Plato's Ghost

Introduction 1 I.1 Opening Remarks 1 I.2 Some Mathematical Concepts 16 CHAPTER 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 CHAPTER 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 CHAPTER 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 CHAPTER 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 4.7 The Paradoxes of Set Theory 247 4.8 Anxiety 266 4.9 Coming to Terms with Kant 277 CHAPTER 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. Writing the History of Mathematics 365 CHAPTER 6: Mathematics, Language, and Psychology 374 6.1 Languages Natural and Artificial 374 6.2 Mathematical Modernism and Psychology 388 CHAPTER 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

Jeremy Gray is professor of the history of mathematics and director of the Centre for the History of the Mathematical Sciences at the Open University. His books include "Worlds Out of Nothing" and "Janos Bolyai, Non-Euclidean Geometry", and the "Nature of Space".

In Plato's Ghost, he has ... present[ed] us with an ambitious and in many respects remarkable synthesis of the modern transformation of mathematics via structural and set-theoretic notions, together not only with its logic and philosophy but also with related developments in artificial languages and psychology... I can certainly recommend Plato's Ghost highly as a rich resource and point of departure for readers who want to learn more about this exciting period in the development of modern mathematics. -- Solomon Feferman American Scientist This accessible, rigorous volume belongs in every serious library. -- J. McCleary Choice In a book aimed at the educated public, the author presents an impressive amount of data--both of the kind mathematicians with some awareness of the history of their subject may be aware of, and of an entirely different kind, coming from the outskirts of mathematics, from philosophy, from physics, or from the popularization of mathematics, which will likely be new even to historians of mathematics. -- Victor V Pambuccian Mathematical Reviews It is ... no small assertion to say that the book under review, Plato's Ghost, is [Gray's] most far-reaching and ambitious work to date... [T]here is a wealth of valuable data here which, if not fully processed and pigeonholed, is at least tagged and cataloged in a helpful way. Plato's Ghost provides an insightful and informative resource for anyone doing mathematics today who has wondered how (and perhaps why) the subject has come to possess the features it has today. The book gives us a lot to think about, which is exactly what a good history should do. -- Jeremy Avigad Mathematical Intelligencer In this book Jeremy Gray offers us the fruit of more than a decade reading and thinking about modernism in mathematics. He presents it, in very well written form, to a broad audience interested in mathematics, its history and philosophy. -- Erhard Scholz Metascience What we have here ... is an excellent and