4 ms·

I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the fo

I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the formalist approach lead by David Hilbert and still has little main-stream support despite Gödel and his incompleteness proofs. Veritasium's latest video on Gödel's incompleteness [2] gives a pretty fair account of how we settled on the current fashionable foundations of Math (including nods to Cantor and Hilbert). For a more formal history there is a book "The Philosophy of Set Theory" [3] that sketches out how we got to where we are. I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality. 1. https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism 2. https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo 3. https://www.maa.org/press/maa-reviews/the-philosophy-of-set-theory-an-historical-introduction-to-cantors-paradise https://www.maa.org/press/maa-reviews/the-philosophy-of-set-...

eigenket 5y ago
I am somewhat familiar with the standard foundation of maths in set theory, and slightly less familiar with the program to ground maths in Category theory (although I do know a fair bit of category theory). Maybe I am biased but I do not find the category theory foundations any less obtuse than the standard formulations. Like its pretty cool that you can do this stuff with category theory, but I think there is a reason the set theory was done first.
- reggieband 5y ago
  I didn't meant to imply that Category Theory addressed the obtuseness of Set Theory. Rather, I was alluding to work in Category Theory that helps to redefine our ideas of equality and equivalence. A discussion of that is available in Quanta Magazine [1]. 1. https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/ https://www.quantamagazine.org/with-category-theory-mathemat...

eigenket 5y ago

I am somewhat familiar with the standard foundation of maths in set theory, and slightly less familiar with the program to ground maths in Category theory (although I do know a fair bit of category theory). Maybe I am biased but I do not find the category theory foundations any less obtuse than the standard formulations. Like its pretty cool that you can do this stuff with category theory, but I think there is a reason the set theory was done first.

reggieband 5y ago
I didn't meant to imply that Category Theory addressed the obtuseness of Set Theory. Rather, I was alluding to work in Category Theory that helps to redefine our ideas of equality and equivalence. A discussion of that is available in Quanta Magazine [1]. 1. https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/ https://www.quantamagazine.org/with-category-theory-mathemat...