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Who gave you the epsilon? and other tales of mathematical history. (English)
Spectrum. Washington, DC: The Mathematical Association of America (MAA) (ISBN 978-0-88385-569-0/hbk). x, 429~p. (2009).
Lately the MAA has engaged in the custom of reprinting papers on special topics. This book is a continuation of the bestselling “Sherlock Holmes in Babylon" (2004; Zbl 1035.00005 and ME 2006c.01451), and there have been similar collections, like “The Genius of Euler" (2007; Zbl 1120.01009), “Euler and Modern Science" (translated in 2007; Zbl 1165.01010), and “Musing of the Masters" (2004). The book under review consists of a collection of 41 articles on the history of mathematics in the 19th and 20th centuries. The articles are taken from MAA journals printed between 1900 and 2007. They are written by well-known mathematicians (G. B. Halsted, G. H. Hardy, B. L. Van der Waerden, H. Weyl, and others); the length of the papers runs between one and some 20 pages. The collection is divided into analysis (10); geometry, topology and foundations (11); algebra and number theory (16), and surveys (4). There is no doubt that after decades the history of science has made some progress, and in this book for each group such gaps are bridged by Afterwords (2-3 pages) which give a short outlook whereas Forewords (2-3 pages) for each group embed and introduce the topics dealt with. Although the papers are well chosen and interesting to read today, of course the supplied frame, Fore- and Afterwords, is simply too short to be elucidating in a satisfying way. Most of the articles were reviewed individually earlier. That’s why we will use three papers by way of example in order to demonstrate the collection’s advantage and disadvantage. The book opens with Grabiner’s article that gave the collection its title. This paper was published in 1983, i.e., in a period when non-standard analysis had already supplied other views on Cauchy, but such controversial views (in English J. Cleave or G. Fisher, let alone D. Spalt in German) are omitted by the author and the editors (in their Afterword). Nor is the recently published translation of Cauchy’s “Cours d’Analyse" (1821), the first time in English, mentioned. Grabiner states, “Cauchy did not distinguish between point-wise and uniform convergence" (p. 10) as well as “But it was Cauchy who gave rigorous definitions \dots and the modern rigorous approach to calculus" (p. 12). If these statements (on continuity) were true, how could Cauchy be rigorous on the basis of such vague and obviously confused notions? Furthermore, we find two comments on the 2nd ICM in Paris in 1900, one by Halsted (2 pp.) and the other one by Griffith (12 pp.), written a century apart ‒ an interesting historical combination. Although the intention of the editors is to present above all American papers in the Afterwords, they leave the non-American reader anyhow irritated because this concentration mostly on the U.S. book market overlooks some essential works, like J.-P. Pier’s “Developing of Mathematics" (2000), one that surely should be referenced.
Reviewer: Rüdiger Thiele (Halle)
Classification: A30
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