Love and Math. By Edward Frenkel

Love and Math. The Heart of Hidden Reality

Edward Frenkel

Basic Books. First published 2013.

The book traces the early career path of the author, a professor of mathematics at the University of California, Berkeley. He is well-respected for his mathematics, and popular due to this book, which made it to the New York Times bestseller list, as well due to a movie about art and maths that he produced, directed and acted in. He is the collaborator of the well-known physicist-mathematician Edward Witten. 

Frenkel was born in Russia during the late 1960s. He showed precocious mathematical skills, but was unable to have a clear upward trajectory in Russian academia owing to rampant antisemitism. In this book he describes his encounters with the academic bureaucracy in Russia as he came of mathematical age at the time of Glasnost and Perestroika. Interspersed with the inspiring tales of heroic dedication to maths are his wonderful descriptions of the mathematics. He does not shy away from complete and detailed descriptions of the most abstruse mathematical ideas and theorems, offering many detailed proofs, which, however, are largely consigned to the footnotes - not given at the foot of the pages, but collected together at the back. (I wish he had followed a slightly different format - for example setting these more technical portions in smaller type, but along with the main text.) So he starts with the symmetry of numbers, goes on from there to talk about symmetry groups, and different objects that belong to the same symmetry groups - integers, real numbers, rational numbers, irrational numbers and imaginary numbers - before going on to braids, which are also represented by the same symmetry operations. He talks about the Langlands program, about Galois groups and finite fields, about the Shimura-Taniyama-Weil conjecture and its connection to Fermat's last theorem, about modular forms and automorphic functions, about Riemann surfaces, manifolds, Lie groups, Kac-Moody algebras, automorphic sheaves, gauge theories, quantum duality, fiber bundles and branes. Obviously, in a book read over a week or so, it is impossible to convey anything but the most superficial acquaintance with these ideas. But even with this limitation, Frenkel conveys the beauty, the majesty and the mystery of modern mathematics. The best thing about this book is that excites the reader. It made me confident that the ideas described, while difficult, are not entirely in a different world of understanding, and that with a few years of effort, I, too, could access the concepts in satisfactory depth. Whether I will expend that kind of effort, of course, is an entirely different question. 

Frenkel is a modernist through and through, and has no time to consider, except disparagingly, the idea that the theorems constructed by mathematicians may be one version of the underlying nature of reality. He dismisses post-modernist ideas of the cultural relativism of maths (or of science, in general). In my admittedly extremely humble opinion the Wignerian unreasonable effectiveness of maths is just that - unreasonable. Is maths discovery, or invention?  Frenkel unequivocally calls it discovery. He describes the work mainly of white men (and a few Japanese), but neither acknowledges that fact, nor stops to consider the possible cultural biases that may introduce. 

Frenkel, as a Jew in Russia, had to fight a lot against the kind of prejudice seen, say, by 'lower' castes in India. And while he does not explicitly address politics in this book, his ideas clearly fall into the Universe of the Right. Maybe with good personal reason.