Read This! The MAA Online book review column Mathematics and Common Sense: A Case of Creative Tension by Philip J. Davis Reviewed by Darren Glass

Of all of the mathematicians who have dipped their toes into the waters of writing about mathematics for popular audiences, Philip J. Davis is certainly one of the most successful, whether you measure success by sales, by numbers of honors, or by sheer quality. He is probably best known for The Mathematical Experience, cowritten with Ruben Hersh, which did a magnificent job of explaining what mathematics is and what mathematicians do, and in doing so won the American Book Award in 1983. In the final paragraph of his 2000 book The Education of a Mathematician, Davis writes that "the history of future mathematics will be seen as the increased tension… between the real and the virtual" and Davis' latest book, published this year by AK Peters, picks up this theme of tension, and is even titled Mathematics and Common Sense: A Case of Creative Tension.

The opening section of the book is entitled "Letters to Christina: Answers to Frequently Asked Questions," and in it the author gives his answers to some common questions about mathematics that any of us in the profession of have certainly heard many times. Some of the questions addressed are: What is mathematics? ("The science, art, and a language of quantity, space, and pattern" is how Davis begins his answer) Why should I learn mathematics? ("Mathematics opens up the possibility of rational thought"), How is mathematics research organized? What is the greatest challenge to modern mathematics? Can mathematicians look at an open problem and tell whether this is something deep or something easy? and What can an interested amateur do?

The rest of the book consists of a series of loosely linked essays about mathematics, mathematicians, and their role in society. These essays are almost completely independent of one another, and many of them appeared in other venues before this book was published. This lack of cohesion can be a bit off-putting at times, and at times I found myself wanting Davis to make a stronger overall point, but the individual essays are well-written, often including amusing epigraphs and detailed bibliographies. In almost all of the essays, he concentrates more on the philosophy or history of the mathematics rather than on the mathematics itself, which is almost certainly the best decision for his target audience, but sometimes leaves the mathematically-minded reader unsatisfied and wanting more. (This reviewer was particularly disappointed in the half-hearted treatment of mathematical developments in the study of baseball, but a number of the other essays also ended several pages before I would have liked them to.) The brevity, however, allows the chapters to cover an extremely wide range of topics and on the whole the essays were quite interesting to read. Some of the ones I found most intriguing included:

• The title essay on "Mathematics and Common Sense," in which Davis first discusses what is meant by the term 'common sense' and then contrasts it with ideas about mathematical thinking. For example, while it may seem like common sense that you should not stop in Atlanta on a flight from Boston to Los Angeles, it might still be a mathematically optimal solution.
  
• An essay entitled "If Mathematics Says 'No' Does It Really Mean It?" which discusses some aspects of impossibility in mathematics and what it really means to say that, for example, we cannot square the circle.
  
• A discussion worthy of NPR's This I Believe about why mathematicians in general (and Davis in particular) believe certain things in mathematics, ranging from the fact that 2+2=4 to the Pythagorean Theorem to the idea that there are an infinite number of twin primes, with the reasons for belief ranging from axiomatic to computational to theological.
  
• An essay about how mathematics is treated in the media, both fictional and in news reporting, and what Davis would like to see from the media in the future. In particular, he writes that "I should think that at the very least it should be possible for a paper to educate us to the fact that mathematics is formatting a good portion of today's life and to point out where this is occurring. It need not give the readers a semblance of understanding of the technical mathematics; that is too much to expect. But I should hope that clever writers might point out how mathematics is altering our lifestyles, and do it in a manner that would not lead Garfield the cat to say 'ho hum.'" (Note that an abbreviated version of this essay recently appeared in the Notices of the American Mathematical Society).

Davis has many strong opinions, and is clearly not afraid to share them. I imagine any mathematician who picks up a copy of this book will find themselves shaking their head in disagreement a few times, and at other times feel embarrassed at the idea that non-mathematicians are reading this book and forming opinions of our profession based on Davis' strong opinions. But there were at least as many other times where I was in complete agreement with Davis, and often I found that he articulated quite well opinions that I never knew I had. In all of these situations, Davis once again shows himself to be a fine expositor, and despite its flaws Mathematics and Common Sense is a book well worth reading.

Publication Data: Mathematics and Common Sense: A Case of Creative Tension, by Philip J. Davis. A K Peters, 2006.Hardcover, 242 pages, $34.95. ISBN 1568812701.

Darren Glass (dglass@gettysburg.edu) is an Assistant Professor of Mathematics at Gettysburg College. His research interests include number theory, algebraic geometry, and cryptography.

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