The foundation labs pursuing math problems was purely a commercial flex. LLM's are well suited to the sequential nature of math proofs and they had the tools to turn agents loose in a massive brute force approach. They found the solutions to some specific math problems, because they trained the systems on math problems. All it shows is that they can be good a math problems if you focus on it. The bigger question is not unsolved problems can it solve, what are the problems that are worth solving at all? Don't be misled by the headlines--I doubt if what the did with the NS problem is generalizable beyond a very narrow field that they put a lot of effort into.
I think the danger is that at least until their IPOs, it's in the interest of OpenAI as well as Anthropic to throw tons of resources at all nine millennium problems and other comparable problems - already, there are rumors that a couple more have been solved. It won't matter whether the techniques are narrow or broad, because it will send the signal that any problem is a matter of compute. The question you raise: "what's worth solving in the first place," is far more important, but I am guessing the answer to that question will change as a result of this onslaught, and perhaps be replaced by "what's worth conjecturing in the first place."
One thing I’m thinking - is this just a transition to ‘big math’?
We already have big science in fundamental physics (cern), most of astronomy, biology (human genome project, alphafold, etc) and so on.
In all these fields there is still room for the lone genius. But the big shiny objects attract a lot of people and funding and they realize the only way to tackle the problems is as a team.
Math has largely managed to slip through, until now.
One way to view navier stokes is OpenAI stole the idea and scooped the two geniuses working on it.
Another way to view it is hundreds of engineers and business people and construction teams worked for years and spent hundreds of billions of dollars to solve the problem.
These big shiny theorems seem destined to be the territory of big math going forward. And big math will likely suck up many mathematicians into its vortex.
But there may still be a lot of room for mathematicians to work in a new, but perhaps still somewhat traditional, way, in neglected, but possibly very interesting corners.
That’s one thing I’m thinking.
But the other thing I’m thinking is that there is clearly something lost. And I feel bad for mathematicians. And these systems are only getting better. And they are coming for me next.
Fascinating question. Almost makes me want to start a company called Riemann to solve - you guessed it.
Physics is interesting - while experimental evidence, say, for the Higgs Boson, is gathered by large teams, theory is still done by the lone genius: Higgs.
I think what will change in big Math/big Physics, and I agree with you that it’s coming, is that theory will also become industrial, with the creative aspects falling under the ‘industrial design’ end of theoretical activity and the big shiny theorems being proven by teams of mathematical engineers.
But the skill sets needed to create and run these industrial organizations are very different from those needed to win Fields medals today. Like anything else, the medium is the message, which makes me think a better analogy is with small story vs big story, aka novelists versus movie directors: both are creative pursuits but movies need a large organization while novels don’t.
the real math may be taught in old schools with retired profs where languages and one year pg classes might be taught. big colleges will focus on jobs& the current requirements of markets.
The foundation labs pursuing math problems was purely a commercial flex. LLM's are well suited to the sequential nature of math proofs and they had the tools to turn agents loose in a massive brute force approach. They found the solutions to some specific math problems, because they trained the systems on math problems. All it shows is that they can be good a math problems if you focus on it. The bigger question is not unsolved problems can it solve, what are the problems that are worth solving at all? Don't be misled by the headlines--I doubt if what the did with the NS problem is generalizable beyond a very narrow field that they put a lot of effort into.
I think the danger is that at least until their IPOs, it's in the interest of OpenAI as well as Anthropic to throw tons of resources at all nine millennium problems and other comparable problems - already, there are rumors that a couple more have been solved. It won't matter whether the techniques are narrow or broad, because it will send the signal that any problem is a matter of compute. The question you raise: "what's worth solving in the first place," is far more important, but I am guessing the answer to that question will change as a result of this onslaught, and perhaps be replaced by "what's worth conjecturing in the first place."
One thing I’m thinking - is this just a transition to ‘big math’?
We already have big science in fundamental physics (cern), most of astronomy, biology (human genome project, alphafold, etc) and so on.
In all these fields there is still room for the lone genius. But the big shiny objects attract a lot of people and funding and they realize the only way to tackle the problems is as a team.
Math has largely managed to slip through, until now.
One way to view navier stokes is OpenAI stole the idea and scooped the two geniuses working on it.
Another way to view it is hundreds of engineers and business people and construction teams worked for years and spent hundreds of billions of dollars to solve the problem.
These big shiny theorems seem destined to be the territory of big math going forward. And big math will likely suck up many mathematicians into its vortex.
But there may still be a lot of room for mathematicians to work in a new, but perhaps still somewhat traditional, way, in neglected, but possibly very interesting corners.
That’s one thing I’m thinking.
But the other thing I’m thinking is that there is clearly something lost. And I feel bad for mathematicians. And these systems are only getting better. And they are coming for me next.
Fascinating question. Almost makes me want to start a company called Riemann to solve - you guessed it.
Physics is interesting - while experimental evidence, say, for the Higgs Boson, is gathered by large teams, theory is still done by the lone genius: Higgs.
I think what will change in big Math/big Physics, and I agree with you that it’s coming, is that theory will also become industrial, with the creative aspects falling under the ‘industrial design’ end of theoretical activity and the big shiny theorems being proven by teams of mathematical engineers.
But the skill sets needed to create and run these industrial organizations are very different from those needed to win Fields medals today. Like anything else, the medium is the message, which makes me think a better analogy is with small story vs big story, aka novelists versus movie directors: both are creative pursuits but movies need a large organization while novels don’t.
Small story vs big story. Love it!
the real math may be taught in old schools with retired profs where languages and one year pg classes might be taught. big colleges will focus on jobs& the current requirements of markets.