Read This! The MAA Online book review column Mathematical Mountaintops The Five Most Famous Problems of All Time by John L. Casti Reviewed by David P. Roberts

On the other hand, there are some lapses in the book. Generally well-referenced, there are places where it is badly referenced. Often original, there are places that it fails badly to be original. More about this later.

The book is structured as five fairly independent chapters, each one devoted to a problem and its solution, as follows:

Each chapter is a blend of history, mathematics, and philosophy, all at a level appropriate for say enrichment reading for undergraduate math majors. For example, Chapter 3 takes us mostly chronologically through the Cantor-Kronecker-Schröder-Bernstein-Hilbert-Gödel-Cohen story. Simultaneously it presents various mathematical preliminaries, Cantor's proofs of the countability of Q and the uncountability of R, and some indication of Gödel's and Cohen's work. It concludes by describing how the rather ambiguous resolution of Cantor's continuum problem might be understood by formalists, logicists, platonists, and intuitionists.

Of course, all five chapters truly represent a mathematical mountaintop, in fact a regularly visited mountaintop. For example, consider just Keith Devlin's book Mathematics: The New Golden Age, first published in 1989. Of its eleven chapters, four of them correspond to chapters in Mathematical Mountaintops. There is nothing intrinsically bad that it is the same problems which receive continuing attention. On the contrary, it is good that our profession has some canonical stories to help bind us together, despite our often disjoint specializations.

Now about the referencing and originality in Casti's book. The book concludes with a final section on suggested readings, giving about ten general references for each chapter and describing the level and content of each reference. Certainly that's a great start to a well-referenced book. In the text, there's often appropriate further referencing. For example, page 94 contains two interesting paragraphs comparing Cantor to his contemporary Vincent van Gogh; Casti references William Dunham for this idea. The very next page illustrates a mathematical point both amusingly and well, as "ponks", "lonks", and "zonks" are introduced and later revealed to be interpretable as "points", "lines", and the relation "connects"; Casti attributes this example to James Brown. There is also a lot of originality the overall narrative. For example, Casti is particularly interested in the boundary between mathematics and philosophy. He treats things like the role of computer computations in proofs in his own way. But there are those lapses, too. Here are three.

1. Casti tells a funny anecdote in his chapter on Kepler's problem. He offers as one of his general references the expository article "Cannonballs and honeycombs" by Thomas Hales, the solver of Kepler's problem. But Casti gives no indication that the anecdote he presents comes from Hales' article. Here is what happens to Hales' first paragraph:

The fate of Hales' next three paragraphs is similar: it's almost as if Casti ran them through some sort of automorphism of the English language: "fruit stand" became "grocery store", "greengrocers" became "fruit dealers", and "grocer" became "greengrocer." The story is not improved, just trivially changed. So we have here a problem in referencing; Casti should have treated this anecdote as he did the material from Dunham or Brown.

2. In his chapter on the four color problem, Casti says that he will follow Devlin's presentation of the nineteenth-century solution of the much easier but still instructive five color problem. This accounts for Casti's Figure 10, which describes that proof, being straight from Devlin's chapter. It doesn't account for Figures 1, 3, 5, 9, and 12, also being straight from Devlin's chapter. Also some of the paragraphs in Casti's chapter can easily be recognized as modified versions of paragraphs in Devlin's chapter. In general, there is no particular improvement. For example, Devlin's informative captions are missing on the new Figures 1, 3, 5, 9, and 12. So here we have both a referencing and an originality problem. Simply throwing in some more references to Devlin would not be an adequate fix. We are allowed to demand some more novelty. Casti's book is aimed at a slightly more sophisticated readership than Devlin's book, so Casti had a great number of options. With regard to the solution of the five color problem, for example, he might have presented it not in the original map-theoretic terms but rather in the more streamlined graph-theoretic language.

3. In the references to his chapter on the Hilbert's tenth problem, Casti says he has borrowed some examples from Devlin's chapter on the same subject. Readers who interpret "borrowed" as "reworked", "elaborated on", or "improved" have misled themselves; "borrowed" is being used in a very literal sense. For example, there is paragraph which in the different versions begins,

The two paragraphs each go on to discuss the real and integer solutions to the equations x² + y² = 2 and x² + y² = 3. Devlin did not draw a picture or even use the word "circle," and neither does Casti. So here we can concede that the referencing is acceptable, but still there is a serious shortage of originality. As one of many options, Casti could have asked his more sophisticated readers to consider points with integral coordinates on the circles x² + y² = p for other primes p. The result is simple and classical: there are exactly eight or zero integral points, according to whether p has the form 4k + 1 or 4k + 3. Something of this sort would have illustrated that Hilbert was very much within reason when he hoped one could similarly decide general Diophantine equations.

Mathematical Mountaintops is overall a positive contribution to the literature. But it would be nicer if Casti, one of our most influential popularizers, slowed down a bit on the quantity and focused more effort on truly maximizing quality.

This book has been withdrawn from circulation by Oxford University Press.

References:

Mathematics: The New Golden Age, by Keith Devlin. 2nd edition. Columbia University Press, 2001. Hardcover, 336pp, $25.95. ISBN 0231116381.

Thomas Hales' article Cannonballs and Honeycombs appeared in the April 2000 Notices of the AMS.

Two of Casti's recent books share the five chapter structure of Mathematical Mountaintops but discuss different problems:

Five golden rules: great theories of 20th century mathematics, and why they matter , by John L. Casti. John Wiley & Sons. 1995. Hardcover, 235pp, $26.95. ISBN 0821820702. Also in paperback.

Five More Golden Rules: knots, codes, chaos and other great theories of 20th-century mathematics , by John L. Casti. John Wiley & Sons. 2000. Hardcover, 288pp, $25.00. ISBN 0471322334. Also in paperback.

The problems discussed in the two Golden Rules books are closer to, and sometimes exactly in, Casti's research expertise. They are generally pitched at a slightly higher level than Mathematical Mountaintops, although several chapters from Five More Golden Rules are at a substantially higher level. The latter book has been reviewed in MAA Online.

Publication Data:

Mathematical Mountaintops: The Five Most Famous Problems of All Time, by John L. Casti. Oxford University Press. 2001. Hardcover, 288pp, $25.00. ISBN 0195141717.

This book has been withdrawn from circulation by Oxford University Press.

David Roberts is an assistant professor of mathematics at University of Minnesota, Morris.

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