Thinking About Mathematics: The Philosophy of Mathematics

An introduction to the philosophy of mathematics organised partly historically and partly by position. Shapiro begins with the questions of what mathematical objects are and how we can know about them, then surveys the main answers: Plato and Aristotle, Kant's account of mathematics as synthetic a priori, Mill's empiricism, the logicism of Frege and Russell, Hilbert's formalism, and Brouwer's intuitionism together with Dummett's later defence of it. He then turns to contemporary views, including Quine's argument that mathematics is indispensable to science, nominalist programmes that try to do without mathematical objects, and structuralism, Shapiro's own position, on which mathematics is the study of structures rather than of individual objects. Technical material is kept to a minimum, though a reader will benefit from knowing some logic.

It is a fair and readable map of the competing answers to what mathematics is about and how we know it.