Naturalism in Mathematics

by Maddy, Penelope

• ISBN13: 9780198250753
• ISBN10: 0198250754
• Format: Paperback
• Copyright: 2000-09-14
• Publisher: Clarendon Press

Summary

Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress inwork on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view--realism--is assessed and finally rejected in favour of another-- naturalism--which attends less tometaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomaticstatus in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Author Biography

Penelope Maddy is Professor of Philosophy at the University of California, Irvine, having previously held positions at the University of Illinois, Chicago, and the University of Notre Dame.

Table of Contents

PART I: THE PROBLEM	1	(86)
The Origins of Set Theory	3	(19)
Set Theory as a Foundation	22	(14)
The Standard Axioms	36	(27)
Independent Questions	63	(10)
New Axiom Candidates	73	(9)
V = L	82	(5)
PART II: REALISM	87	(74)
Godelian Realism	89	(6)
Quinean Realism	95	(13)
Set Theoretic Realism	108	(2)
A Realist's Case against V = L	110	(20)
Hints of Trouble	130	(3)
Indispensability and Scientific Practice	133	(25)
Indispensability and Mathematical Practice	158	(3)
PART III: NATURALISM	161	(72)
Wittgensteinian Anti-Philosophy	162	(10)
A Second Godelian Theme	172	(5)
Quinean Naturalism	177	(6)
Mathematical Naturalism	183	(23)
The Problem Revisited	206	(10)
A Naturalist's Case against V = L	216	(17)
Conclusion	233	(2)
Bibliography	235	(14)
Index	249