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*Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever*, Hal Hellman, 2006, 256pp. Hardcover, $24.95, ISBN 978-0-471-64877-2, New York: John Wiley & Sons,Inc., http://www.wiley.com

America seems obsessed with feuds nowadays. The subject was featured in a recent episode of CBS Sunday Morning. To my surprise Hal Hellman was briefly interviewed and mention was made of his series of books on “great feuds”. He has previously written 3 books on the subjects of science(1998), medicine(2001), and technology(2004). His latest, *Great Feuds in Mathematics,* is the subject of this review.

The subtitle of the book is “Ten of the Liveliest Disputes Ever”.They are 1) Tartaglia vs.Cardano 2) Descartes vs. Fermat 3) Newton vs. Leibniz 4) Jakob Bernoulli vs. Johann Bernoulli 5) Sylvester vs. Huxley 6) Kronecker vs. Cantor 7) Borel vs. Zermelo 8) Poincaré vs. Russell 9) Hilbert vs. Brouwer and 10) Platonists vs. Constructivists. A short epilogue follows, although one might say that the epilogue begins with Chapter 10. In my opinion, the book is divided into two sections: before Cantor and after Cantor. Feuds # 7-10 concern themselves with foundational questions influenced by Cantor’s work.

Hellman clearly states in the introduction that the book was not his idea and that he “knew nothing of mathematics history” before he began. The approach taken is clearly as an externalist. He is interested in writing for the non-mathematician. However, the book is well researched. I was particularly impressed by his description of E.T. Bell’s “vivid imagination” with respect to Cantor. In general, I found the book to be very interesting. I can honestly say that I learned something new in each chapter.

Hellman’s lack of expertise in the history of mathematics does influence his choice of coverage. The period 1750-1850 is conspicuously missing from the book. While non-Euclidean geometry is mentioned several times, there is no hint of the priority dispute between Gauss, Bolyai, and Lobachevsky. The dispute over the discovery of the method of least squares is, also, noticeably ignored. Apparently, Gauss is exempt from controversy and gets a “free pass” in Hellman’s book. I think either of these discussions would have made a better choice than the inclusion of Sylvester vs. Huxley. Still for those interested in a quick read on any on the feuds listed above, this book would fit the bill.

Hal Hellman has a website for his writings at www.greatfeuds.com. You can find a quiz on great feuds in mathematics there.

Jim Kiernan, Adjunct Professor, Brooklyn College

Jim Kiernan, reviewer, "Great Feuds in Mathematics," *Loci* (July 2007)