271–280 of 564 posts
Earlier quoted context omitted.
I think you might be thinking of the recent project to start Fermat's Last Theorem? The Riemann hypothesis has been easy to state (given what's in Mathlib) for years.
Yeah lol i don't think either is hard to formalize in lean
This is the real deal. AlphaGeometry solved a very limited set of problems with a lot of brute force search. This is a much broader method that I believe will have a great impact on the way we do mathematics. They are really implementing a self-feeding pipeling from natural language mathematics to formalized mathematics where they can train both formalization and proving. In principle this pipeline can also learn bas…
By contrast, this kind of open ended formalization is something where progress used to be extremely slow and incremental. I worked in an adjacent field 5 years ago and I cannot stress enough that these results are simply out of reach for traditional automated reasoning techniques.
Real automatic theorem proving is also useful for a lot more than pure mathematics. For example, it's simple to write out an axiomatic semantics for a small programming language in lean and pose a question of the form "show that there exists a program which satisfies this specification".
If this approach scales it'll be far more important than any other ML application that has come out in the last few years.
Machines have been better than humans at chess for decades. Yet no one cares. Everyone's busy watching Magnus Carlsen. We are human. This means we care about what other humans do. We only care about machines insofar as it serves us. This principle is broadly extensible to work and art. Humans will always have a place in these realms as long as humans are around.
Actually, I was looking up Elo ratings of the top computer chess players, and learned that it is not that trivial to compare these, due to differences in hardware requirements and whatnot.
Machines have been better than humans at chess for decades. Yet no one cares. Everyone's busy watching Magnus Carlsen. We are human. This means we care about what other humans do. We only care about machines insofar as it serves us. This principle is broadly extensible to work and art. Humans will always have a place in these realms as long as humans are around.
Sure but if an AI can prove e.g the Goldbach conjecture then that is a bfd.
Machines have been better than humans at chess for decades. Yet no one cares. Everyone's busy watching Magnus Carlsen. We are human. This means we care about what other humans do. We only care about machines insofar as it serves us. This principle is broadly extensible to work and art. Humans will always have a place in these realms as long as humans are around.
magnus carlsen basically quit because computers ruined chess. As did kasparov. Fischer was probably the last great player who was unassisted by tools.
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Formalization is in principle just a translation process and should be a much simpler problem than the actual IMO problem. Besides, they also trained a Gemini model which formalizes natural language problems, and this is how they generated training data for AlphaProof. I would therefore expect that they could have also formalized the IMO problems with that model and just did it manually because the point is not to de…
Yet the facts at hand are the opposite of what you say. Reliable formalizer was the more difficult problem than solving formalized IMO problems, because they have not produced one.
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> a lot of brute force search Don't dismiss search, it might be brute force but it goes beyond human level in Go and silver at IMO. Search is also what powers evolution which created us, also by a lot of brute forcing, and is at the core of scientific method (re)search.
Also AlphaProof had to search for 60 hours for one of the IMO problems it solved.
This is the real deal. AlphaGeometry solved a very limited set of problems with a lot of brute force search. This is a much broader method that I believe will have a great impact on the way we do mathematics. They are really implementing a self-feeding pipeling from natural language mathematics to formalized mathematics where they can train both formalization and proving. In principle this pipeline can also learn bas…