Path induction and the indiscernibility of identicals

Abstract: Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction. This talk introduces a closely related proof technique called "path induction," which can be thought of as an expression of Leibniz's "indiscernibility of identicals": if x and y are identified, then they must have the same properties, and conversely. What makes this interesting is that the notion of identification referenced here is given by Per Martin-Löf's intensional identity types, which encode a more flexible notion of sameness than traditional equality because an identification can carry data, for instance of an explicit isomorphism or equivalence. The nickname "path induction" for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations. Path induction is then justified by the fact that based path spaces are contractible.

Mathematics

Audience: general audience

VCU Mathematics and Applied Mathematics Colloquium Series

Series comments: A series of public lectures in Mathematics and Applied Mathematics from Virginia Commonwealth University.

To attend virtually, please connect via Zoom using the following coordinates:

Zoom Meeting ID: 822 7853 4531 Password: VCUMATH101