Reuben Hersh - What Is Mathematics, Really?

What Is Mathematics, Really?

What Is Mathematics, Really?

Reuben Hersh

Oxford New York
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Copyright 1997 by Reuben Hersh

First published by Oxford University Press, Inc., 1997

First issued as an Oxford University Press paperback, 1999

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All rights reserved. No part of this publication may be reproduced,
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Cataloging-in-Publication Data
Hersh, Reuben, 1927
What is mathematics, really? / by Reuben Hersh.
p. cm. Includes bibliographical references and index.
ISBN 0-19-511368-3 (cloth) / 0-19-513087-1 (pbk.)
1. MathematicsPhilosophy. I. Title.
QA8.4.H47 1997 510.1dc20 96-38483

Illustration on dust jacket and p. vi Arabesque XXIX courtesy of Robert Longhurst. The sculpture depicts a minimal surface named after the German geometer A. Enneper. Longhurst made the sculpture from a photograph taken from a computer-generated movie produced by differential geometer David Hoffman and computer graphics virtuoso Jim Hoffman. Thanks to Nat Friedman for putting me in touch with Longhurst, and to Bob Osserman for mathematical instruction. The Mathematical Notes and Comments has a section with more information about minimal surfaces.

Figures 1 and 2 were derived from Ascher and Brooks, Ethnomathematics, Santa Rosa, CA.: Cole Publishing Co., 1991; and figures 617 from Davis, Hersh, and Marchisotto, The Companion Guide to the Mathematical Experience, Cambridge, Ma.: Birkhauser, 1995.

10 9 8 7 6 5 4

Printed in the United States of America
on acid free paper

to Veronka

Robert Longhurst, Arabesque XXIX

... , So long lives this, and this gives life to thee.

Shakespeare, Sonnet 18

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.

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 philosophyPlatonism. Finally, a radically different viewhumanism.

What should we require of a philosophy of mathematics? Some standard criteria arent essential. Some neglected ones are essential.

Anecdotes from mathematical life show that humanism is true to life.

All are subjects of long controversy. Humanism shows them in a new light.

Is mathematics created or discovered? What is a mathematical object? Object versus process. What is mathematical existence? Does the infinite exist?

From Pythagoras to Descartes, philosophy of mathematics is a mainstay of religion.

From Spinoza to Kant, philosophy of mathematics and religion supply mutual aid.

A crisis in set theory generates searching for an indubitable foundation for mathematics. Three major attempts fail.

Mainstream philosophy is still hooked on foundations.

Humanist philosophy of mathematics has credentials going back to Aristotle.

Modern mathematicians and philosophers have developed modern humanist philosophies of mathematics.

Mathematicians and others are contributing to humanist philosophy of mathematics.

Philosophy and teaching interact. Philosophy and politics. A self-graded report card.

Mathematical issues from chapters 113. A simple account of square circles. A complete minicourse in calculus. Booloss three-page proof of Gdels Incompleteness Theorem.

Forty years ago, as a machinists helper, with no thought that mathematics could become my lifes work, I discovered the classic, What Is Mathematics? by Richard Courant and Herbert Robbins. They never answered their question; or rather, they answered it by showing what mathematics is, not by telling what it is. After devouring the book with wonder and delight, I was still left asking, But what is mathematics, really?

This book offers a radically different, unconventional answer to that question. Repudiating Platonism and formalism, while recognizing the reasons that make them (alternately) seem plausible, I show that from the viewpoint of philosophy mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. I call this viewpoint humanist.

I use humanism to include all philosophies that see mathematics as a human activity, a product, and a characteristic of human culture and society. I use social conceptualism or social-cultural-historic or just social-historic philosophy for my specific views, as explained in this book.

This book is a subversive attack on traditional philosophies of mathematics. Its radicalism applies to philosophy of mathematics, not to mathematics itself. Mathematics comes first, then philosophizing about it, not the other way around. In attacking Platonism and formalism and neo-Fregeanism, Im defending our right to do mathematics as we do. To be frank, this book is written out of love for mathematics and gratitude to its creators.

Of course its obvious common knowledge that mathematics is a human activity carried out in society and developing historically. These simple observations are usually considered irrelevant to the philosophical question, what is mathematics? But without the social historical context, the problems of the philosophy of mathematics are intractable. In that context, they are subject to reasonable description and analysis.

The book has no mathematical or philosophical prerequisites. Formulas and calculations (mostly high-school algebra) are segregated into the final Mathematical Notes and Comments.

Theres a suggestive parallel between philosophy of mathematics today and philosophy of science in the 1930s. Philosophy of science was then dominated by logical empiricists or positivists (Rudolf Carnap the most eminent). Positivists thought they had the proper methodology for all science to obey (see ).

By the 1950s they noticed that scientists didnt obey their methodology. A few iconoclastsKarl Popper, Thomas Kuhn, Imre Lakatos, Paul Feyerabendproposed that philosophy of science look at what scientists actually do. They portrayed a science where change, growth, and controversy are fundamental. Philosophy of science was transformed.

This revolution left philosophy of mathematics unscratched. Its still dominated by its own dogmatism. Neo-Fregeanism is the name Philip Kitcher put on it. Neo-Fregeanism says set theory is the only part of mathematics that deserves philosophical consideration. Its a relic of the Frege-Russell-Brouwer-Hilbert foundationist philosophies that dominated philosophy of mathematics from about 1890 to about 1930. The search for indubitable foundations is forgotten, but its still taken for granted that philosophy of mathematics is aboutfoundations!