Around 1970, Alexander Grothendieck, the greatest of all modern mathematicians and arguably the greatest mathematician of all time, announced — at the age of 42 — the official end of his research career. Another great mathematician once told me that he thought he knew why. Following two decades of discoveries and insights that, one after the other, stunned the mathematical world, Grothendieck had, for the first time, achieved an insight so unexpected and so consequential that he himself was stunned. Grothendieck had discovered his own mortality.

I am told that just a few hours ago, his vision proved accurate. But the notion of Grothendieck as a mortal seems hard to swallow. He dominated pure mathematics not just through the force of his ideas — ideas that seemed eons ahead of everyone else’s — but through the force of his personality. When, around 1960, he announced his audacious plan to solve the notoriously difficult Weil conjectures by first rewriting the foundations of geometry, dozens of superb mathematicians put the rest of their careers on hold to do their parts. The project’s final page count, including the twelve volumes known as SGA (Seminaire de Geometrie Algebrique) and the eight known as EGA (Elements de Geometrie Algebrique) approached 10,000 pages. The force and clarity of Grothendieck’s unique vision scream forth from nearly every one of those pages, demanding that the reader see the mathematical world in a new and completely original way — a perspective that has proved not just compelling, but unspeakably powerful.

In Grothendieck, modesty would have been ridiculous, and he was never ridiculous. Here, in his own words — words that ring utterly true — is Grothendieck’s own assessment of how he stood apart (translated from 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 ….

Sometime in the near future, I hope to fashion a blog post or two that will convey at least a scintilla of the Grothendieckian worldview to a non-mathematical audience. For now, I point you to a few good biographical articles and some of Grothendieck’s own most influential writing, which I’ve posted here. I particularly recommend the piece by Colin McLarty, the first few pages of which, at least, are mostly non-technical.

Edited to add: See my followup post here.