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How Not to Solve It

William Thompson

The Random Walks of George Pólya. Gerald L. Alexanderson. xii + 303 pp. Mathematical Association of America, 2000. $29.95 (paper).

Mathematical Fallacies, Flaws, and Flimflam. Edward J. Barbeau. xvi + 167 pp. Mathematical Association of America, 2000. $23.95 (paper).

George Pólya (1887–1985) is well known among mathematicians for his insightful techniques for solving a wide range of mathematical problems. Among scientists and others he is famous for his book How to Solve It: A New Aspect of Mathematical Method, first published by Princeton University Press in 1945 and currently available in at least 22 languages. Gerald Alexanderson, who was Pólya's student at Stanford, has now written a carefully researched hagiography, which has been published as part of the Spectrum Series of the Mathematical Association of America. Pólya emphasized clear understanding and presentation of mathematical proofs, so it is appropriate to review his biography with Mathematical Fallacies, Flaws, and Flimflam, another title in the same series. Both books will appeal to scientists interested in the scientific method and problem solving.

Alexanderson's book presents Pólya's work chronologically and details his sojourns in Hungary, Austria, Germany, France, Switzerland (where he obtained citizenship), England and finally the United States. Along the way, Pólya encountered many well-known mathematicians and physicists and collaborated with several of them. With Gábor Szegö he collected and published in 1925 Aufgaben und Lehrsätze aus der Analysis [Theorems and Problems from Analysis], a very influential book in the education of mathematicians who read German; it does not seem to have been translated into English.

Pólya drew inspiration for his work from many sources, as Alexanderson emphasizes. For example, Pólya's studies of the random walk (a term he coined) were prompted by several impromptu encounters with a couple while he was strolling in the woods near Zürich. A polyglot as well as a polymath, Pólya was the sole author of papers in six languages; his first work in Hungarian was published in 1912, and his first paper in English appeared in 1923.

I would have liked to have found in Alexanderson's book more analysis of Pólya's methods and less emphasis on chronology and sociology. Pólya's approaches to problem solving are at least as important in the physical sciences as in mathematics. Indeed, Pólya wrote several papers and a book in mathematical physics with engineering applications. In the preface of How to Solve It he remarks that it "should interest anybody concerned with the ways and means of invention and discovery."

The last third of The Random Walks of George Pólya contains a bibliography of Pólya's publications and 14 appendices, including seven essays on Pólya's work in mathematics and math education by associates of his and five short articles by Pólya himself, which should be accessible to those without advanced mathematical training. These pieces illustrate the clarity of his thinking and the breadth of his interests in math and science. There is a complete index of proper names, but very few subjects are indexed, making it difficult to find material in the text. Overall this is a good book, well worth the price and the time needed to read it.

Mathematical Fallacies, Flaws, and Flimflam is a collection of columns by Edward Barbeau that have appeared in The College Mathematical Journal over the past 11 years. It contains more than 170 examples of often plausible assertions that are discovered to be false or ambiguous when examined carefully; these are drawn from finite mathematics, algebra, trigonometry, geometry, probability, calculus and set theory. Anyone with a year of university mathematics can appreciate the flaws in most of these assertions once the author has explained them.

Many of the erroneous mathematical results in the collection provide a counterpoint to the naive use of heuristics discussed (with many cautions) in Pólya's How to Solve It. Indeed, it would be very interesting to read How to Solve It and Mathematical Fallacies together in order to understand the power and pitfalls of Pólya's methods.

Because Barbeau's book is replete with mathematical errors, I recommend it wholeheartedly for students and teachers of college mathematics who wish to learn and teach clearly.—William J. Thompson, Physics and Astronomy, University of North Carolina at Chapel Hill

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