There are two styles in mathematics, said Alexander Grothendieck, a titan of twentieth-century math. Picture a theorem as a hard nut, the mathematician’s task to open it. One way would be to hit it with a hammer and chisel until it cracks, but another way, and this was his preferred way, was to sink the nut into water. “From time to time you rub so the liquid penetrates better, and otherwise you let time pass,” he wrote. Eventually the nut opens easily, practically on its own. Math advancing through a series of imperceptible chemical reactions.
Or, he added, you could think of a sea rising, washing over hard earth until it softens.
It was Grothendieck who softened the shell of one part of the Weil conjectures by developing the right sort of mixture in which to dissolve them. Where Weil was fastidious and distrustful of big machinery, Grothendieck was the field’s abstract expressionist, an otherworldly, romantic figure who emerged from a lonesome disaster of a childhood and made a haven of mathematics, mounting large, revolutionary canvases, only to resign at forty-two and return to a life of isolation. Another mathematician would compare him to Simone Weil, noting that “his life was burned by the fire of the spirit.” Like her, he was an ascetic, and like her, he believed that math was an attribute of God.
André Weil wasn’t an Alexander Grothendieck (who left math to become a hermit), or a Georg Cantor (who was in and out of asylums), or a John Nash (a.k.a. the guy from A Beautiful Mind), not one of those troubled men who seem to arouse the most curiosity from outside the field and who contribute to the image of the great mathematician as an unhinged genius. André was a great mathematician who also happened to be sane—and irascible, prank-loving, imperious—a married father of two who lived a long and productive life.
But the more I learn about him and his sister, the more I begin to wonder whether his sanity somehow implicated his sister’s extremity, whether in the Weil family, the two roles were divided between them: he would be the great mathematician, and she would come unhinged.
In Kyoto, André Weil gave an acceptance speech. I picture, at the podium, the chassis of a human thinker. A cranium in glasses, the mound of his forehead made larger by the retreat of his hairline, the rest of his face thinner than ever. In his tuxedo, with his little black bat of a bow tie.
He’d been asked to give a lecture of a personal rather than a technical character, he said. And so he recalled his childhood, the early love for math that had possessed him—when it came to his career, he said, “There was no choice on my part”—his great luck in being let into Hadamard’s seminar, his travels. Then he spoke of Bourbaki, how it came about and conducted its business, even though by then the grandest dream of Bourbaki, to produce the modern equivalent of Euclid’s Elements, had receded.
In his later years, he said, he had come to dwell more and more in history, which had not only occupied his time but given him a kind of social life. An imaginary social life, that is: as he immersed himself in the writings and correspondence of the great mathematicians of the past, men like Pierre de Fermat and Leonhard Euler had become “personal friends,” he said. Their companionship had brought him happiness in his old age.
“Will such a statement edify and enlighten the present audience?” he asked at the close. “I am inclined to doubt it, but at my age, I fear it is the best I can do.”
The mathematician walks off the stage.
The Weil conjectures were invoked by Barry Mazur, in a 2005 dialogue published in a humanities journal, to illustrate a larger point he wanted to make about mathematics: “Sixty years ago, André Weil dreamt up a striking way of very tightly controlling and counting the number of solutions of systems of polynomial equations over finite fields,” he said, referring to the conjectures. Weil did so by proposing that a method could be developed in number theory analogous to one in to pology for counting intersections of geometric subspaces. And this, said Mazur, goes to show that no mathematics, not even number theory, is divorced from our geometrical intuition. That is to say no mathematics is entirely cut off from the sensual.
Nor is it entirely cut off from the people who devised it. What Mazur’s language suggests, beyond the broadest outline of a mathematical result, is conveyed in that phrase “very tightly controlling and counting”—hinting at Weil’s mathematical personality, which was at once disciplined and visionary. I see him as a commander on the battlefield, a great strategist if at times a difficult person. “What makes his work unique in the mathematics of the twentieth century,” wrote his friend and peer Jean-Pierre Serre after Weil’s death, “is its prophetic aspect . . . combined with the utmost classical precision.”
Maybe the dream of pristine writing, in which the writer is present but not present, masked behind the light of her brilliant transmission, is realized in these works I only dream of reading.
One August evening, a few months after sending the e-mail to the mathematician who never replied, I spot him at my local supermarket, headed for the bulk foods section. I push my cart in that direction but can’t muster the will to go up to him, not just because of the unanswered e-mail but because a store employee is vacuuming the trays under the bulk food dispensers with a very noisy machine, also the mathematician is wearing earbuds, so I can only imagine an interaction with him as a desperate exchange of hand gestures. I hesitate, realize he’s no longer among the bulk bins, try to turn around, but a wheel of my cart gets stuck on the vacuum cord, and by the time I work it free I figure him for lost.
I give up—for half a minute or so—and then I think, No, dammit, I’m going to find that guy and talk to him. I begin scanning the aisles, one by one, and finally corner the poor man by the tortillas. If there’s one thing I had plenty of chances to practice during years of work as a journalist, it’s approaching somebody I feel quite shy about approaching. Yet I’m not sure whether repetition has made it any easier. My smile feels like a sticker slapped onto my face.
Yes, he says, you sent me an e-mail. I’m sorry that I never replied—
Oh that’s okay, I say. E-mails! Maybe there’s someone else in your department that I could to talk to—
I’m probably the one, he says, slowly, not with any evident enthusiasm.
In other words, it’s pretty awkward. What do I even want from him? I’m about to leave town for a couple of weeks, I tell him, and we agree, tentatively, that we’ll have coffee once I’m back.
A very broad, bird’s-eye overview of the development of twentieth-century mathematics (which I’m mostly borrowing from a lecture given by the mathematician Michael Atiyah) goes like this: In the first half of the century, the prevailing concern was to define and formalize things, as in the Bourbaki effort, and to pursue work specific to the different subject areas within mathematics. Then in the second half, mathematicians looked for ways to tie together those different areas, to transfer techniques from one to the other, and math became more global, even as it expanded so much that it became impossible for anyone to fully comprehend all that was happening in the field.
So one can faithfully call André Weil a pivotal figure in twentieth-century mathematics. He helped to found Bourbaki in his early career, emblematic of that era’s determination to ground and make rigorous the subfields of math, and not long after that he began to envision connections between mathematical land masses, which his successors would build out.
It’s very hard to conceive of how mathematicians from a long time ago thought about things, Atiyah said in that same lecture, because subsequent discoveries have become so ingrained. “In fact if you make a really important discovery in mathematics you will get omitted altogether!” said Atiyah in that same lecture. “You simply get absorbed into the background.”
Another reckless dash through twentieth-century math might emphasize that as time went on (at least in some camps), identity became less important than relatedness. Theories were developed in which knowing relationships among mathematical objects matters more than knowing about the mathematical objects themselves. Or you could even say that knowing an object itself is the same thing as knowing its relations to other objects of the same kind.
A mathematics increasingly globalized, increasingly concerned with links from one thing to another, increasingly aided by computers, a highly connected world in which it’s still very difficult to see much more than your own small part—remote as the world of math is, this all begins to seem familiar. Like that forest of links in which I keep losing my way.
But there’s a second part to my supermarket chat with the mathematician, in which I learn that he has a son the same age as my son, and we take refuge in that, as parent-strangers will do. His son (go figure) likes math too. We compare notes about math-related books for kids, and somehow that makes it seem more possible that we will actually at some point get together and talk about the Weil conjectures.
“I can remember Bertrand Russell telling me of a horrible dream,” wrote Godfrey Harold Hardy in his book A Mathematician’s Apology. “He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of [Russell’s own] Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated . . .”
The mathematician is required to disappear.
Goro Shimura saw André Weil for the last time on a December afternoon in 1996, at the Institute for Advanced Study. Shimura, by then a professor at Princeton, had tried to speak to Weil by phone the day before, but Weil couldn’t hear very well, and he’d asked Shimura to meet him at the institute instead. Come tomorrow, he said, otherwise I won’t remember.
It had been drizzling endlessly, Shimura would later write. Forty-one years had passed since they first met in Tokyo. When he arrived at the institute, Shimura discovered that Weil had forgotten his hearing aid, and they drove to his house for it. Yet even with the device Weil couldn’t hear well, and at lunch their conversation was halting and uncomfortable.
Shimura was working on a problem; he’d had a new idea about the Siegel mass formula, a topic that Weil had once studied, but when Shimura asked about the history of the subject, all Weil could say was, “I don’t remember.” He kept saying the same thing over and over: That was a long time ago, I don’t remember.
Shimura asked whether he was still writing history.
I cannot write anymore, he said.
They left the dining hall and walked through the drizzle to Shimura’s car.
You are certainly disappointed, but I am disappointed too, Weil said, with myself.
And then he said again, I cannot write anymore.
This episode appears in a reminiscence that Shimura wrote after Weil died, for the Bulletin of the American Mathematical Society. Like his earlier “Yutaka Taniyama and His Time,” it’s a beautiful tribute, and just based on those two essays I’m ready to pronounce Shimura the great elegist of twentieth-century mathematics.
Is that what I am writing, I wonder, some sort of elegy for math, or for my own entanglement with math? At times it feels that way, but I don’t think that’s what this is. As it turns out, one stilted encounter in the supermarket is enough to send me back to the algebra lectures, which for no good reason I still want to finish. And so it’s on to ring theory, which is of course nothing I need to remember, nothing I need to know.
A mathematician might dream of an afterlife in which all is revealed, mathematical structures extending as far and wide and high as the eye can see, their nature and relations made transparent, and look, there are Galois and Archimedes strolling by, there Germain and Taniyama deep in conversation. And in that paradise, maybe, Simone and André would be reunited; they would find each other and argue in ancient languages all through the never-ending day.
Their earthly afterlives are, in a sense, opposites: the image of Simone the person weighs upon her writings, inseparable from them, even overtaking them, while André grows ever more attenuated. His person will vanish—be absorbed into the background—and leave behind just a name, a quartet of letters attached to other symbols, theorems and conjectures.
Where are numbers? my son asks. And where, for that matter, are all the unknown theorems, all the hidden proofs, all the math not yet discovered?
Thirty-five hundred years ago, a Babylonian presses a wedge into clay. One, two, three times. Then leaves a gap. Then presses again.