Comment by wizzwizz4 | Hacker News Reader

Comment by wizzwizz4 | Hacker News Reader

Human mathematicians have been being out-counterexampled for at least two decades. The main difference, as I understand, is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so. But the sizes of counterexample we're seeing are around about what I'd expect pre-generative-AI counterexample search systems to be able to find.

It's not easy to find a counterexample to the Jacobian conjecture, by any means – by which I mean to say that naïve brute-force search will take too long – but the scope of existing searches listed on Wikipedia[0] suggest that many tricks are already known, and that people just hadn't looked, systematically, for a counterexample in three variables before. Wikipedia writes:

> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.[17][18]

where reference 17 is from 1983, and reference 18 is a preprint with no given date. Knowing very little about this problem, my impulse is to side with the unnamed faculty member cited in the article:

> [who] said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem,

For context, the auto-generated counterexample is in three variables, has degree 7, and was discovered in 2026.

> about what I'd expect pre-generative-AI counterexample search systems to be able to find.

The difference today (and the reason why everyone is excited about it) is that the same system that does advanced math can write poetry, play an above average game of chess, code frontend/backend stuff and do cybersec. These are not "expert systems", nor are they trained for each task individually. That's the catch.

> just indicated that humans had not spent enough time thinking about the problem

Heh, this is a weak excuse. We've seen variations on this theme every time something cool gets solved by the models.

> nor are they trained for each task individually.

They are explicitly trained for each task individually.

They are not. Pretraining just dumps every piece of content in the mix, and only has one objective - next token prediction. And you can get pretty good results even with base models, you just have to manage context differently. Later stages (mid, post training) involve RL that "surfaces" the right "traces" out of the pre-training. But they are not trained individually, as we used to do.

Have you looked at the what data companies (e.g. Scale, Mercor) hire for? Why do you think Meta records their employees every keystroke/mousestroke/eye-movement?

EDIT: just re-read your comment. I don't think you have a good understanding here, no offence.

None taken, but it would be odd, since I've been training GOFAI models since 2010s and have had LMs in production since before chatgpt came out (before RLHF), so I think I have a pretty good understanding :) But I'm always open to learning.

I think the misunderstanding comes from "individually". You are thinking about diverse datasets, but that's not what individually means in this context. In ML individually trained means that for each task you prepare an architecture, dataset and eval and train that model on that data. And each model has its own objective that you train for. In LMs the objective is singular, for every data point - next token prediction. And, importantly, you train on every datapoint, the more diverse the better, but not independently. The cool thing is that training on diverse datasets improves scores on other downstream tasks, while the training objective is the same.

So you can have a training run on common crawl + programming that improves scores on logic puzzles, or common crawl + novels that improves scores on planning tasks. But the important thing is that it's all trained together, not independently.

I think we're saying the same thing? I said they are explicitly trained on these tasks, not that they are some separate models during programming RL, or business tasks RL.

> The cool thing is that training on diverse datasets improves scores on other downstream tasks, while the training objective is the same.

Maybe for some language modelling tasks, i.e. it learns some internal representation that is transferable. However I would find it quite odd if a model becomes good at Bio while not explicitly going through Bio training.

> I said they are explicitly trained on these tasks, not that they are some separate models during programming RL, or business tasks RL.

You said:

> > nor are they trained for each task individually.

> They are explicitly trained for each task individually.

And that's the main misunderstanding.

Collins says:

individually in American English (ˌɪndəˈvɪdʒuəli, ˌɪndəˈvɪdʒəli) adverb 1. as an individual or individuals rather than as a group; one at a time; separately; singly

Which is precisely what LMs don't do (in contrast to previous "AI" models, which did do that). They are trained on every datapoint at the same time. So long as we agree on that, I think we are saying the same thing :)

Your username indeed checks out

The word "just" is the mark of a coward.

AI is only as good as a human mathemetician, which we don't have enough of? "just"?

https://en.wikipedia.org/wiki/Jacobian_conjecture

> The conjecture was first stated for two variables by Ludwig Kraus in 1884 [...] an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.

> The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. It was notorious for the large number of published and unpublished false proofs that turned out to contain subtle errors.

This is a counterexample for the three-variable case, which hasn't been studied as much. The long-standing two-variable case remains open.

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