> 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.
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.
>It is up to us to coordinate and make use of the technology.
1) Coordination is hard, and 2) "us" is a hopelessly nebulous term that appears inclusive but is almost always exclusive.
I think people would have said the creation of social media platforms like Facebook was neither good or bad, but rather it was "up to us to coordinate and make use of".
But did our society really have a say over how Facebook and other social media sites were able to embed themselves in our everyday lives? I would argue the answer is no.
I think it is easy to just roll of a local maximum, however, you may get stuck in a local minimum.
I kid, of course, but I do wonder where the use of local "maximum" comes from, what is maximum there? Why do you not see this as a landscape of hills and valleys where marbles with certain energies may indeed get stuck in deep enough holes... Of course, I just assume and picture gravity pointing down in that landscape, but hey. I'm human, I feel it is expected of me.
There's some interesting stuff in a book on Katathymic Imaginative Psychotherapy where patients are asked to picture a mountain. As far as I remember it's about using the process as diagnostic tool. What does the mountain look like? Is it a steep rising, rugged massif or a shallow hill. Are you looking up, do you picture yourself climbing it, or are you on it's top etc. While it said that it's always a contextual matter and in the conversation there was a suggestion of imagining to climb the steep massif, being potentially related to narcisim. Whenever I think of it i associatr that with casper david friedrichs painting.
as a society of researchers we've tended to cultivate pretty effective strategies for escaping local maxima. I think of it like ants, where you can see if you place an obstacle in between their nest and a foot source, they develop a path that loops around it. if you remove the obstacle, for some time they continue to follow the old looped path. However, some ants deviate and go around at random, exploring. eventually by chance one happens to find a quicker route. he gets a couple of his friends to follow him, by pheremone, and over time more and more take the quicker route, and they end up abandoning the old route
It works this way with research, with most following the current trends, and some curious souls searching around for other ideas, be they contrarians, dreamers, or just convinced of some strange truth. But if we're right, signs tend to slowly begin to point their way, and we can shift the whole hulking edifice of science towards their point of view.
The problem of llms is that while they may be able to find a shorter route, we can't follow them unless we understand the route. So the forces that slowly begin to change everyone's behavior are lost
The way you get out of a local maximum is to force yourself to try something radically new every so often, even if what you were doing before was working just fine.
This is the 'stochastic' part of 'stochastic gradient descent,' and it's as important for human minds as it is for ANNs. These math wizards seem to be stuck in a rut of their own digging.
What do you mean by "fundamental truth"? If LLMs produce a collection of proved facts this doesn't sound like what I would call fundamental. It may be the case that there is no such thing as "fundamental truth", that the world is just a blizzard of facts, but in that case we would need to give up on using it as a guiding star.
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.
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)
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
> math problems are really there to solve a real world problem
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
My point is that when the consumer application became apparent we already had the required concepts and built the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks.
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
It might. It might not though. Would a 1300s superintelligence advocate for heliocentricism in the face of all institutional players advocating for geocentrism, or would it create the most refined epicycle model imaginable? I’d guess the latter.
Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
Why is it always a false dichotomy between two ridiculous extremes? Maybe inventing some dangerous technology benefits from human understanding for a bunch of obvious reasons, like human beings being responsible it goes well?
>If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
This depends on unanswered questions on what math actually is and it's causal connectivity.
Imagine we have problem A that needs to connect to math solution Z.
The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.
As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.
The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.
I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing..
don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels..
Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?
If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere.
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
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.
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.
"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
>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.
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.
vs
> 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 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.
> 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).
> 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).
> 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.
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].
Looks like Timothy is joining Scott Aaronson [1] and other academics in viscerally feeling the AGI!
Perhaps he will follow Jacob Tsimerman and try to work on AI safety. Or alternatively lend his influence to politics. It's an interesting time for sure.
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”).
> 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.
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.
Now, if an LLM proves a theorem, it's like discovering a new mountain and 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.
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
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 to 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 to 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.
> A significant amount of poverty could be alleviated through improved resource distribution and policy, without really needing improved productivity/increased resource production.
reply
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.
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.
When society gets to point where creating things like food, clean water, and medication becomes near-zero and over-abundant, just like the amount of clothing waste in the photo, then the tech bro overlords won't hesitate to dump a bunch of free useful shit to the masses to sedate them. Right now, societies in developed countries throw out 25% of all food. There is no one in the west starving, the poorer you are the more likely you are to be obese. We are being sedated by youtube and tiktok every day, everyone has a phone. Poor in the first world is like king 200 years ago, and we continue to up our standards more and more. Once giving someone a house and land becomes genuinely dirt cheap due to interplanetary expansion- people will get houses and LAND as welfare.
Meanwhile there are kids in Africa that would consider their prayers answered if we developed AGI so they could stop sending their 6 year old son out in the dunes to mine cobalt.
"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)."
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:
"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."
Note that there isn't any attempt to establish a possible horizon as to how that coule happen.
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).
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.")
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 convinc...
> 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.
Your productive friction (eutripsis? ~ negentropy) seems like a wonderful concept that the original letter should have rallied around to unify the community (seems like Gowers would not have missed this point if they had a new name for it!)
I think Tao had a relevant talk about the paradox of efficiency..
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 that 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
Eg, KPIs, metrics, but of productivity, of "veracity", not understanding
Anecdotes--> better friction than data, sometimes, though :)
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)
Just taxidermy, stamp collecting, and vibe-anthropologizing here TT
> 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.
This has not yet been explored with AI only because solving hard problems is where everyone, practicing mathematician or layperson, understands 99.99% of the prestige to be.
AI's attention will not be directed towards generating interesting new conjectures until all the low-hanging prestige-rich fruit of famous decades-old conjectures have been mined, because it makes no economic sense for frontier AI companies to do so.
There won’t be anybody around to understand and care about the new open problems if the AI companies proceed with totally destroying the culture of mathematics. Gowers doesn’t care, apparently because it’s just about winning for him. Same with Tsimerman.
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
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’ve never read it but I did look up the Wikipedia summary after your comment.
I don’t see the similarity? It’s a dystopia run by a maniacal computer and everyone is forced to live underground until the machine collapses for some reason?
> I don’t see the similarity? It’s a dystopia run by a maniacal computer and everyone is forced to live underground until the machine collapses for some reason?
I read that same summary. The similarity is pretty clear: the first step was letting a machine take over all economic and intellectual functions, leaving humanity useless and dependent it.
The question, of course, is why should it inevitably lead to that kind of world, rather than The Culture? Or, to give a more realistic and likely example, something like the Colligatarch from Alan Dean Foster's "The I Inside".
> 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 the mother in the story:
> Vashti is content with her life, which, like most inhabitants of the world, she spends producing and endlessly discussing second-hand 'ideas'. She enjoys talking to friends but uses her work to defend herself against their invitations to be more social, remaining in her ‘room’ where all her basic needs are met.
She's basically something like an academic reduced to something more like wikipedia researcher while something else has the first-hand ideas, communicating with her circle through facetime/vision pro persona.
This is practically impossible, because the "economically necessary" things to do are never fixed but instead are in a state of constant fluctuation.
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.
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.
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.
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.
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.
Why isn't this going to change?
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.
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.
It’s like asking why we don’t just run all the programs instead of making users decide what to run. There’s an infinite number of problems that could be solved, and the question of which ones are important to solve is a question about us and what we want.
Nobody can avoid it. You can RL things in a box until the box is mapped perfectly but that doesn't mean you can go forever on out of distribution data. Our number of boxes will increase but model collapse is an open problem.
Astra goes farther than any model before but it cannot go forever.
>AI didnt break through stavier-nokes until a human set the initial direction. That's not going to change.
That's not what happened. There was no 'initial direction' to set because OpenAI didn't know whatever tristan et al were working on or what direction they took. They heard rumors thta navier stokes was solved and set their best model on it.
> 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.
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.
> 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.
That's true only if you manage to implement communism.
If the economy remains capitalist, it means you'll suffer. There's no pleasant future of self-driven hobbies when you've got nothing to offer the capitalists, so they have no reason to give you any money. You'll have to work hard to scrape by on scraps and meager handouts to merely survive.
And in an odd way, I wouldn't be surprised if society becomes essentially communist for the nepo babies of the wealthy; after a long, slow holocaust of the now-unnecessary working classes. Under capitalism, the only people who are permitted comfort without work are the wealthy.
I'm surprised there's no actual commie coming out and claiming the end of capitalism is near. (except maybe me in my personal socials, only larping as a hobby)
I'm toying with the idea that until people realize capitalism is based on an increasingly invalid assumption that "things are scarce", the capitalist-ish economical system will eventually realize that the only scarce things are new, fun and creative ideas, and slowly value them more and more, which kind of might work out if nothing drastic happens between the times of transition.
And by surprised you probably mean "not really surprised" hm?
Just in case btw.: the world tends to get poorer in material ressources since around the 70s e.g. regarding habitable space, harvestable energy and the destruction of the integrity of earths biosphere - lots of things that establish value that is not bound to labour.
Knowledgework isn't everything, some tend to forget that.
> I'm surprised there's no actual commie coming out and claiming the end of capitalism is near. (except maybe me in my personal socials, only larping as a hobby)
I'm a (libertarian, free-market) socialist, so not quite communist, but I think the differences are not substantial as far as everyone outside far left circles is concerned. And I've been saying this exact thing since this whole AI wave started.
If there was ever a time to seize the means of production and make them commons, this is it.
> I'm surprised there's no actual commie coming out and claiming the end of capitalism is near.
Aren't they doing that like literally from the very beginning? I clearly remember Lenin said that capitalism will end in a couple of decades because of electricity.
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.
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.
> I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
What stops you? I can fathom it alright.
I am always suspicious of these arguments like "a machine won't do thing x which humans can". This always basically pre-suppose that humans aren't machines. If that's what you're arguing for, say it outright.
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 or lean. Maybe in 5 years (when lean attains 10% popularity of rust, and/or gpt7 level subs cost ~USD5 on average, per month) these submissions will make those sites as fun/educational as mathoverflow [has always been for me]
Such forums can then replace math grad school, if the pros 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..
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)
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.
> 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.
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.
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.
In my experience it's the opposite. AI tends to make interesting things boring. You spend less time building a deep understanding by doing the hard but rewarding work yourself, and you focus more on developing a more superficial understanding by studying someone else's work. There are some productivity gains, depending how well defined the tasks are and how well the ability to produce more translates to creating more value. But at the same time, an increasingly large part of the work feels like bureaucracy with no inherent value.
Use AI to deepen my understanding and learning some interesting math. I studied a lot of combinatorics in high school, but my understand at that time was very adhoc (that most of combinatorics problems are just a bunch of elementary tricks applying here and there). With AI, I could categorize the bag of tricks into systematic topics, and discover formal concepts.
For example, here are two topics that came up during my time solving combinatorics problems in the past:
> 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.
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.
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.
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
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.
I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.
> Most of modern pure mathematics doesn't look like it has any chance of being that.
The stuff powering current tech (incl LLMs), the very foundational math, could have been described in exactly that way when it was new. Number theory, basis of most cryptography, was "pure math" not too long ago. It's only useful in hindsight.
Yes, there's so many applications that we couldn't even imagine until we got modern computers, and most of us couldn't imagine them either until we got them. What didn't sit right with me is that these concepts are a very small subset of mathematics, and look like "basic" stuff, while the branches got developed much further than what the applications could catch up with. You don't need Fermat's last theorem for cryptography, for example. I have no idea if we'll ever need homotopy groups of high-dimensional spheres, etc.
Either way it seems like nobody is in a position to say "this branch of math will surely stay arcane abstract nonsense forever", so the argument for continuing specialized math research comes down to "we might lose out on some cool stuff if we stop", even if most of it will indeed remain useless for a long time.
Exactly. Most results will stay "useless" in the sense of not having a direct application. But the process of having figured out those results is a necessary step on the way of figuring out those few things that eventually do have a direct application. And you don't know which is which beforehand.
(Besides, it's not more useless than playing the piano. But that's a whole other type of conversation. Though it shares resemblance if you think about it long enough.)
There's been some knot theory applied to molecule analysis chemistry as well as things like topological quantum field theory, with both of those being examples of fields that benefit greatly from previously unapplied mathematics introducing tools to use.
Thanks, that sounds interesting. As I said in a parallel comment after some thought on this, nobody can really say with absolute certainty that a particular branch will forever stay useless, so that should be good enough to keep going.
Sometimes one can be convinced that their mathematics will be useful, too. I've just recalled a rather motivating quote by Grassmann, the inventor of modern linear algebra, whose mathematical work was not appreciated during his lifetime.
"I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me (as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments. For truth is eternal and divine."
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.
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.
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.
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.
My vote is for interpretive dance. Give the laity something for their money.
Lest someone gets the wrong impression, I think that any mention of the a priori strange-looking resolution to the Jacobian conjecture should be accompanied with a read of pages 160–161 of https://arxiv.org/abs/2609.05746. The tl;dr being that the very same construction featuring in the resolution appeared earlier in a draft paper that was accidentally made publically available for a travel award application. The same story has accompanied several other of the major announcements made by now. We haven't have a move 37 for maths yet, despite what OpenAI's marketing department might want you to believe.
Similarly, if all you would ever read were OpenAI blog posts, you would get a very wrong impression of the usefulness of large language models of today in maths. For a working researcher, it's not a magic wand that you point at any given proposition and it tells you whether that proposition is true or not. It does appear to help if, while pointing your wand and utter the magical formula “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
Instead, the reality seems to be closer to this, projecting a fair bit: a given mathematician will have a collection of propositions that they care about, and that they'll use as their own internal benchmark as new models come out. Very rarely will anything come out of it, but sometimes, in particular if you make sure to provide the wand with all relevant context, papers that could be relevant, proof strategies and lemma structures that you suspect are useful, something (which may or may not be plagiarism) will pop out, and that's really nifty. Moreover, it is not unimportant what the proposition and the relevant proof is like. It's still very unclear what kind of maths the models are good at, but it seems to certainly be an advantage if what you're looking for is a counterexample hidden in a pile of otherwise similar-looking non-counterexamples, if your proof is one that requires considering 36 different cases, each of which are so tedious that no researcher would have the patience to go through them by hand, or if the proof is an amalgamation of several existing structures, some of which are only documented in Georgian.
The gold rush, more than anything else, seems to be populating the convex hull of existing maths.
This can all change. The $15 million wand requirement today will be less tomorrow. Whether we ever get a move 37 is less clear, or whether we will eventually reach stagnation as all low-hanging fruit is picked, and the convex hull is populated; call this cope if you like. But maybe we do get move 37s all over the place, and it's fine that people think about what that future will look like.
Until then, let us rather have a think about what we can do to fix the incentive mismatch, to ensure that we increase the prestige of digestion over being the first to convince the LLM to do it up.
> It does appear to help if, while pointing your wand and utter the magical incantation “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
That's like where chess was when Deep Blue was built by IBM. Productivity improved. There was someone complaining on here recently that the seat-back entertainment system on some airline had a chess program set to "trounce all humans".
I don't think the prevailing notion of mathematicians is that LLMs are generating proofs that a human couldn't have come up with (like, the Move 37 of AlphaGo). Rather, the concern of those Fields medalists was the opposite - that LLMs are solving problems that humans definitely could have solved, and through solving would have gained a greater understanding of the underlying structures.
> You must explain what you give back to these people.
Ha-ha. Sorry, but it is not "these people", who is paying to the mathematician. Government is doing it. So they must explain to the government what they cold give back to the government. Usually it is loyalty and the use of their social position to confirm the correctness of the government’s actions and political programs.
Goverment is extracting taxes from these people "for their own benefit" and giving it to the mathematician. When a farmer farms the food of the mathematician and sells it to the mathematician for money, he is simply earning back money that already used to be his.
Which is all well and good -- if it is actually in his own benefit.
Tao's blog is a nice example of human interpretation of math. He explains in an understandable but rigorous way beyond what would just be in a journal paper (or the output of an AI).
I would like to ask a question to you as a math professor: I think we all agree we do not know what the discipline will look like in ten years. But doesn't the rapid surge in mathematical proofs and methods imply that - at least for the coming years - there will be more, not less, work for mathematics?
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
In practice, mathematicians' workloads have been a function of their work ethic, motivation, and competing demands on their time. There's no big-picture question of "how much math there is to do now"; the amount of remaining math to discover has long been presumed to be, for all intents and purposes, infinite. (Similar questions are relevant on a much smaller scale -- for example in case of someone who has specialized in a narrow specialty which goes dead.)
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
Where I see models having a huge impact is in simulation code development.
One blocker for years now has been the adoption of GPUs. LLMs can fairly successfully and very quickly port to GPU and suggest/implement useful optimisations. Once it's verified, a code can go from anywhere between 2x to 1000x faster (mainly because CPU codes are so poorly optimised). Some science can reach much greater problem sizes, while some can run the same problems in hours rather than months and both can be revolutionary. Even more than that, LLMs seem to be finding fundamental performance bugs in both open and closed source core libraries so there's a bit of a whole-ecosystem uplift.
Can't comment on the more theoretical, less computational applied maths impacts!
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.
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.
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.
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.
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 lawyers to lack ethics, and 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.
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. I don't think Greg Brockman's wife can shed enough tears to convince him to donate enough money to save math departments and throttle Astra access.
This is meant to be a stylized and colorful comment.
What is the recent history of theoretical math? So much turf has been ceded to applied physics and computer science, it is firmly a humanities department now, maybe since polar codes in 2008 and compression schemes in 2006 were discovered. There's nothing wrong with this, it's not meant to be applied, the applications are serendipitous, whatever.
What about in the rest of the humanities? Lots of people, maybe nearly all, who study literature or history are really nice and beautiful. Many become lawyers or historians and win elections or become actors or whatever. There's a significant amount of people skills concentrated in the humanities. In math, you can name Terence Tao, okay, he was on the compressed sensing paper, and he's a great writer and a nice guy. But there are fewer than 10 freely elected politicians worldwide with a math background. There are thousands who were lawyers or historians.
So do people like mathematicians? I do, but you can see how regularly the answer is, no. In fact, instead of taking a great mathematician and giving him a well-rounded liberal education that confers academy-preserving people skills - many mathematicians from the recent past were big personalities, and they were actually very influential in politics - there are instead, loud 24 year olds suing universities for admissions policies they faced when they were 18 that they regard as discriminatory. The entire basis of that is: they got good test scores in math and other exams, and in their opinion, it doesn't matter how big of an asshole they are. Merits of their claims aside, suing someone over diversity policies is a dick move. The entrance into the math academy is crowded with people who are really good at math and who have made it their mission that having people skills is the opposite of a great math education.
The people who destroyed math - the call was from inside the house! The president of OpenAI studied math, he was math camp. He became president by alienating the actual PhDs who invented everything, and the technology is the ultimate academic cheating machine, and OpenAI and Anthropic just deployed its powers most forcefully against their own, stealing proofs from end users!
We talk about math because it makes a ton of money. Like that's why many assholes like it. The diary entries from the president of OpenAI talk about, how do you become a billionaire? That's what this is about. It aligns. Jim Simons created that (along with Edward Thorp). That's where this slide started. Jim Simons and Edward Thorp and a few others made a ton of money - Albert Einstein and Richard Feynman and other applied sciences people with great math skills had never made as much money - and they donated a ton of money to math departments and changed the ethos of mathematics permanently. Renaissance Technologies is not a nice company. Two Sigma and Jane Street and Citadel are not nice places, they are full of not nice people, you could erase what they do, and literally nothing of value would be lost. FTX and Situational Awareness and all the math people they've hired and all the money they've lost has made a negative impact on daily life for most people - the nice people who work at these places are victims of their own neurodivergence, they are unintentionally blind to ethics, which is really why the people that run these places like math people with no people skills. Basically because it lets them gamble with huge amounts of money in kind of dumb ways. Not because they are smart or because they "solve problems." I cannot name a single good thing any of these banks have done for anyone, which is a bit of a blowhard thing to say, but on the other hand, they have had ample opportunity to try to justify anything positive to anyone and the best they can come up with some abstract, absolute utter bullshit about liquidity or vulture capitalism.
There are unethical lawyers, but Barack Obama was a good guy. There are tons of lawye...
Throughout the history, we had made some of the best mathematicians abandon math. In turn, they apply their math skills somewhere else (hedge fund), and the companies they founded absorbed all the people from academia.
This has happened numerous times in the past, and it will happen over and over again.
It's inherently political for so many reasons including the fight for taking action that apparently has no meaning. This very debate in other words. So I disagree. It's like philosophy (or opera) and there's nothing wrong with that.
What's interesting to me is that this can be completely true (I believe it is), but why can seemingly nobody see the value and prioritize saving it. Opera is beautiful, it's one of the many things that make life worth living.
More efficiently allocating capital ought to be just a way to generate more resources for producing opera. And yet...
>why mathematicians should widely receive funding for merely understanding things
imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
all maritime engineering and trade was done with geometry and arithmetic at the time. there were no practical applications, not for likely at least a century until hydrodynamics were incorporated into shipbuilding
now look at today. how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
there's no KPI to be derived from any academic field of study at the bleeding edge of theory. theoretical underpinnings lead to practical applications much further down the line after many paradigm shifts
semiotics and cultural capital as theoretical concepts is another example - at the time they were purely seen as navel-gazey literary theory work. these days, half a century later, they're in wide use (for better or worse) in marketing and advertising - they birthed the whole concept of 'branding'
not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
True, but this is also a stunning example of survivorship bias.
Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
It's difficult to draw the conclusion that "every possible branch of learning ought to be funded" by appealing to "later practical value" based on this cherrypicked example.
On the other hand, clearly _some_ novel theoretical work with no apparent immediate value _does_ yield real world benefit later.
Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
You can call it survivorship bias, but another way to phrase it is that it's very difficult to forecast the practical benefit of any one piece of mathematics even while the long-term impact of mathematics as a whole is undeniable. And any schemes to further ration resources among mathematicians ignores the reality that for such foundational subject, math research already one of the least funded compared to other disciplines or domestic priorities.
> Countless other mathematical curiosities were developed in the 1700s -- and forgotten.
Really? Like what?
edit: this is either an unfalsifiable claim, because by "forgotten" you meant here is no extant knowledge or remaining record of it, or it's almost certainly nonsense and anything you could cite would be foundational to some area of modern mathematics, even if as a disproven counter theory.
A maybe example that springs to mind is how Gauss discovered the FFT in the early 1800s (predating even Fourier analysis) but didn't find it interesting enough to publish, so while his work was important, it was also forgotten and had to be rediscovered.
Really hardcore spherical trigonometry comes to mind, but I may be off by a century. It used to be considered fundamental, but almost nobody besides maybe a few historians of mathematics knows the methods anymore.
"... but this margin is not large enough to contain it." Sound familiar?
Now, it's possible that Fermat thought he had a proof of his theorem, but that it would have turned out to be wrong. That happens sometimes. But he did not write it down on any document that has survived to this day, so we know that he had something that has been lost.
It's similar to history. We know that there are books, plays, etc. that used to exist, but that nobody knows today. Because other writings that have survived to this day quote from them. But beyond the quote, we don't know anything else about the play or the book or whatever.
Fermat's Last Theorem almost certainly isn't the only case of mathematical theories that we know were published somewhere but that we have no record of today. It's just the only one that I happen to know about, not being a mathematician myself. I'm sure there are some people on HN who know of others.
> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Are you proposing that humanity should rely on AI determine the fields of study that should be perused and those that should be defunded?
Why don't you first come up with an AI can predict if the stock market will go up or down tomorrow, with 99.999% accuracy. Should be really simple as it only needs to answer what will happen tomorrow.
After that, come up with the AI that will predict how actions or non-actions today may affect outcomes 100 years into the future.
> Are you proposing that humanity should rely on AI to determine the fields of study that should be perused and those that should be defunded?
We already have examples of politicians relying on chatbots to understand problems. I wouldn't be at all surprised if, given a few more years, the above will be effectively the case without any deliberate effort.
The sad thing is that I would actually trust a chatbot more. Not because they are that good, but because the bar for good governance is that low, especially of late.
>> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus
The direct use of that math to biuld better weapons. That math was/is essential to the development of modern artillery. The first tasks assigned most early computers were to calculate ballistic trajectories, and also tide tables which also have immense military applications. Math was and is a weapon.
> Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
1. You may not agree, but some people will argue that knowledge is worth accumulating in itself. We fund astronomy well beyond the solar system despite there being no real prospect of practical applications.
2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
in general, yes. but specifically studying exoplant A or exoplanet B? or understanding dark matter? or driving a little science robot on Mars? what's the immediate utility of that?
The same thing could have been asked of radio waves when they were first discovered. It was actually considered purely an academic curiosity with no practical application. You can't know ahead of time which discoveries will be the lynchpin for some as yet unimagined future advance. Much like VC you need to spread out your bets and expect that most won't make direct return, but that every now and then something with 10000x return will surface.
right, which is all the more reason to continue studying them in spite of the lack of immediate returns. our capitalist mode of quarter-driven roadmaps means we no longer care about the long horizon, it's all 'results this, metrics that, now now now' - meanwhile any student of history who's ever looked at the history of computing, or data science, or virtually any of our modern marvels knows how much purely abstract stone had to first be cut to get us to the results we have today
can you imagine chucking all of computer science away because Babbage's thinking machines and Lovelace's programming cards didn't really result in meaningful capital gains here in FY26Q3? and yet that's the basis for so much of the widespread funding cuts across all fields of inquiry
Agreed. For a long time it felt like most governments/funders accepted that pure research is a necessary part of feeding the applied research machine. And now the funders seem particularly short-sighted.
In saying that, in the spirit of a good faith argument, is there value in temporarily deprioritising pure research in (current) times of funding drought?
> 2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
That is totally not true. I am sure you can hardly name even one of those "curiosities" that later turn out to be useful and not "free" (i.e. if not existed, wouldn't be invented on the spot as a tool for solving those particular problem that they end up solving).
1. is debatable, but in any case that is not the argument that the GP post made -- the post specifically argued that it's worth developing knowledge with no immediate known application, because some application may be developed later.
That's a different argument than "knowledge ought to be accumulated for its own sake."
2. is unclear; the economic funding model of the 18th century was very different. At that time, there were very few academics, and nearly all were independently wealthy and self-funding. That is, they chose to spend their own money on mathematical curiosities to amuse themselves.
Now, in modern academia, we are discussing the case that public tax money ought to be appropriated and allocated to pure knowledge / curiosity discovery _at scale_, for work which -- by definition -- has no known application or use. This is a very different economic and ethical proposal. This is no longer the question of how one rich man chooses to spend his own money; we are discussing the social application of common resources -- compulsory tax payments for all.
It's easy to imagine other ways to spend that tax money that have much less debatable social value -- how about free healthcare for all? Better roads, or high speed trains? No famines? Flood control? There are a million other ideas with immediate benefit.
I appreciate the value of pure knowledge as much as anyone, but it's also hard to make the case that it should be pursued indefinitely, whereever possible, and without limit or boundary when so many competing priorities exist. It's a question of competition for very scarce resources, versus unlimited wants. How do we decide appropriate allocations?
I actually just think every person should have their basic survival provided for efficiently by society. My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily and be more inclined to share their work. My claim then, is that society would net a benefit from this arrangement which would more than pay for itself. And you don’t have to try to pick winners and losers.
It's very unlikely anyone with the intelligence to contribute to math research can't already find a non-hectic job that provides for their basic survival. People choose hectic jobs because they want to do better than basic survival.
This is such an insane take I don’t even know where to start. Even in the western world people increasingly need to work more than one job / have multiple incomes to live a life where basic necessities are given, and some luxuries are attainable (read: vacation, not rolex). That’s without children, in rich countries. That isn’t even touching on the problems someone e.g. from a lower caste in India might face. Asserting that „everyone with the intelligence to contribute to math“ would be able to just “find a job” that would them also contribute to the field is completely detached from reality.
People working multiple jobs in the western world have some reason beyond "basic survival". Supporting other people, living in a desirable location, or like the vast majority of people they want to make enough money to go beyond basic survival because basic survival is low status. Which is why the idea that all we have to do is give people enough money for a basic survival lifestyle and they'll be happy and spend the rest of their time on things with perceived aesthetic value like math research is the insane take.
You’re leaving out the niceties like living in a neighborhood where you don’t wake up and see if your child is still alive every time you hear gunshots in the night. Or working a job that doesn’t risk your limbs or even your life if you make a mistake. Or working for someone who will not just fire you the instant you stop being able to work due to an on-the-job injury.
Most of these aren’t even new inventions; read about the working conditions in the 19th century.
> My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily and be more inclined to share their work
We have natural experiments of this: retirees, both early and otherwise. Do retirees during the first five years of their retirement "find productive work voluntarily" (your choice of words, not mine) in quantities that are comparable to what they did at work during the five years prior to their retirement? Not exceptional individuals, but on average.
I retired circa five years ago and spend my time doing stuff like walking the dog, drinking coffee and reading books, much like other retirees. My productivity is as close to zero as I can muster, and it is the rule, not the exception.
There's an obvious bias in focusing on retirees though. I mean, the whole notion of retirement exists for a reason.
But also, in all honesty, if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor, what's the problem with most people walking the dog, drinking coffee, and reading books?
> if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor
But that is not the premise of the discussion at hand, though. My comment was written in this specific context:
>>> My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily
If I could truly live decently enough without working another day in my life, I'd happily spend all day doing the bare minimum I had to so that I could enjoy most of my days doing things outside with the people I love.
Of course, in such a world I'd be competing with the millions of other people doing the same — and going to most of the same places, and doing the same things.
I suspect the waiting list for an actual vacation would be about a decade long, as we saw in the Soviet Union.
I think this mostly comes down to what work is productive for a person. People who are fascinated by research topics are likely to work on them and people who are enamoured with creating art will do so. But people who's productivity was largely being some cog in some machine will be all too happy to say good riddance.
I think everyone deserves the opportunity. Most will still veg out even if they had no minimum wage blue collar monotonous job. But everyone deserves a chance. Some people are born into it and have no chance. While I was lucky to have a supportive financially stable family through no work of my own yet I wasted it all scrolling my phone.
> I actually just think every person should have their basic survival provided for efficiently by society
Define basic survival. Tantalum mine worker in the Congo who earns 22 cents per hour and raises his three children on that money, is he under "basic survival" level or below?
And to be clear, as a part of a society, are you ready to forgo your salary, except maybe 1 dollar per hour, to provide to other people? Why haven’t you done it yet?
It's not survivorship bias, because math isn't a bunch of independent, parallel things, where one turned out to be useful and the rest was junk. There is no known way to advance only the portions of math that, centuries later, will turn out to be economically useful. Even with AI, we only know how to advance the entire subject.
Also it ignores the fact that the core of calculus and the idea of dealing with infinitesimal quantities was independently discovered for many centuries. eg. Archimedes use techniques that were eerily similar to integrals.
The work that Newton and Leibniz did was in formalizing it and coming up with the notation that would end up being more widely accepted and applicable. It's very likely that the techniques would have been developed later to solve a practical problem.
And if calculus was the only useful thing to come out of 1600s mathematical research, it would have been worth it. The other dead ends don't need to justify themselves. Getting one thing of this magnitude justifies it all
On how to select what to fund in the future: find the brightest minds, fund whatever they want to do. That tends to work out in aggregate
> And if calculus was the only useful thing to come out of 1600s mathematical research, it would have been worth it. The other dead ends don't need to justify themselves. Getting one thing of this magnitude justifies it all
It would have been worth what? Doing mathematical research at all? Nobody is suggesting we shut down mathematical research.
It doesn't justify a funding system that didn't exist at the time.
> It doesn't justify a funding system that didn't exist / didn't fund those researchers. We need to come up with better reasons to fund such a thing.
Do we?
For most of us, it seems like the EV (expected value) from such funding is quite positive. The work pays for itself in the long run. That's justification enough.
> But I would suggest replacing "seems" with the best numbers available.
I know there's research on it for applied and basic science in general, and the rewards were massively EV+. But I can't recall whether similar research covers applied and basic mathematics research.
> this is also a stunning example of survivorship bias.
No, it's an example of why you have to allow people to pursue what at the time look like "curiosities", even though most of them don't go anywhere--because the very small portion that do go somewhere, end up changing the world, and we don't know in advance which ones those are going to be.
It's quite true that the funds we have for this are a finite resource. But that doesn't mean that "foreseeable practical applications" is a useful filter for how to deploy that resource.
You know how drunken debates with friends over trivia became pointless with the advent of Google and Wikipedia. The debates didn't get better. But they could. They did. Today we have HN mods and HN citations and upvotes
>useful filter
There's also "useful friction" as _almost_ imagined by Tao here
Yes you have to allow people to pursue these things. That's not a compelling argument for funding, especially when nobody was funding the people that did invent calculus.
> nobody was funding the people that did invent calculus.
Of course someone was. Newton and Leibniz weren't living in autarky an some island. They had an income stream that was taken away from somewhere else and given to them so that they could afford working on their pet projects.
The fact that it's not publicly funded doesn't change the fact that there's still resource allocation happening.
> every mathematician in the current day is funded,
This is tautological, “mathematician” being defined in the modern world as someone paid to do mathematics…
Notice that the modern world is able able to sustain significantly more mathematicians than when “mathematicians” used to mean “someone well-off enough to work on mathematics in their ample amount of free time without worrying about their subsistence”.
> This is tautological, “mathematician” being defined in the modern world as someone paid to do mathematics…
Note that I'm including all the ones that a normal definition would not call paid, but your definition does.
But yes when a definition declares that everyone doing a thing is "funded", it does get pretty tautological and get in the way of arguments for actual funding.
During the years when Newton made his great discoveries, of calculus and universal gravitation, he had a scholarship from the University of Cambridge. (Note that it was not until a couple of decades later that Newton published this work.) After he got his MA he became a Fellow of Trinity College. So society was basically subsidizing him.
Newton was able to attend Cambridge at all only because his parents were prosperous yeoman farmers who owned substantial agricultural estates and held private incomes.
Nearly all schools at the time were not subsidized by society and required substantial tuition payments. Most children had to get jobs. School was rare.
Newton's parents, while not extremely wealthy, could afford to send him to expensive private schools, instead of requiring him to get a job at age 8. This was a prerequisite for entering Cambridge.
His early career was subsidized by his upper middle class parents and inherited capital, not by society at large. He became a professor at age 26, and was paid a salary in exchange for work after that.
Newton and Leibniz both inherited substantial estates from their upper middle class parents.
Neither was fabulously wealthy, but they were both able to bootstrap their academic careers with inherited money until they became famous enough to receive patronage.
"Individuals choosing voluntarily to spend their own money researching mathematical puzzles and curiosities and pure knowledge" is a very different ethical proposition than modern academia, which depends heavily on compulsory tax payments levied on everyone.
That's an empirical question. And you also need to justify why your country in particular should finance that.
For the kind of research where benefits accrue to the inventor (or her employer) and others can be excluded so that the benefits don't 'spill over', then private companies can fund it.
For the kind of research that spills over, you can just let the tax payers of that other country foot the bill. Eg the US can free-ride on Chinese research, and if having mathematicians in the population is good, the US can offer green cards to whatever has a math degree from a good enough university.
There might be some intermediate research that has just enough spill over that a company won't do it, but a country might, should be a rare creature. And differently sized countries should have different sweet spots: some companies are bigger than some countries after all. But I don't think we see different countries select the research (mathematical or otherwise) with an eye towards exactly tailoring spill over.
> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Ah yes, lets take the existing system that's already barely holding up and flood it with slop. What could go wrong?
> Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Do away with the perpetual uncertainty of the "will we or won't we continue to fund you" grant treadmill for practicing academics. Hold a yearly competition of academic prowess and intelligence open to any adult US citizen under retirement age. Award the winners a grant good for 40 years. If we allocated slots equal to 0.005% of the population each year (ie ~17k) that would represent 0.2% (ie 1 per 500) of the population in total at any given time.
Obviously I say that (mostly) in jest but clearly there are workable solutions if we approach things from a new angle instead of determinedly clinging to the status quo.
Not if the billionaire who has been funding the endowment for their university's math department wakss up one morning to read a three sentence Claude summary of this thread, and says "hmm... sounds like what we really need is a ChatGPT moment for math" and decides to redirect his cash to the cutting edge of next gen slop generation tech instead.
> True, but this is also a stunning example of survivorship bias.
It's not “survivorship bias”, it's merely the illustration that mathematics are a strong link problem: it's the strongest result that determines the impact of the field (not the weakest, like in weak-link problems).
I agree, I really do. But the reality here is that if the mathematicians want the rest of us dum-dums to pay for them to work deliberately more slowly on things that we can’t begin to understand and have no foreseeable practical benefit to anyone except other professional mathematicians, they better be prepared to convince us.
Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
Today, science funding comes out of the tax collected from everyone. The taxpayers want an explanation for how their money is spent. It could be, of course, vanity, just like it was before (taxpayer money is spent on sporting events, for example because people root for the athletes who represent them)... but, maybe there's a better way?
> Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
This is not the full story, but fits today's cartoon history caricature. In reality, there were also salaried employees who were teaching or working on developing practical applications, there were church fundings, the universities like Göttingen etc. Not all were particularly rich, Euler came from a modest background etc.
Erm... not really. Well, since you mentioned Euler: most of his funding came from the St. Petersburg Academy, which, in turn, was bankrolled by the Russian Czar (Catherine the Great, in this instance). The crown had absolutely no need to report back to its subjects on how and why it spends its money. The primary reason academies like that existed was that his or her highness wanted to impress their royal cousins with how intellectual they were and how much money they could spend on the matters they believed themselves to be responsible for.
That's why if you look back at scientific publications or organizations, a lot of them have a "royal" in their name "the royal society of this", "the royal college of that". This is because of their sponsors.
Since we are talking about Russia here, basically, until Alexander II (second half of 18 hundreds), there weren't any government institutions that could, in principle, manage the tax collected from the royal subjects and had to report and justify their activity. Until then, by and large, all the tax collected belonged to the crown and the crown was free to do with it however their heart desired. The initial ministry of education created by Alexander I (early 18 hundreds).
If you are interested in the reasons: both the liberation of the peasants and the defeat in the Crimean campaign created a large hole in the crown's finances. If, until then, the crown disbursed some undisclosed amount to the minister, who would use the money however he wished (mostly stole), the changes in the 1860s made it harder to steal (the minister had to report the expenditures). One could say that at this point, the current approach to financing the education began. A lot of things, however, were still subsidized directly by the crown or various elites, rather than in centralized government-sanctioned way. Another large reform, that made the system more transparent and rule-based followed in 1904, but since the Communists completely reorganized the government in 1917, it's hard to tell what the results might have been. It's only really since 1917 that education and science became exclusively financed by the tax and not through private donors.
It's a bit tautological that rich men funded it to impress other rich men. A lot of things can be bent to that mold. Yes obviously someone with money had to fund it by definition. A peasant could not. Ultimately it came from someone wealthy. But the Bailey is some kind of good for nothing lazy rich evil white patriarchal man oppressively doing some hobby science whatnot. And I just roll my eyes, sorry.
Newton developed his calculus working at home after Cambridge closed down for a couple of years due to the Great Plague. I don't think he received funding for it.
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
That is a pretty weak point. Calculus from the very beginning was an applied math.
> how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
Idk, about zero? I mean almost all of our modern technologies could exist perfectly fine even without previous decades-worth refinements and extrapolations of math theory. All the math tools that they would be needed would be created on the spot if they are required.
And Newton's calculus is rather an argument that is support that view: it is not like there was calculus, that had make describing of the world possible. No calculus was born as a tool to describe physical world from the very beginning.
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.
Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.
Not relevant to this discussion. If you are funded by patronage networks, you need to justify your work to your patron. If you are funded by grants or public research funds, you need to justify your work to the general public.
This doesn't explain why we need to pay humans to do it. What if it were more efficient to let the machines do the research? I don't think this is a great idea, since humans presently need jobs, but the argument may come up.
I think most of the long hanging fruit has been discovered, and even then you'll have better ROI focusing funding on applied math instead of pure math.
Reminds me of Michelson's (of Michelson-Morley) famous statement in 1900 that all of physics had essentially already been discovered, so the only remaining work was to apply what was known to new experiments. Similar statements were made about chemistry after Mendeleev and history after the cold war.
I didn't claim you wouldn't discover things in pure math that would decades later turn out to be useful in other fields. I'm sure you would. My claim was that the ROI is lower compared to applied math, which matters when funding is limited.
1. Those early discoveries were much closer to today’s applied mathematics (which many of the pure math academic types sneer at FWIW), and 2. They made those discoveries without public funding.
If AI can bring forth mathematical discoveries much faster than academics, wouldn’t it behove society to use AI? It seems the me the tradeoff here is the benefit of all vs the ego of a few. I empathize with the struggle many math PhDs must be going through, and maybe I don’t fully understand the tradeoff as they see it, but I’m not convinced by the letter written by a few career, tenured academics.
The more pertinent question is not the binary "should we fund mathematicians or not?" It's "how many?" The money you dole out to them needs to come from someone else.
Easy yes: Let's impose a tax on everyone to support at least one mathematician.
Easy no: Let's impose a tax on everyone to support one billion mathematicians.
Where are you going to draw that line? How are you going to convince a majority of voters that you're drawing it at the right place?
Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Do we support people who aren't using vaccines and are intentionally infecting themselves directly (measles parties)?
Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
With civilization and technology we do loose some mental capabilities - like say the part of the brain Amazonian people use for extremely skillful face recognition we do use for reading/writing while loosing that face recognition skill level that those people in the jungle have. So goes the medieval way of doing mathematics (it is a pity though - I liked it and hoped that mathematics will fall among the last, yet it happened to be among the firsts)
Assuming paper maps were the only alternative - absolutely yes. I probably would, and least on a pro rata basis for bigger weekend trips in unfamiliar areas.
> Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Yes. We do - it's a sport from way back and ongoing - a good many people want to retain non GPS navigational skills.
> Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
Yes. We do. Proficiency in multiple weapons and non weapon combat is a basic part of service training. Axes and wedges still have their place in tree felling.
You are making your argument on the basis that LLMs and future AI system will be just as good or batter as current mathematicians at all the relevant processes involved in academic math and mathematics in general. I'd like to see an experiment where an AI system was trained as a 1870s mathematician and see if it can come up with set theory, if that is not possible, then current AI systems are not fit to replace mathematicians.
100%. You're at the place that I was trying to lead the parent comment. There's a wide range of reasonable positions on how much to invest in any given thing like science, the arts, etc.
The public is generally bad at debates around taxing and spending, sadly. Voters struggle to accept the idea that more of X means less of Y. They want to believe they can have it all, for free, and there are plenty of politicians willing to tell them they can.
If you trust the system and you think things are going to work out, then you have nothing to worry about. But people are saying this is a crisis, and to solve a crisis you have to have at least a vague plan.
> not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
This.
And ironically enough, the obsession for measurable successes at all cost is what drove 20th century communists regimes to their most catastrophic failures.
Thinking that mathematicians are now useless because they cannot continue publishing results that an AI couldn't is akin to Mao's claim that “bourgeois” intellectuals were worthless because they weren't busy driving agricultural yields up. We know where that ended…
> why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work under these circumstances.
This is blinders on thinking. The future we are looking at with the help of AI is full of abundance. Money is not an issue in that world.
> The future we are looking at with the help of AI is full of abundance.
Why would you think that, when the gains of productivity increases haven't been distributed uniformly during the past several decades? What could disrupt the current trend towards more concentration?
Mostly the fact that the ultra-rich always inevitably give in to greed over caution, reach too far, and trigger a revolt.
And in this case it can go as far as "y'all can starve, we don't need you anymore". Which is perfectly true from a pure capitalist point of view, but also I can't think of a better way to get people to move on from Flock cameras to bigger things.
The only question is, how bad it will get before society snaps. And you can already see many prominent figures among the tech elites promoting UBI as a palliative - basically giving people just enough of the wealth that they won't revolt. And it would even work if they kept it up, but, again, greed always wins over prudence in the long term with these people.
> failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things
Mathematicians receive almost no funding, as it is pennies coming from the already anemic NSF budget. I don't even know why it's a question as to why liberal arts needs more funding. They don't get any to begin with!
Crazy take. Whatever justification existed before exists now. Why is generation of proofs justified but not understanding? What's the point of a proof if no one understands?
And anyway a mathematician's lifetime salary is basically nothing. I think total worldwide mathematics budget is less than a tenth of a percent of GDP. Meanwhile, openai cumulative loss stands at tens of billions?
I guess there is a bit of a realization from the laymen (and I'm speaking as a member of that group) that a lot of these problems are toy problems.
It's clear of Fermat's last theorem. But I figured that the Millennium Prize Problems (of which there are only 10) would have some real spillover into either physics or chemistry.
Yet when one of them gets solved, it seems to be that the argument is, "yeah but not like that".
Imagine if an AI solved cancer, or a vastly cheaper way to get into space, we wouldn't react like that.
I think they'll end up more like scientists. Landau famously advised his students to be proficient in math so that when they're working on a problem it's the scientific understanding that's the rate limiting step, not the math. Now mathematicians will be freed of some of the burden of proving things and their challenge will instead be to find interesting new things worth proving.
Yes - but then we may well wind up with a hoard of hungry AI labs that are just waiting for interesting problems to show up so that they can devour it before any human mathematician gets a chance to solve it.
Reminder that the federal government spends over $1.5 trillion annually on Social Security, the vast majority to people who 1) didn't put in as much as they are now getting and 2) spend their day watching TV or socializing and cooking food. 25% of our federal budget is paid to people that contribute nothing while the rest of us work our asses off to pay into a system which will likely collapse by the time we retire.
Mathematicians getting funded only do 0.0001% of the leeching that retirees do.
This argument is a bit nonsensical to me: the practical “value” of having mathematician has always been very indirect anyway, it's not like mathematical proofs of most problems mathematicians are working on have any practical value.
The reason why you want, as a society, to have a pool of human mathematicians sitting around solving problem nobody's asking but them, is to have this pool of people who understand math deeply enough to teach a digestible version to all your engineers and physicists (who are the one who actually use math for actually productive stuff sometimes) and to have them nearby to help if a physician or engineer has a question about the math they are using. And it's through these two means that over the long time a tiny fraction of mathematical progress ends up in the actual world. And it's fine because many thing can end up depending on a single piece of progress that happened 80 ago (for instance without the theory of numbers, there would be no modern cryptography, but at the same time most results of the theory of numbers will never be useful in any way).
And I don't think we have good reasons to believe an AI is going to be a good replacement for these two just because it knows how to solve well-studied hard problems.
More generally, I think the history of the Cultural Revolution should make us very reluctant to try accessing the “usefulness” of certain groups of intellectuals based on their measurable first-order output.
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.
There's definitely the possibility that in just a few years, human mathematicians will largely become irrelevant in the face of extremely capable models.
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.
Interesting how many commenters here can (regretfully) no longer justify humanity's existence.
Prediction: In a hundred years, some humans will survive. Hopefully 8 billion or more. But by that time, actively anti-human attitudes and behavior will become unpopular [1]. Even carelessly anti-human behavior ("throw the dice, maybe+ we all die") will seem ill-considered [1].
[1] I mean really very, very unpopular. Untenable even.
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.
Few of them mention research theft, none of them mentions concentration of capital and resources.
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'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.
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.
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.
There is an iq requirement to doing good mathematics and science, there is no way other than emergence of experts. Unless you believe a random foaming red hat wearer could have solved this or that millenium problem.
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.
This is the state of affairs today and how everyday people will talk about it.
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[ 25.1 ms ] story [ 943 ms ] threadThis 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.
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.
1) Coordination is hard, and 2) "us" is a hopelessly nebulous term that appears inclusive but is almost always exclusive.
I think people would have said the creation of social media platforms like Facebook was neither good or bad, but rather it was "up to us to coordinate and make use of".
But did our society really have a say over how Facebook and other social media sites were able to embed themselves in our everyday lives? I would argue the answer is no.
I kid, of course, but I do wonder where the use of local "maximum" comes from, what is maximum there? Why do you not see this as a landscape of hills and valleys where marbles with certain energies may indeed get stuck in deep enough holes... Of course, I just assume and picture gravity pointing down in that landscape, but hey. I'm human, I feel it is expected of me.
It works this way with research, with most following the current trends, and some curious souls searching around for other ideas, be they contrarians, dreamers, or just convinced of some strange truth. But if we're right, signs tend to slowly begin to point their way, and we can shift the whole hulking edifice of science towards their point of view.
The problem of llms is that while they may be able to find a shorter route, we can't follow them unless we understand the route. So the forces that slowly begin to change everyone's behavior are lost
This is the 'stochastic' part of 'stochastic gradient descent,' and it's as important for human minds as it is for ANNs. These math wizards seem to be stuck in a rut of their own digging.
> 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?
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)
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.
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.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
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.
If we give it a real problem to solve, it may just have to invent a new form of math to solve.
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.
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.
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
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.
Would this slow technological progress? Or make it go faster?
That math problems are found and solved as we run into real physical problems.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
problem specificity, concreteness, engineering relevance, or even "empiricity" seems (vaguely)
proportional to how much "good friction" can be generated.
There's also bad friction related to "meta-ness", "bad names", "aesthetics", etc, I presume. Like bikeshedding and its relatives. Is yakshaving?
Eutripsis? Vs just tripsis
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 eutripstic.
Probably you're off somewhere and never return, but AI might also provide that "Coasean floor" ;)
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.
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...
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.
But yes, none of those are very tangible, until applied to problem solutions that are tangible.
> However, math problems are really there to solve a real world problem.
vs
> 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?"
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!
> none of the millennium problems have anything to do with a "real" problem
P vs NP has a lot to do with real problems.
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
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).
Yeah and they should start seeing it differently
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].
[1] I mean really very, very unpopular.
Perhaps he will follow Jacob Tsimerman and try to work on AI safety. Or alternatively lend his influence to politics. It's an interesting time for sure.
[1]: https://scottaaronson.blog/?p=10062
> 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.
Now, if an LLM proves a theorem, it's like discovering a new mountain and 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.
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
Here the worry is the social structures of mathematics are eroded such that fewer humans become able to to 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 to 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.
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.
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.
When society gets to point where creating things like food, clean water, and medication becomes near-zero and over-abundant, just like the amount of clothing waste in the photo, then the tech bro overlords won't hesitate to dump a bunch of free useful shit to the masses to sedate them. Right now, societies in developed countries throw out 25% of all food. There is no one in the west starving, the poorer you are the more likely you are to be obese. We are being sedated by youtube and tiktok every day, everyone has a phone. Poor in the first world is like king 200 years ago, and we continue to up our standards more and more. Once giving someone a house and land becomes genuinely dirt cheap due to interplanetary expansion- people will get houses and LAND as welfare.
Meanwhile there are kids in Africa that would consider their prayers answered if we developed AGI so they could stop sending their 6 year old son out in the dunes to mine cobalt.
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:
"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."
Note that there isn't any attempt to establish a possible horizon as to how that coule happen.
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).
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.")
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 convinc...
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.
I think Tao had a relevant talk about the paradox of efficiency..
https://youtu.be/svl_1upFpQo
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 that 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.
Eg, KPIs, metrics, but of productivity, of "veracity", not understanding
Anecdotes--> better friction than data, sometimes, though :)
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)
Just taxidermy, stamp collecting, and vibe-anthropologizing here TT
> 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.
This has not yet been explored with AI only because solving hard problems is where everyone, practicing mathematician or layperson, understands 99.99% of the prestige to be.
AI's attention will not be directed towards generating interesting new conjectures until all the low-hanging prestige-rich fruit of famous decades-old conjectures have been mined, because it makes no economic sense for frontier AI companies to do so.
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 don’t see the similarity? It’s a dystopia run by a maniacal computer and everyone is forced to live underground until the machine collapses for some reason?
I read that same summary. The similarity is pretty clear: the first step was letting a machine take over all economic and intellectual functions, leaving humanity useless and dependent it.
We're talking about taking that step too.
> 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 the mother in the story:
> Vashti is content with her life, which, like most inhabitants of the world, she spends producing and endlessly discussing second-hand 'ideas'. She enjoys talking to friends but uses her work to defend herself against their invitations to be more social, remaining in her ‘room’ where all her basic needs are met.
She's basically something like an academic reduced to something more like wikipedia researcher while something else has the first-hand ideas, communicating with her circle through facetime/vision pro persona.
This is practically impossible, because the "economically necessary" things to do are never fixed but instead are in a state of constant fluctuation.
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.
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.
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.
AI didnt break through stavier-nokes until a human set the initial direction. That's not going to change.
Why isn't this going to change?
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.
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.
If you solve it then yes humans are toast mental usefulness wise
Astra goes farther than any model before but it cannot go forever.
Sythetic data works in a bounded box we have mapped you can't just make up novel data and it works.
Model collapse is completely unsolved.
That's not what happened. There was no 'initial direction' to set because OpenAI didn't know whatever tristan et al were working on or what direction they took. They heard rumors thta navier stokes was solved and set their best model on it.
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.
Already predicted. How much Whuffie you got?
https://www.gutenberg.org/ebooks/8086
> 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.
That's true only if you manage to implement communism.
If the economy remains capitalist, it means you'll suffer. There's no pleasant future of self-driven hobbies when you've got nothing to offer the capitalists, so they have no reason to give you any money. You'll have to work hard to scrape by on scraps and meager handouts to merely survive.
And in an odd way, I wouldn't be surprised if society becomes essentially communist for the nepo babies of the wealthy; after a long, slow holocaust of the now-unnecessary working classes. Under capitalism, the only people who are permitted comfort without work are the wealthy.
I'm surprised there's no actual commie coming out and claiming the end of capitalism is near. (except maybe me in my personal socials, only larping as a hobby)
I'm toying with the idea that until people realize capitalism is based on an increasingly invalid assumption that "things are scarce", the capitalist-ish economical system will eventually realize that the only scarce things are new, fun and creative ideas, and slowly value them more and more, which kind of might work out if nothing drastic happens between the times of transition.
Just in case btw.: the world tends to get poorer in material ressources since around the 70s e.g. regarding habitable space, harvestable energy and the destruction of the integrity of earths biosphere - lots of things that establish value that is not bound to labour. Knowledgework isn't everything, some tend to forget that.
I'm a (libertarian, free-market) socialist, so not quite communist, but I think the differences are not substantial as far as everyone outside far left circles is concerned. And I've been saying this exact thing since this whole AI wave started.
If there was ever a time to seize the means of production and make them commons, this is it.
Aren't they doing that like literally from the very beginning? I clearly remember Lenin said that capitalism will end in a couple of decades because of electricity.
What stops you? I can fathom it alright.
I am always suspicious of these arguments like "a machine won't do thing x which humans can". This always basically pre-suppose that humans aren't machines. If that's what you're arguing for, say it outright.
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.
>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 or lean. Maybe in 5 years (when lean attains 10% popularity of rust, and/or gpt7 level subs cost ~USD5 on average, per month) these submissions will make those sites as fun/educational as mathoverflow [has always been for me]
Such forums can then replace math grad school, if the pros 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..
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)
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.
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.
This is exactly the activity Tao and others worry will be hollowed out by instant "true/false" responses to novel questions.
finding new conjectures is the second interesting part
"instant "true/false" responses to novel questions."
Fair enough. Then we can add ordering and analyzing theorems with the help of AI as a third activity, admittedly the least interesting one.
If a conjecture can be one-shotted by OpenAI Astra, I have a hard time believing it is interesting.
For example, here are two topics that came up during my time solving combinatorics problems in the past:
https://en.wikipedia.org/wiki/Parking_function
https://www.sciencedirect.com/science/article/pii/S0012365X0...
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.
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.
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.
I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.
The stuff powering current tech (incl LLMs), the very foundational math, could have been described in exactly that way when it was new. Number theory, basis of most cryptography, was "pure math" not too long ago. It's only useful in hindsight.
Either way it seems like nobody is in a position to say "this branch of math will surely stay arcane abstract nonsense forever", so the argument for continuing specialized math research comes down to "we might lose out on some cool stuff if we stop", even if most of it will indeed remain useless for a long time.
(Besides, it's not more useless than playing the piano. But that's a whole other type of conversation. Though it shares resemblance if you think about it long enough.)
Sometimes one can be convinced that their mathematics will be useful, too. I've just recalled a rather motivating quote by Grassmann, the inventor of modern linear algebra, whose mathematical work was not appreciated during his lifetime.
"I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me (as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments. For truth is eternal and divine."
Yes, same as any other field. No one is immune
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.
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.
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.
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
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.
My vote is for interpretive dance. Give the laity something for their money.
Similarly, if all you would ever read were OpenAI blog posts, you would get a very wrong impression of the usefulness of large language models of today in maths. For a working researcher, it's not a magic wand that you point at any given proposition and it tells you whether that proposition is true or not. It does appear to help if, while pointing your wand and utter the magical formula “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
Instead, the reality seems to be closer to this, projecting a fair bit: a given mathematician will have a collection of propositions that they care about, and that they'll use as their own internal benchmark as new models come out. Very rarely will anything come out of it, but sometimes, in particular if you make sure to provide the wand with all relevant context, papers that could be relevant, proof strategies and lemma structures that you suspect are useful, something (which may or may not be plagiarism) will pop out, and that's really nifty. Moreover, it is not unimportant what the proposition and the relevant proof is like. It's still very unclear what kind of maths the models are good at, but it seems to certainly be an advantage if what you're looking for is a counterexample hidden in a pile of otherwise similar-looking non-counterexamples, if your proof is one that requires considering 36 different cases, each of which are so tedious that no researcher would have the patience to go through them by hand, or if the proof is an amalgamation of several existing structures, some of which are only documented in Georgian.
The gold rush, more than anything else, seems to be populating the convex hull of existing maths.
This can all change. The $15 million wand requirement today will be less tomorrow. Whether we ever get a move 37 is less clear, or whether we will eventually reach stagnation as all low-hanging fruit is picked, and the convex hull is populated; call this cope if you like. But maybe we do get move 37s all over the place, and it's fine that people think about what that future will look like.
Until then, let us rather have a think about what we can do to fix the incentive mismatch, to ensure that we increase the prestige of digestion over being the first to convince the LLM to do it up.
That's like where chess was when Deep Blue was built by IBM. Productivity improved. There was someone complaining on here recently that the seat-back entertainment system on some airline had a chess program set to "trounce all humans".
I would not presume this at all.
Every brick in the house you live in has been put there by a worker. The food you eat has been cultivated by a farmer. etc. etc.
You must explain what you give back to these people. It's fine if it's in a roundabout way, but it can't be nothing.
And the presumption isn't that there's something. The presumption is that there's nothing, and you must prove there's something.
Ha-ha. Sorry, but it is not "these people", who is paying to the mathematician. Government is doing it. So they must explain to the government what they cold give back to the government. Usually it is loyalty and the use of their social position to confirm the correctness of the government’s actions and political programs.
Goverment is extracting taxes from these people "for their own benefit" and giving it to the mathematician. When a farmer farms the food of the mathematician and sells it to the mathematician for money, he is simply earning back money that already used to be his.
Which is all well and good -- if it is actually in his own benefit.
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
Where I see models having a huge impact is in simulation code development.
One blocker for years now has been the adoption of GPUs. LLMs can fairly successfully and very quickly port to GPU and suggest/implement useful optimisations. Once it's verified, a code can go from anywhere between 2x to 1000x faster (mainly because CPU codes are so poorly optimised). Some science can reach much greater problem sizes, while some can run the same problems in hours rather than months and both can be revolutionary. Even more than that, LLMs seem to be finding fundamental performance bugs in both open and closed source core libraries so there's a bit of a whole-ecosystem uplift.
Can't comment on the more theoretical, less computational applied maths impacts!
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.
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.
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.
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 lawyers to lack ethics, and 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.
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. I don't think Greg Brockman's wife can shed enough tears to convince him to donate enough money to save math departments and throttle Astra access.
I don’t understand, can you explain this line?
What is the recent history of theoretical math? So much turf has been ceded to applied physics and computer science, it is firmly a humanities department now, maybe since polar codes in 2008 and compression schemes in 2006 were discovered. There's nothing wrong with this, it's not meant to be applied, the applications are serendipitous, whatever.
What about in the rest of the humanities? Lots of people, maybe nearly all, who study literature or history are really nice and beautiful. Many become lawyers or historians and win elections or become actors or whatever. There's a significant amount of people skills concentrated in the humanities. In math, you can name Terence Tao, okay, he was on the compressed sensing paper, and he's a great writer and a nice guy. But there are fewer than 10 freely elected politicians worldwide with a math background. There are thousands who were lawyers or historians.
So do people like mathematicians? I do, but you can see how regularly the answer is, no. In fact, instead of taking a great mathematician and giving him a well-rounded liberal education that confers academy-preserving people skills - many mathematicians from the recent past were big personalities, and they were actually very influential in politics - there are instead, loud 24 year olds suing universities for admissions policies they faced when they were 18 that they regard as discriminatory. The entire basis of that is: they got good test scores in math and other exams, and in their opinion, it doesn't matter how big of an asshole they are. Merits of their claims aside, suing someone over diversity policies is a dick move. The entrance into the math academy is crowded with people who are really good at math and who have made it their mission that having people skills is the opposite of a great math education.
The people who destroyed math - the call was from inside the house! The president of OpenAI studied math, he was math camp. He became president by alienating the actual PhDs who invented everything, and the technology is the ultimate academic cheating machine, and OpenAI and Anthropic just deployed its powers most forcefully against their own, stealing proofs from end users!
We talk about math because it makes a ton of money. Like that's why many assholes like it. The diary entries from the president of OpenAI talk about, how do you become a billionaire? That's what this is about. It aligns. Jim Simons created that (along with Edward Thorp). That's where this slide started. Jim Simons and Edward Thorp and a few others made a ton of money - Albert Einstein and Richard Feynman and other applied sciences people with great math skills had never made as much money - and they donated a ton of money to math departments and changed the ethos of mathematics permanently. Renaissance Technologies is not a nice company. Two Sigma and Jane Street and Citadel are not nice places, they are full of not nice people, you could erase what they do, and literally nothing of value would be lost. FTX and Situational Awareness and all the math people they've hired and all the money they've lost has made a negative impact on daily life for most people - the nice people who work at these places are victims of their own neurodivergence, they are unintentionally blind to ethics, which is really why the people that run these places like math people with no people skills. Basically because it lets them gamble with huge amounts of money in kind of dumb ways. Not because they are smart or because they "solve problems." I cannot name a single good thing any of these banks have done for anyone, which is a bit of a blowhard thing to say, but on the other hand, they have had ample opportunity to try to justify anything positive to anyone and the best they can come up with some abstract, absolute utter bullshit about liquidity or vulture capitalism.
There are unethical lawyers, but Barack Obama was a good guy. There are tons of lawye...
This has happened numerous times in the past, and it will happen over and over again.
What's interesting to me is that this can be completely true (I believe it is), but why can seemingly nobody see the value and prioritize saving it. Opera is beautiful, it's one of the many things that make life worth living.
More efficiently allocating capital ought to be just a way to generate more resources for producing opera. And yet...
imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
all maritime engineering and trade was done with geometry and arithmetic at the time. there were no practical applications, not for likely at least a century until hydrodynamics were incorporated into shipbuilding
now look at today. how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
there's no KPI to be derived from any academic field of study at the bleeding edge of theory. theoretical underpinnings lead to practical applications much further down the line after many paradigm shifts
semiotics and cultural capital as theoretical concepts is another example - at the time they were purely seen as navel-gazey literary theory work. these days, half a century later, they're in wide use (for better or worse) in marketing and advertising - they birthed the whole concept of 'branding'
not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
It's difficult to draw the conclusion that "every possible branch of learning ought to be funded" by appealing to "later practical value" based on this cherrypicked example.
On the other hand, clearly _some_ novel theoretical work with no apparent immediate value _does_ yield real world benefit later.
Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Survivorship bias is literally just how invention happens.
No one strikes gold on the first swing.
Really? Like what?
edit: this is either an unfalsifiable claim, because by "forgotten" you meant here is no extant knowledge or remaining record of it, or it's almost certainly nonsense and anything you could cite would be foundational to some area of modern mathematics, even if as a disproven counter theory.
Now, it's possible that Fermat thought he had a proof of his theorem, but that it would have turned out to be wrong. That happens sometimes. But he did not write it down on any document that has survived to this day, so we know that he had something that has been lost.
It's similar to history. We know that there are books, plays, etc. that used to exist, but that nobody knows today. Because other writings that have survived to this day quote from them. But beyond the quote, we don't know anything else about the play or the book or whatever.
Fermat's Last Theorem almost certainly isn't the only case of mathematical theories that we know were published somewhere but that we have no record of today. It's just the only one that I happen to know about, not being a mathematician myself. I'm sure there are some people on HN who know of others.
Are you proposing that humanity should rely on AI determine the fields of study that should be perused and those that should be defunded?
Why don't you first come up with an AI can predict if the stock market will go up or down tomorrow, with 99.999% accuracy. Should be really simple as it only needs to answer what will happen tomorrow.
After that, come up with the AI that will predict how actions or non-actions today may affect outcomes 100 years into the future.
We already have examples of politicians relying on chatbots to understand problems. I wouldn't be at all surprised if, given a few more years, the above will be effectively the case without any deliberate effort.
As is almost all of human endeavor.
The direct use of that math to biuld better weapons. That math was/is essential to the development of modern artillery. The first tasks assigned most early computers were to calculate ballistic trajectories, and also tide tables which also have immense military applications. Math was and is a weapon.
1. You may not agree, but some people will argue that knowledge is worth accumulating in itself. We fund astronomy well beyond the solar system despite there being no real prospect of practical applications.
2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
There are a lot of immediate practical benefits to learning more about the physics and chemistry of the universe beyond our solar system.
can you imagine chucking all of computer science away because Babbage's thinking machines and Lovelace's programming cards didn't really result in meaningful capital gains here in FY26Q3? and yet that's the basis for so much of the widespread funding cuts across all fields of inquiry
In saying that, in the spirit of a good faith argument, is there value in temporarily deprioritising pure research in (current) times of funding drought?
That is totally not true. I am sure you can hardly name even one of those "curiosities" that later turn out to be useful and not "free" (i.e. if not existed, wouldn't be invented on the spot as a tool for solving those particular problem that they end up solving).
Granted for the sake of argument. But our AI overlords are really good at accumulating that knowledge, and very patient in explaining it to us.
That's a different argument than "knowledge ought to be accumulated for its own sake."
2. is unclear; the economic funding model of the 18th century was very different. At that time, there were very few academics, and nearly all were independently wealthy and self-funding. That is, they chose to spend their own money on mathematical curiosities to amuse themselves.
Now, in modern academia, we are discussing the case that public tax money ought to be appropriated and allocated to pure knowledge / curiosity discovery _at scale_, for work which -- by definition -- has no known application or use. This is a very different economic and ethical proposal. This is no longer the question of how one rich man chooses to spend his own money; we are discussing the social application of common resources -- compulsory tax payments for all.
It's easy to imagine other ways to spend that tax money that have much less debatable social value -- how about free healthcare for all? Better roads, or high speed trains? No famines? Flood control? There are a million other ideas with immediate benefit.
I appreciate the value of pure knowledge as much as anyone, but it's also hard to make the case that it should be pursued indefinitely, whereever possible, and without limit or boundary when so many competing priorities exist. It's a question of competition for very scarce resources, versus unlimited wants. How do we decide appropriate allocations?
Most of these aren’t even new inventions; read about the working conditions in the 19th century.
We have natural experiments of this: retirees, both early and otherwise. Do retirees during the first five years of their retirement "find productive work voluntarily" (your choice of words, not mine) in quantities that are comparable to what they did at work during the five years prior to their retirement? Not exceptional individuals, but on average.
I retired circa five years ago and spend my time doing stuff like walking the dog, drinking coffee and reading books, much like other retirees. My productivity is as close to zero as I can muster, and it is the rule, not the exception.
But also, in all honesty, if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor, what's the problem with most people walking the dog, drinking coffee, and reading books?
But that is not the premise of the discussion at hand, though. My comment was written in this specific context:
>>> My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily
I see this also among my early retiree friends. People are quite happy just "playing" - gardening, joining a book club, making art, etc.
And I would be, too! There are times I enjoy doing bigger work, but man, I could build quite a delightful life living like a hobbit.
Of course, in such a world I'd be competing with the millions of other people doing the same — and going to most of the same places, and doing the same things.
I suspect the waiting list for an actual vacation would be about a decade long, as we saw in the Soviet Union.
Define basic survival. Tantalum mine worker in the Congo who earns 22 cents per hour and raises his three children on that money, is he under "basic survival" level or below?
And to be clear, as a part of a society, are you ready to forgo your salary, except maybe 1 dollar per hour, to provide to other people? Why haven’t you done it yet?
The amount of money that current local economy can provide to everybody sustainably, given the appropriate arrangements in the system.
The work that Newton and Leibniz did was in formalizing it and coming up with the notation that would end up being more widely accepted and applicable. It's very likely that the techniques would have been developed later to solve a practical problem.
On how to select what to fund in the future: find the brightest minds, fund whatever they want to do. That tends to work out in aggregate
It would have been worth what? Doing mathematical research at all? Nobody is suggesting we shut down mathematical research.
It doesn't justify a funding system that didn't exist at the time.
Do we?
For most of us, it seems like the EV (expected value) from such funding is quite positive. The work pays for itself in the long run. That's justification enough.
But I would suggest replacing "seems" with the best numbers available.
I know there's research on it for applied and basic science in general, and the rewards were massively EV+. But I can't recall whether similar research covers applied and basic mathematics research.
No, it's an example of why you have to allow people to pursue what at the time look like "curiosities", even though most of them don't go anywhere--because the very small portion that do go somewhere, end up changing the world, and we don't know in advance which ones those are going to be.
It's quite true that the funds we have for this are a finite resource. But that doesn't mean that "foreseeable practical applications" is a useful filter for how to deploy that resource.
>useful filter
There's also "useful friction" as _almost_ imagined by Tao here
https://youtu.be/svl_1upFpQo?t=25m11s
We can't know in advance, but something like the "weird cadence" that arises when 2 skilled humans try to do a threesome with an AI..
could be reified into a local measure (eutripsis) of how likely advances (conceptual or otherwise) are in the midterm
Of course someone was. Newton and Leibniz weren't living in autarky an some island. They had an income stream that was taken away from somewhere else and given to them so that they could afford working on their pet projects.
The fact that it's not publicly funded doesn't change the fact that there's still resource allocation happening.
This is tautological, “mathematician” being defined in the modern world as someone paid to do mathematics…
Notice that the modern world is able able to sustain significantly more mathematicians than when “mathematicians” used to mean “someone well-off enough to work on mathematics in their ample amount of free time without worrying about their subsistence”.
Note that I'm including all the ones that a normal definition would not call paid, but your definition does.
But yes when a definition declares that everyone doing a thing is "funded", it does get pretty tautological and get in the way of arguments for actual funding.
This is an anachronism, you're projecting modern social organization to an era that worked very differently.
Nearly all schools at the time were not subsidized by society and required substantial tuition payments. Most children had to get jobs. School was rare.
Newton's parents, while not extremely wealthy, could afford to send him to expensive private schools, instead of requiring him to get a job at age 8. This was a prerequisite for entering Cambridge.
His early career was subsidized by his upper middle class parents and inherited capital, not by society at large. He became a professor at age 26, and was paid a salary in exchange for work after that.
Neither was fabulously wealthy, but they were both able to bootstrap their academic careers with inherited money until they became famous enough to receive patronage.
"Individuals choosing voluntarily to spend their own money researching mathematical puzzles and curiosities and pure knowledge" is a very different ethical proposition than modern academia, which depends heavily on compulsory tax payments levied on everyone.
For the kind of research where benefits accrue to the inventor (or her employer) and others can be excluded so that the benefits don't 'spill over', then private companies can fund it.
For the kind of research that spills over, you can just let the tax payers of that other country foot the bill. Eg the US can free-ride on Chinese research, and if having mathematicians in the population is good, the US can offer green cards to whatever has a math degree from a good enough university.
There might be some intermediate research that has just enough spill over that a company won't do it, but a country might, should be a rare creature. And differently sized countries should have different sweet spots: some companies are bigger than some countries after all. But I don't think we see different countries select the research (mathematical or otherwise) with an eye towards exactly tailoring spill over.
Ah yes, lets take the existing system that's already barely holding up and flood it with slop. What could go wrong?
> Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Do away with the perpetual uncertainty of the "will we or won't we continue to fund you" grant treadmill for practicing academics. Hold a yearly competition of academic prowess and intelligence open to any adult US citizen under retirement age. Award the winners a grant good for 40 years. If we allocated slots equal to 0.005% of the population each year (ie ~17k) that would represent 0.2% (ie 1 per 500) of the population in total at any given time.
Obviously I say that (mostly) in jest but clearly there are workable solutions if we approach things from a new angle instead of determinedly clinging to the status quo.
If it’s slop they can just safely ignore it and keep on doing business as usual
It's not “survivorship bias”, it's merely the illustration that mathematics are a strong link problem: it's the strongest result that determines the impact of the field (not the weakest, like in weak-link problems).
Today, science funding comes out of the tax collected from everyone. The taxpayers want an explanation for how their money is spent. It could be, of course, vanity, just like it was before (taxpayer money is spent on sporting events, for example because people root for the athletes who represent them)... but, maybe there's a better way?
This is not the full story, but fits today's cartoon history caricature. In reality, there were also salaried employees who were teaching or working on developing practical applications, there were church fundings, the universities like Göttingen etc. Not all were particularly rich, Euler came from a modest background etc.
That's why if you look back at scientific publications or organizations, a lot of them have a "royal" in their name "the royal society of this", "the royal college of that". This is because of their sponsors.
Since we are talking about Russia here, basically, until Alexander II (second half of 18 hundreds), there weren't any government institutions that could, in principle, manage the tax collected from the royal subjects and had to report and justify their activity. Until then, by and large, all the tax collected belonged to the crown and the crown was free to do with it however their heart desired. The initial ministry of education created by Alexander I (early 18 hundreds).
If you are interested in the reasons: both the liberation of the peasants and the defeat in the Crimean campaign created a large hole in the crown's finances. If, until then, the crown disbursed some undisclosed amount to the minister, who would use the money however he wished (mostly stole), the changes in the 1860s made it harder to steal (the minister had to report the expenditures). One could say that at this point, the current approach to financing the education began. A lot of things, however, were still subsidized directly by the crown or various elites, rather than in centralized government-sanctioned way. Another large reform, that made the system more transparent and rule-based followed in 1904, but since the Communists completely reorganized the government in 1917, it's hard to tell what the results might have been. It's only really since 1917 that education and science became exclusively financed by the tax and not through private donors.
That is a pretty weak point. Calculus from the very beginning was an applied math.
> how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
Idk, about zero? I mean almost all of our modern technologies could exist perfectly fine even without previous decades-worth refinements and extrapolations of math theory. All the math tools that they would be needed would be created on the spot if they are required.
And Newton's calculus is rather an argument that is support that view: it is not like there was calculus, that had make describing of the world possible. No calculus was born as a tool to describe physical world from the very beginning.
They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.
Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.
Leibniz was a diplomat, then a librarian.
Since the state was the wealthy privates, for all practical purposes there was by patronage (or the wealthy elite themselves).
Assuming this is true, how did they justify their work, which had, at the time, no practical application?
So... network effects... same way most people get jobs, these days.
Art got patrons, too - what practical use is it?
If AI can bring forth mathematical discoveries much faster than academics, wouldn’t it behove society to use AI? It seems the me the tradeoff here is the benefit of all vs the ego of a few. I empathize with the struggle many math PhDs must be going through, and maybe I don’t fully understand the tradeoff as they see it, but I’m not convinced by the letter written by a few career, tenured academics.
Easy yes: Let's impose a tax on everyone to support at least one mathematician.
Easy no: Let's impose a tax on everyone to support one billion mathematicians.
Where are you going to draw that line? How are you going to convince a majority of voters that you're drawing it at the right place?
Do we support people who aren't using vaccines and are intentionally infecting themselves directly (measles parties)?
Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
With civilization and technology we do loose some mental capabilities - like say the part of the brain Amazonian people use for extremely skillful face recognition we do use for reading/writing while loosing that face recognition skill level that those people in the jungle have. So goes the medieval way of doing mathematics (it is a pity though - I liked it and hoped that mathematics will fall among the last, yet it happened to be among the firsts)
> Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Yes. We do - it's a sport from way back and ongoing - a good many people want to retain non GPS navigational skills.
> Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
Yes. We do. Proficiency in multiple weapons and non weapon combat is a basic part of service training. Axes and wedges still have their place in tree felling.
a car or a future starship wouldn't climb a tree better than ancient human.
The public is generally bad at debates around taxing and spending, sadly. Voters struggle to accept the idea that more of X means less of Y. They want to believe they can have it all, for free, and there are plenty of politicians willing to tell them they can.
This.
And ironically enough, the obsession for measurable successes at all cost is what drove 20th century communists regimes to their most catastrophic failures.
Thinking that mathematicians are now useless because they cannot continue publishing results that an AI couldn't is akin to Mao's claim that “bourgeois” intellectuals were worthless because they weren't busy driving agricultural yields up. We know where that ended…
This is all they were receiving funding for previously? Nothing has particularly changed there.
This is blinders on thinking. The future we are looking at with the help of AI is full of abundance. Money is not an issue in that world.
Why would you think that, when the gains of productivity increases haven't been distributed uniformly during the past several decades? What could disrupt the current trend towards more concentration?
And in this case it can go as far as "y'all can starve, we don't need you anymore". Which is perfectly true from a pure capitalist point of view, but also I can't think of a better way to get people to move on from Flock cameras to bigger things.
The only question is, how bad it will get before society snaps. And you can already see many prominent figures among the tech elites promoting UBI as a palliative - basically giving people just enough of the wealth that they won't revolt. And it would even work if they kept it up, but, again, greed always wins over prudence in the long term with these people.
Mathematicians receive almost no funding, as it is pennies coming from the already anemic NSF budget. I don't even know why it's a question as to why liberal arts needs more funding. They don't get any to begin with!
And anyway a mathematician's lifetime salary is basically nothing. I think total worldwide mathematics budget is less than a tenth of a percent of GDP. Meanwhile, openai cumulative loss stands at tens of billions?
It's clear of Fermat's last theorem. But I figured that the Millennium Prize Problems (of which there are only 10) would have some real spillover into either physics or chemistry.
Yet when one of them gets solved, it seems to be that the argument is, "yeah but not like that".
Imagine if an AI solved cancer, or a vastly cheaper way to get into space, we wouldn't react like that.
Let’s be honest, Gauss and Euler are a statistical phenomenon.
Mathematicians getting funded only do 0.0001% of the leeching that retirees do.
The reason why you want, as a society, to have a pool of human mathematicians sitting around solving problem nobody's asking but them, is to have this pool of people who understand math deeply enough to teach a digestible version to all your engineers and physicists (who are the one who actually use math for actually productive stuff sometimes) and to have them nearby to help if a physician or engineer has a question about the math they are using. And it's through these two means that over the long time a tiny fraction of mathematical progress ends up in the actual world. And it's fine because many thing can end up depending on a single piece of progress that happened 80 ago (for instance without the theory of numbers, there would be no modern cryptography, but at the same time most results of the theory of numbers will never be useful in any way).
And I don't think we have good reasons to believe an AI is going to be a good replacement for these two just because it knows how to solve well-studied hard problems.
More generally, I think the history of the Cultural Revolution should make us very reluctant to try accessing the “usefulness” of certain groups of intellectuals based on their measurable first-order output.
There's definitely the possibility that in just a few years, human mathematicians will largely become irrelevant in the face of extremely capable models.
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.
Prediction: In a hundred years, some humans will survive. Hopefully 8 billion or more. But by that time, actively anti-human attitudes and behavior will become unpopular [1]. Even carelessly anti-human behavior ("throw the dice, maybe+ we all die") will seem ill-considered [1].
[1] I mean really very, very unpopular. Untenable even.
Few of them mention research theft, none of them mentions concentration of capital and resources.
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.
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.
This is the state of affairs today and how everyday people will talk about it.