THE GENETIC VERSUS THE AXIOMATIC METHOD: RESPONDING TO FEFERMAN 1977

Abstract

AbstractFeferman (1977) argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro (2005), we can be structuralists all the way down; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his (Hilbert, 1900a) distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up.