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 logicGödel'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.
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About the Author
Charles Parsons is Edgar Pierce Professor of Philosophy, Emeritus, at Harvard University.
Table of Contents
Part I Some Mathematicians as Philosophers
1 The Kantian Legacy in Twentieth-Century Foundations of Mathematics 11
2 Realism and the Debate on Impredicativity, 1917-1944 40
Postscript to Essay 2 62
3 Paul Bernays' Later Philosophy of Mathematics 67
4 Kurt Gödel 93
5 Gödel's "Russell's Mathematical Logic" 103
Postscript to Essay 5 123
6 Quine and Gödel on Analyticity 127
Postscript to Essay 6 147
7 Platonism and Mathematical Intuition in Kurt Godel's Thought 153
Postscript to Essay 7 188
Part II Contemporaries
8 Quine's Nominalism 199
9 Genetic Explanation in The Roots of Reference 220
10 Hao Wang as Philosopher and Interpreter of Gödel 243
11 Putnam on Existence and Ontology 267
12 William Tait's Philosophy of Mathematics 290
Copyright Acknowledgments 343