American Scientist

Mathematics and Common Sense: A Case of Creative Tension. Philip J. Davis. xlvi + 242 pp. A K Peters, Ltd., 2006. $34.95.

What sets esteemed mathematician Philip J. Davis apart from many of his colleagues is that he has spent a good part of his career trying to explain mathematics to the person on the street. He does this well: His 1981 book, The Mathematical Experience (written in collaboration with Reuben Hersh), received an American Book Award. For many years he has been a regular contributor to SIAM News, the newsletter of the Society for Industrial and Applied Mathematics. He loves doing math and explaining the subject on many different levels to many different audiences.

Davis addresses his recent book, Mathematics and Common Sense, "to all who are curious about the nature of mathematics and its role in society," adding that he aims to convey "an appreciation of the flights of human imagination that both join and transcend common sense and that have created the mathematical world we live in." The book's 33 essays, some adapted from Davis's SIAM News columns, show a first-rate mathematician thinking hard about his craft and its relevance to the world at large.

These essays offer, among other things, a bird's-eye view of the world of professional mathematics. The more interesting material derives from the fertile, if somewhat uneasy or even confusing, relationship mathematics has with the real world and real people. In essence, Davis makes the point that mathematics and common sense spring from the same source—a human, if not primal, inclination to organize and communicate experience—but that mystery, confusion and even magic can arise out of these humble human origins.

For example, the property of "fiveness," which could be common to a small flock of sheep, the members of the shepherd's family and the fingers on the shepherd's hand, is more generally a concept of number that comes out of the penchant for and necessity of identifying one-to-one correspondences. Geometry can be seen as the natural outcome of the search for a means of communicating size and shape. The irony is that from such "common sense" and concrete inclinations, mysteries are born. Considerations of number lead naturally to the primes, still a source of simple-to-state but difficult-to-solve problems. Contemplating distance, we come quickly to irrational numbers (note the name!) and, over time, to the mind-bending pursuits and puzzles of modern geometry and topology.

Thus does mathematics readily move from common sense to something that is beyond the senses. Charles Darwin, who regretted his adolescent impatience with algebra, famously referred to those with an understanding of the "leading principles of mathematics" as having an "extra sense." Of course, sadly, for some, math also sometimes descends into nonsense.

Davis attempts to humanize the discipline, trying to say in English what its concerns, processes and tools are, going so far as to take on the subjects of mathematical intuition, proof, evidence, epistemology and beauty. He argues passionately for the relevance of mathematics, yet he places it not just in the world of endeavors, but in the world of ideas. I especially enjoyed an essay on the difficulty of finding honest or at least sympathetic representations of mathematics in the media.

The book contains a wealth of wonderful ideas. However, the tone seems a bit uneven overall,  and I often found myself unsure of the intended audience. One would have to be more than "curious" to make it through a number of the essays. Even the opening section, "Letters to Christina," in which Davis answers questions posed by a nonmathematician friend, contains an integral, as well as a brief excerpt from the classic text of Edward Charles Titchmarsh on the behavior of an analytic function in a closed domain! So at many points nonmathematicians may be left scratching their heads, forced to leapfrog over phrases or paragraphs, hoping for (but not always finding) a safe landing on the other side. The book might have been more approachable—and more useful for anyone wanting to follow up on intriguing points—had it contained an index, or a better bibliography and footnotes.

Mathematicians interested in communicating something of their profession to outsiders are perhaps the best audience for these essays; they will appreciate Davis's exploration of the kinds of questions they might encounter on a plane trip or at a cocktail party as much as his answers. Other readers will find that in sampling the collection (which is best enjoyed in small bites) they get a taste of what mathematics is, as well as a sense of its sometimes complicated relation to experience.

I do wish, though, that Davis had said more about the experience of working on a problem, turning it over and over, first in search of any solution at all and then in search of the "right" one. I also wish he had discussed the notion that, in mathematics, the technique of a solution or proof is usually as important as the result itself. Taking different routes to the same truth can, in the best cases, be like looking at the same painting under different lighting—each view revealing a new connection or aspect of beauty. But ultimately Davis is demonstrating these characteristics rather than discussing them. Reading these essays, we see a mathematician finding 33 ways to prove that mathematics is a never-ending source of mystery, utility, beauty and pleasure.