The Art of the Infinite: The Pleasures of Mathematics.

ROBERT KAPLAN AND ELLEN KAPLAN

In The Nothing That Is, Robert Kaplan considered the relevance of zero to the history of mathematics. Now, along with his wife, Ellen Kaplan, he addresses infinity. Beginning with natural numbers and covering the realms of infinity from prime numbers to parallel lines, the authors explore how mathematicians have tried to grasp the ungraspable, Profiles of individuals range from Pythagoras, who theorized about irrational numbers, to Georg Cantor, who proved that infinity can come in different sizes. These stories illustrate the two modes of mathematical thinking: the "intuitionist" notion that mathematical truth is discovered as it exists and the "formalist" belief that math is true because we invent consistent rules for it. These seemingly opposed approaches actually intersect in the concept of infinity, the Kaplans write, In making such points, the authors don't shy away from equations, theorems, and graphs. They provide hundreds of them to describe the role of infinity in various areas of mathematics. However, most topics in this book require little more of a reader than a background in algebra and geometry. OUP, 2003, 324 p., b&w illus., hardcover' $26.00.