Plato's Ghost. The Modernist Transformation of Mathematics

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. – Introduction (Opening Remarks; Some Mathematical Concepts); – Chapter 1: Modernism and Mathematics (Modernism in Branches of Mathematics; Changes in Philosophy; The Modernization of Mathematics); – Ch. 2: Before Modernism (Geometry; Analysis; Algebra ; Philosophy; British Algebra and Logic; The Consensus in 1880); – Ch. 3: Mathematical Modernism Arrives (Modern Geometry: Piecemeal Abstraction; Modern Analysis; Algebra; Modern Logic and Set Theory; The View from Paris and St. Louis); – Ch. 4: Modernism Avowed (Geometry; Philosophy and Mathematics in Germany; Algebra; Modern Analysis; Modernist Objects; American Philosophers and Logicians; The Paradoxes of Set Theory; Anxiety; Coming to Terms with Kant); – Ch. 5: Faces of Mathematics (Introduction; Mathematics and Physics; Measurement; Popularizing Mathematics around 1900; Writing the History of Mathematics); – Ch. 6: Mathematics, Language, and Psychology (Languages Natural and Artificial; Mathematical Modernism and Psychology); – Ch. 7: After the War (The Foundations of Mathematics; Mathematics and the Mechanization of Thought; The Rise of Mathematical Platonism; Did Modernism'"Win"?; The Work Is Done). M.-M. V.