Friday, 20 January 2017 in Leacock 927 (see note below)
3:30 - 5:30   M. Makkai (McGill)
Meta-mathematics in Bourbaki's structuralism and in more recent developments in mathematical structuralism

[Abstract] Mathematical structuralism is summarized by the somewhat oversimplified statement that "mathematics is the study of abstract structures". "Meta-mathematics" is meant here in a sense that is more general than, but still closely related to, S. C. Kleene's classic "Introduction to Metamathematics" (1952; with many later editions). Bourbaki's concept of structure, published in 1969, but formulated much earlier, places the concept of isomorphism of structures in the center of the concept, by requiring the provable invariance under isomorphism for each of the formally stated concepts of the species of structures. Here the expressions "provable" and "formally stated" are the ones that show that Bourbaki needs a context in which an underlying formal language (formalized set-theory in their case) has to be discussed -- and not just used -- by "exact", "mathematical", in fact, meta-mathematiical methods. What "meta-mathematical methods" are is one of the things I will have to try to explain - but if you want to get an idea about it, read the introductory "Part I. The Problem of Foundations" in Kleene's book. The "more recent developments" I am alluding to in the title are in several new formal systems, related to category theory in various ways, for set theory, in which Bourbaki's concept of structure is much more naturally formulated than in classical set theory.

Note: This is not part of this seminar's activities, but may be of interest to our regular participants.
Location: The talk will be held in the Dept of Philosophy, McGill, Leacock 927.