I often hear that category theory (and/or homotopy type theory) can serve as an alternative to set theory as a foundation for mathematical thinking.
How mainstream is this idea among mathematicians and philosophers of mathematics?
Perhaps @johncarlosbaez can weigh in?
Thank you!
Yes, I'm wondering if this is a widespread view in math departments. Has it influenced how mathematics is taught to undergraduates? I studied physics two decades ago: we were taught informal set theory and real analysis but no one mentioned category theory. Has the situation changed?
I was reading philosophy from the mid 20th century that filed sets into the ontologically "real" because of their usefulness in foundations. Wondering how cat theory changes things.
"Yes, I'm wondering if this is a widespread view in math departments."
I mainly talk to people who know category theory, so I have a biased sample. If I try to account for this, I imagine most mathematicians don't know or care much about foundations, much less alternative foundations. On the other hand, category theory is widely accepted as useful in algebraic topology and algebraic geometry, which are very popular subjects.
"Has it influenced how mathematics is taught to undergraduates?"
Not much, though I keep meeting bright undergrads who have studied category theory on their own, or want to. Category theory is very popular on the internet!
When it comes to "foundations", it's important to remember that most mathematicians couldn't recite the ZF axioms if their lives depended on it. They tend to think of sets as fundamental, and know a bunch of things you're allowed to do with them, but don't care much about reducing these things to a limited set of axioms. I suspect philosophers are much more enamored of the axiomatic approach.
Yohan John 🤖🧠
John Carlos Baez
Antonio Patriarca