I thought one of the implicit points of the open letter was that unsolved problems are not something that falls out of the sky, they are a curated resource that people have spent time on and shared for the benefit of like-minded peers and humanity as a whole. And the AI companies treat them like they treat absolutely everything else: natural resources, literature, art, code etc. as something to be chucked into the ravening maw and pooped out the back as profit. They don't care if mathematics advances, they don't care if they strip-mine the available problems and damage the field. In fact, as with programming I think they see that as in their long-term interest - soon there will be no intelligence or creativity but the one that Sam Altman bills you for.
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
> math problems are really there to solve a real world problem
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere.
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
problem specificity, concreteness, engineering relevance, or even "empiricity" seems (vaguely)
proportional to how much "good friction" can be generated.
There's also bad friction related to "meta-ness", "bad names", "aesthetics", etc, I presume. Like bikeshedding and its relatives. Is yakshaving?
AI can also introduce its own "bad friction", or it (ppl?) can bypass good friction without necessarily directly be removed by ai, eg bikeshedding, slowly pivoting towards directions and problems that ai is better in tackling, because that can produce these accelerated results vs fields and problems that ai may not be able tackle as well and thus the output there is poor, demotivating people from following these fields and missing important insights from them. Of course this could have the opposite effect, depending on which direction the whole hype can go, or not happen at all if ai will be able to tackle everything equally well.
I rewatched the Tao video, he indeed overlooked at one point that collaborating with AI could produce good friction
25m12s
It's easy to get into "the flow" with roughly equally-skilled humans, but the weird cadence of 2 humans+1 AI seems to be potentially eutriptic.
Probably you're off somewhere and never return. AI might also provide that "Coasean floor", that HN doesn't seem to ;)
Technological progress is bottlenecked by the fact that there are many mathematical problems for which currently there are no known practical methods of solution.
Because even with supercomputers the equations that describe many physical systems cannot be solved, research and development is still based on a lot of empirical methods, i.e. things must be physically built and measured, because mathematical computations cannot predict their properties with sufficient accuracy.
So if some miraculous algorithms would be discovered for the approximate solution of the systems of equations that are insoluble for now, that could accelerate technological progress a lot in certain domains, especially for the discovery of new materials or chemical substances with desirable properties.
fruitl00p · · focus · HN ↗
aurareturn · · focus · HN ↗
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
tempfile · · focus · HN ↗
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
aurareturn · · focus · HN ↗
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
freehorse · · focus · HN ↗
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
aurareturn · · focus · HN ↗
Would this slow technological progress? Or make it go faster?
That math problems are found and solved as we run into real physical problems.
freehorse · · focus · HN ↗
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
oliculipolicula · · focus · HN ↗
problem specificity, concreteness, engineering relevance, or even "empiricity" seems (vaguely)
proportional to how much "good friction" can be generated.
There's also bad friction related to "meta-ness", "bad names", "aesthetics", etc, I presume. Like bikeshedding and its relatives. Is yakshaving?
Eutripsis? Vs just tripsis
freehorse · · focus · HN ↗
oliculipolicula · · focus · HN ↗
25m12s
It's easy to get into "the flow" with roughly equally-skilled humans, but the weird cadence of 2 humans+1 AI seems to be potentially eutriptic.
Probably you're off somewhere and never return. AI might also provide that "Coasean floor", that HN doesn't seem to ;)
adrian_b · · focus · HN ↗
Because even with supercomputers the equations that describe many physical systems cannot be solved, research and development is still based on a lot of empirical methods, i.e. things must be physically built and measured, because mathematical computations cannot predict their properties with sufficient accuracy.
So if some miraculous algorithms would be discovered for the approximate solution of the systems of equations that are insoluble for now, that could accelerate technological progress a lot in certain domains, especially for the discovery of new materials or chemical substances with desirable properties.