question, “What is the speed of light?”

If that is anywhere near being correct, then it is one step from there to asking, “What is the fabric of the cosmos?” But maybe I am daydreaming (I know I am sleep deprived).

PS: Maybe it is just my PC, but I couldn’t get the arrow keys or copy/paste to work in Firefox. Had to resort to IE to post this *shudder*

Posted by: Eric on September 2, 2006 2:15 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

I can give some more clues about Grothendieck’s question if people want.

So can I.

Posted by: Kea on September 2, 2006 12:05 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

Sam Leithe wrote:

In his correspondence with Leila Schneps, he told her he would be willing to share his research into physics with her if she could answer one question: ‘What is a metre?’

There are, of course, many ways to take this question, and in some sense it’s a Rorschach ink blot test - one can assume Grothendieck is asking something profound and search for a profound answer, or assume he has gone crazy and dismiss the question, or anything in between.

Pierre Cartier writes:

If I can believe his most recent visitors, he is obsessed with the Devil, whom he sees at work everywhere in the world, destroying the divine harmony and replacing 300,000 km/sec by 299,887 km/sec as the speed of light!

(In fact the speed of light is 299,792.458 km/sec, exactly - by definition.)

Alain Connes writes:

It is sad tht while Grothendieck was asking the right question: “what is the metre” and rightly saying that the convention c = 300,000,000 m/s would have been simpler the standard reaction to his query was to see there the clear symptom of a deranged mind.

I will not try to guess what Grothendieck was thinking, but simply answer his question in a very literal-minded way:

A meter is the distance that light emitted by a cesium 133 atom transitioning between the two hyperfine levels of its ground state will travel as the light vibrates exactly 9,192,631,770 / 299,792,458 times.

Of course the definition of the meter has changed repeatedly and may change again, but this is the definition now. Clearly such a complicated definition is the work of the Devil.

Posted by: John Baez on September 2, 2006 2:14 AM | Permalink | Reply to this
Weblog: leuschke.org
Excerpt: Letter from Grothendieck | The n-Category Café some recently uncovered correspondence between Grothendieck and Ronald Brown (tags: grothendieck toread) Cornell Library Historical Mathematics Monographs 200,000 pages on file. amazing. some great thing...
Tracked: September 3, 2006 6:17 AM

### Re: Letter from Grothendieck

I understand that Grothendieck disliked physics (though many of his ideas have found application in physics today) for its military uses, much in the same way Lev Landau disliked topology although for other reasons.

Can anybody elaborate on that comment of his regarding to “share all his research in physics”?

Posted by: Comentator on September 6, 2006 9:11 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Here a nice new article from W. Scharlau on Grothendieck.

Posted by: Thomas Riepe on March 29, 2008 8:59 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

The obvious question - how do we know AG is still alive, if at all? I was under the impression someone keeps an indirect eye on him.

And what about the caption on the photo in the article?

“Alexander Grothendieck in May 1998 in the south of France. Three years later he disappeared” (translation provided by Google).

Posted by: David Roberts on March 31, 2008 4:10 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

As far as I know, he is not entirely without contact to other people. IMHO there are several other interesting questions, e.g. on his specific way of thinking, the content and meaning of his non-mathematical ideas etc. Much of the seemingly strange statements and attitudes ascribed to him appear to me as resulting from his growing up in the spiritual environment of Heydorn and the Weimar way to think of intellectuals in the 1920s.

Posted by: Thomas Riepe on March 31, 2008 6:30 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

>As far as I know, he is not entirely without contact to other people.

Is this definite? In his age he may be easily in need for medical help.

### Re: Letter from Grothendieck

Just to be correct : the photo in the Scharlau article is already from 1988…

but it’s good to see that you are still thinking about him:-)

Posted by: Iris on April 6, 2008 9:55 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

…from 1988…

I suspected that might be the case, since that would coincide with his 1991 disappearance.

Posted by: David Roberts on April 7, 2008 12:08 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

Posted by: T.R. on December 15, 2008 4:36 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Here is a lecture on Grothendieck’s anabelian ideas.

Posted by: Thomas on January 15, 2009 3:23 PM | Permalink | Reply to this
Read the post What to Make of Mathematical Difficulties
Weblog: The n-Category Café
Excerpt: How to make the most of obstructions to a theory
Tracked: December 16, 2008 2:47 PM

### Re: Letter from Grothendieck

Andreas Holmstrom just posted a link to videos of the talks on the Grothendieck conference this january on his blog.

Posted by: Thomas on March 28, 2009 2:11 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

Grothendieck worked in functional analysis and then algebraic geometry/number theory. His vision cannot be meaningfully compared with the vision of countless other people working in other, nearly disjoint, areas of math. Therefore the ubiquitous claim, parroted here, that he was “the most this-or-that mathematician of such-and-such time period” are without much content.

There is perhaps only one phrase of this sort that is true, and it’s “Grothendieck was the mathematician with the greatest personality cult of the 20th century”. This actually says more about the people who have been working in this kind of math than about Grothendieck himself. (Great as he was.)

Mathematicians tend to have quite unsophisticated views of social structures and very little perception of social subtleties. The number theory / algebraic geometry lot is particularly bad. It’s sad to see adults immersed in such childish games, which they can’t even tell they’re playing.

Cue indignant responses from people who spend all their mental energy trying to get ahead in the game, and have never really thought about this for more than 5 minutes.

Posted by: Anonymous on April 17, 2010 11:24 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

> worked in functional analysis and then algebraic geometry/number theory

This narrowing is uninformed, at least. While it has some correlation with his finished work and from about 1952-1969, the statement undermines the difference between working out some particular result and going to the bottom of the things. Algebraic geometry in 1952 was, like in 19th century, algebraic equations, and due Serre, by 1954 sheaves entered the scene. In only 4 years after that the very concept of a space had to change (with seeming purpose of just getting something about solving algebraic equation, which you reduce the vision to); one has scheme, TOPOS, abelian category and the idea of motive; in doing this he went to the bottom, even the concept of the equivalence of categories had to be invented. All by 1958, 4 years after functional analysis thesis. The things which came later like triangulated category, homotopy hypothesis, dessins d’enfants, derivateurs, regular differential operators, crystals, stacks, are NOT limited to algebraic geometry.

For example the Kodaira, Spencer etc. theory of differential systems, prolongations, deformations etc. (we talk an area in differential equations, complex geometry, integrable systems) borrowed and adapted all basic concepts from Grothendieck’s work on crystals, deformation, and differential operators; this went on with fundamental works of Malgrange, Deligne etc. on meromorphic systems etc. What stacks, homotopy hypothesis, topoi, functor of points viewpoint, Grothendieck topologies do for logics, homotopy theory, differential geometry, geometric Langlands etc. is hard to underestimate. This is not algebraic geometry/number theory. Let me also mention that AG in his high school rediscovered Lebesgue measure theory, being unhappy with his school teaching of surface and area concepts and uninformed of 30 years old theory. This shows a way of thinking which in most major efforts came straight to the right concept, regardless of the area.

It is probably true that many mathematicians (even unknown) are smarter than A.G. was, and make more complicated constructions, work faster, or learn quicker (G. says he had hard time reading papers of other people and understand them). You are right, it is difficult to compare SUCH things. The difference which makes only few people in the whole history of mathematics (e.g. Gauss) comparable to AG is going to the bottom of the things: finding as a rule the right concept behind the mess, abstracting the problem to a universal question, and then many of those can live their life far beyond the original field. So when John says visionary and radical that points to a extreme sense of fundamental universality; when you say “nearly disjoint, areas of math” you seem to imply that you do not believe much in existence of such things in mathematics and that the “fields” will obscure comparison. I do not see such a problem as a physicist, why do you do ?

Posted by: Zoran Skoda on April 17, 2010 10:37 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Let me also mention that AG in his school rediscovered Lebesgue measure theory, being unhappy with his school teaching of surface and area concepts and uninformed of 30 years old theory.

This sounds interesting! Did he really just rediscover it exactly as Lebesgue did it, or (as I would guess) did he come up with something which is trivially equivalent to Lebesgue's theory (once you compare them) but comes from a different angle? If the latter, is his approach written up anywhere?

Posted by: Toby Bartels on April 19, 2010 12:09 AM | Permalink | Reply to this

### Re: Letter from Grothendieck

I do not believe that there is a preserved write up. The case was explained in “Recoltes et Semailles”; when AG after his high school came to the university he explained his theory to his new university professor who told him that this is known since Lebesgue. AG recalls that this moment was not a disappointment for him, namely the professor’s statement has shown him that there were other people sharing his way of thinking, what was a comforting thought.

It has been often said of I. M. Geljfand that he had similarly rediscovered the infinitesimal calculus as a pupil at the age of 14.

Posted by: Zoran Skoda on April 19, 2010 3:57 PM | Permalink | Reply to this

### Re: Letter from Grothendieck

Ah, rediscovered while in high school. Yes, I did some of that too, although nothing as important as Lebesgue integration.

Posted by: Toby Bartels on April 25, 2010 12:12 AM | Permalink | Reply to this

Post a New Comment