An Excerpt from “The Weil Conjectures” by Karen Olsson
Jul 15 ● BY Karen Olsson
The following is an excerpt from Karen Olssons’s The Weil Conjectures: On Math and the Pursuit of the Unknown (Farrar, Straus and Giroux, 2019). We hope you enjoy.
An Excerpt from The Weil Conjectures
Having forgotten whatever I once knew about complex functions Fourier series field extensions compact surfaces hyperbolic spaces random walks et cetera, I am left with memories of the Science Center at Harvard, a building with a facade like stair-stepped boxes, constructed around the time I was born, in the early seventies. I attended classes there most days and spent I don’t know how many nights working on problem sets in the library or in vacant rooms. I remember tracking slush through an entry already muddied by hundreds of boot prints, coming and going, descending to the basement computer center or landing in one of the main-floor lecture halls or making my way upstairs to a classroom empty of charm or even the notion of charm. Though elsewhere literature and history were taught in stately old rooms softened by high windows and wainscoting, historically appropriate paint colors, the mossy aura of textual study accompanied by a certain whiff of wealth, the sciences had been paired with austere minimalism, that is to say white walls and black chalkboards and silver conduit pipes leading to clusters of heavy-duty switches. I sat among my fellow students (whizzes, immigrants, nerds, with all their anxious, humming brainpower) in a plastic chair, my backpack at my feet, always afraid that I was about to fall hopelessly behind but also proud that I’d so far managed to hold on to the fast horse of a difficult class. I who, as a white girl from private school, would’ve seemed marked for the humanities but who had wandered over there instead, as though by mistake. Not a boy, not Asian or Indian or Jewish, not from Russia or eastern Europe, not a child of scientists. It seemed as if all the other math kids belonged to one or more of those categories.
As for why I touched down in their midst: I could blame my erratic curiosity, a tendency to follow my nose no matter how many times my nose has led me astray. Or I could say that I was trying to prove myself, to no one other than myself. But after more than two decades, if anything my dalliance with math seems like just that, a past love, one I remember with nostalgia and the kind of echo feeling that adheres to the memory of an old romance. I mean, I had always liked math, but just how it came to consume me in college is a question that I produce a different answer to each time I’m asked—whether it’s somebody else who’s asking or whether I’m asking myself, as I still sometimes do.
An Italian ship takes André from Genoa to Bombay, a two-week voyage, and when the weather is good he strolls back and forth on the deck, reciting lines from a Sanskrit poem. To shield himself from the southern sun, he wears a cork helmet he bought in Paris, and yet, surrounded by sparkling ocean, he burns in no time. His skin peels, and over the course of two weeks he turns the red-brown color of a fox.
mandam mandam nudati pavanah
(gently gently blows the wind upon you)
From Bombay he goes by train to Delhi and then on to Aligarh, where the two men who’ve been sent to greet him titter at his helmet. He shares an adobe house with a zoologist from Germany: no electricity or plumbing, but high ceilings and a roof terrace and furniture that he has had custom-made, according to French designs. His office at the university overlooks a courtyard, which every day fills with students and empties out again, then just before sunset fills with long-tailed birds that speckle the grounds with their shit.
A few doors down is a gnarled old chemistry professor who complains about his idiotic students, about his paltry wages, and especially about the lowlife employees of the railway station who, he’s sure of it, have stolen the shipment of guavas that he ordered from a faraway orchard. He claims he can smell the fruit in the station, that he has found guava seeds near the tracks.
Some of the oldest recorded mathematics comes to us from ancient Egypt, in documents such as the Rhind papyrus (named after a Scottish archeologist who purchased it, in Luxor, in 1858), which dates from around 1700 B.C. and contains various calculations. The opening line of the papyrus has sometimes been translated as “Directions for Attaining the Knowledge of All Dark Things.”
It was a dark thing, perhaps, to find the volume of a truncated pyramid, as in this problem: “If you are told: A truncated pyramid of 6 for the vertical height of 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take a third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right.”
My freshman year, during the first week of classes, when you could attend a lecture or two before committing yourself, I visited a daunting, a yearlong math course meant for prospective math majors. By the time I found the room, a narrow auditorium with a sloped floor, the seats were already taken. I stood in the back and peered down at the professor, who seemed really far away, not only because he was in the front of the lecture hall and I was in the rear but because of his thick glasses, and the way he spoke in damp, guttural torrents inflected by what might’ve been a mild speech impediment, not to mention the very energy in the room. A geeky electricity.
CONSIDER A BALL IN N DIMENSIONS, the professor said.
Below me, a pack of heads bobbing, nodding, while he chalked a mysterious inequality up on the blackboard. A ball in n dimensions? I had no idea what that meant, much as I would’ve liked to think it referred to a fancy-dress occasion in an alternate universe. I left and took a computer science class instead.
But I turned out to be lousy at computer science—I had no patience for debugging programs—and at the beginning of my sophomore year I went back to the same math class. Which was muleheaded of me, since in the prior semester, in addition to bombing out of second-semester computer science, I had not done especially well in a multivariable calculus class meant for physics and engineering majors. I’d pretty much decided to leave all that behind and major in philosophy. But I can remember reading Hamlet at nineteen and understanding even that play as essentially a story of a fellow adolescent who, like me, was indecisive. And there were other factors, forces that drew me back. Before that second semester came to its sorry end, I’d signed up for a math summer program, and there I learned that the math that professional mathematicians do has a different tenor from multivariable calculus for physics majors, more abstract and more rigorous. That summer I also began an epistolary romance via a platform called Pine, an early form of e-mail used mostly by science people (and all too perfectly named, for a facilitator of epistolary romance), which would lead to an actual romance with someone studying physics and math. In other words, I found some social support for the whole endeavor. In other words, I fell in love. And then there was the very fact that I’d felt defeated—I wanted to prove those other classes wrong!
By that time, I had overcome my fear of the ball in n dimensions, and I knew that the course split into two classes after the first exam: one faster-paced and directed at the sort of kids who seemed to have all met one another already at International Math Olympiad competitions; the other, while still challenging, accessible to more ordinary people who happened to like math. I took that other class, which as it turned out, was the best class I ever took in anything.
Though André is a twenty-three-year-old foreigner in his first proper job, he’s nonetheless been made chairman of the department. Right away he is saddled with a complicated subtraction problem: he must produce a report on the staff of his department and in effect choose which of his three colleagues should be fired. None had made a good impression. “Pathetic characters,” he’ll call them in his memoir, “devoid of merit.”
One of them short and obsequious. One of them with a long beard he dyes red, known for his willingness to help students. One who claims to be studying a copy of an ancient Arabic manuscript, yet no one besides the man himself has ever laid eyes on the putative document. If André knew where to find decent replacements, he would happily fire them all.
The best class I ever took in anything, not just because I was entranced by the math itself but because we were encouraged to work on the difficult weekly assignments in groups, and I had never collaborated like that before: arguing my way through problems with three or four or five other people into the wee hours of the night. We were a small band of students giddily, exhaustedly trekking through an abstract moonscape, helping one another across patches of ice or fighting over which direction to head next. The egos, the insecurities, the unabashed nerdiness! I miss it still.
Also, at nineteen, so much is up in the air, open to question, unreliable. I think part of what I liked about math was simply that it seemed like a sure thing, as sure as a thing could be, a solid mass of true and rigorous and irreproachable knowledge that I could grab like a pole on a bus.
See, it is 56. You will find it right.
I’ll just go ahead and say, in case it’s not already clear, that André can be pretty abrasive. He is arrogant, impatient, short with people. Wound tight.
As the house has no electricity and hence no electric fans, a boy is paid a pittance to stand on the veranda and pull at the string that sets in motion the panka, a piece of cloth hung from the ceiling, so that the air might circulate while André naps. Sometimes the boy dozes off, and André wakes up in a sweat and shouts “Pankevale!” to rouse him.
(His sister not only would’ve refused to nap under those conditions but surely would’ve insisted upon yanking the string herself, while someone else—the boy—slept.)
But he loves India. Everything is brighter or else darker than it is in France, louder or else more opulently silent, more fragrant or else more foul. Oleander bursting in front of the house, mangoes rotting in the back. He loves it. He loves the spicy food. He loves to read the railway timetable in bed and take trips on the weekends.
On a full-moon night, he and two friends are lent a car and driver by a university benefactor, and they travel to Fatehpur Sikri, where in the latter part of the sixteenth century the Mughal emperor Akbar built magnificent palaces of red sandstone for himself and his courtiers, only to abandon them because of the lack of water. There are no gates or guards or hours of operation, and under the light of the moon André and his friends wander through the abandoned imperial city, through courtyards and galleries and harems where, in the blazing moonlight, lattices within the windows produce honeycombs of shadow on the floors.
Another time, he goes by train to a nearby village, and when he returns to Aligarh he finds the station strangely deserted, without a single employee in sight. Finally, in what passes there for a toilet, he finds one of them hunched miserably over the latrine. Later he’ll discover that the chemistry professor, determined to prove his hypothesis in the case of the missing guavas, traveled to his cousin’s orchard and injected the entire shipment with a strong purgative. In this way, the professor identifies the thieves as the railway employees he suspected all along.
Then there is the fact that I had a serious boyfriend for the first time, I was really in love for the first time, and my elation seeped into everything, I saw the world through infatuation-tinted lenses. Part of loving math, for me, was loving a person who also loved math, who walked with such long strides, at once forceful and awkward, and called bullshit whenever he saw it—which was often—and in whose company I felt let in on a truer and more powerful and more beautiful mode of being in the world, even if in retrospect it seems to me that sheltered as we were inside the concentric bubbles of our relationship and math and college itself, we were barely in the world at all.
I mimicked the ironic but heartfelt praise that he and his roommates would give some new revelation from math or science class: Dude, that is so rad.
The earliest recorded Hindu math makes its appearance in the Sulvasutras, which give rules for the building of altars. A design may incorporate squares, circles, and semicircles, but every shape used in an altar is required to have the same area. Thus every altar is a geometry problem.
Ancient Greek legend gives us another altar exercise. According to Eratosthenes and Theon of Smyrna, who both recorded the story, the city of Delos was once afflicted by a plague, and its leaders traveled to the oracle at Delphi to ask what they might do to halt the epidemic. The oracle—supposedly speaking for Apollo but sounding more like a math teacher than a god—instructed them to double the volume of Apollo’s altar, which was in the shape of a cube. Hence the problem of doubling a cube.
“Astonishing Phenomenon,” André begins a letter to his sister in November 1931. “Word of your exploits has reached me here.”
Simone has by now graduated and become a lycée teacher in the town of Le Puy, southwest of Lyon. She has also made herself an advocate for the city’s jobless men, whom she has observed from the windows of the girls’ high school, as they break stones in the Place Michelet. Grubby and gaunt, swinging scratched-up hammers—all day she can hear the thwacks and clanks and every so often the cry of someone who has been hit in the foot or has thrown out his back. If the men work the entire day and produce enough, they earn six francs from the town. Simone accompanies them to meetings of the city council and mayor, to ask for better work and pay.
She is described in a newspaper report as a “bespectacled intellectual lady, with her legs sheathed in sheer silk” not that she’s ever worn fancy stockings or fancy anything. The “miserable, unemployed worker” deserves our sympathy, notes the author. “It is for him that such feelings should be reserved and not for those intellectuals who want to ‘make a splash’ and who flourish on the misery of the poor like mushrooms on humus.”
“Mushroom on the Humus,” begins another letter from André. “I send you my wholehearted congratulations and encourage you to continue down this road. Unqilab Zindabad!”—Urdu for Vive la révolution!
The Hindus were the first culture to use negative numbers. It’s sometimes said that they discovered them, though it’s unlikely that they thought of them as a discovery, that is to say as something new they’d unearthed about the essence of number (if number can be said to have an essence). Evidently they considered negatives more of a trick, an accounting device: when negative numbers first appear, in the work of Brahmagupta, circa a.d. 628, they represent debts, as against positive-valued assets. It took centuries for them to be seen as legitimate numbers. Even five hundred years later, by which time, you might suppose, negatives should’ve settled unremarkably into the mix, the Hindu mathematician Bhaskara notes that although one may find a negative solution to a problem, “people do not approve of negative solutions.”
When recording an equation with more than one unknown in it, the Hindus used the same words they used to denote colors. Writes Morris Kline, a historian of mathematics: “The first one was called the unknown and the remaining ones black, blue, yellow, and so forth.”
The unknown plus the square of blue minus three times red equals zero.
Kline again: “It is noteworthy that they found pleasure in many mathematical problems and stated them in fanciful or verse form, or in some historical context, to please and attract people.”
“Dear Noumenon,” Simone writes back. “Thank you for the congratulations and encouragement…. It’s been established here that I am an agent of Moscow.”
Agent of Moscow, because another article has called her the “Red virgin of the tribe of Levi, bearer of the Muscovite gospels,” though in fact she’s skeptical of the Communist Party (as she is of all political parties) and has soured on Stalin, well ahead of many of her left-leaning friends. But since she writes articles for radical newspapers, she is thought to be a Communist and is followed to school by police.
Noumenon, or a thing whose existence can be reasoned but never perceived—like God, like the soul. Like mathematics? As opposed to phenomenon, an appearance, a thing apprehended by the senses.
The sister who could be perceived directly. The brother a distant god.
Un qil ab Zindibad!!! she signs off.
I remember walking home from the Science Center after midnight, a layer of new snow underfoot. No wind, no one else around. I slipped into an enfolding stillness. Although it was the darkest part of a winter night, the streetlamps were ablaze and the snow was shining; it didn’t seem dark at all but like I was walking through a lit passage back to my room, a tunnel of light cushioned by an endless black sky. I experienced then, experienced from time to time, a kind of pleasure that came only after having thought hard about math, the mental equivalent of having gone for a long run. A gentle euphoria.
Negative numbers infiltrated Europe during the Middle Ages, were imported like viruses by international voyagers, who brought from the Middle East certain Arab mathematical texts that elaborated on the advances made by the Hindus. Yet in the sixteenth century, people still questioned whether negative numbers were truly numbers. Michael Stifel called them “absurd numbers.” Blaise Pascal said they were utter nonsense. John Wallis reasoned that they must be larger than infinity. And Gerolamo Cardano—a.k.a. Jerome Cardan, a notorious scoundrel, professional gambler, physician, and caster of horoscopes, who happened also to be an exceptional mathematician—declared them impossible solutions. “Fictitious numbers,” he called them.
Negatives weren’t the only shady characters. There was a whole rogues’ gallery of irrationals, like √2—they kept poking their heads up in equations, but were they numbers? “We find that they flee away perpetually,” Stifel wrote, before concluding that an irrational was “not a true number, but lies hidden in a kind of cloud of infinity.”
Worse yet, along came the square roots of negatives—Descartes gave them the name “imaginary numbers”—and the hybrid creatures we now call complex numbers, composed of a real number added to an imaginary number. Cardano associated them with “mental tortures.” A complex root of an equation is “as refined as it is useless,” he wrote.
André’s mathematical investigations dead-end, and dead-end again: He tries to expand upon the work he did for his thesis, on Diophantine equations, without making any progress. He has an idea about making use of John von Neumann’s work on unitary operators in Hilbert spaces to attack the problem known as the ergodic hypothesis, but the idea isn’t specific enough. He flirts with celestial mechanics, drops it.
In the spring he orders the boy servant to drag his bed up to the roof terrace. It’s almost too bright to fall asleep there, the tropical sky is so clear, the stars sublime, but truth be told, he had just as much trouble falling asleep inside the house. He reads the railway timetables restlessly, he’s a young man with no wife, no girlfriend, in a foreign country, and so naturally he has a lot of pent-up energy that hurls itself into his weekend sojourns.
Might there have been some girl in Kashmir? A Dutch lady traveler on one of those trains? Some colleague’s cousin, who mocks him but later slips him a note? Or was he untouched, untouchable for those two years? He lies up there on the roof and moans at the stars.
The madness of reason.
On other winter nights I would work in my dorm room, and when it grew too cold I would run my electric kettle as a heater, boiling the water down and refilling it and boiling more water until I’d made a sauna of the area around my desk and futon, until the lone window wept. I would sit there, in a fog of my own making, trying to demonstrate small truths. Prove this, disprove that, describe such and such explicitly.
In hear yearning to inhabit the life of the worker, Simone wangles a visit to a coal mine, normally off-limits to women. There she is not only escorted down into the shaft but allowed to try her hand at the compressed-air drill, a deafening machine that sends continual tremors through her small body. She hold on for dear life. Had someone not stopped her, a companion later reports, she would’ve kept on using the air drill until she collapsed.
She asks the boss to hire her.
The coal miner, she writes afterward, is a pawn in a titanic struggle between coal and compressed air: “Clinging to the pickax or drill, his entire body being shaken, like the machine, by the rapid vibrations of compressed air, he confines himself to keeping the machine applied at each instant to the wall of coal, in the required position.” The miner winds up becoming a part of the machine, “like a supplementary gear.”
She asks, under what conditions could a revolution possibly be successful, given that the machinery of labor is itself oppressive? It would have to be a technological revolution, as well as an economic and political one.
What André most desires, during those long nights in India, is to set his own head spinning. He once, during a stay in Germany, entered a mathematical fugue state that lasted for hours and lit the way for his dissertation, but he doesn’t know whether it will ever come to him again. “Every mathematician worthy of the name,” he’ll write in his memoir, “has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets this word) seems to play a role.”
At last it comes the spark, a new idea about certain functions. Functions of several complex variables—in the realm of Cardano’s mental tortures. It hits him on his rooftop bed as he is half asleep, a silk-threaded structure spinning its way out from a worm in his mind, and it shocks him awake, he rolls onto his stomach and gropes for the timetable, wanting the stub of a pencil he tucked between its pages. The pencil rolls away, and now he is down on his knees, patting the tiles ever so gently in hopes of finding the pencil and not a scorpion. At last there it is, and he starts to write over the schedule of the Delhi–Calcutta line. His ideas arrive as clear and bright as the sky, there are routes among the stars he never noticed before, it’s as though he doesn’t have to produce the thoughts any longer, they are simply supplied to him, and he is the scribe who turns them into symbols. His mind has been razed and there are barely any words in it. Only gratitude, only functions.
Does the value lie in what he’s writing down or in this moment of lucid exaltation itself, this obscure bliss? Could it be a new porthole on reality, the theorem that is taking form, or is it better considered as a kind of access to the innermost architecture of thought? An infinitely nested diagram, unfolding itself. Arguments and computations, variables falling away, functions suspended between floating fields of numbers. Cool flames. The mathematician’s early harvests.
The legend of Archimedes, struck by insight during a bath and then running naked through the streets of Syracuse, shouting, Eureka! Eureka!
“I have reviewed the circles on the basis of the demonstrations you have given me,” writes a young worker to Simone. When she’s not teaching or going to union meetings, she volunteers to teach free lessons, convinced as she is that workers, in order to advance, must be better educated—and that geometry lies at the heart of a proper education.
“In the three lessons I had with you, you have given me almost all the elementary facts of geometry; it is a pity that I cannot see you more often, for I would have ended by becoming a truly learned person,” the letter continues. “With you as the teacher I was never bored for a second; and these few instants exalt all the noble thoughts that inhabit me. If I could see you more often, I would make double progress, intellectual as well as moral.”
I wonder whether this young man was in love with her, can’t help wishing that he was and that she would’ve loved him back. When Simone was a child, her mother had a strong fear of germs and obsessed over hygiene. Simone absorbed that worry and exaggerated it to the point that she shrank from others’ touch. She didn’t hug or kiss people. As far as anyone knows, she never had a lover.
So there’s no telling whom she might’ve been thinking of, if she was thinking of anyone, in this passage from her notebooks: “All our desires are contradictory, like the desire for food. I want the person I love to love me. If he is, however, totally devoted to me he does not exist any longer and I cease to love him.”
Cardano, by the way, was one of those men of the Renaissance whose polymathic, credulity-straining lives gave us the whole notion of the Renaissance man. He was an outlandishly brilliant thinker and, it seems, a total dick: “high-tempered, devoted to erotic pleasures, vindictive, quarrelsome, conceited, humorless, incapable of compunction, and purposely cruel in speech,” writes Kline in his history. Born in 1501, the illegitimate son of a lawyer and a woman who’d tried and failed to abort him, Cardano was imprisoned for heresy because he’d committed the ecclesiastical faux pas of casting a horoscope for Jesus. Yet after he finished his prison term, the pope hired him as an astrologer.
In addition to gambling, playing chess for money, and practicing medicine, Cardano would, for a fee, detect your character and fate based on your facial irregularities. He wrote an entire book on this kind of conjecturing, with some eight hundred labeled diagrams of faces.
“A great enquirer of truth, but too greedy a receiver of it,” Sir Thomas Browne would later say of him. During his lifetime, Cardano published 131 works, encompassing not only mathematics, astronomy, physics, and medicine but also astrology, dreams, portents, and charms. He wrote on angels and demons. He plagiarized his father’s friend Leonardo da Vinci. He wrote a memoir, De Vita Propria Liber, in which he lamented his wretched boyhood and the extreme poverty he endured as a young man. His son, Giambatista, was executed for poisoning his wife.
Cardano died on September 20, 1576, the very day he’d predicted for himself. In his final year, he had fourteen good teeth.
The more than seven thousand pages he left behind are a monument to the encyclopedic tendency, to the idea that a powerful intellect could be in possession of the whole scope of human learning. When did that idea expire? Now there is too much to know, not to mention the whole problem of the unreliability of the knower. Now we have machines for knowing.
Or we turn knowledge into an ornament, a bauble. There’s a welcome tingle brought on by the flotsam of scholarship, the odd facts that wash up onto the shore. The fourteen good teeth.
Simone works in defiance of her own body. She goes for long stretches without food and rest, and that’s not the only way in which she flirts with extinction. She teaches a course for union members called Insights into Marxism, and her students fear for her. “This Simone,” complains one man attending the class, “just look, five times she lights her cigarette and throws the matches and sparks on her blouse. She’ll end up by setting herself on fire.”
André returns to Europe in May 1932 and stops in Rome to pay a visit to Vito Volterra, a distinguished older mathematician. After he explains to Volterra his progress in functions of complex variables, the Italian abruptly stands and runs toward the back of the apartment, calling to his wife, Virginia! Virginia! Il signor Weil ha dimostrato un gran bel teorema!
Virginia, Virginia, Mr. Weil has demonstrated a very beautiful theorem!