A Radical Approach to Real Analysis, second edition by David M. Bressoud, 2007. 323+xvi pp., illustrations, $49.95 (MAA members: $39.95) cloth. ISBN 0-88385-747-2. Mathematical Association of America, Washington, DC 20090-1112. 1-800-331-1622 or www.maa.org

A radical approach involves, by definition, digging down to the roots, and that is bound to unearth history, too. However, the purpose of Bressoud’s text is to teach not history but mathematics specifically, an introduction to real analysis that culminates in proving the convergence of the Fourier series of a suitably well-behaved function. In his Preface, the author states the pedagogical problem:

The traditional course begins with a discussion of properties of the real numbers, moves on to continuity, then differentiability, integrability, sequences, and finally infinite series, culminating in a rigorous proof of the properties of Taylor series and perhaps even Fourier series. This is the right way to view analysis, but it is not the right way to teach it.

I happen to agree, and I also think that the text is singularly successful in teaching real analysis. Part of that success can be traced to its ontogeny-recapitulates-phylogeny historical approach, in which the reader contends anew with the puzzles, paradoxes, and contradictions that faced the 19th century explorers of this new mathematics. The upshot of this approach is that, throughout  the text, mathematical argument and historical exposition are entwined, with each illuminating the other. The result is both good mathematics and an intellectual history that t is remarkably thorough and satisfying.

James Callahan, Professor of Mathematics, Smith College, Northampton, MA