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.

**PART I: THE PROBLEM**

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

**PART II: REALISM**

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

**PART III: NATURALISM**

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

Conclusion

Bibliography

Index

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