The Weil Conjectures

by Karen Olsson

  At around this time my daughter, aged two, is learning to count by rote, so that part of our household soundtrack is her calling out numbers in the way that toddlers do, drawing out the vowels, her voice dropping into a lower pitch before ascending: oooone, twoo-ooo, three-ee, seee-ven, elee-ven . . .

  A day or two later my son asks, “Can we talk about that thing again that we were talking about the other night? About ‘Where are numbers?’”

  Wanted for draft evasion, André is transported by ship to Le Havre, where an agent of British intelligence turns him over to French gendarmes. They take him to a dirty jail and lock him in a cell by himself, where he doesn’t have access to math books or scratch paper. His only diversion is to peruse the obscene graffiti on the walls.

  In a note he sends to his lawyer, he describes himself as at his wit’s end. Alone in a cell, without the materials he needs to work, he says, there’s no telling what he might do. In fact these histrionics are for the benefit of the prison administrators, who will read the postcard before mailing it; he hopes to scare them into treating him better. But his sister and his parents and his wife don’t pick up on the ruse. They rush to Le Havre, only to learn that they won’t be allowed to visit. They find the lawyer and haunt the parlor outside his office—which is stuffed full of red plush furniture, absolutely repulsive to Simone, like being inside the stomach of a pig.

  She hectors the lawyer in her low flat voice to do something. She writes to an acquaintance in government: “You know what effect solitude, silence, and a lack of work can have on any human being; but my brother, especially, has never, I think, remained idle for an hour in his whole life, and he is accustomed to uninterrupted intellectual activity.”

  In other words, a war has begun without altering the Weil family belief that the genius son must be allowed to continue his work, without interruption, never mind the fact that he’s in prison. Never mind that soon enough they’ll all need to solve a more pressing problem than any in mathematics, namely, how to get out of France. Then again, maybe the Weils’ belief in the importance of André’s work is only a more extreme variation, distributed among family members, of the wool that any would-be intellectual or artist must keep pulled over her eyes, shutting out enough of the world and its strife to stay at least somewhat convinced that her work could be worth doing.

  With the help of one of the guards, André sends a message to his family that he’s not as unhinged as the postcard might have implied. The prison officials move him into the neighboring cell, with two other men: one who was caught with the dead quail he’d poached from a private estate, another who doesn’t reveal his offense but takes from his pocket a tarnished sou he’s managed to keep and starts to trace new graffiti into the wall. André looks on in envy, sitting on his hands, dying to grab the coin and write something himself.

  6.

  Not long ago, I discovered that all the lectures from a certain Harvard math class are on YouTube. These were filmed, in 2003, so that they could be watched by online viewers as well as by the enrolled students. (An early foray into e-learning, it seems.) The class was an iteration of one that I’d taken in 1993, a one-semester course in abstract algebra, and the textbook was that same book my son and his friend had dug out of my office, though a different person was teaching. My class had been taught by a beautiful Italian woman, a visiting professor—I remember hearing rumors that at least one male mathematician in the department had lately taken up the study of Italian. Her lectures had been clear and well punctuated and imbued with a kind of wholehearted gravity, and as she stood there laying out some theorem about symmetry groups or Euclidean domains, I used to marvel at her, wondering what this woman would have become had she been born fifty years earlier. A bookkeeper in some Italian store?

  I started to watch these online lectures and to half-ass the problems assigned as homework, looking to recapture some of what I’d been occupied with in college. But when you’re just out of high school, chances are you’ve studied some sort of math almost every year that you can remember being alive, and so even the more abstract realms of mathematics seem connected to something you’re used to doing. Twenty-some years later, twenty-some years in which I’d become a writer and rarely thought about abstract math or had even a glancing encounter with abstract math, the stuff of university-level algebra seemed very, very remote.

  And still, it was beautiful. I’m ambivalent about expressing it that way—“beauty” in math and science is something people tend to honor rather vaguely and pompously—instead maybe I should say that still, it was very cool. (This is something the course’s professor, Benedict Gross, might say himself, upon completing a proof: “Cool? Very cool.”) A quality of both good literature and good mathematics is that they may lead you to a result that is wholly surprising yet seems inevitable once you’ve been shown the way, so that—aha!—you become newly aware of connections you didn’t see before.

  Yet in math these surprises break loose from their creators. They find a place in the firmament of what’s been discovered. To me, one thing math always had going for it was its solidity; its theorems seemed not only ingenious but true. They were facts of the universe. As a teenager I always felt the ground moving under my feet, and there was something fixed and unassailable about math. At the same time, math seemed to be a domain unto itself, at a comfortable distance from daily life. It was as though there were another world besides this one—private, constructed mentally, a symbol world—and yet it turned out to house a map of this one, in some sense it was this one too.

  “What’s the ontology of mathematical things? How do they exist?” the mathematician John Conway once said. “There’s no doubt that they do exist but you can’t poke and prod them except by thinking about them. It’s quite astonishing, and I still don’t understand it, despite having been a mathematician all my life. How can things be there without actually being there?”

  Everything is number, according to the Pythagoreans, a secret brotherhood in the sixth century B.C. that wove mysticism into mathematics and vice versa, cultivating ideas of order that had begun with the scratching of figures into the sand, the press of a wedge into clay.

  Pythagoras himself is a hazy figure, his story and his teachings recorded only elliptically by his contemporaries. Born around 570 B.C. on the island of Samos, he traveled widely, consulting with Egyptian priests and Babylonian architects before settling down in Croton, in southern Italy, and founding a school. He and his followers subscribed to a doctrine they kept secret, one that seems to have combined mathematics and religious teaching and to have contained in embryo the concept that the objects of mathematics are mental objects, abstractions—but abstractions believed to be in intimate correspondence with the essence of the universe. The Pythagoreans thought that the only numbers were the whole numbers (1, 2, 3 . . .) and their ratios.

  They also thought that numbers were friendly, perfect, sacred, lucky, or evil.

  Ultimately their theory fell apart. They realized that if you have a square in which the sides measure, say, one unit, the length of its diagonals cannot be expressed as a ratio of whole numbers. There had to be other kinds of numbers, what they called incommensurables and we know as irrational numbers.

  Supposedly the cult tried to suppress this truth. Meanwhile, agitators in Croton attacked the Pythagoreans, and, as different sources have it, Pythagoras himself perished in a fire, or died of starvation, or committed suicide, or was murdered.

  Some more number types of the ancient world: perfect, excessive, defective, and amicable. Another story has it that one Hippasus of Metapontum discovered incommensurables, and that as a result he was thrown off a ship and drowned.

  Professor Gross, that is to say the 2003 version of Gross preserved on YouTube, has gone half gray, meaning his beard is gray but the hair on his head and eyebrows remains dark. He is no longer young, not yet old. Every so often he strikes an avuncular note, reassuring the students that this material might seem hard at first but they
’ll get it, they’ll do fine. I’m fond of him, the miniature mathematician inside my laptop.

  One day, he turns and looks straight at the camera. “Hi, people online!” he calls out.

  That’s me! Hi!

  What we know as the Pythagorean theorem—for a right triangle, the square of the length of the longest side is the sum of the squares of the shorter sides’ lengths—was discovered well before Pythagoras. Hundreds of proofs of the theorem have been devised over the centuries, one of them by James A. Garfield, who was a member of Congress when his proof was published in 1876 and who five years later would become the first left-handed U.S. president. And who died later that same year, after having been shot by Charles J. Guiteau.

  André is transferred from Le Havre to a military prison at Rouen, where he is allowed paper and pencil and books. My dear sister, André writes in February. “I received your letters, the one that went to Le Havre and the one that you sent me here. I’m angry that you can’t see me right now. But I don’t think that will last.” He tells her that he’s been spending the daylight hours correcting a draft of an article, an exacting and mechanical labor that serves him well after so long without work. It’s as though he’s relighting his mind, one candle at a time. He occasionally takes a break and reads a novel, and he stops for the day at sundown, since there’s nothing but a window to read by.

  His cell is long and narrow, its consolation a small writing table attached to the wall. On one side of the cell is the window, high and barred; on the opposite side is a thick, heavy door with a circular peephole. Every so often, when he happens to glance up from his work and look in that direction, André sees some portion of a guard’s face pressed right up to the hole, staring at him.

  Finally he is permitted to see visitors twice a week, though no more than two at a time. Because there are four Weils (Bernard, Selma, Eveline, Simone) who come by train, sometimes with Alain, Eveline’s son by her first marriage, one or two of them will go to visit André while the others remain at the station, sitting on the wooden benches of the waiting room there. The Weils are an anxious and bumptious and hyperintelligent family unit, with an artillery of wicker baskets and packages and sausages and books. The five of them, muttering and worrying and occasionally reciting a line of poetry, come and go from that waiting room like birds to and from a tree. One pair flies off, then returns; another pair departs.

  At visits Simone is told where to stand, separated from her brother by two iron grilles, between them a passage where a prison guard marches back and forth. She calls to André in Greek, and the guard barks that they must speak only French. They trade off, a halting chant of practical concerns and empty but heartfelt assurances. The sister, holding tightly to the bars, has the same eager face she had at age three whenever her six-year-old brother invited her to play with him. The brother, paler than usual but in decent spirits, says that he has instructed the editor of a certain journal to send page proofs of his article to her so that she can copyedit them. In Greek, forgetting herself, she says she’ll do it gladly. The guard informs them the visit is over.

  They carry on a separate, deeper dialogue in the letters they send back and forth during this period, which are long and cerebral and discuss at length the mathematics of the Babylonians and the Pythagoreans. It’s a throwback to the intellectual closeness they had as children, the older brother once again playing tutor to the younger sister—but now her devotion has a sharper edge to it. She’ll keep dogging him to tell her about his research. She’ll ask, What is the value of work so abstract and specialized that it has no meaning to the common person?

  Certain later Pythagoreans believed that the human soul is reincarnated every 216 years. Probably they favored the number 216 because it’s the cube of 6—but then why the cube of 6?

  Embedded within the shaggy narrative of Don DeLillo’s 1976 novel Ratner’s Star—about scientists in a trippy ’70s sort of future—are several wonderful depictions of mathematical reasoning. About a mathematician solving a problem, DeLillo writes: “He scribbled calmly, oblivious to everything but one emerging thought, feeling the idea unerase itself, most evident of notions, an idea with a history . . . What breathless ease, to fall through oneself.” I love that coinage of “unerase” to describe how something created might seem already to have existed. And the way that thought and feeling here combine in a spell of self-forgetting. Again breathlessness. Again I wonder, was it the truth André Weil was after, or this feeling? Maybe it doesn’t matter, since they go hand in hand.

  Though it was not the same thing as making a new discovery, the closest I ever came to such a feeling was while working on a homework problem from that algebra class I took in college. I don’t remember the specific problem but I remember being stumped for a long time, trying versions of the same doomed attack and not getting anywhere, and then, while I was doing something else, it came to me that I could get at the answer by constructing an entity not given in the original problem, a family of functions that behaved in a certain way, which would form a kind of bridge to a solution. I didn’t trust myself, because it seemed that I had just arbitrarily made something up, but the method seemed to work. It was the middle of the night and I was delighted. The next day I ran into a guy also taking the class, who was small and soft-spoken and from Hungary, a country that seems to export mathematicians as one of its principal products. He was ahead of me in math generally, but he hadn’t figured out the problem yet, and I showed him my solution.

  Nice, he said, nodding.

  That was probably the high point of my mathematical career.

  Simone bides her time in that train station waiting room, her face eclipsed behind glasses and hair, her body draped in heavy clothes, her spirit likewise cloaked in heavy intellectual armor, in argument, in the whole weight of Western civilization. Let’s say she’s writing now, while sitting on a hard bench with her books strewn around her. Scribbling in anger. She wishes she were the one suffering in jail, and somehow this makes her regret even more the fact that her brother’s great passion is something she can’t understand, though in her letters she tempers her frustration. She writes that she hopes to be able to visit him soon, given that it’s impossible for them to change places as she would prefer.

  In the meantime she proposes to him that he try to clarify the nature of his mathematical research. Could he explain it to her, since after all he has some extra time on his hands? She would like to know “what exactly is the interest and significance of your work.” Even if ultimately there’s no way to convey it fully, she says, he might benefit from the effort, and she would surely find it interesting.

  “Inasmuch as I am less interested in mathematics than in mathematicians,” she writes.

  Her own capacity for work has lately been low, she says, and so she’s undertaken what she considers light duty: learning the language of the Babylonians. She studies from a bilingual text, and also reads the Epic of Gilgamesh, with its story of a friendship cut short by death.

  In her next letter, she again presses him to explain what he’s doing, claiming that she can’t remember whether she’s brought this up already. “What would be the risk?” she writes. “You don’t risk losing time, since you have time to lose.” Maybe there’s a way to account for what you do, to make it clear to nonspecialists, she suggests. “This makes me passionate.”

  The trains come and go, heaving their way back to Paris. She pauses to help Alain with his ancient Greek.

  She has managed to get her hands on a book by Otto Neugebauer, a scholar who’s undertaken the labor of translating the mathematics contained in cuneiform tablets into German. Simone copies one of Neugebauer’s Babylonian algebra problems in her letter to André. The problem gives the dimensions of a canal to be dug, the amount of dirt that a worker can dig per day, and finally the sum (but only the sum) of the number of days worked plus the number of workers. The problem requires the solver to find the two numbers that make up that sum: how many workers, how many days wor
ked. “Funny people, these Babylonians,” Simone writes, for it’s a ridiculous calculation. In no actual canal-digging scenario would something like the sum of days worked plus workers be known, without knowing the two quantities individually—and this, to her, speaks to the Babylonian way of thinking. Such a problem is only abstract, a manipulation without any reality behind it.

  “Me, I don’t so much like this spirit of abstraction,” she writes. “But you must be descended directly from the Babylonians.” In her letters Simone has a bone to pick with algebra and with abstraction itself; for her, there is something distasteful about mathematical thinking untethered from any study of nature. This type of math, in her eyes, is merely a game, referring only to itself. She prefers geometry, by which she means the geometry of the ancient Greeks. (“I think that God, as the Pythagoreans said, is a geometer—but not an algebraist,” she writes.) Given that the Pythagoreans and their successors surely would’ve known about Babylonian algebra but didn’t incorporate it into their work until centuries later, she infers that her beloved Greeks objected to algebra just as she herself does. She goes so far as to suggest that there must have been a religious injunction that caused them to steer clear.

  As in her thesis about Descartes, Simone interprets history in a way that seems eccentric, if not outright bonkers. And this animosity toward algebra—where did it come from? Was it that she resented her brother for leaving her behind, as he entered into his ethereal vocation? Was it something that stewed in her as she wandered around those Bourbaki conferences, listening to grown men speaking in jargon she had no way of understanding?

  But I don’t think it was quite that, since even as she questioned her brother about his work, she praised him and saw his research as very valuable—in other letters she urged him to keep it up, and when he was jailed in Le Havre, no one was more frantic than she to get him his materials. My guess is that while her distrust of abstraction can’t be entirely separate from her relationship to her brother, it’s more directly tied to her search for meaning. Everywhere people were suffering, under threat. Mathematicians, she felt, should continue their work regardless, but at the same time the work, all intellectual work, should tell us something true about the world, shouldn’t it?