Review: The Weil Conjectures by Karen Olsson

The Weil Conjectures by Karen Olsson

I. 

I liked that this was a book full of questions, and I also liked the questions. Here are some of the questions she asks in the book:

  • Why do people like math?
  • Why did I (the author Karen Olsson) like math when I studied it in college? Even though I was an aspiring novelist?
  • Why was Simone Weil — philosopher, writer, mystic — attracted to mathematics?
  • What’s the deal with Simone’s relationship with her brother (famed mathematician) Andre?
  • Where do mathematical ideas come from?
  • What do (the author Karen Olsson) get out of abstract math now that I’m no longer swimming in it?
  • Do analogies for abstract mathematical ideas do a person any good if the math itself isn’t accessible to them?

It wouldn’t be fair to Olsson or the book to reduce it to a neat set of answers to those questions. The book is structured so as to provide an experience that is a lot like the experience of learning abstract math. The Weil Conjectures suppose a connection between two mathematical domains — topology and number theory — and Olsson wants (as far as I understand) to create a literary experience that is analogous to the search for such domain-bridging mathematical connections. So she lays out the Weil biography, her own memoir, mathematics and writing about mathematics for the reader. And I think she really succeeds — as the book goes on it feels a bit like learning some deep bit of theory.

So it’s not fair to reduce the book to a neat set of answers because the book is primarily about the experience of reading it. Some books are like that, and that’s fine. But she does answer some of those questions in interesting ways, and memory is necessarily structured, so it’s worth trying to say a bit about what Olsson says about mathematics itself.

II.

Q: What do people love about abstract mathematics?

A: Attraction to the unknown itself.

That, I think, is as close as we can get to Olsson’s answer in brief. The mathematician is someone who desires to create unknowns and to put obstacles in the way of their knowing so that they can search for answers. So for instance we hear about a Kafka story (“The Top”) about a philosopher who seeks enlightenment by hoping to catch spinning tops in mid-spin (whatever). On this story Anne Carson says “he has become a philosopher (that is one whose profession is to delight in understanding) in order to furnish himself with pretexts for running after tops.” That’s what a mathematician is — they love the chase.

Q: Is this so different from what a writer does?

A: No.

And so there is a connection between Olsson’s 2.5 years studying mathematics in college and her life as a writer.

I love this take on mathematics — that it’s about this love of being in the dark and searching for light. So it’s more about finding light than the light itself, if that makes sense.

This should be seen (I think) in contrast with writers who make much of the beauty of mathematics, or the search for beauty. Olsson is good on this. Twenty years after finishing her degree she decides to go back and watch some online lectures for an Abstract Algebra class. She finds it remote and foreign, but also:

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.

Still, the mathematician’s next move is to plunge themselves into darkness. This comes from a desire towards something that cannot be grasped.

This part was hard for me to understand. A key for Olsson seems to be Anne Carson’s Eros the Bittersweet but the theory isn’t entirely clicking for me. “A mood of knowledge is emitted by the spark that leaps in the lover’s soul,” she writes but I don’t quite get. Olsson’s take: “It’s not the knowledge itself, not consummation but the mood, the excitement when you are on the verge of grasping.”

What I understand Olsson to be saying is that the main fun of math isn’t the understanding but the feeling that understanding might be near. And that explains the pleasure we non-experts get out of mathematical analogies. There’s nothing unusual about the idea that analogies give us the thrill of desire — what’s more novel is saying this isn’t so different than the usual state of a mathematician.

For this point she goes to the act of mathematical creation itself, and how unpredictable it is:

What does mathematical creation consist of? asks Poincare, who blazed his way through a large territory of mathematics and physics by relying on his remarkable geometric intuition. It requires not only the combining of existing facts but the avoiding of useless combinations: making the right choices. The facts worthy of study are those that reveal unsuspected relationships between other facts. Moreover, much of this combining and discarding and retrieving goes on without the mathematician’s full awareness, occurring instead behind the scrim of consciousness.

Since the mathematician is dependent on their unconscious associations, mathematical discovery is not entirely in their control. In fact, many suspected relationships don’t work out. And so the mathematician spends most of their time afflicted with that same desire for the unreachable that afflicts we, the non-technical lay audience, who only get analogies.

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.

It’s a neat picture, I think!

I’ve left out all the connections to mysticism, the biographical details of Simone and Andre, and (nearly) all the connections to writing, but that’s in there too. Again, very neat stuff.

III.

I also learned from this book that Brouwer retired early and practiced nudism, that Flannery O’Connor didn’t particularly care for Simone Weil’s writing, and of Hadamard’s fascinating book “The Psychology of Invention in the Mathematical Field.”

One last good quote: “Honestly I think I understand anyone else’s dislike of math better than I understand whatever hold math has had on me.”

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