Wed Oct 15, 2008
234 Moses Hall, 6–8 PM
|Working Group in the History and Philosophy of Logic, Mathematics, and Science
Elain Landry (UC Davis)
Reconstructing Hilbert to Construct Category-Theoretic Algebraic Structuralism
Recently Shapiro has used the Frege-Hilbert debate to argue that category theory cannot be used to frame the position of the algebraic structuralist. More generally, he argues that one cannot be a structuralist all the way down; either, like Frege, one is forced to accept the existence of an assertory background theory, or, like Hilbert, one is forced to appeal to “philosophy”. In this paper I argue that this dichotomy is false. One can rationally reconstruct the components (conceptual, logical and contentual) of Hilbert’s structuralism in a way that shows that one can be a structuralist all the way down. Using category-theory to then frame the position of the Hilbert-inspired algebraic structuralist, one can use the various category axioms to mathematically analyze the concepts of mathematics, one can use various categorical logics to logically analyze the “criterion of acceptability” for axiom systems themselves, and one can use various category-theoretic structures and methods to analyze the content (semantic, proof-theoretic, or finitistic) of mathematical reasoning itself. Thus, one need not appeal to either an assertory, meta-mathematical, background theory or to “philosophy”; one can use category theory itself to frame the position of the algebraic structuralist.