The Spatial Content of Logic — Lingyuan Ye

Abstract:

Descartes completely changed our mathematical vision, by finding a correspondence between spaces (geometry) and equations (algebra). Starting from high school, we have learnt that x^2+y^2-1 can be viewed as the the unit circle in the Euclidean plane, and a lot of algebraic arguments we do have very natural geometric meaning behind them, and this geometric intuition serves as a guide for many algebraic solutions. Modern algebraic geometry extends this picture to also allow a geometric study of solutions on any rings other than R or C, like Z or Q. The powerful tools developed in topology suddenly applies to number theory, and derives powerful results.

What I intend to convey in the talk is that the situation in logic is exactly the same, by viewing theories as algebras and models as spaces. Notice in logical terms, we may view x^2+y^2-1 as a formula, and the points in the unit circle are models for this formula, in the sense that they satisfy it (being a root). Similarly, models of a theory will be understood as a space, and each model will be a point in this space. This allows a similar geometric understanding of logical arguments. In fact, this picture for (modal) propositional logic is well known for a long time, in the form of Stone duality (and various other duality results for modal algebras). The interesting development is a corresponding theorem for first-order logic, called conceptual completeness theorem, first proven by Michael Makkai, and recently further developed by Steve Awodey and some of his students, and Jacob Lurie. It is really exciting to see whether the highly developed geometric tools can be applied to logic and model theory, once we understand logic in this way.