Mathematical Evolutions

Mathematical Evolutions, Abe Shenitzer and John Stillwell, eds., 2001. 304 pp., $38.95 paper. ISBN 0-88385-536-4. www.maa.org.

The editors have assembled an excellent collection of surveys of the history and of the state of major fields of mathematics. Many of the articles are also translated by the editors. The volume opens with Michael Atiyah’s general survey, "Mathematics in the 20th Century," transcribed from his Fields Lecture in 2000, and closes with notes from a talk given by Wilhelm Magnus on "The Significance of Mathematics: The Mathematicians' Share in the General Human Condition."  In between the articles focus on analysis, algebra and number theory, geometry and topology, logic and foundations, and applications.

For example, the section on algebra and number theory begins with two articles sketching the development of the concepts of algebra from the Babylonians to the end of the 19th century. Next come discussions of algebraic integers, rings, field theory, elliptic curves, and modular functions. The section closes with an article by Herman Weyl on "Topology and Abstract Algebra as Two Roads of Mathematical Comprehension."

Through the volume one can follow many different threads. One is the notion of infinitesimal. In the analysis section are two letters from N. N. Luzin to M. Ya. Vygodskiĭ, discussing Vygodskiĭ’s Foundation of Infinitesimal Calculus. Luzin recounts his attraction to the notion of an infinitesimal and the harsh reaction to his attempts to discuss the idea with his teachers. This theme is echoed in Doob’s article on "The Development of Rigor in Mathematical Probability (1900-1950)." He writes, "In the 1930’s Banach spaces were sneered at as absurdly abstract, later it was the turn of locally convex spaces, and now it is the turn of nonstandard analysis."

The volume is a valuable resource for any teacher of mathematics. The articles are well written, but not too detailed, making this an ideal companion for the traveling mathematician.

Lang Moore, Associate Professor Emeritus, Duke University

See also the MAA Review by David Graves.