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
"I'm really bugged that linear Algebra is not taught over Qbar"
In an upper-level undergraduate course where you prove stuff, linear algebra is often done over an arbitrary field. This is basically just as easy as working over the algebraic completion of the rationals, and it covers a lot more ground, since finite fields, the rational numbers, and more exotic fields like the computable reals and definable reals and hyperreals are all covered.
@johncarlosbaez @jesusmargar @caten @internic :
The Linear Algebra course is where a lot of people will first see abstract algebra (in the coverage of arbitrary vector spaces), but I still expect them to stick to the real and complex numbers, without doing arbitrary fields. (And while any field of characteristic 0 might as well be a subfield of ℂ, finite characteristic has some surprises.) Do they do arbitrary fields in the course at UCR? (If so, I didn't notice while I was there, but I never TAed it so could have missed that).
By your testimony further down, they did arbitrary fields in the course at Princeton, but that's a more prestigious school that might be more willing to go its own way. I too went to a prestigious undergrad school (Caltech), and even they didn't do that (although they still did something special with it, which was to put it early, before multivariable calculus, where everybody would have to take it). Bottom line, I think that arbitrary fields are rare in American undergraduate Linear Algebra courses.
@TobyBartels @johncarlosbaez @caten @internic in my non-Russell group, minimum entry B UK public university we teach over arbitrary k. In top 10 UK British (top 1 Scottish) Edinburgh university they chose (choose?) R or C, according to my recollection. My point is that the choice has more to do with opinions, priorities and preferences than a real need for one or the other from a pedagogical viewpoint. My preference for Q bar over C is that it has all the needed properties, not more and not less
@jesusmargar @TobyBartels @johncarlosbaez @caten When I first took linear algebra (as a senior-level math course "for scientists and engineers" in the US) I believe it was specifically limited to vector spaces over the reals. The book may have technically addressed vector spaces over a more general field, but that was certainly never discussed in the class (nor was the definition or properties of a field).
Looking back, that seems like an odd choice; since scientists and engineers using linear algebra are often concerned with eigen-problems and have to work with complex numbers anyway, teaching it over the complex numbers would seem to me like the more useful and ultimately simpler road to take. But I think it was an issue of prerequisites and where complex numbers were taught in the math sequence. There was also a version of the course for mathematicians, and I don't know how that may have differed.
@internic @jesusmargar @johncarlosbaez @caten : I understand an emphasis on real numbers, but it seems shocking to me to have no complex vector spaces at all. After all, real matrices can have imaginary eigenvalues, so imaginary eigenvectors, imaginary diagonalizations, etc. And science and engineering majors should certainly be familiar with complex numbers by their senior year! I suppose that if you kludge it enough, you can talk about matrices with complex entries without formally introducing vector spaces over the complex field, but it seems like a bad approach.
@TobyBartels @jesusmargar @johncarlosbaez @caten Agreed. As I said, looking back it seems like a really odd choice. Especially because it's not like you have to teach complex analysis, just the basics.
I'm not sure who all "science and engineering majors" was supposed to cover. Certainly physics and EE students should know complex numbers well by senior year. If they were trying to accommodate other sciences and things like software engineering then I'm less sure. In my professional life I've found that people with a CS degree (from a US university) often have little to no familiarity with complex numbers.
@internic @TobyBartels @johncarlosbaez @caten interesting. In Spain you see complex numbers in compulsory high school (age 15) including De Moivre's Theorem. I believe in the UK at least within A-levels (18yo, not compulsory) Mathematics (if not in Further Maths) they are seen too.
@jesusmargar @TobyBartels We may have seen complex numbers in high school (around age 15--16); but they definitely weren't covered in significant depth. I only really learned them well in my early physics courses on wave mechanics. In any case, if people only saw them briefly as teenagers I wouldn't count on them remembering that much by the 3rd or 4th year of university.
@internic @jesusmargar @johncarlosbaez @caten : I teach at an American community college. We cover complex numbers in an Intermediate Algebra course that's considered remedial by the 4-year schools that we transfer into. In every other course, complex numbers are treated only briefly, essentially as an auxiliary to real numbers, but we expect the students to already know about them. (We don't teach Linear Algebra, since that's considered upper-level and so wouldn't be accepted for transfer credit.)
@internic @TobyBartels @johncarlosbaez @caten yep, same thing when I taught this version in Hopkins. I'm not sure if there was an 'honours' version for mathematicians, probably yes (there was one for calculus). If it was it was probably done over arbitrary field.