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We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domain

by nilkn 28d ago

We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains. Why does this tend to be the case, even when some of the smartest people in the world have historically predicted incorrectly that certain branches of math would forever be useless (e.g., number theory)? I can only offer my own theory on that, but my guess is that mathematics is simply a predictive framework based on pattern compression. A more powerful pattern compression framework accelerates every single field that relies on pattern recognition or prediction of the unknown based on patterns.

skybrian 28d ago
It sounds like the idea is to turn on a math generator and keep running it until it generates something interesting. And it might be fun to try it. But if it’s too much output to read and we don’t understand the output either, how does anyone recognize when it’s done something that’s practically interesting? The output might make a cool screen saver as-is, but we probably need a way to evaluate it somehow.
- nilkn 28d ago
  Yes, of course we'd need a way to evaluate it. I don't right now have a fully conceived answer to what that will look like. But I'm confident at least in saying we would not evaluate it, like Tao is suggesting, by only accepting something once a human can easily teach it unassisted to another human. That sets the bar dramatically too low and would quickly become an extraordinary impediment to progress. You'd have to think of yourself less like a researcher and more like the director of the world's largest research institute. It's highly unlikely you'll understand or even care about every single paper every one of your researchers is producing, but you'll care about the overall research direction and whether the intermediate results are accumulating into outcomes you consider meaningful. How to do this where the institute is based on superhuman AI mathematicians is an unsolved problem, but I see no reason to imagine it's unsolvable. Let me make up an example of where I could imagine this going. Something we essentially cannot do right now is predict coarse-grained phenomena from systems that involve millions or trillions or more of interacting components. Over hundreds/thousands of years of experiment and theory we've derived laws that essentially do this in a few special cases, but we have no systematic theoretical way of doing it in general, and frankly I think it's beyond human ability. Whatever deep patterns or structures exist for doing this in a general way I think are simply out of reach for us.
- energy123 28d ago
  This is relevant, but only after AI has solved all the open problems including Millenium problems. Until then, as AI keeps solving harder open problems, people will pay attention and be interested.
chongli 28d ago
We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains. That's a misconception. Only a tiny percentage of mathematics has seen any applications whatsoever. There are vast libraries full of mathematics no one (in this discussion, anyway) has ever heard of that no one reads anymore and has never been applied to anything. This idea of trying to "prove all the math" with AI makes as much sense to me as using chess engines to try to "solve chess."
- nilkn 28d ago
  > There are vast libraries full of mathematics no one (in this discussion, anyway) has ever heard of that no one reads anymore and has never been applied to anything. And that's an issue why? It would seem to me that producing that also produced the mathematics that revolutionized the world repeatedly for centuries. I would go further and claim that, if you want the mathematics that revolutionizes the world, there's no way to get it without advancing mathematics as a field broadly. Those are not two separate activities, and thinking that they are is indeed a misconception. > This idea of trying to "prove all the math" with AI makes as much sense to me as using chess engines to try to "solve chess." You're right: "prove all the math" does not make sense on any level, and nobody serious would phrase any of this in that way. I certainly didn't.
  - chongli 28d ago
    And that's an issue why? The issue is SNR: signal to noise ratio. Generating exponentially more mathematics, particularly if the process is indiscriminate or optimized for something other than usefulness or mathematical relevance (such as optimizing for machine-provability), does not imply that we get exponentially more applications. We may end up halting the progress of applications altogether as the entire capacity of the world's mathematical apparatus is consumed by the interpretation and investigation of machine-generated proofs. You can already visit arXiv and find vast numbers of not-yet-published mathematical papers. Most should never be published. None of this junk is benefitting humanity in the slightest.

by nilkn 28d ago

skybrian 28d ago

It sounds like the idea is to turn on a math generator and keep running it until it generates something interesting. And it might be fun to try it. But if it’s too much output to read and we don’t understand the output either, how does anyone recognize when it’s done something that’s practically interesting? The output might make a cool screen saver as-is, but we probably need a way to evaluate it somehow.

nilkn 28d ago
Yes, of course we'd need a way to evaluate it. I don't right now have a fully conceived answer to what that will look like. But I'm confident at least in saying we would not evaluate it, like Tao is suggesting, by only accepting something once a human can easily teach it unassisted to another human. That sets the bar dramatically too low and would quickly become an extraordinary impediment to progress. You'd have to think of yourself less like a researcher and more like the director of the world's largest research institute. It's highly unlikely you'll understand or even care about every single paper every one of your researchers is producing, but you'll care about the overall research direction and whether the intermediate results are accumulating into outcomes you consider meaningful. How to do this where the institute is based on superhuman AI mathematicians is an unsolved problem, but I see no reason to imagine it's unsolvable. Let me make up an example of where I could imagine this going. Something we essentially cannot do right now is predict coarse-grained phenomena from systems that involve millions or trillions or more of interacting components. Over hundreds/thousands of years of experiment and theory we've derived laws that essentially do this in a few special cases, but we have no systematic theoretical way of doing it in general, and frankly I think it's beyond human ability. Whatever deep patterns or structures exist for doing this in a general way I think are simply out of reach for us.
energy123 28d ago
This is relevant, but only after AI has solved all the open problems including Millenium problems. Until then, as AI keeps solving harder open problems, people will pay attention and be interested.

chongli 28d ago

We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains. That's a misconception. Only a tiny percentage of mathematics has seen any applications whatsoever. There are vast libraries full of mathematics no one (in this discussion, anyway) has ever heard of that no one reads anymore and has never been applied to anything. This idea of trying to "prove all the math" with AI makes as much sense to me as using chess engines to try to "solve chess."

nilkn 28d ago
> There are vast libraries full of mathematics no one (in this discussion, anyway) has ever heard of that no one reads anymore and has never been applied to anything. And that's an issue why? It would seem to me that producing that also produced the mathematics that revolutionized the world repeatedly for centuries. I would go further and claim that, if you want the mathematics that revolutionizes the world, there's no way to get it without advancing mathematics as a field broadly. Those are not two separate activities, and thinking that they are is indeed a misconception. > This idea of trying to "prove all the math" with AI makes as much sense to me as using chess engines to try to "solve chess." You're right: "prove all the math" does not make sense on any level, and nobody serious would phrase any of this in that way. I certainly didn't.
- chongli 28d ago
  And that's an issue why? The issue is SNR: signal to noise ratio. Generating exponentially more mathematics, particularly if the process is indiscriminate or optimized for something other than usefulness or mathematical relevance (such as optimizing for machine-provability), does not imply that we get exponentially more applications. We may end up halting the progress of applications altogether as the entire capacity of the world's mathematical apparatus is consumed by the interpretation and investigation of machine-generated proofs. You can already visit arXiv and find vast numbers of not-yet-published mathematical papers. Most should never be published. None of this junk is benefitting humanity in the slightest.