The Weil Conjectures
Page 16
I give up—for half a minute or so—and then I think, No, dammit, I’m going to find that guy and talk to him. I begin scanning the aisles, one by one, and finally corner the poor man by the tortillas. If there’s one thing I had plenty of chances to practice during years of work as a journalist, it’s approaching somebody I feel quite shy about approaching. Yet I’m not sure whether repetition has made it any easier. My smile feels like a sticker slapped onto my face.
Yes, he says, you sent me an e-mail. I’m sorry that I never replied—
Oh that’s okay, I say. E-mails! Maybe there’s someone else in your department that I could to talk to—
I’m probably the one, he says, slowly, not with any evident enthusiasm.
In other words, it’s pretty awkward. What do I even want from him? I’m about to leave town for a couple of weeks, I tell him, and we agree, tentatively, that we’ll have coffee once I’m back.
A very broad, bird’s-eye overview of the development of twentieth-century mathematics (which I’m mostly borrowing from a lecture given by the mathematician Michael Atiyah) goes like this: In the first half of the century, the prevailing concern was to define and formalize things, as in the Bourbaki effort, and to pursue work specific to the different subject areas within mathematics. Then in the second half, mathematicians looked for ways to tie together those different areas, to transfer techniques from one to the other, and math became more global, even as it expanded so much that it became impossible for anyone to fully comprehend all that was happening in the field.
So one can faithfully call André Weil a pivotal figure in twentieth-century mathematics. He helped to found Bourbaki in his early career, emblematic of that era’s determination to ground and make rigorous the subfields of math, and not long after that he began to envision connections between mathematical land masses, which his successors would build out.
It’s very hard to conceive of how mathematicians from a long time ago thought about things, Atiyah said in that same lecture, because subsequent discoveries have become so ingrained. “In fact if you make a really important discovery in mathematics you will get omitted altogether!” said Atiyah in that same lecture. “You simply get absorbed into the background.”
Another reckless dash through twentieth-century math might emphasize that as time went on (at least in some camps), identity became less important than relatedness. Theories were developed in which knowing relationships among mathematical objects matters more than knowing about the mathematical objects themselves. Or you could even say that knowing an object itself is the same thing as knowing its relations to other objects of the same kind.
A mathematics increasingly globalized, increasingly concerned with links from one thing to another, increasingly aided by computers, a highly connected world in which it’s still very difficult to see much more than your own small part—remote as the world of math is, this all begins to seem familiar. Like that forest of links in which I keep losing my way.
But there’s a second part to my supermarket chat with the mathematician, in which I learn that he has a son the same age as my son, and we take refuge in that, as parent-strangers will do. His son (go figure) likes math too. We compare notes about math-related books for kids, and somehow that makes it seem more possible that we will actually at some point get together and talk about the Weil conjectures.
“I can remember Bertrand Russell telling me of a horrible dream,” wrote Godfrey Harold Hardy in his book A Mathematician’s Apology. “He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of [Russell’s own] Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated . . .”
The mathematician is required to disappear.
Goro Shimura saw André Weil for the last time on a December afternoon in 1996, at the Institute for Advanced Study. Shimura, by then a professor at Princeton, had tried to speak to Weil by phone the day before, but Weil couldn’t hear very well, and he’d asked Shimura to meet him at the institute instead. Come tomorrow, he said, otherwise I won’t remember.
It had been drizzling endlessly, Shimura would later write. Forty-one years had passed since they first met in Tokyo. When he arrived at the institute, Shimura discovered that Weil had forgotten his hearing aid, and they drove to his house for it. Yet even with the device Weil couldn’t hear well, and at lunch their conversation was halting and uncomfortable.
Shimura was working on a problem; he’d had a new idea about the Siegel mass formula, a topic that Weil had once studied, but when Shimura asked about the history of the subject, all Weil could say was, “I don’t remember.” He kept saying the same thing over and over: That was a long time ago, I don’t remember.
Shimura asked whether he was still writing history.
I cannot write anymore, he said.
They left the dining hall and walked through the drizzle to Shimura’s car.
You are certainly disappointed, but I am disappointed too, Weil said, with myself.
And then he said again, I cannot write anymore.
This episode appears in a reminiscence that Shimura wrote after Weil died, for the Bulletin of the American Mathematical Society. Like his earlier “Yutaka Taniyama and His Time,” it’s a beautiful tribute, and just based on those two essays I’m ready to pronounce Shimura the great elegist of twentieth-century mathematics.
Is that what I am writing, I wonder, some sort of elegy for math, or for my own entanglement with math? At times it feels that way, but I don’t think that’s what this is. As it turns out, one stilted encounter in the supermarket is enough to send me back to the algebra lectures, which for no good reason I still want to finish. And so it’s on to ring theory, which is of course nothing I need to remember, nothing I need to know.
A mathematician might dream of an afterlife in which all is revealed, mathematical structures extending as far and wide and high as the eye can see, their nature and relations made transparent, and look, there are Galois and Archimedes strolling by, there Germain and Taniyama deep in conversation. And in that paradise, maybe, Simone and André would be reunited; they would find each other and argue in ancient languages all through the never-ending day.
Their earthly afterlives are, in a sense, opposites: the image of Simone the person weighs upon her writings, inseparable from them, even overtaking them, while André grows ever more attenuated. His person will vanish—be absorbed into the background—and leave behind just a name, a quartet of letters attached to other symbols, theorems and conjectures.
Where are numbers? my son asks. And where, for that matter, are all the unknown theorems, all the hidden proofs, all the math not yet discovered?
Thirty-five hundred years ago, a Babylonian presses a wedge into clay. One, two, three times. Then leaves a gap. Then presses again.
ACKNOWLEDGMENTS
Writing this book was an unexpected journey down a rabbit hole, and I’ve been extremely grateful for the help and support I received along the way. Early reads and encouragement from Cecily Parks, Louisa Hall, and Andrew Bujalski were crucial. Amy Williams’s enthusiasm was heartening and infectious. I am also indebted to the following people who read drafts of the book and offered insightful comments: Ric Ancel, David Ben-Zvi, Pam Colloff, Lauren Meyers, Dominic Smith, and Kirk Walsh. And to Jessica Halonen for her wonderful art.
I count myself very fortunate to have Emily Bell as an editor, and I’d also like to thank everyone else at FSG, in particular Jackson Howard, Lottchen Shivers, Karla Eoff, Scott Auerbach, and Thomas Colligan.
Thanks also to the Ucross Foundation, and to Sharon Dynak and Tracey Kikut and Cindy Brooks in particular, for the gifts
of time and space and sustenance.
Many books and articles informed this project, and of those I’d like to acknowledge The Apprenticeship of a Mathematician, by André Weil; Simone Weil’s essays and letters; At Home with André and Simone Weil, by Sylvie Weil; Simone Weil, by Simone Pétrement; Remarkable Mathematicians, by Ioan James; Bourbaki: A Secret Society of Mathematicians, by Maurice Mashaal; Mathematical Thought from Ancient to Modern Times, by Morris Kline; and Mathematics Without Apologies, by Michael Harris. Excerpts from André’s 1940 prison letter to his sister are from Martin Krieger’s translation.
Though I’m not able to name them all, I am also thankful for every schoolteacher, professor, teaching assistant, and fellow student who taught me math over the years.
A Note About the Author
Karen Olsson is the author of the novels Waterloo and All the Houses. She has written for The New York Times Magazine, Slate, Bookforum, and Texas Monthly, among other publications, and is a former editor of the Texas Observer. She graduated from Harvard University with a degree in mathematics and lives in Austin, Texas, with her family.
First published in Great Britain 2019
This electronic edition published in 2019 by Bloomsbury Publishing Plc
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