Plato's Ghost : The Modernist Transformation of Mathematics

by Gray, Jeremy

• ISBN13: 9781400829040
• ISBN10: 1400829046
• Copyright: 2009-01-15
• Publisher: Princeton Univ Pr

Summary

Plato's Ghostis the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions. Plato's Ghostevokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Julius Dedekind, Henri Lebesgue, Jules Henri Poincareacute;, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naiuml;ve set theory and the revived axiomatic method--debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism. Plato's Ghostis essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.

Table of Contents

Introduction	p. 1
Opening Remarks	p. 1
Some Mathematical Concepts	p. 16
Modernism and Mathematics	p. 18
Modernism in Branches of Mathematics	p. 18
Changes in Philosophy	p. 24
The Modernization of Mathematics	p. 32
Before Modernism	p. 39
Geometry	p. 39
Analysis	p. 58
Algebra	p. 75
Philosophy	p. 78
British Algebra and Logic	p. 101
The Consensus in 1880	p. 112
Mathematical Modernism Arrives	p. 113
Modern Geometry: Piecemeal Abstraction	p. 113
Modern Analysis	p. 129
Algebra	p. 148
Modern Logic and Set Theory	p. 157
The View from Paris and St. Louis	p. 170
Modernism Avowed	p. 176
Geometry	p. 176
Philosophy and Mathematics in Germany	p. 196
Algebra	p. 213
Modern Analysis	p. 216
Modernist Objects	p. 235
American Philosophers and Logicians	p. 239
The Paradoxes of Set Theory	p. 247
Anxiety	p. 266
Coming to Terms with Kant	p. 277
Faces of Mathematics	p. 305
Introduction	p. 305
Mathematics and Physics	p. 306
Measurement	p. 328
Popularizing Mathematics around 1900	p. 346
Writing the History of Mathematics	p. 365
Mathematics, Language, and Psychology	p. 374
Languages Natural and Artificial	p. 374
Mathematical Modernism and Psychology	p. 388
After the War	p. 406
The Foundations of Mathematics	p. 406
Mathematics and the Mechanization of Thought	p. 430
The Rise of Mathematical Platonism	p. 440
Did Modernism'"Win"?	p. 452
The Work Is Done	p. 458
Appendix: Four Theorems in Projective Geometry	p. 463
Glossary	p. 467
Bibliography	p. 473
Index	p. 503
Table of Contents provided by Publisher. All Rights Reserved.