Read This!

The MAA Online book review column

Briefly Noted

September 2000

While the publishers still advertise this fine book as "one of the most popular mathematical books ever written for a general audience" its not at all likely that it could be used today in mathematics for liberal arts courses. In fact, a recent Mathematics Magazine review (vol. 73, no. 3, June 2000, p. 247) even claims the original edition was a "loss-leader `prestige' book" and that "this book was not widely used in liberal-arts mathematics courses, much less by high-schools."

While it might not live up to the publishers claim, and is not likely to find a wide audience outside of the mathematical community, I concur with the Mathematics Magazine review which closes as follows: "What a magnificent living fossil of a book! So, where's today's audience? You yourself will find the book entertaining, enlightening, and delightful, and so will curious mathematics students at all levels who admire abstract thought and are eager to indulge in it. (Julian Fleron)

From Kant to Hilbert was originally published in 1996 as a much-too-expensive two-volume set in hard covers. The fact that it is now available in two paperback volumes at a much lower price is great news for anyone interested in reading and/or teaching from original sources. This is a very rich selection of material indeed. The first volume opens with a long section devoted to George Berkeley's philosophy of mathematics, including both the obvious (The Analyst, matched with selections from Newton's Principia Mathematica) and the non-obvious (selections from Alciphron and other writings). This is followed by material from Colin MacLaurin and Jean D'Alembert on the foundations of the calculus, then selections from Kant, Lambert, Bolzano, Gauss, etc. It's quite an impressive list, and goes far beyond the standard list of sources on foundational issues. I particularly like having material from Cayley, Clifford, and Sylvester. The last section in the first volume is named for Charles Sanders Pierce, but it also includes a selection Benjamin Pierce's famous "Linear Associative Algebra." The second volume includes material from Riemann, Helmholtz, Dedekind, Cantor, Kronecker, Klein, Poincaré, Borel, Baire, Hilbert, Brouwer, Zermelo, Hardy, and Bourbaki. Having material from Helmholtz is particularly welcome, as he is not often given his due as a philosopher of mathematics. The middle portion is more standard, but here and there one notices something really nice. Overall, the selection of material is very intelligent, and most of the texts are definitely worth reading. This is Very Good Stuff, a book that you should make sure your library owns and which is a wonderful source of texts for a course in the history and philosophy of modern mathematics or for reading seminar for upper-level undergraduates. (Fernando Q. Gouvêa)

By now, everyone knows that Wiles' proof of Fermat's Last Theorem is based on the theories of elliptic curves and of modular forms. What most people don't know is that after Ribet's 1991 work which established the connection between the modularity conjecture and Fermat's Last Theorem, most of the work that remained to be done (and that Wiles and Taylor did) had to do with the connection between modular forms and Galois representations. In this book, Hida explores exactly this fundamental connection between modular forms and Galois representations (including an account of Wiles' crucial theorem right in the middle of the book). Along the way, he explores both the theory of Galois representations and the cohomology theory of Galois groups. The final chapter, on "Modular L-values and Selmer Groups," includes a number of Hida's own results. The book is based on a course taught by Hida at UCLA, and it looks to be a good source for people with the appropriate background who want to learn this material. (Fernando Q. Gouvêa)

Publication Data

Excursions into Mathematics, Millennium Edition, by Anatole Beck, Michael N. Bleicher, and Donald W. Crowe. A K Peters, Natick, MA., 2000. Softcover, xxv + 499 pp., $34.00. ISBN 1-56881-115-2.

From Kant to Hilbert: a Source Book in the Foundations of Mathematics, ed by William Ewald. Oxford University Press, 1996 (paperback edition, 1999). Softcover, 1340 pages in two volumes, $85.00. ISBN 0-19-850537-X.

Modular Forms and Galois Cohomology, by Haruzo Hida. Cambridge University Press, 2000. Hardcover, 343pp, $69.95. ISBN 0-521-77036-X.

Julian F. Fleron (j_fleron@foma.wsc.mass.edu) is Associate Professor of Mathematics at Westfield State College. He has broad mathematical interests which he tries to share with his students and both students and citizens in the local community - hoping to change rampant negative perceptions of mathematics.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.

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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.