Sam Leith

The Einstein of maths

Sam Leith on Alexandre Grothendieck, the revolutionary number-cruncher who was last heard of in the Pyrenees raging about the Devil

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The odds are that the name Alexandre Grothendieck will mean little or nothing to most Spectator readers. It’s a name I heard for the first time in high summer two years or so ago, not long, as I remember it, after the film A Beautiful Mind had come out. I was in the garden of my friend Umar’s house in Cambridge, and we were waiting for his ancient cast-iron barbecue, Camp Freddie, to cook some sausages.

Umar is a mathematician of considerable braininess, and when we are together we often end up talking maths. That is, I tend to ask him to explain what he does, and he tends to try, and I tend not to understand. But sometimes we strike gold. An entire afternoon was once passed happily playing logic games involving prisoners with different-coloured hats. I have giggled ignorantly at maths jokes (‘What’s purple and commutes?’ ‘An Abelian grape’), hummed and hawed over the question of whether maths is discovered or invented, and been mindboggled for a week after he explained the concept of the ‘cardinality of infinities’ (some infinities are bigger than others, it turns out).

By Grothendieck I was riveted. The story, in short, is of a mathematician of staggering accomplishment (in one of the hens’-teeth-rare public references to him he is described as ‘the mathematician whose work was to lead to a unification of geometry, number theory, topology and complex analysis’) who has retreated, like a Salinger or a Pynchon of number theory, into utter isolation. The most recent cutting dismisses him as ‘last heard of raging about the Devil somewhere in the Pyrenees’.

It would not be an exaggeration, I think, to describe Grothendieck as a legendary figure in the mathematical world. Yet if you run a search through the cuttings library of the mainstream press, you will find barely a single mention of his name. Until very recently, even the Internet, that repository of all arcane knowledge, contained little verging on nothing. There was one blurred and out-of-date photograph, a couple of fragments. Nothing at all for the lay reader to go on.

There are, I’d say, two good reasons why this would be so. In the first place, the luminous brilliance of Grothendieck’s mathematical achievement — his 1966 Fields Medal, the Nobel Prize of maths, is an indicator — is matched only by the near impossibility of explaining it to anyone without a background in pure maths.

His work — especially in the golden period between 1955 and 1970 — is described as being ‘maximally deep’; that is, he was interested in stating his solutions to mathematical problems in the most general way possible, and applying congruences across discrete fields. He used, for example, algebraic geometry to crack number theory — and as Umar puts it, ‘number theory was the bigger fish to crack’. The mathematician Leila Schneps — custodian of the newly established website — describes him simply as ‘the Einstein of mathematics’.

But there is another reason why he has retreated below the radar of the non-mathematical world. Grothendieck doesn’t write popularising books like Stephen Hawking; he doesn’t tour American universities lecturing undergraduates; and he doesn’t, any more, publish his researches and discoveries in mathematical journals. In fact, 13 years ago, Grothendieck more or less disappeared altogether.

In the Pyrenean village in which he had lived since the 1970s, he burned thousands of pages of manuscript in the garden of his then girlfriend, left on her kitchen table an enormous manuscript copy of a memoir by his mother, and vanished.

The biographical section of the Grothendieck Circle ends in 1991: ‘In August, Grothendieck leaves his home suddenly, without warning anyone, for an unknown location. He spends his time writing an enormous work on physics and philosophical meditations on themes such as free choice, determinism and the existence of evil. He refuses practically every human contact.’

Among the few images of him with which we are left, the most recent show a shaven-headed, bespectacled man — in looks not a million miles from Foucault — with the austere grace of a Buddhist monk. He used, indeed, to sleep on the floor instead of a bed, but is said to have mellowed to the extent that he now owns a bed and tends a garden with enthusiasm. A vegetarian, he presses on his rare visitors armfuls of apples and figs.

His later work survives in mimeographs and photocopies; his lettres fleuves — correspondence in French or English, often running to hundreds of pages, mixing philosophical invective, attacks on rival mathematicians and, seeded like nuggets in the texts, insights into maths on a very, very high level.

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 subseque nt 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 never once doubted,’ Grothendieck writes in an early chapter of Récoltes et Semailles, his philosophical autobiography, ‘that I would eventually succeed in getting to the bottom of things.’

A more important mathematician than John Nash, a more extraordinary and horrific back story, and a beautiful, beautiful mind. If anyone can figure out what a metre is, who is to say that we will not one day have it back?