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>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

Math professor here.

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.

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 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.

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.

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.

What if a world of “vibe coding math” creates 10x as many mathematicians? Like, good ones?

Let’s be honest, Gauss and Euler are a statistical phenomenon.

Well, we can reduce every human activity to 0 value and take it from there: if there’s no value in understanding there’s no place for humans in the process, we can just go back to worshipping stones and let the AIs burn tokens deluding themselves chasing their hallucinations

One of the failings of modern society is its inability to consider that not everything can be quantified.

> 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.

> for why mathematicians should widely receive funding for merely understanding things

This is all they were receiving funding for previously? Nothing has particularly changed there.

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 politicians and lawyers to lack ethics, which is pure projection, nearly all of our best presidents and congressmen were lawyers. Even the academic environment that supports mathematicians and the STEM community generally - all of that concentrated neurodivergence and lack of people skills has led to less political power, which means less funding and less new students, which has been much more threatening to mathematics than automated proofs.

I love math, but my honest POV is, the value of theoretical math cannot get much lower. The crisis is insurmountable. The community made its deal with the devil (Jim Simons) long ago, it thought it was a STEM discipline like bio and it's really a philanthropic humanities discipline like opera. Math is having its opera moment. Someone would have to step up as the rich person who saves math, and unfortunately, all the best candidates are right now destroying it with chatbots.

> 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!

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.

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.

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Real mathematicians will continue their research, whether they receive funding or not.

> 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.

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.

> 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 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.

>Everything therefore becomes a hobby or a game.

Already predicted. How much Whuffie you got?

https://www.gutenberg.org/ebooks/8086

Kinda relevant here: https://jonnegroni.com/2015/04/15/the-humans-of-wall-e-were-... I kinda expect/hope for something like this myself (minus being fat, GLP-1s have that taken care of now, so a future of fat people seems unlikely heh)

The Machine Stops by E. M. Forester is basically this world.

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

"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 convinced that the displacement of the problem into some sort of pedagogy (and it should be appreciated that the author implicate themselves in the responsibility to establish it) instead of finding solutions in the realm of policy and regulation. Seems realistic though, that they see that as an effort unlikely to succeed in a meaningful way.

Can someone argue against that reading? Am I missing something? My conclusion as such is that not only are they not opposing the authors of the fields medalist letter, but also that their evaluation of the situation is much more dire without them giving in to resignation.

> 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.

>that AI can do everything economically necessary

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.

> 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, the only people left will be people so rich money doesn't matter. Under capitalism, the only people who are permitted comfort without work are the wealthy.

I don't understand this logic.

Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.

However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.

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?

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Yup, exactly. The problem looks like this:

(1) OpenAI is merely abusing math problems for their PR stunt. That is, once the PR value dries up, they'll leave.

(2) OpenAI, and any AI companies, can incur serious harms to the social dynamics in the field of mathematics by taking away big prizes away from mathematicians.

(3) The current gen AI is too inefficient -- requires a lot of money to operate -- so it's near impossible for academia to replicate what OpenAI invested in Navier-Stokes.

(4) Also, human experts are still economically much more efficient in searching critical directions, as we saw in the case of Navier-Stokes. The best working model is human-AI collaboration, with AI taking over the grinding part. This is what OpenAI failed to recognize.

Thus, the aggressive campaign from OpenAI is socially harmful -- disrupting a working social system without offering any alternate working sustainable system.

It's a great analogy.

In short: explore versus exploit. Both are necessary, both are valuable.

I largely agree with the OP, because I can only see an explosion in both camps.

The explore side might be automated by compute, and that breaks the social contract of the existing academic incentive structure.

But people who are inherently curious will continue to be inherently curious. Nothing will stop anyone from exploring their curiosity, and the speed and depth at which they explore will only increase. We will build tools that enhance and automate our pedagogical compression.

And for the people who want to cook to feed the family, that automation is coming too.

Or perhaps we'll find a new ceiling: the frontier failure modes that still require a human mathematician in the loop, because discovering some class of novel insight doesn't scale with current AI architecture.

Err... society will pay your wife reliably because it needs to eat while you need rich benefactors to indulge your experiments?

Progress is more about the destination with the experience of the journey being a luxury. Enjoy your luxuries while they still last.

Your position is compatible with the letter.

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

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.

As the article had said there are two cultures in mathematics. At one end of the spectrum are mathematicians who are primarily motivated by the wish to solve problems, and at the other are mathematicians who are primarily motivated by the wish to attain conceptual understanding.

You are clearly on one end of the spectrum and do not seem to understand the other end.

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> 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". These always basically pre-suppose that humans aren't machines. If that's what you're arguing for, say it outright.

A cynical could say that AI could not interpolate the Dedekind cut but it could extrapolate to create a lot of money but just using an imaginary extra point.

The ironic part is that generating a joke like this requires jumping across three distant fields: real analysis, machine learning limits, and VC market cynicism. For an AI to discover that specific overlap through statistical search, the combinatorial space is absurdly huge. Yet a human brain connects them in a fraction of a second. This tiny joke is a micro-proof of the macro-argument: human conceptual leaps routinely bypass exponential search spaces.

Just to add that the Dedekind cut example seems to stand beyond any RL policy used in chess or go, AlphaZero or AlphaProof. In the classical RL there is an state-action space. If the solution requires jumping to a totally difference action space (that must be created) the local policy stalls. Dedekind cut is an example of a out-of-distribution state-space generation.

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

We built the helicopter? Why not use it and find a bigger mountain.

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.

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

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

I had someone here saying that Trump should sign an executive order to ban all AI generated art, it's all the same crap lol

Read “the feel of power” by Asimov. Written 1958. Very prescient.

We can steelman Uber, or, less emotionally, but more productively, the idea of "dynamic pricing".

Dynamic pricing is already less controversial than "rent seeking" but Elizabeth Warren is already getting pushback from Pg who is getting pushback from the centre-left.

Your comment will likely fall off due to the HN voting algorithm, but I'd like to make a quick hattip to the idea of productive friction (eutripsis) which AI-augmented debates can produce.

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.

Yes there are really only these two strictly separated groups in the AI talking point.

Or perhaps the second group has always exploited and stolen from real open source developers? Maybe the second group is a bit dumb, just like the AI addicts? But the second group is always ready for ideological justifications for their theft.

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.

[1]: https://scottaaronson.blog/?p=10062

Tokens want to be paid for

why stop at mathematics. this applies to everything and is the logical conclusion of ASI. singularity.

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

>the stepping stones required for human growth are disappearing.

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

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

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

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

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

Sabine would say that as she left academia.