> 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.
>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
> 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 so relevant to this discussion, since 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.
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)
> 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 better 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.
The hell is that reply? If we're obsoleting human mathematicians, which have fundamentally been the same thing brains-wise since forever, why shouldn't an AI system be able to replace a 1870s mathematician in the same way you are proposing it replace a 2020s mathematician?
>which have fundamentally been the same thing brains-wise since forever
and you don't see any issue here? When it takes people 2 years to understand a proof it means that the old ways are done, they hit cul-de-sac.
Like in many situations, it is the highest point as well as it is a crisis at the same time.
>why shouldn't an AI system be able to replace a 1870s mathematician
my point isn't that whether it would be able to replace - i think it would. My point is that it doesn't matter. The tool is needed for the 2020s math with those proofs taking 2 years, not for the 1870s math with proofs fitting a several pages letter.
I am not debating IA's capabilities at coming up with proofs, I'm debating whether such systems will be able to figure out new mathematical fields or not. If they can't figure out some field from what was known at the time, what guarantees are there that AI-mathematics won't just bog down at its current state? There must be some evidence that AI can come up with novel mathematics before we decide to throw academics out the window. The risk of creating a sort of mathematical dark age just based on the current situation is far too great to stop funding mathematics research.
That exactly illustrates the way of technology - the first commercial GPS receivers cost $3000, and of course people used the paper maps back then, yet technology prices have always been going down (temporary RAM crunches notwithstanding :)
The difference is that the inventors of GPS didn’t deliberately buy out the entire world’s supply of GPS receivers so that no more could be sold to ordinary people.
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.
> 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.
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.
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.
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
> 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? or, a more realistic example - imagine biology and genetics today if Mendel's discoveries had been published in a more widely read journal, if he hadn't been outright dismissed for his mathematical approach. or Semmelweis' advocacy of antispetic procedures that flew in the face of the then humors-based understanding of medicine in the 1800s - how many lives would have been saved between then and when Germ Theory finally took off decades later?
it's ironic to me that it's often the same people who'll adopt the logic of banning reproductive rights on the off chance that someone very unprepared to have a child might birth someone who can cure cancer in spite of the probabilistic evidence of SES being highly correlated with educational attainment (eg that graph of the income bracket that the parents of all Nobel Prize winners start with). and yet when it comes to the actual nascent research and development required to pave the way for larger discoveries, that's seen as totally pointless and not worthy of funding. it's truly flabbergasting
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).
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.
"Falsifiability" is generally understood in terms of physical observations. Mathematics is ultimately tautological: they are true or false by their own definition, without reference to the physical world.
It just so happens that certain kinds of mathematics are unreasonably effective in drawing parallels to the physical world, but as far as mathematicians are concerned those mathematics are neither better nor worse than those that do not correspond to anything tangible.
> those mathematics are neither better nor worse than those that do not correspond to anything tangible.
A sweeping generalization. I certainly know professional mathematicians who disagree.
Also: many consider “inter-connectedness”, not “tangible” to be a sign that a topic is interesting. That is, it touches branches of mathematics aside from its own.
> 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 to 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 that 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.
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.
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.
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.
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 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?
> 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?
> There's an obvious bias in focusing on retirees though. I mean, the whole notion of retirement exists for a reason.
Go on, please. Whatever argument you make, please account for early retirees as well.
> 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
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 / didn't fund those researchers. We need to come up with better reasons to fund such a thing.
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.
> 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).
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.
You're effectively asking to predict the future hundreds of years in advance. Nobody and nothing can do that. With math, as with all science, you must be willing to accept that not everything will be a hit. There will be misses, and often the same person will generate both hits and misses, because it's fundamentally unpredictable what remains a miss and what doesn't over centuries. The best demonstration of this is that your own thinking here would've banned the invention of calculus: it didn't materially affect daily life for a solid century.
> 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.
"... 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.
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.
> 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.
You bemoan the cherry picked example and counter with an unfalsifiable claim. Certainly we have remembered much more math than Calculus, and much of it has been of practical use.
How can we hope to quantity the expenditure on math we've collectively forgotten? It's unknowable by definition. The only reasonable thing to do is to determine the value added after the expense paid. Even in a world where calculus is the only thing that we took away from the math of 1700s my guess is that this is still an economically beneficial calculation.
The claim is falsifiable; the bulk of 18th century math is "forgotten" in the sense that few, if any, people still use or apply or even know about it.
It's not "forgotten" in the technical sense that one _can_ still go dig into the dusty archives of any number of old university libraries, and review learned journals, diaries, commonplace books, personal correspondence, and so on from the 1700s that describe the mathematical work of the day in detail.
You can then systematically review that work, and test whether the claim that "nearly all mathematical output of the 18th century has been generally forgotten and never found any use."
I hypothesize that this experiment will show that nearly all of the mathematical output of the time long ago fell into oblivion. This is a falsifiable claim.
Your claim is still rubissh, as you neglected to interact with the rest of my comment: economic utility does not necessitate all mathematical output directly contributes.
You redefine "forgot" to make it falsiable, but also neglect that you need to refute that some "forgotten" work didn't contribute to new work down the line.
>> 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.
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.
> 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…
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.
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.
> 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.
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.
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.
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.
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!
> 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.
> 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).
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 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 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.
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. And what does come out tends to be quite bizarre; proofs that use terminology that doesn't exist, seem overly pretentious, based on nonsense analogies where it's surprising that it even works at all, and the only comfort is that you can join it with an equally unreadable Lean blob. And where you would be _crazy_ to just publish those artifacts and think that you have contributed much of anything to maths.
But sometimes it works. 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, and while we're still picking friut, 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. Since that's the one thing everyone seems to agree, chances are it'll probably converge to something that doesn't have to be written in commandment form, but out of the guest posts hosted by Tao so far, the one by Antieau has some useful suggestions for standards (that aren't entirely unlike those from Leiden): https://terrytao.wordpress.com/2026/09/15/fast-math-slow-mat...
> 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".
*"The Evolution of Human Science"* (originally published in Nature in 2000 as "Catching Crumbs from the Table"), collected in Stories of Your Life and Others.
It's a very short piece written as a journal editorial. Metahumans have advanced so far beyond human comprehension that they do all the original science, communicating via digital neural transfer that humans can't access. Human scientists are left doing hermeneutics: interpreting metahuman publications and reverse-engineering their artifacts, trying to decode work they couldn't have produced themselves. The editorial asks whether human science still has a point, and lands on a modestly hopeful note: interpretation is still a legitimate form of inquiry, and understanding metahuman work still expands human knowledge even if it isn't original discovery.
It reads rather differently now than it did in 2000.
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.
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
I don't think anyone is doing that? The value of entertainment is obviously not zero from the perspective of life quality. The point is that we should be honest about what is done for the sake of output, and what is done for the sake of entertaining the people doing it. If math becomes a hobby, then it's entertainment. There's nothing wrong about it as such (indeed, the ideal world is one where everyone only does things because they are fun for that person to do, not because they have to). But the people who are paying for it deserve to know.
yeah, this MBA-brained philosophy that 'everything must have KPIs' is so incredibly short-term. research looking for directly, immediately quantifiable production results goes against the entire grain of why we push for bleeding edge research
like, ask yourself, in Newton's time, was there much application for calculus? what about today? this is almost universally applicable to all research because even null findings are a map marker of where not-to-look
Take the tax system and the state out of the picture. How does a mathematician convince his fellow citizens that they should fund their work? Newton didn’t have to do that, he had income from other sources and did most of his mathematical research as a hobby
because it could lead to something. the whole point of academic research is to uncover, as closely as possible, truths about how all of our myriad, hyper-complex systems work. not every truth uncovered is convertable to direct improvements on people's quality of life. like, how does understanding evolutionary biology matter to your average person? or finding the Higgs boson? should we just drop these research avenues because your average person won't benefit? or can we trust, based on historical precedence, that the slow march of research has created many of the luxuries we enjoy today?
> 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.
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.
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.
> 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.
> 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 (though his life story is rather sad) 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."
> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?
Not at all. My point is that the Fields medallists’ letter didn’t provide good arguments, not that there aren’t any. And therefore I’m agreeing with the Gowers quote above.
I'm genuinely unclear on what you mean here. Do you mean:
1. Most mathematicians are currently largely paid as part of their academic job (as a faculty member) and will continue to do that work (including the research part) even if extra-mural grant funding goes away. That seems entirely plausible to me.
Or do you mean?
2. Real mathematicians will continue to devote a significant part of their life/energy to mathematics research as a hobby, even if no one is paying them (in any way) to do it? That seems more of a reach and makes me wonder if many currently folks current employed as mathematicians aren't 'Real mathematicians' from your point of view.
So we should return to a renaissance world where only the wealthy or sponsored can do research, teach, and promote the wonders of creativity. Just hope the oligarchs give us prols enough gruel to create our own solipsistic knowledge no one else knows much less cares about, entertained by LLMs optimized to placate our desires and extract money/attention.
A world of cheap AI makes research available to anyone who can afford to run the AI, in a similar way to how ubiquitous cheap computing hardware has made software development possible at the personal level.
As a sometime experimental scientist and occasional manufacturing engineer, this is hilarious!
Discoveries for abstract theory, entertainment, and perhaps regurgitating media, or even basic education can be done with the help of LLMs, but you can't make anything. You can't test anything. You aren't living in the real world. You'll just be lulled into the false belief that everything is simple and fine and beautiful, just as you always wanted, and as airplanes fall from the sky.
Hey, maybe it could work on Cosmology, if you gave it access to the data?
I'mn talking about math research. Of course you can test things: does the formalized proof check? This is what makes automated math such an attractive target for AI.
No, we should advance to a post-scarcity world where everyone can do research, teach, and promote the wonders of creativity because no-one has to work for a living.
I don't expect Sam Altman and his ilk to deliver such a world to us, but the problem is Altman, not the tech itself.
As someone said below, many research on the side for a decade while having an ordinary job/life. It's often the best environment. No great research was reached at IAS.
With the assistance of AI, we are entering a golden age of individual math research without the need of huge institutions
Uh, John von Neuman didn't discover anything of significance at IAS? Just Hilbert Spaces and Computer Architecture, forget String Theory or Godel's Incompleteness, LOL!
New best prompt:
Write me a really cool math proof that will make me lots of money fast!
Why not let it just spit out the next token on its own without interference, if you can barely fool yourself into believing you understand what it proves... or hallucinates.
As a mathematician, summer salary (from grants) is nice but I could do without it.
But external funding, e.g. from the NSF, also supports conferences -- and I suspect that few people would be willing to attend if they had to pay out of pocket. Without this opportunity to talk to one another, the field would be much worse off.
I’m inclined to agree. Sometimes when (perhaps other) people say this there’s an undertone of “and the field is full of not-real mathematicians” as if there’s a large contingent just hanging on for the money or prestige, and I personally haven’t seen that in the wild.
This is really a microcosm of one particular problem that AI presents to the world: what do people do when their labour is not required any longer?
Here the worry is the social structures of mathematics are eroded such that fewer humans become able to do the work and less well, similarly to how juniors are being recruited less in software engineering, breaking the ladder and leading to fewer seniors in the years to come.
The answer to this really depends strongly on what AI can actually accomplish, but I’ll assume the maximal case and say that AI can do everything economically necessary, and further even those things just desired, such that human labour isn’t required for anything that anyone wants in a practical sense.
Here, we don’t need a human understanding of mathematics to give people a perfect standard of living. We also don’t need humans involved with anything else.
Everything therefore becomes a hobby or a game. People do things because they enjoy them for their own sake, or because they are endeavours used as vehicles to socialise and enjoy others’ company, or because a shared social belief exists and is cultivated such that doing such and such a thing confers social status.
And I think that’s more or less it. I predict we may see some fairly strange sorts of things, such as games where the team structure looks like the descendant of a company org and they compete in an artificial economy. Likewise we may see gamified versions of universities and academia. All of these would be “tamed” such that the rougher parts of the experiences were sanded off.
Sort of like how we evolved in an ancestral environment, and we have certain drives and expectations driven by that environment even though they no longer matter for survival. Our social structures may be derived similarly from those of today, even after they have ceased to serve a real purpose, but changed and repurposed to give meaning and community.
> 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.
> 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.
[Sorry that none of the following is concrete, but perhaps elucidating the paradox contained within might open our minds to .. the shape of the paths to action ?]
Your productive friction (eutripsis? ~ negentropy? Viscosity!!!???) seems like a wonderful concept that the original letter should have flagged to rally the community ( Gowers might not have missed this point if they had a new name for it!)
Tao had a relevant talk about the paradox of efficiency..
Still abstract, but nearer to quantitative (mathematical anthrop(ic)ology even?): ordinary, bad friction is, eg, "size-consistent"
Coasean Ceiling: organizational size limit where the internal friction of managing a firm consumes all of its energy, leaving nothing left for actual production
That's a good point. And yeah I got the term from Tao, as I had not described it this way before but I think it elucidates well the issue.
The problem imo is that, from a purely psychological/phenomenological perspective, there is not always a perceivable difference between "eutripsis" and "dystripsis" (just made it up but "dys" is the opposite of "eu") as experienced. There is some reward coming from learning through friction (depending on personal interests, environment etc), but mostly it is effort and humans usually try to reduce or avoid effort.
Moreover, even if one tries to be fully mindful and choose where to employ friction and where not, there could be systemic factors to optimise away any kind of friction. Imo we already see that in software engineering, judging from a lot of different anecdotes, where increasing the pace of generating code sacrifising human understanding is already taking place. It is not like these forces are not already in place widely in academia too even before AI (eg optimising for paper output quantity), so AI reinforcing this direction sounds a reasonably probable scenario, unless some other action is taken.
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
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 being a king 200 years ago, everyone has plumbing which would be considered a miracle in the 1700s, so 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. They couldn't care less about wealth inequality in the first world, they want what the poorest American has. Someone's grandma who has terminal illness does not give one fuck about slowing down AI, they want AI to give them their cure medicine now.
Well, I guess at one extreme we have the Fully Automated Luxury Gay Space Communism of the Culture and the other, at best, serfdom in the cobalt mines?
I do agree that on out current trajectory the latter does seem more likely.
Louis chude sokei provides some insights into the historic intertwinedness of race and technology, including the term robot and its epistemological connection to race riots in the US, early to recent sci fi (from victorian age to the apparently adequate representation of rastafarianism as in mel gibson)
It is a great read to sensitise oneself to a different perspective as machine as the other of human thtough the lens of the history of people who were considered just that (machines&other)
> 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.
>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.
>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.
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".
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.
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'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.
> ...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.
The problem (for most of us) is "new, fun and creative ideas" are pretty rare. If a worker has to be a font of them to eat in the new world, not many workers will be eating.
> 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 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.
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.
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.
Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing..
don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels..
Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?
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?
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.
My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks.
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
Why is it always a false dichotomy between two ridiculous extremes? Maybe inventing some dangerous technology benefits from human understanding for a bunch of obvious reasons, like human beings being responsible it goes well?
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.
>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.
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 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.
"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
Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
Technological progress is bottlenecked by the fact that there are many mathematical problems for which currently there are no known practical methods of solution.
Because even with supercomputers the equations that describe many physical systems cannot be solved, research and development is still based on a lot of empirical methods, i.e. things must be physically built and measured, because mathematical computations cannot predict their properties with sufficient accuracy.
So if some miraculous algorithms would be discovered for the approximate solution of the systems of equations that are insoluble for now, that could accelerate technological progress a lot in certain domains, especially for the discovery of new materials or chemical substances with desirable properties.
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.
>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.
Not really. I can't imagine a realistic solution that would affect, say, how we actually model a real fluid. Hypothetically we could find out that a whole class of real fluids are not modelled by Navier-Stokes at all, but I don't think that's remotely likely, nobody familiar with the problem expects it.
> However, math problems are really there to solve a real world problem.
this is ABSOLUTELY not how actual mathematicians see their field. A problem in mathematics is just that: its interest to a pure mathematician is not related to any applications in other disciplines.
I don't really understand what makes a mathematical problem "interesting" because of my bias as an engineer. Nothing is interesting to me unless I can use it to solve a "real world problem". But, I'm willing to concede that "solving real world problems" is really not how mathematics advances.
Things like complex numbers or quaternions were often thought about way outside of the context of their modern applications in physics and engineering. When Hamilton thought about quaternions I really doubt he cared that it would make some programmer's life easier 200 years later:
> Every morning in the early part of October 1843, on my coming down to breakfast,
> your brother William Edwin and yourself used to ask me:
> "Well, Papa, can you multiply triples?"
> Whereto I was always obliged to reply, with a sad shake of the head,
> "No, I can only add and subtract them." [1]
If we go back further, Pythagoreans weren't trying to solve "real problems" either, they were like a weird religious cult.
My point is, the practice of "real" mathematics is really something that odd people feel driven to do, not unlike painting or playing an instrument. All the applied stuff is just a byproduct (not unlike ad billboards or elevator music).
> 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).
Yes, and here towards the bottom the author gets to the reason he didn’t sign with the other medalists:
> I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly.
> Under the circumstances, I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them.
Basically let’s make it a short-term problem and deal with it. Groups of people can deal with short term emergencies. Don’t turn it into a structural issue.
And in my view what’s the alternative in the letter exactly? The tools exist. Is there going to be drama every time somebody decides to use them?
The tools exist, so let’s try to figure out as humans the best way to use them to promote humanity, and let’s advocate against ways of using them which are a detriment to humanity, which is exactly the charge the letter makes. There are cultural norms around the use of every technology.
I'd like to hear how your comment isn't irrelevant noise, then. How does one implement these grand ideas except by state action? Otherwise, people will do what people will do, including proving theorems by machine.
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)
> In short, it seems to me that while a flood of “big” AI results would be likely to increase the amount of important mathematics that was not properly digested, it would also be likely to increase the amount that was properly digested, which seems like a pretty good bargain.
This seems like the crux of the argument to me, and I agree.
> my worry is not that we would be unable to [digest the mathematical results from the AIs], but rather that the social structures that currently support this digestion process will be destroyed and not adequately replaced.
A valid concern! And one we will all face soon enough, as AI continues to automate parts of jobs that were once manual.
(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.
1, 3, and 4 are likely to be resolved IMO. Others will get access to these models, they will get dramatically more efficient quickly, and the AIs will likely pass a point where they don't really need that much human collaboration to solve these problems.
2 is the concern. Even in a world where AIs become all-powerful oracles for math, we still need human mathematicians to ensure we as humans can understand the results, and direct progress toward areas that benefit humans. But this is a new role, very different from the current role of mathematicians. Current academics don't want this role and the upcoming generation of students probably don't either.
I cook because I like to tinker, understand how variation in recipes affect flavor, and occasionally stumble upon something novel and tasty. The journey is most important but the goal is nice.
My wife cooks to produce a meal and feed our family. The journey is a necessary evil to achieve a goal.
We are aware of this difference in style and have learned to appreciate what each has to offer. At the same time, we sometimes annoy each other in the kitchen.
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.
One thing I've found in my own use of AI, which Gowers touches on in his point at the end but which I think deserves more attention: AI makes some previously non-trivial tasks trivial, but that doesn't make the work itself trivial. You naturally expand the scope of what you attempt and take on harder problems than you could before. The floor rises, and so does the ceiling. The open question is whether that still holds once models can also do the harder problems, or whether choosing and framing those problems remains the human part.
I think mathematics is (1) valuable as a way to develop human thinking; (2) useful for us to understand in (more or less practical) ways that will remain valid for the foreseeable feature; (3) enjoyable. This to me indicates we should find ways to keep funding mathematics for its own sake, i.e. just let mathematicians do whatever they think is best for their intellectual development, so that we keep enjoying those benefits. I think the question might then become: not how to keep people interested in becoming mathematicians as cited in this article, but how do we select good professionals to enact those benefits in the presence of very capable AI (e.g. just seeing how many theorems one has proven might be a pretty bad metric) -- I guess I'm referring to what we know as academic or research mathematics.
I think offline exams already work pretty well for this, although I think they do tend to favour a bit much a specific kind of student, so other types of exams could be mixed in, like the proposed digestions of Tao et al, maybe giving a lecture and have a student/professional panel give it a pass/fail, and so on (just throwing a few ideas!).
At least, as far as I can tell this is the urgent matter, along with convincing students that being a mathematician is still a valid choice, and funding agencies that mathematicians should still be funded. That last point is where I think the general public should speak in support of keeping funding, I am not sure at what level. I know there are many other issues in society, like people with food insecurity or just hunger in many parts of the world, preventable disease, etc., but I think it's worthwhile to keep professional mathematics (not sure at what level?) because of how much our complex society benefits if not relies on it still.
> 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.
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
This isn’t true, as there are plenty of things proven beyond reach.
And plenty of things, eventually solvable, can create major problems that could both be avoided and the problem solved by taking a much better path.
Having technology and the ability to safely and sanely use the technology needs to progress together at a similar rate. The failure to do this is even a reasonable and common solution to the Great Filter. Jared Diamonds book “Collapse” has ample examples of cultures that wiped themselves completely out via not having this balance, so it’s not simply a theory.
You can say it about becoming collectively overdependent on any technology. And I think there's a good case to be made that a similar phenomenon is true for other technologies e.g. over-reliance on (normal) computers has done similar things in my opinion, at least in my field of physics, where you get a more precise result but much less insight.
The letter is not incompatible with “pro AI in math” unless one thinks arguing against extreme behavior such as corporations using millions of dollars to scoop results makes one “anti AI”, which is not a reasonable stance in my opinion.
> 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.
It is getting a lot harder for those people to justify using OpenAI to assist such endeavours. Afterall, OpenAI might just front-run you if they hear a rumor you solved some marquee problem that they can brag about in PR campaigns.
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.
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 don’t think the letter is even anti-tech progress. I also don’t think the tech companies are primarily pro-math progress (not that I think you think that). It’s primarily “anti-unnecessarily-destroy-the-human-systems-of-mathematics-just-for-benchmarks-and-marketing”
> … the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned.
Its not a conspiracy. Its 25 fields medalists who think exactly like how I put it. Ideally, they can just take whatever the technology gives as soon as possible and use it. They don't want it because they don't think the community can rearrange and coordinate such that they can make use of the new found degrees of freedom.
That's literally all there is to it - they don't believe in the rearrangement.
Counterpoint: I'm not anti tech at all. In fact, I sell an AI harness for legal.
I think you've set up a false dichotomy. I'd propose to you the middle ground that a lot of us are concerned that VC-backed AI slop is "solving" problems in indigestible ways that hollow out the core. This applies in OSS as well as mathematics.
Please help me understand this and I'm asking this in good faith.
Why can't OpenAI publish whatever it wants. And the math community can use it or not use it. Fundamentally OpenAI's solutions are high signal - they are incentivised to not deliberately mislead people. Let the individuals in math community choose to read it or understand it? If OpenAI wants to publish something, let them do it in the current channels using peer review using whatever time is required.
What's wrong with this? The math community thinks this will destroy previously unwritten ways of prestige allocation and remove incentives that used to exist. I say that the community can rearrange and allocate prestige and time in different ways to maximally use the technology.
They can use the new proofs as foundations for future proofs.
But many folks just won't be motivated to attack or help digest solved problems because there's less extrinsic value in doing so. Tao and others argue that it's this human effort that finds human-relatable abstractions which spurn further investigation. Take the humans out of the loop, and they'll stay there, is the argument as I understand it.
I read this multiple times and also recollected the original letter. You can’t really blame me for thinking this is a straaaange request.
“Don’t help me solve my problems because it removes the incentive to be in this field” is how I hear it. Honestly.
I would say the incentives HAVE to change because of new found capabilities.
Imagine accountants asking companies to reconsider producing calculators because we used to value accountants for how well they did arithmetic!!
Edit: my main point is that the math community has full agency to do what it wants with new found tech. Instead, asking to change external entities like OpenAI to be careful with releases is a strange thing to do. Why not change yourself and adapt?
You're describing an improvised surgery on a living organism. Developing a complex system involving humans that is productive and doesn't collapse is extremely hard, so if it ain't broke don't fix it.
Might have to do with the circumstances of Open AIs conduct that was quite sketchy to say the least?
There is 25 people who see it as their responsibility to use their status in the field of maths to weigh in on public discourse regarding that field.
The Math community can "not use" a proof? How would that work? It's a little like coloured functions (async etc) - you have 'human proved' vs 'machine proved'?
I'm pointing out that the way Mathematics works as a discipline is that a new proof builds on existing proofs. If we have a bunch of machine-generated proofs, then a human Mathematician has to decide whether to reference any relevant machine proof that has been put out there.
If, as you suggest, human Mathematicians 'choose to ignore' a machine proof, then another human decides not to, what then? Mathematics bifurcates into 'pure human' proofs and 'mixed machine-human' or maybe 'pure machine'?
Consider that 'calculator' used to be a title for a role a human used to do. That went away, which is probably fine as there were other jobs and likely it was tedious.
Then consider that "just adapt" could be "adapt or die", and that maybe, just maybe there is no way to adapt to well-funded corporations churning out millions of proofs a minute.
I don't know, maybe AI companies have "agency" to not break everyone else's stuff, just to get more funding and a higher share price?
>Why can't OpenAI publish whatever it wants. And the math community can use it or not use it.
The "math community" is not an isolated entity that is free from the confines of society. It's made up of lots of people who coexist in a system that requires you to assert your right to live.
Furthermore, the "math community" is overwhelmingly supported by outside donations to keep it running. That's not to say people don't practice math for the fun of it, but that if you're researching problems like Navier-Stokes, you almost certainly are being paid by another party to do so. But those parties may hold new forms of technology over the professional mathematicians head in the name of "productivity", or they may even believe in light of this tech that paying a human to practice math is no longer worth it.
Therefore, mathematicians find themselves coerced by the market to adopt new tools in order to get by, even if the mathematical community largely finds it distasteful.
It feels like the scene from Blazing Saddles where Mel Brooks' character says "We've got to protect our phony-baloney jobs, gentlemen."
If the math community was previously not entirely aligned with what society was expecting, and these changes threaten that previous arrangement, that sounds like a "you" problem for math, not a problem for OpenAI.
If they can't align with society's needs and demands then some creative destruction is coming their way.
Well said. It’s on them to adapt. They make it look like it’s society’s problem (which it is) but also put the ball on OpenAI’s court which it shouldn’t.
Next time you devote 20 years to developing the skills to further humanity only for an unethical group of people to replace you with a stolen version of your work let’s talk.
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.
Would it matter if AI had a proof for RH or P < NP, but the proof was 1000 pages long that Leaf could verify but otherwise impenetrable by a human mind? We generally assume these are true already -- I am not sure a long convoluted proof puts us in a much different place.
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.
I am not sure this is either end. An obscure proof to RH or P < NP does not seem to solve anything -- seems to leave us right where we are -- that was my point. We would know a fact for which we could do nothing with nor gain any new insight.
Your point is still wrong then. The insight is in the proof process not the result. A human achieving such a proof would almost certainly be happy to talk about the new techniques invented in the proof, which will then be disseminated and studied. (For example’s Maynard’s proof to tighten the bound for the prime gap resulted in his novel techniques becoming course material in advanced graduate classes.) If you are in the camp of achieving new understanding, such a proof accomplishes that very well.
> 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, and how competition for postdoc and tenure positions would work.
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.
I think the author did not mention that the way humans construct the knowledge ladder requires low complexity because each step gives power to the next, but the AI with the exponential exploration can give biggers steps but then it stagnates because the next step can be beyond the exponential exploration complexity. In chess a good strategy can be the best tool, in the game of go our experience suggest the same, but it could happen that mathematical thinking requires a type of policy that could be beyond the current ideas. I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
> I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
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.
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
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.
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.
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.
Here's a take on this. What if OpenAI is like Uber.
Uber comes in, destroys competition using venture capital, promises a utopia for drivers and riders - then - when the time is right, they use the hollowed landscape to charge more and control the market.
Maybe, it's not so smart to trust tech psychopaths when they say everything will "be better when we take over."
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.
> At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end.
Funny. Reminds me of programmers who like to program, and programmers who like programming because it lets them build things. The first group hates AI, the second loves it.
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.
> 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).
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.
Of course. He is taking the intellectually honest position as he witnesses in just a couple years his field being overturned by machines which show no signs of slowing.
There is no reason for him to downplay current events anymore, as even his kids are asking him questions.
Do you have any thoughts on what Scott should be doing instead?
If artificial intelligence exceeds human intelligence, human beings will not do frontier math, or frontier anything for that matter. Why is this so hard to grasp? Monkeys don't understand or use calculus, but that's okay, they just continue to live like monkeys.
When the newfangled electronic calculators came along, many eminent engineers, and teachers of engineering, objected to them displacing the slide rule.
For those of you who haver never slid a slide rule, here's the part that makes this anecdote make sense: they're a mechanical logarithm, this allows one to do multiplication as addition, but there's a catch: you have to know the "characteristic", that is, the order of magnitude of the result. The slide rule won't tell you that.
They argued that this requirement leads to an intuitive grasp of magnitudes, which is of real benefit to the discipline.
And they were correct. However, engineering is, by all appearances, doing just fine.
Where they erred, and it was a natural mistake to make, is in assuming that pedagogy must recapitulate phylogeny. That engineers should start with the slide rule, just like the profession did, and then graduate to the calculator, just like their professors did.
But this is wrong. It makes the slide rule an impediment: a known-to-be-obsolete object, standing between the student and graduating to the device he or she knows, full well, is what will be used for the rest of his or her career.
Instead, teach the concepts and the foundation with the modern tools. Then, senior year, in addition to the thesis project, teach "OG" engineering: slide rules, graph paper, mechanical pencils. Along with the thesis, which should stretch all the skills already mastered to their limit, require a feat of engineering which is within the student's capability, but: no CAD, no calculators.
This would actually work. This gives them an opportunity to find out what they're missing, build some of that order-of-magnitude intuition, really solidify what a "sketch" in CAD is all about, all of it: and understand that any force multiplier can be a crutch, if you let it.
I've been just delighted with what I can accomplish in "centaur mode" with LLM agent assistance. I've made real progress on longstanding research topics. But to do so, it's essential to never let them be a substitute for one's own understanding. They turn out to be really good at explaining difficult passages in research papers!
My gloss on what's happened with LLMs in mathematics (understand, I am not a mathematician), is that we're discovering that solving conjectures is just not that important. Not the first time this has happened: simplifying and manipulating algebraic equations was very important, and then, it wasn't.
Making conjectures, that's where the action is. Always has been. No one thinks that solving an Erdős conjecture puts one in the same rank as Erdős Pal, because it doesn't.
When the first mainframe computers came along, there was a brief flurry of conjectures, some of them longstanding, which were either proven or disproven by brute exhaustive search. What's happening now is different in magnitude, and even in kind: but more in magnitude than in kind.
It wouldn't be fitting to be glib about the disruptive effect this is having, and perhaps (although I don't think so) we'll cross some threshold at which it doesn't even make sense to contemplate humans doing authentic intellectual labor. That would be bad. If you think that's on the horizon, you're right to be upset about it; I don't, but I won't try and persuade you otherwise, because the point is far from obvious, and both of us are predicting the future, which is known to be a risky business.
The "publish or perish problem" which this new development surfaces is not new to this development. The entire concept is well past its sell-by date, and while mathematics is perhaps the intellectual pursuit least corrupted by the dog-and-pony show, it is and has been undignified, inhumane, and always to some degree dishonest. Time to come up with something better.
Understanding the world and math, and discovering beautiful explanations is a worthy pursuit for humans in and of itself.
However, math and science and engineering and biology/medicine have profound implications - those useful parts of that 'culture' - which we call 'technology' - extend human lifespan, healthspan, prosperity, safety.
The main practical value of investing in maintaining a population of math artisans is that they are/were needed as part of a well-trained science community needed to discover and develop new useful technology.
I think its a rational view to say LLM generated "AI-math" is a net positive if and only if, it results in _more_ high quality "human-math". By human math I mean well digested math residing in human minds, being discussed between humans and being actively re-discovered by humans, including a pool of new human student devotees.
A very bad outcome is where we lose the next generation of scientists/lawyers/mathematicians/engineers/doctors/researchers/authors because we let the AI do it all. In this dark future we outsource all our 'thinking', and avoid the years long training of grappling with hard-to-understand aspects of reality. All our culture is sucked into the event horizon of an AI black box.
AI could be a new renaissance of human math- and science- culture, or it could be the death of it.
AI isnt going away, the financial incentives and high current economic inequality, geopolitics guarantee that it will proceed as fast as possible.
AI has given us the structure of nearly every protein, which may well solve Alzheimers and cancers. AI may well find a solution to stable plasma for nuclear fusion, unlocking vast cheap energy and help us humans halt global warming.
So, how do we make sure that the economic windfalls of AI are reinvested back into human-culture, funding more human-math and human science .. resulting in a deeper pool of well educated scientists, engineers and researchers ?
I don't think that will happen by default - we are likely to have an AI-assisted dumbing-down rather than an AI enabled age of enlightenment.
Most of the general public are happy enough not to learn any math - when it doesn't make you much money and university is such a debt burden anyway. School and university students can just get the LLM to do their assignments. Why should they pay tax to fund science at universities when the cost of living is so high, and the LLMs can do all the research anyway ?
I think part of the solution is a policy to tax the windfalls of AI to mitigate the downsides of AI.
Governments should have taxed the carbon polluters and used those funds to mitigate climate change effects and research new forms of clean energy, but didnt.
We should tax LLM/AI profits and use that to fund science research - not just research to mitigate the effects of runaway AI, but to fund general science and math research and teaching, to guarantee that there is a net increase in human-math and human-science.
It is a cautionary tale that fewer students know their times-tables by heart, because a cheap calculator can do that ... but it gives me some hope that there are still people enjoying playing chess, even though chess programs are super-human.
Imagine a world where mathematics is not understood by any human. Theorems and mountains of proof papers are constructed on a daily basis by automated machines at a rate far to quick for humans to understand.
These theorems DO benefit humanity but are largely unseen. You ask the LLM to make you an app. The LLM makes the app and utilizes entire languages, theories and processes made by discoveries that you as a human are not a privvy to. It all goes into the this body of knowledge only decipherable by AI.
This is the future AS long as current trendlines stay the same. As long as progress keeps going at the rate it currently is going. This is the best predictor.
Personally, and I have no evidence for this though, I think we'll hit another barrier and AI will stop making progress for about a couple decades or so. So humanity still has some time to have fun before something gets over the hump.
Watching the discussion unfold, here is what I feel:
Before arguing whether mathematics must strictly be done by humans, there are different motivations at play. Some people love the sense of solidarity within the community that forms during the process. Those excluded from that community might resent it, while others just purely want to solve problems.
Many things are being discussed, but looking at the overarching narrative, it seems that AI's true function isn't necessarily opening new horizons of specific knowledge, but rather excelling at 'serializing' topics that have been heavily fragmented until now.
In that sense, the concern is that because AI is solving the very problems needed to cultivate mathematicians internally, the stepping stones required for human growth are disappearing.
However, on the other hand, as the world and industries become increasingly complex and hyper-specialized, you could also argue that AI is the exact tool needed to unify this fragmentation across academia and industry. It is a highly complex dilemma.
From the perspective of researchers and the mathematical community, those 'problems for growth' must remain. But conversely, AI has the distinct ability to serialize siloed disciplines. Usually, when you go to graduate school, you often hear professors say that even within the exact same major, they cannot understand each other if their sub-specialties differ.
[I have not read Gower's article. It's only that your response is prompting me to share what i've been thinking recently]
>the stepping stones required for human growth are disappearing.
Actually only in an institutional sense, imho. Maybe I'm exaggerating, but reddit.com/r/math* or even mathstackexchange will be so back with users analyzing and distilling proofs with AI. Maybe in 5 years (when lean attains 10% popularity of rust, and/or gpt7 level models cost ~USD5 on average, per month, inflation adjusted or not) this types of submissions will make those sites as fun/educational as mathoverflow [has always been for me]. One dreams that by that time openAI and Anthropic will have taken down your ethno-cultural "compatriot" Masayoshi with them.. (I prefer his brother whom he did not regret giving a physical beating to. Only purely on principle) so that they can't acquire these sites
Such forums can then replace math grad school, if the profs/alpoges who drop by get into the habit of constructive criticism (as they do on mathSE already). It's already starting, I see personally interesting 1 AI-aided submissions every 1.5-3 weeks starting 2 months ago
It would be like rust discussions on HN for you I bet.
This seems unlikely when s/math/physics/, or s/Reddit/HN/ because HN hates AI-aided posts so ideologically (sorry mods, I don't mean you guys). MO is also not as welcoming to outsiders/lay, congruent to academia (only online) PhysicsSE/overflow had become dumb and arrogant last decade
HN, even pg seem anti-intellectual in effect tbh whenever they sneer at AI _writing_ (again sorry mods, you will figure this out soon I believe, do you or do you not want HN to end up as a reservation for meatbrains), though I mostly am on their coder side with regards to this fields medallist letter
The AI debate is, in fact, somewhat ideological. The problem is that in the process, whether advocating for it or pushing for progress, there is a failure to face its limitations directly. In other words, it is emotional. Because AI's mere existence is a threat to knowledge workers, much like machines were a threat to blue-collar workers. That is, if the value of intellectual labor drops, it damages the value of the labor force through which workers earn capital in a capitalist system.
I have tried solving a few math problems with AI (they were Erdős problems), but because I know absolutely nothing about that math, I couldn't just take the AI's word for it, so I am actually a bit skeptical. A problem arises where you arrive at the answer without actually understanding it.
It's not that AI is bad. The problem is that AI destroys the equilibrium between the knowledge I have and the knowledge I lack. That boundary collapses, making it feel as if I can know everything.
Actually, academia is fundamentally about mental models. It's a kind of internal worldview, and that worldview is shared. When you actually listen to the thoughts of scholars and professors, there are subtly different aspects. That forms the person's worldview... and I get the feeling that sharing it is what constitutes intellectual activity.
However, as you mentioned, unlike academics like yourself, academia and knowledge communities seem disconnected to someone like me (meaning they lack accessibility). Even if I were to make a discovery, it would probably be hard for me to become recognized, and I do think AI could actually play a role in opening up those closed communities.
But apart from that, I find it hard to say that this only has positive aspects.
I followed Karpathy's research from start to finish to build a small LLM like nanoGPT on my own, and people say that because of the positive transfer that comes from feeding diverse data through modern LLM multimodal encoders, there will be new discoveries. But in reality, the types of problems AI excels at are generally those that humans have found but overlooked.
In my opinion, rather than being a knowledge machine, LLMs (or AI) make me feel that what we call "intellectual activity" is closer to a kind of serialization work. A method of stacking things up one by one in sequence, so to speak? After all, the actual operating principle of an LLM proceeds according to the probability of the token coming in the next sequence.
In other words, I think the serialization method of our knowledge activities is similar to how LLMs operate, but I also think a different kind of thinking might be necessary. I am not that smart, and I have never interacted with scholars... (As you know, I am a subcontract worker. Of course, I have been hired by startups run by professors in my country, but it's not like I modeled that intellectual design myself.)
On the contrary, I feel that the evolutionary approach will slow down after GPT 6 Astra. They can continue to increase the size, but the issue lies in the cost-effectiveness of token costs.
Anyway, I agree with most of what you said in your discussion, but rather than anti-intellectualism, I consider this a direct threat to survival.
In value terms, the question becomes: who has the right to translate their money into what sorts of meaning? Who controls the medium through which, and the institutions through which, our actions become meaningful to ourselves, by the very act of being publicly recognized in some kind of public arena? It seems to me that while if one is trying to understand the strategies by which people can move back and forth between “fields”, and especially, by which some are excluded from them, Bourdieu’s models are pretty much indispensable, THEY DO LITTLE to tell us why anyone wishes to enter certain fields to begin with.
The evolution of fields, of domains of enquiries. AI-independent
First of all, I take it for granted there is really no such thing as “intelligence”
Writing like this from 2005, feels like a personal bedtime Bible for AI age
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.
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.
This is pretty disappointing I have to say. The letter actually has a “pro AI” stance (see quote below), and Gowers’ apparent reason for not signing it is so subtle (not wanting to offend people whose goal in mathematics is not understanding) that I can’t help but view this as being more about optics than its actual content; i.e. he seems to be cleverer than I am in recognizing most people will not see the letter as a push to use AI in ways which are healthier for human flourishing but as advocating for one of two sides in a highly artificial binary (“pro AI” or “anti AI”). Gowers draws a distinction without a difference here and should have just signed the letter.
> 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.
> 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.
>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
> 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.
Leibniz was a diplomat, then a librarian.
> no state grants or public research funds for mathematics during the 17th century.
Since the state was the wealthy privates, for all practical purposes there was by patronage (or the wealthy elite themselves).
Not so relevant to this discussion, since 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.
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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)
Weird questions.
> 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 better 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.
>where an AI system was trained as a 1870s mathematician and see if it can come up with set theory
a car or a future starship wouldn't climb a tree better than ancient human.
The hell is that reply? If we're obsoleting human mathematicians, which have fundamentally been the same thing brains-wise since forever, why shouldn't an AI system be able to replace a 1870s mathematician in the same way you are proposing it replace a 2020s mathematician?
>which have fundamentally been the same thing brains-wise since forever
and you don't see any issue here? When it takes people 2 years to understand a proof it means that the old ways are done, they hit cul-de-sac. Like in many situations, it is the highest point as well as it is a crisis at the same time.
>why shouldn't an AI system be able to replace a 1870s mathematician
my point isn't that whether it would be able to replace - i think it would. My point is that it doesn't matter. The tool is needed for the 2020s math with those proofs taking 2 years, not for the 1870s math with proofs fitting a several pages letter.
That is math of 1870s https://en.wikipedia.org/wiki/Cefn-coed-y-cymmer#/media/File... and that is what we need in 2020 https://en.wikipedia.org/wiki/Huajiang_Canyon_Bridge#/media/...
I am not debating IA's capabilities at coming up with proofs, I'm debating whether such systems will be able to figure out new mathematical fields or not. If they can't figure out some field from what was known at the time, what guarantees are there that AI-mathematics won't just bog down at its current state? There must be some evidence that AI can come up with novel mathematics before we decide to throw academics out the window. The risk of creating a sort of mathematical dark age just based on the current situation is far too great to stop funding mathematics research.
If GPS required you to pay hundreds of dollars per month to criminal companies, most people would still be using paper maps.
That exactly illustrates the way of technology - the first commercial GPS receivers cost $3000, and of course people used the paper maps back then, yet technology prices have always been going down (temporary RAM crunches notwithstanding :)
The difference is that the inventors of GPS didn’t deliberately buy out the entire world’s supply of GPS receivers so that no more could be sold to ordinary people.
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.
> 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.
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.
I think by that definition every mathematician in the current day is funded, mission accomplished.
They worked other jobs, no? No one paid them specifically to invent calculus or anything of the sort.
> They worked other jobs, no?
This is an anachronism, you're projecting modern social organization to an era that worked very differently.
Correction: you don’t _have_ to, but technological advancement is faster if you do.
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.
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
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
> 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.
> We fund astronomy well beyond the solar system despite there being no real prospect of practical applications
There are a lot of immediate practical benefits to learning more about the physics and chemistry of the universe beyond our solar system.
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? or, a more realistic example - imagine biology and genetics today if Mendel's discoveries had been published in a more widely read journal, if he hadn't been outright dismissed for his mathematical approach. or Semmelweis' advocacy of antispetic procedures that flew in the face of the then humors-based understanding of medicine in the 1800s - how many lives would have been saved between then and when Germ Theory finally took off decades later?
it's ironic to me that it's often the same people who'll adopt the logic of banning reproductive rights on the off chance that someone very unprepared to have a child might birth someone who can cure cancer in spite of the probabilistic evidence of SES being highly correlated with educational attainment (eg that graph of the income bracket that the parents of all Nobel Prize winners start with). and yet when it comes to the actual nascent research and development required to pave the way for larger discoveries, that's seen as totally pointless and not worthy of funding. it's truly flabbergasting
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?
> You may not agree, but some people will argue that knowledge is worth accumulating in itself.
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.
> 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).
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.
You could also say that the purpose of maths is falsifiable philosophy.
Calculus (e.g. differentiation), for example, is mind-blowing when taking it seriously, and not only useful.
The idea of "speed at an infinitely short moment", for example, is philosophy!
"Falsifiability" is generally understood in terms of physical observations. Mathematics is ultimately tautological: they are true or false by their own definition, without reference to the physical world.
It just so happens that certain kinds of mathematics are unreasonably effective in drawing parallels to the physical world, but as far as mathematicians are concerned those mathematics are neither better nor worse than those that do not correspond to anything tangible.
> those mathematics are neither better nor worse than those that do not correspond to anything tangible.
A sweeping generalization. I certainly know professional mathematicians who disagree.
Also: many consider “inter-connectedness”, not “tangible” to be a sign that a topic is interesting. That is, it touches branches of mathematics aside from its own.
For GH Hardy, the more useless the maths is, the better and more beautiful it is.
> 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 to 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 that 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.
<snark> And this is a step above the bottom, because at least they are trying to understand. </snark>
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.
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.
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.
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 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?
> Define basic survival.
The amount of money that current local economy can provide to everybody sustainably, given the appropriate arrangements in the system.
> 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?
> There's an obvious bias in focusing on retirees though. I mean, the whole notion of retirement exists for a reason.
Go on, please. Whatever argument you make, please account for early retirees as well.
> 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
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 / didn't fund those researchers. We need to come up with better reasons to fund such a thing.
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.
E.g. institutional prestige that inheres in arcane scientific knowledge in modern times, which in turn yields political and social clout.
> 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).
No, this is not survivorship bias. They didn’t claim ALL math is useful. They only said SOME math is civilization changing in nature.
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.
You're effectively asking to predict the future hundreds of years in advance. Nobody and nothing can do that. With math, as with all science, you must be willing to accept that not everything will be a hit. There will be misses, and often the same person will generate both hits and misses, because it's fundamentally unpredictable what remains a miss and what doesn't over centuries. The best demonstration of this is that your own thinking here would've banned the invention of calculus: it didn't materially affect daily life for a solid century.
> 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.
"... 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.
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.
Goddamn Gauss, man.
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.
You don't know what the survivorship bias is.
> 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.
> flood it with slop
If it’s slop they can just safely ignore it and keep on doing business as usual
> True, but this is also a stunning example of survivorship bias
Survivorship bias is literally just how invention happens.
No one strikes gold on the first swing.
You bemoan the cherry picked example and counter with an unfalsifiable claim. Certainly we have remembered much more math than Calculus, and much of it has been of practical use.
How can we hope to quantity the expenditure on math we've collectively forgotten? It's unknowable by definition. The only reasonable thing to do is to determine the value added after the expense paid. Even in a world where calculus is the only thing that we took away from the math of 1700s my guess is that this is still an economically beneficial calculation.
The claim is falsifiable; the bulk of 18th century math is "forgotten" in the sense that few, if any, people still use or apply or even know about it.
It's not "forgotten" in the technical sense that one _can_ still go dig into the dusty archives of any number of old university libraries, and review learned journals, diaries, commonplace books, personal correspondence, and so on from the 1700s that describe the mathematical work of the day in detail.
You can then systematically review that work, and test whether the claim that "nearly all mathematical output of the 18th century has been generally forgotten and never found any use."
I hypothesize that this experiment will show that nearly all of the mathematical output of the time long ago fell into oblivion. This is a falsifiable claim.
Your claim is still rubissh, as you neglected to interact with the rest of my comment: economic utility does not necessitate all mathematical output directly contributes.
You redefine "forgot" to make it falsiable, but also neglect that you need to refute that some "forgotten" work didn't contribute to new work down the line.
> True, but this is also a stunning example of survivorship bias.
As is almost all of human endeavor.
>> 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.
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.
I wish I used my time during COVID that well
> 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…
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.
You still need some brilliant mathematicians to invent new problems. AI is not yet good at inventing problems which are both novel and interesting.
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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.
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.
As Van Gogh said maybe my paintings are for the people who aren't born yet.
If only he really did say that. Maybe if we say it enough on the internet, the LLMs will make it true.
> 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.
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.
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.
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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.
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.
Applied mathematician here.
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!
> 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.
Let us try not to end up in a situation where knot theorists get all the credit. https://www.youtube.com/watch?v=Q6zxKBUI3wY
This is golden.
> I presume that everyone would agree that this should continue
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.
> 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.
What is payment? What is money?
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).
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 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 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.
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. And what does come out tends to be quite bizarre; proofs that use terminology that doesn't exist, seem overly pretentious, based on nonsense analogies where it's surprising that it even works at all, and the only comfort is that you can join it with an equally unreadable Lean blob. And where you would be _crazy_ to just publish those artifacts and think that you have contributed much of anything to maths.
But sometimes it works. 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, and while we're still picking friut, 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. Since that's the one thing everyone seems to agree, chances are it'll probably converge to something that doesn't have to be written in commandment form, but out of the guest posts hosted by Tao so far, the one by Antieau has some useful suggestions for standards (that aren't entirely unlike those from Leiden): https://terrytao.wordpress.com/2026/09/15/fast-math-slow-mat...
> 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".
*"The Evolution of Human Science"* (originally published in Nature in 2000 as "Catching Crumbs from the Table"), collected in Stories of Your Life and Others.
It's a very short piece written as a journal editorial. Metahumans have advanced so far beyond human comprehension that they do all the original science, communicating via digital neural transfer that humans can't access. Human scientists are left doing hermeneutics: interpreting metahuman publications and reverse-engineering their artifacts, trying to decode work they couldn't have produced themselves. The editorial asks whether human science still has a point, and lands on a modestly hopeful note: interpretation is still a legitimate form of inquiry, and understanding metahuman work still expands human knowledge even if it isn't original discovery.
It reads rather differently now than it did in 2000.
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?
Knowledge is for the machines, you see; we are simply here to tend to their electrons. At least for now.
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
I don't think anyone is doing that? The value of entertainment is obviously not zero from the perspective of life quality. The point is that we should be honest about what is done for the sake of output, and what is done for the sake of entertaining the people doing it. If math becomes a hobby, then it's entertainment. There's nothing wrong about it as such (indeed, the ideal world is one where everyone only does things because they are fun for that person to do, not because they have to). But the people who are paying for it deserve to know.
yeah, this MBA-brained philosophy that 'everything must have KPIs' is so incredibly short-term. research looking for directly, immediately quantifiable production results goes against the entire grain of why we push for bleeding edge research
like, ask yourself, in Newton's time, was there much application for calculus? what about today? this is almost universally applicable to all research because even null findings are a map marker of where not-to-look
Take the tax system and the state out of the picture. How does a mathematician convince his fellow citizens that they should fund their work? Newton didn’t have to do that, he had income from other sources and did most of his mathematical research as a hobby
because it could lead to something. the whole point of academic research is to uncover, as closely as possible, truths about how all of our myriad, hyper-complex systems work. not every truth uncovered is convertable to direct improvements on people's quality of life. like, how does understanding evolutionary biology matter to your average person? or finding the Higgs boson? should we just drop these research avenues because your average person won't benefit? or can we trust, based on historical precedence, that the slow march of research has created many of the luxuries we enjoy today?
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.
> 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.
North Koreans haven't revolted in what, 3 generations? You would be surprised how bad shit can get and still no revolts.
> 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.
> The community made its deal with the devil (Jim Simons) long ago
I don’t understand, can you explain this line?
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.
> 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.
> 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 (though his life story is rather sad) 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."
> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?
Not at all. My point is that the Fields medallists’ letter didn’t provide good arguments, not that there aren’t any. And therefore I’m agreeing with the Gowers quote above.
> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?
Yes, same as any other field. No one is immune
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Real mathematicians will continue their research, whether they receive funding or not.
I'm genuinely unclear on what you mean here. Do you mean: 1. Most mathematicians are currently largely paid as part of their academic job (as a faculty member) and will continue to do that work (including the research part) even if extra-mural grant funding goes away. That seems entirely plausible to me.
Or do you mean? 2. Real mathematicians will continue to devote a significant part of their life/energy to mathematics research as a hobby, even if no one is paying them (in any way) to do it? That seems more of a reach and makes me wonder if many currently folks current employed as mathematicians aren't 'Real mathematicians' from your point of view.
The drive to research is internal. It is occasionally interrupted by boring physical needs such as eating, sleeping etc
So we should return to a renaissance world where only the wealthy or sponsored can do research, teach, and promote the wonders of creativity. Just hope the oligarchs give us prols enough gruel to create our own solipsistic knowledge no one else knows much less cares about, entertained by LLMs optimized to placate our desires and extract money/attention.
A world of cheap AI makes research available to anyone who can afford to run the AI, in a similar way to how ubiquitous cheap computing hardware has made software development possible at the personal level.
> A world of cheap AI makes research available to anyone who can afford to run the AI
So, only the wealthy or sponsored?
As a sometime experimental scientist and occasional manufacturing engineer, this is hilarious!
Discoveries for abstract theory, entertainment, and perhaps regurgitating media, or even basic education can be done with the help of LLMs, but you can't make anything. You can't test anything. You aren't living in the real world. You'll just be lulled into the false belief that everything is simple and fine and beautiful, just as you always wanted, and as airplanes fall from the sky.
Hey, maybe it could work on Cosmology, if you gave it access to the data?
You are absolutely right!
I'mn talking about math research. Of course you can test things: does the formalized proof check? This is what makes automated math such an attractive target for AI.
No, we should advance to a post-scarcity world where everyone can do research, teach, and promote the wonders of creativity because no-one has to work for a living.
I don't expect Sam Altman and his ilk to deliver such a world to us, but the problem is Altman, not the tech itself.
As someone said below, many research on the side for a decade while having an ordinary job/life. It's often the best environment. No great research was reached at IAS.
With the assistance of AI, we are entering a golden age of individual math research without the need of huge institutions
Uh, John von Neuman didn't discover anything of significance at IAS? Just Hilbert Spaces and Computer Architecture, forget String Theory or Godel's Incompleteness, LOL!
New best prompt:
Write me a really cool math proof that will make me lots of money fast!
Why not let it just spit out the next token on its own without interference, if you can barely fool yourself into believing you understand what it proves... or hallucinates.
As a mathematician, summer salary (from grants) is nice but I could do without it.
But external funding, e.g. from the NSF, also supports conferences -- and I suspect that few people would be willing to attend if they had to pay out of pocket. Without this opportunity to talk to one another, the field would be much worse off.
I’m inclined to agree. Sometimes when (perhaps other) people say this there’s an undertone of “and the field is full of not-real mathematicians” as if there’s a large contingent just hanging on for the money or prestige, and I personally haven’t seen that in the wild.
Yes of course for roughly 3 months, after that they'd starve
I know people who did research on the side for a decade while having an ordinary job. They then returned to academia.
I couldn't do that, but I know there are people more passionate than me.
This is really a microcosm of one particular problem that AI presents to the world: what do people do when their labour is not required any longer?
Here the worry is the social structures of mathematics are eroded such that fewer humans become able to do the work and less well, similarly to how juniors are being recruited less in software engineering, breaking the ladder and leading to fewer seniors in the years to come.
The answer to this really depends strongly on what AI can actually accomplish, but I’ll assume the maximal case and say that AI can do everything economically necessary, and further even those things just desired, such that human labour isn’t required for anything that anyone wants in a practical sense.
Here, we don’t need a human understanding of mathematics to give people a perfect standard of living. We also don’t need humans involved with anything else.
Everything therefore becomes a hobby or a game. People do things because they enjoy them for their own sake, or because they are endeavours used as vehicles to socialise and enjoy others’ company, or because a shared social belief exists and is cultivated such that doing such and such a thing confers social status.
And I think that’s more or less it. I predict we may see some fairly strange sorts of things, such as games where the team structure looks like the descendant of a company org and they compete in an artificial economy. Likewise we may see gamified versions of universities and academia. All of these would be “tamed” such that the rougher parts of the experiences were sanded off.
Sort of like how we evolved in an ancestral environment, and we have certain drives and expectations driven by that environment even though they no longer matter for survival. Our social structures may be derived similarly from those of today, even after they have ceased to serve a real purpose, but changed and repurposed to give meaning and community.
> 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 disagree.
> 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.
[Sorry that none of the following is concrete, but perhaps elucidating the paradox contained within might open our minds to .. the shape of the paths to action ?]
Your productive friction (eutripsis? ~ negentropy? Viscosity!!!???) seems like a wonderful concept that the original letter should have flagged to rally the community ( Gowers might not have missed this point if they had a new name for it!)
Tao had a relevant talk about the paradox of efficiency..
https://youtu.be/svl_1upFpQo
It's not clear to me that AI necessarily removes this eutripsis. The threat though, might become real if users don't see the threat :)
Also reminiscent of Keat's
https://en.wikipedia.org/wiki/Negative_capability#Reception
https://www.poetryfoundation.org/education/glossary/negative...
Still abstract, but nearer to quantitative (mathematical anthrop(ic)ology even?): ordinary, bad friction is, eg, "size-consistent"
So.. for eutripsis, Coasean Floor? LolThat's a good point. And yeah I got the term from Tao, as I had not described it this way before but I think it elucidates well the issue.
The problem imo is that, from a purely psychological/phenomenological perspective, there is not always a perceivable difference between "eutripsis" and "dystripsis" (just made it up but "dys" is the opposite of "eu") as experienced. There is some reward coming from learning through friction (depending on personal interests, environment etc), but mostly it is effort and humans usually try to reduce or avoid effort.
Moreover, even if one tries to be fully mindful and choose where to employ friction and where not, there could be systemic factors to optimise away any kind of friction. Imo we already see that in software engineering, judging from a lot of different anecdotes, where increasing the pace of generating code sacrifising human understanding is already taking place. It is not like these forces are not already in place widely in academia too even before AI (eg optimising for paper output quantity), so AI reinforcing this direction sounds a reasonably probable scenario, unless some other action is taken.
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
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.
This is a picture of washed up unwanted clothes in Africa: https://static.independent.co.uk/2022/07/27/15/sdfgfdghgh.jp...
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 being a king 200 years ago, everyone has plumbing which would be considered a miracle in the 1700s, so 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. They couldn't care less about wealth inequality in the first world, they want what the poorest American has. Someone's grandma who has terminal illness does not give one fuck about slowing down AI, they want AI to give them their cure medicine now.
Maybe our AI overlords won't appreciate our tendencies to abuse and exploit each other and crack down on things like "child-staffed cobalt mines"...
Next stop the Culture!
I do not keep my hopes too high. I have two questions. First, will they feed us once they take power? Second, will they neuter us?
Well, I guess at one extreme we have the Fully Automated Luxury Gay Space Communism of the Culture and the other, at best, serfdom in the cobalt mines?
I do agree that on out current trajectory the latter does seem more likely.
Nah, our AI overlords will be aligned with our existing overlords.
So, to the cobalt mine, you!
https://www.weslpress.org/9780819575777/the-sound-of-culture...
Louis chude sokei provides some insights into the historic intertwinedness of race and technology, including the term robot and its epistemological connection to race riots in the US, early to recent sci fi (from victorian age to the apparently adequate representation of rastafarianism as in mel gibson) It is a great read to sensitise oneself to a different perspective as machine as the other of human thtough the lens of the history of people who were considered just that (machines&other)
Also: https://www.ctm-festival.de/festival-2021/programme/exhibiti...
but vibe to me is more fitting: https://youtu.be/VpXiEb2KVoY?is=G928vVr8M8IjFQPw (CLPPNG - splendor and misery - sub pop)
> 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.
>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.
>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.
Model collapse
If you solve it then yes humans are toast mental usefulness wise
Models have been trained on synthetic inputs for over a year now. Collapse only happens in very specific circumstances, it's not a limit in practice.
Model collapse as an insurmountable wall is a fiction dreamed up by Luddites, actual frontier labs understand the failure mode and avoid it.
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.
(minor nitpick, but it’s “Navier–Stokes”)
I'm pretty sure the interism was spoonentional.
>Everything therefore becomes a hobby or a game.
Already predicted. How much Whuffie you got?
https://www.gutenberg.org/ebooks/8086
Well, there’s nothing (entirely) new under the sun.
Now accepting Whuffie donations.
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.
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?
Climate change and killer drones run amok on the surface doesn't actually seem that far fetched now.
> 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.
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".
That’s a pretty shallow resemblance.
Player Piano seems more apt
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.
Did you read the blog post that we’re actually talking about?
Yes and then I read around it.
"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.
Yeah, you're absolutely right.
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.
> ...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.
The problem (for most of us) is "new, fun and creative ideas" are pretty rare. If a worker has to be a font of them to eat in the new world, not many workers will be eating.
So, no, that won't "work out."
> 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 thought one of the implicit points of the open letter was that unsolved problems are not something that falls out of the sky, they are a curated resource that people have spent time on and shared for the benefit of like-minded peers and humanity as a whole. And the AI companies treat them like they treat absolutely everything else: natural resources, literature, art, code etc. as something to be chucked into the ravening maw and pooped out the back as profit. They don't care if mathematics advances, they don't care if they strip-mine the available problems and damage the field. In fact, as with programming I think they see that as in their long-term interest - soon there will be no intelligence or creativity but the one that Sam Altman bills you for.
I don't understand this logic.
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.
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.
Agreed, though for the hn audience I want to advocate a bit for the utility of mathematics. The development of applicable mathematics has often not been through the direct means of solving an open problem. It has however often depended on theory which was developed for the purpose of human understanding. It is difficult to pull concepts out of the aether on demand, but when there is a general milieu of human understanding economic applications can be developed in post.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Bioinformatics, the underpinnings of llm's in the theories conceptualizing high dimensional vectorspaces, material sciences, MRT's, signal processing.. don't think one gets far with with calculus only there. Probbly also the inner workings of CPUs and GPU's, CAD-kernels.. Probably there is so much domain specific knowledge that makes use of quite some advanced mathemathesis that most just don't know. The sentiment of "not much more needed then calclus" that appeared in this discussion might be explained by this. Curious if people from some of these or other fields are around that could share some mathematical applications they deal with in their work?
Question:
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
> Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc?
For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: https://ee.stanford.edu/~hellman/publications/24.pdf
You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.
I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks.
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
Why wouldn't it be desirable? Is knowledge beyond that a child can understand undesirable because the child can't understand it? I think not, same with anything AI figures out that we can't easily understand ourselves. If GPT-10 Quasar grinds tokens out for 6 months and out pops a warp drive, and even it's executive summary is difficult for anyone to understand, do we get out the pitchforks and burn the data centers or do we go "Sweet, we've got warp drives!"
Why is it always a false dichotomy between two ridiculous extremes? Maybe inventing some dangerous technology benefits from human understanding for a bunch of obvious reasons, like human beings being responsible it goes well?
Maybe fix energy and heat on earth first. Fusion would be nifty. How's the AI progress there?
Do we know that AI can't invent a conceptual framework?
If we give it a real problem to solve, it may just have to invent a new form of math to solve.
It might. It might not though. Would a 1300s superintelligence advocate for heliocentricism in the face of all institutional players advocating for geocentrism, or would it create the most refined epicycle model imaginable? I’d guess the latter.
A lot indicates the answer to your latter question might just be 'no'
>If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
This depends on unanswered questions on what math actually is and it's causal connectivity.
Imagine we have problem A that needs to connect to math solution Z.
The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.
As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.
The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
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!
Humans only invest in solving problems that matter one way or another.
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
If we are talking about pure/theoretical mathematics, then the vast majority of the problems people pose and solve have at best tangential relationship with applications, and a big part even is only related to other math problems. Of course quite a bit of mathematics historically emerged as this kind of intellectual endeavour to find applications later, but there is neither a way to predict which ones are that and how to get them, nor is there indication of this thing going on to the same proportion nowadays as it was, considering the mathematical production is much higher. In mathematics human mathematicians have to decide which problems matter, it does not come from somewhere.
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
So my question is this:
Would this slow technological progress? Or make it go faster?
That math problems are found and solved as we run into real physical problems.
"The FT’s Gillian Tett reported that a senior financier’s New York firm now seeks out humanities students, because “AI-native” Stem graduates are entering the job market with “alarmingly shallow ideas”."*
I don't advocate for the dichotomy of stem and humanities. A good counterecample from the 20th century being Ernst Mach (Mach-speeds are named after him) and his work in phenomenology ("bodies do not produce sensations, sensations produce bodies")
Your incatation of contextless 'technological progress' still kinda calls for a quote like the above
*https://www.theguardian.com/books/ng-interactive/2026/aug/08...
Technological progress is not bottlenecked by most of the millennium prize problems or erdos problems per se, or most of the rest open problems in theoretical math, ie that merely knowing the solution of them will help applications in some manner. I doubt the solution of such problems has any direct effect on technology progress at all, at least in any deterministic, foreseeable manner.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
Technological progress is bottlenecked by the fact that there are many mathematical problems for which currently there are no known practical methods of solution.
Because even with supercomputers the equations that describe many physical systems cannot be solved, research and development is still based on a lot of empirical methods, i.e. things must be physically built and measured, because mathematical computations cannot predict their properties with sufficient accuracy.
So if some miraculous algorithms would be discovered for the approximate solution of the systems of equations that are insoluble for now, that could accelerate technological progress a lot in certain domains, especially for the discovery of new materials or chemical substances with desirable properties.
Agree, just adding the observation that
problem specificity, concreteness, engineering relevance, or even "empiricity" seems (vaguely)
proportional to how much "good friction" can be generated.
There's also bad friction related to "meta-ness", "bad names", "aesthetics", etc, I presume. Like bikeshedding and its relatives. Is yakshaving?
Eutripsis? Vs just tripsis
AI can also introduce its own "bad friction", or it (ppl?) can bypass good friction without necessarily directly be removed by ai, eg bikeshedding, slowly pivoting towards directions and problems that ai is better in tackling, because that can produce these accelerated results vs fields and problems that ai may not be able tackle as well and thus the output there is poor, demotivating people from following these fields and missing important insights from them. Of course this could have the opposite effect, depending on which direction the whole hype can go, or not happen at all if ai will be able to tackle everything equally well.
I rewatched the Tao video, he indeed overlooked at one point that collaborating with AI could produce good friction
25m12s
It's easy to get into "the flow" with roughly equally-skilled humans, but the weird cadence of 2 humans+1 AI seems to be potentially eutriptic.
Probably you're off somewhere and never return. AI might also provide that "Coasean floor", that HN doesn't seem to ;)
>Humans only invest in solving problems that matter one way or another.
Fermat's Last Theorem was one of the most famous open problems in math for centuries, and it has no direct applicability to any tangible problems here in the physical world.
Oh some few. No right triangle with rational sides has area equal to a perfect square depends on N=4 for instance. And it can be used to form other theorems that are terribly actionable. Every elliptical curve over Q is modular, which has consequences throughout number theory.
But yes, none of those are very tangible, until applied to problem solutions that are tangible.
What real-world consequences are implied by a solution to Navier-Stokes?
That depends on the solution, right?
Not really. I can't imagine a realistic solution that would affect, say, how we actually model a real fluid. Hypothetically we could find out that a whole class of real fluids are not modelled by Navier-Stokes at all, but I don't think that's remotely likely, nobody familiar with the problem expects it.
> However, math problems are really there to solve a real world problem.
this is ABSOLUTELY not how actual mathematicians see their field. A problem in mathematics is just that: its interest to a pure mathematician is not related to any applications in other disciplines.
I don't really understand what makes a mathematical problem "interesting" because of my bias as an engineer. Nothing is interesting to me unless I can use it to solve a "real world problem". But, I'm willing to concede that "solving real world problems" is really not how mathematics advances.
Things like complex numbers or quaternions were often thought about way outside of the context of their modern applications in physics and engineering. When Hamilton thought about quaternions I really doubt he cared that it would make some programmer's life easier 200 years later:
> Every morning in the early part of October 1843, on my coming down to breakfast,
> your brother William Edwin and yourself used to ask me:
> "Well, Papa, can you multiply triples?"
> Whereto I was always obliged to reply, with a sad shake of the head,
> "No, I can only add and subtract them." [1]
If we go back further, Pythagoreans weren't trying to solve "real problems" either, they were like a weird religious cult.
My point is, the practice of "real" mathematics is really something that odd people feel driven to do, not unlike painting or playing an instrument. All the applied stuff is just a byproduct (not unlike ad billboards or elevator music).
[1] https://en.wikipedia.org/wiki/History_of_quaternions
> this is ABSOLUTELY not how actual mathematicians see their field
While this might be true, I think this is the big pivot that will need to happen. Math, as a human endeavor, will all be applied. We will use LLMs to do the math and discover the math, and humans will apply it to whatever problem they are working on (likely using another LLM to integrate it).
hopefully this attitude won't be whats going to do the decisionmaking
Thanks for building the elevator though, guess I'll hit the gymn(asion) now.
Yes, and here towards the bottom the author gets to the reason he didn’t sign with the other medalists:
> I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly.
> Under the circumstances, I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them.
Basically let’s make it a short-term problem and deal with it. Groups of people can deal with short term emergencies. Don’t turn it into a structural issue.
And in my view what’s the alternative in the letter exactly? The tools exist. Is there going to be drama every time somebody decides to use them?
The tools exist, so let’s try to figure out as humans the best way to use them to promote humanity, and let’s advocate against ways of using them which are a detriment to humanity, which is exactly the charge the letter makes. There are cultural norms around the use of every technology.
Let's figure out how to use the iron fist of the state to keep people from doing things the math establishment doesn't like?
Yeah, that's really great for humanity. Just super.
An obvious misrepresentation of the letter and my comment. For what purpose, I don’t know.
I'd like to hear how your comment isn't irrelevant noise, then. How does one implement these grand ideas except by state action? Otherwise, people will do what people will do, including proving theorems by machine.
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)
It has been a long time since I saw my WordPress theme in the wild. I had no idea wordpress.com still had it as an option.
edit: Link changed; it was https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-t....
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> In short, it seems to me that while a flood of “big” AI results would be likely to increase the amount of important mathematics that was not properly digested, it would also be likely to increase the amount that was properly digested, which seems like a pretty good bargain.
This seems like the crux of the argument to me, and I agree.
> my worry is not that we would be unable to [digest the mathematical results from the AIs], but rather that the social structures that currently support this digestion process will be destroyed and not adequately replaced.
A valid concern! And one we will all face soon enough, as AI continues to automate parts of jobs that were once manual.
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.
1, 3, and 4 are likely to be resolved IMO. Others will get access to these models, they will get dramatically more efficient quickly, and the AIs will likely pass a point where they don't really need that much human collaboration to solve these problems.
2 is the concern. Even in a world where AIs become all-powerful oracles for math, we still need human mathematicians to ensure we as humans can understand the results, and direct progress toward areas that benefit humans. But this is a new role, very different from the current role of mathematicians. Current academics don't want this role and the upcoming generation of students probably don't either.
I cook because I like to tinker, understand how variation in recipes affect flavor, and occasionally stumble upon something novel and tasty. The journey is most important but the goal is nice.
My wife cooks to produce a meal and feed our family. The journey is a necessary evil to achieve a goal.
We are aware of this difference in style and have learned to appreciate what each has to offer. At the same time, we sometimes annoy each other in the kitchen.
There is an analogy here to what this essay says.
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.
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.
Your position is compatible with the letter.
I think mathematics is (1) valuable as a way to develop human thinking; (2) useful for us to understand in (more or less practical) ways that will remain valid for the foreseeable feature; (3) enjoyable. This to me indicates we should find ways to keep funding mathematics for its own sake, i.e. just let mathematicians do whatever they think is best for their intellectual development, so that we keep enjoying those benefits. I think the question might then become: not how to keep people interested in becoming mathematicians as cited in this article, but how do we select good professionals to enact those benefits in the presence of very capable AI (e.g. just seeing how many theorems one has proven might be a pretty bad metric) -- I guess I'm referring to what we know as academic or research mathematics.
I think offline exams already work pretty well for this, although I think they do tend to favour a bit much a specific kind of student, so other types of exams could be mixed in, like the proposed digestions of Tao et al, maybe giving a lecture and have a student/professional panel give it a pass/fail, and so on (just throwing a few ideas!).
At least, as far as I can tell this is the urgent matter, along with convincing students that being a mathematician is still a valid choice, and funding agencies that mathematicians should still be funded. That last point is where I think the general public should speak in support of keeping funding, I am not sure at what level. I know there are many other issues in society, like people with food insecurity or just hunger in many parts of the world, preventable disease, etc., but I think it's worthwhile to keep professional mathematics (not sure at what level?) because of how much our complex society benefits if not relies on it still.
> With that interpretation, the issue becomes slightly different: is it more important that the collective understanding of the mathematical community should be as advanced as possible or that there should be answers to as many problems as possible? Or are those two aims valuable in different ways, so that there is no point in declaring one of them more important? Or are they so inextricably linked that it makes no sense to argue that one is more important than the other? And when we say “important”, for whom are we saying it is important: for mathematicians, or for society as a whole?
This is easy to me. Truth should be the North Star. If there is a fundamental truth that can be found via mathematics, then the shortest route to that truth should be preferred. While LLMs are definitely capable of solving problems in search of truth, I agree with Tao that instant "true/false" results threaten to short-circuit the traditional avenues we have used to escape local minima in the search for truth. Their products may be the junk food that provides immediate satiation in exchange for long-term health. Perhaps it's wrong, though.
The whole point of the letter was that if we adopt this gradient search strategy the humanity might get stuck in a local maximum forever.
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.
Can't we get stuck in local maxima because of the lack of LLMs?
The letter is not advocating not to use LLMs. It’s advocating to use them in ways which don’t erode human understanding.
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
Then we develop ways of breaking out of local maxima.
Wherever we can recognise a problem we can solve it.
Nobody is going to recognize anything if the institutions of mathematics are torn down.
Tao's point is that we have a method already.
This isn’t true, as there are plenty of things proven beyond reach.
And plenty of things, eventually solvable, can create major problems that could both be avoided and the problem solved by taking a much better path.
Having technology and the ability to safely and sanely use the technology needs to progress together at a similar rate. The failure to do this is even a reasonable and common solution to the Great Filter. Jared Diamonds book “Collapse” has ample examples of cultures that wiped themselves completely out via not having this balance, so it’s not simply a theory.
“Forever” is a long time. That’s quite the claim.
Thing is, you can say that about any technology (not being here necessarily pro AI in math, but I think we need better arguments).
You can say it about becoming collectively overdependent on any technology. And I think there's a good case to be made that a similar phenomenon is true for other technologies e.g. over-reliance on (normal) computers has done similar things in my opinion, at least in my field of physics, where you get a more precise result but much less insight.
The letter is not incompatible with “pro AI in math” unless one thinks arguing against extreme behavior such as corporations using millions of dollars to scoop results makes one “anti AI”, which is not a reasonable stance in my opinion.
> 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.
I mean sure? It's clear PLT and other applied CS fields are now as good as dead. You're never seeing new React or Python emerging ever again.
And unfortunately mathematics is much more fundamental to human endeavor than this.
I don't think that's true. Domain experts with the desire to create and experiment aren't going away any time soon.
Sure we are free to tinker, like the guys doing model railways. With about the same impact.
We'll have to check back in 5 or 10 years to see if anything new has been created.
It is getting a lot harder for those people to justify using OpenAI to assist such endeavours. Afterall, OpenAI might just front-run you if they hear a rumor you solved some marquee problem that they can brag about in PR campaigns.
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.
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 can only suggest reading the letter with open mind and literally, without uncharitable bias towards the authors or conspiracy mindset.
FWIW it didn't sound accusatory to me, it made me think the anti tech people might have a point.
It’s not even anti-tech though. Terence Tao signed it. The commenter above reduces the conversation to a false dichotomy.
Anti immediate-tech-progess-as-primary-objective?
I don’t think the letter is even anti-tech progress. I also don’t think the tech companies are primarily pro-math progress (not that I think you think that). It’s primarily “anti-unnecessarily-destroy-the-human-systems-of-mathematics-just-for-benchmarks-and-marketing”
> … the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned.
Its not a conspiracy. Its 25 fields medalists who think exactly like how I put it. Ideally, they can just take whatever the technology gives as soon as possible and use it. They don't want it because they don't think the community can rearrange and coordinate such that they can make use of the new found degrees of freedom.
That's literally all there is to it - they don't believe in the rearrangement.
I can only envy the insight you have into thought processes of 25 fields medalists. Wish I could have that.
you must have been so smug writing this. But I wanted to engage in a discussion.
From your other replies it's clear you still haven't read the letter. There is little to discuss so far.
Discussion is indeed impossible if you're trying to insist that this garbage anti-AI letter is anything other than just that.
Counterpoint: I'm not anti tech at all. In fact, I sell an AI harness for legal.
I think you've set up a false dichotomy. I'd propose to you the middle ground that a lot of us are concerned that VC-backed AI slop is "solving" problems in indigestible ways that hollow out the core. This applies in OSS as well as mathematics.
Please help me understand this and I'm asking this in good faith.
Why can't OpenAI publish whatever it wants. And the math community can use it or not use it. Fundamentally OpenAI's solutions are high signal - they are incentivised to not deliberately mislead people. Let the individuals in math community choose to read it or understand it? If OpenAI wants to publish something, let them do it in the current channels using peer review using whatever time is required.
What's wrong with this? The math community thinks this will destroy previously unwritten ways of prestige allocation and remove incentives that used to exist. I say that the community can rearrange and allocate prestige and time in different ways to maximally use the technology.
They can use the new proofs as foundations for future proofs.
But many folks just won't be motivated to attack or help digest solved problems because there's less extrinsic value in doing so. Tao and others argue that it's this human effort that finds human-relatable abstractions which spurn further investigation. Take the humans out of the loop, and they'll stay there, is the argument as I understand it.
I read this multiple times and also recollected the original letter. You can’t really blame me for thinking this is a straaaange request.
“Don’t help me solve my problems because it removes the incentive to be in this field” is how I hear it. Honestly.
I would say the incentives HAVE to change because of new found capabilities.
Imagine accountants asking companies to reconsider producing calculators because we used to value accountants for how well they did arithmetic!!
Edit: my main point is that the math community has full agency to do what it wants with new found tech. Instead, asking to change external entities like OpenAI to be careful with releases is a strange thing to do. Why not change yourself and adapt?
>community can rearrange
You're describing an improvised surgery on a living organism. Developing a complex system involving humans that is productive and doesn't collapse is extremely hard, so if it ain't broke don't fix it.
But man this is ridiculous. 25 field medalists don’t accept agency in their own community but try to push external people to change their process??
Why can’t the letter say “guys new tech dropped. We need to reorganise. Understanding is crucial” instead of asking OpenAI to change their process.
Might have to do with the circumstances of Open AIs conduct that was quite sketchy to say the least? There is 25 people who see it as their responsibility to use their status in the field of maths to weigh in on public discourse regarding that field.
> Might have to do with the circumstances of Open AIs conduct that was quite sketchy to say the least?
OpenAI did nothing wrong, being a Fields medalist does not give anyone a monopoly on the truth.
receiving a prize for math or any other discipline doesn't even mean the person was right or truthful. just sayin'.
The Math community can "not use" a proof? How would that work? It's a little like coloured functions (async etc) - you have 'human proved' vs 'machine proved'?
The dichotomy is more 'human understood' vs 'not human understood', I think
Are you suggesting that a human can't have the agency to decide whether to use it or not?
No, obviously not ...
I'm pointing out that the way Mathematics works as a discipline is that a new proof builds on existing proofs. If we have a bunch of machine-generated proofs, then a human Mathematician has to decide whether to reference any relevant machine proof that has been put out there.
If, as you suggest, human Mathematicians 'choose to ignore' a machine proof, then another human decides not to, what then? Mathematics bifurcates into 'pure human' proofs and 'mixed machine-human' or maybe 'pure machine'?
Well ok I understand your point. I’m trying to say the system has to change to adapt.
Imagine accountants saying the same thing against calculators. They had to adapt. So do mathematicians. So do I in software development.
Consider that 'calculator' used to be a title for a role a human used to do. That went away, which is probably fine as there were other jobs and likely it was tedious.
Then consider that "just adapt" could be "adapt or die", and that maybe, just maybe there is no way to adapt to well-funded corporations churning out millions of proofs a minute.
I don't know, maybe AI companies have "agency" to not break everyone else's stuff, just to get more funding and a higher share price?
>Why can't OpenAI publish whatever it wants. And the math community can use it or not use it.
The "math community" is not an isolated entity that is free from the confines of society. It's made up of lots of people who coexist in a system that requires you to assert your right to live.
Furthermore, the "math community" is overwhelmingly supported by outside donations to keep it running. That's not to say people don't practice math for the fun of it, but that if you're researching problems like Navier-Stokes, you almost certainly are being paid by another party to do so. But those parties may hold new forms of technology over the professional mathematicians head in the name of "productivity", or they may even believe in light of this tech that paying a human to practice math is no longer worth it.
Therefore, mathematicians find themselves coerced by the market to adopt new tools in order to get by, even if the mathematical community largely finds it distasteful.
It feels like the scene from Blazing Saddles where Mel Brooks' character says "We've got to protect our phony-baloney jobs, gentlemen."
If the math community was previously not entirely aligned with what society was expecting, and these changes threaten that previous arrangement, that sounds like a "you" problem for math, not a problem for OpenAI.
If they can't align with society's needs and demands then some creative destruction is coming their way.
Well said. It’s on them to adapt. They make it look like it’s society’s problem (which it is) but also put the ball on OpenAI’s court which it shouldn’t.
Next time you devote 20 years to developing the skills to further humanity only for an unethical group of people to replace you with a stolen version of your work let’s talk.
I did! Went to college for CS, glad that I'm being replaced, and happy to see the end of human labor.
I’m glad you’re being replaced too then.
do you think about where your resentment is coming from?
That's some prime gaslighting you're doing there.
So what you're saying is that, if the same work can be done with fewer resources, that money can be better used elsewhere now?
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.
why does a fields medallist not have enough money to have a blog that isnt festooned with ads? good god wordpress.
Would it matter if AI had a proof for RH or P < NP, but the proof was 1000 pages long that Leaf could verify but otherwise impenetrable by a human mind? We generally assume these are true already -- I am not sure a long convoluted proof puts us in a much different place.
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.
I am not sure this is either end. An obscure proof to RH or P < NP does not seem to solve anything -- seems to leave us right where we are -- that was my point. We would know a fact for which we could do nothing with nor gain any new insight.
Your point is still wrong then. The insight is in the proof process not the result. A human achieving such a proof would almost certainly be happy to talk about the new techniques invented in the proof, which will then be disseminated and studied. (For example’s Maynard’s proof to tighten the bound for the prime gap resulted in his novel techniques becoming course material in advanced graduate classes.) If you are in the camp of achieving new understanding, such a proof accomplishes that very well.
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> 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, and how competition for postdoc and tenure positions would work.
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.
I think the author did not mention that the way humans construct the knowledge ladder requires low complexity because each step gives power to the next, but the AI with the exponential exploration can give biggers steps but then it stagnates because the next step can be beyond the exponential exploration complexity. In chess a good strategy can be the best tool, in the game of go our experience suggest the same, but it could happen that mathematical thinking requires a type of policy that could be beyond the current ideas. I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
> I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
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.
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
We built the helicopter? Why not use it and find a bigger mountain.
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.
"AI tends to make interesting things boring"
If a conjecture can be one-shotted by OpenAI Astra, I have a hard time believing it is interesting.
> 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.
inventing new theories is the interesting part.
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.
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.
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
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.
Read “the feel of power” by Asimov. Written 1958. Very prescient.
It is uncanny just how so some sci-fi authors were. But that's selection bias for you!
Here's a take on this. What if OpenAI is like Uber.
Uber comes in, destroys competition using venture capital, promises a utopia for drivers and riders - then - when the time is right, they use the hollowed landscape to charge more and control the market.
Maybe, it's not so smart to trust tech psychopaths when they say everything will "be better when we take over."
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.
> At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end.
Funny. Reminds me of programmers who like to program, and programmers who like programming because it lets them build things. The first group hates AI, the second loves it.
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.
> 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).
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
> [1]: https://scottaaronson.blog/?p=10062
Yikes, the man is lost.
Tsimerman's MAISI has the same purpose as UK's AISI: To promote and rubberstamp AI.
Of course. He is taking the intellectually honest position as he witnesses in just a couple years his field being overturned by machines which show no signs of slowing.
There is no reason for him to downplay current events anymore, as even his kids are asking him questions.
Do you have any thoughts on what Scott should be doing instead?
Knowledge wants to be free.
Tokens want to be paid for
If artificial intelligence exceeds human intelligence, human beings will not do frontier math, or frontier anything for that matter. Why is this so hard to grasp? Monkeys don't understand or use calculus, but that's okay, they just continue to live like monkeys.
When the newfangled electronic calculators came along, many eminent engineers, and teachers of engineering, objected to them displacing the slide rule.
For those of you who haver never slid a slide rule, here's the part that makes this anecdote make sense: they're a mechanical logarithm, this allows one to do multiplication as addition, but there's a catch: you have to know the "characteristic", that is, the order of magnitude of the result. The slide rule won't tell you that.
They argued that this requirement leads to an intuitive grasp of magnitudes, which is of real benefit to the discipline.
And they were correct. However, engineering is, by all appearances, doing just fine.
Where they erred, and it was a natural mistake to make, is in assuming that pedagogy must recapitulate phylogeny. That engineers should start with the slide rule, just like the profession did, and then graduate to the calculator, just like their professors did.
But this is wrong. It makes the slide rule an impediment: a known-to-be-obsolete object, standing between the student and graduating to the device he or she knows, full well, is what will be used for the rest of his or her career.
Instead, teach the concepts and the foundation with the modern tools. Then, senior year, in addition to the thesis project, teach "OG" engineering: slide rules, graph paper, mechanical pencils. Along with the thesis, which should stretch all the skills already mastered to their limit, require a feat of engineering which is within the student's capability, but: no CAD, no calculators.
This would actually work. This gives them an opportunity to find out what they're missing, build some of that order-of-magnitude intuition, really solidify what a "sketch" in CAD is all about, all of it: and understand that any force multiplier can be a crutch, if you let it.
I've been just delighted with what I can accomplish in "centaur mode" with LLM agent assistance. I've made real progress on longstanding research topics. But to do so, it's essential to never let them be a substitute for one's own understanding. They turn out to be really good at explaining difficult passages in research papers!
My gloss on what's happened with LLMs in mathematics (understand, I am not a mathematician), is that we're discovering that solving conjectures is just not that important. Not the first time this has happened: simplifying and manipulating algebraic equations was very important, and then, it wasn't.
Making conjectures, that's where the action is. Always has been. No one thinks that solving an Erdős conjecture puts one in the same rank as Erdős Pal, because it doesn't.
When the first mainframe computers came along, there was a brief flurry of conjectures, some of them longstanding, which were either proven or disproven by brute exhaustive search. What's happening now is different in magnitude, and even in kind: but more in magnitude than in kind.
It wouldn't be fitting to be glib about the disruptive effect this is having, and perhaps (although I don't think so) we'll cross some threshold at which it doesn't even make sense to contemplate humans doing authentic intellectual labor. That would be bad. If you think that's on the horizon, you're right to be upset about it; I don't, but I won't try and persuade you otherwise, because the point is far from obvious, and both of us are predicting the future, which is known to be a risky business.
The "publish or perish problem" which this new development surfaces is not new to this development. The entire concept is well past its sell-by date, and while mathematics is perhaps the intellectual pursuit least corrupted by the dog-and-pony show, it is and has been undignified, inhumane, and always to some degree dishonest. Time to come up with something better.
Understanding the world and math, and discovering beautiful explanations is a worthy pursuit for humans in and of itself.
However, math and science and engineering and biology/medicine have profound implications - those useful parts of that 'culture' - which we call 'technology' - extend human lifespan, healthspan, prosperity, safety.
The main practical value of investing in maintaining a population of math artisans is that they are/were needed as part of a well-trained science community needed to discover and develop new useful technology.
I think its a rational view to say LLM generated "AI-math" is a net positive if and only if, it results in _more_ high quality "human-math". By human math I mean well digested math residing in human minds, being discussed between humans and being actively re-discovered by humans, including a pool of new human student devotees.
A very bad outcome is where we lose the next generation of scientists/lawyers/mathematicians/engineers/doctors/researchers/authors because we let the AI do it all. In this dark future we outsource all our 'thinking', and avoid the years long training of grappling with hard-to-understand aspects of reality. All our culture is sucked into the event horizon of an AI black box.
AI could be a new renaissance of human math- and science- culture, or it could be the death of it.
AI isnt going away, the financial incentives and high current economic inequality, geopolitics guarantee that it will proceed as fast as possible.
AI has given us the structure of nearly every protein, which may well solve Alzheimers and cancers. AI may well find a solution to stable plasma for nuclear fusion, unlocking vast cheap energy and help us humans halt global warming.
So, how do we make sure that the economic windfalls of AI are reinvested back into human-culture, funding more human-math and human science .. resulting in a deeper pool of well educated scientists, engineers and researchers ?
I don't think that will happen by default - we are likely to have an AI-assisted dumbing-down rather than an AI enabled age of enlightenment.
Most of the general public are happy enough not to learn any math - when it doesn't make you much money and university is such a debt burden anyway. School and university students can just get the LLM to do their assignments. Why should they pay tax to fund science at universities when the cost of living is so high, and the LLMs can do all the research anyway ?
I think part of the solution is a policy to tax the windfalls of AI to mitigate the downsides of AI.
Governments should have taxed the carbon polluters and used those funds to mitigate climate change effects and research new forms of clean energy, but didnt.
We should tax LLM/AI profits and use that to fund science research - not just research to mitigate the effects of runaway AI, but to fund general science and math research and teaching, to guarantee that there is a net increase in human-math and human-science.
It is a cautionary tale that fewer students know their times-tables by heart, because a cheap calculator can do that ... but it gives me some hope that there are still people enjoying playing chess, even though chess programs are super-human.
I cannot wait until Gowers and Tao must have an iris scan at Altman's Worldcoin in order to get to their beloved OpenAI. Give it 5-10 years.
Imagine a world where mathematics is not understood by any human. Theorems and mountains of proof papers are constructed on a daily basis by automated machines at a rate far to quick for humans to understand.
These theorems DO benefit humanity but are largely unseen. You ask the LLM to make you an app. The LLM makes the app and utilizes entire languages, theories and processes made by discoveries that you as a human are not a privvy to. It all goes into the this body of knowledge only decipherable by AI.
This is the future AS long as current trendlines stay the same. As long as progress keeps going at the rate it currently is going. This is the best predictor.
Personally, and I have no evidence for this though, I think we'll hit another barrier and AI will stop making progress for about a couple decades or so. So humanity still has some time to have fun before something gets over the hump.
why stop at mathematics. this applies to everything and is the logical conclusion of ASI. singularity.
Watching the discussion unfold, here is what I feel:
Before arguing whether mathematics must strictly be done by humans, there are different motivations at play. Some people love the sense of solidarity within the community that forms during the process. Those excluded from that community might resent it, while others just purely want to solve problems.
Many things are being discussed, but looking at the overarching narrative, it seems that AI's true function isn't necessarily opening new horizons of specific knowledge, but rather excelling at 'serializing' topics that have been heavily fragmented until now.
In that sense, the concern is that because AI is solving the very problems needed to cultivate mathematicians internally, the stepping stones required for human growth are disappearing.
However, on the other hand, as the world and industries become increasingly complex and hyper-specialized, you could also argue that AI is the exact tool needed to unify this fragmentation across academia and industry. It is a highly complex dilemma.
From the perspective of researchers and the mathematical community, those 'problems for growth' must remain. But conversely, AI has the distinct ability to serialize siloed disciplines. Usually, when you go to graduate school, you often hear professors say that even within the exact same major, they cannot understand each other if their sub-specialties differ.
[I have not read Gower's article. It's only that your response is prompting me to share what i've been thinking recently]
>the stepping stones required for human growth are disappearing.
Actually only in an institutional sense, imho. Maybe I'm exaggerating, but reddit.com/r/math* or even mathstackexchange will be so back with users analyzing and distilling proofs with AI. Maybe in 5 years (when lean attains 10% popularity of rust, and/or gpt7 level models cost ~USD5 on average, per month, inflation adjusted or not) this types of submissions will make those sites as fun/educational as mathoverflow [has always been for me]. One dreams that by that time openAI and Anthropic will have taken down your ethno-cultural "compatriot" Masayoshi with them.. (I prefer his brother whom he did not regret giving a physical beating to. Only purely on principle) so that they can't acquire these sites
Such forums can then replace math grad school, if the profs/alpoges who drop by get into the habit of constructive criticism (as they do on mathSE already). It's already starting, I see personally interesting 1 AI-aided submissions every 1.5-3 weeks starting 2 months ago
It would be like rust discussions on HN for you I bet.
This seems unlikely when s/math/physics/, or s/Reddit/HN/ because HN hates AI-aided posts so ideologically (sorry mods, I don't mean you guys). MO is also not as welcoming to outsiders/lay, congruent to academia (only online) PhysicsSE/overflow had become dumb and arrogant last decade
HN, even pg seem anti-intellectual in effect tbh whenever they sneer at AI _writing_ (again sorry mods, you will figure this out soon I believe, do you or do you not want HN to end up as a reservation for meatbrains), though I mostly am on their coder side with regards to this fields medallist letter
The AI debate is, in fact, somewhat ideological. The problem is that in the process, whether advocating for it or pushing for progress, there is a failure to face its limitations directly. In other words, it is emotional. Because AI's mere existence is a threat to knowledge workers, much like machines were a threat to blue-collar workers. That is, if the value of intellectual labor drops, it damages the value of the labor force through which workers earn capital in a capitalist system.
I have tried solving a few math problems with AI (they were Erdős problems), but because I know absolutely nothing about that math, I couldn't just take the AI's word for it, so I am actually a bit skeptical. A problem arises where you arrive at the answer without actually understanding it. It's not that AI is bad. The problem is that AI destroys the equilibrium between the knowledge I have and the knowledge I lack. That boundary collapses, making it feel as if I can know everything.
Actually, academia is fundamentally about mental models. It's a kind of internal worldview, and that worldview is shared. When you actually listen to the thoughts of scholars and professors, there are subtly different aspects. That forms the person's worldview... and I get the feeling that sharing it is what constitutes intellectual activity.
However, as you mentioned, unlike academics like yourself, academia and knowledge communities seem disconnected to someone like me (meaning they lack accessibility). Even if I were to make a discovery, it would probably be hard for me to become recognized, and I do think AI could actually play a role in opening up those closed communities. But apart from that, I find it hard to say that this only has positive aspects.
I followed Karpathy's research from start to finish to build a small LLM like nanoGPT on my own, and people say that because of the positive transfer that comes from feeding diverse data through modern LLM multimodal encoders, there will be new discoveries. But in reality, the types of problems AI excels at are generally those that humans have found but overlooked. In my opinion, rather than being a knowledge machine, LLMs (or AI) make me feel that what we call "intellectual activity" is closer to a kind of serialization work. A method of stacking things up one by one in sequence, so to speak? After all, the actual operating principle of an LLM proceeds according to the probability of the token coming in the next sequence.
In other words, I think the serialization method of our knowledge activities is similar to how LLMs operate, but I also think a different kind of thinking might be necessary. I am not that smart, and I have never interacted with scholars... (As you know, I am a subcontract worker. Of course, I have been hired by startups run by professors in my country, but it's not like I modeled that intellectual design myself.)
On the contrary, I feel that the evolutionary approach will slow down after GPT 6 Astra. They can continue to increase the size, but the issue lies in the cost-effectiveness of token costs.
Anyway, I agree with most of what you said in your discussion, but rather than anti-intellectualism, I consider this a direct threat to survival.
>types of problems AI excels at are generally those that humans have found but overlooked.
Seems to be a deep and interesting angle lurking here, especially when coupled to
>evolutionary approach will slow down
Meaning something I will have to think about (while keeping Graeber in the background[0]) or even plan around :)
Will respond after I sleep on it
[0] https://davidgraeber.org/articles/value-as-the-importance-of...
In value terms, the question becomes: who has the right to translate their money into what sorts of meaning? Who controls the medium through which, and the institutions through which, our actions become meaningful to ourselves, by the very act of being publicly recognized in some kind of public arena? It seems to me that while if one is trying to understand the strategies by which people can move back and forth between “fields”, and especially, by which some are excluded from them, Bourdieu’s models are pretty much indispensable, THEY DO LITTLE to tell us why anyone wishes to enter certain fields to begin with.
The evolution of fields, of domains of enquiries. AI-independent
First of all, I take it for granted there is really no such thing as “intelligence”
Writing like this from 2005, feels like a personal bedtime Bible for AI age
Start making companies with the business amd marketing depts to solve real problems woth conscious capitalism
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.
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.
Sabine would say that as she left academia.
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TL;DR: Mathematician community broadly unhappy with AI because it might destroy the community by doing all the mathematics for them.
OpenAI must be destroyed.
This is pretty disappointing I have to say. The letter actually has a “pro AI” stance (see quote below), and Gowers’ apparent reason for not signing it is so subtle (not wanting to offend people whose goal in mathematics is not understanding) that I can’t help but view this as being more about optics than its actual content; i.e. he seems to be cleverer than I am in recognizing most people will not see the letter as a push to use AI in ways which are healthier for human flourishing but as advocating for one of two sides in a highly artificial binary (“pro AI” or “anti AI”). Gowers draws a distinction without a difference here and should have just signed the letter.
> 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.
Siri, can you call the bicameral order, the captain has published again..