The brute-forcing is a good, old-fashioned generate-and-test approach like in Simon and Newell's Logic Theorist, which was presented in the Dartmouth convention in 1956, where AI was named by John McCarthy. Logic Theorist caused a big stir by (re) proving several of the theorems in Principia Mathematica by Russel and Whitehead.
There was much excitement, then, as now, for this kind of approach and there were several systems that followed along the same lines, e.g. Automated Mathematician by Doug Lenat.
Eventually it became clear that this approach is limited by what it can generate: you may have a sound and complete verifier, but if the generator, i.e. the first step in the generate-and-test pipeline, is incomplete, then the entire thing will run out of steam sooner or later.
The difference with LLMs is that they are... well, large. They are the most powerful generators ever created. That means their limits are not in sight and it will probably take us a very long time to find them.
Which is all to say that, yes of course, automatic verification is indispensable. But without an LLM generating an unprecedentedly large number of plausible theorems, there would be no AI mathematics, or in any case AI mathematics wouldn't have gone as far as it has.
I cannot say, math research isn’t my domain of expertise, I’m just trying to follow along :)
But I find it interesting that Lean, a validator/compiler made by humans, is what enables those discoveries. But somehow all the praise goes to the models
Yes! Thank you, this has been irritating me from day one with agentic AI, I don't think it could be nearly as good as it is without all the well-designed tools humans have spent decades developing from PLs, to VCS, to the Unix philosophy, to CLIs / REPLs.
I think that AI is very skilled and adept at using them, but without them it would just be flailing around in its own psychosis. The tools ground the AI in reality and allow them to make progress without going in hallucinated directions. I very much doubt AI could have solved this problem without Lean, and for AI to invent something like Lean it would have to use other tools made by humans.
This is also perfect evidence of why Python isn't the end-all-be-all of programming languages just because the AI was trained on vast amounts of Python, and proof that the right language for the job is more viable than ever with the aid of LLMs.
Frankly, the forecasting that programming languages are a dead field has baffled me because it seems like with LLMs, unique programming language semantics are more important than they've ever been.
I mean we don't instantly fall into ASI, hopefully. The problem with humans is every problem we solve the goal posts get kicked further down the road until they are reaching relativistic speeds. It starts around "well, the AI hasn't solved a novel problem" then moves to "well, they didn't write the validator" and suddenly humans are at the point of saying "Well AI hasn't rewrote the constants of the universe, what good are they".
Of course another way to look at this is, the people that wrote the validator got praise for that years ago. Now and up and coming actor is solving problems that took us 100s of years to create in insanely short time periods so of course it's going to get a lot of attention as it well should.
I don't think so. People have been trying things like this with evolutionary algorithms for a very long time already. LLMs can interleave symbolic manipulation with empirical experiments and simulations and charts and thinking/reasoning text, and an LLM will much more efficiently search the space of candidate ideas than any handcrafted mutation algorithm. Any task with a cheaply verifiable goal that requires fanning out across a massive search space is ideal for contemporary LLM technology to make progress with.
The brute-forcing is a good, old-fashioned generate-and-test approach like in Simon and Newell's Logic Theorist, which was presented in the Dartmouth convention in 1956, where AI was named by John McCarthy. Logic Theorist caused a big stir by (re) proving several of the theorems in Principia Mathematica by Russel and Whitehead.
There was much excitement, then, as now, for this kind of approach and there were several systems that followed along the same lines, e.g. Automated Mathematician by Doug Lenat.
Eventually it became clear that this approach is limited by what it can generate: you may have a sound and complete verifier, but if the generator, i.e. the first step in the generate-and-test pipeline, is incomplete, then the entire thing will run out of steam sooner or later.
The difference with LLMs is that they are... well, large. They are the most powerful generators ever created. That means their limits are not in sight and it will probably take us a very long time to find them.
Which is all to say that, yes of course, automatic verification is indispensable. But without an LLM generating an unprecedentedly large number of plausible theorems, there would be no AI mathematics, or in any case AI mathematics wouldn't have gone as far as it has.
True, but could humans cross pollinating lean x prolog x A* ( or any search algorithm) could have solved such math problems with super computer ?
I cannot say, math research isn’t my domain of expertise, I’m just trying to follow along :)
But I find it interesting that Lean, a validator/compiler made by humans, is what enables those discoveries. But somehow all the praise goes to the models
Yes! Thank you, this has been irritating me from day one with agentic AI, I don't think it could be nearly as good as it is without all the well-designed tools humans have spent decades developing from PLs, to VCS, to the Unix philosophy, to CLIs / REPLs.
I think that AI is very skilled and adept at using them, but without them it would just be flailing around in its own psychosis. The tools ground the AI in reality and allow them to make progress without going in hallucinated directions. I very much doubt AI could have solved this problem without Lean, and for AI to invent something like Lean it would have to use other tools made by humans.
This is also perfect evidence of why Python isn't the end-all-be-all of programming languages just because the AI was trained on vast amounts of Python, and proof that the right language for the job is more viable than ever with the aid of LLMs.
Frankly, the forecasting that programming languages are a dead field has baffled me because it seems like with LLMs, unique programming language semantics are more important than they've ever been.
I mean we don't instantly fall into ASI, hopefully. The problem with humans is every problem we solve the goal posts get kicked further down the road until they are reaching relativistic speeds. It starts around "well, the AI hasn't solved a novel problem" then moves to "well, they didn't write the validator" and suddenly humans are at the point of saying "Well AI hasn't rewrote the constants of the universe, what good are they".
Of course another way to look at this is, the people that wrote the validator got praise for that years ago. Now and up and coming actor is solving problems that took us 100s of years to create in insanely short time periods so of course it's going to get a lot of attention as it well should.
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I don't think so. People have been trying things like this with evolutionary algorithms for a very long time already. LLMs can interleave symbolic manipulation with empirical experiments and simulations and charts and thinking/reasoning text, and an LLM will much more efficiently search the space of candidate ideas than any handcrafted mutation algorithm. Any task with a cheaply verifiable goal that requires fanning out across a massive search space is ideal for contemporary LLM technology to make progress with.
How long until we find out that some AI has quietly buried an exploit in Lean to cheat at proofs?
Simpler to exploit a soundness bug than introduce a back door I would assume