In a moment, mathematicians merge probability and number theory | Hacker News Reader

In a moment, mathematicians merge probability and number theory

quantamagazine.org

57 pointsrbanffy16 comments

There has been some confusion in the discussion here about whether the work https://arxiv.org/abs/2210.06279 is actually about category theory. To clarify the confusion, here is a snippet from the abstract.

> The moment problem in probability theory asks for criteria for when there exists a unique measure with a given tuple of moments. We study a variant of this problem for random objects in a category, where a moment is given by the average number of epimorphisms to a fixed object.

So the work is about category theory. The confusion comes in because the Quanta article focuses on what is expected to be the most important application of the work, which is in the study of ideal class groups from number theory. That is, the authors will try to (but have not yet) use their theory on the category of groups in order to make progress on the Cohen-Lenstra heuristics.

I'm not qualified to judge the significance of their work here since all of the math goes over my head. Taking the authors' words,

> This is a significant advance, as previous work has always relied on a priori knowing a distribution with certain moments. Being able to construct a distribution from its moments opens new avenues such as in our subsequent paper mentioned above

I guess we will find out if/when their subsequent paper on the Cohen-Lenstra heuristics comes out.

Coming from category theory, and having followed the work studying probability theory categorically (e.g. Fritz, Perrone, Lucyshyn-Wright’s stuff, Leinster’s notion of the magnitude of a category), is this actually interesting? The actual category theory seems pretty rudimentary, and I’m not sure if they’ve invented anything new on that side (they don’t reference any of the authors I’ve mentioned, so it doesn’t seem like there was much scholarship on that end).

It feels like Quanta just takes press releases from a few big departments and asks a writer to get a few quotes to flesh out an article.

As far as I can tell, the article says nothing about category theory and doesn't claim to. It's a popularization of recent probabilistic investigations in number theory, popularization of math is what Quanta does.

When it says "Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories" I'm pretty sure it means category in the non-technical sense, category as in classifying into categories. The term doesn't appear again in the article.

Oh, I’m talking about the paper they’re hyping up. I’m just not terribly impressed, and I think if was written by people at UCSD instead of Harvard nobody would have even noticed it.

The title of the post promises to much.

"In a Moment, Mathematicians Merge Probability and Number Theory" - I really expected some deep connections between Probability and Number Theory.

I do not like the quanta magazine title as well. "Probability and Number Theory Collide — in a Moment"

Guess it should be funny, I would call it "bemüht" (german).

I don't know if this work is particularly ground breaking, but this looks to be more concerned with applications towards number theory (specifically cohen-lenstra heuristics) rather than adding to the category theory literature.

My issue is that I can’t see a cursory literature review into the established category literature on these methods - how do they even know they have something new/novel if they haven't checked?

This is a bit like the recent news about the “ML discovered a new fast matrix multiplication” press push out of Google, where researchers in the field just sort of rolled their eyes.

> In work published in the Annals of Mathematics in 2016, Ellenberg, along with Akshay Venkatesh and Craig Westerland, used moments to study the statistics of class groups in a slightly different setting than Cohen and Lenstra had considered. This idea was reused several times. But each time researchers used the moments, they would lean on the quirks of their particular problem to prove that the infinite set of equations had a solution. That meant their techniques were not transferable. The next mathematician who needed to use moments would have to solve the moment problem all over again.

Are you suggesting that this need to solve the moment problem all over again could have been avoided by searching Category Theory developments?

As I said in another comment, this research isn't about category theory. [Edit: Well, not about the foundations anyway.]

The use of moments as a method for understanding probability distributions isn't particularly new. What is new, is the introduction and adaptation of this framework for trying to understand the Cohen-Lenstra heuristics -- a smaller-scope problem in number theory that has seen a lot of attention recently.

The people quoted are among the world's leading experts in this topic, and I would say that Melanie Wood is the foremost expert. This is definitely novel.

What is your definition of 'these methods' and what is the established literature on them that you'd want to see cited? I'd be surprised if the specific construction and results that they need have been published.

The two authors are no slouches... I certainly wouldn't roll my eyes at them.

So, category theory is the new gospel, I guess, that from which all mathematics is derived?

“I’ve discovered a new thing about Y in X’s! Oh, no I haven't actually looked at the X literature about Y.” This is hardly specific to category theory.

> this looks to be more concerned with applications towards number theory (specifically cohen-lenstra heuristics) rather than adding to the category theory literature.

(I'm a specialist in this area of number theory)

This is correct. The word "category" is being used in its colloquial English sense; this is not work on category theory.

[Edit: I'm mistaken in the above point; see comment below]

It's weird. It definitely reads like the Quanta author either didn't know what a mathematical category is, or didn't think it merited explanation, despite explaining simpler things in excruciating detail in the article.

The paper is absolutely a construction on categories.

Okay, looking at the paper more carefully (I'd just looked at the article before, and was arguing from my knowledge of the field more generally), I take it back -- there is definitely some formal category theory here.

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