Review of The Mathematician’s Brain

Birth of a Theorem

The title is slightly misleading, as it might lead one to expect an analysis of how mathematicians work based on psychology or neuroscience. In fact, there is very little of that, especially if one discounts a short chapter on Freud’s views (which don’t really have much of value on the subject). So it would be justifiable to be disappointed with the book if one had that expectation.

While it would be great if someday scientific study of the brain could reveal the mechanisms of creativity in mathematics or in general, that’s not the situation now. Ruelle does report the ideas on mathematical creativity of Poincaré and Hadamard, but there hasn’t been much to add in the last 100 years. Nevertheless, the present book is an excellent follow-on to Ruelle’s Chance and Chaos (reviewed here). These two have much in common: they are relatively short, comprise a number of brief chapters on diverse topics, and are quick reads. Both books cover a lot more ground than their titles would suggest. But, happily, there’s little overlap between them.

In one respect the present book is even shorter than its 130 pages (exclusive of the excellent notes at the end) would suggest, because it’s really two books in one. The first part, about 60 pages, is mostly nontechnical and quite accessible to general readers. It gives, in that short space, a clear picture of the general subject matter of concern to mathematicians since the ancient Greeks. (Which is far more than just arithmetic, geometry, algebra, and calculus.) The remainder of the book deals with more technical topics. There are some more detailed explanations in the notes. However, a reader without some college-level math courses would need to be satisfied to accept the technical terminology without much explication.

But the reward from reading the later chapters, for readers with any amount of mathematical background, is a clear picture of how professional mathematicians actually work. I’ve given the book a top rating for its coverage of topics such as sketches of one of the 20th century’s most original mathematicians, Alexander Grothendieck (whom Ruelle knew personally), set theory as a foundation for mathematics, Gödel’s incompleteness theorem (very succinctly explained), and the Riemann Hypothesis (and the intriguing idea that if RH could be proven to be undecidable within standard arithmetic, it should be provable in a broader theory).

There’s a lot more. The general idea of mathematical structures is explained, and how new concepts are created. This leads to a discussion of whether mathematical concepts are “created” or “discovered” – a very old debate. (There seems to be consensus now that the truth is “some of both”.) There’s a discussion of a theorem, the “Circle Theorem”, which was actually discovered by physicists T. D. Lee and C. N. Yang (best known as winners of a Nobel Prize for discovering CP parity nonconservation). The Circle Theorem, which isn’t well-known to mathematicians, is surprising but simple to state, and has a relatively simple proof.

Finally, there is a discussion of the nitty-gritty details of proving some of the complicated theorems of modern mathematics. In general, what makes this activity difficult is the huge number of possible approaches to finding proofs. It’s like finding one’s way through a high-dimensional labyrinth to a valid proof. There are a huge number of possible paths to explore, almost all of which lead nowhere. Yet, somehow, a process of enlightened “tinkering” with the details of a proof can succeed. This involves proving just the right set of lemmas with a variety of slightly different hypotheses and conclusions in order to reach the desired result. The chapter on “The Strategy of Mathematical Invention” gives a sketch of how this is done, but there are no explicit examples. Fortunately, for anyone who’s curious about this, Cédric Villani’s Birth of a Theorem (reviewed here) provides excruciating details of one example of this process.

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