> we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.<p>This is the main issue, and while I fully agree with that value sentiment, the Fields medallists’ letter failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work under these circumstances.
Math professor here.<p>Different academic disciplines are very different, and most don't engage in the same kind of black-and-white problem solving that mathematicians do, where you either have solved a problem or you haven't.<p>But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, <i>something</i>.<p>We haven't yet figured out what that should be, but I presume that everyone would agree that this should continue. As one possible model, check out this blog post of Terry Tao's, where he gives his own perspective on the recently proved Jacobian conjecture.<p><a href="https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/" rel="nofollow">https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...</a><p>When computers can solve the underlying actual problem, this sort of work seems likely to rise in value, and be something which a greater number of mathematicians engage in.
> But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.<p>My vote is for interpretive dance. Give the laity something for their money.
I don't think that the main issue either discussed by Gowers or the others who have signed the letter is simply that mathematicians should be paid "for merely understanding things". The main issue is outsourcing, laziness, and learned helplessness.<p>Outsourcing understanding, teaching, and proving maths to LLMs is just as dumb as outsourcing food production, manufacturing, or entertainment to a foreign power. It's not that we haven't already done most of those things in the pursuit of temporary profit optimization, but each of them has obviously bad impacts on both the individuals in society who perform those tasks, and on risks to everyone when that outsourcing fails for any reason.<p>As described, using LLMs is like navigating with GPS. It's fine when you're using it to optimize a path due to traffic conditions, but it can be deadly if you're flying a plane and suddenly don't have it, or any training to deal with that situation. That is where we're headed. I don't really have a take on signing the letter in particular, because as mentioned it seems a bit pointless. At the same time it also seems a bit pointless to spend $10s of millions proving a conjecture a few months sooner than a group of mathematicians already were. I don't imagine they were going to make millions doing it. The definition of material waste.<p>Perhaps it serves a temporary marketing win for OpenAI, or a medium term improvement in LLM performance on some productive output, but longer term it certainly risks ceding whole swaths of human endeavor to a tool we may not always have. Just as falling demand for farmers, or skilled machinists, or writers, or artists is not an immediate crisis, in the long term there is no one left to transfer the knowledge to the next generation, if something goes wrong.
This is meant to be a funny, stylized comment. If you can't bother to read to the end, just ask a chatbot to explain it to you.<p>Most people do not use any math after school in their lives. All of their real problems are political. Kids spend significant time in school being told that math teaches them "problem solving" and then leave school and discover, since math solves no political problems, math doesn't solve problems at all. In other places around the world, where by many measures people are much more numerate than Americans, people are actually poorer and live under more authoritarian governments. In industry, math has been reduced to a shibboleth for a neurodivergent lack of ethics, which is why mathematics PhDs go into banking, and never politics - and we imagine politicians and lawyers to lack ethics, which is pure projection, nearly all of our best presidents and congressmen were lawyers. Even the academic environment that supports mathematicians and the STEM community generally - all of that concentrated neurodivergence and lack of people skills has led to less political power, which means less funding and less new students, which has been much more threatening to mathematics than automated proofs.<p>I love math, but my honest POV is, the value of theoretical math cannot get much lower. The crisis is insurmountable. The community made its deal with the devil (Jim Simons) long ago, it thought it was a STEM discipline like bio and it's really a philanthropic humanities discipline like opera. Math is having its opera moment. Someone would have to step up as the rich person who saves math, and unfortunately, all the best candidates are right now destroying it with chatbots.
Well, we can reduce every human activity to 0 value and take it from there: if there’s no value in understanding there’s no place for humans in the process, we can just go back to worshipping stones and let the AIs burn tokens deluding themselves chasing their hallucinations
Depends. Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics? If not then we'd be advised, imo, to keep funding which is inherently just a training pipeline.<p>The person who might have made a mathematics breakthrough that results in some miraculous medical or other discovery will likely have decided it's not economically prudent to pursue a career (pure math academia) that the tech sages, in their all encompassing wisdom (exclusively over the next 1 to 2 financial quarters), have deemed worthless, and that instead he or she should just respond to that pesky Big Four recruiter who keeps dangling a cushy six figure internship.
> mathematics breakthrough that results in some miraculous medical or other discovery<p>Do you believe it is possible or have any recent examples? I feel like most of the stuff that could be applied to something like this has probably already been developed 100 years ago. This hope for some miracle math that cures cancer sounds like the thing they tell the government to keep the funding going. Most of modern pure mathematics doesn't look like it has any chance of being that. It's getting more specialized every day with more and more papers on obscure topics being published that 2 people in the world read.<p>I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.
> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?<p>Not at all. My point is that the Fields medallists’ letter didn’t provide good arguments, not that there aren’t any. And therefore I’m agreeing with the Gowers quote above.
Real mathematicians will continue their research, whether they receive funding or not.
I'm genuinely unclear on what you mean here. Do you mean:
1. Most mathematicians are currently largely paid as part of their academic job (as a faculty member) and will continue to do that work (including the research part) even if extra-mural grant funding goes away. That seems entirely plausible to me.<p>Or do you mean?
2. Real mathematicians will continue to devote a significant part of their life/energy to mathematics research as a hobby, even if no one is paying them (in any way) to do it? That seems more of a reach and makes me wonder if many currently folks current employed as mathematicians aren't 'Real mathematicians' from your point of view.
As a mathematician, summer salary (from grants) is nice but I could do without it.<p>But external funding, e.g. from the NSF, also supports conferences -- and I suspect that few people would be willing to attend if they had to pay out of pocket. Without this opportunity to talk to one another, the field would be much worse off.
I’m inclined to agree. Sometimes when (perhaps other) people say this there’s an undertone of “and the field is full of not-real mathematicians” as if there’s a large contingent just hanging on for the money or prestige, and I personally haven’t seen that in the wild.
Yes of course for roughly 3 months, after that they'd starve
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It has been a <i>long</i> time since I saw my WordPress theme in the wild. I had no idea wordpress.com still had it as an option.<p>edit: Link changed; it was <a href="https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-the-fields-medallists-letter/" rel="nofollow">https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-t...</a>.
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?<p>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.<p>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.<p>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.<p>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.<p>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.<p>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.
>Everything therefore becomes a hobby or a game.<p>Already predicted. How much Whuffie you got?<p><a href="https://www.gutenberg.org/ebooks/8086" rel="nofollow">https://www.gutenberg.org/ebooks/8086</a>
> what do people do when their labour is not required any longer<p>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.<p>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 ?]<p>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!)<p>Tao had a relevant talk about the paradox of efficiency..<p><a href="https://youtu.be/svl_1upFpQo" rel="nofollow">https://youtu.be/svl_1upFpQo</a><p>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 :)<p>Also reminiscent of Keat's<p><a href="https://en.wikipedia.org/wiki/Negative_capability#Reception" rel="nofollow">https://en.wikipedia.org/wiki/Negative_capability#Reception</a><p><a href="https://www.poetryfoundation.org/education/glossary/negative-capability" rel="nofollow">https://www.poetryfoundation.org/education/glossary/negative...</a><p>Still abstract, but nearer to quantitative (mathematical anthrop(ic)ology even?): ordinary, bad friction is, eg, "size-consistent"<p><pre><code> Coasean Ceiling: organizational size limit where the internal friction of managing a firm consumes all of its energy, leaving nothing left for actual production
</code></pre>
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.<p>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.<p>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.
Ah, contrary to what I mused elsewhere, concreteness can also lead to bad friction<p>Eg, KPIs, metrics, but of productivity, of "veracity", not understanding<p>Anecdotes--> better friction than data, sometimes, though :)<p>How about Inverse Metrics. of simplicity? Parsimony? Shortness of code? (Efficiency/compressibility is a sort of "intensive" metric, so it might not be especially relevant, thermodynamically speaking)<p>Just taxidermy, stamp collecting, and vibe-anthropologizing here TT
I somewhat disagree. we still need humans to provide direction and context and expanded the concept of... well everything. But we will certainly need less of them. AI will afford the ability to more quickly disseminate state-of-the-art. As soon as a breakthrough is discovered, you no longer need to read 300 whitepapers to hopefully stumble upon it, AI will identify relevancy quicker.<p>AI didnt break through stavier-nokes until a human set the initial direction. That's not going to change.
>AI didnt break through stavier-nokes until a human set the initial direction. That's not going to change.<p>Why isn't this going to change?<p>Right now it's easy to see why don't set AI lose on every problem, someone human or AI has to delegate rare resources between competing interests. But if we look at general compute it was no different in the past. In 1980 you had to ask for permission to get CPU compute time. In 2026 you run it on your own computer, or maybe pay for it on Amazon. The constraints are much different.<p>If we keep pumping out chips and increasing efficieny someone will just make the LLM into an agentic loop (build the harness in) and set it lose on problems.
> that AI can do everything economically necessary<p>Who gets to define this? Those with power are famously bad at understanding long-term implications, how things work, and what matters - especially when it comes to funding things like research.
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.
> A significant amount of poverty could be alleviated through improved resource distribution and policy, without really needing improved productivity/increased resource production.
reply<p>The concern is that AI will further reduce the leverage we have to improve resource distribution, and that we'll return to something that more closely resembles feudalism (IIRC, wealth inequality is already close to where it was in feudal times, albeit we're all richer overall). And/or that (because people won't accept that) we'll end up in a war situation.<p>You'd think the people who own the technology would be smart enough to avoid that outcome, but so far they aren't showing any signs of it.
Maybe our AI overlords won't appreciate our tendencies to abuse and exploit each other and crack down on things like "child-staffed cobalt mines"...<p>Next stop the Culture!
> 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.<p>There is actually a big leap in logic you’re doing here that it seems you are naive to. The world looks the way it does today because human societies, businesses and individuals are fundamentally in competition with one another for scarce resources, and having more resources leads to more capacity to survive, so there is always an incentive to acquire more. If you can no longer compete, you can and will be exploited. You don’t have to look very hard, far or back in time to realize that this is what society looks like. If everyone is incapable of competing except those who can afford to run a datacenter, that will not end up in socialist utopia.
The Machine Stops by E. M. Forester is basically this world.
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.
>that AI can do everything economically necessary<p>This is practically impossible, because the "economically necessary" things to do are never fixed but instead are in a state of constant fluctuation.<p>Furthermore, a chain is only strong as it's weakest link. AI alone is not sufficient to do all the "economically necessary" things. We would need vast numbers of highly advanced robots to do the rest of the work going on in the world (which is >> than the work currently done by AI). So if we restrict the definition of "economically necessary" to mean the subset of all possible tasks that we currently need to do, then we are sorely lacking in the required infrastructure. We may never get there either.<p>In the meantime, the labor shock knowledge workers experience could lead to large unemployment. Even those that find work doing non-knowledge work, the decrease in pay and benefits may be substantial.<p>I think these realities are worth understanding to help ground these pie in the sky ideas about what the future will be like. Not to be mean, but your idea of the future is basically a fantasy.
"And maybe I could have tried to gain that respect in a different way, such as thinking very hard about an area of mathematics until I was able to demonstrate to others just how well I understood it. But I’m not sure how motivating that would have been for me. I very much hope that there is a pool of young people for whom it will be a powerful motivation, because I think the survival of a human mathematical tradition may well depend on it. [...]
Thus, the primary risk, as I see it, is that a lot of people who would have done a PhD in mathematics and gone on to become custodians of the mathematical tradition will no longer wish to do so. Those of us who have PhD students, including me, need to try as hard as we can to come up with imaginative ways for them to use their time productively (in consultation with the students themselves, obviously)."<p>When I read this I'm trmpted to read "imaginary" instead of imaginative - that is: in the sense lacanian psychoanalysis uses the term imaginary in contrast to symbolic - the latter of which would mean that it's having consequences within the symbolic social order. From their own, quite selfhonest evaluation the author assumes that without those they would most likely not have persued the mathematical profession. Which also relates to:<p>"A related risk is that the perception among policy-makers will be that mathematicians are no longer needed and that funding will become much harder to come by: we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems."<p>Note that there isn't any attempt to establish a possible horizon as to how that coule happen.<p>It seems to me that the end of the article has a strong tendency to a somewhat stoic attitude of one might phrase as: it is going to happen anyways - the systemic context in which these corporations act makes it inevitable ("However much we might regret that, there is no chance that the impact of such models on mathematics will persuade AI companies to stop their release, though perhaps concerns about safety will lead to some delay and give us a bit more time to work out how to adapt. Assuming that they are released, there will be a flood of new results, whether we like it or not, and it will no longer be the AI companies producing them, though perhaps the pattern will continue that the AI companies will have access to more powerful models and so will obtain more than their fair share of headline results.")
- which makes me kind of wonder if it is not some sort of cognitive dissonance within the authors view of the situation to not also think 'it's not going to happen' in regard to their stated requirements (students that don't need to be motivated by a desire to be symbolically valuated as explorers of the mathematical frontier, and policy makers that acknowledge the value of funding the social "production" of human mathematical experts /enable such a community through funding).<p>So basically they see the same problems as the authors of the letter (which they also state in their letter - their should be no room for a misrepresentation of that fact: "I felt that I could not sign the letter, despite agreeing with much of what it said. Instead, it seemed better to do what I did with the Leiden Declaration and set out my own position in a blog post. But it should be understood that by doing that I am not setting myself up as a member of some opposing camp: indeed one of my worries at the moment is that the mathematical community might become bitterly divided, something I would very much like to avoid.")<p>So what stays is mostly their disagreement on that conceptual understanding is more important than the solving of problems for all mathematicians. - which kind of makes the problemfield of ai in mathematics somehow more urgent, as far as I have thought it through. And I'm not really convinced that the displacement of the problem into some sort of pedagogy (and it should be appreciated that the author implicate themselves in the responsibility to establish it) instead of finding solutions in the realm of policy and regulation. Seems realistic though, that they see that as an effort unlikely to succeed in a meaningful way.<p>Can someone argue against that reading? Am I missing something? My conclusion as such is that not only are they not opposing the authors of the fields medalist letter, but also that their evaluation of the situation is much more dire without them giving in to resignation.
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.
I don't understand this logic.<p>Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.<p>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<p>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.<p>I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Question:<p>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?
>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?<p>This depends on unanswered questions on what math actually is and it's causal connectivity.<p>Imagine we have problem A that needs to connect to math solution Z.<p>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.<p>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.<p>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.
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.<p>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!"
Do we know that AI can't invent a conceptual framework?<p>If we give it a real problem to solve, it may just have to invent a new form of math to solve.
Humans only invest in solving problems that matter one way or another.<p>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.<p>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.
>Humans only invest in solving problems that matter one way or another.<p>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.<p>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?
> However, math problems are really there to solve a real world problem.<p>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.<p>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.<p>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:<p>> Every morning in the early part of October 1843, on my coming down to breakfast,<p>> your brother William Edwin and yourself used to ask me:<p>> "Well, Papa, can you multiply triples?"<p>> Whereto I was always obliged to reply, with a sad shake of the head,<p>> "No, I can only add and subtract them." [1]<p>If we go back further, Pythagoreans weren't trying to solve "real problems" either, they were like a weird religious cult.<p>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).<p>[1] <a href="https://en.wikipedia.org/wiki/History_of_quaternions" rel="nofollow">https://en.wikipedia.org/wiki/History_of_quaternions</a>
Yes, and here towards the bottom the author gets to the reason he didn’t sign with the other medalists:<p>> I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly.<p>> 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.<p>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.<p>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.
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'"*<p>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?<p>*(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)
My Mom was worried I wouldn't finish my thesis so instead of encouraging me she encouraged an LLM. She says now I don't have to worry about submitting a proof anymore, there is a counter-example.<p>This is the state of affairs today and how everyday people will talk about it.
> 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?<p>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.
I cannot wait until Gowers and Tao must have an iris scan at Altman's Worldcoin in order to get to their beloved OpenAI. Give it 5-10 years.
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.
I'm not sure what the primary signatories want, exactly. "No mathematics" safeguards added to frontier models, like the "no cybersecurity or bio" ones that currently ship? That seems like a bleak future. Protectionism like tariffs.
> we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.<p>This is the main issue, and while I fully agree with that value sentiment, the referenced letter failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work.
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.
> I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.<p>What stops you? I can fathom it alright.<p>I am always suspicious of these arguments like "a machine won't do thing x which humans can". These always basically pre-suppose that humans aren't machines. If that's what you're arguing for, say it outright.
A cynical could say that AI could not interpolate the Dedekind cut but it could extrapolate to create a lot of money but just using an imaginary extra point.<p>The ironic part is that generating a joke like this requires jumping across three distant fields: real analysis, machine learning limits, and VC market cynicism. For an AI to discover that specific overlap through statistical search, the combinatorial space is absurdly huge. Yet a human brain connects them in a fraction of a second. This tiny joke is a micro-proof of the macro-argument: human conceptual leaps routinely bypass exponential search spaces.
Just to add that the Dedekind cut example seems to stand beyond any RL policy used in chess or go, AlphaZero or AlphaProof. In the classical RL there is an state-action space. If the solution requires jumping to a totally difference action space (that must be created) the local policy stalls. Dedekind cut is an example of a out-of-distribution state-space generation.
Solving a theorem is like climbing a new mountain. The mountain is already there, and there is a list of the hardest known mountains to climb. The point of climbing them and not just dropping with a plane on top of the peak is to help develop human climbing skills and expand our knowledge and abilities. Also, already conquered mountains are climbed all the time to test new strategies.<p>Now, if an LLM proves a theorem, it's like discovering a new mountain <i>and</i> knowing what its peak looks like. Does that mean the problem is finished? No, we still need climbers to actually do the work and advance the field with human understanding.
Why? Why should a government invest money into this? Here you see human understanding as an ends, while historically in society it has been applied as a means to ends like social power, resource accumulation, etc.<p>Now these means are generated. If the proof yields some improvement somewhere, it can be used and there needs to be no human in the loop
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.
This was also guest-posted on Tao's blog. Tao has a flood of guest posts that cope in a varying degree. Most start with empathy, move on to inevitability and emphasize human understanding.<p>Few of them mention research theft, none of them mentions concentration of capital and resources.<p>The fact that Gowers quotes Tsimerman (OpenAI employee) says it all. A couple of mathematicians are determined to bulldoze forward with AI, and I suspect Tao will also continue after the meek concern letter and the cleverly worded guest posts that are subliminal ads for AI.
I hate to say it, but if AI keeps advancing at the current rate, the justification for keeping humans in the loop will grow increasingly hard to make.<p>There's definitely the possibility that in just a few years, human mathematicians will largely become irrelevant in the face of extremely capable models.<p>I don't like it, I think there are a lot of bad side effects to it, but I think it's important to face that possibility.
> One way that might happen is that AI disrupts society so much, or even kills vast numbers of us, that the preservation of something like the current mathematical tradition ceases to be of any concern: all that will matter is the survival of the human race. But that again is a topic for a different blog post (which in fact I am in the middle of writing).
Yeah. It's interesting how many commenters here can (regretfully) no longer justify humanity's existence. Trolling or not, that's becoming familiar.<p>Prediction: In a hundred years, some humans will survive. Hopefully 8 billion or more. But among the survivors, actively anti-human attitudes and behavior will become unpopular [1].<p>[1] I mean really very, very unpopular.
Knowledge wants to be free.
This is pretty disappointing I have to say. The letter actually has a “pro AI” stance (see quote below), and Gowers’ apparent reason for not signing it is so subtle (not wanting to offend people whose goal in mathematics is not understanding) that I can’t help but view this as being more about optics than its actual content; i.e. he seems to be cleverer than I am in recognizing most people will not see the letter as a push to use AI in ways which are healthier for human flourishing but as advocating for one of two sides in a highly artificial binary (“pro AI” or “anti AI”). Gowers draws a distinction without a difference here and should have just signed the letter.<p>> AI offers the potential of enhancing and accelerating genuine mathematical study and understanding. Mathematics as a profession will need to adapt to these changes in several ways. However, whether these changes ultimately benefit the field or have a destructive effect will in large part be determined by the decisions of the humans in control of this new technology.
Watching the discussion unfold, here is what I feel:<p>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.<p>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.<p>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.<p>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.<p>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]<p>><i>the stepping stones required for human growth are disappearing.</i><p>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<p>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<p>It would be like rust discussions on HN for you I bet.<p>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<p>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.<p>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.<p>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.<p>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.<p>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.<p>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.)<p>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.<p>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.
><i>types of problems AI excels at are generally those that humans have found but overlooked.</i><p>Seems to be a deep and interesting angle lurking here, especially when coupled to<p>><i>evolutionary approach will slow down</i><p>Meaning something I will have to think about (while keeping Graeber in the background[0]) or even plan around :)<p>Will respond after I sleep on it<p>[0] <a href="https://davidgraeber.org/articles/value-as-the-importance-of-actions/" rel="nofollow">https://davidgraeber.org/articles/value-as-the-importance-of...</a><p><i>In value terms, the question becomes: who has the right to translate their money into what sorts of meaning? Who controls the medium through which, and the institutions through which, our actions become meaningful to ourselves, by the very act of being publicly recognized in some kind of public arena? It seems to me that while if one is trying to understand the strategies by which people can move back and forth between “fields”, and especially, by which some are excluded from them, Bourdieu’s models are pretty much indispensable, THEY DO LITTLE to tell us why anyone wishes to enter certain fields to begin with.</i><p>The evolution of fields, of domains of enquiries. AI-independent<p><i>First of all, I take it for granted there is really no such thing as “intelligence”</i><p>Writing like this from 2005, feels like a personal bedtime Bible for AI age
The problem with the Fields medallists petition was that its central argument relied on reiterating a very fashionable, neoliberal-academic trope, that science as a whole is "done by a culture of experts", implying that there is no value or even science existentially outside of "the social practices of science". It is a predictable extension of centrist-pseudoleft political ideologies (~ ivory tower belief system) which Thomas Piketty for instance has conceptualized as the Brahmin left. Indeed the whole AI/ASI/AGI issue is hugely political and it cannot be dismissed and analyzed as a neutral topic in isolation from people's political premises.<p>That said, I think there is an articulable concern about the recent AI events as it may threaten human's autonomy and relation to science and mathematics. Just not one premised on that. What's interesting is seeing different academics having predictable reactions, for example Sabine Hossenfelder is more politically libertarian so in this case she said DGAF about human mathematicians at all. Scott Aaronson is another interesting example, he wrote a blog post a couple days ago as well.
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TL;DR: Mathematician community broadly unhappy with AI because it might destroy the community by doing all the mathematics for them.
Siri, can you call the bicameral order, the captain has published again..