The n-Category Café

Answers to Grothendieck’s meter question?

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 Schneps and Lochak in due course staked out the marketplace of the town, carrying an out-of-date photograph of Grothendieck, and waited for the greatest mathematician of the 20th century to show up in search of beansprouts.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 He thought his unpublished work would all have been long forgotten.’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.

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*

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.

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).

Here is a lecture on Grothendieck’s anabelian ideas.

> 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 ?