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
@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.
"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.
@andrejbauer
In the comments of your post, you answered marks question by saying that the absolute value is definable as a function from {𝑥 ∈ ℝ | 𝑥 ≤ 0 ∨ 𝑥 ≥ 0} . But for some infinitesimal dx, since 𝑑𝑥 ≤ 0, it is still in the domain of the function, and you can still prove that dx = -dx = 0. So unless I am missing something, the domain of the absolute value function should be something like {𝑥 ∈ ℝ | 𝑥 < 0 ∨ 𝑥 = 0 ∨ 𝑥 > 0} instead.
@anshthewad I don't understand your comment. What is the problem? The entire argument has nothing to do with infinitesimals, as it works equally well for any ordered ring. Here are some details.
We would like to define the absolute value in terms of a functional relation A(x,y), i.e., we want y = |x| iff A(x, y). Our candidate for A(x, y) is
(x ≤ 0 ⇒ y = -x) ∧ (x ≥ 0 ⇒ y = x)
Here y ranges over ℝ, and x ranges over some subset D ⊆ ℝ such that we can prove ∀ x ∈ D . ∃! y ∈ ℝ . A(x, y). We want D to be as large as possible.
I claim that taking D := {x ∈ ℝ | x ≤ 0 ∨ x ≥ 0} does the job. Indeed, consider any x ∈ D. Then either x ≤ 0 or x ≥ 0. In the first case the unique y satisfying A(x, y) is -x. In the second case the unique y satisfying A(x, y) is x.
(The domain {x ∈ ℝ | x < 0 ∨ x = 0 ∨ x > 0} works as well, but is contained in D anyhow.)
We now appeal to the fact that the functional relation A ⊆ D × ℝ determines a function abs : D → ℝ satisfying A(x, abs(x)) for every x ∈ D.
No infinitesimals anywhere, and the construction is purely intuitionistic. (Yes, I know I secretely used the principle of unique choice – but synthetic differential geometry happens in a topos. In toposes unique choice is valid.)
@andrejbauer I'm sorry if I wasn't clear enough. Let me rephrase my argument. I am trying to say that your relation A is not functional on the domain D.
This is because, for some d such that 𝑑 ≤ 0 and 𝑑 ≥ 0, if there exists a y such that (d ≤ 0 ⇒ y = -d) ∧ (d ≥ 0 ⇒ y = d), then y = d and y = -d. That would mean d = -d and hence d must equal to zero.
For the relation A to be functional, d would have to be zero for all d such that 𝑑 ≤ 0 and 𝑑 ≥ 0, or in other words ≤ would have to be antisymmetric. However, this doesn't hold in R. This is because there are infinitesimals d in R such that 𝑑 ≤ 0 and 𝑑 ≥ 0, and it is not the case that all such infinitesimals are zero. That is why the domain D := {x ∈ ℝ | x ≤ 0 ∨ x ≥ 0} does not do the job.
Did that make sense?
@anshthewad So if I understand you correctly, you are saying that:
1. My proof uses the fact that ≤ is antisymmetric, and
2. On R the relation ≤ is not antisymmetric.
I can well believe that 2 is the case. After all, the infinitesimals are all contained in the {x ∈ R | 0 ≤ x ≤ 0}.
Can you pinpoint where in the proof that A is functional I used antisymmetry of ≤? Aha, showing that the two cases agree on the overlap! Very nice. Thanks for the correction.
@anshthewad This leads to a natural question. Can we have a reasonable "gluing" principle on R that allows us to define functions piece-wise on some subsets of R? For example, suppose we have f : (-∞, 0] → R and g : [0, ∞) → R. What conditions must they satisfy so that we can glue them together to h : (-∞,0] ∪ [0,∞) → R? On the face of it, it has to be: ∀ x . 0 ≤ x ≤ 0 ⇒ f x = g x. Can we do better?
@andrejbauer It would be necessary to have all their nth derivatives agree at 0, but that is not sufficient. For it to be sufficient, we could additionally assume the axiom that ∀ x . 0 ≤ x ≤ 0 ⇒ x is nilpotent (in other words, \( D_\infty = [0, 0] \) )
@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.