I have a friend who thinks she does not understand maths (apologies to the American readers, I will use “maths” in this review rather than the “math” in the book title). She thinks maths is “all about numbers”. She thinks she’s not particularly good at it. She is wrong. Actually, I think she is remarkably gifted, primarily because she asks amazingly prescient questions. She has an inquisitive nature that I fear is often drilled out of most people by our “rigorous” schooling in the tedium of official school maths.
Over the past few months, I have spent quite a bit of time discussing maths with this friend, introducing her to set theory and trying to overcome a fear of algebra. (It is remarkable how many super bright people can reason about insanely complex general phenomena in their head without algebra in a way I never could, yet stumble over a simple equation!)
I saw this book reviewed elsewhere – I can’t quite remember where. It was a short but positive review, and the title is attractive to me, given my recent discussions. The author, Edward Frenkel, is a professor of mathematics at UC Berkeley, has made considerable contributions to mathematics himself, and also has a direct personal knowledge of many of the larger-than-life characters he talks about in the book. I knew nothing about his work, or his life, both of which turn out to be rather interesting, before reading this book.
The book is part autobiography and part introduction to the joy of mathematics. We learn about the author’s early career, the anti-Semitism he faced in Russia and his conversion as a teen from physics to mathematics, before the two come back together through quantum physics later in his career. We learn about his immersion in the Langlands Program.
As a book about “doing mathematics”, I think he does quite well at getting his feelings across. The detail of much of the mathematics is missing; sketches of subjects are given: Galois Groups, the Shimura-Taniyama-Weil Conjecture, Lie groups and Lie algebras, etc., but only very brief sketches. While this gives the reader insight into the depth of the subject areas covered by modern mathematics (and certainly helps to show the breadth of maths beyond school arithmetic) it doesn’t really show the reader mathematics. We read about proof, but we don’t experience proof. We read about revelation, but we don’t feel the revelation ourselves. We are told that mathematics rests on clear and unequivocal definitions, but these are absent from the book. Initially, this left me feeling unsatisfied. Slowly, I put aside my usual desire to understand everything I was reading, and began to enjoy the book as a novel, a love affair between the author and his subject, in which specific theorems appear only as partially sketched characters. From this perspective, the bulk of the book was very enjoyable, and provides insight not only into the beauty of mathematics, but also the ways in which mathematicians think. As an engineer, albeit one who has a lot of time for – and use for – pure mathematics, my way of thinking remains quite distinct from some of the methods and approaches outlined in this book. However, late in the book, where the author covers the links between this mathematics and quantum physics, I feel the unexplained mathematics again gets in the way. The author seems to acknowledge this, when he says (p.221), “All this stuff, as my dad put it, is quite heavy: we’ve got Hitchin moduli spaces, mirror symmetry, A-branes, B-branes, automorphic sheaves […] But my point is not for you to learn them all…”
The final chapter of the book takes a violent switch in direction as we learn of Frenkel’s artistic projects, in particular his film Rites of Love and Math. Had I known about this work beforehand, the book would have made a lot more sense! In this chapter, Frenkel provides his philosophical opinions on the nature of the material world, the world of our consciousness, and – in his view – the entirely separate Platonic world of mathematics. He says his approach to making the film was “let the viewers first feel rather than understand” – much the same could be said of this book.
The experience of reading this book has got me thinking in two directions.
Firstly, this is not the book I initially expected from the first few chapters. But how easy would it be to write a book which both conveys the joy of solving problems, say in Lie algebras, which Frenkel explores in some depth, and at the same time covers enough mathematics for a reasonably rigorous approach to the problem for a general reader? Perhaps such a task is impossible, and I ask too much. Does this mean that the beauty of modern mathematics is necessarily harder to access than than the beauty of modern art? Does this mean that we must necessarily pay the price of “boring maths” in order to be able to access “beautiful maths”? Or is there actually no “boring maths” and the prerequisite knowledge can also be described in a way that appeals and excites, even if not in one volume. I hope for the latter.
Secondly, the book has made me realise that I am far less certain of my own philosophical viewpoint on reality, Platonism, and knowledge than I used to be. It has made me want to explore these issues again and read more. Perhaps I will start with Penrose.
Is this the book I wanted for my friend? Probably not, but it’s certainly stimulated my thinking, and probably would also stimulate the thinking of others, regardless of their current level of mathematical maturity.