Nick @internic@social.cleverlibre.org

@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.

@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.

@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.

@rudyharrelson I'm currently using Authy but one must beware of setting a password for syncing credentials and disabling "add more devices" when you're not actively adding more, so that's far from ideal. For important stuff that doesn't support FIDO/webauthn, I use my Yubikey to store a TOTP secret (which I then use via the Yubikey authenticator app).

@cholling@hachyderm.io @mcc I don't know that I think the advertising model is going away any time soon, but I do think that until we find workable ways to sustainably fund the upkeep of online spaces at scale things will remain pathological; users will remain the product rather than the customer (as has often been observed). Paying people to lie to us so that we'll buy things and then hoping they may support useful online spaces as a byproduct was never a good model, and the only widespread alternative is generally VC-funded services operating at a loss who just haven't started down that path *yet*.

This seems like a key problem that Mastodon and other non-commercial services haven't really solved either. As far as I can tell, most servers just rely on the sporadic charity of a small minority of their users.

These days I'm primarily using #Firefox on my #Linux desktop (specifically Mint), and I'm curious if #passkey might be a reasonable option for me to use on sites that support it. I have an #Yubikey that I already use for #FIDO #2FA on sites that support it. I also have an Pixel phone with a titan chip, but I'm running #grapheneOS on there, and I'm not sure if that will complicate matters. Any relevant experience on either the phone or desktop front or pointers to relevant information would be appreciated.

Near as I can tell, it sort of sounds like passkey can work on Linux but maybe isn't supported in Firefox yet?

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