ALS: E. Landry
March 26 @ 3:30 pm - 5:30 pm
University of California, Davis, Department of Philosophy
As-ifism: Mathematics and Method without Metaphysics
Zoom webinar. Pre-registration is required. Please register here: https://pitt.zoom.us/webinar/register/WN_iYWEBu-gRMCVQwzUtfYIDg
ABSTRACT: I aim to carve out an as-if interpretation of mathematical structuralism by disentangling methodological considerations from metaphysical ones. I begin first with Plato and draw important lessons from his account of mathematics. More specifically, my aim will be to show that much philosophical milk has been spilt owing to our confusing the method of mathematics with the method of philosophy, and that, as a result, mathematical considerations are conflated with metaphysical ones. I further use my reading of Plato to develop what I call as-if-ism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the aim of solving mathematical problems. I then extend this view to modern mathematics wherein the method of mathematics becomes the axiomatic method, noting that this engenders a shift from as-if hypotheses to as-if axioms and from the investigation of kinds of objects to the consideration of systems that have a structure. This structuralist perspective is then set within a Plato-inspired methodological context to argue for an as-if interpretation of mathematical structuralism. Again, I pause to note that confusion of the method of mathematics with the method of philosophy, witnessed well by the Frege-Hilbert debate, has led to the continued conflation of mathematics with metaphysics. Finally, I combine Plato’s as-if account of applicability with Maddy’s more recent enhanced if-thenist approach to show how such conflations can and should be avoided in the current structural realist debates. My overall lesson then is this: when we shift our focus from philosophical problems to mathematical ones, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of both the practice and the applicability of mathematics whilst avoiding the conflation of mathematical and metaphysical considerations.