I never met Alexander Grothendieck. I was never in the same room with him. I never even saw him from a distance. But whenever I think about math — which is to say, pretty much every day — I feel him hovering over my shoulder. I’ve strived to read the mind of Grothendieck as others strive to read the mind of God.
Those who did know him tend to describe him as a man of indescribable charisma, with a Christ-like ability to inspire followers. I’ve heard it said that when Grothendieck walked into a room, you might have had no idea who he was or what he did, but you definitely knew you wanted to devote your life to him.
And people did. In 1958, when Grothendieck (aged 30) announced a massive program to rewrite the foundations of geometry, he assembled a coterie of brilliant followers and conducted a seminar that met 10 hours a day, 5 days a week, for over a decade. Grothendieck talked; others took notes, went home, filled in details, expanded on his ideas, wrote final drafts, and returned the next day for more. Jean Dieudonne, a mathematician of quite considerable prominence in his own right, subjugated himself entirely to the project and was at his desk every morning at 5AM so that he could do three hours of editing before Grothendieck arrived and started talking again at 8:00. (Here and elsewhere I am reporting history as I’ve heard it from the participants and others who followed developments closely as they were happening. If I’ve got some details wrong, I’m happy to be corrected.) The resulting volumes filled almost 10,000 pages and rocked the mathematical world. (You can see some of those pages here).
I want to try to give something of the flavor of the revolution that unfolded in that room, and I want to do it for an audience with little mathematical background. This might require stretching some analogies almost to the breaking point. I’ll try to be as honest as I can. In the first part, I’ll talk about Grothendieck’s radical approach to mathematics generally; after that, I’ll talk (in a necessarily vague way) about some of his most radical and important ideas.
Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.
Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.
It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.
Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…
Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.
Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.
There are, of course, times when it does pay to examine the inner workings of things. Jean-Pierre Serre, another titan of modern mathematics with whom Grothendieck had an intimate working relationship, was often, in Grothendieck’s words “the yang to my yin”. If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”
In other words, the philosophy was this: If a phenomenon seems hard to explain, it’s because you haven’t fully understood how general it is. Once you figure out how general it is, the explanation will stare you in the face.
It is, I believe, just plain impossible, without trying to teach you a lot of mathematics, to convey the extremes to which Grothendieck carried this philosophy, or the magnificence of its success. Of course it might still have had its limits. In a 1986 letter to Grothendieck, Serre raised the question of why Grothendieck had largely stopped working on mathematics:
One might ask oneself, for example, if there is not a deeper explanation than simply being tired of having to bear the burden of so many thousands of pages. Somewhere, you describe your approach to mathematics, in which one does not attack a problem head-on, but one envelops and dissolves it in a rising tide of general theories. Very good: this is your way of working, and what you have done proves that it does indeed work. For topological vector spaces or algebraic geometry at least…. It is not so clear for number theory….Whence this question: Did you not come, around 1968-1970, to realize that the “rising tide” method was powerless against this type of question, and that a different style would be necessary—which you did not like?
As far as I know, Grothendieck never responded to this letter.
My friend Bob Thomason once told me that the reason Grothendieck succeeded so often where others had failed was that while everyone else was out to prove a theorem, Grothendieck was out to understand geometry. So when Grothendieck set out to attack the notoriously difficult Weil Conjectures, the goal wasn’t so much to solve the problems as to use them as a test for the philosophy that if you generalize sufficiently, all difficult problems become easy.
In that sense, the Grothendieck seminar was both a magnificent success and in the end, to Grothendieck, a disappointment The last and hardest of the Weil conjectures was settled in 1974 by Grothendieck’s student Pierre Deligne, who solved the problem largely by immersing it in the vast Grothendieckian sea, but then, at the last minute, bringing out the mathematical equivalent of a chisel. Grothendieck never forgave him.
What are the Weil conjectures? [Feel free to skip this and the next paragraph if you don't care!] Very very roughly: Start with an equation (or, if you prefer, a family of equations) — say, for example, Y2=x3+10x-3. One solution is (X=2,Y=5). How many others are there? Well, it depends on what you count as a solution. Do you only allow integers? Or are you willing to consider solutions like (X=1,Y=√8) ? The more liberal you are about what kind of numbers you allow, the more solutions (or approximate solutions) you’re going to have. The Weil conjectures make some very precise quantitative claims about how the number of (approximate) solutions grows as you become increasingly liberal.
As we all know from high school, an equation itself defines a curve. Counting “allowable” solutions means counting the “allowable” points on this curve. That, in turn, turns out to require a very subtle understanding of the curve’s geometric properties — does it, for example, have any sharp kinks? Does it surround any “holes”? How many? Et cetera. The Weil conjectures say (in a very precise way) that these purely geometric properties control the answer to our purely arithmetic questions about the growth rate of the number of solutions.
Bottom line: To study the Weil conjectures, you have to think very hard about the subtle properties of curves, surfaces and higher-dimensional objects. When you do this, you find yourself mentally “moving around” the curve, trying to hop from point to point. And — in a sense that I cannot hope to make precise here — you sometimes find that your mental exploration is hampered by the fact that there somehow aren’t enough points to hop to.
So life would be easier if these curves (and other objects) had more points. A normal person might say “Well, life’s not always easy. We’ll just have to get by somehow with the points we’ve got”. But Grothendieck lived by the conviction that everything is easy if you look at it right — which means there have got to be enough points. And if we think there aren’t, it must be because we haven’t yet figured out what a point is.
So — what is a point? Grothendieck’s insight was roughly that a point is a landscape with only one place to stand — or, a little more precisely, a point is a space where all functions are constant.
But wait a minute — what’s a function? It’s something with a domain — which in this case is our point — and a range, which is — what? Sometimes the range of a function is the rational numbers. Sometimes it’s the real numbers. Sometimes it’s the complex numbers.
This means, then, that there are many different kinds of points, depending on what kinds of (constant) functions they support. There are “real points”, where every function is equal to some constant real number. There are “complex points”, where every function is equal to some constant complex number. There are even points where every function is equal to some expression like (3x2+1)/(7x3+4). You might object that that’s not a constant — but it is, because the x in that expression is not a variable; it’s just a symbol, and that symbol always remains just x.
To Euclid, a point was just a point. Post-Grothendieck, a point has a great deal of internal structure, determined by the sorts of (always constant) functions that live on that point.
When you look at, say, the ordinary Euclidean plane, the points you see — the points that stretch out to infinity in all directions, the ones you familiarized yourself with in high school — are just the real points. But from a Grothendieckian perspective, that’s not the whole plane. There are also plenty of (invisible) complex points, (and those points, incidentally, can “spin in place”, ultimately because the complex numbers contain two square roots of minus one, which can be interchanged.) And there are plenty of far more complicated points besides. The plane is teeming with points you never learned about in high school.
It turns out that when you have all those extra points to work with, a lot of technical problems melt away, and you can solve a lot of problems you couldn’t solve before. Generalize sufficiently — allow the possibility that your notion of a point was always too specific and too cramped — and hard problems suddenly get easy.
Grothendieck rewrote the foundations of geometry not just once, but three times, first replacing classical geometry with his “theory of schemes” — the material we just touched on in Section II — and then going on to the “theory of toposes”, and finally the great unfinished “theory of motives”. One of the great goals of contemporary mathematics is to complete that final theory, which, if our expectations are correct, would give us the tools to settle many of the hardest outstanding problems in several areas of mathematics.
In this section, I’ll talk (once again in a vague sort of way) about the theory of toposes.
Let’s start over: What’s a point? Answer: A point is a landscape from which there is only one point of view. What’s a curve? It’s a landscape from which there are many points of view. What’s a surface? A landscape from which there are even more points of view. But what are we viewing?
The theory of schemes posits that we’re viewing the values of functions, which are constant on points, but can vary on curves. The theory of toposes posits that we’re viewing entire mathematical universes.
In classical mathematics, questions have unambiguous answers. Is 7 a prime number? Yes, unambiguously. Must the angles of a euclidean triangle add up to 180 degrees? Yes, unambiguously. How many fundamentally different kinds of symmetry are possible in a 2-dimensional design? Exactly 17, unambiguously. How many possible configurations are there for a Rubik’s cube? Exactly 43,252,003,274,489,856,000 as a matter of fact. Unambiguously.
That’s how the world looks when you’re standing on a point and therefore have only one point of view. That, then, tells us what a point is: It’s a place where the mathematical universe looks the way we expect it to. In some sense, we might as well say that a point is the classical mathematical universe.
(One little glitch: Points themselves belong to the mathematical universe, so if a point is the mathematical universe, then we’re in danger of something very like circularity. Grothendieck sidestepped this problem by defining the mathematical universe to contain all mathematical structures up to some unfathomably large size; the universe itself, being larger than that unfathomable size, does not count as a member of itself.)
What, then, is a curve? A curve is a place where you can move around, and look at things from many points of view. Is 7 a prime number? It might depend on where you’re standing. So we can identify a curve with a different mathematical universe, a universe that admits a certain amount of ambiguity — not at all the same as the classical universe we’re used to, but still a perfectly valid object of study.
And of course a different curve gives us still another mathematical universe, and a surface is yet another…
What’s the point (pardon the pun)? Two quite extraordinary things fall out of this viewpoint:
First, it turns out, miraculously enough, that when we see points and curves and surfaces as the homes for entire mathematical universes, we are able to use that insight to solve hard problems in classical geometry and arithmetic. We’re still studying the same old points and curves, but by recognizing that each of these points and curves supports an entire Universe, and by making good use of that insight, we can (rather incredibly) learn new things about the ordinary geometry of those points and curves.
Second, and quite independently of that, we now have a whole new family of mathematical universes to explore. Not only is that cool in its own right, but it sheds a lot of new light on the good old classical universe. For example, all of these universes satisfy many of the same basic axioms. So whenever we find a statement that’s true in one universe and false in another, we can conclude that our axioms are not enough to settle it. This yields vast new insights into the relative power of various axiom systems.
What I want to stress, of course, is not just the success of this viewpoint, but how daring it was. Who, before Grothendieck, would have dared to redefine a point, let alone identify it with the entire universe of classical mathematics? Yet this turned out to be exactly what was needed, both for solving old questions and for raising new ones.
There are at least three major themes running through all of Grothendieck’s mathematics. One, as we’ve seen, is his commitment to the principle that all problems become easy if only you can find the right generalizations. Another, as we’ve also seen, is his willingness to redefine classical objects like points and curves in order to make them more susceptible to being generalized. The third, which is equally central, is Grothendieck’s lifelong insistence that mathematical objects are intrinsically uninteresting — instead it’s the relations between mathematical objects that matter. The internal structure of a line or a circle is boring; the fact that you can wrap a line around a circle is fundamental.
Perhaps consistent with that philosophy — or, depending on how you look at things, perhaps in direct contradiction to it — Grothendieck seems to have devoted vast energy to meditating on his own place in the Universe, his role in history, his relations with other mathematicians, and the influences that made him the extaordinary man he was. Long after his retirement (and his disappearance to a remote village in the Pyrenees), he was producing 1000-page autobiographies, transcribing his dreams, and writing long letters, some of them devoted to visionary new mathematics, and others to the sort of rambling that led some old friends to believe he’d completely lost his mind. There are, apparently, another 20,000 pages of writings in a locked box at the University of Montpelier that might soon be opened.
If we can extrapolate from the fact that the unexamined life is not worth living, then Grothendieck led the most worthwhile life in history. In his early twenties, he wrote a doctoral thesis in the subject of “functional analysis” that provided new tools so powerful that they left almost no problems in that field left to solve. He then went off searching for a new field vast enough to contain his talent, and found it in algebraic geometry. In the days of the Seminaire de Geometrie Algebrique, he reportedly worked 18 hours a day, 7 days a week, 10 years per decade. And when he needed an even bigger subject, he brought that same vast energy to studying himself.
It has been the great privilege of my life to understand some small fraction of the mind of Grothendieck. The world is an infinitely more beautiful place because he lived.