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The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. "Perfect numbers" certainly never did any good, but then they never did any particular harm.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
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Everything and More
Everything and More: A Compact History of ∞, by David Foster Wallace, 2003. Hardcover, 319pp., $23.95. ISBN 0-393-00338-8. New York: W. W. Norton & Company,http://www.wwnorton.com .
In Everything and More: A Compact History of ∞, David Foster Wallace relates the fascinating story of how mathematicians through the ages have grappled with the concept of infinity. Beginning with the ancient Greeks and Zeno’s paradoxes of motion, Wallace goes through two thousand years of mathematics, focusing on the problems and dilemmas that arose from an unsatisfactory understanding of the nature/existence of infinite sets. Of particular interest is his discussion of how the calculus evolved in the seventeenth century in spite of this lack of a clear understanding of the infinitely large and small which are at the heart of the subject.
The hero of the story is Georg Cantor, whose (at the time) controversial theory succeeded in defining and legitimizing infinite sets and established consistent rules for dealing with them. Although the book contains many interesting biographical details about the different characters that contributed to the history of infinity, the main emphasis is on the mathematical achievements. Wallace’s aim, stated in the foreword, is to “... discuss these achievements in such a way that they’re vivid and comprehensible to readers who do not have pro-grade technical backgrounds and expertise” (pp. 1-2). In fact, the mathematics gets quite technical and will be hard going even for a talented mathematics major. However, Wallace does a superb job of conveying the central ideas and how they fit into and motivate the overall story.
The book has several annoying features, among which are lack of a table of contents and index, excessive use of abbreviations, and a multitude of footnotes. In addition, Wallace often gets the mathematical details wrong. For example, he applies the Extreme Value Theorem to “prove” that in any time interval [t1, t2], there is a “very next instant” after t1 (p. 190). There are many other examples of mathematical missteps, all of which could easily have been avoided by having the manuscript checked by a professional mathematician. Nevertheless, Wallace’s conversational style is engaging and entertaining. Anyone with an interest in the history of mathematics will enjoy this book. Readers with little mathematical training will most likely skip over the technical details. Others will need to have some tolerance for occasional errors in the mathematics. Mathematics teachers in particular will find much material in this book that will increase their understanding and knowledge of the history of infinity, as well as enrich a Calculus or Real Analysis class.
Gabriela Sanchis, Professor of Mathematics, Elizabethtown College, Elizabethtown, PA