Structuralism in the Philosophy of Mathematics

Two related slogans for structuralism in the philosophy of mathematics are that “mathematics is the general study of structures” and that, in pursuing such study, we can “abstract away from the nature of objects instantiating those structures”. (As such, structuralism stands in contrast with several other general views about mathematics, including: the traditional view that mathematics is the science of number and quantity; the view that it is an empty formalism used primarily for calculation; and the view that it is the study of a basic set-theoretic universe.) As the present survey aims to show, these slogans, while suggestive, are ambiguous and in need of clarification. Indeed, they have been interpreted in various different, even conflicting ways.

The introduction of structuralist views in the philosophy of mathematics is often assumed to have happened in the 1960s, in works by Paul Benacerraf and Hilary Putnam; the trend picked up steam in the 1980s–90s, when Michael Resnik, Stuart Shapiro, Geoffrey Hellman, Charles Parsons, and others entered the fray; and these debates have been reshaped again during the last 20 years, by several philosophical challenges to structuralism and by the introduction of further variants, including category-theoretic forms of structuralism. Besides introducing the reader to the general topic of “structuralism in the philosophy of mathematics”, a second main goal of the present essay will be to provide a novel, broader, and relatively comprehensive taxonomy for the varieties of structuralism on offer today.