The Weil Conjectures

by Karen Olsson

  A rough draft. A trial balloon. It seems to me laced with optimism, a bullishness about what could, in the future, come more fully to light.

  André, now a professor at the University of Strasbourg, travels back to Paris and convenes a meeting of young mathematicians at the Café A. Capoulade. They all teach similar undergraduate courses in analysis—higher-level calculus—and they are dissatisfied with the existing textbooks, so they’ve decided to form what they call the Committee for Writing a Treatise on Analysis.

  This rather dry mission soon takes a strange turn. The Committee for Writing a Treatise on Analysis will evolve into a semisecret society, a sort of wry mathematical mystery cult.

  My math fever dream lasted for two and a half years, which were spent messing around on the lower rungs of a tall lad der that stretched into the clouds, that led to a cloud land of tantalizing abstract structures, curves and surfaces and fields and vector spaces, accessible only to those who learn the elaborate cloud language, a vehicle for truths that cannot be expressed in any other tongue.

  Then I stepped off the ladder and walked away. It’s not often that I experience even a passing wish to go back, and even in those moments the allure of math is much fainter than it was in college, since now I have no illusions that I would ever make it very far up—I’m left to imagine that land, and what I wish for now is less the specific math knowledge than a certain constellation of feelings that came with it.

  Simone goes to work in a factory because she wants to investigate directly what she hasn’t been able to figure out theoretically, namely: How can an industrial society be organized in a way that its workers are not oppressed? And then there’s that self-mortifying impulse, the fact that she has wanted to do hard physical labor since she was a teenager. She tells her friend Simone Pétrement (who will later become her biographer) that although she’s scared—she is notoriously clumsy, not good with her hands—she is determined to kill herself if she can’t manage the work.

  She is twenty-five. Her confidence, Pétrement would write, “was quite terrifying, especially when one knew her almost inhuman energy and her lack of self-pity.”

  Into a den of machines. A woman regularly overcome by headaches assigns herself to the factory floor with its stamping press, the fly press, the iron crank, the compressed-air hose, the screws, the blades, the mallets. The screech and whine and hammer and hiss. She is assigned to fit copper plates into magnetic circuits and must take care not to ruin the pieces in the process.

  The days are short, and when the windows go dark the wan lamplight hardly compensates. She can barely see. It’s loud and it’s dark and she can’t keep up with what the foremen demand, the required rate of work. Holding a flashlight in one hand and adjusting newly assembled parts with the other. “Manual labor,” she’ll write later. “Time entering into the body.”

  Wednesday, she doesn’t make the rate. Thursday, she doesn’t make the rate.

  She imagines that the others pity her, but then she sees how they side with the foreman when another woman is fired. The fired woman was tubercular, her husband unemployed—but she botched a job and hundreds of pieces had to be done over. She ought to have known better, the others comment. “You’ve got to be more conscientious when you have to make a living,” they say. Even on the night shift, even in the near darkness.

  “What I went through there marked me in so lasting a manner that still today when any human being . . . speaks to me without brutality, I cannot help having the impression that there must be a mistake,” Simone will recall.

  She is assigned to a small workshop, separate from the rest of the factory, and instructed to insert copper bobbins into a furnace and then take them back out again. She burns her hands and her arms, comes out blistered and scarred. Yet it’s her favorite part of the factory, because the workers there are decent to one another.

  The young mathematicians hold their first conference in the summer of 1935, near Lac Pavin, a volcanic crater collared by pine forest in central France. Seven months after forming the Committee for Writing a Treatise on Analysis, they’ve advanced well beyond their original intention to improve undergraduate math education. This has been supplanted by a much grander goal, no less than laying a formal, consistent, and comprehensive foundation for all of modern mathematics—or at least all of it that they find interesting.

  Taking a break from a stalled discussion of analytic functions, several members of the group flee to Lac Pavin and dive naked into the water, yelling out a Greek name: Bourbaki! Bourbaki! Bourbaki! Over and over again, their shouts echoing across the surface of the lake: Bourbaki! Bourbaki! Bourbaki! Bourbaki!

  One seed of this happy outburst was planted in 1923, back when André and many of the others at the conference were studying at the École Normale. A third-year student named Raoul Husson played a prank on the first-years by posing as one Professor Holmgren. Wearing a false beard and speaking in an indefinite foreign accent, he gave a lecture on a series of made-up results, which culminated in the presentation of “Bourbaki’s theorem.”

  And, when he was living in India, André had advised a young mathematician friend by the name of Kosambi, who was caught up in an academic rivalry with another man, to flummox his competitor by publishing an article about an imaginary Russian mathematician with a Greek name. Kosambi did just that; his (spoof) paper “On a Generalization of the Second Theorem of Bourbaki” appeared in the Bulletin of the Academy of Sciences of the United Provinces of Agra and Oudh.

  The group at the summer conference adopts the name Bourbaki, a mischievous pseudonym that also serves a purpose, in that they’ll avoid having to sign every single one of their names to publications. In the fall, André will submit to the journal Comptes Rendus a perfectly serious article concerning the theory of integration, yet he’ll report that it was passed on to him by a man named Bourbaki. “I’m sure you’ll recall that Mr. Bourbaki is the former professor of the Royal University of Besse-en-Poldévie whom I met some time ago at a café where he spends most of his day and even the night, having lost both his job and most of his fortune amid the troubles that caused the unfortunate Poldévian nation to disappear from Europe,” André will write to the journal editor, a fellow mathematician who is in on the joke. “Now he earns his living at the café by giving lessons in belote, the card game he plays so brilliantly.”

  There was, by the way, an actual Charles Bourbaki, a nineteenth-century French general of Greek heritage who distinguished himself in battle and as an inspector general of the infantry before he was sent, in 1870, to command a miserable, half-starved French army already losing to the Prussians. After a defeat at Héricourt, in eastern France, he was forced to retreat through Switzerland, where his soldiers’ guns were confiscated and he tried to kill himself. The attempt failed, and he eventually returned to France.

  The brief Kafka story “The Top” is about a philosopher who loiters around children because he is fascinated by their tops and hopes to catch one in mid-spin, convinced that a spinning top could lead him to enlightenment—that “the understanding of any detail, that of a spinning top for instance, was sufficient for the understanding of all things.”

  As a top spins, the philosopher goes “running breathlessly after it,” full of hope, but upon catching it he is disappointed, for “when he held the silly piece of wood in his hand he felt nauseated.” It’s the pursuit, the running after knowledge, that takes his breath away.

  Commenting on this story in the preface to her book Eros the Bittersweet, Anne Carson doubts that the desire to gain knowledge is what really motivates Kafka’s philosopher. “Rather,” she writes, “he has become a philosopher (that is, one whose profession is to delight in understanding) in order to furnish himself with pretexts for running after tops.”

  Running breathlessly after tops, in that heightened state that comes of being deprived not only of the thing you’re chasing but of the air you need to chase it. Sometimes I think of writing in this light, too: as running af
ter tops.

  A partial list of women Simone encounters at the factory, as recorded in her notebook: Mme Forestier, Mimi, Cat, Mimi’s sister, a coworker on the iron bars (Louisette), a blonde from the munitions factory, a redhead (Josephine), a divorced woman, a mother of a burned child, a woman who gave her a roll to eat, an Italian woman.

  Some of the men: a violinist, a conceited blond, an old man with glasses, the singer at the furnace, the worker in drilling goggles, the boy with the mallet, a young blond Italian, a welder, a coppersmith.

  The grooves in the forehead of her supervisor, Mouquet, look to have been carved there by one of the machines. He is tormented by invisible forces and torments the workers in turn. One day Simone is told to redo a long metal-polishing job, which makes her furious. Is the demand justified, or is it bullying? She can’t even tell. The way Mouquet instructs her to go about it, she has to duck a counterweight that swings toward her every time a piece is polished.

  Hindering her productivity, she believes, is her contemplative nature. “I am still unable to achieve the required speeds, for many reasons: my unfamiliarity with the work, my inborn awkwardness, which is considerable, a certain natural slowness of movement, headaches, and a peculiar inveterate habit of thinking, which I can’t shake off.”

  She feels disgusted by the exhaustion and by how little she’ll earn from her slow work. She’s disgusted by herself, her identity worn down as though in the polishing machine. Oppression, she concludes, doesn’t foster a spirit of rebellion but a kind of docile slavery.

  And one’s inner life, she writes in her notebook, is merely a kind of temptation. One shouldn’t indulge in emotions unless they are useful. Can a person experience feelings without imagination, without projecting them into the past or the future? This seems to be her aim. “Above all, never allow oneself to dream of friendship. Learn to reject friendship, or rather the dream of friendship.”

  Determined to live off what she earns, she insists on paying her parents when she dines with them. Her mother responds by stashing coins around Simone’s apartment, knowing that Simone is absentminded and won’t realize she didn’t leave them there herself.

  “She’s killing herself, and she doesn’t listen to me,” her father complains.

  “It is not by chance you have never been loved,” she writes in her notebook.

  André’s memoir doesn’t offer any details about how he met and wooed his eventual wife, Eveline, perhaps because when the two of them became lovers she was married to—and had a son by—one of his colleagues, René de Possel, another of the original members of the Bourbaki group. The story of how André courted her is, must be, a story of betrayal.

  Let’s say it happens at that first conference, in the summer of 1935. Young professors wandering through fields of wildflowers, sunning themselves in chairs. Mountains in the distance. He and she cross paths in the dining room, in the garden. She is short and broad-shouldered, curvy, with wide eyes and an easy smile. Maybe she always wanted to marry a genius and has not quite found one in her first husband. Is it that, André’s intellect? Or maybe his genius in combination with his nearsightedness, the way he is so cocky but can’t figure out where he’s left his pen. Wiry in a way that makes him look taller than he is, worldly, imperious and mocking around the other men, but when she finally works up the nerve to say hello one morning, he blushes. He stares at her as he talks to her.

  Stray hairs blowing forward, into her mouth, distract him.

  He isn’t wearing a tie, and she likes his open shirt, his skin.

  Words come and go, which is to say they speak them while taking the measure of each other’s bodies, shyly looking and then looking away.

  He isn’t wearing socks either, she notices.

  He eyes one of the silver buttons of her blouse, thinks about those buttons all afternoon.

  I suppose I’ll see you at dinner, he says.

  It was good to meet you, she says.

  As he walks away, he stumbles over his own feet.

  A second-rate mind like de Possel doesn’t deserve her—that’s what he decides. Over the next few days he keeps a furtive eye on Eveline and René and convinces himself that there is no real warmth between them, only schedules and habits, dry kisses on the cheek.

  Within days André has lured Eveline out to the lake. Sneaking off in the dark, they ride bicycles to the edge of the crater, where the lake is ritzy with starlight, and he tells her this and he tells her that, how much brighter the stars are in India, let’s say, and then next thing they know they’ve entered the water. Her camisole becomes transparent, an aquatic creature daubed across her heavy breasts. Their wet skin, the warm wet rippling night. Afterward, on the shore, she wants to wait until her hair dries before they go back. As though wet hair were the only giveaway.

  In eight months of line work at two different enterprises, Simone finds no answer to the problem of social organization—that is to say, she concludes that there is no answer, that she has become a slave among slaves. By the time she leaves the factory for good, she’s utterly spent. She hatches a plan to travel by freighter down the Spanish coast, but along the way her health declines, and her parents take her to a seaside town in Portugal to rest. There, one evening as she wanders through the village alone, she has the first of several mystical encounters that impel her toward Christianity.

  For the festival of the local patron saint, the village women have formed a procession. They wend their way around the docks, carrying candles and singing hymns permeated with “heart-rending sadness,” as Simone will later write. A thought—a conviction—comes to her: that Christianity is the religion of slaves.

  She will say that not only the Christian religion but elements of other religious traditions and the beauty of the world and its reflection in great works of art delivered her “into Christ’s hands as his captive.” She becomes an idiosyncratic almost-Catholic, at once mystical and scholarly, skeptical of the church and declining to be baptized but believing herself possessed by Christ, believing the invisible world to be more real than the visible one.

  Carson’s Eros the Bittersweet takes up the idea that when ancient Greece became literate, developed a culture of writing and reading, which set in around the seventh or sixth century B.C., its collective mind was blown. Literacy comes with its own psychology, Carson suggests: to become a reader you must learn a new kind of self-control. You have to shut out the world in order to focus your attention, and in so doing you develop a new awareness of the interior self.

  Distant objects or people, represented by symbols on a page: we are so accustomed to this that it’s hard to conceive of a time in human history when it was a new phenomenon. Carson compares this representing, the conjuring of things by written words, to the way that a lover constructs a mental image of an absent beloved. Desire spans the distance between the abstract thought and the actual person. Everything is triangulated—the lover, the beloved, the image. The writer, the thing, the word.

  “A mood of knowledge is emitted by the spark that leaps in the lover’s soul,” she writes. “He feels on the verge of grasping something not grasped before.” It’s not the knowledge itself, not consummation but the mood, the excitement when you are on the verge of grasping.

  Skeptical though she is of political idealism, Simone never relinquishes her passion for self-sacrifice. In 1936 she travels to Spain to join the republican fighters in the civil war, less out of devotion to the cause than in the belief that once war cannot be prevented, one must submit to one’s share of misfortune. Posing as a journalist, she reaches the front and manages to join a small international commando group.

  Simone is nearsighted and inept with a rifle, terrifying the others every time she has the weapon in hand. Her captain assigns her to kitchen duty, but she still finds herself, one afternoon, stretched out on her back while nationalist planes fly overhead, gun pointed toward bombers in the sky. Once again she’s put herself in a situation that’s nearly intolerable yet brings it
s own hard thrills. Absolute visibility: she’s pierced by the sight of the summer sky as she lies there, grappling with a heavy gun she can barely operate and beset by a fierce headache, by beauty and terror and pain. The planes are too high to shoot, and the soldiers scramble back to camp.

  Even the kitchen has its hazards. The comrades start their cooking fire in a hole, so that the flames won’t give their position away, and one morning Simone steps right into the hole, her leg landing in a pot of boiling oil. As another woman removes Simone’s stocking, her skin peels off with the wool. She is taken to Barcelona and then to a hospital south of the city, which likely saves her life, as several weeks later most of the international group, including all of its remaining women, are killed in an engagement at Perdiguera.

  During the siege of Syracuse by the Romans, circa 212 B.C., a Roman foot soldier encountered Archimedes at the shore, studying a diagram he’d drawn—according to one account, he was so engrossed by a problem he hadn’t even registered the Roman invasion. The soldier ordered Archimedes to follow him, but he was seventy-five by then, and maybe he didn’t hear so well, or maybe he told the soldier that he would come along once he finished working through the problem. Although the Roman command had put out word that the famous mathematician should be spared, the angry soldier slew him on the spot.

  Strategies for tackling problems, from Pólya’s How to Solve It: Do you know a related problem? Look at the unknown! Here is a problem related to yours and solved before. Could you use it?

  But what about the problem of too many related problems? My weakness for juxtaposition: I’ll sense that one thing might be illuminated by another thing and go chasing the other thing (Thing Two, like one of Dr. Seuss’s little devils in The Cat in the Hat) off in another direction. For better or worse, a light paranoia goads me along. Maybe it’s all connected! This and this and this and—look, over there—that. There’s the bringing together of disparate elements that informs a conjecture, and then there’s the mental nausea brought on by the fact that there’s too much out there to know. Not grasping but googling. I can’t always tell one from the other.