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.
Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks.
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
It might. It might not though. Would a 1300s superintelligence advocate for heliocentricism in the face of all institutional players advocating for geocentrism, or would it create the most refined epicycle model imaginable? I’d guess the latter.
Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
Why is it always a false dichotomy between two ridiculous extremes? Maybe inventing some dangerous technology benefits from human understanding for a bunch of obvious reasons, like human beings being responsible it goes well?
>If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
This depends on unanswered questions on what math actually is and it's causal connectivity.
Imagine we have problem A that needs to connect to math solution Z.
The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.
As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.
The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
> Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc?
For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: <a href="https://ee.stanford.edu/~hellman/publications/24.pdf" rel="nofollow">https://ee.stanford.edu/~hellman/publications/24.pdf
You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.
I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing..
don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels..
Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?
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.
GPerson · · focus · HN ↗
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
aurareturn · · focus · HN ↗
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
GPerson · · focus · HN ↗
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
aurareturn · · focus · HN ↗
If we give it a real problem to solve, it may just have to invent a new form of math to solve.
GPerson · · focus · HN ↗
mrngld · · focus · HN ↗
NonHyloMorph · · focus · HN ↗
GPerson · · focus · HN ↗
pixl97 · · focus · HN ↗
This depends on unanswered questions on what math actually is and it's causal connectivity.
Imagine we have problem A that needs to connect to math solution Z.
The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.
As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.
The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
j2kun · · focus · HN ↗
For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: <a href="https://ee.stanford.edu/~hellman/publications/24.pdf" rel="nofollow">https://ee.stanford.edu/~hellman/publications/24.pdf
You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.
I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
NonHyloMorph · · focus · HN ↗
NonHyloMorph · · focus · HN ↗