A long time ago, a friend and fellow undergraduate math major said of a world-famous professor at our university, “He doesn’t teach you any math; he teaches you what it means to be a mathematician.” The enthralling book under review can be said to be the proverbial “further reading,” provided that “further” does not connote second-class status. It is also true that, as a side effect, I did learn some math from this book, as its clarity (and outstanding economy of expression in matters mathematical) is unsurpassed; the learning was real, but at 30,000 feet. (It is tangential here that I wound up in physics; this as a result of an epiphany wherein Kepler’s three laws of planetary motion were shown to follow from a single equation of Newton. This personal experience purports to tie in with Harris’s use of “pathos” in extension, as in the pathos of mathematics, of which a little more below, and a lot more in the book.)

I have always characterized G. H. Hardy’s evocative A mathematician’s apology [1] as an innocuous, allegorical book, and am ashamed of having failed to connect the present book’s title with Hardy’s title. Harris’s chapter 10, “No Apologies” (which I spotted first upon initial, nonlinear browsing) provided me with that remedial insight. Nobel laureate Frederick Soddy [2] certainly would disagree with my judgment of Hardy’s Apology as being benign: “From such cloistral clowning the world sickens.” Be that as it may, my internal embarrassment was exacerbated when I subsequently read (linearly) in Harris’s preface that “[the] title is a transparent [ouch!] allusion to” Hardy’s book. I’ll dig the hole deeper by pedantically nit-picking Hardy’s “apology” as in reality being an apologia, which is an elaborate explication, explanation, and ecce homo without the “I’m sorry.” The present book’s author is, of course, well aware of the apology-apologia distinction, as have been many readers of Hardy’s book.

I imagine Paul Halmos to be the originator of the term “automathography” [3]. Harris’s book qualifies as such, but focuses on the spiritual (pathos) and social/sociological dimensions of pure mathematics. If there is any meaning to the term “literary work,” that term must apply here, modulo the essentially irrelevant nonfiction category. Though I share Hardy’s contempt for critics (a peculiar attitude for a reviewer!), I’ll sound like one in saying that the writing is evocative of Thomas Mann’s The magic mountain [4], in that Harris places the reader body-and-soul into any situation that he is describing. This is a most impressive feat in the case of pure mathematics: its psychology, philosophy, sociology, economics, and pathos. To overdo my literary-critic shtick, Mann’s locale is Davos, one of today’s venues of Harris’s powerful people (who have sway over pure mathematics’ funding and public agenda), and the place where Mann’s fictional Settembrini and Naphta indulged in endless, impotent sophistry regarding elimination of human suffering. Harris, in contrast, sprinkles subchapters alpha through delta (there’s a delta-point-five) that carry a continuing conversation between a number theorist and a performing artist (actress) titled “How to Explain Number Theory at a Dinner Party.” This is as effective as an ostensive definition of number theory can be for non-mathematicians or non-fellow-travelers.

A sentence in chapter 2, “How I Acquired Charisma,” that illustrates Harris’s sophistication in style and content reads: “One of the premises of this chapter is that the generous license granted hieratic figures is of epistemological as well as ethical import.” (Harris is modest in hardly, if at all, alluding to his prestigious 2007 Clay Research Award for his contribution to the Langlands Program, which latter he summarizes in the book, the award certainly contributing to his charisma.) A slightly-off-target case in point from my experience is charismatic physicist Julian Schwinger’s suggestion that the parity-violation experiment, motivated in 1956 by Lee and Yang’s theory of weak interactions, not be “bothered with,” since the result would surely be negative. The result was maximally positive, sending shock waves through physics almost as large as those of 30-year-young quantum mechanics itself. This is of epistemological as well as normative (if not ethical) import. Of course, pure mathematics ultimately appeals only to (dis)proofs, and these latter are modulo theories (axioms).

The book lists good, true, and beautiful as the conventionally enumerated attributes of mathematics, with good being connected to useful, Hardy’s nemesis. If I may be allowed a trite mathematical analogy, the author views the nature and pathos of pure mathematics as transcending any linear combination of those three basis vectors, into what I inadequately characterize as the realm of je ne sais quoi, the intangible, undefined, transcendent. John von Neumann, in his “The Mathematician” [5], gently warned of this as art for art’s sake, which he ever-so-tactfully deprecated. On the other hand, Donald Knuth’s [6] Turing Award speech comprised a substantial explication of his choice of the term “art” for the eminently practical activity of computer programming.

This unique book should not and cannot be speed-read. It will challenge and ultimately reward the reader along many dimensions, one of which is, of course, mathematics.

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