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 worked. “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?
“After this his fame grew great,” the ancient Roman philosopher Porphyry of Tyre wrote of Pythagoras, “and he won many followers from the city itself (not only men but women also, one of whom, Theano, became very well known too) and many princes and chieftains from the barbarian territory around. What he said to his associates, nobody can say for certain; for silence with them was of no ordinary kind.”
Neugebauer, the scholar of ancient Babylonia, was an Austrian who fought in World War I and wound up in the same prisoner-of-war camp as Ludwig Wittgenstein. After Hitler rose to power, Neugebauer found a job in Denmark and over the course of several years published three volumes of his translations. Then, in 1939, he emigrated to America, where in collaboration with another scholar he published an English version of his magnum opus, as Mathematical Cuneiform Texts.
He would also write and publish a journal article called “The Study of Wretched Subjects.”
In his lectures, the Professor Gross of 2003 manages to impart something like suspense and drama to a subject without much inherent narrative tension. There’s an urgency to his presentation, a vigor born of logic itself. “Now I claim,” he says, raising his voice and drawing out the I and the claim, like a magician announcing his next trick—“Now I CLAIM . . . that r = 0.”
He has a fondness for historical digressions (as do I, obviously), and during the first week of class, he presents one as though it were a theorem: “If you go to Paris and if you take the ligne de sud to the Parisian suburb of Bourg-la-Reine, an absolutely disgusting suburb, and you go to the main intersection of Bourg-la-Reine, an absolutely disgusting intersection, you will find a plaque that says, Ici est née Évariste Galois, mathematicien.”
And if you give an introductory algebra course, surely you must mention Galois. A misunderstood, mournful prodigy, his short career a series of arguments and rebuffs, he bequeathed to the world not only the fundamentals of what is called group theory but also the romantic image of the mathematician in its purest, most distilled form. He died in 1832, when he was twenty, as the result of a duel: pistols at dawn, the whole shebang. His father, a provincial mayor slandered by enemies, had committed suicide. The papers he’d submitted to the Académie Française had been lost or rejected. He’d been jailed for making treasonous remarks while drunk at a dinner. The night before the duel, or so the legend goes, he tried to write down everything he knew but hadn’t yet committed to paper, all the math swirling around in his head, pausing to scribble in the margins, “I have not time, I have not time.”
This tale circulated for many years, though it was steeped in exaggeration. (For instance, Galois wrote up his major mathematical ideas in those papers he submitted to the Académie Française, rather than on the eve of the duel.) Galois’s story aside, his insights helped transform the study of equations, which in the nineteenth century would be flipped on its head. He and others developed an inspired new approach: Instead of trying to understand an equation by finding the solutions, one could presume the solutions and examine the domain in which those solutions exist. To say it another way, given the polynomial equations we learned about in school (equations like x2 – 5x + 6 = 0) and were tasked with finding the roots of (here x = 2 and x = 3), you could take a step back into a more abstract realm: instead of solving for those roots, you could assume they existed, designate them by symbols (u and v rather than 2 and 3), and consider the smallest set containing u, v, and all the numbers you could obtain from them by adding, subtracting, multiplying, and/or dividing. By investigating the properties of such sets of numbers, rather than the original polynomials, you could cast the study of equations in a new light.
“Gradually the attachment of the symbols to the world of numbers loosened, and they began to drift free, taking on lives of their own,” the historian and novelist John Derbyshire writes about this shift in algebraic thinking. What the x might represent mattered less than what the x might do in various circumstances. It was the sort of turn toward abstraction that Simone would come to hate but was hugely fruitful in a way that she would refuse to see.
When the Pythagoreans claimed that the universe was number, they meant it, literally; for them creation had begun with a unit, a number somehow possessed of spatial magnitude. First there was a single unit, then it split into two units, and as it did so, the universe drew into itself a void that would keep the two units apart—as though taking in breath, they said. The universe inhaling the void. The two units begat the first line, and then, by more splitting, a triangle, then a tetrahedron. From a single point all of geometry spilled forth, and this geometry constituted the material world.
Much of this belief system was noted by Aristotle, who wrote about Pythagorean thought and, more often than not, dismissed it as absurd. “All . . . suppose number to consist of abstract units, except the Pythagoreans; but they suppose the numbers to have magnitude.”
Every day André is permitted a thirty-minute walk inside the circular prison yard. The yard is divided into sectors, and the prisoner is restricted to one of them, as the guards watch him from a tower in the center. Though André normally wears sandals even in winter, here he has only the ill-fitting, bulky work shoes that he was given in Finland. Nonetheless he tries to walk fast, clopping along in his brogans, around and around the sector’s edge. When he walks toward the guard tower he thinks about complex functions. When he walks away from it he cranes his neck to see the birds, the clouds beyond the fence.
“My dear sister,” he writes, when he is back in his cell, “Telling nonspecialists of my research or of any other mathematical research, it seems to me, is like explaining a symphony to a deaf person. It could be attempted, you could talk of images and themes, of sad harmonies or triumphant dissonances, but in the end what would you have?
“A kind of poem, good or bad, unrelated to the thing it pretends to describe.”
You might compare mathematics to an art, he goes on, to a type of sculpture in a very hard, resistant material. The grains and countergrains of the material, its very essence, limit the mathematician in a manner that gives his work the aspect of objectivity. But just like any work of art, it is inexplicable: the work itself is its explanation.
As for Babylonian algebra, it did in fact infiltrate Greek mathematics, but it was algebra translated into geometric terms, he writes. Take for example the work of Apollonius, in which algebraic equations became conic sections: parabolas, hyperbolas, ellipses. Yet the most original thing about Greek mathematics is that the Greeks didn’t deal in approximations; they killed number for the benefit of Logos. In other words the Greeks, rather than just using numbers to calculate, considered them as pure quantities, and this abstraction—that is, this conceiving of the whole numbers as such—constituted the leap forward from which everything else followed.
So here we have André in prison, Simone at loose ends, all Europe going to hell—and they take to arguing about the nature of ancient mathematics.
(But who’s to say, when the world is going to hell, that you shouldn’t argue about ancient mathematics?)
When it was discovered that the whole numbers couldn’t fully account for even simple geometric relationships, the Greeks had to start over, he writes, at the foot of the hill. Since one could no longer be sure of anything.
Another morsel from Eros the Bittersweet: our word symbol comes from the ancient Greek symbolon, which was half of a knucklebone, “carried as a token of identity to someone who has the other half,” like one of those cheap heart-shaped lockets that break into two pieces. I imagine André and Simone as having such half knucklebones. Each sibling carrying one, as a symbol.
Although I’ve never been susceptible to phantom sense impressions, one day as I watched an algebra lecture on my laptop I began to smell chalk dust so strongly that I wondered whether my kids hadn’t stashed some sidewalk chalk in my office. (They hadn’t.) Thick in my nostrils, I could’ve sworn: vintage chalk dust from 2003.
That year, the class reminds me, was a long time ago. In one lecture, a phone starts to ring, and Professor Gross, with a very 2003 self-consciousness about the technology, pulls out a flip phone, opens it, and without preamble asks, “Is the governor of Puerto Rico here?”
In 2003 he is also the dean of Harvard College, and as he explains after the call, he is supposed to receive the visiting governor of Puerto Rico later that day.
He turns to face the class. “Moocho goosto!” he says with relish—mucho gusto, pleased to meet you. Then: “I’m practicing my Spanish.”
In a postscript to that letter in which André tells his sister that it would be pointless to explain his work, he relates a story about the nineteenth-century Norwegian mathematician Sophus Lie—for whom I myself have a sympathetic fondness, because he was a tall Scandinavian who for a long time couldn’t make up his mind about what to study or which career to pursue. As a university student, Lie had first tried his hand at languages, then switched to science, after failing Greek. He would later say that he found the road to mathematics long and difficult. But he had a thing for long roads—one weekend he walked the sixty kilometers from the capital to the town where his parents lived, only to find that they were not at home, at which point he turned around and walked back—and when he was twenty-six, he encountered certain new ideas in geometry that led him to embrace math for good.
“The powerful Northman with the frank open glance and the merry laugh, a hero in whose presence the common and the mean could not venture to show themselves,” wrote Anna Klein about Lie. Anna was the wife of Felix Klein, another renowned mathematician and a friend of Lie’s. As young men they both lived in Paris, in adjacent rooms. Felix Klein would recall that on one morning he had risen early and was headed out for the day when Lie, still in bed, called to him. During the night, Lie explained, he had discovered a connection between lines of curvature and minimal lines.
I did not understand one word, Klein would write. Yet later on, after he had left and was going about his business, he perceived in a flash what Lie had meant and how to prove it geometrically. That afternoon he returned home and, finding that Lie had gone out, wrote a letter containing the proof.
I imagine the excitement he must’ve felt as he slid it under Lie’s door.
Dude, that is so rad.
In another of his lectures, Gross pauses to consider a particular function (if you’re curious: the function f(t) = e2πit, along with its derivative f' = 2πif), and then he invokes Lie.
“This is what Sophus Lie discovered,” Gross says, with feeling, “that the study of group homomorphisms in the context of continuous groups is intimately related to the solution of differential equations!”
Leaving aside the details here, the gist of the matter is that Lie, by expanding a theory in algebra, namely the study of groups, found a way to shed light on an entirely separate area of math—or one that had seemed entirely separate, namely differential equations. It’s as though he located a wormhole from one mathematical realm to another.
“Is that amazing?” Gross exclaims. A pause, and then: “But I digress.”
The Franco-Prussian War forced Klein and Lie to flee Paris. Lie, naturally, left on foot. Near the town of Fontainebleau, he was arrested and (as André wrote to Simone) accused of spying for the Prussians. His mathematical notes were thought to contain coded military secrets, just as André’s papers would draw the attention of the Finnish police seventy years later.
“Occupied unceasingly with the ideas which were fermenting in his head, he walked in the forest each day, stopping at the places furthest from the beaten track, taking notes, drawing diagrams in pencil,” recalled one of his French colleagues, Gaston Darboux. “At that time this was quite enough to awaken suspicions.” Darboux appealed to the imperial prosecutor, insisting that the notes had nothing to do with national security, and after a month in prison, Lie was released.
He returned to Norway, though in his native country he eventually came to feel isolated, ignored by mathematicians working on the Continent. In a letter to Klein he complained that Darboux “plunders my work”—that is to say, the Frenchman who’d pleaded to save Lie’s neck was stealing his results and publishing them as his own.
Simone imagines that a young paratrooper has landed on the terrace of the Weil family apartment, a bewildered German teenager for whom, in her reverie, she feels nothing but tenderness. At dinner she asks her parents whether they would offer shelter to a German soldier.
Absolutely not, says her father. I would turn him in, of course.
She refuses to eat another bite until he promises her that he would help the young man, this hypothetical soldier she saw in a daydream.
Selma Weil, who has always fretted about her daughter, now worries about her son, too, not to mention the fate of France itself. She goes all around the neighborhood sharing gossip and advice, buying up more food than her kitchen can hold, huffing her way upstairs and back down again, and if she happens to sit still, she has the twitchy look of a hare. Her dark eyes are anxious, vigilant. She’s had her bangs cut very short, which has the contradictory effect of making her look girlish while exposing her lined forehead. One day she absently pulls a button off her sweater, unaware that she’s been tugging at it until she finds it in her hand. She stares at it in surprise, then mutters to herself in Russian, much as she claims to have forgotten every word of the language she spoke as a young girl. She writes to cousins in California who have invited the family to come live with them. She rereads André’s letters. She plays her beloved Beethoven sonatas, too loud and too fast.
“What can I say about myself?” André writes to Eveline. “I am like the snail, I have withdrawn inside my shell; almost nothing can get through it, in either direction.”
Only paper, that of his correspondence, his manuscripts, his books. Because he is allowed those things, it is a tolerable existence, that of a snail who executes the following actions: work, eat, write, read, sleep. While his life is stripped-down and constrained, he strides ahead in his mathematical investigations, making progress in the area of algebraic curves.
His colleagues, who’d previously written him to offer their sympathy, now envy him instead, reminding him that not everyone is so lucky to sit and work undisturbed. “I’m beginning to think that nothing is more conducive to the abstract sciences than prison,” he tells Eveline. Her letters back to him are reassuringly full of activity; she says that she’s been cultivating pea plants, trying to teach Alain about fractions. In a sense this arrangement merely heightens a division of labor they’d already adopted, André off by himself, contemplating mathematics, and Eveline out in the world, with her child and later with their children.
“A heavy, opaque, suffocating atmosphere has settled over the country,” Simone writes to André. “People are downhearted, discontent, but they also tend to swallow whatever they are served without protest or surprise. Characteristic situation of a period of tyranny. Unhappiness is joined to an absence of hope. France is and will be for a long time (unless there’s a social convulsion) in a state of torpor and resignation.”
She sends a short letter while working on a draft of a longer one, funneling a fury of thoughts into her neat, upright script. She sets the finished pages next to her on the train-station bench and tucks the edges under her thigh while she prosecutes her argument about the ancients. Countering her brother, she insists that the Greek stance toward algebra can’t be explained merely by saying that they assimilated algebra into their geometry. It must have been taboo—algebra must have seemed impious, she writes. Mathematics, for the Greeks, was not just a mental exercise but a key to nature. It illuminated a structural identity between the human mind and the universe.
As for André’s comparison of mathematics to art in a hard material, Simone has her doubts: What material? The proper arts work on material that exists physically. Even poetry has the material of language regarded as an ensemble of sounds. The material of mathematical art is a metaphor, and to what does it correspond?
The material of Greek geometry was space, but space in three dimensions, actual space, a constraint imposed on all men. It is no longer this way. Your material, she says to André, isn’t it just the ensemble of previous mathematical work, a system of signs? Rather than a point of contact between man and the universe, Simone argues, present-day mathematics has become inaccessible.
In 1886 Lie moved to Leipzig to take over a position from Klein, who’d been hired by the University of Göttingen. There he became known for his informal way of teaching (Felix Hausdorff was one of his students), yet he began to chafe at the workload and the fact that his colleagues treated him as Klein’s disciple. He found it harder and harder to sleep at night, and finally he suffered a nervous breakdown. At a psychiatric hospital near Hanover, he resisted the prescribed opium treatment. Although he returned to teaching the following year, it took that year and the next before his insomnia passed. “With sleep,” he wrote, “the pleasure of life and work has returned.”
But he was a changed person, touchy and irritable. He wrote to friends in Norway that he longed to return there. He clashed with Klein, his friend of more than twenty years, then publicly attacked him. (“I am no pupil of Klein, nor is the opposite the case, although that might be nearer to the truth,” he wrote.) Klein found this “both painful and incomprehensible,” Anna would recall, but “soon my husband understood that his best friend was ill and could not be held responsible for his acts.”
A few years passed, and then one summer evening, as Felix and Anna were returning home from an excursion, they found their estranged friend waiting for them. “There, in front of our door, sat the pale sick man,” Anna wrote. “‘Lie!’ we cried, in joyful surprise.”
Felix Klein and Sophus Lie “shook hands, looked into one another’s eyes, all that had passed since their last meeting was forgotten.”
Lie went back to Norway in 1898, but he could only lec ture for a few months before his deteriorating health forced him to retire. He died, of pernicious anemia, the following year.
At last André goes on the offensive, that is to say, he answers Simone’s repeated requests with a long, technical description of some of his mathematical work, a treatise in the form of a letter. He knows full well that she won’t understand these “thoughts,” as he calls them: “I decided to write them down, even if for the most part they are incomprehensible to you.” He plunges into a density of terms she wouldn’t know, with only minimal efforts to say what he means by quadratic residues, nth roots of unity, extension fields, elliptic functions.
In the first half of his letter he sketches a historical context for his work, starting with the nineteenth-century watershed in algebra, that leap by which mathematicians inverted the problem of solving equations within given domains by constructing domains in which given equations had solutions. He alludes to a time when questions about numbers began to rub up against questions about equations or functions in new ways. “Around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle . . . and the division of elliptic functions,” he writes.
Anguish and delight! As he’s laying out his none too explanatory explanation of his research, André emphasizes the role of analogy in mathematics—which his sister might appreciate, even if the rest of it flies right over her head. Here, analogy is not merely cerebral. The hunch of a connection between two different theories is something felt, a shiver of intuition. For as long as the connection is suspected but not entirely clear, the two theories engage in a kind of passionate courtship, characterized by “their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels,” he writes. “Nothing is more fecund than these slightly adulterous relationships.”
Analogy becomes a version of eros, a glimpse that sparks desire. “Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often.” This, of course, describes more than mathematics; it expresses an aspect of thinking itself—how creative thought rests on the making of unlikely connections. The flash of insight, how often it leads us off course, and still we chase after it.
Pernicious anemia, the cause of Lie’s death, is a decrease in normal red blood cells that results when the intestines cannot absorb enough vitamin B12. Symptoms include confusion, depression, loss of balance, and numbness of the hands and feet.
David Hilbert, one of the leading mathematicians of the early twentieth century, also suffered from pernicious anemia, and although he benefited from a new treatment developed in the 1920s, the disease contributed to his decline at the same time as his beloved Mathematical Institute in Göttingen, for decades a hothouse of mathematical progress, was drained of its talent by the Nazis. One evening Hilbert attended a banquet where he was seated next to the Nazi minister of education, who asked him whether it was true that the institute had suffered following the removal of its Jewish faculty and their supporters. “It hasn’t suffered, Herr Minister,” Hilbert replied. “It just doesn’t exist anymore.”
Dementia set in, and he came to believe he was living again in Königsberg, the Prussian city of his childhood. He died in 1943; hardly anyone went to his funeral.
But I digress.
Simone studies her brother’s letter closely, so closely that it hurts. She pulls some of his books down from their shelves and begins to wade through a German text on complex functions, until her head threatens to split open.
Yet might she be right, or at least not altogether wrong, to think her brother’s work should be explicable? That it should do more than just extend the work that preceded it, that it should reveal something about the world?
Meanwhile André sends her another letter, two days later, continuing their debate about ancient mathematics. In their correspondence both he and Simone propose ways in which the Greeks might’ve discovered irrational numbers, drawing geometrical diagrams in the margins to illustrate their theories. Did the discovery trouble the Greeks or inspire them? André suggests that they were disturbed by it, while Simone counters that it would’ve brought them joy.
In one, two, three drafts that she composes in reply to the second letter, again there are figures traced in the margins, as well as more geometrical ideas, discussions of Platonic dialogues, musings on the relationship between an artist’s worldview and her art, speculations about mysticism in ancient Greece.
Simone conceives of a civilization in which mathematical reasoning, mystical belief, and existential loneliness formed an energetic triangle. The Greeks, she writes, experienced intensely the feeling that the soul is in exile: exiled in time and space. Mathematics could bring some ease to the exiled soul, she says. Doing math could free you from the effects of time, and your soul could come to feel almost at home in its place of exile.
She also writes, regarding André’s explanation of his research, “I understood nothing of your sixteen-page letter (which I read several times).” She has nonetheless sent excerpts to his lawyer, thinking perhaps a description of his research might be useful for his court case. And she tells him, “Send me more things like that if the inspiration strikes. I like it very much.”
It wasn’t lost on me, as I watched math lectures on my computer, that my return to (casually) studying math, much as it was not quite intentional, much as I stumbled into it, had come to resemble one of those self-improvement projects that middle-aged people embark on, like learning a language or taking piano lessons or rereading the classics. And yet studying German or piano or Homer seemed downright practical, relative to Math E-222.
What lies behind these efforts? Nostalgia? Wanting to stave off the inevitable, already perceptible decline? I suppose both, and then of course (almost too obvious to mention, and I feel like a scold and a cliché for even bringing it up) there is simply the enterprise of attention itself, the goal of concentrating on anything at all, in this era of distraction technologies. Surely Simone Weil, as odd as some of her beliefs and proposals were, was right to emphasize the importance of sustained attention, which is something we are letting slip away, or really giving away, with little more than mild, fleeting second thoughts.
Great titles in math:
The Whetstone of Witte
What Are and What Should Be the Numbers?
New Invention in Algebra: As Much for the Solution of Equations as for Recognizing the Number of Solutions That They Have with Several Other Things Necessary for the Perfection of This Divine Science
The Sand-Reckoner
Logarithmatechnica
The Mathematical Science Reduced to Tables
Essay on Fire
Simone understands more of André’s disquisition than she lets on. She includes a detailed summary of it in her letter to his lawyer, presented in the unlikely hope that the military tribunal could be convinced of the importance of André’s work and that her brother would thus be freed.
I picture this lawyer as a harried sad sack, exhaling as he pulls from its envelope an overly long missive sent by the same pestering angular female who has been unrelenting in her visits to his office. He reads the opening lines: “Number theory (putting aside some propositions discovered by the Greeks, notably the Pythagoreans) begins in France, in the seventeenth century, with Fermat. It was then the German mathematician Gauss, in the beginning of the nineteenth century, who made the most decisive progress . . .”
He blinks and sets the letter down on his desk.
Sophie Germain, born in 1776 in Paris, could not, as a woman of that era, be admitted to the École Normale or the École Polytechnique, but she studied mathematics as best she was able, borrowing from other students’ lecture notes and studying the books in her father’s library. She wrote letters to the great mathematicians of her day under a pseudonym, M. Le Blanc.
“But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur Le Blanc metamorphosed into this celebrated person,” replied Carl Friedrich Gauss, after she was compelled to reveal her true identity. By then she had withdrawn from social life, weary of being regarded as a curiosity because of her mathematical talent. “The taste for the . . . mysteries of numbers is very rare; this is not surprising, since the charms of the sublime science, in all their beauty, reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men, in familiarizing herself with knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.”
He challenged her to come up with her own proofs for three of his latest theorems, which she duly sent back to him. Six months later, he sent his last letter to her. He had accepted a professorship at Göttingen and, he explained, had become too busy to keep up their correspondence.
“Remain always happy, my dear friend,” Gauss wrote to Germain before terminating their exchange for good. “The rare qualities of your heart and mind deserve it, and continue from time to time to renew the gentle assurance that I may count myself among your friends, a title of which I will always be proud.”
At trial, André is defended poorly by his lawyer and sentenced to five years in prison, and so he agrees to serve in a combat squad, in return for a suspended sentence. One ordeal ends and another begins; he joins a machine-gun unit, alternately lifting chests full of ammunition and sneaking off to read math books. During a German offensive, the men are evacu ated to England, where, because he misses curfew one evening, André is put in “prison,” that is to say he’s confined to an area of tents surrounded by barbed wire. Separated from the rest of his company, he literally misses the boat when the others are sent to Morocco. He becomes, instead, an interpreter for the guards who’ve been his jailers, and as such enjoys a good deal of spare time, free to go to the library and visit colleagues at the University of Bristol. He’s managed to turn military service into a rather leisurely, math-centered existence.
The Germans, meanwhile, have reached Paris. Simone and Bernard and Selma squeeze onto the last southbound train and arrive in Nevers, in central France, barely ahead of the occupiers. They continue on foot toward Vichy, buying themselves baskets in an effort to pass as peasants, though this only makes them look like three well-off urban Jews carrying baskets.
André makes his way to London, where he flirts with English women and persuades a friend in the camp office to fill out a card stating that he has pneumonia. A hospital ship takes him to Marseilles, and he reunites with Eveline and Alain. The three of them board another ship, sailing across the Atlantic to Martinique, and from there he secures an American teaching position and a visa.
His parents and sister remain in the south of France, in Vichy and then in Marseilles, trying to obtain visas to enter Morocco or Portugal, with the idea that from there they’ll find a way to follow André to the United States.