jesparentAug 3, 2023
Yohan John 🤖🧠

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?

#Philosophy #Mathematics

Aug 3, 2023 at 10:57pmWeb
Show threadRichard ZachAug 4, 2023

@DrYohanJohn @johncarlosbaez not directly an answer to your question but relevant:

Awodey's From sets to types to categories to sets

Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby.
https://kilthub.cmu.edu/articles/journal_contribution/From_Sets_to_Types_to_Categories_to_Sets/6491651

From Sets, to Types, to Categories, to Sets

Department of Philosophy technical report

figshare
Show threadYohan John 🤖🧠Aug 4, 2023

@rrrichardzach @johncarlosbaez

Thanks! I've read some of Awodey's papers... I'll definitely have a look.

Show threadJohn Carlos BaezAug 4, 2023
@DrYohanJohn - it's hard to say how mainstream it is. To me the most important thing is that it's true: various approaches to category can serve as alternatives to traditional set theoretic foundations, and homotopy type theory does this in a truly elegant way. But maybe you're mainly interested in how many people believe this?
Show threadYohan John 🤖🧠Aug 4, 2023

@johncarlosbaez

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.

Show threadJohn Carlos BaezAug 4, 2023

@DrYohanJohn

"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.

Show threadAntonio PatriarcaAug 4, 2023
@johncarlosbaez @DrYohanJohn I attended a topology course in my BSc containing some basic algebraic topology and category theory and it was a requirements for the MSc. It was however in Italy and most of the courses we did in the BSc from the second year were from books in the graduate math category of the various editors. On the other hand I think I never had a course on set theory foundations. I was however specialized in geometry and topology.
Show threadJon SterlingAug 4, 2023
@DrYohanJohn @johncarlosbaez @rrrichardzach I would not say this viewpoint is widespread, but it kind of doesn’t matter: professional mathematics is done in an informal language that could be encoded formally in set theory, type theory, or category theory. In this sense, everyday mathematics is abstracted away from the choice of one of these foundational styles.