What is Mathematics Really
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Abstract
historically evolved, and intelligible only in a social context.” Hersh describes some of the standard issues of philosophy of mathematics, such as existence of finite and infinite mathematical entities, intuition, proof, and truth, and tries to show that his philosophy deals with these issues better than do the philosophies he rejects. Although I found the book very interesting and informative in many ways, I am not sure Hersh succeeds in making his case for his humanist philosophy. Part One The opening gambit of the book is presented as both “a worked exercise in Pólya’s heuristic” and “an inquiry into mathematical existence.” The problem is to count the various parts of a 4-dimensional cube and reflect on what kind of sense the calculations could make. In true pólyaesque spirit, Hersh switches immediately to the 3-cube and counts its vertices, edges, and faces. He does the same for the 2-cube and the 1-cube. The three sets of formulas show a clear pattern that is easily generalized to four dimensions. This leads to a list of questions about the existence of a 4-cube. If it exists, where is it? If it does not exist, how could we obtain such detailed information about it? What about a 3-cube? Does it exist in ordinary space, given that we can’t produce a perfect 3cube as a physical object? A little bit later, Hersh uses possible answers to these questions to help explain various philosophies of mathematics, including his own humanism. What Is Mathematics, Really? Reuben Hersh Oxford University Press, 1999 ISBN 0-19513-087-1 Softcover, 368 pages, $16.95