# Grothendieck: the rising sea

Alexander Grothendieck is one of the greatest mathematicians of all time, certainly the greatest by sheer repute of the second half of the century. Grothendieck the man was as mythical a figure as any mathematician I’ve ever come across. Pierre Cartier, in the introduction to A Country Known Only By Name:

Grothendieck’s journey? A childhood devastated by Nazism and its crimes, a father who was absent in his early years and then disappeared in the storm, a mother who kept him in her orbit and long disturbed his relationships with other women. He compensated for this with a frantic investment in mathematical abstraction until psychosis, kept at bay through this very involvement, caught up with him and swallowed him in morbid anguish.

Grothendieck once described two styles in mathematics. If you think of a theorem to be proved as a nut to be opened, so as to reach “the nourishing flesh protected by the shell”, then the hammer and chisel principle is: “put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks—and you are satisfied”. He says:

I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration. . . the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it. . . yet it finally surrounds the resistant substance. [Grothendieck 1985–1987, pp. 552-3]

His mathematical vision pierced to the downright philosophical: how deeply can we understand the notion of a (geometric) point? Cartier:

His originality lay in deepening of the concept of a geometric point.1 Such research may seem trifling, but the metaphysical stakes are considerable; the philosophical problems it engenders are still far from solved. In its ultimate form, this research, Grothendieck’s proudest, revolved around the concept of a motive, or pattern, viewed as a beacon illuminating all the incarnations of a given object through their various ephemeral cloaks. But this concept also represents the point at which his incomplete work opened to a void. Most scientists are somewhat keener to erase their footprints from the sand, silence their fantasies and dreams, and devote themselves to the statue within, as François Jacob puts it. Grothendieck’s idiosyncrasy prompted him fully to accept this flaw.

His mathematical coming of age was the quintessential story of genius flowering in isolation, the kind that makes for narratives steeped in myth but are also hopelessly intractable to future generations of otherwise able thinkers not blessed by his tremendous originality and singular vision:

His first explicitly mathematical episode occurred when he was an undergraduate. He described himself as very unhappy with the education given at the time. His professor told him that a certain Lebesgue had already solved all (!) the problems in mathematics, but that it would be too difficult to teach. So alone, with almost no guidance, Grothendieck reconstructed a very general version of the Lebesgue integral. In Récoltes et Semailles, he describes in detail the genesis of this first mathematical work, achieved in isolation; he honestly believed that he was the only mathematician in the world.

Grothendieck most certainly didn’t work alone. Cartier again, recounting the Golden Age spanning the years 1958-70:

His mathematical work in Nancy had established his renown, and he might well have sailed along on that momentum. But he described himself well when he said that he was a builder of houses in which he was not meant to live. He embarked upon the classical career of a researcher, was quickly recruited to the CNRS (Centre National de la Recherche Scientifique, National Center for Scientific Research), promoted, and then spent a few years abroad after writing his dissertation. When he returned from São Paulo, he closed the chapter on Functional Analysis. That was the beginning of his master period, 1958 to 1970, which coincided with the Bourbaki group’s prime. The springboard that allowed him to do this phenomenal work was given to him by Léon Motchane, a brilliant businessman who had thrown himself into the creation of the IHÉS (Institut des Hautes Études Scientifiques, Institute for Advanced Scientific Studies) in Bures-sur-Yvette. Motchane offered Dieudonné, who had just completed his theory of formal groups, the future institute’s first chair in mathematics. Dieudonné accepted on the condition that he hire Grothendieck as well. The duo then recruited Jean-Pierre Serre, who with his keen sense of the unity of mathematics, high scientific culture, quickness of mind, and technical prowess, would keep them on their toes. Serre acted as an intermediary between Weil and Grothendieck when they no longer wished to communicate directly, and contributed greatly to the clarification of the Weil conjectures. Serre was the perfect beater of mathematical pheasants (I was going to say matchmaker), scaring the quarry straight into Grothendieck’s nets, and in nets as strong as those, the quarry barely struggled.

Grothendieck was then moved to create one of the most prestigious mathematics seminars the world has ever seen. Surrounded by young talent, he threw himself with a passion into mathematical discovery, in sessions lasting from ten to twelve hours!7 He formulated a formidable program intended to fuse arithmetic, algebraic geometry, and topology. A builder of cathedrals, as he put it in his own allegory, he distributed the work to his teammates. Every day, he sent interminable and illegible mathematical feuilletons to Dieudonné, who, sitting at his worktable from five to eight each morning, transformed the scribbles into an imposing collection of volumes co-signed by Dieudonné and Grothendieck and then published in the IHÉS’ Publications Mathématiques. Dieudonné abjured all personal ambition and consecrated himself to this service with the same self-abnegation he had demonstrated under Bourbaki.

Here’s Grothendieck describing his vision; it’s what prompted me to write this post in the first place. (I’ve wanted to write about the man for years now so I was looking for a good excuse to do so.) It’s translated from the French by Roy Lisker:

Most mathematicians take refuge within a specific conceptual framework, in a “Universe” which seemingly has been fixed for all time – basically the one they encountered “ready-made” at the time when they did their studies. They may be compared to the heirs of a beautiful and capacious mansion in which all the installations and interior decorating have already been done, with its living-rooms , its kitchens, its studios, its cookery and cutlery, with everything in short, one needs to make or cook whatever one wishes. How this mansion has been constructed, laboriously over generations, and how and why this or that tool has been invented (as opposed to others which were not), why the rooms are disposed in just this fashion and not another – these are the kinds of questions which the heirs don’t dream of asking . It’s their “Universe”, it’s been given once and for all! It impresses one by virtue of its greatness, (even though one rarely makes the tour of all the rooms) yet at the same time by its familiarity, and, above all, with its immutability.

When they concern themselves with it at all, it is only to maintain or perhaps embellish their inheritance: strengthen the rickety legs of a piece of furniture, fix up the appearance of a facade, replace the parts of some instrument, even, for the more enterprising, construct, in one of its workshops, a brand new piece of furniture. Putting their heart into it, they may fabricate a beautiful object, which will serve to embellish the house still further.

Much more infrequently, one of them will dream of effecting some modification of some of the tools themselves, even, according to the demand, to the extent of making a new one. Once this is done, it is not unusual for them make all sorts of apologies, like a pious genuflection to traditional family values, which they appear to have affronted by some far-fetched innovation.

The windows and blinds are all closed in most of the rooms of this mansion, no doubt from fear of being engulfed by winds blowing from no-one knows where. And, when the beautiful new furnishings, one after another with no regard for their provenance, begin to encumber and crowd out the space of their rooms even to the extent of pouring into the corridors, not one of these heirs wish to consider the possibility that their cozy, comforting universe may be cracking at the seams. Rather than facing the matter squarely, each in his own way tries to find some way of accommodating himself, one squeezing himself in between a Louis XV chest of drawers and a rattan rocking chair, another between a moldy grotesque statue and an Egyptian sarcophagus, yet another who, driven to desperation climbs, as best he can, a huge heterogeneous collapsing pile of chairs and benches!

The little picture I’ve just sketched is not restricted to the world of the mathematicians. It can serve to illustrate certain inveterate and timeless situations to be found in every milieu and every sphere of human activity, and (as far as I know) in every society and every period of human history. I made reference to it before , and I am the last to exempt myself: quite to the contrary, as this testament well demonstrates. However I maintain that, in the relatively restricted domain of intellectual creativity, I’ve not been affected by this conditioning process, which could be considered a kind of ‘cultural blindness’ – an incapacity to see (or move outside) the “Universe” determined by the surrounding culture.

I consider myself to be in the distinguished line of mathematicians whose spontaneous and joyful vocation it has been to be ceaseless building new mansions.

We are the sort who, along the way, can’t be prevented from fashioning, as needed, all the tools, cutlery, furnishings and instruments used in building the new mansion, right from the foundations up to the rooftops, leaving enough room for installing future kitchens and future workshops, and whatever is needed to make it habitable and comfortable. However once everything has been set in place, down to the gutters and the footstools, we aren’t the kind of worker who will hang around, although every stone and every rafter carries the stamp of the hand that conceived it and put it in its place.

The rightful place of such a worker is not in a ready-made universe, however accommodating it may be, whether one that he’s built with his own hands, or by those of his predecessors. New tasks forever call him to new scaffoldings, driven as he is by a need that he is perhaps alone to fully respond to. He belongs out in the open. He is the companion of the winds and isn’t afraid of being entirely alone in his task, for months or even years or, if it should be necessary, his whole life, if no-one arrives to relieve him of his burden. He, like the rest of the world, hasn’t more than two hands – yet two hands which, at every moment, know what they’re doing, which do not shrink from the most arduous tasks, nor despise the most delicate, and are never resistant to learning to perform the innumerable list of things they may be called upon to do. Two hands, it isn’t much, considering how the world is infinite. Yet, all the same, two hands, they are a lot ….