The Weil Conjectures

by Karen Olsson

  They don’t venture far from their hotel until the next morning, when they board a train to Kyoto. As Sylvie is dressed all in black, André cracks that she is properly attired for his professional funeral. He then stares at a timetable, though he can’t read the Japanese, until his eyes start to close. Much as he insisted on coming to receive his prize in person, now the thought of sitting through the ceremony enervates him. Lately he has a way of slowing down and then reviving, unpredictably; a strong smell or a flash of light will cause the unoiled gears of his mind to turn more quickly, and he becomes himself again, the learned professor. It’s as if he’s been given some temporary drug that makes him ten years younger, and for thirty minutes, an hour, until it wears off, he’ll be witty, erudite, expansive.

  At their next hotel, a retinue of local mathematicians and interpreters meets them at the door and welcomes them into the lobby—a giant clamshell of medicine-pink marble. André takes it all in, not just the marble but the gilded furniture, the heavy drapes, the sweeping staircase, the uniformed women at the reception desk, and in that moment Simone returns, it’s Sylvie whose arm he’s clutching but he can feel Simone there too, and his outrage is also hers, he’s protesting on her behalf.

  This is out of the question! he bellows, then initiates the arduous procedure of turning himself back toward the doors. Absolutely not, he says. I could not possibly linger for even an hour in an establishment for nouveau riche businessmen such as this.

  He squeezes his daughter’s arm even more tightly. His hosts smile, obviously panicked. One of them goes straight for a telephone closet. But Simone—not Simone, Sylvie—stops him.

  Yes, it’s ridiculous, she murmurs back, very crass, but I would find it so amusing to stay in an overpriced hotel. As an anthropological study.

  A girl again, asking him for chocolate.

  Please?

  Later on, considering the menu of the hotel restaurant, Sylvie wonders whether it was the right tactic. They can’t afford to eat there, so she asks a receptionist to recommend something more modest, which initiates an excruciating round of giggling and bowing and missed signals, then at last the drawing of a small, perfect map, to a place across the river. Father and daughter creep down a dark, narrow street of red lanterns. He reminisces about his past trips to Japan, and for a spell she is reminded of walks they took when she was a girl, her father’s voice the very sound of security. The restaurant, though, seems as dark as the street, and he squints and scowls and is displeased to be told the seating is on the floor. First he has to get himself all the way down there, and then it’s uncomfortable for him to sit like that, and although the food costs less than at the hotel restaurant, the prices are still out of reach. They dine on soup and rice.

  He refuses to bow to anyone, lashes out at the waiter.

  Would it be so hard for him to be polite? Sylvie asks.

  May I remind you, he says, that your own talents would never get you an invitation such as the one that has brought us here.

  In the silence that follows, she thinks of how he used to send her long letters during her student days in Paris, typically full of advice about her studies, though some of them were made up entirely of airline and ocean-liner schedules, which he’d copied so that she could plan her vacation trips to Princeton. When she came home, he would be holed up in his office in the basement, not to be interrupted, the eminent mathematician at his desk. Only the cat was allowed to keep him company.

  They struggle back to the hotel and do it all over again the next day.

  The idea that math is immortal, that its discoveries accumulate over time but that its truths are outside of time, is implicit in its everyday language of theorem and proof, all those statements made in the eternal present tense. But how can that be? How can math be timeless even as everything that underlies it—the historically specific ways that concepts are described, manipulated, and proved—shifts over the years?

  Proof, that seeming ironclad warranty, is at the end of the day a rhetorical device, a method of persuading others of your conclusion. Proof in itself is hardly immune to history. It has evolved over the centuries, finding different means of expression and adhering to different standards of rigor.

  But then again I’m not going to sit here and say that math is not timeless.

  L.E.J. Brouwer, he of the nudism and the fixed-point theorem, had a change of heart regarding proofs. Influenced by his philosophical investigations—beginning in the 1910s, he argued that math was a mental construct independent of language—he decided to reject the method of proof by contradiction, upon which a vast number of math theorems rely, including his own fixed-point theorem. This method, a.k.a. reductio ad absurdum, depends on what Aristotle termed the law of excluded middle: something is either true or not true. You assume a proposition is not true, and if that leads to an absurd or self-contradicting consequence, then you conclude that the proposition must be true.

  There was something unsatisfying about that kind of argument, Brouwer came to believe. He wished to establish math on a purely constructive basis. Which meant that if you wanted to prove that something (such as a fixed point) exists, you couldn’t merely show that its nonexistence would imply the impossible, you had to give a prescription for somehow finding the thing.

  His program attracted a few followers, but in the main, proof by contradiction lives on. And proofs have become even more oblique; nowadays it’s not uncommon for them to run to hundreds of pages. Some require the verification of so many different cases that you need a computer to complete the task, so that the role of the human is to write the code rather than the proof.

  Might there be some kind of wiggle room between true and false; in other words, can something not false be, at the same time, not quite true? Logicians have their undecidable propositions and what they call many-valued logics, while if we’re talking about ordinary life, then easily yes, this is the case for all sorts of exaggerations and understatements, not to mention innumerable “I love you”s uttered in fraught circumstances. And this can pertain to ideas too, can’t it? To conjectures that will never be proved? Maybe even to the idea that math is timeless.

  I imagine the ghost of André Weil would have something acid to say about all this, that he would disdain my having ventured to write about him. In his later years he was clear about which people he thought were qualified to tell the story of mathematics and its pioneers, and those were mathematicians themselves—if not active mathematicians, then people of “better than average mathematical talent” who were in close contact with active mathematicians. Their proper purpose, moreover, was to aid working mathematicians in their research, insofar as revisiting the classics could inspire further advances. During his time as a member of Bourbaki, Weil had contributed historical notes to Éléments de mathématique, and when he neared retirement age, recognizing that his mathematical creativity had diminished, he took up the writing of history in earnest.

  Around this same time—the 1970s—he assailed at least two historians of math who weren’t trained as mathematicians. The first was a Princeton professor and the author of a book about Fermat, which Weil demolished in a review for a math journal; the second was a Romanian-born Israeli historian named Sabetai Unguru who could be every bit as vicious as Weil. Late in the decade, Unguru and several mathematicians threw punches back and forth in a forum that I’d guess was normally rather more genteel: a scholarly journal called Archive for History of Exact Sciences.

  Mathematicians who resorted to writing history after they’d become “professionally sterile” in their own fields, proclaimed Unguru, would never see past their own biases; they couldn’t help turning their chronicles into oversimplified narratives of progress. Men like Weil, in other words, were sorry has-beens and bad historians. Weil counterattacked, describing Unguru and his ilk as parasites, pretenders equally ignorant of history and mathematics.

  The mathematicians’ view of the past tended to be slanted by, and served to reinforce, the Platonic
ideal of math as an eternal structure somehow immune to historical forces, Unguru wrote. I think that was probably a fair criticism of how Weil regarded math’s history, however low the potshots at older mathematicians.

  But he always relished a good fight. After Weil died, Shimura would remember an incident from 1957 or 1958, when they both had positions at the Centre National de la Recherche Scientifique in Paris—Weil on sabbatical from the University of Chicago, Shimura a visiting researcher. One day he sought out Weil in his office to deliver a message. At the door of the office Shimura heard Weil and another man yelling at each other. Shimura knocked, the shouting stopped, the door opened, and Weil introduced him to his shouting partner, a visiting mathematician from Johns Hopkins. When Shimura left, Weil and the other man started up again. Shimura went on to the library, where he spent about half an hour. Upon his return he could hear that Weil and the other man were shouting still.

  In what way do geometrical demonstrations in the books of Euclid belong in the same category as byzantine proofs devised by twenty-first-century computers? Maybe we could think about mathematics, the historian Moritz Epple has suggested, by analogy with music. If we hear a musical composition from centuries ago performed, we identify it as the same piece of music, although everything but the notes— the instruments, the interpretation, the context, our whole manner of listening—has changed.

  An audience with a princess. A dull bus ride to look at a temple, marvelously restored, they tell him, though all he can make out is that the building has been smothered in gold leaf like the girl in that James Bond movie. An interminable reception, where the champagne is too cold and he can’t locate a chair.

  André recalls a story about his contemporary Paul Erdős (a childlike vagabond, too fond of amphetamines, about whom there are many stories): that he once attended a party along with some colleagues at the house of another mathematician, and when they were well into the evening the others realized that nobody had seen Erdős for the last hour or so. A search began, and they discovered him upstairs, talking with the host’s blind father, who had been sitting alone in a room when Erdős came upon him and decided to keep him company.

  Now André is the old man, not blind but weary of looking, and though he and Erdős were never close, what he wouldn’t give for his companionship just then.

  If only. If only he’d cracked the Riemann hypothesis. If only Eveline were here.

  Then the awards ceremony, inside a grand hall where women’s bright kimonos stream like fireworks through a sky of tuxedos. A multitude of dignitaries and translators and assistants and photographers navigate to and fro. The floodlights, they tell him, are for the sake of the photographs. They are blinding me, he says, and someone digs up a pair of huge sunglasses, which he puts on over his other glasses.

  The prize has been granted to three men: André, a hulking American chemist, and the film director Akira Kurosawa, escorted by his own daughter, who wears a suit even pinker than the hotel marble. Kurosawa and his daughter are a head taller than André and Sylvie, and elegant, at ease in their own country, seeming to preside over the table where they are all sitting, even over the ceremony itself, where the director is the crowd favorite, at least for the half of the crowd that stays awake. Maybe that’s why, at the reception afterward, André leans over to Kurosawa and says, “I have a great advantage over you. I can love and admire your work, but you cannot love and admire my work.”

  Suspended from their necks by thick blue ribbons are gold medals inlaid with synthetic jewels, sapphires and emeralds and rubies. In a photograph taken of André at the event, Sylvie will later write, he looks like an entry at a livestock fair—like a prizewinning pig or cow.

  On their final night in Japan, André and Sylvie dine with a long-ago colleague, Shokichi Iyanaga, and a Mr. and Mrs. Satake who’d been helping the Weils in Tokyo. After dinner, they wish him a safe trip and hope there will be an opportunity to invite him back to their country. André moves his jaw a bit without speaking. Iyanaga, who is eighty-eight, the same age as André, waits for him, remembering how he used to pounce before Iyanaga had finished a sentence.

  At last André says, “The next time perhaps in another world.”

  14.

  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 de
veloped 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.