The Once-Mathematicians

Henry Reichard

The Weil Conjectures, by Karen Olsson, Farrar, Strauss & Giroux, 2019, 224 Pages, $13.99

My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.

-Hermann Weyl

Among mathematicians, there is a truth universally acknowledged: that numbers are beautiful, but that most people are either too stupid or too mean-spirited to appreciate them. There is a sense that mathematics – more than poetry or music or scripture – is the language of the universe. “Let no one ignorant of geometry enter here,” Plato wrote above the doors of the Academy. Those who understand mathematics are the witnesses to absolute truth. The rest of us — poor, illiterate souls that we are — are the ones left behind.

“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way,” G.H. Hardy wrote in A Mathematician’s Apology. “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

“Why are numbers beautiful?” Paul Erdős repeated, incredulous, when a non-mathematician once asked him the question. “It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.” 

“My dear sister,” André Weil wrote in a letter to his sister, Simone, after she asked him to explain his work to her in simple terms. “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.”

“In my mind,” Karen Olsson writes, “I am that deaf person, nagging a composer to explain a symphony.”

Images and themes, sad harmonies, triumphant dissonances. This, in a nutshell, is the content of Olsson’s The Weil Conjectures: On Math and the Pursuit of the Unknown – the latest in a small collection of popular math books that try to democratize the beauty of numbers. Such books are, as a rule, few and far between. Few non-mathematicians consider numbers to be beautiful, and only a handful of mathematicians care enough about what the rest of the world thinks to explain why they are. In 1940, the world-famous mathematician G.H. Hardy attempted to describe mathematical beauty in a long and passionate essay titled A Mathematician’s Apology. Yet Hardy also claimed that such apologies are written only by old, burnt-out mathematicians. “Exposition, criticism, appreciation, is work for second-rate minds,” he wrote. “It is a melancholy experience for a professional mathematician to find himself writing about mathematics.” In 2013, an algebraic geometer named Edward Frenkel made a similar attempt in a book called Love and Math about his own career as a mathematician (it later became a New York Times bestseller). And now, in 2019, Olsson (a journalist and fiction writer, not a mathematician) has written The Weil Conjectures.

Reading the first few paragraphs, you might conclude that Olsson’s book is a biography of the Weil siblings: André — a famous mathematician renowned for a series of eponymous conjectures and Simone — a philosopher, writer, and activist who lived most of her short life under her brother’s shadow. But The Weil Conjectures isn’t a mere biography. It’s an eclectic mesh of biography, autobiography, anecdote, philosophical rumination, and novelistic invention. “If this were a fable,” Olsson writes, “I might begin: Once there were a brother and sister who devoted themselves to the search for truth. A brother who spent his long life solving problems. A sister who died before she could solve the problem of life.” Olsson’s book isn’t a fable, or a novel, or a popular math book. It’s hard to tell what it is.

Unlike Frenkel and Hardy, Olsson doesn’t try to teach her readers higher-level mathematics. She doesn’t even make a serious attempt to explain the content of Andre’s conjectures (she admits to not fully understanding them herself). She studied mathematics decades ago, in college, and throughout the book she seems to be looking back at mathematics from a great distance, with a mixture of awe and nostalgia. She describes trying to relearn abstract algebra via an online course, reminisces on the discomfort she felt in math classes, describes walking up to a former classmate – who has since become a professional mathematician – and asking him, haltingly, to explain Weil’s conjectures to her. These little vignettes aren’t ancillary to the book’s main narrative. They are the book’s main narrative. Whereas Frenkel’s and Hardy’s books are both confident (and sometimes condescending) discourses on mathematical beauty delivered by authors fully in command of their subject, Olsson’s book is a faltering attempt to describe that beauty from an outsider’s perspective, delivered by an author who has forgotten most of the mathematics she once knew.

Many professional mathematicians – perhaps even many students of mathematics – will become irritated with Olsson’s half-explanations, mysticism, and personal anecdotes. I advise such readers to find a book written by a practicing mathematician. But to the layman, Olsson offers a rare glimpse into the world of abstract mathematics, and she also describes why an aspiring mathematician might turn away from that world, even after she has seen the beauty within it. I suspect that The Weil Conjectures will interest anyone who has turned away from an academic career or a childhood passion, if only because such readers will hear an echo of their own lives in Olsson’s story. 

I heard more than an echo. For me, Olsson’s book resembled a mirror. 

* * *

In certain people’s lives, there is a book, an essay, a poem, or even a single sentence that inspires them to take up a new line of thought which, in adulthood, will entirely enthrall and consume them. A young girl, reading Don Quixote for the first time, may become so entranced by the magic of words that she falls in love with reading, becomes just as mad and quixotic as the famous knight errant, and ends just as badly, by becoming a scholar of literature. A young boy, picking up Pilgrim at Tinker Creek several years before he is ready for it, may become convinced that there is no finer thing in the world than the life of an egomaniacal hermit. I will not even speak of the many promising children whom Frost led astray with “The Road Not Taken.” Such people account for at least half the mass of academia. For most of my time at college, I expected to become one of them. 

I also had been seduced by a book. I read it for the first time on a clear day in the summer before my senior year of high school. The fields, spread in even and endless lines of corn, looked from a distance like sheets of battered gold glinting under the sun; the trees were strong and full-leafed, verdant, ecstatic in fresh foliage; birds sang; flowers bloomed everywhere; dandelion seeds drifted in the wind like flecks of light swimming in a river. Time felt slow and heady, and I leaned back in a weatherworn armchair, reading a book about mathematical beauty with a picture of Van Gogh’s The Starry Night on its cover. It was Edward Frenkel’s Love and Math.

That summer marked the beginning of a love affair with numbers that I maintained, off and on, throughout college. In my first year at Yale, I took a course in pure mathematics taught by a professor who studied mathematics for its own sake: not because he wanted to use it to build a bridge or balance his checkbook, but because he thought that numbers were beautiful. I was inspired. I delved into pure mathematics, did mathematical research in the summer after my sophomore year, began to idolize Erdős and Euler and Gauss. But after three years, something shifted within me. I looked at equations and theorems that other mathematicians assured me were beautiful — equations that, a year before, I also had considered beautiful — and I felt nothing. I did mathematical research again, in the summer after my junior year, and did not enjoy it. And I became more and more obsessed with a sentence written by Clarice Lispector, in her novel Água Viva: “I can still reason – I studied mathematics, which is the madness of reason – but now I want the plasma – I want to eat straight from the placenta.”

I had done well in my classes, impressed my professors, and published a research paper. Yet now it seemed to me that I had spent the past three years imitating my classmates, pretending that I shared their passion for numbers, when in truth I did not. I suspected that I had misjudged myself. This whole business of shutting oneself up in an ivory tower, scribbling on blackboards, writing papers about algebraic varieties and irreducible representations, scarcely ever looking out the window – all of that seemed a musty and dreary sort of life, and it no longer appealed to me. I yearned for something else. I wanted to find my raw matter; I wanted to eat from the placenta. I decided to become a writer.

At this point, I’m afraid that I’ll have to ask for your forgiveness. All writers are afflicted by narcissism – yet my case (you must think to yourself) appears to be terminal. You began reading this piece with the expectation that it would be a review of The Weil Conjectures; instead I’ve started narrating my life history, have turned the book review into a thinly disguised personal essay, and at this rate (you mutter contemptuously to yourself) I’ll soon become so besotted with my own reflection that I’ll drown within it, and good riddance too. All of this is true, and quite reprehensible…but you must understand, I’m in a peculiar situation. I’ve been asked to write about an author, Karen Olsson, who also studied mathematics in college, also was inspired by an early professor, and also became disillusioned in her senior year, left numbers, and turned to writing. Not only that: Olsson herself, in her book, is writing about another long-dead writer, Simone Weil, who studied mathematics as a young girl, loved it at first, later became convinced that she lacked her brother’s genius, and abandoned it in college. My editors seem to find the entire situation rather humorous. You can picture us arrayed in a triptych: Simone in the center, me on the left, and Olsson on the right.

In truth, I would rather not tell you about my personal reformation. I’m worried about what others will say. Those who turn away from pure mathematics – physicists, statisticians, engineers, writers – are reviled by mathematicians with the same scorn a Catholic feels for an apostate. My former classmates and instructors, no doubt, will judge me. At best, they will pity me. 

* * *

Anyone who writes about mathematics must confront a single great dilemma: if one writes technically, then the result will be comprehensible only to mathematicians, but if one discards mathematical notation, then the result will be only a facsimile of mathematics —“A kind of poem, good or bad, unrelated to the thing it pretends to describe.” When Simone asked André to explain his research to her, he chose the technical approach. “I understood nothing of your sixteen-page letter (which I read several times),” she later wrote back to him. Olsson chooses the facsimile. “Having forgotten whatever I once knew about complex functions Fourier series field extensions compact surfaces hyperbolic spaces random walks et cetera,” she writes, “I am left with memories of the Science Center at Harvard, a building with a facade like stair-stepped boxes, constructed around the time I was born, in the early seventies.” It’s these memories — the after-images of mathematics, the hazy reflections, the fragments — that Olsson gives us. 

André Weil, in his old age, argued that only professional mathematicians should be allowed to write about mathematics. I suspect that he would have little regard for The Weil Conjectures. But I can’t dismiss the book so easily. It seems to me that such a critique misses the point. Yes, Olsson is a novelist, a journalist, not a real mathematician. Yes, one could learn more mathematical theorems by skimming the first chapter of a high-school algebra textbook than by reading the The Weil Conjectures. Yet dismissing her book on these grounds is like dismissing Walden because Thoreau’s narrative doesn’t give a sufficiently detailed account of farming and carpentry. 

I’ve heard that some over-credulous readers, inspired by Walden, decided to use it as a handbook for wilderness survival. Such readers were often discovered deep in the woods many months later: cold, half-starved, and rather disillusioned with Thoreau. Anyone who tried to use The Weil Conjectures as a handbook for mathematical research would soon become just as angry with Olsson. Her woods are, to most of us, more foreign and perhaps less hospitable than Thoreau’s. Yet mathematics is, in its way, just as mystical and elusive as anything Thoreau tried to find in nature. “If only we had more access to the untranslated thoughts, to the mystery of how the mind churns,” Olsson writes in one chapter. In another, she continues: “The Weil siblings both undertook to translate into language something beyond words, beyond symbols, in Simone’s case maybe a realm beyond thought itself. I can only follow either of them so far, reading their words and making guesses as to what lay beyond articulation.”

I am drawn to these phrases: beyond symbols, beyond thought, beyond articulation. In my view, they approach the heart of the matter. They describe what Olsson glimpsed at Harvard, what André saw fully but could not communicate to Simone: the untranslated thoughts, the beauty under chalk. But how to express it? How to put it into a book? How, in short, to articulate those things beyond articulation?

No one can fully appreciate the poverty of language until they have endured a transcendent experience. It’s only at this point that writers begin straining with words, wrestling with sentences. They become poets; they invent new words or join together old ones in oxymoronic pairs; they turn to simile and metaphor and allusion, omit punctuation, become incomprehensible. The moments when Olsson strains against words are the moments when she comes closest to expressing that beauty beyond articulation. They’re also the moments when she becomes the most mystical. 

Consider the following passage: 

 “Simone dreams of Crete. She travels to a village of ancient geniuses, who harvest grains and triangles, combine words and figures into a form of pure expressive speech, a giving back to the air in exchange for the gift of breath. They light fires on the beach. Fish swarm their feet in the shallows. Cherishing geometry and the ocean, they arrange their bodies in certain set configurations before they begin their open-air dances, which are cued by the rhythms of the waves. They draw pentagons in the sand, sing stories, wail for the dead, all forms of praise. The practice of an ecstatic order. They tell of the god who endowed them with speech as though he came by a few days ago. Yes, yes, they tell her, he was just here.”

I expect that most professional mathematicians will become irritated with Olsson when reading such paragraphs. They will say that Olsson (like Simone) indulges in mysticism, that she invokes mathematics without understanding it, in the same way that certain orators, hoping to give their speeches a note of righteousness, invoke God despite having never read the Bible. This is a shortcoming of which Olsson is well aware. She even wonders whether she’s still capable of caring about pure mathematics, so many years after college: “In one sense, I myself don’t care. I don’t understand it well enough to care.” 

In a book supposedly dedicated to Weil’s conjectures, such an admission of apathy is rather damning. Yet still I’m not quite able to dismiss Olsson’s book, and maybe the reason is this. Olsson isn’t really writing about André Weil. There are older, better biographies available. She isn’t writing about his groundbreaking research. A real mathematician could talk about that far more intelligently. No. She’s writing a book about Simone: the younger sister who couldn’t keep up with her brother, the woman who, long after she had abandoned a formal study of mathematics, was still haunted by it. Olsson is writing about those left behind. The deaf people trying to understand a symphony.

And the testimony of such a deaf person is, to me, far more captivating than the report of a seasoned concertgoer asked to review another symphony. Olsson notices things that professional mathematicians, too familiar with numbers, pass over. Few mathematicians would say that their field is “beyond articulation” — it is, after all, their job to articulate it. Yet I think that Olsson is right. Whatever part of mathematics we have managed to articulate must be a microscopic speck of the infinite whole: no larger than a drop of ink floating in the ocean. When I think of Olsson, I think of those churchgoers who stand at the edge of the congregation, humming the hymns because they can’t hear the words. And I can’t disdain her mysticism.

I’ve heard that Paul Erdős, one of the greatest mathematicians of the twentieth century, claimed that god keeps a Book in which all of the most beautiful mathematical proofs are tightly bound. Erdős said that god is jealous of this treasure and has permitted his creation to discover only a handful of proofs: the entirety of The Book is infinite, the portion discovered by man, infinitesimal. Is this not also mystical? Or Hardy’s claim — “Beauty is the first test: there is no permanent place in the world for ugly mathematics.” Is that not a sentence one might find in a holy text? 

All mathematicians are mystics. The ancient Pythagoreans practiced asceticism and worshipped the Pythian Apollo; modern-day mathematicians have discarded such rituals, but we have not let go of our beliefs. Mathematics is a holy art. We are all striving to articulate that which lies beyond articulation; we are all chasing after God, trying to snatch pages from the Book; we are all grasping at fragments, listening to echoes of the melody. Sometimes we hear it in an elegant equation or a powerful theorem. Sometimes it visits us of a sudden, as we walk down the street, in a moment when we understand how to complete a difficult proof. And sometimes — very rarely, but sometimes — an echo of that music can be bound within words, within a sentence about numbers.

All of us try to put numbers in our words, even as a musician tries to imbue her words with music, or a painter tries to give them color. That is all Olsson and I can do. On the second-to-last page of The Weil Conjectures, Olsson asks herself: “Is that what I am writing, I wonder, some sort of elegy for math, or for my own entanglement with math?” Yes, it is. The words and the sentences, those distorted echoes of mathematical beauty – they’re all that’s left to us, the once-mathematicians. 

Henry Reichard is a recent graduate of Yale currently reporting in Chile.