An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice.

In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes—numbers, rigor, geometry, proof, computability, incompleteness, and set theory—that give rise again and again to philosophical considerations.

Hamkins shows, for example, how number systems set the stage for discussions of such philosophical issues as platonism, logicism, and the nature of abstraction. Consideration of the rise of rigor in the calculus leads to a discussion of whether the indispensability of mathematics in science offers grounds for mathematical truth. Sophisticated technical developments in set theory give rise to a necessary engagement with deep philosophical concerns, including the criteria for new mathematical axioms. Throughout, Hamkins offers a clear and engaging exposition that is both accessible and sophisticated, intended for readers whose mathematical backgrounds range from novice to expert.

Joel David Hamkins is Professor of Logic at Oxford University and Sir Peter Strawson Fellow in Philosophy at University College, Oxford. He has published widely in refereed research journals in mathematical logic and set theory and is the creator of the popular blog Mathematics and Philosophy of the Infinite. He is a prominent contributor to MathOverflow, where he has posted more than 1,000 mathematical arguments.