by Karen Olsson
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 the 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 was 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 a 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’s 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’s 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 article’s 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 mathemat ics? 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 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 c
lear, the stars sublime, but truth be told he has 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 her keenness to inhabit the life of a 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 holds 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.
He declines.
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 happen 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 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 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, 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 give 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 Eur
ope 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!
4.
The word conjecture derives from a root notion of throwing or casting things together, and over the centuries it has referred to prophecies as well as to reasoned judgments, tentative conclusions, whole-cloth inventions, and wild guesses. “Since I have mingled celestial physics with astronomy in this work, no one should be surprised at a certain amount of conjecture,” wrote Johannes Kepler in his Astronomia Nova of 1609. “This is the nature of physics, of medicine, and of all the sciences which make use of other axioms besides the most certain evidence of the eyes.” Here conjecture allows him to press past the visible, to sacrifice the certainty of witnessing for the depth and predictive power of theory. There’s another old definition of conjecture that means something inferred from signs or omens (for example, from a Renaissance work on occult philosophy: “Whence did Melampus, the Augur, conjecture at the slaughter of the Greeks by the flight of little birds . . .”).
Elsewhere it’s hokum, claptrap, bull: “Conjecture, which is only a feeble supposition, counterfeits faith; as a flatterer counterfeits a friend, and the wolf the dog,” wrote one early Christian theologian. So it’s a word with contradictory meanings, since at times conjecture carries the weight of reasoning behind it, and at other times it’s a wild statement, an unfounded claim. Good thinking or bad, clever speculation or a reckless mental leap.
In contemporary mathematics, conjectures present blueprints for theorems, ideas that have taken on weight but haven’t been proved. Couched in the conditional, they establish a provisional communication between what can be firmly established and what might turn out to be the case. More than a guess, conjecture in this sense is a reasoned wager about what’s true.