In this thread with @johncarlosbaez I was mentioning how when I first encountered Category Theory it seemed like little more than a curiosity (for my purposes, as a physicist), even though mathematicians seemed excited about it.
I had almost the opposite experience with Nonstandard Analysis (i.e. the hyperreal numbers), in the sense that I bumped into this notion, read a bit about it, and it sounded potentially quite useful. Physicists tend to talk in terms of infinitesimals anyway, so a framework where that could be done rigorously seemed useful, and I was curious if it might provide nice ways to think about other things such a path integrals or even renormalization. But the only mathematician I talked to about it dismissed it as basically a curiosity. I believe the way he put it was that it was "just a trick to avoid an extra quantifier in in proofs."
Now I'm curious if that is the consensus among the #physics and #mathstodon folks around here or if people see it as a practically useful tool.
#math #maths
QT: https://qoto.org/@internic/110639879113598032
@internic @johncarlosbaez I had an interesting conversation with Mike Gage (of the Gage-Hamilton-Grayson Theorem) about this one time. If I remember correctly, he had done some kind of REU on nonstandard analysis (then a new topic) and thought he might go into the area. He didn't end up doing this however, and he told me that people already think in terms of infinitesimals when doing analysis, so as long as there is at least one workable formalism it doesn't really matter which one. I also met the author of a nonstandard analysis textbook (although I confess I no longer remember the guy's name or the name of the book) and he told me that part of the utility of nonstandard analysis was the way in which one could construct completions of spaces to create objects like the hypperreals. I still don't kmow enough about the subject to say more about what he was getting at here.
@caten @johncarlosbaez Right, it definitely seems to be a general recipe that can be applied to make nonstandard versions of many things, not just the reals. But I never delved into further applications.
