Plato's ghost..The modernist transformation of mathematics

526 Pages·2008·2.96 MB·English

by  Jeremy Gray

PLATO’S GHOST
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PLATO’S GHOST
THE MODERNIST TRANSFORMATION OF MATHEMATICS
JEREMY GRAY
P R I N C E T O N U N I V E R S I T Y P R E S S
PRINCETON AND OXFORD
Copyright ª 2008 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton,
New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street,
Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Gray, Jeremy, 1947–
Plato’s ghost : the modernist transformation of mathematics = Jeremy Gray.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-13610-3 (alk. paper)
1. Mathematics–History–19th century. 2. Mathematics–Philosophy.
3. Aesthetics, Modern–19th century. I. Title.
QA26.G73 2008
510.9'034–dc22 2007061027
This book has been composed in
Printed on acid-free paper.?
press.princeton.edu
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
WHAT THEN?
His chosen comrades thought at school
He must grow a famous man;
He thought the same and lived by rule,
All his twenties crammed with toil;
‘‘What then?’’ sang Plato’s ghost. ‘‘What then?’’
Everything he wrote was read,
After certain years he won
Sufficient money for his need;
Friends that have been friends indeed;
‘‘What then?’’ sang Plato’s ghost. ‘‘What then?’’
All his happier dreams came true—
A small old house, wife, daughter, son,
Grounds where plum and cabbage grew,
Poets and Wits about him drew;
‘‘What then?’’ sang Plato’s ghost. ‘‘What then?’’
‘‘The work is done,’’ grown old he thought,
‘‘According to my boyish plan;
Let the fools rage, I swerved in naught,
Something to perfection brought’’;
But louder sang that ghost, ‘‘What then?’’
—William Butler Yeats
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CONTENTS
Introduction 1
I.1 Opening Remarks 1
I.2 Some Mathematical Concepts 16
1 Modernism and Mathematics 18
1.1 Modernism in Branches of Mathematics 18
1.2 Changes in Philosophy 24
1.3 The Modernization of Mathematics 32
2 Before Modernism 39
2.1 Geometry 39
2.2 Analysis 58
2.3 Algebra 75
2.4 Philosophy 78
2.5 British Algebra and Logic 101
2.6 The Consensus in 1880 112
3 Mathematical Modernism Arrives 113
3.1 Modern Geometry: Piecemeal Abstraction 113
3.2 Modern Analysis 129
3.3 Algebra 148
3.4 Modern Logic and Set Theory 157
3.5 The View from Paris and St. Louis 170
4 Modernism Avowed 176
4.1 Geometry 176
4.2 Philosophy and Mathematics in Germany 196
4.3 Algebra 213
4.4 Modern Analysis 216
4.5 Modernist Objects 235
4.6 American Philosophers and Logicians 239
viii Con ten ts
4.7 The Paradoxes of Set Theory 247
4.8 Anxiety 266
4.9 Coming to Terms with Kant 277
5 Faces of Mathematics 305
5.1 Introduction 305
5.2 Mathematics and Physics 306
5.3 Measurement 328
5.4 Popularizing Mathematics around 1900 346
5.4 Writing the History of Mathematics 365
6 Mathematics, Language, and Psychology 374
6.1 Languages Natural and Artificial 374
6.2 Mathematical Modernism and Psychology 388
7 After the War 406
7.1 The Foundations of Mathematics 406
7.2 Mathematics and the Mechanization of Thought 430
7.3 The Rise of Mathematical Platonism 440
7.4 Did Modernism ‘‘Win’’? 452
7.5 The Work Is Done 458
Appendix: Four Theorems in Projective Geometry 463
Glossary 467
Bibliography 473
Index 503
PLATO’S GHOST