Steve Vickers: "The Fundamental Theorem of Calculus: point-free"

Topos Institute Colloquium, 30th of May 2024.
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Point-free topology is known in various guises (locales, formal topology), but can be boiled down to a procedure of defining the points of a space not (“point-set”) as elements of a set, but as models of a logical theory. This is for a constrained “geometric” logic, so that the opens of the space correspond to propositional formulae derived from the theory: thus the theory defines both the points and the topology. Then continuity of maps just means that they are constructed in accordance with the constraints of the logic.
Why bother to do topology that way? After all, the logic is even more constrained than constructive reasoning, and we still don’t know how far it reaches.
The first reason is that it quite painlessly extends to toposes, viewed as generalized spaces. This uses the machinery of classifying toposes, but only in an unobtrusive way [1]. Many proper classes can then be viewed as point-free spaces - just write down a geometric theory whose models are the elements of the class.
The second reason follows from the first but can then be applied back to ungeneralized spaces in a very natural treatment of bundles. Theory presentations can themselves be described as the models of a geometric theory, and this allows us to view bundles as continuously mapping base points to spaces (the fibres). For physics in particular, this invites exploration of how much can be done using geometric methods. See eg [2].
Meanwhile, there is the question of how much ordinary mathematics, and in particular real analysis, can be done in this style. I present as a case study the Fundamental Theorem of Calculus [3]. This illustrates some typically geometric features of the reasoning, such as attention paid to one-sided reals and the use of hyperspaces and their analogues, and some exploitation of the geometric fact that everything is continuous, as well as a cute new trick using uniform probability measures.
[1] Steven Vickers “Topical categories of domains”, Mathematical Structures in Computer Science 9 (1999)
[2] Bas Spitters, Steven Vickers and Sander Wolters “Gelfand spectra in Grothendieck toposes using geometric mathematics", Electronic Proceedings in Theoretical Computer Science 158 (2014)
[3] Steven Vickers “The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms", arXiv:2312.05228

[@Kram1032]

8:10 and everybody woke up

[@MDNQ-ud1ty]

That was the moment Satan tried to take over Vickers soul but failed because Vickers Top-fu was too to free.

[@dmc2925]

For some reason, it wouldn't let me reply again on the Livestream, but Ming blew it out the park with his answer.