by Karen Olsson
From André Weil, whose work I don’t understand, I have nevertheless gained an indirect appreciation of number theory, one that begins with my picturing him in his prison cell, staking out his own colonies among the fields and functions, or later on in his basement study in Princeton (he joined the Institute for Advanced Study there in 1956), his cat perched on the edge of his desk, his family’s footsteps sounding against the floors above him.
The closest I can really get to the Weil conjectures, as much as I can grasp, comes courtesy of our man of mathematics Pierre de Fermat. In a letter written in October 1640 to one Bernard Frénicle de Bessy, an official at the French mint who had a gift for mental arithmetic, Fermat proposed what later became known as Fermat’s little theorem—not to be confused with the more famous assertion known as Fermat’s last theorem, though in both cases Fermat himself did not provide a proof: “I would send you the demonstration, if I did not fear its being too long,” he informed Frénicle de Bessy. (Did he really know one? Or was this what Bell called a “certain hint of vanity”? The first proof was published almost a century later.)
This theorem is fairly straightforward. In brief: Consider a restricted set of numbers, these being the whole numbers less than some prime number p. If p were 5, for instance, we’d be talking about {0, 1, 2, 3, 4}. Any number larger than p is defined to be equivalent to one of the smaller ones, by taking the remainder when you divide that larger number by p. So 5 is equivalent to 0, 6 is equivalent to 1, 7 to 2, 8 to 3, 9 to 4, 10 to 0 again, and so on.
Fermat’s little theorem states that in these realms {0, 1, 2, . . . , p–1}, you can take any number a, raise it to the pth power, and get a back. Another way of saying this would be to say that all the numbers in our little domain are fixed points of the map that takes x to x^p. And this turns out to be a forerunner of the Weil conjectures, which hinge on an analogy between, on the one hand, counting the fixed points of continuous functions that relate points in a mathematical object called a topological space to other points in that space, and, on the other hand, counting solutions to systems of polynomial equations over finite fields of numbers.
Even Fermat’s relatively simple theorem starts to grow hair when I try to lay it out in ordinary language, I realize, and it’s hard to articulate why it’s interesting without invoking more math. At the end of the day, why would it matter to a nonmathematician that André Weil figured out how to count solutions to polynomial equations in finite number fields? In one sense, it doesn’t, not to me. I don’t understand it well enough for it to matter. But at the same time there’s a flicker of fascination, a door that cracks open just a sliver when I learn about these constructed realms and the relations within and among them, whether the realm is as simple as the numbers from 0 to p–1, or something too complicated for me to fathom. It’s not so much the particular result as the intricate mesh of them that moves me: models nested within models, labyrinths built on top of labyrinths, the unlikely connections—the eros that André wrote about in his letter to Simone—in this mental universe.
And again I picture the prickly mathematician in his basement, surveying one section of his landscape while the cat looks on.
Simone lands in the hospital and never recovers.
Since leaving France she’s eaten less and less, wanting to consume no more than what is rationed to children in France—a quantity concerning which she has no actual information. She makes up an amount, then restricts herself to less than that, in the same way that in the past she thought (mistakenly) that the working class was forced to live without heat and so refused to heat her own rooms.
Despairing, hardly eating, working herself to the bone. In April a friend finds her collapsed in her room and brings her to the hospital, where she is diagnosed with tuberculosis. She needs rest and food, they say. She spends three months in Middlesex Hospital, resting but not eating enough to recover. Her digestion is shot and she has no appetite; some days she’s too weak to hold a spoon. A friend who comes to see her is appalled by her condition. Another is touched by what he perceives to be a spirit on the brink of releasing itself from the flesh. She implores everyone who visits not to inform her parents that she’s in the hospital, and writes her former address on the letters she sends them, misrepresenting where she is. Her letters are one long lie full of tenderness, her friend and biographer Simone Pétrement would write.
The less she eats, the stronger her wish to take Communion becomes. In Simone’s final months the Abbé de Naurois, chaplain of the Free French forces, makes three trips to see her. I picture him, no doubt wrongly, with a pasty but smooth complexion and a fine wool suit, the clever third son of an industrialist, let’s say, his intelligence palpable enough for Simone to grab and shake. That is to say, they argue the way she argues with every man of the cloth. She claims she’s only trying to find out whether he would consider her eligible to be baptized, but under that pretext she rattles on without listening to his responses, criticizing Catholic dogma, zeroing in on the church’s doctrine of salvation and wanting to know precisely whom it includes or excludes.
The abbé sits next to the bed with his hands on his knees as this woman, this febrile figure in spectacles, half covered by a sheet, barely strong enough to move her legs, her arms even, binds him in a long, baffling chain of logic, then pauses and is silent for a while. Then she lets fly another knotty sentence.
The abbé interrupts, or tries to—her thinking is confused, he finds, contorted, swerving this way and that. “The acrobatics of a squirrel in a revolving cage,” is how he’ll one day describe her ruminations during those visits. He sits there, and as her flood of argument washes over him, he begins to doubt the very worth of the intellect, that it could spew this hairball of thought that, it seems to him, could only interfere with the spiritual contact she longs to experience.
Yet at the end of each meeting he blesses her, and she goes silent and is suddenly so gentle. Docile even, a wide-eyed young girl. Here is an extraordinarily pure and generous soul, he realizes. He will remember her that way, as a paragon of seeking.
The Weil siblings both undertook to translate into language something beyond words, beyond symbols, in Simone’s case maybe beyond thought itself. I can only follow either of them so far, reading their words and making guesses as to what lay beyond articulation. Each had the run of an elaborate mental (or mental-spiritual) universe, each subjected perceptions to a ruthless accounting.
They thought their way into esoteric domains, found purpose in concentrated inquiry and likewise in the glimpse, the pursuit, the almost there, the exhilarations, the frustrations, of being partially shown and at the same time denied the dangling fruits of their searches.
Weeks or months, in wartime there’s no telling how long it might take for a letter to cross the Atlantic, and so the correspondence between André and Simone is erratic. Their dialogue dwindles, and in what will turn out to be their last letters, they hardly know what to say.
Try to write us sometime and let us know how you are doing, André asks.
I haven’t written until now because it’s truly difficult to know what to write, Simone begins her reply, this on the heels of a period during which she seemed unable to stop writing, when about subjects other than herself.
London is full of fruit trees and flowers, she writes to André.
I have lately made the acquaintance of several charming young girls, she reports to her parents, omitting the fact that these girls are nurses in a hospital where she is a patient.
Here nothing new, André writes in July. Your niece is growing normally and continues to have a happy character. She seems to find life pleasant and agreeable.
A month later, he receives a telegram, out of the blue. YOUR SISTER DIED PEACEFULLY YESTERDAY, it says. SHE NEVER WANTED TO LET YOU KNOW.
She was thirty-four. “The deceased did kill and slay herself by refusing to eat while the balance of her mind was disturbed,” reported the coroner. It’s not entirely incorrect to say tha
t she starved herself to death, but the full story is stranger than that, both more and less gruesome, or maybe it is better said that the details of self-starvation are not necessarily what we would imagine, or that we wouldn’t truly be able to imagine self-starvation at all. What were her intentions? She may not have willed her death, yet it seems as though she might have been able to will her survival, or at least made more of an effort to survive.
The friends who visited her didn’t believe that she wanted to die. Still she wrote, in a letter to another friend, “I am done, broken. Perhaps the object might be provisionally reassembled, but even this provisional reassembly can only be done by my parents”—her parents who had always swooped in to rescue her when she was on the brink but who were now stranded in New York, unaware of her condition and without visas.
She tried to eat, requested mashed potatoes prepared in a certain French way, consumed an egg yolk mixed with sherry, after that refused food for fear she could not tolerate it.
Blind man’s stick, she wrote in her notebook.
To perceive one’s own existence not as itself but as part of God’s will.
A supernatural faculty.
Charity.
The eternal part of the soul feeds on hunger.
Nurses.
PART FIVE
11.
Simone’s notebooks are sent to her parents, who lug them from one country to another, all through their wartime migrations, from New York to Brazil, then to Switzerland, and then at last back to Paris, where the apartment on rue Auguste-Comte has been stripped of its furniture. All that remains is a large framed wall mirror, so tarnished it barely returns their reflections, not that they’re inclined to look. They ease into the old rooms, now slowly molting a fur of dust and vacancy, everywhere shadows—unexpected traces, seen from the corner of an eye if never dead-on, of the other rooms they’ve taken and left since they departed these, the hotels and apartments and the camp in Morocco where Simone sat all day in that wooden chair.
There’s a trail of dead leaves along a windowsill, a desiccated scrap of orange peel, a smell of kerosene. They heave open the casements, lift the limp clothing from their trunks. They buy beds and tables and lamps. And then, once everything is in order, to the extent that it can be in order, they start on the notebooks: every day Bernard and Selma, not quite ruined, sit across from each other at the wooden dining table and copy their daughter’s thoughts, line by line, from her original notebooks into a series of blank accounting ledgers. Two bent heads, two pairs of eyeglasses. They follow along with their fingers, mouth her words as they write.
Having outlived her, having to swallow daily this fact that won’t stay down, everywhere they go they find the same bleak streets, the same wrecking balls of memory. Maybe they wouldn’t even remember whose idea it was to transcribe the notebooks. They’re so steady in their efforts that for a long time their granddaughter Sylvie, when she comes to visit, concludes that this work of copying is their job. And really she’s right, it is their job, though no one has hired them to do it. They’ve become their daughter’s most diligent students, reading her, glimpsing her between the lines.
(How she went stomping through doorways.)
They keep the original notebooks and the copies in a cabinet in the living room, where other people might keep linens or vases or playing cards.
(A stub of pencil between her lips as she read the newspaper. Or at work in the other room, calling to them that she would come out to eat in just a moment. J’arrive!)
But would you, could anyone, want this? That in the event of your death your parents would slowly, laboriously copy the contents of your notebooks into other notebooks?
(Her face when she listened to music.)
(Her little hands.)
Day after day.
Philolaus of Croton was said to have been the first to reveal in writing the beliefs of the Pythagoreans. He was also said to have conjectured the existence of a planet no one had seen: since the number ten was, in the Pythagorean view, the most perfect number, and only nine heavenly bodies were known (the earth, the moon, the sun, and six planets), he postulated a tenth one, an invisible planet. A counter-earth.
According to one story his work was plagiarized, by none other than Plato. There is also a theory that many of the surviving fragments of his writing may not be his writing but instead a forgery, produced much later by someone familiar with Aristotle’s account of Pythagorean thought.
One way or another, posterity has linked him to stolen ideas.
Counter-children, counter-trees, counter-courtyards, counter-dishes, counter-dogs?
Simone would allude to him repeatedly in her last notebooks, that is to say, she invoked the “Philolaus” who authored the texts, who may or may not have been the true Philolaus. She had a way in the notebooks of cycling through what she knew, returning again and again to certain lodestones of her thinking. Passages from the Bible, from the ancient Greeks, fragments by which she recalled herself.
Sometimes she just wrote the name “Philolaus”—an invocation of the idea (stolen or not) that mathematical truth was, in those ancient times, inseparable from spiritual truth, that mathematics was a bridge to the divine.
“And all things that can be known contain number,” wrote Philolaus, or the imposter writing as Philolaus. “Without this nothing can be thought or known.”
She kept turning over math in her mind. In those late writings there are not only notes on the history of algebra but pages of actual calculations in trigonometry and combinatorics. At the same time, she considered math from the outside, wrote of math as a model of certainty, math as an image of divine things.
She repeated to herself what she’d written privately to her brother—that contemporary mathematicians had roamed too far off course. The reign of algebra, she would say, is like the reign of finance: Just as money has gummed up the relationship between work and its products, so math too has become divorced from the material world. “The relation of the sign to the thing being signified is being destroyed, the game of exchanges between signs is being multiplied of itself and for itself,” she wrote.
Ultimately she wanted to think her way past thinking. For her, the ultimate goal, the thing for which math trains us, is to surpass the part of our brain that does math, to transcend reason, even to transcend time.
“One needs to have traversed the perpetual duration of time within a finite period of time,” she wrote. “In order that this contradiction may be possible it is necessary that the part of the soul which is on the level of time—the part that reasons discursively and measures—should be destroyed.”
Logic sometimes makes monsters, Poincaré said.
In the final chapter of his memoir, André tells of receiving the cable that announced his sister’s death, a telegram that would forever remain, as he put it, etched in his mind. All he says of his reaction is that it was suppressed.
“How can I describe my grief?” he wrote. “But I did not have the luxury of indulging it; it was up to me to inform my parents, and I did not feel equal to the task.”
It could be an accident of timing that the memoir, though he wrote it late in life, concludes soon after Simone’s death, that the years of his apprenticeship, as he construed it, happened to roughly coincide with the years she was alive. Or not an accident and yet not the sole reason he would’ve bracketed his memories in that way: it was the sum of everything he experienced over the course of the war—prison, the military, emigration, the birth of his first child, and the death of his sister—that caused him to divide his life into before and after.
Still, I wonder whether his very mode of remembering might have altered after he lost her, once his first and best witness was gone. Decades later, responding to an interviewer who asked him why he had confined his autobiography to his first forty years, he said, “I had no story to tell about my life after that.”
No story, even though after that, in 1946, he and Eveline had a second daught
er, Nicolette, and after that, in 1949, he published a paper, “Numbers of Solutions of Equations in Finite Fields,” laying out what would become known as the Weil conjectures. Not just an achievement but a landmark, setting a course for algebraic geometry to follow for the rest of the century. (A fresh set of equipment had to be invented in order to verify the conjectures—the proofs were assembled, over time, by Bernard Dwork, then by Alexander Grothendieck, and finally by Pierre Deligne.)
Yet he had nothing more to say about himself and sneered at the “very boring” autobiographies some of his peers had written, which seemed to him mere litanies of their academic appointments and the theorems they’d proved.
Simone was “naturally bright and full of mirth,” André would write, “and she retained her sense of humor even when the world had added on a layer of inexorable sadness.” As adults, he noted, they’d had few serious conversations.
“But if the joys and sorrows of her adolescence were never known to me at all, if her behavior later on often struck me (and probably for good cause) as flying in the face of common sense, still we remained always close enough to one another so that nothing about her really came as a surprise to me—with the sole exception of her death. This I did not expect, for I confess that I had thought her indestructible.”
When does the story of a life end? Simone went on to a posthumous existence that she wouldn’t have anticipated: she became famous, an intellectual celebrity in France and abroad, her work published and translated and admired by the likes of T. S. Eliot, Mary McCarthy, Albert Camus, Elizabeth Hardwick, Czesław Miłosz, Susan Sontag, and Iris Murdoch. Having written so relentlessly and died so young, she acquired, after death, the burnish of genius cut short, an Elliott Smith for the Partisan Review set. Her fans were intellectuals unmoored from tradition and shaken by the atrocities of war; maybe she spoke to a desire for some (necessarily cryptic, necessarily tragic) sense of what it all meant, a wish, even as they distanced themselves from the religions of their parents, to conceive of a philosophically respectable spiritual life, to rehabilitate the idea of the soul.