Foundations of Mathematical Structuralism

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures.