The Fundamental Theorem of Calculus: Point-Free Analysis

Explore a detailed lecture from the Topos Institute Colloquium that examines point-free topology and its application to the Fundamental Theorem of Calculus. Discover how point-free topology defines points of a space through logical theory models rather than traditional point-set elements, with opens corresponding to propositional formulae. Learn about the advantages of this approach, including its seamless extension to toposes as generalized spaces and natural treatment of bundles. Examine a case study of the Fundamental Theorem of Calculus using geometric reasoning, featuring discussions on one-sided reals, hyperspaces, and uniform probability measures. Delve into the implications for physics and real analysis while understanding how geometric methods can be applied to mathematical concepts through this alternative topological perspective.