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Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever

John Wiley
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It will come as no surprise to those who have read widely in the history of mathematics to learn that disagreements between mathematicians on matters of priority, philosophy, and the acceptability of new ideas can get every bit as nasty as similar spats in other fields. It will come as even less of a surprise to discover that the average layperson, however well educated, thinks of mathematics as the least likely place in which to find such immature goings-on. Hal Hellman, the author of a series of books on famous feuds in technology, science, and medicine, admits in the introduction to Great Feuds in Mathematics that he originally scoffed at the notion of doing a book on mathematical misunderstandings; after all, mathematics is supposedly "a cold, logical discipline where questions can be decided, if not quickly, [then] at least objectively and decisively." A little research convinced him otherwise, and the result is this general-audience effort — a bit light on details and (occasionally) a bit loose with terminology, but a worthwhile read nonetheless.

Great Feuds could almost be considered two narratives in one. The first four feuds (and they really are feuds, in the sense that personal animosity and jealousy played as significant a role as disagreement over content) cover familiar ground: the cubic and calculus priority disputes, Descartes' several quarrels with Fermat, Jakob and Johann Bernoulli's general crankiness and inability to deal with one another. Hellman does a good job of highlighting the highs and (mostly) lows of these affairs, but the overall narrative lacks a clear throughline, despite the author's repeated attempts to indicate how mathematics as a whole benefited from each disagreement. The latter discussions are somewhat hampered by the unfortunate fact that Hellman is not what one would call a profound mathematical thinker. He does his best with the limited tools at his disposal, but those with some knowledge of the situation will probably be disappointed in the overall quality of his summary statements.

Starting with the (in truth, rather decorous) difference of opinion between J. J. Sylvester and T. H. Huxley on the utility of mathematics vis-à-vis other scientific disciplines, the book begins to acquire a firmer backbone. Fully half of the work is taken up with various aspects of the "foundations debates" of the late 19th and early 20th centuries, including the dispute between Poincaré and Russell over the role of logic in mathematics, the dust-up between Borel and Zermelo over the utility of the axiom of choice, the Cantor/Kronecker "transfinite troubles," and, both last and least (in terms of how the participants and their followers behaved), the head-on collision between Hilbert's formalism and Brouwer's intuitionism.

Hellman quotes all parties at some length, and, while one could have asked for a bit more authorial "mortar" between the "bricks" of offset quotations, this reader appreciated the decision to let the combatants speak for themselves. (In many of the history texts I have read, by contrast, the authors have seemed content to simply paraphrase the views of the various "foundational schools" and their stalwarts.) As in the earlier part of the book, Hellman's descriptions of important results and his grasp of terminology could have been clearer and surer, but the discussion of foundational matters (which climaxes with a brief discussion of the dispute between "Platonists" and constructivists and its possible implications for mathematics education) is sufficiently lively and interesting that it could also have served as the basis for an entire volume in and of itself.

One of the more interesting features of this book is Hellman's treatment of other writers' opinions of the disputes he discusses. Even those who know all the facts may appreciate his brief but enlightening descriptions of others' viewpoints. In his discussion of the Cantor/Kronecker dispute, for example, he does a particularly good job of presenting various alternatives to the standard viewpoint (first "established" by E. T. Bell in Men of Mathematics) that Cantor was the unfortunate victim of Kronecker's vicious persecution.

Hellman quotes psychologists, sociologists, and other non-mathematicians who have studied the disputes from the point of view of the personalities involved, thereby making the book more palatable for the intended audience.

Great Feuds suffers from occasionally slipshod editing (I found the inability to decide between "aleph" and א0 to be particularly irritating) and such occasionally inscrutable phraseology as "infinitistic [sic] mathematics." I freely admit to letting out an audible groan upon reading, in Hellman's discussion of the cubic controversy, that "the term 'Renaissance Man' could have been invented to describe" Cardano. In the main, however, this is a very engaging effort and a good introduction to the human side of mathematics. At the very least, it may stimulate curious students to seek out more information about what, exactly, got people so riled up over "meaningless marks on paper" so many years ago.

Christopher E. Barat ( is an associate professor of mathematics at Villa Julie College. He received a B.S. in Mathematics from the University of Notre Dame and Sc.M. and Ph.D. degrees in Applied Mathematics from Brown University. He has previously taught at Randolph-Macon College, J. Sargeant Reynolds Community College, and Virginia State University.

Date Received: 
Thursday, September 21, 2006
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Hal Hellman
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Christopher E. Barat




1. Tartaglia versus Cardano • Solving Cubic Equations.

2. Descartes versus Fermat • Analytic Geometry and Optics.

3. Newton versus Leibniz • Credit for the Calculus.

4. Bernoulli versus Bernoulli • Sibling Rivalry of the Highest Order.

5. Sylvester versus Huxley • Mathematics: Ivory Tower or Real World?

6. Kronecker versus Cantor • Mathematical Humbug.

7. Borel versus Zermelo • The “Notorious Axiom”.

8. Poincaré versus Russell • The Logical Foundations of Mathematics.

9. Hilbert versus Brouwer • Formalism versus Intuitionism.

10. Absolutists/Platonists versus Fallibilists/Constructivists • Are Mathematical Advances Discoveries or Inventions?





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Tuesday, February 20, 2007