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Comment by fruitl00p | Hacker News Reader

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.

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Yes, and here towards the bottom the author gets to the reason he didn’t sign with the other medalists:

> I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly.

> Under the circumstances, I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them.

Basically let’s make it a short-term problem and deal with it. Groups of people can deal with short term emergencies. Don’t turn it into a structural issue.

And in my view what’s the alternative in the letter exactly? The tools exist. Is there going to be drama every time somebody decides to use them?

The tools exist, so let’s try to figure out as humans the best way to use them to promote humanity, and let’s advocate against ways of using them which are a detriment to humanity, which is exactly the charge the letter makes. There are cultural norms around the use of every technology.

Let's figure out how to use the iron fist of the state to keep people from doing things the math establishment doesn't like?

Yeah, that's really great for humanity. Just super.

An obvious misrepresentation of the letter and my comment. For what purpose, I don’t know.

I'd like to hear how your comment isn't irrelevant noise, then. How does one implement these grand ideas except by state action? Otherwise, people will do what people will do, including proving theorems by machine.

Your comments are irrelevant noise because you’re just making up a bunch of nonsense out of nowhere. Are you a vegetable?

Well I think in response to 'what are you going to do', what people are probably going to do is stop sharing on-going work, stop bothering to curate problems that are just going to inflate someone's IPO valuation, basically accelerating the tragedy of the commons that the AI companies are driving.

Seems legit. I speculate their are emergent mechanisms for the development of memory that are not dependent on what ever mechanism is behind the "improve $model for 'everyone'"*

Is someone knowlegeable on research that engages the question on the development of something that could be considered as some emergent mechanism of memory, that is independent of instances and their contextwindow an specifically also independent of the use of conversations for trainingsdata?

*(This is regarding the value of selfhosted models - maybe small selforganisations that share ressources to do though to do so? Generally the value of machine learning seems to be to big to reject)

I don't understand this logic.

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…

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?

Question:

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.

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?

Maybe fix energy and heat on earth first. Fusion would be nifty. How's the AI progress there?

Do we know that AI can't invent a conceptual framework?

If we give it a real problem to solve, it may just have to invent a new form of math to solve.

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.

> 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: 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.

A lot indicates the answer to your latter question might just be 'no'

>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.

If this is the viewpoint of mathematics, why does it make a difference that human or AI solve it? And forget about "understanding", because "understanding" in mathematical sense means in a very narrow way: top experts of math in certain field would understand and accept it (my estimation is that ~100 people in the world would understand Fermat's Last Theorem proof). Mathematicians could spend the whole year digesting FLT and "convince" the public that this is correct, and for most people (including math PHDs and professors), FLT is correct because some smart people say it is.

As an aside, it's amusing that this conversation is a re-statement of a main point in TFA:

> However, math problems are really there to solve a real world problem.

> That theory might be inspired by the real world, but the problem itself is purely theoretical.

From TFA:

> In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.

The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"

I only just skimmed the referenced essay, but a priori I don’t think “problem-solving” as Gowers uses it has anything to do with real-world practicality. The problems under consideration are entirely theoretical, regardless of which “culture” a mathematician belongs to.

You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture.

The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect.

I may have misinterpreted either or both of them though!

If solving a problem is "purely" theoretical it's of no use outside the math community's enjoyment. Otherwise it's not purely theoretical.

> none of the millennium problems have anything to do with a "real" problem

P vs NP has a lot to do with real problems.

Humans only invest in solving problems that matter one way or another.

I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.

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.

So my question is this:

Would this slow technological progress? Or make it go faster?

That math problems are found and solved as we run into real physical problems.

"The FT’s Gillian Tett reported that a senior financier’s New York firm now seeks out humanities students, because “AI-native” Stem graduates are entering the job market with “alarmingly shallow ideas”."*

I don't advocate for the dichotomy of stem and humanities. A good counterecample from the 20th century being Ernst Mach (Mach-speeds are named after him) and his work in phenomenology ("bodies do not produce sensations, sensations produce bodies")

Your incatation of contextless 'technological progress' still kinda calls for a quote like the above

*https://www.theguardian.com/books/ng-interactive/2026/aug/08...

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.

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.

Agree, just adding the observation that

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

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 ;)

>Humans only invest in solving problems that matter one way or another.

Fermat's Last Theorem was one of the most famous open problems in math for centuries, and it has no direct applicability to any tangible problems here in the physical world.

Oh some few. No right triangle with rational sides has area equal to a perfect square depends on N=4 for instance. And it can be used to form other theorems that are terribly actionable. Every elliptical curve over Q is modular, which has consequences throughout number theory.

But yes, none of those are very tangible, until applied to problem solutions that are tangible.

What real-world consequences are implied by a solution to Navier-Stokes?

That depends on the solution, right?

Not really. I can't imagine a realistic solution that would affect, say, how we actually model a real fluid. Hypothetically we could find out that a whole class of real fluids are not modelled by Navier-Stokes at all, but I don't think that's remotely likely, nobody familiar with the problem expects it.

> However, math problems are really there to solve a real world problem.

this is ABSOLUTELY not how actual mathematicians see their field. A problem in mathematics is just that: its interest to a pure mathematician is not related to any applications in other disciplines.

I don't really understand what makes a mathematical problem "interesting" because of my bias as an engineer. Nothing is interesting to me unless I can use it to solve a "real world problem". But, I'm willing to concede that "solving real world problems" is really not how mathematics advances.

Things like complex numbers or quaternions were often thought about way outside of the context of their modern applications in physics and engineering. When Hamilton thought about quaternions I really doubt he cared that it would make some programmer's life easier 200 years later:

> Every morning in the early part of October 1843, on my coming down to breakfast,

> your brother William Edwin and yourself used to ask me:

> "Well, Papa, can you multiply triples?"

> Whereto I was always obliged to reply, with a sad shake of the head,

> "No, I can only add and subtract them." [1]

If we go back further, Pythagoreans weren't trying to solve "real problems" either, they were like a weird religious cult.

My point is, the practice of "real" mathematics is really something that odd people feel driven to do, not unlike painting or playing an instrument. All the applied stuff is just a byproduct (not unlike ad billboards or elevator music).

[1] https://en.wikipedia.org/wiki/History_of_quaternions

> this is ABSOLUTELY not how actual mathematicians see their field

While this might be true, I think this is the big pivot that will need to happen. Math, as a human endeavor, will all be applied. We will use LLMs to do the math and discover the math, and humans will apply it to whatever problem they are working on (likely using another LLM to integrate it).

hopefully this attitude won't be whats going to do the decisionmaking

> this is ABSOLUTELY not how actual mathematicians see their field

Yeah and they should start seeing it differently

Thanks for building the elevator though, guess I'll hit the gymn(asion) now.

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