Enhanced descriptions from Syndetics:

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science.

Table of contents provided by Syndetics

• Preface Aims and Goals (p. xi)
• Acknowledgments (p. xvii)
• Dialogue with Laura (p. xxi)
• Part 1 (p. 1)
• 1 Survey and Proposals (p. 3)
• 2 Criteria for a Philosophy of Mathematics (p. 24)
• 3 Myths/Mistakes/ Misunderstandings (p. 35)
• 4 Intuiton/Proof/Certainty (p. 48)
• 5 Five Classical Puzzles (p. 72)
• Part 2 (p. 89)
• 6 Mainstream Before the Crisis (p. 91)
• 7 Mainstream Philosophy at Its Peak (p. 119)
• 8 Mainstream Since the Crisis (p. 137)
• 9 Foundationism Dies/ Mainstream Lives (p. 165)
• 10 Humanists and Mavericks of Old (p. 182)
• 11 Modern Humanists and Mavericks (p. 198)
• Twelve Contemporary Humanists and Mavericks (p. 220)
• Summary and Recapitulation (p. 233)
• Thirteen Mathematics is a Form of Life (p. 235)
• Mathematical Notes/Comments (p. 251)
• Bibliography (p. 317)
• Index (p. 335)

Reviews provided by Syndetics

Library Journal Review

Hersh, mathematician and coauthor of The Mathematical Experience (1983), attempts to answer here the philosophical question, "What is mathematics?" Many practitioners think of themselves as "platonists," discovering truths about ideal, eternally existing, abstract objects. The principal alternative to this concept is the "formalist" notion that mathematics is a game in which theorems are developed logically, starting from a set of axioms chosen almost arbitrarily. Hersh's humanistic position is, in essence, that mathematics is what mathematicians do. This is hard to disagree with but does not really explain how the subject evolves. Many feel that all mathematics begins with real-world applications, from which we try to extract the common properties and thence create a universe of abstract objects that can reveal unexpected beauty. In this somewhat disjointed book, Hersh reviews the history of the philosophy of mathematics, discusses the major players, and convincingly sets forth his thesis while undermining those of the competitors. For academic collections.‘Harold D. Shane, Baruch Coll., CUNY (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

CHOICE Review

By Hersh's own admission, his book is a mathematician's "subversive attack on traditional philosophies of mathematics." Countering historical arguments raised within Platonism, formalism, or neo-Fregeanism, Hersh suggests that "humanism" is the proper philosophical approach to mathematics, whereby mathematics is viewed as a human activity and a product of human culture and society. Thus, the author begins his personal response to the primary question, "What is mathematics, really?" In the process, the humanistic view is used to reexamine the age-old controversies regarding proof, intuition, infinity, existence, meaning, invention versus discovery, and truth-values. In turn, a major portion of the book is a fascinating historical examination of different philosophies and philosophers within the context of mathematics. Throughout, the author contends that "Mathematics comes first, then philosophizing about it, not the other way around." To help the reader focus on the philosophical arguments, the development of the necessary mathematical symbolism and formulas are relegated to the extensive "Notes and Comments" section. This thought-provoking, enjoyable work is complemented by a great bibliography and a thorough index. It succeeds in an enticing manner similar to the author's previous The Mathematical Experience, written jointly with Philip Davis (1981). General readers; undergraduates through faculty. J. Johnson Western Washington University

Author notes provided by Syndetics

About the Author: Reuben Hersh taught at several distinguished colleges and universities around the country. Now retired, he resides in Santa Fe, New Mexico.