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Price: $66.50

Paperback 262 pp.
138 mm x 216 mm



Publication date:
June 2000

Imprint: OUP UK

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Naturalism in Mathematics

Penelope Maddy

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 in work 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 to metaphysical 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 axiomatic status 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.

Readership : Philosophers, mathematicians, and logicians; graduate students in these subjects.


  • `An excellent book ... The philosopher's task is not to provide external criticism, but rather to clarify debates in this so-called mathematical community. And Ms Maddy makes some fascinating (and very technical) steps in this direction.'
    The Economist Review
  • `'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 and original 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.'

1. The origins of set theory
2. Set theory as a foundation
3. The standard axioms
4. Independent questions
5. New axiom candidates
6. V = L
1. Godelian realism
2. Quinean realism
3. Set-theoretic realism
4. A realist's case against V = L
5. Hints of trouble
6. Indispensability and scientific practice
7. Indispensability and mathematical practic
1. Wittgensteinian anti-philosophy
2. A second Godelian theme
3. Quinean naturalism
4. Mathematical naturalism
5. The problem revisited
6. A naturalist's case against V = L

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

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Special Features

  • The latest work from a leading figure in the field
  • An original new theory of the foundations of mathematics
  • Written accessibly and engagingly for both philosophers and mathematicians
  • Technicalities are kept to a minimum
  • Maddy explains set theory in a way that non-mathematicians can understand
  • The follow-up to the highly successful Realism in Mathematics