The Weil Conjectures

by Karen Olsson

  The ancient world left a record of at least one female mathematician: Hypatia, the daughter of Theon of Alexandria. She wrote commentaries on the mathematical works of Diophantus and Apollonius and then died in A.D. 415 at the hands of Christian fanatics who, because she refused to give up her pagan beliefs, dragged her through the streets and scraped the flesh from her bones with pottery shards or possibly—the relevant Greek word has more than one translation—with oyster shells.

  Simone attends Bourbaki conferences, though she can’t follow most of the technical discussion. A photo taken in 1938, in Dieulefit, shows her standing in front of a doorway with her brother and five other mathematicians. She is the only one looking away from the camera, as though she knows better than to claim a place there.

  The Bourbakians often yell back and forth, cut one another down. Shut up! No, you shut up! During one of their meetings, at a hotel in the French Alps, an employee wonders whether to call the police after hearing one of them threaten to throw another out the window.

  She does her best to keep up, smoking, thinking, pacing back and forth in the rear of the room while the men attack their math, attack one another. During the breaks she hits up André with questions, while he stirs sugar into his coffee, maybe wishing a little that she would go home and at the same time finding a familiar satisfaction in continuing to teach his first pupil. (Besides, what home does she have?) Then again sometimes her questions jab serendipitously in the right direction, cause him to reconsider some ignored assumption. To the others it’s startling to see his same glasses, his same face attached to this body clothed in an unstylish dress and an off-kilter brown beret, carrying on in that odd monotone as she argues, via the château’s telephone, with the editors who publish her political articles.

  Although André’s work is beyond her, Simone still thinks that a better social order depends somehow on making mathematics accessible to the masses. “Mathematics above all,” she writes. “Indeed, unless one has exercised one’s mind seriously at the gymnastic of mathematics one is incapable of precise thought, which amounts to saying that one is good for nothing.”

  Good for nothing! And yet her life as she lives it during the 1930s, trying and failing to find a toehold in practical activity, suggests a contrary principle—that too much knowledge, too much bowing over Descartes, leaves a person incapable in other respects.

  Sometime late in the decade, her father reads an article in a medical journal describing the symptoms of larval sinusitis, which remind him of symptoms (the frequent headaches, the congestion) that Simone has. He wonders whether she could have somehow contracted this condition, even though larval sinusitis is primarily a disease of sheep, in which the larvae of a particular kind of fly make their way into an animal’s nostrils. The reported human cases have occurred mainly in shepherds.

  Nevertheless, Simone reads the article herself, agrees it might be possible, and tries the recommended treatment: doses of cocaine applied to the sinuses.

  For all their lofty intelligence, the men behind Bourbaki are goofballs: They adamantly maintain the fiction that their papers and books are the work of a man named Bourbaki, member of the Royal Academy of Poldévia. They refute reports that the name Bourbaki refers to a group in France. After Ralph P. Boas, the editor of the American journal Mathematical Reviews, writes a paragraph about Bourbaki for the Britannica Book of the Year, explaining that N. Bourbaki is not a man’s name but a pseudonym, the group propagates a rumor that it is Ralph P. Boas who does not exist—that B.O.A.S. is in fact an acronym for the cohort that edits Mathematical Reviews.

  Playfulness aside, their project is to systematically ground all of contemporary mathematics in a series of abstract, rigorous volumes titled Éléments de mathématique. This endeavor is a distant cousin to one undertaken earlier in the century by the British logicians Bertrand Russell and Alfred North Whitehead, who tried to put all of mathematics in the form of a consistent logical system—only to see Kurt Gödel, a brilliant Austrian, pull the rug out from under them by showing that any such system would contain the propositions that couldn’t be proved within the system. But Bourbaki doesn’t want to build the same kind of logically watertight apparatus. Its members, working mathematicians, don’t tend to care that much about the logicians and their fancy tricks, those snakes swallowing their own tails. Bourbaki aims to put everything on the same footing, to unify it by way of an axiomatic language. To become the Euclid of the twentieth century.

  They meet and hash out the contents of a volume, assign to one or two members the task of writing it, then meet again and subject the draft to the harshest criticism. As textbooks they aren’t terribly useful. More than a hundred pages of set theory must be laid out before Bourbaki can define the number 1. But they conjure a vision of modern mathematics, a vision of structures and maps, according to which the intrinsic qualities of mathematical objects (numbers, sets, spheres, or what have you) matter less than the relationships among them.

  They find a congenial publisher in Enrique Freymann, a descendant of German immigrants to northern Mexico, who was a painter and a diplomat before he married the daughter of a French publisher and agreed to run the company out of a cluttered office near the rue de la Sorbonne.

  A squat man, let’s say, who loves to talk and loves to read and is fascinated by science. He drives a big Renault convertible, badly, around the streets of Paris. He is as surprised as anybody when the company, under his direction, begins to turn a profit. He spends the better part of his days perusing the rare books he picks up during his lunch hour from the stalls along the Seine; or he rambles on to whoever walks in, addressing miscellaneous themes, many of which eventually lead him back to his desolate yet eventful childhood in a Mexican town near the U.S. border during the volatile years after the revolution. Or else he tells of his initial puzzlement by and subsequent education in French decorum and social hierarchies, or he’ll speak of the irony of having traded the dust of the Chihuahuan desert for the dust of an out-of-the-way publishing office.

  But good God, did Simone Weil really have tiny worms lodged in her nose? Hard to take that seriously, it’s like a story one kid tells another kid on the playground. Yet it seems she and her father thought that it was possible—and that the remedy was cocaine.

  Her attraction to suffering was deep, and some of her experiences beggar belief. I’ll be reading along in a biography of her, and everything is what it is, but when I close the book and try to imagine what I’ve just read, or later on, while I’m driving maybe, I’ll remember what I learned earlier and be baffled by it. Wait, what?

  Toward the end of his life André would be called “the last universal mathematician” by Scientific American, which is to say the last to work in so many of its subdisciplines, though by then it may have been too late for anyone to really be universal in scope. You could, at least, call him a descendant of the universalists, bearer of the deteriorating mantle of David Hilbert and Henri Poincaré—turn-of-the-century omnivores who are also referred to as the last universalists. Like them, André took a broad view, probing for connections linking different parts of mathematics.

  Or you could instead make a case that the last universal mathematician, the last to really have an aerial view of the field, was not any one person but the group known as Bourbaki. With them the dream of the encyclopedia survived: Knowledge as a neat set, a series of ordered volumes. Fine print and thin paper. A committee of surveyors taking the measure of the kingdom.

  It’s not as though André lives in a cave of pure abstraction. He buys the newspapers and follows the mounting threat of war (how could he not?) and scorns the leaders of France, whom he’ll compare to sheep, “resigned to follow wherever they were led—be it to the slaughterhouse.” Yet there are days when he dives into his mathematical work, loses himself in it, and maybe it’s that tendency that inspires Simone, one day in 1939, to send him a telegram that says, RECOMMEND READ NEWSPAPERS. The occasion is Hitler’s invasion of Czechoslovakia. The wa
r has begun.

  PART TWO

  5.

  How I would like to write something as clean and powerful as the best kind of mathematical proof. In pen, on quadrille paper: lines of black script conforming neatly, inevitably, to the faint blue squares. The sun would sly its way out from behind the clouds and beam right at your head, warm up some inner lobe of the walnut. A hint, a glimpse into the nature of things.

  André Weil is arrested in Finland, on suspicion of spying, in December 1939. He’s been in Scandinavia for months, initially on vacation with Eveline—a trip they take to escape the impending storm at home—and then prolonging his stay after France enters a war in which he doesn’t intend to fight. The army is not his destiny, his dharma, he’s convinced, and so he feels justified in avoiding the draft. He has taken from the Bhagavad Gita an idea that everyone has a purpose in life, and his is mathematics, he believes, not soldiering.

  He doesn’t realize, at this stage, that he is entering into a life of professional exile—but then again, would he behave any differently if he knew?

  When the Finnish police take him into custody, he expects that he’ll be questioned and released, that he’ll go home that very afternoon to the slice of meat he’s just bought—and so by clinging to the image of his lunch he holds on, for a few hours, to the life he’s been leading in Helsinki, returning in a pictured future to a place and a routine, a life he’ll never actually resume.

  Or if I could recover that feeling I had when I first learned algebra and geometry, subjects that didn’t need to be taken in and memorized, the way you had to take in and retain other things. They were, it seemed, already right at hand. As though there were simply some latent machine I could turn on with logic, and then! An entire world I never suspected.

  Two men on opposite sides of a desk, in a small, cold, dark room. André can’t suppress his didactic habit, even under strain. The interrogator, off-blond and husky, proceeds through a series of questions, and because neither knows the other’s native language, the two men speak in German. After the officer accuses André of having lied, André corrects his grammar.

  Sie haben gelugt, says the officer.

  Man sagt, “Sie haben gelogen,” André replies.

  The officer nods stiffly and continues. He produces a stack of suspicious documents the police have found in a search of André’s apartment, including a long letter from Lev Pontryagin, a colleague in Leningrad who, because he’d been blinded in a childhood accident, dictated all his correspondence. And then it’s as though the Russian’s blindness invades the chamber where the interview is taking place. The city of Helsinki has declared a blackout, and the room grows darker and darker, until André can barely see his interrogator.

  But who is this Pontryagin? the officer asks.

  A mathematician.

  And when did you first meet him?

  At a conference in Brussels.

  What was the date of the conference?

  I don’t recall the date. It was several years ago—August?

  A young cadet arrives to paint the windowpanes blue, a job he must complete in that near-total darkness. He stands inches from the glass and feels his way with one hand while painting with the other. The officer asks nothing of significance while the young man is there. He lights a cigarette and, after some mumbling response from the cadet, stubs it out again. He asks about André’s work, which André describes in more detail than he would normally share with a nonmathematician, hoping to convince the officer that he is only who he claims to be, even though he knows that being a French mathematician would not, in the other man’s mind, disqualify him from being a Russian spy.

  Once the window has been painted and the cadet leaves, the officer lights a gas lamp, hardly bright except by comparison with the preceding darkness. As their eyes adjust, both men drop their heads.

  André spends the night in jail. The next morning, when he is escorted out, he thinks he’s going to be executed. According to a story he would hear many years later, the chief of police did intend to kill him for being a spy, but the night before the planned execution, the chief attended a state dinner and there ran across a Finnish mathematician who knew André and who suggested he be deported instead. So he’s put on a train, locked in a compartment with three other inmates, and delivered to the Swedish border, close to the Arctic Circle. There he is interrogated again, detained again. He has nothing with him but the clothes on his back, his passport, and his wallet. He is housed at a jail, with the local drunks, though he’s free to go for walks and dine at a restaurant.

  And then come a series of further journeys: by train to Stockholm and then on to Bergen, by ship from Bergen to Newcastle, this last leg a terribly rough crossing in an old behemoth that plows through a section of the North Sea that has been seeded with floating mines. André ignores the instructions to wear a life belt day and night, for he’s sure he doesn’t stand a chance of surviving a shipwreck in those waters in January.

  The sea is calmer by the time the ship makes it to Newcastle, but he’s chilled by the sight of the harbor, so clotted with boats at anchor, ships that have been immobilized because of the fighting elsewhere, that it takes three days before his boat is allowed to dock. It’s the first time that the war hits him so squarely in the face.

  There’s a certain kind of young kid for whom the word algebra has a magical shimmer, portending the enigmas of grades not yet reached, all the unimaginable revelations of junior high and high school. I remember this from around fourth or fifth or sixth grade, or whenever it was that I became aware that some of the eighth graders were learning an exotic math involving letters, and I see this even more strongly in my six-year-old son. Give me an algebra problem, he begs, and I’ll comply, if somewhat reluctantly: 2x + 4 = 12. Give me one with x’s and y’s, he’ll say.

  This isn’t just him. One day a classmate comes over to play, and before I even notice that they’ve snuck into my office, they come marching back out, having dug up a treasure that my son’s friend now presses to his chest: a raggedy old textbook. The royal-blue cover has worn down to cardboard at the corners, and the outermost layer of the spine has begun to peel away, exposing a strip of dried glue. The title, Algebra, appears in gold along the bowed binding and also on the front, which is otherwise undecorated except for the author’s name and an image of a black rectangle with thin gold vertical stripes, as though algebra were a dark jail with golden bars.

  They carry this book of mysteries, along with pencil and paper, out to the back porch, set everything down on the couch there, and, after briefly paging through the book and scribbling obliquely on the paper, proceed to climb all over the couch, up above the book and back down and then around it. They circle the couch with its totem as though in ritual worship. They might as well be constructing one of those Hindu altars, an invisible altar that is also an invisible math problem.

  The word cuneiform comes from cuneus, the name for a wedge-shaped stamp used by the Babylonians to mark a clay tablet before it dried.

  Dating from the second millennium B.C., the Babylonian “problem texts” reveal not only practical mathematics, calculations concerning inheritances for instance, but early stabs at algebra. The Babylonians used terms that expressed unknowns, and they posed and solved problems apart from any application. For example, “The igibum exceeded the igum by seven”—here igibum and igum mean reciprocals, n and 1/n, though in other contexts they mean other things. As unknowns, igibum and igum may have lacked the economy of Descartes’s x, but they indicate that even then, thousands of years ago, this practice of representation was under way.

  I like the thudding sounds of these terms, in my invented pronunciation of them; to me they seem like bungling syllables trying to form words but not quite succeeding. I ought to put igibum on a plaque and hang it near my desk.

  I’m acquainted with a pair of twins, one of whom is considerably larger, both taller and bigger, while the other is a more modest presence and slightly concave in the
chest, which was presumably the result, they explained to me, of this smaller brother having been head-butted or jostled in the womb by the bigger one as he made room for himself.

  Though Simone and André were born three years apart, they seem to me like those twins, yin-yang children who shaped and were shaped by each other. André was so sure of himself, while his younger sister was his first peer, but too young and too differently inclined to ever catch up; he could enjoy an older sibling’s inflated belief in his own superiority while Simone, it seemed, was bent from birth by a sense of shame. That cavity in her chest, was he the first cause? Did her brother lean too hard against her ribs?

  She once characterized friendship as “a miracle by which a person consents to view from a certain distance, and without coming any nearer, the very being who is as necessary to him as food,” and maybe she was also describing the relationship of younger sister to older brother, the brother always leaping ahead and going away. The food she couldn’t eat.

  She believed that she had been poisoned, in her first year of life, by her mother’s milk. “That’s why I’m such a failure,” she told people.

  One evening my six-year-old wonders aloud whether some person long ago came up with the numbers, or whether they already existed before anybody thought about them.

  People have been arguing about this for a long, long time! I say all too excitedly. Are numbers real or not? Were they discovered or invented? We pursue this question for a couple of minutes, by which I mean that my son thinks out loud with a logic I can’t quite follow. Then we hit a wall. Neither of us knows how to take the matter any further. We move on.