Why I didn’t sign the Fields medallists’ letter
gowers.wordpress.com
gowers.wordpress.com
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.
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
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.
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 believe it is possible or have any recent examples? I feel like most of the stuff that could be applied to something like this has probably already been developed 100 years ago. This hope for some miracle math that cures cancer sounds like the thing they tell the government to keep the funding going. Most of modern pure mathematics doesn't look like it has any chance of being that. It's getting more specialized every day with more and more papers on obscure topics being published that 2 people in the world read.
I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.
The stuff powering current tech (incl LLMs), the very foundational math, could have been described in exactly that way when it was new. Number theory, basis of most cryptography, was "pure math" not too long ago. It's only useful in hindsight.
Either way it seems like nobody is in a position to say "this branch of math will surely stay arcane abstract nonsense forever", so the argument for continuing specialized math research comes down to "we might lose out on some cool stuff if we stop", even if most of it will indeed remain useless for a long time.
(Besides, it's not more useless than playing the piano. But that's a whole other type of conversation. Though it shares resemblance if you think about it long enough.)
Sometimes one can be convinced that their mathematics will be useful, too. I've just recalled a rather motivating (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."
Yes, same as any other field. No one is immune
I couldn't do that, but I know there are people more passionate than 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.
With the assistance of AI, we are entering a golden age of individual math research without the need of huge institutions
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.
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!
So, only the wealthy or sponsored?
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.
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.
Different academic disciplines are very different, and most don't engage in the same kind of black-and-white problem solving that mathematicians do, where you either have solved a problem or you haven't.
But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.
We haven't yet figured out what that should be, but I presume that everyone would agree that this should continue. As one possible model, check out this blog post of Terry Tao's, where he gives his own perspective on the recently proved Jacobian conjecture.
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
When computers can solve the underlying actual problem, this sort of work seems likely to rise in value, and be something which a greater number of mathematicians engage in.
My vote is for interpretive dance. Give the laity something for their money.
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.
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...
That's like where chess was when Deep Blue was built by IBM. Productivity improved. There was someone complaining on here recently that the seat-back entertainment system on some airline had a chess program set to "trounce all humans".
I would not presume this at all.
Every brick in the house you live in has been put there by a worker. The food you eat has been cultivated by a farmer. etc. etc.
You must explain what you give back to these people. It's fine if it's in a roundabout way, but it can't be nothing.
And the presumption isn't that there's something. The presumption is that there's nothing, and you must prove there's something.
Ha-ha. Sorry, but it is not "these people", who is paying to the mathematician. Government is doing it. So they must explain to the government what they cold give back to the government. Usually it is loyalty and the use of their social position to confirm the correctness of the government’s actions and political programs.
Goverment is extracting taxes from these people "for their own benefit" and giving it to the mathematician. When a farmer farms the food of the mathematician and sells it to the mathematician for money, he is simply earning back money that already used to be his.
Which is all well and good -- if it is actually in his own benefit.
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
Where I see models having a huge impact is in simulation code development.
One blocker for years now has been the adoption of GPUs. LLMs can fairly successfully and very quickly port to GPU and suggest/implement useful optimisations. Once it's verified, a code can go from anywhere between 2x to 1000x faster (mainly because CPU codes are so poorly optimised). Some science can reach much greater problem sizes, while some can run the same problems in hours rather than months and both can be revolutionary. Even more than that, LLMs seem to be finding fundamental performance bugs in both open and closed source core libraries so there's a bit of a whole-ecosystem uplift.
Can't comment on the more theoretical, less computational applied maths impacts!
Outsourcing understanding, teaching, and proving maths to LLMs is just as dumb as outsourcing food production, manufacturing, or entertainment to a foreign power. It's not that we haven't already done most of those things in the pursuit of temporary profit optimization, but each of them has obviously bad impacts on both the individuals in society who perform those tasks, and on risks to everyone when that outsourcing fails for any reason.
As described, using LLMs is like navigating with GPS. It's fine when you're using it to optimize a path due to traffic conditions, but it can be deadly if you're flying a plane and suddenly don't have it, or any training to deal with that situation. That is where we're headed. I don't really have a take on signing the letter in particular, because as mentioned it seems a bit pointless. At the same time it also seems a bit pointless to spend $10s of millions proving a conjecture a few months sooner than a group of mathematicians already were. I don't imagine they were going to make millions doing it. The definition of material waste.
Perhaps it serves a temporary marketing win for OpenAI, or a medium term improvement in LLM performance on some productive output, but longer term it certainly risks ceding whole swaths of human endeavor to a tool we may not always have. Just as falling demand for farmers, or skilled machinists, or writers, or artists is not an immediate crisis, in the long term there is no one left to transfer the knowledge to the next generation, if something goes wrong.
Most people do not use any math after school in their lives. All of their real problems are political. Kids spend significant time in school being told that math teaches them "problem solving" and then leave school and discover, since math solves no political problems, math doesn't solve problems at all. In other places around the world, where by many measures people are much more numerate than Americans, people are actually poorer and live under more authoritarian governments. In industry, math has been reduced to a shibboleth for a neurodivergent lack of ethics, which is why mathematics PhDs go into banking, and never politics - and we imagine 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.
I don’t understand, can you explain this line?
This has happened numerous times in the past, and it will happen over and over again.
What's interesting to me is that this can be completely true (I believe it is), but why can seemingly nobody see the value and prioritize saving it. Opera is beautiful, it's one of the many things that make life worth living.
More efficiently allocating capital ought to be just a way to generate more resources for producing opera. And yet...
imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
all maritime engineering and trade was done with geometry and arithmetic at the time. there were no practical applications, not for likely at least a century until hydrodynamics were incorporated into shipbuilding
now look at today. how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
there's no KPI to be derived from any academic field of study at the bleeding edge of theory. theoretical underpinnings lead to practical applications much further down the line after many paradigm shifts
semiotics and cultural capital as theoretical concepts is another example - at the time they were purely seen as navel-gazey literary theory work. these days, half a century later, they're in wide use (for better or worse) in marketing and advertising - they birthed the whole concept of 'branding'
not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
It's difficult to draw the conclusion that "every possible branch of learning ought to be funded" by appealing to "later practical value" based on this cherrypicked example.
On the other hand, clearly _some_ novel theoretical work with no apparent immediate value _does_ yield real world benefit later.
Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
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.
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.
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.
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!
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.
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.
Survivorship bias is literally just how invention happens.
No one strikes gold on the first swing.
Really? Like what?
edit: this is either an unfalsifiable claim, because by "forgotten" you meant here is no extant knowledge or remaining record of it, or it's almost certainly nonsense and anything you could cite would be foundational to some area of modern mathematics, even if as a disproven counter theory.
Now, it's possible that Fermat thought he had a proof of his theorem, but that it would have turned out to be wrong. That happens sometimes. But he did not write it down on any document that has survived to this day, so we know that he had something that has been lost.
It's similar to history. We know that there are books, plays, etc. that used to exist, but that nobody knows today. Because other writings that have survived to this day quote from them. But beyond the quote, we don't know anything else about the play or the book or whatever.
Fermat's Last Theorem almost certainly isn't the only case of mathematical theories that we know were published somewhere but that we have no record of today. It's just the only one that I happen to know about, not being a mathematician myself. I'm sure there are some people on HN who know of others.
Are you proposing that humanity should rely on AI 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.
We already have examples of politicians relying on chatbots to understand problems. I wouldn't be at all surprised if, given a few more years, the above will be effectively the case without any deliberate effort.
As is almost all of human endeavor.
The direct use of that math to biuld better weapons. That math was/is essential to the development of modern artillery. The first tasks assigned most early computers were to calculate ballistic trajectories, and also tide tables which also have immense military applications. Math was and is a weapon.
1. You may not agree, but some people will argue that knowledge is worth accumulating in itself. We fund astronomy well beyond the solar system despite there being no real prospect of practical applications.
2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
There are a lot of immediate practical benefits to learning more about the physics and chemistry of the universe beyond our solar system.
can you imagine chucking all of computer science away because Babbage's thinking machines and Lovelace's programming cards didn't really result in meaningful capital gains here in FY26Q3? 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
In saying that, in the spirit of a good faith argument, is there value in temporarily deprioritising pure research in (current) times of funding drought?
That is totally not true. I am sure you can hardly name even one of those "curiosities" that later turn out to be useful and not "free" (i.e. if not existed, wouldn't be invented on the spot as a tool for solving those particular problem that they end up solving).
Granted for the sake of argument. But our AI overlords are really good at accumulating that knowledge, and very patient in explaining it to us.
That's a different argument than "knowledge ought to be accumulated for its own sake."
2. is unclear; the economic funding model of the 18th century was very different. At that time, there were very few academics, and nearly all were independently wealthy and self-funding. That is, they chose to spend their own money on mathematical curiosities to amuse themselves.
Now, in modern academia, we are discussing the case that public tax money ought to be appropriated and allocated to pure knowledge / curiosity discovery _at scale_, for work which -- by definition -- has no known application or use. This is a very different economic and ethical proposal. This is no longer the question of how one rich man chooses to spend his own money; we are discussing the social application of common resources -- compulsory tax payments for all.
It's easy to imagine other ways to spend that tax money that have much less debatable social value -- how about free healthcare for all? Better roads, or high speed trains? No famines? Flood control? There are a million other ideas with immediate benefit.
I appreciate the value of pure knowledge as much as anyone, but it's also hard to make the case that it should be pursued indefinitely, whereever possible, and without limit or boundary when so many competing priorities exist. It's a question of competition for very scarce resources, versus unlimited wants. How do we decide appropriate allocations?
Most of these aren’t even new inventions; read about the working conditions in the 19th century.
We have natural experiments of this: retirees, both early and otherwise. Do retirees during the first five years of their retirement "find productive work voluntarily" (your choice of words, not mine) in quantities that are comparable to what they did at work during the five years prior to their retirement? Not exceptional individuals, but on average.
I retired circa five years ago and spend my time doing stuff like walking the dog, drinking coffee and reading books, much like other retirees. My productivity is as close to zero as I can muster, and it is the rule, not the exception.
But also, in all honesty, if we can feed and shelter everyone on the planet with only a small proportion of the population engaged in productive labor, what's the problem with most people walking the dog, drinking coffee, and reading books?
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
I see this also among my early retiree friends. People are quite happy just "playing" - gardening, joining a book club, making art, etc.
And I would be, too! There are times I enjoy doing bigger work, but man, I could build quite a delightful life living like a hobbit.
Define basic survival. Tantalum mine worker in the Congo who earns 22 cents per hour and raises his three children on that money, is he under "basic survival" level or below?
And to be clear, as a part of a society, are you ready to forgo your salary, except maybe 1 dollar per hour, to provide to other people? Why haven’t you done it yet?
The amount of money that current local economy can provide to everybody sustainably, given the appropriate arrangements in the system.
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.
The work that Newton and Leibniz did was in formalizing it and coming up with the notation that would end up being more widely accepted and applicable. It's very likely that the techniques would have been developed later to solve a practical problem.
On how to select what to fund in the future: find the brightest minds, fund whatever they want to do. That tends to work out in aggregate
It would have been worth what? Doing mathematical research at all? Nobody is suggesting we shut down mathematical research.
It doesn't justify a funding system that didn't exist / didn't fund those researchers. We need to come up with better reasons to fund such a thing.
Do we?
For most of us, it seems like the EV (expected value) from such funding is quite positive. The work pays for itself in the long run. That's justification enough.
But I would suggest replacing "seems" with the best numbers available.
I know there's research on it for applied and basic science in general, and the rewards were massively EV+. But I can't recall whether similar research covers applied and basic mathematics research.
No, it's an example of why you have to allow people to pursue what at the time look like "curiosities", even though most of them don't go anywhere--because the very small portion that do go somewhere, end up changing the world, and we don't know in advance which ones those are going to be.
It's quite true that the funds we have for this are a finite resource. But that doesn't mean that "foreseeable practical applications" is a useful filter for how to deploy that resource.
>useful filter
There's also "useful friction" as _almost_ imagined by Tao here
https://youtu.be/svl_1upFpQo?t=25m11s
We can't know in advance, but something like the "weird cadence" that arises when 2 skilled humans try to do a threesome with an AI..
could be reified into a local measure (eutripsis) of how likely advances (conceptual or otherwise) are in the midterm
Of course someone was. Newton and Leibniz weren't living in autarky an some island. They had an income stream that was taken away from somewhere else and given to them so that they could afford working on their pet projects.
The fact that it's not publicly funded doesn't change the fact that there's still resource allocation happening.
This is tautological, “mathematician” being defined in the modern world as someone paid to do mathematics…
Notice that the modern world is able able to sustain significantly more mathematicians than when “mathematicians” used to mean “someone well-off enough to work on mathematics in their ample amount of free time without worrying about their subsistence”.
Note that I'm including all the ones that a normal definition would not call paid, but your definition does.
But yes when a definition declares that everyone doing a thing is "funded", it does get pretty tautological and get in the way of arguments for actual funding.
This is an anachronism, you're projecting modern social organization to an era that worked very differently.
Nearly all schools at the time were not subsidized by society and required substantial tuition payments. Most children had to get jobs. School was rare.
Newton's parents, while not extremely wealthy, could afford to send him to expensive private schools, instead of requiring him to get a job at age 8. This was a prerequisite for entering Cambridge.
His early career was subsidized by his upper middle class parents and inherited capital, not by society at large. He became a professor at age 26, and was paid a salary in exchange for work after that.
Neither was fabulously wealthy, but they were both able to bootstrap their academic careers with inherited money until they became famous enough to receive patronage.
"Individuals choosing voluntarily to spend their own money researching mathematical puzzles and curiosities and pure knowledge" is a very different ethical proposition than modern academia, which depends heavily on compulsory tax payments levied on everyone.
For the kind of research where benefits accrue to the inventor (or her employer) and others can be excluded so that the benefits don't 'spill over', then private companies can fund it.
For the kind of research that spills over, you can just let the tax payers of that other country foot the bill. Eg the US can free-ride on Chinese research, and if having mathematicians in the population is good, the US can offer green cards to whatever has a math degree from a good enough university.
There might be some intermediate research that has just enough spill over that a company won't do it, but a country might, should be a rare creature. And differently sized countries should have different sweet spots: some companies are bigger than some countries after all. But I don't think we see different countries select the research (mathematical or otherwise) with an eye towards exactly tailoring spill over.
Ah yes, lets take the existing system that's already barely holding up and flood it with slop. What could go wrong?
> Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Do away with the perpetual uncertainty of the "will we or won't we continue to fund you" grant treadmill for practicing academics. Hold a yearly competition of academic prowess and intelligence open to any adult US citizen under retirement age. Award the winners a grant good for 40 years. If we allocated slots equal to 0.005% of the population each year (ie ~17k) that would represent 0.2% (ie 1 per 500) of the population in total at any given time.
Obviously I say that (mostly) in jest but clearly there are workable solutions if we approach things from a new angle instead of determinedly clinging to the status quo.
If it’s slop they can just safely ignore it and keep on doing business as usual
It's not “survivorship bias”, it's merely the illustration that mathematics are a strong link problem: it's the strongest result that determines the impact of the field (not the weakest, like in weak-link problems).
Today, science funding comes out of the tax collected from everyone. The taxpayers want an explanation for how their money is spent. It could be, of course, vanity, just like it was before (taxpayer money is spent on sporting events, for example because people root for the athletes who represent them)... but, maybe there's a better way?
This is not the full story, but fits today's cartoon history caricature. In reality, there were also salaried employees who were teaching or working on developing practical applications, there were church fundings, the universities like Göttingen etc. Not all were particularly rich, Euler came from a modest background etc.
That's why if you look back at scientific publications or organizations, a lot of them have a "royal" in their name "the royal society of this", "the royal college of that". This is because of their sponsors.
Since we are talking about Russia here, basically, until Alexander II (second half of 18 hundreds), there weren't any government institutions that could, in principle, manage the tax collected from the royal subjects and had to report and justify their activity. Until then, by and large, all the tax collected belonged to the crown and the crown was free to do with it however their heart desired. The initial ministry of education created by Alexander I (early 18 hundreds) was the first step towards transparency, but it wasn't much.
If you are interested in the reasons: both the liberation of the peasants and the defeat in the Crimean campaign created a large hole in the crown's finances. If, until then, the crown disbursed some undisclosed amount to the minister, who would use the money however he wished (mostly stole), the changes in the 1860s made it harder to steal (the minister had to report the expenditures). One could say that at this point, the current approach to financing the education began. A lot of things, however, were still subsidized directly by the crown or various elites, rather than in centralized government-sanctioned way. Another large reform, that made the system more transparent and rule-based followed in 1904, but since the Communists completely reorganized the government in 1917, it's hard to tell what the results might have been. It's only really since 1917 that education and science became exclusively financed by the tax and not through private donors.
That is a pretty weak point. Calculus from the very beginning was an applied math.
> how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
Idk, about zero? I mean almost all of our modern technologies could exist perfectly fine even without previous decades-worth refinements and extrapolations of math theory. All the math tools that they would be needed would be created on the spot if they are required.
And Newton's calculus is rather an argument that is support that view: it is not like there was calculus, that had make describing of the world possible. No calculus was born as a tool to describe physical world from the very beginning.
They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.
Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.
Leibniz was a diplomat, then a librarian.
Since the state was the wealthy privates, for all practical purposes there was by patronage (or the wealthy elite themselves).
I'm another part of the general public, and I accept their justification.
We can all vote, or use whatever levers our political system gives us to show our acceptance/rejection of basic research, but IMO basic research is absolutely necessary.
Assuming this is true, how did they justify their work, which had, at the time, no practical application?
So... network effects... same way most people get jobs, these days.
Art got patrons, too - what practical use is it?
If AI can bring forth mathematical discoveries much faster than academics, wouldn’t it behove society to use AI? It seems the me the tradeoff here is the benefit of all vs the ego of a few. I empathize with the struggle many math PhDs must be going through, and maybe I don’t fully understand the tradeoff as they see it, but I’m not convinced by the letter written by a few career, tenured academics.
Easy yes: Let's impose a tax on everyone to support at least one mathematician.
Easy no: Let's impose a tax on everyone to support one billion mathematicians.
Where are you going to draw that line? How are you going to convince a majority of voters that you're drawing it at the right place?
Do we support people who aren't using vaccines and are intentionally infecting themselves directly (measles parties)?
Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
With civilization and technology we do loose some mental capabilities - like say the part of the brain Amazonian people use for extremely skillful face recognition we do use for reading/writing while loosing that face recognition skill level that those people in the jungle have. So goes the medieval way of doing mathematics (it is a pity though - I liked it and hoped that mathematics will fall among the last, yet it happened to be among the firsts)
> Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Yes. We do - it's a sport from way back and ongoing - a good many people want to retain non GPS navigational skills.
> Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
Yes. We do. Proficiency in multiple weapons and non weapon combat is a basic part of service training. Axes and wedges still have their place in tree felling.
a car or a future starship wouldn't climb a tree better than ancient human.
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/...
Edit in response to the comment below as i hit my post limit :
New mathematical fields require many years of work by many people. What seems to be coming is people would be able to explore various ideas in say days instead of those years. It means more fields will be discovered and built up. And by wider array of people, ie. by the less capable than the top dog people. Mathematics will be more like engineering. Including more practical applications - like exploring transformers trajectories mapped into manifolds in p-adic spaces and applying the dynamic systems chaos theory there - borderline gibberish that no human would spend meaningful time on, yet a one would be able to throw a couple bucks at Claude++ for it to play with it. That would mean more money into the math, not less.
We draw the line, where the voting public, and elected representatives decide. Our political system is squishy and doesn't require upfront reasoning, or absolutely correct decisions.
We often make wrong, sometimes even horrifically wrong decisions, and adjust later.
The public is generally bad at debates around taxing and spending, sadly. Voters struggle to accept the idea that more of X means less of Y. They want to believe they can have it all, for free, and there are plenty of politicians willing to tell them they can.
This.
And ironically enough, the obsession for measurable successes at all cost is what drove 20th century communists regimes to their most catastrophic failures.
Thinking that mathematicians are now useless because they cannot continue publishing results that an AI couldn't is akin to Mao's claim that “bourgeois” intellectuals were worthless because they weren't busy driving agricultural yields up. We know where that ended…
This is all they were receiving funding for previously? Nothing has particularly changed there.
This is blinders on thinking. The future we are looking at with the help of AI is full of abundance. Money is not an issue in that world.
Why would you think that, when the gains of productivity increases haven't been distributed uniformly during the past several decades? What could disrupt the current trend towards more concentration?
And in this case it can go as far as "y'all can starve, we don't need you anymore". Which is perfectly true from a pure capitalist point of view, but also I can't think of a better way to get people to move on from Flock cameras to bigger things.
The only question is, how bad it will get before society snaps. And you can already see many prominent figures among the tech elites promoting UBI as a palliative - basically giving people just enough of the wealth that they won't revolt. And it would even work if they kept it up, but, again, greed always wins over prudence in the long term with these people.
Mathematicians receive almost no funding, as it is pennies coming from the already anemic NSF budget. I don't even know why it's a question as to why liberal arts needs more funding. They don't get any to begin with!
And anyway a mathematician's lifetime salary is basically nothing. I think total worldwide mathematics budget is less than a tenth of a percent of GDP. Meanwhile, openai cumulative loss stands at tens of billions?
It's clear of Fermat's last theorem. But I figured that the Millennium Prize Problems (of which there are only 10) would have some real spillover into either physics or chemistry.
Yet when one of them gets solved, it seems to be that the argument is, "yeah but not like that".
Imagine if an AI solved cancer, or a vastly cheaper way to get into space, we wouldn't react like that.
Let’s be honest, Gauss and Euler are a statistical phenomenon.
Mathematicians getting funded only do 0.0001% of the leeching that retirees do.
The reason why you want, as a society, to have a pool of human mathematicians sitting around solving problem nobody's asking but them, is to have this pool of people who understand math deeply enough to teach a digestible version to all your engineers and physicists (who are the one who actually use math for actually productive stuff sometimes) and to have them nearby to help if a physician or engineer has a question about the math they are using. And it's through these two means that over the long time a tiny fraction of mathematical progress ends up in the actual world. And it's fine because many thing can end up depending on a single piece of progress that happened 80 ago (for instance without the theory of numbers, there would be no modern cryptography, but at the same time most results of the theory of numbers will never be useful in any way).
And I don't think we have good reasons to believe an AI is going to be a good replacement for these two just because it knows how to solve well-studied hard problems.
More generally, I think the history of the Cultural Revolution should make us very reluctant to try accessing the “usefulness” of certain groups of intellectuals based on their measurable first-order output.
> 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?
Yeah, that's really great for humanity. Just super.
Is someone knowlegeable on research that engages the question on the development of something that could be considered as some emergent mechanism of memory, that is independent of instances and their contextwindow an specifically also independent of the use of conversations for trainingsdata?
*(This is regarding the value of selfhosted models - maybe small selforganisations that share ressources to do though to do so? Generally the value of machine learning seems to be to big to reject)
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
I have in mind GPS, cryptography, numerical fluid simulation, lasers, etc…
Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc? Or did we encounter a real physical problem, then we found that someone had done some theoretical math before that would be useful for this application? If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?
Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.
If we give it a real problem to solve, it may just have to invent a new form of math to solve.
This depends on unanswered questions on what math actually is and it's causal connectivity.
Imagine we have problem A that needs to connect to math solution Z.
The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.
As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.
The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.
For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: https://ee.stanford.edu/~hellman/publications/24.pdf
You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.
I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.
I also disagree that none of them solve "real" problems. They clearly do. Solving them have implications on real world problems.
Solving "real" problems in theoretical mathematics (as in problems directly related to applications) is a very small proportion compared to the vast majority of math work that does not. So if we are discussing about the future of mathematics as a field, we have to understand what the field of theoretical mathematics is actually about.
Would this slow technological progress? Or make it go faster?
That math problems are found and solved as we run into real physical problems.
In fact, the relationship between theoretical mathematics and "real physical problems" is bidirectional, as in "real physical problems" informs to some degree some problems that may be interesting to research on in theoretical math, and at the same time pure mathematical research that is developed completely independent may find applications at some point. And even theoretical mathematicians working close to applications are mostly dealing with problems not directly addressing applications. Eg maybe they study properties of a certain function that arises often in application without any view to solving a specific "real physical problem" with it, and somebody after may find that useful for some application after some point, but that could be one out of 50 papers (random number) and it is hard to predict that. There is of course some work more related to specific real problems, but that's most often not what theoretical math is about, and not what these new developments with erdos problems, navier stokes etc are about.
So what could (in a chaotic sense) have effect in application is mathematical theories developed along the way of solving these pure math problems, which brings us back to the question of what happens if we remove this friction and if AI can do more than construct examples and proofs, ie actually build theories (autonomously or humans+AI). If anything, it is through building theories that mathematical progress germinates applied sciences, as this is the process that develops mathematical tools that can be taken up later, including whole mathematical fields. Building mathematical theories is a heavily social process, and it is the community that basically decides which directions are important to follow.
problem specificity, concreteness, engineering relevance, or even "empiricity" seems (vaguely)
proportional to how much "good friction" can be generated.
There's also bad friction related to "meta-ness", "bad names", "aesthetics", etc, I presume. Like bikeshedding and its relatives. Is yakshaving?
Eutripsis? Vs just tripsis
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 ;)
Because even with supercomputers the equations that describe many physical systems cannot be solved, research and development is still based on a lot of empirical methods, i.e. things must be physically built and measured, because mathematical computations cannot predict their properties with sufficient accuracy.
So if some miraculous algorithms would be discovered for the approximate solution of the systems of equations that are insoluble for now, that could accelerate technological progress a lot in certain domains, especially for the discovery of new materials or chemical substances with desirable properties.
I don't advocate for the dichotomy of stem and humanities. A good counterecample from the 20th century being Ernst Mach (Mach-speeds are named after him) and his work in phenomenology ("bodies do not produce sensations, sensations produce bodies")
Your incatation of contextless 'technological progress' still kinda calls for a quote like the above
*https://www.theguardian.com/books/ng-interactive/2026/aug/08...
Fermat's Last Theorem was one of the most famous open problems in math for centuries, and it has no direct applicability to any tangible problems here in the physical world.
But yes, none of those are very tangible, until applied to problem solutions that are tangible.
> However, math problems are really there to solve a real world problem.
vs
> That theory might be inspired by the real world, but the problem itself is purely theoretical.
From TFA:
> In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.
The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"
The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect.
I may have misinterpreted either or both of them though!
> none of the millennium problems have anything to do with a "real" problem
P vs NP has a lot to do with real problems.
this is ABSOLUTELY not how actual mathematicians see their field. A problem in mathematics is just that: its interest to a pure mathematician is not related to any applications in other disciplines.
I don't really understand what makes a mathematical problem "interesting" because of my bias as an engineer. Nothing is interesting to me unless I can use it to solve a "real world problem". But, I'm willing to concede that "solving real world problems" is really not how mathematics advances.
Things like complex numbers or quaternions were often thought about way outside of the context of their modern applications in physics and engineering. When Hamilton thought about quaternions I really doubt he cared that it would make some programmer's life easier 200 years later:
> Every morning in the early part of October 1843, on my coming down to breakfast,
> your brother William Edwin and yourself used to ask me:
> "Well, Papa, can you multiply triples?"
> Whereto I was always obliged to reply, with a sad shake of the head,
> "No, I can only add and subtract them." [1]
If we go back further, Pythagoreans weren't trying to solve "real problems" either, they were like a weird religious cult.
My point is, the practice of "real" mathematics is really something that odd people feel driven to do, not unlike painting or playing an instrument. All the applied stuff is just a byproduct (not unlike ad billboards or elevator music).
While this might be true, I think this is the big pivot that will need to happen. Math, as a human endeavor, will all be applied. We will use LLMs to do the math and discover the math, and humans will apply it to whatever problem they are working on (likely using another LLM to integrate it).
Yeah and they should start seeing it differently
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.
Next stop the Culture!
So, to the cobalt mine, you!
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)
I do agree that on out current trajectory the latter does seem more likely.
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.
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.
I don't think this is the main issue (if at all) discussed here or in the field medalist letter. If anything, the majority of pure mathematics graduates are absorbed from the industry and a lot of exodus happens along the different scales of academia to there (though industry is also dealing with this issue but that's not what's concerned here). The issue discussed is what mathematics itself will be, which is much deeper. AI finding proofs to open problems does not solve at all the question of how to produce new problems, and there is no indication imo that there is way to go with that with AI.
Pure mathematics is not like applied sciences, as it is only tangentially influenced by external applications. Deciding which problems to tackle is a social process and a matter of taste/aesthetics, and is built largely through the exact friction that is more and more removed with AI. This is what makes it unclear how one can find problems without this friction, and none of these posts/letters have an answer really. Each one seems to describe just different standpoints than concrete, practical ideas.
Your productive friction (eutripsis? ~ negentropy? 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..
It's not clear to me that AI necessarily removes this eutripsis. The threat though, might become real if users don't see the threat :)
Also reminiscent of Keat's
https://en.wikipedia.org/wiki/Negative_capability#Reception
https://www.poetryfoundation.org/education/glossary/negative...
Still abstract, but nearer to quantitative (mathematical anthrop(ic)ology even?): ordinary, bad friction is, eg, "size-consistent"
Coasean Ceiling: organizational size limit where the internal friction of managing a firm consumes all of its energy, leaving nothing left for actual production
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.
Eg, KPIs, metrics, but of productivity, of "veracity", not understanding
Anecdotes--> better friction than data, sometimes, though :)
How about Inverse Metrics. of simplicity? Parsimony? Shortness of code? (Efficiency/compressibility is a sort of "intensive" metric, so it might not be especially relevant, thermodynamically speaking)
Just taxidermy, stamp collecting, and vibe-anthropologizing here TT
> AI finding proofs to open problems does not solve at all the question of how to produce new problems, and there is no indication imo that there is way to go with that with AI.
This has not yet been explored with AI only because solving hard problems is where everyone, practicing mathematician or layperson, understands 99.99% of the prestige to be.
AI's attention will not be directed towards generating interesting new conjectures until all the low-hanging prestige-rich fruit of famous decades-old conjectures have been mined, because it makes no economic sense for frontier AI companies to do so.
Who gets to define this? Those with power are famously bad at understanding long-term implications, how things work, and what matters - especially when it comes to funding things like research.
I don’t see the similarity? It’s a dystopia run by a maniacal computer and everyone is forced to live underground until the machine collapses for some reason?
I read that same summary. The similarity is pretty clear: the first step was letting a machine take over all economic and intellectual functions, leaving humanity useless and dependent it.
We're talking about taking that step too.
> Everything therefore becomes a hobby or a game. People do things because they enjoy them for their own sake, or because they are endeavours used as vehicles to socialise and enjoy others’ company, or because a shared social belief exists and is cultivated such that doing such and such a thing confers social status.
And the mother in the story:
> Vashti is content with her life, which, like most inhabitants of the world, she spends producing and endlessly discussing second-hand 'ideas'. She enjoys talking to friends but uses her work to defend herself against their invitations to be more social, remaining in her ‘room’ where all her basic needs are met.
She's basically something like an academic reduced to something more like wikipedia researcher while something else has the first-hand ideas, communicating with her circle through facetime/vision pro persona.
This is practically impossible, because the "economically necessary" things to do are never fixed but instead are in a state of constant fluctuation.
Furthermore, a chain is only strong as it's weakest link. AI alone is not sufficient to do all the "economically necessary" things. We would need vast numbers of highly advanced robots to do the rest of the work going on in the world (which is >> than the work currently done by AI). So if we restrict the definition of "economically necessary" to mean the subset of all possible tasks that we currently need to do, then we are sorely lacking in the required infrastructure. We may never get there either.
In the meantime, the labor shock knowledge workers experience could lead to large unemployment. Even those that find work doing non-knowledge work, the decrease in pay and benefits may be substantial.
I think these realities are worth understanding to help ground these pie in the sky ideas about what the future will be like. Not to be mean, but your idea of the future is basically a fantasy.
AI didnt break through stavier-nokes until a human set the initial direction. That's not going to change.
Why isn't this going to change?
Right now it's easy to see why don't set AI lose on every problem, someone human or AI has to delegate rare resources between competing interests. But if we look at general compute it was no different in the past. In 1980 you had to ask for permission to get CPU compute time. In 2026 you run it on your own computer, or maybe pay for it on Amazon. The constraints are much different.
If we keep pumping out chips and increasing efficieny someone will just make the LLM into an agentic loop (build the harness in) and set it lose on problems.
If you solve it then yes humans are toast mental usefulness wise
Astra goes farther than any model before but it cannot go forever.
Sythetic data works in a bounded box we have mapped you can't just make up novel data and it works.
Model collapse is completely unsolved.
That's not what happened. There was no 'initial direction' to set because OpenAI didn't know whatever tristan et al were working on or what direction they took. They heard rumors thta navier stokes was solved and set their best model on it.
There is actually a big leap in logic you’re doing here that it seems you are naive to. The world looks the way it does today because human societies, businesses and individuals are fundamentally in competition with one another for scarce resources, and having more resources leads to more capacity to survive, so there is always an incentive to acquire more. If you can no longer compete, you can and will be exploited. You don’t have to look very hard, far or back in time to realize that this is what society looks like. If everyone is incapable of competing except those who can afford to run a datacenter, that will not end up in socialist utopia.
Already predicted. How much Whuffie you got?
Now accepting Whuffie donations.
> 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.
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."
Just in case btw.: the world tends to get poorer in material ressources since around the 70s e.g. regarding habitable space, harvestable energy and the destruction of the integrity of earths biosphere - lots of things that establish value that is not bound to labour. Knowledgework isn't everything, some tend to forget that.
I'm a (libertarian, free-market) socialist, so not quite communist, but I think the differences are not substantial as far as everyone outside far left circles is concerned. And I've been saying this exact thing since this whole AI wave started.
If there was ever a time to seize the means of production and make them commons, this is it.
Aren't they doing that like literally from the very beginning? I clearly remember Lenin said that capitalism will end in a couple of decades because of electricity.
edit: Link changed; it was https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-t....
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.
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.
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.
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.
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.
And unfortunately mathematics is much more fundamental to human endeavor than this.
> 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.
Wherever we can recognise a problem we can solve it.
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.
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.
That's literally all there is to it - they don't believe in the rearrangement.
> … 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.
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.
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.
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.
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.
OpenAI did nothing wrong, being a Fields medalist does not give anyone a monopoly on the truth.
Mathematicians were somewhat wiser in the 1960s but the biologists and social sciences never were.
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'?
Imagine accountants saying the same thing against calculators. They had to adapt. So do mathematicians. So do I in software development.
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?
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.
“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?
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.
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.
How is a chatbot going to end human labor? Where are the millions of robots required to do all the various other jobs in the economy.
You and everyone else with this rhetoric must see AI as the rapture or something.
You're kidding, right?
1) Coordination is hard, and 2) "us" is a hopelessly nebulous term that appears inclusive but is almost always exclusive.
I think people would have said the creation of social media platforms like Facebook was neither good or bad, but rather it was "up to us to coordinate and make use of".
But did our society really have a say over how Facebook and other social media sites were able to embed themselves in our everyday lives? I would argue the answer is no.
I kid, of course, but I do wonder where the use of local "maximum" comes from, what is maximum there? Why do you not see this as a landscape of hills and valleys where marbles with certain energies may indeed get stuck in deep enough holes... Of course, I just assume and picture gravity pointing down in that landscape, but hey. I'm human, I feel it is expected of me.
It works this way with research, with most following the current trends, and some curious souls searching around for other ideas, be they contrarians, dreamers, or just convinced of some strange truth. But if we're right, signs tend to slowly begin to point their way, and we can shift the whole hulking edifice of science towards their point of view.
The problem of llms is that while they may be able to find a shorter route, we can't follow them unless we understand the route. So the forces that slowly begin to change everyone's behavior are lost
This is the 'stochastic' part of 'stochastic gradient descent,' and it's as important for human minds as it is for ANNs. These math wizards seem to be stuck in a rut of their own digging.
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.
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.
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.
Is that really the case? The cost to OpenAI for the Navier-Stokes problem has been estimated in the millions of dollars, anything between 6 and 40 million depending on who you ask. Now, I know that OpenAI has a lot of money stashed under the mattress but we have to remember that they by no means had any guarantee that the money they spent would actually solve the problem. In fact, they have probably burned similar amounts of money for many other problems that we have never heard anything about, for the simple reason that their super-secret in-house special models didn't solve them.
What I'm saying is that you need to think of the monetary constraints to solving major open problems in maths etc. If it costs millions of dollars a pop for a shot in the dark that has a small chance to succeed, that's not something that's sustainable. Letting the users take on some of that cost by burning through their own token budget so that they can claim to solve a problem that "has stumped mathematicians for 80 years" and the like, can make reduce the cost a bit but by how much? Remember that OpenAI had 10,000 agents running for 88 hours on Navier-Stokes. Who, outside of AI companies, has that many tokens? What power user is going to spend thousands of millions to try and solve the Millennium Prize problems?
So, really, maybe mathematicians who are looking forward to a new era of AI-generated results that advance mathematics may end up being disappointed. AI companies are not charities, they do not really care about mathematics and they aren't doing things that they don't think will bring them more money, somehow. And when did solving mathematical problems bring anyone any money?
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.
Yikes, the man is lost.
Tsimerman's MAISI has the same purpose as UK's AISI: To promote and rubberstamp AI.
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?
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.
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.
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.
Before arguing whether mathematics must strictly be done by humans, there are different motivations at play. Some people love the sense of solidarity within the community that forms during the process. Those excluded from that community might resent it, while others just purely want to solve problems.
Many things are being discussed, but looking at the overarching narrative, it seems that AI's true function isn't necessarily opening new horizons of specific knowledge, but rather excelling at 'serializing' topics that have been heavily fragmented until now.
In that sense, the concern is that because AI is solving the very problems needed to cultivate mathematicians internally, the stepping stones required for human growth are disappearing.
However, on the other hand, as the world and industries become increasingly complex and hyper-specialized, you could also argue that AI is the exact tool needed to unify this fragmentation across academia and industry. It is a highly complex dilemma.
From the perspective of researchers and the mathematical community, those 'problems for growth' must remain. But conversely, AI has the distinct ability to serialize siloed disciplines. Usually, when you go to graduate school, you often hear professors say that even within the exact same major, they cannot understand each other if their sub-specialties differ.
>the stepping stones required for human growth are disappearing.
Actually only in an institutional sense, imho. Maybe I'm exaggerating, but reddit.com/r/math* or even mathstackexchange will be so back with users analyzing and distilling proofs with AI. 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
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.
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
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.
This makes sense to me - intuitively, there is a lot more good mathematics to be done than humans are currently able to produce, so focusing human labor freed from production on verification of AI-produced results instead is more valuable now.
You are clearly on one end of the spectrum and do not seem to understand the other end.
This is the state of affairs today and how everyday people will talk about it.
Now, if an LLM proves a theorem, it's like discovering a new mountain and knowing what its peak looks like. Does that mean the problem is finished? No, we still need climbers to actually do the work and advance the field with human understanding.
Now these means are generated. If the proof yields some improvement somewhere, it can be used and there needs to be no human in the loop
https://openai.com/index/advisory-group-on-mathematics-and-a...
This is exactly the activity Tao and others worry will be hollowed out by instant "true/false" responses to novel questions.
finding new conjectures is the second interesting part
"instant "true/false" responses to novel questions."
Fair enough. Then we can add ordering and analyzing theorems with the help of AI as a third activity, admittedly the least interesting one.
If a conjecture can be one-shotted by OpenAI Astra, I have a hard time believing it is interesting.
For example, here are two topics that came up during my time solving combinatorics problems in the past:
https://en.wikipedia.org/wiki/Parking_function
https://www.sciencedirect.com/science/article/pii/S0012365X0...
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.
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."
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.
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.
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.
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.
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.
> 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.