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 - A few people have tried to use nonstandard analysis to prove theorems in analysis that haven't succumbed yet to traditional techniques. I've mainly seen this in attempts to make interracting quantum field theories rigorous - this subject is full of unsolved problems in analysis. But none of the nonstandard attempts have made much progress. And that makes sense to me, since I don't see how nonstandard analysis would help much here.

@johncarlosbaez @internic For that matter, how much does physics really need the real numbers themselves? Of course the real field has lots of nice properties, but I wouldn't be shocked if you could say everything that needed to be said over smaller fields like the definable numbers or larger fields like the hyperreals or surreals, maybe with some slight kludges for when completeness is needed. Perhaps aliens would this it's strange we think we need noncomputable numbers to talk about reality.

@caten @johncarlosbaez @internic i understand the reasons for physicists to use the reals but I'm really bugged that linear Algebra is not taught over Qbar

@jesusmargar @caten @johncarlosbaez If physics is conceived of as an empirical science, which exclusively concerns itself with measurements of finite precision, it seems to me an argument can be made that one never really needs to deal with more than the rational numbers. Added to that, even conceptually it is becoming more an more popular to assume that spacetime consists of discrete constituents of some sort.

Yet all the way from Newton and Leibniz, physics has been integrally based on calculus (pardon the pun), which (at least in my simplistic understanding) requires completeness to formulate*. So it seems like ultimately what one wants is to make statements about quantities that are rational, but the most economical way to express and prove those relationships is via completion of that field...which I guess is ironically somewhat comparable to how one uses NSA to make statements about the reals.

* IIRC (if you define calculus concepts via NSA) the hyperreals are not complete, but ultimately you're seeking to make statements about the reals, which are.