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But the work the Mochizuki case generated can also be done by AI. AI could generate a landmark proof and then people could use it to solve or simplify intermedi

But the work the Mochizuki case generated can also be done by AI. AI could generate a landmark proof and then people could use it to solve or simplify intermediate problems and you could use a different AI prompt to try to disprove it if you were really skeptical. From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it. I don't like nuance here. I think progress is really measured by what humans are able to do and understand, not machines. It is significant if we find problems we struggle to solve. That tells us something. What does it take for humans to solve these problems is related. The best analogy I can give is if you wanted to climb Mt. Everest you might ask someone for guidance. Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak? This is like the AI versus human gap to me. The helicopter is like using AI to generate a proof. The person who actually climbed Mt. Everest has firsthand knowledge of the experience. Same thing for a difficult proof. The struggle people have is actually valuable here. Likewise, we know people are actually capable of climbing Mt. Everest but if they had only ever rode a helicopter to the top, the knowledge of climbing it would not exist, and surely that is meaningful knowledge given the risks. So if we rely on AI for proofs I think we lose a sense of what is difficult and why. We lose a sense of what human achievement is. Surely climbing Mt. Everest means more than taking a helicopter up? For students, why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now? This would have the affect of destroying knowledge. (please do not nitpick the analogy because it's the best but perhaps a clumsy way to describe my thoughts)

visarga 6d ago
> Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak? Depends on if I want to go by helicopter myself.
NateEag 6d ago
Tangent: > From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it. Keep in mind those ~88 hours were spread across ~10,000 simultaneous agent instances. So, roughly 880,000 hours of compute. Assuming a fifty-year career, and forty-hour workweeks, a human mathematician's career is about 100,000 hours of "compute". I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already. The perverse incentives of academia mean this has never occurred. The perverse incentives of industry mean OpenAI intentionally scooped researchers who were getting close (granted, with AI help). I'm not trying to dismiss the achievement - if the proof turns out to be solid, it's quite impressive (though much less so if the training data included the recent human breakthrough, which seems pretty plausible). I'm just pointing out that "88 hours" is a very misleading way of framing this.
- curt15 6d ago
  > I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already. > The perverse incentives of academia mean this has never occurred. This. Mathematicians in their most energetic years are trying to get tenure or land a tenure-track job. They are disincentivized to go all-in on ultra high risk, high-reward problems. The potential downside is just too forbidding. It's much safer to develop a research program in a mainstream field that affords many opportunities for partial progress that can translate to a robust publication record.
- onetimeusename 6d ago
  ok I realize this is a tangent but you're saying my post is very misleading and then also saying that a human mathematician's career is about 100,000 hours of compute and that Navier-Stokes could've had a solution by now if not for perverse incentives. You may be right but I don't think this is a great argument because in a year I would bet that those numbers change since computing power tends to increase or get cheaper over time. So I am taking the stance AI can outdo people if not now, perhaps soon.
  - NateEag 5d ago
    I wasn't trying to say that genAI won't beat humans. It arguably already has, much as that may fill me with horror and revulsion. I'm just trying to point out that economically, we have not yet reached the point where AI mathematics research is a no-brainer hands-down win, no consideration required. It might already be a win, and certainly the ability to compress those 900,000 hours of effort into an actual week of linear time is mind-boggling and potentially a huge game-changer for all kinds of open research questions. It's not obvious that human math research is dead yet. Maybe soon, but not yet.
- contubernio 5d ago
  "I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already" There were more than six doing that and it's essentially why it was ripe for AI to finish it off. But the finishing off was quicker than anyone expected
CamperBob2 6d ago
I think progress is really measured by what humans are able to do and understand, not machines. Building a machine that solves Millennium problems is pretty cool too. You wouldn't know it from reading these stories, though.
tmhn2 6d ago
I agree entirely with what you're saying, right up until your final question: > why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now? I think you answered this yourself earlier: > I think progress is really measured by what humans are able to do and understand People want to make this progress. Therefore people will "grind Everest" as a mathematical community, and that is maybe not so hugely different from a lot of previous mathematical work. There's still ample room for creativity: simplifying, generalizing, asking new questions humans are interested in, ...
- _superposition_ 6d ago
  That grind is emotional. And it's something AI will never have. The desire to solve a problem.
  - unknownunknown7 6d ago
    Citation needed. Who are you to say large enough clusters of neurons can't develo emotions?

visarga 6d ago

> Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak? Depends on if I want to go by helicopter myself.

NateEag 6d ago

Tangent: > From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it. Keep in mind those ~88 hours were spread across ~10,000 simultaneous agent instances. So, roughly 880,000 hours of compute. Assuming a fifty-year career, and forty-hour workweeks, a human mathematician's career is about 100,000 hours of "compute". I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already. The perverse incentives of academia mean this has never occurred. The perverse incentives of industry mean OpenAI intentionally scooped researchers who were getting close (granted, with AI help). I'm not trying to dismiss the achievement - if the proof turns out to be solid, it's quite impressive (though much less so if the training data included the recent human breakthrough, which seems pretty plausible). I'm just pointing out that "88 hours" is a very misleading way of framing this.

curt15 6d ago
> I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already. > The perverse incentives of academia mean this has never occurred. This. Mathematicians in their most energetic years are trying to get tenure or land a tenure-track job. They are disincentivized to go all-in on ultra high risk, high-reward problems. The potential downside is just too forbidding. It's much safer to develop a research program in a mainstream field that affords many opportunities for partial progress that can translate to a robust publication record.
onetimeusename 6d ago
ok I realize this is a tangent but you're saying my post is very misleading and then also saying that a human mathematician's career is about 100,000 hours of compute and that Navier-Stokes could've had a solution by now if not for perverse incentives. You may be right but I don't think this is a great argument because in a year I would bet that those numbers change since computing power tends to increase or get cheaper over time. So I am taking the stance AI can outdo people if not now, perhaps soon.
- NateEag 5d ago
  I wasn't trying to say that genAI won't beat humans. It arguably already has, much as that may fill me with horror and revulsion. I'm just trying to point out that economically, we have not yet reached the point where AI mathematics research is a no-brainer hands-down win, no consideration required. It might already be a win, and certainly the ability to compress those 900,000 hours of effort into an actual week of linear time is mind-boggling and potentially a huge game-changer for all kinds of open research questions. It's not obvious that human math research is dead yet. Maybe soon, but not yet.
contubernio 5d ago
"I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already" There were more than six doing that and it's essentially why it was ripe for AI to finish it off. But the finishing off was quicker than anyone expected

CamperBob2 6d ago

I think progress is really measured by what humans are able to do and understand, not machines. Building a machine that solves Millennium problems is pretty cool too. You wouldn't know it from reading these stories, though.

tmhn2 6d ago

I agree entirely with what you're saying, right up until your final question: > why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now? I think you answered this yourself earlier: > I think progress is really measured by what humans are able to do and understand People want to make this progress. Therefore people will "grind Everest" as a mathematical community, and that is maybe not so hugely different from a lot of previous mathematical work. There's still ample room for creativity: simplifying, generalizing, asking new questions humans are interested in, ...

_superposition_ 6d ago
That grind is emotional. And it's something AI will never have. The desire to solve a problem.
- unknownunknown7 6d ago
  Citation needed. Who are you to say large enough clusters of neurons can't develo emotions?