The Awe and Shock of Mathematicians

September 18, 2011

Book Review
Philip J. Davis

Loving + Hating Mathematics. By Reuben Hersh and Vera John-Steiner, Princeton University Press, Princeton, New Jersey, 2011, 416 pages (including many illustrations, extensive notes, and bibliographic material), $29.95.

Ageométretos medeìs eisíto.
Let no one ignorant of geometry enter.
---Sign reputedly displayed over the entrance to Plato's Academy.

Who is the audience for this book? Readers in the general educated laity will have the chance---to parallel Marlene Dietrich's song in Destry---to "see what formulas the boys [and now the gals] in the back room" are having. The inner circle of pure mathematicians will respond to the book with delight.

Reuben Hersh is an emeritus professor of mathematics at the University of New Mexico. He is also a friend and co-author of mine (The Mathematical Experience,1981). Vera John-Steiner, also a friend, is a professor emerita of linguistics and education at the University of New Mexico. She has written extensively on creative collaboration. I must say immediately and immodestly that in this book they have very kindly given me some nice notices.

Loving + Hating cuts a wide swath through many topics, and contains a mélange of material. It presents histories, biographies, lots of mathematical anecdotes, folklore, and gossip. The authors reveal eccentricities of their subjects, and they mention problems connected with discrimination, age, gender, ethnicity, religion, academic tenure, responses to political and social movements. All this presents challenges for a reviewer.

There are about 35 pages of one-paragraph mini-biographies of the sort readily available on the web. More extensive biographies run from a few to many pages. Thus, there are four pages on Kurt Gödel, twenty on Alexander Grothendieck. Over the centuries, mathematicians have not been bashful about writing full-length biographical or autobiographical material. Loving + Hating lists about sixty such works. And, of course, this is the mere tip of the iceberg---the book makes no mention, for example, of the wonderful Brief Lives of John Aubrey (1626–1695), which has notices of easily twenty mathematical adepts.

The spotlight of the book is focused most intensely on the Mathematical Great---how they got their start; what mathematical friendships they forged; how they collaborated; in what scientific communities they flourished; what frustrations, secrecies, or jealousies plagued them; whom they married.

Who are the Great? They are the Geniuses, of course, but who are the Geniuses? Mark Kac gave an answer to this question in his autobiography, Enigmas of Chance:

"There are two kinds of geniuses: the ‘ordinary' and the ‘magicians.' An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. The working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark."

Who, then, according to Hersh and John-Steiner, are the Great, the Geniuses? The authors have their list, and they name names, including Jean Dieudonné, the collectively pseudonymous Nicolas Bourbaki, Alexander Grothendieck, John von Neumann, Srinivasa Ramanujan, Andrei Kolmogoroff, Sonya Kovalevskaya, David Hilbert, Samuel Eilenberg, Emmy Noether, I.M. Gelfand.

The Geniuses in the book will elicit readers' awe and shock. The reasons for the awe and adulation are clear. But what about the shock? It may be the "shock of Pythagoras," who reputedly sacrificed a hecatomb of oxen on discovering his formula. Updated, it may be the shock of numerous mathematical absurdities, such as the Banach–Tarski–Hausdorff paradox, that emerge as pure logical inference.

A second sort of shock is that experienced by people who are very good at mathematics when they come to realize that, as Mark Kac implied, there are out there people who are much better than they. One inference I derive from reading Loving + Hating is that if you are a young person who has passed Plato's Entrance Exam and are thinking of entering the field of professional mathematics, and if you're not a genius of either of Kac's types, then you need to resign yourself to the reality that you are unlikely to cut any new ice. You may do useful scientific and socially important work, but you will not be among the mathematical makers and shakers.

Loving + Hating displays mathematics as culture, as solace, as escape, as addiction, as intellectual achievement of the highest order, as madness. Many mathematicians are shown to be skillful musical instrumentalists. Venturing into the authors' private Academy or into their Mathematical Hall of Fame, the uninitiated and the mathophobes may be confused and bewildered, but the Old Timers in pure, abstract mathematics will say, "Yes. They've pretty much hit the nail on the head."

The readership of SIAM News, on the other hand, specializing in what in an old German joke is called abgewandte und unreine Mathematik (turned away and dirty mathematics), will wonder why, if our age is totally mathematized, from product stripes to search engines, from baseball stats to investment strategies, from the chipification of weapons to the Meter Maids' hand-held gizmos that make drivers miserable, there is only a brief glimpse of these applications in the book. The book gives computers short shrift---while asserting that computers have become a central feature of our lives, the authors say nothing about the many talented mathematicians who have brought this about. The "mathematics" of the title is consequently a severely limited discipline.

The book is mostly about loving mathematics. As to the hating, perhaps some of the Greats have been turned off by the inhumane applications of what they themselves created. Where else is the hating? The antepenultimate chapter implies that the widespread aversion to mathematics is caused by the teaching of elementary mathematics as rote. As an antidote, an ideal to be sought, a colorful and optimistic dust jacket depicts two young boys separated by a loving cup and thriving in the best of all mathematical worlds.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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