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
@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.
@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 @johncarlosbaez @internic I'd say that you might as well do it over the reals since, in addition to introducing students to abstract algebra, linear algebra also introduces them to Euclidean n-space. It would be a little jarring to learn in topology or analysis the following semester that your favorite vector space is full of infinitesimal "holes".
@caten @johncarlosbaez @internic ah, but in most places (my university) we don't do this.
"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 @caten @internic that certainly was not my experience. I studied in a very analysis oriented department and it was all done over R or C. I vaguely remember that was also the case in Edinburgh when I TAed there. Upon reflection I realised that the only reason we used these two fields is because they/we saw them in school (and in the case of my UG because we didn't treat fields till the 4th year of 5 of a joint a degree).
@jesusmargar @caten @internic - in some Spanish universities they have a separate Department of Analysis and Department of Algebras. I found this shocking, since it makes me imagine some professors who are only allowed to use ℝ and ℂ while others are only allowed to use other fields. 🙃
However, when I was an undergrad in the US we did linear algebra over an arbitrary field. I remember getting very confused about fields where 1+1 = 0, but it was good for me.
@johncarlosbaez I studied in a Spanish university and while there was only one department de jure, there was such a division de facto. When the division does exist on paper they double all administration (e.g. two dept meetings, two administrators, two sets of paperwork, etc), which sounds insane to me, ring-fence courses (which makes some sense) and positions.
@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 @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.
@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.)
@jesusmargar @caten @johncarlosbaez @internic : If you pretend that ℂ is just the course's idiosyncratic symbol for Qbar, you'll never be contradicted.
@TobyBartels @caten @johncarlosbaez @internic exactly my point.
@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.
@jesusmargar Sorry, in this context NSA = nonstandard analysis (others were using it elsewhere in threads responding to the original post).
@internic ah, I see, thanks. I ignored most of those comments because I never heard of hyper reals before (and I'm a professional mathematician but I guess most of those working on NSA have never heard of flips, flops or klt singularities so each to their own).
@jesusmargar I certainly don't know what any of those things are! 
But it was my mistake for transposing the abbreviation between sub-threads. If I had been clever, I would have defined it in the original post to make everyone's life easier.
@internic i actually feel my UG shoudk have been broader and I should have studied elsewhere. No computer engineering, a faculty with more optional courses in Algebra and force myself to learn beyond getting an A+, which I could easily have done.
@jesusmargar I relate to your last sentence myself.
"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."
One could certainly do this. Currently most mathematical physicists don't feel it's worth the extra bother. It would become worth the bother if something of interest to physicists could be accomplished using one of these alternative formalisms but not the usual one. Merely philosophical advantages, like the supposed advantages of avoiding uncomputable or undefinable numbers, etc., are not going to make most mathematical physicists go to the trouble of learning a new formalism and reproving a lot of theorems they already know. I'd be happy to see alternative traditions emerge, but so far progress is slow because it's hard to get a lot of people to agree to work in an alternative framework. The most progress I've seen is in the framework of computable analysis, e.g. the stuff summarized in Pour-El and Richards' book. This has the sociological advantage of using traditional logic and set theory and the traditionally defined real numbers, but analyzing them using ideas from computability:
https://en.wikipedia.org/wiki/Computability_in_Analysis_and_Physics
@johncarlosbaez @internic That sounds totally reasonable. It's really neat that people have considered aspects of computability in physics enough to write a book, but it also makes sense that one wouldn't want to learn a whole new formalism when the results it obtains are about the same as with the established system. Thanks for sharing this!
@johncarlosbaez Are there areas where you would expect (at least intuitively) there to be more likely to be of some practical use or not really?
As one with a physics background who really got into both nonstandard analysis and category theory, I think the views you describe (category theory is wildly useful and NSA is a curiosity) are predominant from what I’ve seen. However, I have some fairly uncommon views that might be interesting as to why this is and how I interpret the situation.
Over the years, I have grown some strong ultrafinitist tendencies. Since we can only make finite distinctions / measurements in finite time, we can never validate infinite models. Models with continuous elements (like spacetime) are only ever “computationally useful” and not indicative of any revealed truth of verifiable / refutable “reality”. There are always many finite models available to fit any observations we may ever collect.
Because of this, I actually don’t see the bias against NSA as having any merit. Sure it’s a different model of number than the standard reals - but having many models to choose from isn’t an argument against any of them. We will never be able to choose between any of them. But we may grow fruitful ideas by knowing and considering multiple models, where a focus on a single model may cause ideas to stagnate.
And this idea of model pluralism is important in physics generally. There is a tendency in physics to want a single “right” model, and this often turns to mockery and other negative behaviors towards alternate-yet-entirely-isomorphic models. This is huge in quantum mechanics, for instance. There is a long history of proposals like Bohmian mechanics (which is just a rewrite of the evolution equation into polar form and separation into real and imaginary components to reveal a kinetic equation for worldlines), or Many Worlds (simply a reevaluation of the complex wave function into Kripke frames of possible worlds modality), or Consistent Histories, or… All are easily provable to have the exact same predictions as standard interpretations (because they are fundamentally the same equations), but have been met with horrifying abuse over the years. What happened to Everett was utterly cruel.
NSA is “just another model”, but that’s a good thing. It can offer ideas on how to formulate new theories of space and, at the same time, offer illustrative examples of where we could use some model tolerance and be less abusive in our interactions.
As for category theory, it’s similar, but there is an angle. Yes, algebra can capture all the same relationships described by categories, commutative diagrams, etc. But the key usefulness is the intensional definition behind the formalism. When you talk about a certain set of relationships, you are talking about all things that obey them. You aren’t just grabbing an object and proving things about it, you are grabbing definitions and proving things about all things that obey them. And you can move out and look at fragments of definitions and these are obeyed by larger sets. And so theorems grow in usefulness maximally, and you are led to find greater abstractions that capture the essence…
Of course, you can do that without category theory, and people have. But it was really useful to have a framework that built it in. Much of math before category theory was done extensionally, looking at specific objects like sets and building specific structures and concretizations. So there was a lot of duplication, and although much of it was acknowledged metaphorically, some was missed. The discipline of intensional definition is why so many find category theory useful. Again, though, some model pluralism is always healthy.
@NathanHarvey @internic @johncarlosbaez :
It's interesting that you mention ultrafinitism, since this is an ultra-constructive philosophy (historically it was called ultraintuitionism), and there is no constructive nonstandard analysis (so far). Famously (or notoriously), Errett Bishop gave a scathing review to Jerome Keisler's nonstandard calculus textbook, and I was almost expecting you to say that ultrafinitism led you *away* from NSA.
But as someone who has also developed some ultrafinitist tendencies, I think that your philosophy is correct. Mathematics needs a variety of approaches, and not just a variety of mathematical models for physics, but even a variety of models for mathematics itself. Besides classical mathematics (with its infinitary, impredicative, and nonconstructive reasoning) and even fragments of it like Bishop's constructive mathematics, there are whole underexplored worlds of alternative nonclassical mathematics, places where every number is computable, every function is continuous, or every set is measurable, etc. (Fortunately, category theory can help with this.)
Besides, even if ultrafinitist mathematics is the only ontologically true core; it's hard, and since standard approaches are going to violate that anyway, why limit yourself?
@TobyBartels @internic @johncarlosbaez Yeah, I came at my ultrafinitist tendencies through a general radical constructivism based in model theory and an epistemic realizability, so I definitely have a side that questions the “meaningfulness” of certain models like that of nonstandard analysis. But my attraction to model theory (and Tarski’s work in general, who I stan quite widely) originated from strong reactions against the rigid views that sought a single model for reality, and models with untestable assertions may still have use in their observable fragments. Real number distances can be useful even if we only ever really measure rational distances.
But the one thing that also tempers my ultrafinitist tendencies is the possibility of things like hypercomputation. Sure, we may not know how to access infinities in any meaningful way today, and I think it’s good to acknowledge that. But maybe in the future someone finds ways to send information content between finite and infinite worldlines with fancy black hole configurations or other exotic spacetime goodness. Who knows? What is meaningful might change as our sources of meaning grow.
@internic @johncarlosbaez I advocated smooth infinitesimal analysis as a better alternative to nonstandard analysis. https://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/
@andrejbauer Interesting. I'm unfamiliar with that, so I'll have to check it out.
@internic
I have a suspicion that nonstandard analysis might be pedagogically superior in teaching high school calculus. But it is no more than a suspicion -- I have zero evidence to back that up.
@johncarlosbaez
@jswphysics @johncarlosbaez I had run into this book
https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach
that was seemingly developed with that intent. From the article, it sounds like there were some limited attempts in that direction that had positive results but it was never widely attempted/assessed.
I assume there must be a fair amount of research into the pedagogy of teaching calculus, and I don't even know what the most common areas of student difficulty are.
@johncarlosbaez
@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.