Naturalism in Mathematics

Paperback | June 1, 2000

byPenelope Maddy

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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.

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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, esp...

Penelope Maddy is Professor of Philosophy at the University of California, Irvine.

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Format:PaperbackPublished:June 1, 2000Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198250754

ISBN - 13:9780198250753

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Table of Contents

PART I: THE PROBLEM1. The origins of set theory2. Set theory as a foundation3. The standard axioms4. Independent questions5. New axiom candidates6. V = LPART II: REALISM1. Godelian realism2. Quinean realism3. Set-theoretic realism4. A realist's case against V = L5. Hints of trouble6. Indispensability and scientific practice7. Indispensability and mathematical practicPART III: NATURALISM1. Wittgensteinian anti-philosophy2. A second Godelian theme3. Quinean naturalism4. Mathematical naturalism5. The problem revisited6. A naturalist's case against V = LConclusionBibliographyIndex

Editorial Reviews

`'This book represents the culmination of Penelope Maddy's recent work in the philosophy of mathematics. . . . The book is beautifully written, tightly argued and makes compelling reading. I believe the position Maddy introduces and defends - set theoretic naturalism - is a significant andoriginal addition to the philosophy of mathematics landscape, and one that will certainly attract a great deal of attention. . . . In sum, this is a very important book covering some fascinating terrain on the border between philosophy and mathematics.'Mind