Plato’s Ghost

Plato's Ghost is 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 Ghost evokes 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 Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve 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 Ghost is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.

Introduction 1I.1 Opening Remarks 1I.2 Some Mathematical Concepts 16CHAPTER 1: Modernism and Mathematics 181.1 Modernism in Branches of Mathematics 181.2 Changes in Philosophy 241.3 The Modernization of Mathematics 32CHAPTER 2: Before Modernism 392.1 Geometry 392.2 Analysis 582.3 Algebra 752.4 Philosophy 782.5 British Algebra and Logic 1012.6 The Consensus in 1880 112CHAPTER 3: Mathematical Modernism Arrives 1133.1 Modern Geometry: Piecemeal Abstraction 1133.2 Modern Analysis 1293.3 Algebra 1483.4 Modern Logic and Set Theory 1573.5 The View from Paris and St. Louis 170CHAPTER 4: Modernism Avowed 1764.1 Geometry 1764.2 Philosophy and Mathematics in Germany 1964.3 Algebra 2134.4 Modern Analysis 2164.5 Modernist Objects 2354.6 American Philosophers and Logicians 2394.7 The Paradoxes of Set Theory 2474.8 Anxiety 2664.9 Coming to Terms with Kant 277CHAPTER 5: Faces of Mathematics 3055.1 Introduction 3055.2 Mathematics and Physics 3065.3 Measurement 3285.4 Popularizing Mathematics around 1900 3465. Writing the History of Mathematics 365CHAPTER 6: Mathematics, Language, and Psychology 3746.1 Languages Natural and Artificial 3746.2 Mathematical Modernism and Psychology 388CHAPTER 7: After the War 4067.1 The Foundations of Mathematics 4067.2 Mathematics and the Mechanization of Thought 4307.3 The Rise of Mathematical Platonism 4407.4 Did Modernism'"Win"? 4527.5 The Work Is Done 458Appendix: Four Theorems in Projective Geometry 463Glossary 467Bibliography 473Index 503