With that interpretation, the issue becomes slightly different: is it more important that the collective understanding of the mathematical community should be as advanced as possible or that there should be answers to as many problems as possible? Or are those two aims valuable in different ways, so that there is no point in declaring one of them more important? Or are they so inextricably linked that it makes no sense to argue that one is more important than the other? And when we say “important”, for whom are we saying it is important: for mathematicians, or for society as a whole?

This is easy to me. Truth should be the North Star. If there is a fundamental truth that can be found via mathematics, then the shortest route to that truth should be preferred. While LLMs are definitely capable of solving problems in search of truth, I agree with Tao that instant "true/false" results threaten to short-circuit the traditional avenues we have used to escape local minima in the search for truth. Their products may be the junk food that provides immediate satiation in exchange for long-term health. Perhaps it's wrong, though.

The whole point of the letter was that if we adopt this gradient search strategy the humanity might get stuck in a local maximum forever.

Thing is, you can say that about any technology (not being here necessarily pro AI in math, but I think we need better arguments).

I mean sure? It's clear PLT and other applied CS fields are now as good as dead. You're never seeing new React or Python emerging ever again.

And unfortunately mathematics is much more fundamental to human endeavor than this.

I don't think that's true. Domain experts with the desire to create and experiment aren't going away any time soon.

Sure we are free to tinker, like the guys doing model railways. With about the same impact.

Then we develop ways of breaking out of local maxima.

Wherever we can recognise a problem we can solve it.

This pattern of argument keeps repeating in every place.

Pro tech people: technology removes bottlenecks. Sometimes we use those bottlenecks as a side effect to build muscle and so on. But removing bottlenecks gives us much higher degrees of freedom. It is up to us to coordinate and make use of the technology.

Anti tech people: bottlenecks are fundamentally useful. They should remain and technology shouldn't remove those so easily. Humans cannot coordinate as well when the bottlenecks are removed, so lets not remove them so quickly.

I can only suggest reading the letter with open mind and literally, without uncharitable bias towards the authors or conspiracy mindset.

Its not a conspiracy. Its 25 fields medalists who think exactly like how I put it. Ideally, they can just take whatever the technology gives as soon as possible and use it. They don't want it because they don't think the community can rearrange and coordinate such that they can make use of the new found degrees of freedom.

That's literally all there is to it - they don't believe in the rearrangement.

I can only envy the insight you have into thought processes of 25 fields medalists. Wish I could have that.

you must have been so smug writing this. But I wanted to engage in a discussion.

From your other replies it's clear you still haven't read the letter. There is little to discuss so far.

FWIW it didn't sound accusatory to me, it made me think the anti tech people might have a point.

It’s not even anti-tech though. Terence Tao signed it. The commenter above reduces the conversation to a false dichotomy.

Anti immediate-tech-progess-as-primary-objective?

Counterpoint: I'm not anti tech at all. In fact, I sell an AI harness for legal.

I think you've set up a false dichotomy. I'd propose to you the middle ground that a lot of us are concerned that VC-backed AI slop is "solving" problems in indigestible ways that hollow out the core. This applies in OSS as well as mathematics.

Please help me understand this and I'm asking this in good faith.

Why can't OpenAI publish whatever it wants. And the math community can use it or not use it. Fundamentally OpenAI's solutions are high signal - they are incentivised to not deliberately mislead people. Let the individuals in math community choose to read it or understand it? If OpenAI wants to publish something, let them do it in the current channels using peer review using whatever time is required.

What's wrong with this? The math community thinks this will destroy previously unwritten ways of prestige allocation and remove incentives that used to exist. I say that the community can rearrange and allocate prestige and time in different ways to maximally use the technology.

community can rearrange

You're describing an improvised surgery on a living organism. Developing a complex system involving humans that is productive and doesn't collapse is extremely hard, so if it ain't broke don't fix it.

The Math community can "not use" a proof? How would that work? It's a little like coloured functions (async etc) - you have 'human proved' vs 'machine proved'?

Are you suggesting that a human can't have the agency to decide whether to use it or not?

No, obviously not ...

I'm pointing out that the way Mathematics works as a discipline is that a new proof builds on existing proofs. If we have a bunch of machine-generated proofs, then a human Mathematician has to decide whether to reference any relevant machine proof that has been put out there.

If, as you suggest, human Mathematicians 'choose to ignore' a machine proof, then another human decides not to, what then? Mathematics bifurcates into 'pure human' proofs and 'mixed machine-human' or maybe 'pure machine'?

They can use the new proofs as foundations for future proofs.

But many folks just won't be motivated to attack or help digest solved problems because there's less extrinsic value in doing so. Tao and others argue that it's this human effort that finds human-relatable abstractions which spurn further investigation. Take the humans out of the loop, and they'll stay there, is the argument as I understand it.

Can't we get stuck in local maxima because of the lack of LLMs?

Knowledge wants to be free.

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.

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?

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.

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…

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.

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.

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.

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

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

This is really a microcosm of one particular problem that AI presents to the world: what do people do when their labour is not required any longer?

Here the worry is the social structures of mathematics are eroded such that fewer humans become able to do the work and less well, similarly to how juniors are being recruited less in software engineering, breaking the ladder and leading to fewer seniors in the years to come.

The answer to this really depends strongly on what AI can actually accomplish, but I’ll assume the maximal case and say that AI can do everything economically necessary, and further even those things just desired, such that human labour isn’t required for anything that anyone wants in a practical sense.

Here, we don’t need a human understanding of mathematics to give people a perfect standard of living. We also don’t need humans involved with anything else.

Everything therefore becomes a hobby or a game. People do things because they enjoy them for their own sake, or because they are endeavours used as vehicles to socialise and enjoy others’ company, or because a shared social belief exists and is cultivated such that doing such and such a thing confers social status.

And I think that’s more or less it. I predict we may see some fairly strange sorts of things, such as games where the team structure looks like the descendant of a company org and they compete in an artificial economy. Likewise we may see gamified versions of universities and academia. All of these would be “tamed” such that the rougher parts of the experiences were sanded off.

Sort of like how we evolved in an ancestral environment, and we have certain drives and expectations driven by that environment even though they no longer matter for survival. Our social structures may be derived similarly from those of today, even after they have ceased to serve a real purpose, but changed and repurposed to give meaning and community.

I agree with your argument of expected societal shift, but it's important to keep in mind that this is largely a developed country issue. It'll be a while before child-staffed cobalt mines or whatever is superseded. A significant amount of poverty could be alleviated through improved resource distribution and policy, without really needing improved productivity/increased resource production.

what do people do when their labour is not required any longer

I don't think this is the main issue (if at all) discussed here or in the field medalist letter. If anything, the majority of pure mathematics graduates are absorbed from the industry and a lot of exodus happens along the different scales of academia to there (though industry is also dealing with this issue but that's not what's concerned here). The issue discussed is what mathematics itself will be, which is much deeper. AI finding proofs to open problems does not solve at all the question of how to produce new problems, and there is no indication imo that there is way to go with that with AI.

Pure mathematics is not like applied sciences, as it is only tangentially influenced by external applications. Deciding which problems to tackle is a social process and a matter of taste/aesthetics, and is built largely through the exact friction that is more and more removed with AI. This is what makes it unclear how one can find problems without this friction, and none of these posts/letters have an answer really. Each one seems to describe just different standpoints than concrete, practical ideas.

[Sorry that none of the following is concrete, but perhaps elucidating the paradox contained within might open our minds to .. the shape of the paths to action ?]

Your productive friction (eutripsis? ~ negentropy? Viscosity!!!???) seems like a wonderful concept that the original letter should have flagged to rally the community ( Gowers might not have missed this point if they had a new name for it!)

Tao had a relevant talk about the paradox of efficiency..

https://youtu.be/svl_1upFpQo

It's not clear to me that AI necessarily removes this eutripsis. The threat though, might become real if users don't see the threat :)

Also reminiscent of Keat's

https://en.wikipedia.org/wiki/Negative_capability#Reception

https://www.poetryfoundation.org/education/glossary/negative...

Still abstract, but nearer to quantitative (mathematical anthrop(ic)ology even?): ordinary, bad friction is, eg, "size-consistent"

   Coasean Ceiling: organizational size limit where the internal friction of managing a firm consumes all of its energy, leaving nothing left for actual production

So.. for eutripsis, Coasean Floor? Lol

That's a good point. And yeah I got the term from Tao, as I had not described it this way before but I think it elucidates well the issue.

The problem imo is that, from a purely psychological/phenomenological perspective, there is not always a perceivable difference between "eutripsis" and "dystripsis" (just made it up but "dys" is the opposite of "eu") as experienced. There is some reward coming from learning through friction (depending on personal interests, environment etc), but mostly it is effort and humans usually try to reduce or avoid effort.

Moreover, even if one tries to be fully mindful and choose where to employ friction and where not, there could be systemic factors to optimise away any kind of friction. Imo we already see that in software engineering, judging from a lot of different anecdotes, where increasing the pace of generating code sacrifising human understanding is already taking place. It is not like these forces are not already in place widely in academia too even before AI (eg optimising for paper output quantity), so AI reinforcing this direction sounds a reasonably probable scenario, unless some other action is taken.

Did you even read the open letter? Mathematicians are annoyed because the proof is only meaningful if people understand it and came to it. I mean I’m basically just summarising it now but it’s annoying you didn’t read it or can’t comprehend its nuance.

I think the author did not mention that the way humans construct the knowledge ladder requires low complexity because each step gives power to the next, but the AI with the exponential exploration can give biggers steps but then it stagnates because the next step can be beyond the exponential exploration complexity. In chess a good strategy can be the best tool, in the game of go our experience suggest the same, but it could happen that mathematical thinking requires a type of policy that could be beyond the current ideas. I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.

One thing I've found in my own use of AI, which Gowers touches on in his point at the end but which I think deserves more attention: AI makes some previously non-trivial tasks trivial, but that doesn't make the work itself trivial. You naturally expand the scope of what you attempt and take on harder problems than you could before. The floor rises, and so does the ceiling. The open question is whether that still holds once models can also do the harder problems, or whether choosing and framing those problems remains the human part.

Watching the discussion unfold, here is what I feel:

Before arguing whether mathematics must strictly be done by humans, there are different motivations at play. Some people love the sense of solidarity within the community that forms during the process. Those excluded from that community might resent it, while others just purely want to solve problems.

Many things are being discussed, but looking at the overarching narrative, it seems that AI's true function isn't necessarily opening new horizons of specific knowledge, but rather excelling at 'serializing' topics that have been heavily fragmented until now.

In that sense, the concern is that because AI is solving the very problems needed to cultivate mathematicians internally, the stepping stones required for human growth are disappearing.

However, on the other hand, as the world and industries become increasingly complex and hyper-specialized, you could also argue that AI is the exact tool needed to unify this fragmentation across academia and industry. It is a highly complex dilemma.

From the perspective of researchers and the mathematical community, those 'problems for growth' must remain. But conversely, AI has the distinct ability to serialize siloed disciplines. Usually, when you go to graduate school, you often hear professors say that even within the exact same major, they cannot understand each other if their sub-specialties differ.

[I have not read Gower's article. It's only that your response is prompting me to share what i've been thinking recently]

the stepping stones required for human growth are disappearing.

Actually only in an institutional sense, imho. Maybe I'm exaggerating, but reddit.com/r/math* or even mathstackexchange will be so back with users analyzing and distilling proofs with AI. Maybe in 5 years (when lean attains 10% popularity of rust, and/or gpt7 level models cost ~USD5 on average, per month, inflation adjusted or not) this types of submissions will make those sites as fun/educational as mathoverflow [has always been for me]. One dreams that by that time openAI and Anthropic will have taken down your ethno-cultural "compatriot" Masayoshi with them.. (I prefer his brother whom he did not regret giving a physical beating to. Only purely on principle) so that they can't acquire these sites

Such forums can then replace math grad school, if the profs/alpoges who drop by get into the habit of constructive criticism (as they do on mathSE already). It's already starting, I see personally interesting 1 AI-aided submissions every 1.5-3 weeks starting 2 months ago

It would be like rust discussions on HN for you I bet.

This seems unlikely when s/math/physics/, or s/Reddit/HN/ because HN hates AI-aided posts so ideologically (sorry mods, I don't mean you guys). MO is also not as welcoming to outsiders/lay, congruent to academia (only online) PhysicsSE/overflow had become dumb and arrogant last decade

HN, even pg seem anti-intellectual in effect tbh whenever they sneer at AI _writing_ (again sorry mods, you will figure this out soon I believe, do you or do you not want HN to end up as a reservation for meatbrains), though I mostly am on their coder side with regards to this fields medallist letter

The AI debate is, in fact, somewhat ideological. The problem is that in the process, whether advocating for it or pushing for progress, there is a failure to face its limitations directly. In other words, it is emotional. Because AI's mere existence is a threat to knowledge workers, much like machines were a threat to blue-collar workers. That is, if the value of intellectual labor drops, it damages the value of the labor force through which workers earn capital in a capitalist system.

I have tried solving a few math problems with AI (they were Erdős problems), but because I know absolutely nothing about that math, I couldn't just take the AI's word for it, so I am actually a bit skeptical. A problem arises where you arrive at the answer without actually understanding it. It's not that AI is bad. The problem is that AI destroys the equilibrium between the knowledge I have and the knowledge I lack. That boundary collapses, making it feel as if I can know everything.

Actually, academia is fundamentally about mental models. It's a kind of internal worldview, and that worldview is shared. When you actually listen to the thoughts of scholars and professors, there are subtly different aspects. That forms the person's worldview... and I get the feeling that sharing it is what constitutes intellectual activity.

However, as you mentioned, unlike academics like yourself, academia and knowledge communities seem disconnected to someone like me (meaning they lack accessibility). Even if I were to make a discovery, it would probably be hard for me to become recognized, and I do think AI could actually play a role in opening up those closed communities. But apart from that, I find it hard to say that this only has positive aspects.

I followed Karpathy's research from start to finish to build a small LLM like nanoGPT on my own, and people say that because of the positive transfer that comes from feeding diverse data through modern LLM multimodal encoders, there will be new discoveries. But in reality, the types of problems AI excels at are generally those that humans have found but overlooked. In my opinion, rather than being a knowledge machine, LLMs (or AI) make me feel that what we call "intellectual activity" is closer to a kind of serialization work. A method of stacking things up one by one in sequence, so to speak? After all, the actual operating principle of an LLM proceeds according to the probability of the token coming in the next sequence.

In other words, I think the serialization method of our knowledge activities is similar to how LLMs operate, but I also think a different kind of thinking might be necessary. I am not that smart, and I have never interacted with scholars... (As you know, I am a subcontract worker. Of course, I have been hired by startups run by professors in my country, but it's not like I modeled that intellectual design myself.)

On the contrary, I feel that the evolutionary approach will slow down after GPT 6 Astra. They can continue to increase the size, but the issue lies in the cost-effectiveness of token costs.

Anyway, I agree with most of what you said in your discussion, but rather than anti-intellectualism, I consider this a direct threat to survival.

The job of a mathematician just shifts from proving to inventing new theories and finding new conjectures with the help of AI, which should be even more fun.

proving to inventing new theories and finding new conjectures with the help of AI

This is exactly the activity Tao and others worry will be hollowed out by instant "true/false" responses to novel questions.