>> If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.
My bad for not showing my work and inadvertently leading you down the garden path, but the "~100 agent-years" calculation goes like this:
36,666.7 agent-days / 365 days = 100.5 agent-years.
That's what you get for working 24 hours a day, 7 days a week, 365 days a year. Realistically speaking, that's not a work schedule any human can follow.
It's hard to make a realistic estimate because normally even a very dedicated mathematician will not be working exclusively on one problem all their waking time, or even all their working time. But, let's ignore this and assume a pretty standard work schedule of 8 working hours, five working days a week, and 52 working weeks a year.
Now, that's:
8 hours * 5 days = 40 working hours a week
40 hours * 52 weeks a year = 2080 hours a year
880,000 agent-hours / 4 humans = 220,000 hours per human
220,000 hours per human / 2080 hours a year = ~105.8 years
To clarify, that's how I estimate the number of years it would take a mathematician to do a quarter of the work of the 10k OpenAI agents if that mathematician worked only on solving Navier-Stokes and did nothing else in their entire career.
That's just not a realistic work schedule for any human. You can adjust the working hours if you want but I don't believe you'll get any realistic estimate. Don't forget that most academics' careers last around 30 years from PhD to Professor Emeritus. If you want a more realistic estimate of how much time it would take how many humans to do the work of the 10k OpenAI agents, you can start from that assumption and work your way up from that.
>> That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.
Sure. But the thing is agents can run 24/7, 365/365 in parallel and as you see above they can cover 2000 years of human work in much less time. I'm not going to estimate how much because the only bottleneck is the amount of compute and money that an AI company wishes to spend, and that depends on their motivation to solve a particular problem. However, with sufficient motivation 2k years of human research (keeping mind that's not 2k years of continuous work) can be covered in a few ... months? Probably.
> My bad for not showing my work and inadvertently leading you down the garden path
The problem isn't that you didn't show your work, it's that your work was wrong.
I entirely agree with your calculation that 10k agents for 88 hours is about 100 agent-years if we assume 24/7/365 operation. That's not what I was disagreeing with.
But then you said "How many human-years ...?" followed by estimating not the number of human-years that have gone into the problem but merely the number of years.
You can compare elapsed years for humans (26) and elapsed years for AI systems (about 0.01). You can compare agent-years (about 100) and human-years (26 times the average number of humans working on Navier-Stokes at any given time). Either of those is defensible.
But it makes absolutely no sense at all to compare agent-years for the AIs and elapsed years for the humans. Which is what you did.
If a typical human mathematician works 2000 hours a year (actual human mathematicians generally find that they can't do 8 hours a day of focused hard intellectual work, but I think we should count some of their "percolation time" too) then that's about 6 human-years per mathematician. So to get the same amount of mathematician-work as agent-work the average number of mathematicians you need to have been on the job is about 100/6, or about 16.
So when you wrote
> How many mathematicians have worked on the problem? 10k? Not likely.
the 10k figure was a total irrelevance. The number it would actually have to have been is about 16.
(My earlier "as many as four" ignored the fact that humans don't work 24/7/365, as you point out. But my point is that however you slice it the relevant number is more like four than it is like 10,000.)
My guess, for what it's worth is that that is roughly the order of magnitude of the number of human mathematicians working primarily on things that could be classified as "trying to make progress toward resolving the Navier-Stokes problem" during that time. I wouldn't be surprised if the actual figure were 3x bigger or 3x smaller. It probably depends on how broadly you interpret "trying to make progress toward resolving the Navier-Stokes problem", and one important difference is that all those human mathematicians leave behind them a trail of papers proving things that, whether or not they end up on the path to Navier-Stokes, may turn out to be useful later, whereas if OpenAI's agent swarm proved a lot of useful theorems along the way most of them never got published.
I don't, of course, disagree that it's possible for an AI company to put a lot of AI agents to work on a problem, but I'm not sure how that makes what they can do less impressive. The fact that you can do that has always been a major part of why AI could be such a big deal. "A country of geniuses in a datacentre" is the kind of thing people have said; we aren't quite there yet, but the "country" part is as important as the "geniuses" part.
I'm sorry but I'm not sure I understand your argument. I think you're saying I'm comparing apples to oranges. I'm not: I'm comparing apples to apples and oranges to oranges. These are two different questions:
>> But how much time have human brains spent working on the problem in either of those time periods? How many mathematicians have worked on the problem? 10k?
So neither 10k humans worked on Navier-Stokes, nor has any human spent a century of non-stop work on it.
But I could have made the point more clear maybe.
>> I don't, of course, disagree that it's possible for an AI company to put a lot of AI agents to work on a problem, but I'm not sure how that makes what they can do less impressive. The fact that you can do that has always been a major part of why AI could be such a big deal. "A country of geniuses in a datacentre" is the kind of thing people have said; we aren't quite there yet, but the "country" part is as important as the "geniuses" part.
Yes, I see your point, but those are not geniuses. Grigori Perelman proved the Poincaré conjecture alone, though as he has emphasised his work was based on advances made by others, particularly Richard S. Hamilton. That we can call a genius: a single man who solves one of the most interesting problems in all of mathematics building on the work of his predecessors. 10k agents that search blindly and find a result by luck (or by stealing it), I don't agree we can call "genius". That's what I call "brute force". Anyone who wants to call OpenAI's agents "a country of geniuses" has first to deal with the fact that they look a lot like monkeys on typewriters.
My bad for not showing my work and inadvertently leading you down the garden path, but the "~100 agent-years" calculation goes like this:
10,000 agents * 88 hours = 880,000 agent-hours
88,000 agent-hours / 24 hours = 36,666.7 agent-days
36,666.7 agent-days / 365 days = 100.5 agent-years.
That's what you get for working 24 hours a day, 7 days a week, 365 days a year. Realistically speaking, that's not a work schedule any human can follow.
It's hard to make a realistic estimate because normally even a very dedicated mathematician will not be working exclusively on one problem all their waking time, or even all their working time. But, let's ignore this and assume a pretty standard work schedule of 8 working hours, five working days a week, and 52 working weeks a year.
Now, that's:
8 hours * 5 days = 40 working hours a week
40 hours * 52 weeks a year = 2080 hours a year
880,000 agent-hours / 4 humans = 220,000 hours per human
220,000 hours per human / 2080 hours a year = ~105.8 years
To clarify, that's how I estimate the number of years it would take a mathematician to do a quarter of the work of the 10k OpenAI agents if that mathematician worked only on solving Navier-Stokes and did nothing else in their entire career.
That's just not a realistic work schedule for any human. You can adjust the working hours if you want but I don't believe you'll get any realistic estimate. Don't forget that most academics' careers last around 30 years from PhD to Professor Emeritus. If you want a more realistic estimate of how much time it would take how many humans to do the work of the 10k OpenAI agents, you can start from that assumption and work your way up from that.
>> That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.
Sure. But the thing is agents can run 24/7, 365/365 in parallel and as you see above they can cover 2000 years of human work in much less time. I'm not going to estimate how much because the only bottleneck is the amount of compute and money that an AI company wishes to spend, and that depends on their motivation to solve a particular problem. However, with sufficient motivation 2k years of human research (keeping mind that's not 2k years of continuous work) can be covered in a few ... months? Probably.