What Is Mathematics, Really?

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Virtually all philosophers treat mathematics as isolated, timeless, ahistorical, inhuman. In What Is Mathematics, Really? renowned mathematician Reuben Hersh argues the contrary. In a subversive attack on traditional philosophies of mathematics, most notably Platonism and formalism, he shows 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. The humanist standpoint helps him to resolve ancient controversies about proof, certainty, and invention versus discovery. The second half of the book provides a fascinating history of the "mainstream" of philosophy - ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, Hilbert, Carnap, and Quine. Then come the mavericks who saw mathematics as a human artifact - Aristotle, Locke, Hume, Mill, Peirce, Dewey, Wittgenstein. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. Platonism and elitism fit together naturally. Humanism, on the other hand, links mathematics with people, 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 consequences.
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Editorial Reviews

Larry Feldman
This book represents an outstanding resource for helping the mathematics-education community understand the roots of mathematical thought....The book is filled with fascinating mathematical problems and uses very little obscure symbolism.
The Mathematics Teacher
From the Publisher

"Ruben Hersh puts the people into the philosophy and the philosophy into the mathematics: he will take your mind to places it has never dreamed of."--Ian Stewart, The Mathematical Institute, University of Coventry

"Reuben Hersh's insider's view of the big questions about the nature of mathematics is witty, provocative, insightful, and always accessible. It should revitalize discussions among mathematicians, philosophers, and historians--and all those people who have wondered what mathematics is all about."--Philip Kitcher

"Reuben Hersh's What Is Mathematics, Really? is the most thorough, comprehensive survey of all that has been written on the philosophy of mathematics. It will remain the standard reference in the philosophy of mathematics, indispensable to both mathematicians and philosophers. The author combines the experience of a research mathematician with a keen sense of philosophical relevance to give brief, incisive sketches that strikingly summarize every philosopher's contribution (of the lack of it). Reuben Hersh's volume should be required reading of all undergraduate mathematics majors, as well as of any cultivated person with or without a mathematical background."--Gian-Carlo Rota, Department of Mathematics, MIT

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Product Details

  • ISBN-13: 9780195130874
  • Publisher: Oxford University Press, USA
  • Publication date: 7/28/1999
  • Edition description: New Edition
  • Pages: 368
  • Sales rank: 1,009,633
  • Lexile: 1090L (what's this?)
  • Product dimensions: 9.00 (w) x 6.10 (h) x 1.10 (d)

Meet the Author

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

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Table of Contents

Preface: Aims and Goals
Dialogue with Laura
1 Survey and Proposals 3
2 Criteria for a Philosophy of Mathematics 24
3 Myths/Mistakes/Misunderstandings 35
4 Intuition/Proof/Certainty 48
5 Five Classical Puzzles 72
6 Mainstream Before the Crisis 91
7 Mainstream Philosophy at Its Peak 119
8 Mainstream Since the Crisis 137
9 Foundationism Dies/Mainstream Lives 165
10 Humanists and Mavericks of Old 182
11 Modern Humanists and Mavericks 198
12 Contemporary Humanists and Mavericks 220
13 Mathematics Is a Form of Life 235
Mathematical Notes/Comments 249
Bibliography 317
Index 335
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