Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Mathematical exploration and discovery at scale

terrytao.wordpress.com

Post reply on HN
Print view View original

41–50 of 135 posts

Re: Mathematical exploration and discovery at scale

#41

Post by DroneBetter »

wizzwizz4 wrote: post #34
ineedasername wrote: post #26

Earlier quoted context omitted.

Well, there's the goal posts moved and a Scotsman denied. It's got an infrastructure in which it operates and "didn't show its work" so it takes an F in maths.

A random walk can do mathematics, with this kind of infrastructure. Isabelle/HOL has a tool called Sledgehammer, which is the hackiest hack that ever hacked[0], basically amounting to "run a load of provers in parallel, with as much munging as it takes". (Plumbing them together is a serious research contribution, which I'm not at all belittling.) I've yet to see ChatGPT achieve anything like what it's capable of. [0]…

yeah but random walks can't improve upon the state of the art on many-dimensional numerical optimisation problems of the nature discussed here, on account of they're easy enough to to implement to have been tried already and had their usefulness exhausted; this does present a meaningful improvement over them in its domain.

Re: Mathematical exploration and discovery at scale

#42

Post by wizzwizz4 »

dpflan wrote: post #28
swannodette wrote: post #25

Earlier quoted context omitted.

I don't see how anything about what's presented here that refutes such claims. This mostly confirms that LLM based approaches need some serious baby-sitting from experts and those experts can derive some value from them but generally with non-trivial levels of effort and non-LLM supported thinking.

Yes, applied research has yielded the modern expert system, which is really useful to experts who know what they are doing.

It's not the "modern expert system", unless you're throwing away the existing definition of "expert system" entirely, and re-using the term-of-art to mean "system that has something to do with experts".

Re: Mathematical exploration and discovery at scale

#43

Post by throwaway0123_5 »

andrepd wrote: post #12
muldvarp wrote: post #10

There seems to be zero reason for anyone to invest any time into learning anything besides trades anymore. AI will be better than almost all mathematicians in a few years.

I'm very sorry for anyone with such a worldview.

Are you saying this because you think that people should still try to learn things for personal interest in a world where AI makes learning things to make money pointless (I agree completely, though what I spend time learning would change), or do disagree with their assessment of where AI capabilities are heading?

Re: Mathematical exploration and discovery at scale

#44

Post by ants_everywhere »

piker wrote: post #3

That was dense but seemed nuanced. Anyone care to summarize for those of us who lack the mathematics nomenclature and context?

They put an LLM in a loop that mimics how people do real math, and it did research-level math.

Like humans, it wasn't equally capable across all mathematical domains.

The experiment was set up to mimic mathematicians who are excellent at proving inequalities, bounds, finding optimal solutions, etc. So more like Ramanujan and Erdős in their focus on a computationally-driven and problem-focused approach.

Re: Mathematical exploration and discovery at scale

#45

Post by ghm2180 »

nl wrote: post #18

Hopefully this will finally stop the continuing claims[1] that LLMs can only solve problems they have seen before! If you listen carefully to the people who build LLMs it is clear that post-training RL forces them to develop a world-model that goes well beyond a "fancy Markov chain" that some seem to believe. Next step is building similar capabilities on top of models like Genie 3[2] [1] eg https://news.ycombinator.c…

>.. that LLMs can only solve problems they have seen before!

This is a reductive argument. The set of problems they are solving are proposals that can be _verified_ quickly and bad solutions can be easily pruned. Software development by a human — and even more so teams — are not those kind of problems because the context cannot efficiently hold (1) Design bias of individuals (2) Slower evolution of "correct" solution and visibility over time. (3) Difficulty in "testing" proposals: You can't build 5 different types of infrastructure proposals by an LLM — which themselves are dozens of small sub proposals — _quickly_

Re: Mathematical exploration and discovery at scale

#46

Post by stOneskull »

muldvarp wrote: post #10

There seems to be zero reason for anyone to invest any time into learning anything besides trades anymore. AI will be better than almost all mathematicians in a few years.

  But don't you see, I came here to find a new job, a new life, a new meaning to my existence. Can't you help me?

  Well, do you have any idea of what you want to do?

  Yes, yes I have.

  What?

  (boldly) Lion taming.

Re: Mathematical exploration and discovery at scale

#48

Post by omnicognate »

ants_everywhere wrote: post #38
woadwarrior01 wrote: post #37

Earlier quoted context omitted.

Please read section 2 of the paper[1] cited in the blog post. LLMs are used as a mutation function in an evolutionary loop. LLMs are certainly an enabler, but IMO, evolutionary optimization is what deserves credit in this case. [1]: https://arxiv.org/abs/2511.02864

all mathematicians and scientists work with a feedback loop. that's what the scientific method is.

Not one that amounts to a literal, pre-supplied objective function that's run on a computer to evaluate their outputs.

Re: Mathematical exploration and discovery at scale

#49

Post by omnicognate »

qsort wrote: post #6
piker wrote: post #3

That was dense but seemed nuanced. Anyone care to summarize for those of us who lack the mathematics nomenclature and context?

I'm not claiming to be an expert, but more or less what the article says is this: - Context: Terence Tao is one of the best mathematician alive. - Context: AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). Hallucinations are not a problem because the LL…

Important clarification

> search is guided by an LLM

The LLM generates candidates. The selection of candidates for the next generation is done using a supplied objective function.

This matters because the system is constrained to finding solutions that optimise the supplied objective function, i.e. to solving a specific, well-defined optimisation problem. It's not a "go forth and do maths!" instruction to the LLM.

Re: Mathematical exploration and discovery at scale

#50

Post by ants_everywhere »

omnicognate wrote: post #48
ants_everywhere wrote: post #38

Earlier quoted context omitted.

all mathematicians and scientists work with a feedback loop. that's what the scientific method is.

Not one that amounts to a literal, pre-supplied objective function that's run on a computer to evaluate their outputs.

That's exactly how a great deal of research level math is done.

In fact all open conjectures can be cast this way: the objective function is just the function that checks whether a written proof is a valid proof of the statement.

Is there a solution to this PDE? Is there a solution to this algebraic equation? Is there an optimal solution (i.e. we add an optimality condition to the objective function). Does there exist a nontrivial zero that is not equal to 1/2, etc.

I can't tell you how many talks I've seen from mathematicians, including Fields Medal winners, that are heavily driven by computations done in Mathematica notebooks which are then cleaned up and formalized. That means that -- even for problems where we don't know the statement in advance -- the actual leg work is done via the evaluation of computable functions against a (explicit or implicit) objective function.

Post reply on HN