The Weil Conjectures

Joint biographies are not uncommon: we have Gelhorn and Hemingway, Custer and Crazy Horse, Hannibal and Scipio Africanus, for example. Now Karen Olsson has written a joint biography of Simone Weil and her brother, André . Simone will be familiar to many readers, André  to relatively few, I suspect. She was a writer, a religious thinker, and some say, a saint, who was driven to seek out the most painful and laborious undertakings, her chronic ill health notwithstanding. He was a distinguished mathematician whose famous "conjectures" still occupy the attention of learned scholars today.

Do these two exotic spirits have anything much in common, other than the fact that they're siblings? Olsson thinks they might. In any case, she makes use of their letters, writings, and biographies to tell their stories, shuffling back and forth from one to the other, and along the way she also takes time to explore the nether regions of both spirituality and mathematics. 

In the midst of these free-floating segments Olsson also tells us quite a bit about her own experience as a undergrad in the math department at Harvard, where she made it a few rungs up the ladder but soon realized she was not one of those wunderkinds for whom everything came easy—and in the world of math, they're the only ones who matter. She dropped the subject and became a writer, without entirely losing her interest in math's mysterious and often incomprehensible worlds of correspondence, transmutation, speculation, and evident paradox. She compares her efforts to share her interest with readers to that of describing a Beethoven symphony to a deaf person. Often she admits that she doesn't really understand what she's talking about either.

I know nothing about higher math. We didn't even have a calculus class in the small-town high school I attended. I was good at algebra and geometry, and I once sent a note to Martin Gardner offering my proof of the Three-Color-Map problem, which had stumped the experts for centuries. (He sent me a kind postcard explaining that my "proof" was not convincing.  Someday, I will be vindicated.)

My "declared" major as a college freshman was math, but I dropped that pursuit after taking my first freshman history class at the U. History was messier, but much more full of life. Olsson has pursued the subject far enough to offer us a splendid smorgasbord of mathematical fields, problems, and personalities, without expecting us to decipher a single complex formula. She displays a knack for evoking the abstract worlds that numbers, shapes, fields, and theorems describe, and sometimes create for themselves, without confusing us with the details. For example, at one point she describes the work of the nineteenth-century Norwegian mathematician Sophus Lie as follows:

"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 to be entirely separate, namely differential equations. It's as though he had located a wormhole from one mathematical realm to another." 

This is the kind of math that André  Weil's sister Simone was suspicious off. It didn't refer to anything real. It was as if mathematicians were working up half of a structure out of whole cloth, and then attempting to "prove" what the other half would have to look like. Though Simone was good at math herself, up to a point, she was attracted to the physical—factory work, for example—seemingly less out of personal masochism than out of a desire to fell the suffering that those around her often felt.

Olsson admits that in college she owned a copy of The Simone Weil Reader, but can't remember if she ever did more than skim through it. (I have the same book, and I feel the same way about it. I should take a closer look.) And she presents little in the way of quotations to confirm the view of T.S. Eliot, Susan Sontag, and many others, that Simone was an extraordinary thinker, writer, and human being. But the letters Simone exchanged with her brother, to take one example, form an important part of the book. And in the end, along with other details, they expose a fascinating community of personalities rather than a litany of solitary and heroic geniuses.

In one longish passage Ollson refers explicitly to the social aspect of a mathematician's world:    

"The seeming fixedness of mathematics is surely one of the reasons I’ve felt drawn back to it, given our present-day world’s particular instabilities and alternative facts, but an­other reason, a stronger reason, is that my son likes math, which is not to say that I need to relearn abstract algebra for his sake but rather that his excitement has reminded me of my own old excitement, has made me want to blow on the embers—has made me realize there are embers. And as I do, what strikes me are the dialogues, the exchanges, whether it's me talking with my son about numbers or Benedict Gross’s [online] performance of algebra. Even as mathematics presents itself from afar as an austere architecture dreamed up by singular geniuses, up close it’s a torrent of transmissions, teachers lec­turing, college kids trying to solve problems together, col­leagues at conferences, André  writing to his sister. For every solitary discovery there are massive systems of relationships, which I begin to think of as a kind of giant math ant colony, or math hive, and I even begin to wonder whether (or con­jecture that) the desire for mathematical revelation, the wish to dwell in a perfect, abstract world, is secretly, unconsciously twinned by another desire for communion. One the negative imprint of the other. Abstraction the flip side of love."

At this point Olsson adds a remark by Simone. "Nothing which exists is absolutely worthy of love. We must therefore love that which does not exist.” 

No aspect of the tale interests Olsson more the obscure mental process by which mathematicians make their discoveries. Often enlightenment coming only after months or even years of unproductive "head-banging," as she calls it. An insight may come in a dream as a visual image, or during a feverish, sleepless night. She also notes that some mathematicians, including André  Weil, took special pleasure during that phrase when it was clear that some new discovery was "on the tip of one's tongue" but had not yet been fully grasped.

"The cruelty in all this is that the head-banging hardly guarantees the revelation, that to be an ambitious mathe­matician is to spend much if not most of one’s time being stuck... "

André  Weil, in one of his letters to his sister describing the process through which he arrived at the solution to a sticky problem, admitted that “the pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time.” Olsson concludes that "the flicker of a parallel, the suspicion of a connection, excited him, more so than nailing it down, working out the details. As though knowledge itself were a bit of a letdown: it’s being on the cusp that brings the greater thrill."

The Weil Conjectures has the twin virtues of being short and also so loosely organized that the reader could almost pick it up at any point and start reading. But better to begin at the beginning. Its protagonists reappear again and again—not only André and Simone but also historical personages like Fermat, Gauss, Archimedes, Hadamard, and Poincaré.     

The layering of personalities, events, and points of view allows Olsson to raise all kinds of conjectures without arriving at any conclusions about either the "reality" of various fields of math or their possible connections with spiritual enlightenment. We're left with a tantalizing, tingling sense of awe, as if we, too, were on the cusp of some important discovery, not about math, but about the universe, or ourselves.