Infinite Ascent: A Short History of Mathematics

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INFINITE ASCENT: A SHORT HISTORY OF MATHEMATICS by David Berlinski Modern Library, 2005, 197 pp. ISBN: 0-679-64234-X

The first thing that must be said about David Berlinski's Infinite Ascent: A Short History of Mathematics is that it is beautifully written and remarkably non-technical. Infinite Ascent would make a perfect gift or recommendation for anyone who wants to know about the big ideas in mathematics and why they are big.

Nevertheless, this book is not a history of mathematics. It is a collection of ten ideas that Berlinski considers "the ten most important breakthroughs in mathematical history" and some of the people involved in making those breakthroughs happen. The top-ten list includes number (Pythagoras), proof (Euclid), analytic geometry (Descartes), calculus (Newton and Leibniz), complex numbers (Euler, Wessel, Argand), groups (Galois), non-Euclidean geometry (Bolyai, Lobachevsky, Poincare, Riemann), sets (Cantor), incompleteness (Go'del), and algorithms (Turing). This is a traditional list and most mathematicians would agree that the topics included are important ones. Indeed, sets, number, calculus, and algorithms are unquestionably top-ten material. However, other mathematical discoveries, including graph theory, statistical methods, topology, formal languages, and matrices, could make strong claims for being on the top-ten list. Furthermore, many historians of mathematics will be surprised to read in Infinite Ascent that "the history of mathematics begins in 532 B.C." with Pythagoras. The contributions of China and India are conspicuously absent here. Infinite Ascent is a selective series of snapshots of the history of mathematics that highlight certain central ideas and some famous characters. …