Expositions: Their Contents and Discontents

October 18, 2009

Book Review
Philip J. Davis

Who Gave You the Epsilon? & Other Tales of Mathematical History. Marlow Anderson, Victor Katz, and Robin Wilson, editors, Mathematical Association of America, Washington, DC, 2009, 431 pages, $65.00.

The editors of Who Gave You the Epsilon? have put together a collection of more than forty articles, culled from the 20th-century pages of The American Mathematical Monthly and Mathematics Magazine. The selections, which present biographical, expository, and speculative material, cover the century, from George Bruce Halstead's "Gauss and the Non-Euclidean Geometry" (1900) to Phillip A. Griffiths's "Mathematics at the Turn of the Millennium" (2000). For each of the book's sections, a foreword (by the editors) provides a précis of what follows and an afterword presents bibliographic updates.

You will find essays on how the function concept evolved over the centuries; on what monster groups are all about; on how Mary Cartwright and J.E. Littlewood carried out their collaboration. You will also learn which three fundamental number-theoretic problems gave rise to algebraic number theory, and how J.J. Sylvester lashed out against certain "piratical" German mathematicians. You will be able to read one opinion as to what the 21st-century challenges to mathematics will be. As the ad writers say, all this and much, much more for the price of the volume.

The subjects covered are engaging; the editing is well done and informative. But the main thing for me is that all the material was well within my grasp. I felt educated by it, built up. This is so in part because the articles deal largely with older material, in part because the quality of the exposition is so high.

As I leafed through the pages, my vanity was bruised: Why didn't the editors include my nice article on the gamma function (Math Monthly, December 1959)? This would have given them the opportunity to refer in the appropriate afterword to the great Chapter 1 of Special Functions by Andrews, Askey, and Roy. Oh well, editors and authors have the absolute right to select and limit their material in any way that pleases them; no such selection can be regarded as a summing up of 20th-century progress in mathematics. There is much that is missing; the list would be long, and there would be no point in my trying to make a compilation.

If authors and editors have the absolute right to select---subject, of course, to commercial back pressures from their publishers---reciprocally, reviewers have the freedom to write about anything that happens to pop into their minds as they scan (or even read!) the books they're reviewing. So here's what popped into mine.

I should like to know where I can read, with understanding, something about random matrices and then, moving forward in time, about progress made in understanding the mechanics of turbulence, about how the Riemann–Roch theorem morphed into the Attiyah–Singer index theorem. I should also like to learn what the Sato–Tate conjecture is all about, and why I should give it the time of day. I should like to hear the latest from the world of nonlinear dynamics, and would like to learn why Gian-Carlo Rota considered the phenomenology of mathematician and philosopher Edmund Husserl the greatest. Is there a resource I can consult for an introduction to Grothendieck? Have I missed out on the writings of some cranky contemporary critic who, following in the footsteps of the late Dick Hamming, will assure me that there are some mathematical paradises I have no reason to enter?

You may have an answer: Go to Wikipedia. Forget it! The articles there appear to have been written by Ivory Tower specialists with little interest in communicating to the great unwashed mathematical masses, to our hoi polloi, the hewers of wood and the drawers of water---to me, in other words.

What now pops into my mind is a complaint that I've heard repeated again and again, as I'm sure many of my readers have. Over the years, I have gone to many expository talks on so-called hot topics, given, quite often, by the individuals responsible for their development. After the first five minutes, whether the speaker is scribbling with chalk or using PowerPoint, I am lost. I have heard the same criticism not only from a number of mathematicians of my vintage but also from younger colleagues. I have heard that departmental colloquia are in a state of collapse. Are mathematicians constructed, genetically and psychologically, to be non-communicative, to get their jollies and academic brownie points from speaking in private tongues and writing in arcane symbols to a select but diminished audience?

Sadly, it may simply be that the field of mathematics is now so vast, so extensive and varied, so relatively non-intercommunicable, that its well-advertised unity may be in serious doubt. On the plus side, from time to time I hear that a concept from Field A was crucial in treating a question in Field B, miles away from A. But is this a commonplace occurrence? Are mathematicians splitting into twenty-eight separate varieties, with decreasing overlap in their interests, methodologies, and professional output?

I see an opportunity for mathematical societies of all stripes to encourage and develop articles and books that tell the professional "down the hall" what is really going on. This far from easy job requires writers who have not only knowledge of the material, but also the skills, the patience, and the willingness to place themselves in the shoes of the non-specialist. The writers must have some idea of what the non-specialist probably knows and does not know so as to be able to convey the ideas, the excitement, and the significance of the hot new material. It is ironic that the plethora of recent miracles of electronic communication---cellphones, e-mail, iPods, Facebook, Twitter---has been accompanied by a degradation of mutual understanding.

This brings me to my last mental pop-up. What should a professional mathematician now know in order to be considered mathematically "educated" and not merely a brilliant specialist in some sub-sub-field? An answer to this question could serve as a guide for future expositors.

Would anyone care to go out on a limb and make a list? I suspect that any such list would be long, and that I would clearly be swept into the pile of the uneducated.

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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