As Daniel Litt pointed out on Twitter, this was the first time a lot of those problems were hit with a lot of compute. Some of AlphaEvolve's inequalities were beaten rather easily by humans and Moore's law
Mathematical exploration and discovery at scale
51–60 of 135 posts
Re: Mathematical exploration and discovery at scale
#52Earlier 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]…
A random walk could not do the mathematics in this article-- which was essentially the entire starting point for the article.
Re: Mathematical exploration and discovery at scale
#53It's really tiring that LLM fans will claim every progress as breakthrough and go into fantasy mode on what they can do afterwards. This is a really good example of how to use the current capabilities of LLM to help research. The gist is that they turned math problems into problems for coding agents. This uses the current capabilities of LLM very well and should find more uses in other fields. I suspect the Alpha evo…
One could say the same about these kinds of comments. If you don't like the content, simply don't read it?
And to add something constructive: the timeframes for enjoying a hype cycle differ from person to person. If you are on top of things, it might be tiring, but there are still many people out there, who haven't made the connection between, in this case, LLMs and mathematics. Inspiring some people to work on this may be beneficial in the long run.
Re: Mathematical exploration and discovery at scale
#54Earlier quoted context omitted.
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 exis…
Existence problems are not optimisation problems and can't, AIUI, be tackled by AlphaEvolve. It needs an optimisation function that can be incrementally improved in order to work towards an optimal result, not a binary yes/no.
More importantly, a research mathematician is not trapped in a loop, mutating candidates for an evolutionary optimiser loop like the LLM is in AlphaEvolve. They have the agency to decide what questions to explore and can tackle a much broader range of tasks than well-defined optimisation problems, most of which (as the article says) can be approached using traditional optimisation techniques with similar results.
Re: Mathematical exploration and discovery at scale
#55It's really tiring that LLM fans will claim every progress as breakthrough and go into fantasy mode on what they can do afterwards. This is a really good example of how to use the current capabilities of LLM to help research. The gist is that they turned math problems into problems for coding agents. This uses the current capabilities of LLM very well and should find more uses in other fields. I suspect the Alpha evo…
One could say the same about these kinds of comments. If you don't like the content, simply don't read it? And to add something constructive: the timeframes for enjoying a hype cycle differ from person to person. If you are on top of things, it might be tiring, but there are still many people out there, who haven't made the connection between, in this case, LLMs and mathematics. Inspiring some people to work on this…
GP didn’t say they didn’t like it. They criticized it. These things are not the same.
Discussions critical of anything are important to true advancement of a field. Otherwise, we get a Theranos that hangs around longer and does even more damage.
Re: Mathematical exploration and discovery at scale
#56Funny it's actually equivalent to https://news.ycombinator.com/item?id=45834303 except that here content produced is cheaply verifiable and thus usable.
Re: Mathematical exploration and discovery at scale
#57Earlier quoted context omitted.
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 exis…
Existence problems are not optimisation problems and can't, AIUI, be tackled by AlphaEvolve. It needs an optimisation function that can be incrementally improved in order to work towards an optimal result, not a binary yes/no. More importantly, a research mathematician is not trapped in a loop, mutating candidates for an evolutionary optimiser loop like the LLM is in AlphaEvolve. They have the agency to decide what q…
> Existence problems are not optimisation problems
Several of the problems were existence problems, such as finding geometric constructions.
> It needs an optimisation function that can be incrementally improved in order to work towards an optimal result, not a binary yes/no.
This is not correct. The evaluation function is arbitrary. To quote the AlphaEvolve paper:
> or example, when wishing to find largest possible graphs satisfying a given property, ℎ invokes the evolved code to generate a graph, checks whether the property holds, and then simply returns the size of the graph as the score. In more complicated cases, the function ℎ might involve performing an evolved search algorithm, or training and evaluating a machine learning model
The evaluation function is a black box that outputs metrics. The feedback that you've constructed a graph of size K with some property does not tell you what you need to do to construct a graph of size K + M with the same property.
> a research mathematician is not trapped in a loop, mutating candidates for an evolutionary optimiser loop like the LLM is in AlphaEvolve.
Yes they are in a loop called the scientific method or the research loop. They try things out and check them. This is a basic condition of anything that does research.
> They have the agency to decide what questions to explore
This is unrelated to the question of whether LLMs can solve novel problems
> most of which (as the article says) can be approached using traditional optimisation techniques with similar results.
This is a mischaracterization. The article says that an expert human working with an optimizer might achieve similar results. In practice that's how research is done by humans as I mentioned above: it is human plus computer program. The novelty here is that the LLM replaces the human expert.
Re: Mathematical exploration and discovery at scale
#58Re: Mathematical exploration and discovery at scale
#59There 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.
Such an AI will invent plumber robot and welder robot as well.
Robots scale much worse than knowledge work.
But yes, I'm not bullish on trades either. Trades will suck as well when everyone tries to get into them because it's the only way to still earn a living.
Re: Mathematical exploration and discovery at scale
#60There 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.
The underlying assumption here is that you won't have to earn a living anymore. Unless you already own enough to keep living off of it, you'll still have to work. That work will just suck more and pay less.