Philosophy of Mathematics in the Twentieth Century : Selected Essays

In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century. __Philosophy of Mathematics in the Twentieth Century__ insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine's early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work __The Roots of Reference__. Parsons attempts to tease out Putnam's views on existence and ontology, especially in relation to logic and mathematics. Wang's contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait's axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures. In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century. Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl's idea of a "vicious circle" in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell's mathematical logic--Gödel's first mature philosophical statement and an avowal of his Platonistic view. Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine's early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference . Parsons attempts to tease out Putnam's views on existence and ontology, especially in relation to logic and mathematics. Wang's contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait's axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures Contents ......Page 8 Preface......Page 10 Introduction......Page 16 Part I: Some Mathematicians as Philosophers......Page 24 1. The Kantian Legacy in Twentieth-Century Foundations of Mathematics......Page 26 2. Realism and the Debate on Impredicativity, 1917–1944......Page 55 Postscript to Essay 2......Page 77 3. Paul Bernays’ Later Philosophy of Mathematics......Page 82 4. Kurt Gödel......Page 108 5. Gödel’s “Russell’s Mathematical Logic”......Page 118 Postscript to Essay 5......Page 138 6 Quine and Gödel on Analyticity......Page 142 Postscript to Essay 6......Page 162 7. Platonism and Mathematical Intuition in Kurt Gödel’s Thought......Page 168 Postscript to Essay 7......Page 203 Part II: Contemporaries......Page 212 8. Quine’s Nominalism......Page 214 9. Genetic Explanation in The Roots of Reference......Page 235 10. Hao Wang as Philosopher and Interpreter of Gödel......Page 258 11. Putnam on Existence and Ontology......Page 282 12. William Tait’s Philosophy of Mathematics......Page 305 Bibliography ......Page 336 Copyright Acknowledgments......Page 358 Index......Page 360 Frontmatter --CONTENTS --PREFACE --INTRODUCTION --1 THE KANTIAN LEGACY IN TWENTIETH-CENTURY FOUNDATIONS OF MATHEMATICS --2 REALISM AND THE DEBATE ON IMPREDICATIVITY, 1917-1944 --POSTSCRIPT TO ESSAY 2 --3 PAUL BERNAYS'LATER PHILOSOPHY OF MATHEMATICS --4 KURT GÖDEL --5 GÖDEL'S "RUSSELL'S MATHEMATICAL LOGIC" --6 QUINE AND GÖDEL ON ANALYTICITY --POSTSCRIPT TO ESSAY 6 --7 PLATONISM AND MATHEMATICAL INTUITION IN KURT GÖDEL'S THOUGHT --POST SCRIPT TO ESSAY 7 --8 QUINE'S NOMINALISM --9 GENETIC EXPLANATION IN THE ROOTS OF REFERENCE --10 HAO WANG AS PHILOSOPHER AND INTERPRETER OF GÖDEL --11 PUTNAM ON EXISTENCE AND ONTOLOGY --12 WILLIAM TAIT'S PHILOSOPHY OF MATHEMATICS --BIBLIOGRAPHY --COPYRIGHT ACKNOWLEDGMENTS --INDEX In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Goedel, Russell, Quine, Putnam, Wang, and Tait. In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait.