What Is Mathematics, Really?

by Hersh, Reuben

• ISBN13: 9780195113686
• ISBN10: 0195113683
• Format: Hardcover
• Copyright: 1997-08-21
• Publisher: Oxford University Press

Summary

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.

Author Biography

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

Table of Contents

Preface: Aims and Goals Outline of Part One. The deplorable state of philosophy of mathematics. A parallel between the Kuhn-Popper revolution in philosophy of science and the present situation in philosophy of mathematics. Relevance for mathematics education.	xi	(6)
Acknowledgments	xvii	(4)
Dialogue with Laura	xxi
Part One	3	(88)
1 Survey and Proposals Philosophy of mathematics is introduced by an exercise on the fourth dimension. Then comes a quick survey of modern mathematics, and a presentation of the prevalent philosophy--Platonism. Finally, a radically different view-- humanism.	3	(21)
2 Criteria for a Philosophy of Mathematics What should we require of a philosophy of mathematics? Some standard criteria aren't essential. Some neglected ones are essential.	24	(11)
3 Myths/Mistakes/Misunderstandings Anecdotes from mathematical life show that humanism is true to life.	35	(13)
4 Intuition/Proof/Certainty All are subjects of long controversy. Humanism shows them in a new light.	48	(24)
5 Five Classical Puzzles Is mathematics created or discovered? What is a mathematical object? Object versus process. What is mathematical existence? Does the infinite exist?	72	(19)
Part Two	91	(144)
6 Mainstream Before the Crisis From Pythagoras to Descartes, philosophy of mathematics is a mainstay of religion.	91	(28)
7 Mainstream Philosophy at Its Peak From Spinoza to Kant, philosophy of mathematics and religion supply mutual aid.	119	(18)
8 Mainstream Since the Crisis A crisis in set theory generates searching for an indubitable foundation for mathematics. Three major attempts fail.	137	(28)
9 Foundationism Dies/Mainstream Lives Mainstream philosophy is still hooked on foundations.	165	(17)
10 Humanists and Mavericks of Old Humanist philosophy of mathematics has credentials going back to Aristotle.	182	(16)
11 Modern Humanists and Mavericks Modern mathematicians and philosophers have developed modern humanist philosophies of mathematics.	198	(22)
12 Contemporary Humanists and Mavericks Mathematicians and Others are contributing to humanist philosophy of mathematics.	220	(15)
Summary and Recapitulation	235	(14)
13 Mathematics Is a Form of Life Philosophy and teaching interact. Philosophy and politics. A self-graded report card.	235	(14)
Mathematical Notes/Comments Mathematical issues from chapters 1-13. A simple account of square circles. A complete minicourse in calculus. Boolos's three-page proof of Godel's Incompleteness Theorem.	249	(68)
Bibliography	317	(18)
Index	335