Confronting Mathematicians

December 4, 2009

Book Review
Philip J. Davis

Mathematicians: An Outer View of an Inner World. By Mariana Cook (Preface and photographs), Robert Clifford Gunning (Introduction), Brandon Fradd (Afterword), Princeton University Press, Princeton, New Jersey, 2009, 208 pages, $35.00.

Let me begin, as in a playbill, by providing a Dramatis Personae for the people who created this book. Mariana Cook is a photographer, a prot�g�e of Ansel Adams who has gone on to have many gallery shows and books. Robert C. Gunning, a mathematician at Princeton, has written, inter alia, an introduction to holomorphic functions in several variables. Brandon Fradd has an undergraduate degree in mathematics from Princeton and a medical degree; he currently runs a biotech hedge fund.

Mathematicians was born when Fradd met up with Cook and her oeuvre and, having a residual nostalgic hankering for mathematics, suggested the project to her. This elegantly produced book of coffee table size and quality contains splendid photographic portraits---reminiscent of the style of Yousuf Karsh---of 92 distinguished living mathematicians. Accompanying each is a statement, either written by the individual pictured or composed by Cook on the basis of an interview.

The men and women selected vary in age from active researchers and teachers to retired folk, and their statements constitute mini-biographies. Along with the family backgrounds of the individual mathematicians, the reader learns how they got into mathematics, who their teachers and co-workers have been, what they consider to be their major contributions, what has inspired them, what knowledge they have drawn on, what their creative strategies have been, how they see themselves in a world in which mathematicians are but a paltry few and their work is generally incomprehensible or misunderstood.

I believe that the mathematical community will find what is written here attractive and informative, and will be pleased to have the statements of these leaders. The young, should they turn to its pages, will be inspired. The non-mathematical community, on the other hand, may look at the portraits and ask themselves whether mathematicians can be characterized by certain phrenological traits. If they read the subjects' statements, they will often be perplexed by what they read; in numerous cases, clich�d views as to what it is that makes mathematicians tick will be reinforced. On the whole, through the words of the mathematicians included, mathematics comes across as an intellectual subject of great beauty, of unlimited developmental potential, and of mysterious provenance.

A number of the individuals (but by no means the majority) put forward explicit opinions as to the general nature of the mathematical enterprise, quite apart from their experiences in their specialized fields of expertise. It is to these opinions that I now wish to turn. I have selected the 13 clips that follow more or less at random (if such a thing is possible!), and I admit, with apologies to the authors, that presentation of their words out of full context may have caused some loss of overtones.

Peter Lax: "Mathematics is sometimes compared to music, but I find a comparison with painting better. . . . In mathematics there is a creative tension between analyzing the laws of nature and making beautiful logical patterns."

Robert Langlands: "Not satisfied with partial insights and partial solutions, they [Harish-Chandra, Grothendieck] insisted . . . on methods that were adequate to establishing the theories envisaged in their full natural generality."

Gerd Faltings: "It is my experience that I do not need a strategic plan but that interesting problems and new methods tend to come up by themselves. I rarely make conjectures myself."

William Thurston: "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."

Sir Michael Atiyah: "Mathematicans are generally thought of as some kind of intellectual machine, a great brain that crunches numbers and spits out theorems. In fact we are, as Hermann Weyl said, more like creative artists. Although strongly constrained by the rules of logic and by physical experience, we use our imagination to make great leaps into the unknown."

Richard Borcherds: "Seeing the way that such mathematical objects [general relativity, Galois theory] fit together so well sometimes gives me a spooky feeling that they were somehow constructed by some being."

Ingrid Daubechies: "All our mathematics is constructed. It is a construction we make in order to think about the world. . . . [It] is the only way we have to think logically about things we observe. . . . The book of Nature is not written in mathematics; rather, mathematics is the only language we know to explain nature logically."

Barry Mazur: "That there is a sterling architecture behind how we think, an articulation that transcends mood, circumstance, and even culture, is one of the great gifts of being alive. No mode of thought comes closer to this architecture than mathematics---and this is what makes thinking about mathematics both utterly singular as an experience and universally human."

Stephen Smale: "My goal is to understand the world better through mathematics."

Bertram Kostant: "One cannot help having the gut feeling that a real understanding of our universe must somehow involve E_8 [a Lie group comprising a system of 240 vectors in 8-dimensional space]."

Burt Totaro: "In the past, mathematicians always tried to solve problems exactly. Now we realize that most problems will never have an exact solution. Nonetheless, we can hope to understand the general shape of a solution, and topology gives a language for talking about these shapes."

John Forbes Nash, Jr.: "Mathematical thinking is logical and rational thinking. It's not like writing poetry."

Manjul Bhargava: "[Mathematics] is about finding the simple questions and ideas
that lead one to unexpected, unexplored realms---and to deep, elegant, and lasting mathematics."

Anyone who knows some mathematics, who knows something of the history and philosophy of mathematics, will find much to chew on in these clips and---a fortiori---in the full individual statements. Contradictory views are expressed; agreement as well as disagreement lurks behind the verbal arras. But happily, except possibly in Cook's preface, the lightly implied philosophies of these creative people go beyond the standard and somewhat stale questions of whether and why mathematics is true and in what sense mathematical objects exist. Among the clips printed here, the one with which I am most fully in sympathy is that of Ingrid Daubechies, in whose words I detect a whiff of anti-platonism.

What I find distressingly missing from the statements is the idea that mathematicians are part of society, that the development of mathematics takes place within a social context, and that its existence and applications have social consequences that are in some cases beneficial, in others not. Perhaps reflecting the instructions given to the writers, the statements exude more than a bit of in-group solipsism; the writers fail to consider why society should support mathematicians, often handsomely, in what a baffled general public might term the single-minded pursuit of cult activities.

Leo the MGM lion roars out, "Ars Gratia Artis," art for art's sake, a slogan that has been both embraced and rejected. All the arts, seen over the sweep of centuries, teach us what it means to be human. The same might be asserted for mathematics, although here it has been harder to make the case: Paralleling what has been said of architecture, we make our mathematics and then our mathematics makes us.

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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