Everytime I look up of the link between philosophy and mathematics, I see the topics only of the most foundational levels discussed. As in logic, and stuff. When I study higher mathematics theories, say, topology, analysis and so on, the explicit philosophical themes which I was invited in seem to slowly fade out, and it starts turning into something else. I believe, many mathematicians (excp. Logicians) actually think of the mathematical theories outside of the basic logic which they are said in.
This leads to my curiosity, are there developed entry points into higher mathematics from Philsophy? As in, a philosophical view points of mathematical fields excluding those which consider the language of logic that the field written in.
What I know so far:
One book I saw giving "weakly" developed entry point was Sheaf Theory by Rosiak, where it shows how philosophy can be seen as, at the very least, a starting motivation point for many category theoritic concepts. I personally have a pet theory that category theory was influenced by the general themes of structuralism popularized in 1940-1950 time, and maybe, topos theory of post structuralism. For one more interested in post structuralism.
Another thing I learned was modal logic as entry point for topology, where one can understand modal logic through studying some topological spaces.
. For our purposes here, the founding event is Tarski’s topological interpretation of modal logic, culminating in his proof with McKinsey that the simple decidable modal logic S4 is complete for interpreting modal ♦ as topological closure on the reals or any metric space like it.
Modal Logics of Space, Johan van Benthem & Guram Bezhanishvili