New Directions in the Philosophy of Mathematics

by Tymoczko, Thomas

• ISBN13: 9780691034980
• ISBN10: 0691034982
• Edition: Revised
• Format: Paperback
• Copyright: 1998-01-12
• Publisher: Princeton Univ Pr

Summary

The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form. This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

Table of Contents

Preface	ix	(4)
Introduction	xiii
Part I Challenging Foundations	1	(94)
Some Proposals for Reviving the Philosophy of Mathematics	9	(20)
Reuben Hersh
A Renaissance of Empiricism in the Recent Philosophy of Mathematics?	29	(20)
Imre Lakatos
What Is Mathematical Truth?	49	(18)
Hilary Putnam
"Modern" Mathematics: An Educational and Philosophic Error?	67	(12)
Rene Thom
Mathematics as an Objective Science	79	(16)
Nicholas D. Goodman
Interlude	95	(30)
From the Preface of Induction and Analogy in Mathematics	99	(4)
George Polya
Generalization, Specialization, Analogy	103	(22)
George Polya
Part II Mathematical Practice	125	(188)
WHAT IS MATHEMATICAL PRACTICE?	129	(56)
Theory and Practice in Mathematics	129	(24)
Hao Wang
What Does a Mathematical Proof Prove?	153	(10)
Imre Lakatos
Fidelity in Mathematical Discourse: Is One and One Really Two?	163	(14)
Philip J. Davis
The Ideal Mathematician	177	(8)
Philip J. Davis
Reuben Hersh
THE EVOLUTION OF MATHEMATICAL PRACTICE	185	(58)
The Cultural Basis of Mathematics	185	(16)
Raymond L. Wilder
Is Mathematical Truth Time-Dependent?	201	(14)
Judith V. Grabiner
Mathematical Change and Scientific Change	215	(28)
Philip Kitcher
COMPUTERS AND MATHEMATICAL PRACTICE: A CASE STUDY	243	(70)
The Four-Color Problem and Its Philosophical Significance	243	(24)
Thomas Tymoczko
Social Processes and Proofs of Theorems and Programs	267	(20)
Richard A. De Millo
Richard J. Lipton
Alan J. Perlis
Information-Theoretic Computational Complexity and Godel's Theorem and Information	287	(26)
Gregory Chaitin
Part III Current Concerns	313	(72)
Proof as a Source of Truth	317	(20)
Michael D. Resnik
On Proof and Progress in Mathematics	337	(20)
William P. Thurston
Does V Equal L?	357	(28)
Penelope Maddy
Afterword	385	(14)
Bibliography	399	(12)
Supplemental Bibliography of Recent Work	411