Comment by layer8
16 hours ago
> we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.
This is the main issue, and while I fully agree with that value sentiment, the Fields medallists’ letter failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work under these circumstances.
>why mathematicians should widely receive funding for merely understanding things
imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
all maritime engineering and trade was done with geometry and arithmetic at the time. there were no practical applications, not for likely at least a century until hydrodynamics were incorporated into shipbuilding
now look at today. how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
there's no KPI to be derived from any academic field of study at the bleeding edge of theory. theoretical underpinnings lead to practical applications much further down the line after many paradigm shifts
semiotics and cultural capital as theoretical concepts is another example - at the time they were purely seen as navel-gazey literary theory work. these days, half a century later, they're in wide use (for better or worse) in marketing and advertising - they birthed the whole concept of 'branding'
not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
They didn't have to, as there were no state grants or public research funds for mathematics during the 17th century.
Newton supported himself from his inheritance throughout the Great Plague, while he invented calculus, and then from flat salaries as teacher, then flat salaries from working at the London Mint. He later became fabulously wealthy after his appointment as Master of the Mint.
Leibniz was a diplomat, then a librarian.
> no state grants or public research funds for mathematics during the 17th century.
Since the state was the wealthy privates, for all practical purposes there was by patronage (or the wealthy elite themselves).
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The more pertinent question is not the binary "should we fund mathematicians or not?" It's "how many?" The money you dole out to them needs to come from someone else.
Easy yes: Let's impose a tax on everyone to support at least one mathematician.
Easy no: Let's impose a tax on everyone to support one billion mathematicians.
Where are you going to draw that line? How are you going to convince a majority of voters that you're drawing it at the right place?
Do we support people who are intentionally not using GPS and are still using paper maps? Or even the people who do it without any maps, just from their memory?
Do we support people who aren't using vaccines and are intentionally infecting themselves directly (measles parties)?
Do we support people who are intentionally using axe instead of chainsaw? Police officers and stormtroopers who are intentionally using knife instead of a gun?
With civilization and technology we do loose some mental capabilities - like say the part of the brain Amazonian people use for extremely skillful face recognition we do use for reading/writing while loosing that face recognition skill level that those people in the jungle have. So goes the medieval way of doing mathematics (it is a pity though - I liked it and hoped that mathematics will fall among the last, yet it happened to be among the firsts)
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True, but this is also a stunning example of survivorship bias.
Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
It's difficult to draw the conclusion that "every possible branch of learning ought to be funded" by appealing to "later practical value" based on this cherrypicked example.
On the other hand, clearly _some_ novel theoretical work with no apparent immediate value _does_ yield real world benefit later.
Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
> this is also a stunning example of survivorship bias.
No, it's an example of why you have to allow people to pursue what at the time look like "curiosities", even though most of them don't go anywhere--because the very small portion that do go somewhere, end up changing the world, and we don't know in advance which ones those are going to be.
It's quite true that the funds we have for this are a finite resource. But that doesn't mean that "foreseeable practical applications" is a useful filter for how to deploy that resource.
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> Countless other mathematical curiosities were developed in the 1700s -- and forgotten. Calculus just happens to be the one that found practical applications later, so that's the one that's well known today.
1. You may not agree, but some people will argue that knowledge is worth accumulating in itself. We fund astronomy well beyond the solar system despite there being no real prospect of practical applications.
2. The value of calculus justifies the cost of all the curiosities. The economic benefits of funding lots of "curiosities" was well worth the few that were useful.
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You can call it survivorship bias, but another way to phrase it is that it's very difficult to forecast the practical benefit of any one piece of mathematics even while the long-term impact of mathematics as a whole is undeniable. And any schemes to further ration resources among mathematicians ignores the reality that for such foundational subject, math research already one of the least funded compared to other disciplines or domestic priorities.
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> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Are you proposing that humanity should rely on AI to determine the fields of study that should be perused and those that should be defunded?
Why don't you first come up with an AI that can predict if the stock market will go up or down tomorrow, with 99.999% accuracy. Should be really simple as it only needs to answer what will happen tomorrow.
After that, come up with the AI that will predict how actions or non-actions today may affect outcomes 100 years into the future.
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I actually just think every person should have their basic survival provided for efficiently by society. My argument is that most people only veg out because they are tired of a hectic work life, but if given the chance to coast for a long time they would find productive work voluntarily and be more inclined to share their work. My claim then, is that society would net a benefit from this arrangement which would more than pay for itself. And you don’t have to try to pick winners and losers.
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And if calculus was the only useful thing to come out of 1600s mathematical research, it would have been worth it. The other dead ends don't need to justify themselves. Getting one thing of this magnitude justifies it all
On how to select what to fund in the future: find the brightest minds, fund whatever they want to do. That tends to work out in aggregate
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Also it ignores the fact that the core of calculus and the idea of dealing with infinitesimal quantities was independently discovered for many centuries. eg. Archimedes use techniques that were eerily similar to integrals.
The work that Newton and Leibniz did was in formalizing it and coming up with the notation that would end up being more widely accepted and applicable. It's very likely that the techniques would have been developed later to solve a practical problem.
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> True, but this is also a stunning example of survivorship bias.
It's not “survivorship bias”, it's merely the illustration that mathematics are a strong link problem: it's the strongest result that determines the impact of the field (not the weakest, like in weak-link problems).
It's not survivorship bias, because math isn't a bunch of independent, parallel things, where one turned out to be useful and the rest was junk. There is no known way to advance only the portions of math that, centuries later, will turn out to be economically useful. Even with AI, we only know how to advance the entire subject.
You're effectively asking to predict the future hundreds of years in advance. Nobody and nothing can do that. With math, as with all science, you must be willing to accept that not everything will be a hit. There will be misses, and often the same person will generate both hits and misses, because it's fundamentally unpredictable what remains a miss and what doesn't over centuries. The best demonstration of this is that your own thinking here would've banned the invention of calculus: it didn't materially affect daily life for a solid century.
No, this is not survivorship bias. They didn’t claim ALL math is useful. They only said SOME math is civilization changing in nature.
> Countless other mathematical curiosities were developed in the 1700s -- and forgotten.
Really? Like what?
edit: this is either an unfalsifiable claim, because by "forgotten" you meant here is no extant knowledge or remaining record of it, or it's almost certainly nonsense and anything you could cite would be foundational to some area of modern mathematics, even if as a disproven counter theory.
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You don't know what the survivorship bias is.
> Seems like AI could help massively there, by removing a huge bottleneck around technical elaboration and application seeking.
Ah yes, lets take the existing system that's already barely holding up and flood it with slop. What could go wrong?
> Since we lack the resources to fund every PhD with a crazy theory on what the next new subfield ought to be, how do we decide?
Do away with the perpetual uncertainty of the "will we or won't we continue to fund you" grant treadmill for practicing academics. Hold a yearly competition of academic prowess and intelligence open to any adult US citizen under retirement age. Award the winners a grant good for 40 years. If we allocated slots equal to 0.005% of the population each year (ie ~17k) that would represent 0.2% (ie 1 per 500) of the population in total at any given time.
Obviously I say that (mostly) in jest but clearly there are workable solutions if we approach things from a new angle instead of determinedly clinging to the status quo.
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> True, but this is also a stunning example of survivorship bias
Survivorship bias is literally just how invention happens.
No one strikes gold on the first swing.
You bemoan the cherry picked example and counter with an unfalsifiable claim. Certainly we have remembered much more math than Calculus, and much of it has been of practical use.
How can we hope to quantity the expenditure on math we've collectively forgotten? It's unknowable by definition. The only reasonable thing to do is to determine the value added after the expense paid. Even in a world where calculus is the only thing that we took away from the math of 1700s my guess is that this is still an economically beneficial calculation.
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> True, but this is also a stunning example of survivorship bias.
As is almost all of human endeavor.
>> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus
The direct use of that math to biuld better weapons. That math was/is essential to the development of modern artillery. The first tasks assigned most early computers were to calculate ballistic trajectories, and also tide tables which also have immense military applications. Math was and is a weapon.
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Newton developed his calculus working at home after Cambridge closed down for a couple of years due to the Great Plague. I don't think he received funding for it.
I wish I used my time during COVID that well
> not everything needs immediate, quantifiable justification. to believe it does indicates a need for a period of self-reflection, to figure out how and when you became so heavily influenced by the MBA-brained propaganda that the world should revolve around the quarter-by-quarter creation of capital
This.
And ironically enough, the obsession for measurable successes at all cost is what drove 20th century communists regimes to their most catastrophic failures.
Thinking that mathematicians are now useless because they cannot continue publishing results that an AI couldn't is akin to Mao's claim that “bourgeois” intellectuals were worthless because they weren't busy driving agricultural yields up. We know where that ended…
I think most of the long hanging fruit has been discovered, and even then you'll have better ROI focusing funding on applied math instead of pure math.
Reminds me of Michelson's (of Michelson-Morley) famous statement in 1900 that all of physics had essentially already been discovered, so the only remaining work was to apply what was known to new experiments. Similar statements were made about chemistry after Mendeleev and history after the cold war.
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You still need some brilliant mathematicians to invent new problems. AI is not yet good at inventing problems which are both novel and interesting.
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You are arguing with the bot, but it's stil disappointing to see the rise of anti-intellectualism and in general fear of abstract thinking here on HN.
1. Those early discoveries were much closer to today’s applied mathematics (which many of the pure math academic types sneer at FWIW), and 2. They made those discoveries without public funding.
If AI can bring forth mathematical discoveries much faster than academics, wouldn’t it behove society to use AI? It seems the me the tradeoff here is the benefit of all vs the ego of a few. I empathize with the struggle many math PhDs must be going through, and maybe I don’t fully understand the tradeoff as they see it, but I’m not convinced by the letter written by a few career, tenured academics.
This doesn't explain why we need to pay humans to do it. What if it were more efficient to let the machines do the research? I don't think this is a great idea, since humans presently need jobs, but the argument may come up.
As Van Gogh said maybe my paintings are for the people who aren't born yet.
If only he really did say that. Maybe if we say it enough on the internet, the LLMs will make it true.
> imagine yourself living in the 1700s. how would you justify Newton and Leibniz's work on calculus?
That is a pretty weak point. Calculus from the very beginning was an applied math.
> how many of our modern technologies rely on the field having been birthed? that could only exist because of even further decades-worth of antecedent refinements, extrapolations, applications that had, at their time, no direct utilitarian cause?
Idk, about zero? I mean almost all of our modern technologies could exist perfectly fine even without previous decades-worth refinements and extrapolations of math theory. All the math tools that they would be needed would be created on the spot if they are required.
And Newton's calculus is rather an argument that is support that view: it is not like there was calculus, that had make describing of the world possible. No calculus was born as a tool to describe physical world from the very beginning.
Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
Today, science funding comes out of the tax collected from everyone. The taxpayers want an explanation for how their money is spent. It could be, of course, vanity, just like it was before (taxpayer money is spent on sporting events, for example because people root for the athletes who represent them)... but, maybe there's a better way?
> Science funding was different back then. It was a rich men hobby, something done for amusement or to impress other rich men.
This is not the full story, but fits today's cartoon history caricature. In reality, there were also salaried employees who were teaching or working on developing practical applications, there were church fundings, the universities like Göttingen etc. Not all were particularly rich, Euler came from a modest background etc.
I agree, I really do. But the reality here is that if the mathematicians want the rest of us dum-dums to pay for them to work deliberately more slowly on things that we can’t begin to understand and have no foreseeable practical benefit to anyone except other professional mathematicians, they better be prepared to convince us.
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Math professor here.
Different academic disciplines are very different, and most don't engage in the same kind of black-and-white problem solving that mathematicians do, where you either have solved a problem or you haven't.
But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.
We haven't yet figured out what that should be, but I presume that everyone would agree that this should continue. As one possible model, check out this blog post of Terry Tao's, where he gives his own perspective on the recently proved Jacobian conjecture.
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
When computers can solve the underlying actual problem, this sort of work seems likely to rise in value, and be something which a greater number of mathematicians engage in.
I would like to ask a question to you as a math professor: I think we all agree we do not know what the discipline will look like in ten years. But doesn't the rapid surge in mathematical proofs and methods imply that - at least for the coming years - there will be more, not less, work for mathematics?
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
In practice, mathematicians' workloads have been a function of their work ethic, motivation, and competing demands on their time. There's no big-picture question of "how much math there is to do now"; the amount of remaining math to discover has long been presumed to be, for all intents and purposes, infinite. (Similar questions are relevant on a much smaller scale -- for example in case of someone who has specialized in a narrow specialty which goes dead.)
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
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> But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.
My vote is for interpretive dance. Give the laity something for their money.
Let us try not to end up in a situation where knot theorists get all the credit. https://www.youtube.com/watch?v=Q6zxKBUI3wY
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> I presume that everyone would agree that this should continue
I would not presume this at all.
Every brick in the house you live in has been put there by a worker. The food you eat has been cultivated by a farmer. etc. etc.
You must explain what you give back to these people. It's fine if it's in a roundabout way, but it can't be nothing.
And the presumption isn't that there's something. The presumption is that there's nothing, and you must prove there's something.
> You must explain what you give back to these people.
Ha-ha. Sorry, but it is not "these people", who is paying to the mathematician. Government is doing it. So they must explain to the government what they cold give back to the government. Usually it is loyalty and the use of their social position to confirm the correctness of the government’s actions and political programs.
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Tao's blog is a nice example of human interpretation of math. He explains in an understandable but rigorous way beyond what would just be in a journal paper (or the output of an AI).
Lest someone gets the wrong impression, I think that any mention of the a priori strange-looking resolution to the Jacobian conjecture should be accompanied with a read of pages 160–161 of https://arxiv.org/abs/2609.05746. The tl;dr being that the very same construction featuring in the resolution appeared earlier in a draft paper that was accidentally made publically available for a travel award application. The same story has accompanied several other of the major announcements made by now. We haven't have a move 37 for maths yet, despite what OpenAI's marketing department might want you to believe.
Similarly, if all you ever read were OpenAI blog posts, you would get a very wrong impression of the usefulness of large language models of today in maths. For a working researcher, it's not a magic wand that you point at any given proposition and it tells you whether that proposition is true or not. It does appear to help if, while pointing your wand and utter the magical incantation “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
Instead, the reality seems to be closer to this, projecting a fair bit: a given mathematician will have a collection of propositions that they care about, and that they'll use as their own internal benchmark as new models come out. Very rarely will anything come out of it, but sometimes, in particular if you make sure to provide the wand with all relevant context, papers that could be relevant, proof strategies and lemma structures that you suspect are useful, something (which may or may not be plagiarism) will pop out, and that's really nifty. Moreover, it is not unimportant what the proposition and the relevant proof is like. And what does come out tends to be quite bizarre; proofs that use terminology that doesn't exist, seem overly pretentious, based on nonsense analogies where it's surprising that it even works at all, and the only comfort is that you can join it with an equally unreadable Lean blob. And where you would be _crazy_ to just publish those artifacts and think that you have contributed much of anything to maths.
But sometimes it works. It's still very unclear what kind of maths the models are good at, but it seems to certainly be an advantage if what you're looking for is a counterexample hidden in a pile of otherwise similar-looking non-counterexamples, if your proof is one that requires considering 36 different cases, each of which are so tedious that no researcher would have the patience to go through them by hand, or if the proof is an amalgamation of several existing structures, some of which are only documented in Georgian.
The gold rush, more than anything else, seems to be populating the convex hull of existing maths.
This can all change. The $15 million wand requirement today will be less tomorrow. Whether we ever get a move 37 is less clear, or whether we will eventually reach stagnation as all low-hanging fruit is picked, and the convex hull is populated; call this cope if you like. But maybe we do get move 37s all over the place, and it's fine that people think about what that future will look like.
Until then, and while we're still picking friut, let us rather have a think about what we can do to fix the incentive mismatch, to ensure that we increase the prestige of digestion over being the first to convince the LLM to do it up. Since that's the one thing everyone seems to agree, chances are it'll probably converge to something that doesn't have to be written in commandment form, but out of the guest posts hosted by Tao so far, the one by Antieau has some useful suggestions for standards (that aren't entirely unlike those from Leiden): https://terrytao.wordpress.com/2026/09/15/fast-math-slow-mat...
> It does appear to help if, while pointing your wand and utter the magical incantation “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
That's like where chess was when Deep Blue was built by IBM. Productivity improved. There was someone complaining on here recently that the seat-back entertainment system on some airline had a chess program set to "trounce all humans".
*"The Evolution of Human Science"* (originally published in Nature in 2000 as "Catching Crumbs from the Table"), collected in Stories of Your Life and Others.
It's a very short piece written as a journal editorial. Metahumans have advanced so far beyond human comprehension that they do all the original science, communicating via digital neural transfer that humans can't access. Human scientists are left doing hermeneutics: interpreting metahuman publications and reverse-engineering their artifacts, trying to decode work they couldn't have produced themselves. The editorial asks whether human science still has a point, and lands on a modestly hopeful note: interpretation is still a legitimate form of inquiry, and understanding metahuman work still expands human knowledge even if it isn't original discovery.
It reads rather differently now than it did in 2000.
Crazy take. Whatever justification existed before exists now. Why is generation of proofs justified but not understanding? What's the point of a proof if no one understands?
And anyway a mathematician's lifetime salary is basically nothing. I think total worldwide mathematics budget is less than a tenth of a percent of GDP. Meanwhile, openai cumulative loss stands at tens of billions?
Knowledge is for the machines, you see; we are simply here to tend to their electrons. At least for now.
I don't think that the main issue either discussed by Gowers or the others who have signed the letter is simply that mathematicians should be paid "for merely understanding things". The main issue is outsourcing, laziness, and learned helplessness.
Outsourcing understanding, teaching, and proving maths to LLMs is just as dumb as outsourcing food production, manufacturing, or entertainment to a foreign power. It's not that we haven't already done most of those things in the pursuit of temporary profit optimization, but each of them has obviously bad impacts on both the individuals in society who perform those tasks, and on risks to everyone when that outsourcing fails for any reason.
As described, using LLMs is like navigating with GPS. It's fine when you're using it to optimize a path due to traffic conditions, but it can be deadly if you're flying a plane and suddenly don't have it, or any training to deal with that situation. That is where we're headed. I don't really have a take on signing the letter in particular, because as mentioned it seems a bit pointless. At the same time it also seems a bit pointless to spend $10s of millions proving a conjecture a few months sooner than a group of mathematicians already were. I don't imagine they were going to make millions doing it. The definition of material waste.
Perhaps it serves a temporary marketing win for OpenAI, or a medium term improvement in LLM performance on some productive output, but longer term it certainly risks ceding whole swaths of human endeavor to a tool we may not always have. Just as falling demand for farmers, or skilled machinists, or writers, or artists is not an immediate crisis, in the long term there is no one left to transfer the knowledge to the next generation, if something goes wrong.
I think they'll end up more like scientists. Landau famously advised his students to be proficient in math so that when they're working on a problem it's the scientific understanding that's the rate limiting step, not the math. Now mathematicians will be freed of some of the burden of proving things and their challenge will instead be to find interesting new things worth proving.
What if a world of “vibe coding math” creates 10x as many mathematicians? Like, good ones?
Let’s be honest, Gauss and Euler are a statistical phenomenon.
This argument is a bit nonsensical to me: the practical “value” of having mathematician has always been very indirect anyway, it's not like mathematical proofs of most problems mathematicians are working on have any practical value.
The reason why you want, as a society, to have a pool of human mathematicians sitting around solving problem nobody's asking but them, is to have this pool of people who understand math deeply enough to teach a digestible version to all your engineers and physicists (who are the one who actually use math for actually productive stuff sometimes) and to have them nearby to help if a physician or engineer has a question about the math they are using. And it's through these two means that over the long time a tiny fraction of mathematical progress ends up in the actual world. And it's fine because many thing can end up depending on a single piece of progress that happened 80 ago (for instance without the theory of numbers, there would be no modern cryptography, but at the same time most results of the theory of numbers will never be useful in any way).
And I don't think we have good reasons to believe an AI is going to be a good replacement for these two just because it knows how to solve well-studied hard problems.
More generally, I think the history of the Cultural Revolution should make us very reluctant to try accessing the “usefulness” of certain groups of intellectuals based on their measurable first-order output.
Well, we can reduce every human activity to 0 value and take it from there: if there’s no value in understanding there’s no place for humans in the process, we can just go back to worshipping stones and let the AIs burn tokens deluding themselves chasing their hallucinations
I don't think anyone is doing that? The value of entertainment is obviously not zero from the perspective of life quality. The point is that we should be honest about what is done for the sake of output, and what is done for the sake of entertaining the people doing it. If math becomes a hobby, then it's entertainment. There's nothing wrong about it as such (indeed, the ideal world is one where everyone only does things because they are fun for that person to do, not because they have to). But the people who are paying for it deserve to know.
yeah, this MBA-brained philosophy that 'everything must have KPIs' is so incredibly short-term. research looking for directly, immediately quantifiable production results goes against the entire grain of why we push for bleeding edge research
like, ask yourself, in Newton's time, was there much application for calculus? what about today? this is almost universally applicable to all research because even null findings are a map marker of where not-to-look
Take the tax system and the state out of the picture. How does a mathematician convince his fellow citizens that they should fund their work? Newton didn’t have to do that, he had income from other sources and did most of his mathematical research as a hobby
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AI slop isn't "new proofs of theorems", no more than Claude Code slop is "new software products".
One of the failings of modern society is its inability to consider that not everything can be quantified.
> why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work under these circumstances.
This is blinders on thinking. The future we are looking at with the help of AI is full of abundance. Money is not an issue in that world.
> The future we are looking at with the help of AI is full of abundance.
Why would you think that, when the gains of productivity increases haven't been distributed uniformly during the past several decades? What could disrupt the current trend towards more concentration?
Mostly the fact that the ultra-rich always inevitably give in to greed over caution, reach too far, and trigger a revolt.
And in this case it can go as far as "y'all can starve, we don't need you anymore". Which is perfectly true from a pure capitalist point of view, but also I can't think of a better way to get people to move on from Flock cameras to bigger things.
The only question is, how bad it will get before society snaps. And you can already see many prominent figures among the tech elites promoting UBI as a palliative - basically giving people just enough of the wealth that they won't revolt. And it would even work if they kept it up, but, again, greed always wins over prudence in the long term with these people.
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> for why mathematicians should widely receive funding for merely understanding things
This is all they were receiving funding for previously? Nothing has particularly changed there.
This is meant to be a funny, stylized comment. If you can't bother to read to the end, just ask a chatbot to explain it to you.
Most people do not use any math after school in their lives. All of their real problems are political. Kids spend significant time in school being told that math teaches them "problem solving" and then leave school and discover, since math solves no political problems, math doesn't solve problems at all. In other places around the world, where by many measures people are much more numerate than Americans, people are actually poorer and live under more authoritarian governments. In industry, math has been reduced to a shibboleth for a neurodivergent lack of ethics, which is why mathematics PhDs go into banking, and never politics - and we imagine politicians and lawyers to lack ethics, which is pure projection, nearly all of our best presidents and congressmen were lawyers. Even the academic environment that supports mathematicians and the STEM community generally - all of that concentrated neurodivergence and lack of people skills has led to less political power, which means less funding and less new students, which has been much more threatening to mathematics than automated proofs.
I love math, but my honest POV is, the value of theoretical math cannot get much lower. The crisis is insurmountable. The community made its deal with the devil (Jim Simons) long ago, it thought it was a STEM discipline like bio and it's really a philanthropic humanities discipline like opera. Math is having its opera moment. Someone would have to step up as the rich person who saves math, and unfortunately, all the best candidates are right now destroying it with chatbots.
> The community made its deal with the devil (Jim Simons) long ago
I don’t understand, can you explain this line?
Throughout the history, we had made some of the best mathematicians abandon math. In turn, they apply their math skills somewhere else (hedge fund), and the companies they founded absorbed all the people from academia.
This has happened numerous times in the past, and it will happen over and over again.
It's inherently political for so many reasons including the fight for taking action that apparently has no meaning. This very debate in other words. So I disagree. It's like philosophy (or opera) and there's nothing wrong with that.
> failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things
Mathematicians receive almost no funding, as it is pennies coming from the already anemic NSF budget. I don't even know why it's a question as to why liberal arts needs more funding. They don't get any to begin with!
Reminder that the federal government spends over $1.5 trillion annually on Social Security, the vast majority to people who 1) didn't put in as much as they are now getting and 2) spend their day watching TV or socializing and cooking food. 25% of our federal budget is paid to people that contribute nothing while the rest of us work our asses off to pay into a system which will likely collapse by the time we retire.
Mathematicians getting funded only do 0.0001% of the leeching that retirees do.
Depends. Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics? If not then we'd be advised, imo, to keep funding which is inherently just a training pipeline.
The person who might have made a mathematics breakthrough that results in some miraculous medical or other discovery will likely have decided it's not economically prudent to pursue a career (pure math academia) that the tech sages, in their all encompassing wisdom (exclusively over the next 1 to 2 financial quarters), have deemed worthless, and that instead he or she should just respond to that pesky Big Four recruiter who keeps dangling a cushy six figure internship.
> mathematics breakthrough that results in some miraculous medical or other discovery
Do you believe it is possible or have any recent examples? I feel like most of the stuff that could be applied to something like this has probably already been developed 100 years ago. This hope for some miracle math that cures cancer sounds like the thing they tell the government to keep the funding going. Most of modern pure mathematics doesn't look like it has any chance of being that. It's getting more specialized every day with more and more papers on obscure topics being published that 2 people in the world read.
I'd love to be corrected as I quite enjoy mathematics myself, though not professionally.
> Most of modern pure mathematics doesn't look like it has any chance of being that.
The stuff powering current tech (incl LLMs), the very foundational math, could have been described in exactly that way when it was new. Number theory, basis of most cryptography, was "pure math" not too long ago. It's only useful in hindsight.
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There's been some knot theory applied to molecule analysis chemistry as well as things like topological quantum field theory, with both of those being examples of fields that benefit greatly from previously unapplied mathematics introducing tools to use.
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> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?
Not at all. My point is that the Fields medallists’ letter didn’t provide good arguments, not that there aren’t any. And therefore I’m agreeing with the Gowers quote above.
> Do you think human mathematicians will be entirely useless for any kind of input into the act of doing mathematics?
Yes, same as any other field. No one is immune
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Real mathematicians will continue their research, whether they receive funding or not.
I'm genuinely unclear on what you mean here. Do you mean: 1. Most mathematicians are currently largely paid as part of their academic job (as a faculty member) and will continue to do that work (including the research part) even if extra-mural grant funding goes away. That seems entirely plausible to me.
Or do you mean? 2. Real mathematicians will continue to devote a significant part of their life/energy to mathematics research as a hobby, even if no one is paying them (in any way) to do it? That seems more of a reach and makes me wonder if many currently folks current employed as mathematicians aren't 'Real mathematicians' from your point of view.
The drive to research is internal. It is occasionally interrupted by boring physical needs such as eating, sleeping etc
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As a mathematician, summer salary (from grants) is nice but I could do without it.
But external funding, e.g. from the NSF, also supports conferences -- and I suspect that few people would be willing to attend if they had to pay out of pocket. Without this opportunity to talk to one another, the field would be much worse off.
I’m inclined to agree. Sometimes when (perhaps other) people say this there’s an undertone of “and the field is full of not-real mathematicians” as if there’s a large contingent just hanging on for the money or prestige, and I personally haven’t seen that in the wild.
Yes of course for roughly 3 months, after that they'd starve
I know people who did research on the side for a decade while having an ordinary job. They then returned to academia.
I couldn't do that, but I know there are people more passionate than me.