Defending the axioms : on the philosophical foundations of set theory / Penelope Maddy.

Originally published: 2011.

Includes bibliographical references and index.

I. Problem -- 1. Historical reversal -- 2. How applied mathematics became pure -- 3. Where we are now -- II. Proper Method -- 1. Meta-philosophy -- 2. Some examples from set-theoretic practice -- 3. Proper set-theoretic method -- 4. Challenge -- III. Thin Realism -- 1. Introducing Thin Realism -- 2. What Thin Realism is not -- 3. Thin epistemology -- 4. Objective ground of Thin Realism -- 5. Retracing our steps -- IV. Arealism -- 1. Introducing Arealism -- 2. Mathematics in application -- 3. What Arealism is not -- 4. Comparison with Thin Realism -- 5. Thin Realism/Arealism -- V. Morals -- 1. Objectivity in mathematics -- 2. Robust Realism revisited -- 3. More examples from set-theoretic practice -- 4. Intrinsic versus extrinsic.

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments.