Read This!

The MAA Online book review column

Briefly Noted

Short Reviews by Fernando Q. Gouvêa

This page collects brief notes on books that do not fit our review criteria but which nevertheless seem worthy of note. It will be updated periodically. Books mentioned here are not included in the index of MAA Online book reviews. Publication data on each of the books can be found at the bottom of the page. Links to previous "briefly noted" columns are also available below.

May 1999

It also used to be the case that all mathematicians knew lots of physics, or at least theoretical physics. That, too, is no longer the case. Many mathematicians have never even had a course in theoretical mechanics. David Acheson's From Calculus to Chaos is a book that can remedy that lack. It is a friendly introduction to dynamics that uses historical vignettes, well-chosen examples, and computer simulation to survey the field and show us, in the words of the blurb writer, what the calculus is really for. There's not a great amount of detail, but the "big ideas" come through quite well. The book includes a number of programs in QBASIC that illustrate the physical situations under discussion. This is quite nice, though it also turns out to illustrate how quickly the world of computers is changing: today, it would seem much more reasonable to have presented the programs in Java and also to make them available on the web.

Celestial Encounters, by Florin Diacu and Philip Holmes, is a book on the "origins of chaos and stability" in Poincaré's work on celestial mechanics. Using history as a guide, the authors introduce us to celestial mechanics, symbolic dynamics, stability, and KAM theory. The book is both readable and substantive; it differs from many popular accounts of "chaos" in that it actually has some serious mathematical content, presented in its historical context. Their approach is described in the preface: "In spite of chaos's more popular manifestations, including games and toys in airport gift shops, it is primarily a mathematical theory. We hope that a wide range of readers will find the stories that lie behind the ideas as interesting and exciting as we do, and that this will encourage them to probe into the more arcane aspects of the mathematics."

Hurray for them! This is "popular mathematics" with real content, as opposed to the "fireworks in a fog" style that has characterized much popular writing about chaos. Celestial Encounters is a book you can give to a good student as an introduction and an overview of a broad range of beautiful mathematics. We missed the chance to give it a full review it when it came out as a hardcover in 1996, but the new paperback edition in the "Princeton Science Library" gives us a chance to recommend it strongly.

Probably the best way to describe The Quest for Unity, by Éttienne Klein and Marc Lachièze-Rey, is to say that it is a popular history of the various attempts to find unified accounts of the physical world, ranging all the way from the pre-Socratic philosophers to the modern search for a "Theory of Everything". As such, it is more a book about the philosophy of physics than about physics itself, putting heavy emphasis on the contrast between the human desire for unity and the (apparent?) complex multiplicity of the world in which we live. Of course, one can't talk about the history of physics without discussing the history of mathematics (in fact, the two are indistinguishable until very recently), so there's a lot about mathematics and mathematicians here too. In fact, our current dreams of unity are really about a mathematical description of the world in which the bewildering variety of things lies over a fundamental and simple mathematical unity. The authors are quite skeptical of such a view, and their account, at times fascinating and at times pretentious, will get people thinking.

John D. Barrow first showed up on my radar screen with his book Pi in the Sky, which discusses the nature of mathematics, and in particular the curious tendency of mathematicians to behave as if mathematical objects (such as the number Pi) actually exist "out there", independent of human thought. He also co-authored, with Frank J. Tipler, a famous book on The Anthropic Cosmological Principle. Barrow is a professor of Astronomy at the University of Sussex and does research in Cosmology. He is interested both in science and in philosophical issues surrounding science, and most of his books deal with questions of this kind. His latest book, Between Inner Space and Outer Space, is a wide-ranging collection of popular essays and book reviews. The section on mathematics contains four essays. The first three focus mostly on the question of the "unreasonable effectiveness of mathematics," which Barrow wants to relate to philosophical questions about the nature of mathematics and of science. By asking questions not only about the nature of mathematics but also about the nature of science, Barrow renders the problem even more complex, but in a fascinating way. The final mathematics essay is quite different: a popular account of Arrow's theorem, published in a British newspaper on election day, which makes me wonder what the reaction of those reading it was.

There's much else that's interesting in this book. The first couple of essays are about writing popular books about science, and in the introduction to that section Barrow says that this is an area in which there have been great changes since the early twentieth century. At that time, he says, scientists writing for the general public "risked condemnation by friends and foes alike". Things are now different, he tells us. This makes me wonder: have things changed in mathematics also? In particular, does the mathematics community view such writing as significant scholarly work, or is it "extra-curricular" stuff that one may indulge in at times, but not to the detriment of "real work" (i.e., technical papers in mathematics journals)?

In any case, there's much in Barrow's book that makes one think, and that can enrich both our teaching and our scholarly work. The presentation in small bites makes it easy to read bits and pieces. One can then go on to Barrow's other books for more extensive accounts of his point of view. This one is definitely worth your time.

Publication Data

David Acheson, From Calculus to Chaos. Oxford University Press, 1997 (Softcover edition, 1998). ISBN 0-19-850077-7.

Florin Diacu and Philip Holmes, Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, 1996 (Softcover edition, 1999). ISBN 0-691-00545-1.

Éttienne Klein and Marc Lachièze-Rey, The Quest for Unity. Oxford University Press, 1999. ISBN 0-19-512085-X.

John D. Barrow, Between Inner Space and Outer Space. Oxford University Press, 1999. ISBN 0-19-850254-0.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.