by Jonathan Kujawa
Several weeks ago Alexander Grothendieck passed away. It is hardly possible to overstate his influence on twentieth (and twenty-first!) century mathematics. With the help of others he rebuilt vast amounts of mathematics from the ground up. He had a vision that still seems futuristic many decades later [1]. I compare it to Braque, Picasso, and company blowing up the art world with their entirely new vision of what art could be. In Grothendieck's case you'll have it about right if you imagine him as a visiting scholar from an alien civilization whose mathematics is to ours as ours is to one of those Amazonian tribes who can only count to three.
Grothendieck's life was as interesting as his mathematics. It's bound to be turned into one of those movies made to win Oscars [2]. His parents were anarchist political activists and artists, he moved to France as a refugee of Germany in 1938, and for most of his life was legally stateless and traveled with a Nansen passport. Grothendieck was at the peak of his public mathematical life during the 50's and 60's, receiving the Fields medal in 1966. Starting in the 70's he withdrew from the mathematical community and, ultimately, his family and friends as well.
For the last two decades he lived in a village in the south of France and only a very few people knew where he was. Grothendieck issued a letter in 2010 insisting that his work not be published and any existing publications be withdrawn from libraries. He was so isolated that it wasn't immediately clear to many in the math community if he was still alive and, if so, if he was the one who had written the letter.
Since that request seems to no longer be in force we should now have the chance to learn what did with himself for the past thirty years. There are rumors of tens to hundreds of thousands of pages of mathematics and political and philosophical writings. I'm sure I'm not the only one who had idle fantasies of running into Grothendieck at a cafe in France and getting on like gangbusters over espresso while hearing all about what he'd been up to [3].
Grothendieck's tools, language, and point of view are now ubiquitous across a broad spectrum of contemporary mathematics. They are certainly part of the everyday lexicon and mode of thought in my area of research (representation theory). Grothendieck reconsidered such fundamental questions as what is a “point” (there's a lot more to say than you might think!). For an excellent overview of Grothendieck's work I recommend Steve Landsburg's recent essay. It was linked to here on 3QD, but you may have missed it in the shuffle.
I did want to expand upon one point in Landsburg's essay: the high value Grothendieck put on understanding. For him it was completely unsatisfactory to answer a question by brute force or clever tricks. Rather, he took the view that if you truly and deeply understood, then the solution came without effort. The real goal was the understanding and, in the end, successfully answering the question should be viewed as the confirmation of your understanding. If you struggle with a question, then perhaps the real problem is that you don't understand the question well enough yet! You should expand and deepen your understanding like a rising sea until it engulfs the problem.
Grothendieck had a very nice metaphor for this point of view. Let me quote from Colin McLarty's essay “The Rising Sea” in which he quotes from Grothendieck's autobiography:
Grothendieck describes 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.”
In this “rising sea” the theorem is “submerged and dissolved by some more or less vast theory, going well beyond the results originally to be established”.
William Thurston was another greatly influential mathematician. Just before he passed away several years ago he expressed a similar view. Thurston wrote:
The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.
This echoes my own feelings and, I think, is the exact opposite of what most non-mathematicians would expect. Most would say you do math to get an answer. But to say the point of doing math is to do calculations and solve problems is akin to saying that the point of writing is phone books and TV repair manuals. Sometimes it's useful that way, but it's hardly the point!
While both Thurston and Grothendieck fundamentally changed mathematics, and both were known for leaving it to others to flesh out their details, Thurston was also known for being a generous and engaged member of the mathematical community.
The above quote is excerpted from an extended answer Thurston wrote to an anonymous undergraduate student on an online discussion board. The student despaired of having a role to play in mathematics unless they are an absolute genius [4]. Thurston, however, beautifully describes mathematics as a big tent with room for all who care to contribute. Which, of course, is good advice for all of us in whatever we do: find what we value and find a way to meaningfully contribute. As Thurston wrote: “It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics?”
[1] Indeed, 50 years after Grothendieck proposed them, we still don't have a firm grasp of motives. Or even know if they exist!
[2] Let's talk, Harvey Weinstein!
[3] The closest I ever came was when I gave a talk at the same chalkboard and in the same seminar room used sixty years earlier by Grothendieck at the Universty of Kansas.
[4] Sadly, storytelling seems to always prefer the myth of the lone genius (preferably with a touch of madness!) over the reality of the collaborative effort of the many. Call it Ayn Rand and the “Great Men of Science” myth.