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All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.
Quoted in Hilbert's Foundations of Geometry.
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A History of Analysis
A History of Analysis, edited by Hans Niels Jahnke, London Mathematical Society,2003, 422pp.,cloth,ISBN:0899-2428, $89, distributed by The American Mathematical Society,201 Charles Street, Providence RI 02904-2294 www.ams.org .
It is rare to find a book that combines good history with good mathematics, and rarer still to find an anthology that reads as a unified whole. Thus I heartily recommend Jahnke's A History of Analysis as a superb example of a book that accomplishes these tasks.
A History of Analysis begins with the ancient Greeks and concludes in the early 20th century. Every major area is covered; integral and differential calculus, differential equations, foundations, analytical mechanics, and functional analysis all receive extensive coverage. As the topics suggest, the book presupposes a good understanding of the fundamental areas of analysis, equivalent to that of a senior or well-prepared junior mathematics major. Each chapter is written by one of the preeminent historians in the field and ends with an extensive bibliography. The chapters themselves have been skillfully woven together to make the book read as a connected whole, and the cross references are both extensive and useful.
As an example of the difficulties associated with writing a good history of mathematics, and how A History of Analysis avoids them, consider Bell's Men of Mathematics. Though this work is largely hagiographic in character, it does have at least one redeeming quality: Bell constantly refers to the mathematical advances made by his subjects. For example, Bell credits Cauchy with introducing rigor into mathematics, and the modern definitions of limit, continuity, and the convergence of infinite series. Bell's description of the mathematics is a step in the right direction, but he does not go far enough (and in fairness, he could not, since Men of Mathematics was intended for non-specialists).
A History of Analysis, in contrast, is written for mathematicians, so the authors can (and do) go into detail. For example, Lützen's chapter on the foundations of analysis (Chapter 6) includes extensive translations of selections from Cauchy, Fourier, Dirichlet, and other authors, to show precisely how the ideas of limit, continuity, convergence, etc., changed due to the work of Cauchy. The reference to original source material in its original form is a common theme in the book (most of the chapters include excellent selections of translated primary source material) and one of its best features: “study the masters” not only applies to mathematics, but to the history of mathematics as well.
A History of Analysis is one of the best studies available. It would make good reading for advanced students, and it is a valuable addition to any research library.
Jeff Suzuki, Visiting Assistant Professor of Mathematics, Bard College