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Chance-Device 4 hours ago [-]
I think the answer to this is simple: Mathematicians don’t own math. Math is a tool to do useful things. I don’t want this gatekept by people who have a sense of entitlement. And luckily enough, they don’t have any leverage, so the appropriate response is to say no.
hodgehog11 1 hours ago [-]
Math is not merely a tool. The question isn't one of doing the math, it's about publication and release of work. Humans have to follow certain rules, so why should we allow AI labs to behave differently?
Chance-Device 50 minutes ago [-]
Anyone can publish what they want. There’s nothing else to say about it, however anyone may try to pretend otherwise.
hodgehog11 41 minutes ago [-]
No you can't, have you published in top journals before? You can only publish what the community finds valuable, period.
Chance-Device 32 minutes ago [-]
Actually yes, I have my name on a paper in a journal that you’d certainly consider to be top in its field. I’ve also worked in academic publishing.
You have a very institutionalised idea of what publishing means.
AndrewKemendo 19 minutes ago [-]
If a result is replicable that’s the only thing that matters
dannykwells 49 minutes ago [-]
This a thousand times. Thank you.
lucas_t_a 2 hours ago [-]
It would be so much work to follow any of this that the answer is guarateed to be a no
shank-false-rin 24 minutes ago [-]
That is a strange take. Trying to stop the industry/field from becoming a monopoly/oligopoly is gatekeep?
kingstnap 1 days ago [-]
Mostly pretty straight forward. The idea that proofs be desloppified, attribute existing literature properly, and published somewhere expediently where it can be commented on, with artifacts for verification, is all very uncontroversial stuff.
I think the spicy take is definitely this stance that longstanding mathematical problems shouldn't be used as benchmarks for (specifically proprietary) models. Stated right at the very top.
The justification is pretty clear.
> The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline.
It is all fun and games (for non mathematicians) when mathematicians can't compete with AI labs but I think the more dangerous direction is when this starts being true for the rest of everything. For example cybersecurity or whatnot. Hence why I think Anthropics whole stance of being completely against open anything is actually *extremely* dangerous due to the centralization of power which they completely ignore as a risk factor.
The most discussable thing in this is certainly the idea that labs should fund mathematicians to do expositions.
> One of our principles is that AI labs have a responsibility to provide support, including funding, for the development of human understanding of the AI mathematical output that they release.
Obviously this directionally sounds like the role of mathematicians would be shifting towards interpreting AI results instead of making proofs. I'm not sure who would should really be billed for that. Plus how would you decide who gets the grant?
An interesting thing is that by stating that that the lab that dropped the result provide the funding, this is *directly* proposing an example of taxing AI labs for displacing knowledge workers.
zozbot234 6 hours ago [-]
> Mostly pretty straight forward. The idea that proofs be desloppified, attribute existing literature properly, and published somewhere expediently where it can be commented on, with artifacts for verification, is all very uncontroversial stuff.
If Ramanujan could freely disclose and publish results that were arrived at by literal divine inspiration (he certainly didn't bother to write down "proofs" in the modern sense!) why do we suddenly need all this "responsible release" process stuff when it comes to AI-derived results? This makes no sense when compared with existing norms within the mathematical community.
hodgehog11 1 hours ago [-]
He could not freely disclose and publish his results, that was the whole problem. He only got traction when one result was automatically validated because Hardy had already seen it. His findings more broadly were only considered fine once they were understood and verified. This declaration is basically the same thing. Having Lean code doesn't substantially change that.
It's almost like a lot of non-mathematicians are commenting and have no idea about how the field really works. Or at least, how it works at the top level.
meowface 11 hours ago [-]
Am I the odd one, or is this, like... absolutely insane logic?
"The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline."
Are mathematicians' private brains unfair if they're way better at solving open math problems than anyone else? Are they required to share everything they're working on, as they're working on it?
Should Andrew Wiles have been condemned? Wouldn't these arguments all have applied to him? Besides him working at more human speeds.
emil-lp 7 hours ago [-]
It's not insane when you realize that in mathematics, the goal isn't theorems (we have always been able to generate correct theorems, this is a basic programming exercise).
The goal is to expand and extend our understanding.
When an LLM produces a proof, nobody (including the LLM) gained new understanding ... Unless, of course, a human being sits down and understands it, and communicates the new understanding to the broader community.
steinwinde 5 hours ago [-]
Is it not more democratic, if AI companies announce new theorems and proofs, and the discipline of mathematics furthers "our understanding" - instead of that the wealthiest universities in existence (all of them in the US) use their advantageous access to AI to produce them, along with their exclusive furthering of our understanding?
In a world, where expensive AI models running thousands of agents produce the most remarkable results, how should the discipline of mathematics look like, when money and access to compute plays such a pivotal role?
With the Navier-Stokes solutions being out in the wild, any mathematician at an impoverished university can make a stab at it and help our understanding. Assume the solutions being held back by those who produced them at Harvard & Co. (until they have their nice articles ready), what would the role of mathematicians be at less affluent universities?
I do not assume AI companies will be ready to provide equal access to their AIs to worldwide mathematics.
1 hours ago [-]
Calazon 5 hours ago [-]
The insane part is placing the burden on AI labs to advance mathematical understanding. Why should this be their responsibility?
If they want to simply generate theorems and proofs and post them on the internet, why not? Does doing that really inflict "substantial negative externalities ... on the mathematical community"?
meowface 1 hours ago [-]
Also, people would get mad at them (rightly) if they sat on a proof of a major unsolved problem for months while bringing in human mathematicians to convert it into a more human-friendly form
curt15 50 minutes ago [-]
If math is a "burden" on AI labs, why do they willingly spend tens of millions of dollars for a purely theoretical and unphysical result? What's the ROI for that when the frontier labs are already notoriously cash-strapped? Their motives pervade the entire discussion and cannot be ignored here.
Calazon 27 minutes ago [-]
Presumably some combination of marketing existing models and training/improving future models.
My point was about what is and isn't the responsibility of AI labs. Why should they be required to invest more (or fewer) resources into math than they otherwise want to?
(If you ask me, safety and security are what they need to invest more into.)
meowface 42 minutes ago [-]
...Could you not ask this of anyone who works on pure math problems? What on Earth are you talking about??
"Academics are paid paltry salaries, university departments are cash-strapped, why are they studying these abstruse math problems instead of building rockets?"
Successfully advancing fields of math and science is a good thing. Why would they not do it. Why would they not want to prove their increasingly intelligent computers are able to solve unsolved, complex math problems humans have failed to solve for centuries and then release the solutions to the public. What do you want them to do, not try to solve them?
rafaelero 4 hours ago [-]
Isn't easier to understand something backawards? It seems much simpler to understand something once you already have the solution.
octoberfranklin 7 hours ago [-]
Capital can't buy you a better brain. It can buy you GPUs.
The question is whether we want mathematics to be dominated by whoever has the most capital. This is perilous because mathematics doesn't produce profits like commerce does. The AI labs might end up killing something off that they are unable to sustain a replacement for.
BobbyJo 1 days ago [-]
> The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field
This is literally the economic bet of the big labs in the broader economy. You use your relative advantage to front run or outcompete.
I'm not sure this argument will work given it is essentially an argument against the thesis the big labs use to justify their valuations.
genxy 12 hours ago [-]
Once they get good enough, they won't even ship the models, they will just replace all the jobs themselves. This is where Anthropic, OpenAI and Google are heading, esp Google with GSuite, it already is up in your business doing your business.
All your jobs are belong to them.
tehjoker 15 hours ago [-]
Right, the whole idea of these labs is to install themselves as capitalist philosopher kings. They won't tolerate any ideas of "decentralization" despite it benefiting 99% of humans.
OpenAI was started because Elon and Sam thought Demis was going to make himself dictator and they wanted a piece of that action (reading between the lines). All the labs, even Anthropic have the same idea dressed in different marketing gloss. The major exception (I imagine) are Chinese labs, since the Communist Party of China will demand control over these labs and their models. Good! Democratic human control is a good thing. Certainly, there is more democracy in China than in America. That's the idea of socialism, democracy for the workers, not for the capitalists (we do the inverse).
graemep 2 hours ago [-]
You end with some pretty far fetched claims there. Control by an authoritarian government is no democracy. I am sure that the people running AI companies will claim they intend to do it for the good of society, just as the Chinese government already does.
octoberfranklin 7 hours ago [-]
When you used the words "decentralization" and "socialism" in the same post I kinda puked a little bit.
thfuran 1 hours ago [-]
>Anthropics whole stance of being completely against open anything is actually extremely dangerous due to the centralization of power which they completely ignore as a risk factor
I really don’t see how that could be an oversight. It’s the goal.
hodgehog11 55 minutes ago [-]
Nah, it's the whole benevolent dictator mentality that these socially maladjusted individuals believe. If you believe that you are the only good guys, you are likely the bad guys.
killerstorm 4 hours ago [-]
Mathematicians can only make these demands because they believe that AI labs get a lot of credibility from math results. So they propose a trade: we give you credibility, you give us funding and let us to gate-keep.
aaron695 3 hours ago [-]
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throwaway713 1 days ago [-]
> we ask them to stop testing advanced mathematical problems on proprietary models.
Maybe I'm alone on this, but for some reason these sorts of requests strike me as akin to gatekeeping how someone should breathe air. It's math... the numbers and symbols are just out there in the platonic realm available for anyone to do as they like with them. It's patently absurd to request other people to stop.
Ensuring credit where credit is due? That's fine. If your model incorporates the efforts of many others, then it's reasonable to request acknowledgement of everyone who contributed (even indirectly). But that's not what the request states — presumably their ask subsumes any advanced ML model, including those that weren't trained on a giant corpus of text.
isotypic 15 hours ago [-]
To quote Noam Brown, "Our main focus is shipping great models so everyone can use them to make discoveries of their own" [1]. Spending 15 million dollars of compute with an internal model to blitz and scoop a resolution of Navier-Stokes that someone is already on track to resolve (or, from my understanding of why OpenAI did this, had already resolved, as staff at OpenAI stated their attempt was prompted by rumors of a resolution by Anthropic) is in complete opposition to this stated claim. That act is what this portion of the recommendation is in response to: OpenAI perpetually holding out internal models and tooling, using them to solve important problems in math, and thus themselves holding a monopoly on certain aspects of mathematics. Why is it absurd to ask people who are in a position to do an obviously damaging act to not do so, especially when those same people were the ones who asked for your advice in the first place and claimed to not want to do said act previously?
Because it’s not a damaging act, to those who aren’t gatekeepers.
isotypic 15 hours ago [-]
How so? All it does it move who the "gatekeepers" are. OpenAI continuing to pump math problems into their internal models will only mean that if you want to be on the cutting edge of mathematics problem solving, you have to have access to internal OpenAI models. It takes what was previously an incredibly open field -- basically every paper is freely available on the arXiv, any given topic has probably 5 textbooks -- and turns it into one where the frontier is in the hands of a single company. If you are so against "gatekeeping" -- which I assure you, mathematicians are not doing, if you personally use an LLM to solve an important problem and write it up nicely (keyword: nicely) nobody will be mad at you -- why are you so happy to place the field into the hands of a singular entity?
quantumspandex 14 hours ago [-]
How about not trying to be always on the cutting edge ? Is mathematics a sport ?
shank-false-rin 45 minutes ago [-]
Doing research is by definition to be on the cutting edge.
AI labs don't even need models to out compete mathematicians, they can employ enough mathematicians to out compete those who are not affiliated.
The questions is whether if we want research to be held in a oligopoly.
trhway 15 hours ago [-]
>if you want to be on the cutting edge of mathematics problem solving, you have to have access to internal OpenAI models.
that is the way of technology. If i want to be just in the middle of the pack, not even a cutting edge, i have to buy a car, i can't just walk everywhere in reasonable time. I have to use a phone, i go to the doctor for modern medicine, etc. - note that pretty much everything is private companies goods and private services.
>where the frontier is in the hands of a single company.
We did invent anti-monopoly laws though to address such major issues with the private origin of the goods and services, and of course AI companies should be subject to it too (if you noticed, by stocking the scare of AI the BigAI companies are actually trying to get the anti-monopoly laws relaxed for them - and this is where our attention should be, the rest is just red herring)
>It takes what was previously an incredibly open field
It was nice riding a horse among the open rolling hills.
popalchemist 13 hours ago [-]
It directly undermines human achievement - something some people have spent a lifetime in pursuit of - for the sake of marketing. OpenAI gains nothing at all; the mathemeticians on the verge of achieving a lifelong pursuit have their entire career trajectory and perhaps reason for being eviscerated without a thought. If the goal is to further mathematics, the mathemeticians are the ones who would best understand whether what OpenAI is doing is helping or hurting.
fasterik 15 hours ago [-]
Frontier labs aren't under any obligation to release every model they develop. Either way, I don't see much evidence that they're withholding them in perpetuity. Open models aren't far behind so there's no incentive for that.
Difficult math problems are useful benchmarks and milestones. It's well worth throwing money at solving a problem if it drives competition and improves models significantly. The benefits become available to everyone, including the thousands of professional mathematicians who can use them to become more productive.
parisbs 9 hours ago [-]
> Open models aren't far behind so there's no incentive for that.
"But inference costs play a massive role here too, which I think is still the biggest bottleneck in AI development. Running high-impact models for math, medicine, or programming requires a ton of compute. Throw in the politicization of everything AI-related, and it feels like model privatization is just the tip of the iceberg."
trhway 15 hours ago [-]
>was prompted by rumors of a resolution by Anthropic
reminds an old sci-fi story where top engineers were shown a video of a genius inventor who invented anti-gravity and unfortunately died while testing his apparatus which was clearly anti-gravitating in the video. Thus believing that it is a solved problem, just need to rediscover the lost solution, the engineers quickly developed anti-gravity. The original video happened to be a fake specially created for that purpose.
>thus themselves holding a monopoly on certain aspects of mathematics
Monopoly prevents others. In science me knowing something doesn't prevent others from obtaining the same knowledge, especially in math where any knowledge is just a result of thinking.
The math establishment is trying to bring into the math the rules and notions similar to those of the patent and trademark laws. Those attempts should be outright rejected.
skeledrew 1 days ago [-]
> requests strike me as akin to gatekeeping
This is exactly what I was thinking as I read it, along with the bit that AI labs should support human understanding. The whole thing smells as they're trying to place this burden on labs that are just offering a tokens service; them publishing about particular topics is essentially a side quest in the first place IMO. If a community wants to create Math labs dedicated to understanding AI discoveries in the field, then they're free. If they want to petition AI labs for financial support, they're also free. But this wording where they're trying to dictate what AI labs should do (outside of their primary business) just smells.
knuckleheads 1 days ago [-]
I don't see why mathematicians should be protected from AI anymore than any other profession. It's either everybody or nobody, not fair on the face of it otherwise.
seanhunter 1 days ago [-]
TFA isn't asking for mathematicians to be protected from AI. It's asking AI labs to hold themselves to the standards of the mathematical community:
- releasing papers using the normal process to allow peer review
- giving talks etc to disseminate knowledge so humans understand the result
- writing papers in a way (standard terminology etc) that allows mathematicians to digest the result (some AI math papers comprise a huge verbose load of non-standard terminology and waffle and then a massive lean proof. This is very hard for humans to actually understand, and means it's hard for others to take the work forward.)
- giving appropriate credit to results that are used to derive the work
It includes some specific recommendations for situations where the person prompting the model is not in a position to understand the output, and frankly these are really welcome given situations like the recent case at Anthropic where a non-mathematician at Anthropic prompted claude to make a significant improvement to the bounds of a problem related to the Riemann Zeta function[1] which led to widespread misreporting and claims (not by Anthropic themselves notably) that the Riemann hypothesis itself had been proved, which is emphatically not the case.
Research mathematics is fundamentally a collaborative activity and the way in which some of these results are released is done to maximise PR but means a ton of the mathematical value is left on the table.
[1] https://www.anthropic.com/research/riemann-zeta. As I understand it, the Riemann Hypothesis says that all non-trivial zeroes of the zeta function lie on a line called the critical line. Two centuries of previous work had established that at least something like 40.9% of the zeroes lie on the line and noone has ever found a non-trivial zero that does not lie on that line. Claude (with prompting from a non-mathematician to "try harder" etc) improved this bound massively to 67%. Now a lot of people said things like "OK so all we've got to do is to improve that to 100% and we've proved the RH", which is definitely not true unfortunately, because you can say that in the limit the proportion of the zeroes on the line is 100% and still have infinitely many which are not.
T-A 17 hours ago [-]
From the opening paragraph:
we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models
From point 2 under section "1. Background":
AI labs should provide significant support, including funding, to help develop this understanding
Point 3 under the same section:
The development of human understanding must remain organic and community led. It should not be directed by AI labs, even when the labs have produced the results.
If, as you say, they are "asking AI labs to hold themselves to the standards of the mathematical community", I must conclude that the mathematical community
1. Ideally wants a monopoly on mathematics research.
2. Demands money from those who dare violate their ideal monopoly.
3. Insists that the ideal monopoly remain in charge.
curt15 14 hours ago [-]
Your monopoly theory lacks a credible motive and also ignores the fact that your quote specifically targets the frontier labs, not LLMs or computational tools in general. The group (https://agmai.org/) comprises of world-renowned mathematicians, including several Fields medalists, who have nothing left to prove. If they felt like it they could quit mathematics today and take private sector jobs paying far more than their professor salaries.
The basic purpose of mathematics has always been human understanding (see for example
https://mathoverflow.net/a/44213), and the working group's various recommendations simply aim to ensure that computer-generated mathematical activity aligns with that purpose.
T-A 3 hours ago [-]
> Your monopoly theory lacks a credible motive
The motive is obvious and all over the comments. For you convenience, it can be summarized as gatekeeping. Start unpacking it and you will find ego (these are people at the top of a cloistered social pecking order who suddenly see that order threatened), entitlement (their writing reflects a belief that they own mathematics and have the right to dictate how it's done by others) and economics (academic funding follows prestige; being outdone by machines undermines the prestige, hence the funding).
> and also ignores the fact that your quote specifically targets the frontier labs, not LLMs or computational tools in general
That's like saying "your opposition to autonomous weapons ignores the fact that they specifically target [INSERT FAVORITE TARGET]". I ignore what's irrelevant.
If Ken Griffin had not chosen to donate $3 billion to CMU [1] and instead had spent that money on tokens to solve a bunch of open problems [2][3] using publicly available LLMs, do you seriously believe their reaction would have been more positive?
> The group (https://agmai.org/) comprises of world-renowned mathematicians, including several Fields medalists,
I am well aware of who they are, thank you very much.
> who have nothing left to prove.
In the old order now under threat.
> If they felt like it they could quit mathematics today and take private sector jobs paying far more than their professor salaries.
A common delusion among students and maybe even some professors who've never set foot outside academia. The authors of TFA know better, of course. Their current conditions amount to enjoying a comfortable living while pursuing their favorite hobby full time. In the private sector they would have to justify their salary by (gasp) working on something (OMG!) applied with a reasonable prospect of (would you believe it?) meaningfully contributing to their employer's (pardon my French) bottom line. A revolting thought, and prospective employers know that too. Both parties know it wouldn't work.
> the working group's various recommendations simply aim to ensure that computer-generated mathematical activity aligns with that purpose
Yes. It’s remarkably flagrant, and it’s an example of an academic mindset that’s likely been holding progress back across multiple fields. It’s about to be broken rather badly, and it will show where progress has been stifled.
dotancohen 15 hours ago [-]
Math is a tool. It's what people do with the tools after they are invented that matters - and with mathematical tools sometimes that wait is hundreds of years.
The mathematics community has found that the tools have value long after being invented if the inventor leaves his tools in a specific format. The community is doing its best to preserve that - not because they want less toolmakers, but rather because they want the tools to be useful when they become needed.
shank-false-rin 28 minutes ago [-]
The mathematical community isn't really a single entity. It's weird to call that a monopoly.
An (obviously not 1-to-1) analogy is Amazon uses their platform sales data to push out almost identical AmazonBasic branded products. Small businesses can't compete, because Amazon has the economy of scale.
I am not saying that what Amazon doing is "wrong" per se, but I bet if small businesses have a voice they'd also ask Amazon to stop blatantly coping their most popular products.
It'd be weird to say that "the small businesses wants a monopoly on [insert product]."
knuckleheads 1 days ago [-]
Maintaining the existing standards is in fact a form of protection from AI disruption. They are asking the AI companies to follow their norms, instead of them having to conform to new norms created by AI. They don’t want to have to change the way they do things, understandably!, and are asking the companies to accommodate their way of life.
curt15 14 hours ago [-]
New norms are only worth adopting if they are clearly better, and that is far from obvious for whatever norms the proprietary AI companies are trying to push. Also their letter specifically targets proprietary AI companies, not computational tools in general which mathematicians do use when they advance mathematical understanding.
knuckleheads 8 hours ago [-]
Better for who? If I don’t care about the welfare of mathematicians and just the advancement of mathematics, would AI doing the math not be better for me?
As well, if the open models were as good as the proprietary ones, do you think they wouldn’t still have complaints?
munksbeer 1 days ago [-]
Those seem reasonable apart from this one:
> - releasing papers using the normal process to allow peer review
I'm not an academic but I've heard enough stories about how this can very much act as gatekeeping that I don't think it is a good request.
aprilthird2021 1 days ago [-]
Mathematician was never really a "profession" like the others. It doesn't pay well and is largely confined to academia. If you're really good and want to get paid, you don't do the kinds of problems AI have been taking a crack at. You go to a quant firm or some tech company where this kinda math actually matters once in a blue moon
nxpnsv 1 days ago [-]
But on one hand there are labs that pushes tens of millions $ to mine for publicity, and on one hand mostly underfunded researchers trying to improve general understanding. Big ai may seriously harm math and when Pr value diminishes down nobody is there to keep pushing.
dr_dshiv 1 days ago [-]
“researchers trying to improve general understanding”
Pretty sure AI will do better for that.. mathematicians need to be centaurs like the rest of us and stop rhetoric that is going to make existing math centaurs feel like they might get math-cancelled
nxpnsv 1 days ago [-]
My point is that there is a real difference between massive ai company budgets and research mathematicians driven it, it is not an argument against use of ai which seems inevitable.
aeve890 16 hours ago [-]
>mathematicians need to be centaurs like the rest of us and stop rhetoric that is going to make existing math centaurs feel like they might get math-cancelled
Centaurs?
hidroto 14 hours ago [-]
'centaurs' is a term from chess, where people play chess with computer assistance.
(An inseparable fusion of two distinct animals, in this case human/AI. HN uses the word often where "cyborg" would fit. Also: "reverse centaur", where the human part is lacking a head).
api 14 hours ago [-]
I prefer thinking of human AI fusion like a lichen, which is a close symbiosis at the cellular level. That’s what it would look like.
Davidzheng 1 days ago [-]
You're not alone, even among mathematicians.
socializer 16 hours ago [-]
> akin to gatekeeping how someone should breathe air.
Yes, and professions that do that - that vocally insist there is an essential human element to the craft - will likely fare better than the ones that say "whatever, code is code" or "whatever, math is math".
This is smart. We might not like it, but I don't know what's the end game for SWEs with their utilitarian attitudes to AI. We're digging our own grave. Meanwhile, professions such as writers or musicians are positioning themselves better by shunning "artists" that simply pull the lever. If you post gen AI poems or gen AI drawings on any artist forum, you're going to be eaten alive.
fapjacks 14 hours ago [-]
The difference, of course, is that people don't care if their spreadsheet is written by an artisanal human programmer or just some stupid, swarthy Idea Guy banging on a keyboard. It seems that a lot of people do however care if the music they listen to or the art on their living room wall is made by a human or by a machine. Maybe it's about "positioning" but I don't think so. Most people want to pretend they are creatures of finer taste than everyone else around them. In this regard, I'm waiting for the other shoe to drop on AI in (for example) music, and there being kind of lip-syncing scandals with popular musicians.
curt15 1 days ago [-]
The problem with proprietary models is that you don't get to poke inside and see what it is doing and how it arrives at its answer, which is precisely what mathematicians do. Mathematical understanding derives less from any particular result than the insights and methods that pave the road to results. Whenever a theorem is proved, researchers seek to unpack the proof and get inside the author's mind to learn their ways of thinking.
LLM generated results might benefit mathematical understanding if people can inspect their intermediate reasoning traces to discover erroneous human biases or patterns that they might have previously overlooked. Otherwise, the results might as well be produced by oracles.
killerstorm 15 hours ago [-]
The proof itself can be studied, either with AI or not.
"Intermediate reasoning traces" might be interesting but they are not by any means requited.
This letter has nothing to do with reasoning traces anyway: they just don't want math research to be front-run by internal models, that's all.
gus_massa 13 hours ago [-]
Also, in math most "intermediate reasoning traces" are erased and the solution only shows a simplified path that many times is only visible after the proof is complete.
semiquaver 16 hours ago [-]
> The problem with proprietary models is that you don't get to poke inside and see what it is doing and how it arrives at its answer, which is precisely what mathematicians do
Uh, a hypothetical fully open source model would have the exact same problem, because LLMs rely on emergent phenomena and no one understands why they work.
trhway 16 hours ago [-]
>Otherwise, the results might as well be produced by oracles.
no. The math result is a result only when it includes proof. The proof is the value here. The way somebody came to it isn't really important - we don't know how Newton came to his results, whether it was apple or pear, and it isn't really important. Or how Einstein was walking the city streets looking at the tower watches - it is just historic curiosity having no real value for science.
That has been one of the greatest thing about math departments - smooth talkers were always clearly visible as smooth talkers. You're either producing proofs, or you're anything but a mathematician.
I feel for mathematicians. They have similar situation like we have in programming. Well, we all just have to evolve and adjust (in particular reign in our pride as just in a few years - i think once LLMs start hitting 100T+ - we may loose our "top of God's creation" position). Any attempts at gatekeeping, ludditing, organizing in quasi observational/advisory boards really intended to protect their tenures, etc. ... - well, you just can't stop the wave.
It all reminds how Catholic Church insisted on responsible release of the Bible in German. The Church even unleashed the devastating 30 Years War trying to protect its monopoly on religion including the right to sell indulgences, etc.
>AI labs should provide significant support, including funding
And now all those "responsible math" and advisory boards would like to preserve their monopoly on math and would like to sell the indulgences to the AI labs. As usually it is all about money and power, not about science. As a Math PhD dropout myself i feel a bit of a shame and disappointment for that undignified scramble by the mathematics establishment. Being smart they should have led the way and show an example to the rest of humanity ...
curt15 2 hours ago [-]
> The math result is a result only when it includes proof. The proof is the value here. The way somebody came to it isn't really important - we don't know how Newton came to his results, whether it was apple or pear, and it isn't really important. Or how Einstein was walking the city streets looking at the tower watches - it is just historic curiosity having no real value for science.
This mischaracterizes the role of rigourous proofs in mathematical understanding. While undergrads and early grad students focus primarily on proofs, formalism recedes into the secondary role of honing intuition as mathematicians transition to their "post-rigourous" stage of development[1].
The importance of intuition in mathematics cannot be overstated. That's why people go to math talks even when the actual results are codified in papers. In a less formal setting, they get to pick the author's brain to learn their mental pictures and heuristics that don't make their way into papers. Those would be analogous to chain-of-thought traces and agent-to-agent messages for a computer generated result.
When I was a postdoc in genomics, one of my supervisors had a background in mathematics. Current bioinformatics training pipeline didn't exist yet, so most PhD students and postdocs had a background in something like CS, mathematics, statistics, or physics.
From the supervisor's perspective, people coming from pure mathematics were good at thinking about definitions. Coming up with useful definitions was the primary value they created, while theorems and proofs were just technical stuff they did to evaluate the value of proposed definitions.
My own background was in theoretical computer science, specifically algorithms. When you do algorithms without any qualifiers, you are studing them as mathematical objects in a simplified model of computation. The process often starts with a promising algorithmic idea. But if it looks like you can't prove anything nontrivial about the idea, you often stop studying it, regardless of the actual value of the idea. And if you manage to prove something, you start optimizing the algorithm for your theoretical model in order to prove better results. That usually makes it worse in practice.
The end result is that algorithms papers, both good and bad, typically contain theorems and proofs about algorithms nobody cares about. If you are a practicioner, you need to dig through all that noise to find the core algorithmic ideas, so that you can evaluate them in a more realistic setting. And if you are a theoretician, you are probably more interested in the techniques used in the proofs (which may also inspire future algorithmic ideas) than in the actual results.
You can find plenty of other similar situations. The value mathematicians create is rarely in the theorems and the proofs.
trhway 14 hours ago [-]
notice that you're talking about areas other than mathematics. I was talking about mathematics.
jltsiren 14 hours ago [-]
I was talking about the value created by mathematics. Which largely comes from training people to think about technical details. Which typically manifests as new ideas based on deep technical understanding of earlier ideas.
Chance-Device 1 days ago [-]
You’re not alone at all, gatekeeping is exactly what this is and that’s clear from the text almost immediately.
fph 15 hours ago [-]
> akin to gatekeeping how someone should breathe air.
The law is also a sequence of letters available to anyone. Yet you cannot practice without passing the bar exam. And that is not the only example; many professions can only be practiced by licensed individuals: medical doctors, journalists, electricians...
trhway 15 hours ago [-]
>The law is also a sequence of letters available to anyone. Yet you cannot practice without passing the bar exam.
You still can be punished for violating a law even if you hadn't passed the bar exam. So even without bar exam, you're supposed to know that sequence of letters and apply in your life exactly because it is available to anyone (until a law is officially published, it usually has no force)
mathisfun123 15 hours ago [-]
jesus christ the number of asinine contrarian comments here is unbounded apparently
> you cannot practice without passing the bar exam
you cannot represent other people in a court of law without passing the bar. you can absolutely read the law and represent yourself pro se.
> licensed individuals: medical doctors, journalists, electricians
first of all you don't need a license to practice journalism, the press credentials you're thinking are for the news agency itself. second of all doctors and electricians are licensed because there are material liabilities. you should read some more of that law that you're gatekeeping...
EDIT:
> News agencies and media organizations issue internal press credentials to verify a reporter's affiliation and identity, while official access credentials are provided by specific host institutions, government bodies, or event organizers.
EDIT2: it's hilarious to me that people here (again because of pure contrarianism and maybe anti-AI sentiment) are really arguing for a return to mathematical guilds. what's next? destroying all gutenberg presses?
EDIT3: freedom of the press is literally in the first amendment
ComplexSystems 9 hours ago [-]
It seems that, despite spending decades obsessing over their various niche problems, they have little interest in a magic button that can immediately produce the answer. At best they view this button as a mild nuisance they need to contain somehow.
mmaunder 1 days ago [-]
Humans have been trying to outlaw thinking for some time now.
xanderlewis 1 days ago [-]
What (some) mathematicians are asking for, whether one agrees with it or not, is exactly the opposite of 'outlawing thinking'.
12 hours ago [-]
bananaflag 1 days ago [-]
Note that they're saying "proprietary models". Probably open models will reach this level in a year or so and then this discussion will be moot.
omnicognate 1 days ago [-]
> gatekeeping how someone should breathe air
Air is there for all to breathe. It's a more-or-less fungible, free resource for all to use and the consumption of it is a basic requirement for life.
The AI companies, in contrast are using vast financial, human and compute resources to train and operate specialised models that are not available to the public. The advisory group's job is to give non-binding advice on how they can use this privately owned technology in a responsible manner that avoids doing unnecessary harm to the mathematical community. To call that gatekeeping misses the point entirely.
dr_dshiv 1 days ago [-]
The AI models are most definitely available to the public with a maximum of what, a 3 month lead while they do testing?
omnicognate 1 days ago [-]
The very first paragraph of the article says (emphasis mine):
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind.
The Navier Stokes proof came from an internal model that AFAIK still has not been released even in a limited way to scientists, let alone to the general public. Publicly available models are not what these mathematicians are talking about.
eli_gottlieb 15 hours ago [-]
Air is both free as in beer and free as in freedom. Proprietary language models are neither.
psychoslave 1 days ago [-]
Air is not impossible to disrupt in quality as side effect of industrial hubris.
AndrewKemendo 14 hours ago [-]
I agree that this reads like they are clinging to the past for its own sake, rather than to have more predictable systems.
I understand the instinct here, I just disagree with the idea that we should limit mechanical automation to the rate of human understanding. That’s a very low ceiling
Joker_vD 1 days ago [-]
> gatekeeping how someone should breathe air. It's math... the numbers and symbols are just out there in the platonic realm available for anyone to do as they like with them. It's patently absurd to request other people to stop.
The chemical compounds of all kinds are also out there, available for anyone to do as they like with them. Say, mixing ammonium chlorate with peroxide, why not? Or potassium permanganate with powdered aluminum. It's patently absurd to request other people to stop.
pfisch 1 days ago [-]
It's illegal to make weapons out of chemicals because the government has a monopoly on violence.
Mix together whatever you want, and then you can fight the feds when they show up to enforce their monopoly.
mathisfun123 1 days ago [-]
> The chemical compounds of all kinds are also out there,
1. Lol but they're literally not
2. A sample of chemical compound and a piece of math don't share literally any ontological qualities - you might as well have tried to make a comparison between math and nude pictures of the President
Joker_vD 1 days ago [-]
They literally are though.
> you might as well have tried to make a comparison between math and nude pictures of the President.
The latter is but a very large number, interpreted in a particular way. Have you heard of "illegal numbers"? Yeah, apparently they exist. That damn state, treading on literally everything that humans might do as if it's any of its business.
I find somewhat funny and interesting the different posts on how the mathematics community should handle AI. I understand this is entirely new, but I do get a feeling that most of these posts "try to control" something that basically cannot be. How the world will use AI for mathematics is not something any small community of mathematicians can decide. It is a bit like trying to define how language should be used, when in practice, it will evolve whether the group want it to or not. For AI in mathematics, the new equilibrium will be build by a "market" of users, each preferring certain bits of using or not using AI. The attempt to formalize and orient this, like a centralized economy, is I believe doomed to failure. That does not mean people should probably share how they see it for themselves.
hodgehog11 58 minutes ago [-]
It is not deciding control, it is determining agreed values in policy form. At least, that is how I see it.
mchusma 1 days ago [-]
Either mathematical progress helps advance society, in which case progress is a good thing.
Or mathematics is more like a hobby, and while ai may spoil their fun, they need to move on like chess and go players.
xanderlewis 1 days ago [-]
That’s a false dichotomy.
Mathematical progress does (quite obviously, on the whole) help advance humanity, so progress is a good thing. The problem is that defunding mathematicians and handing over control to AI and the companies that create them will cause the subject to stagnate. Sure, for a while we might get progress on existing questions using (perhaps quite novel) combinations of existing techniques, but, so far, given the character of the results we’ve seen, there’s no indication that it will continue indefinitely. Even if it did, what would be the point? Huge textbooks full of work no one can understand or benefit from?
One possible analogy is that humans work to add new points to the space of mathematical knowledge, and AI then fleshes this out to attain the ‘convex hull’ of these points. Essentially, humans ‘invent’ the definitions and pose the questions and AI does the grunt work as well as some creative exploitation of known results and tools to bring down all the low-hanging fruit that follows (important note: what appears to be non-low-hanging fruit to us may in fact be technically low hanging once AI is involved; we saw this for example with the Jacobian conjecture). This seems to be the current situation, and to argue that humans are fully replaced it is necessary to argue that AI is adding points outside the convex hull of human mathematics. A sufficient example would be a first-principles AI proof using alien techniques, and this we haven’t seen so far.
The mathematics-chess comparison is, to put it bluntly, nonsense. I see where it comes from, but, as absolutely anyone with any research experience will tell you, mathematics is orders of magnitude (and this really isn’t strong enough) more open-ended, and doesn’t consist of a game one is seeking to ‘win’. The goal is understanding itself.
DoctorOetker 16 hours ago [-]
I think we need to start looking at another approach, can we create interactive reflex games that upload the knowledge from LLM's or perhaps domain specific ones for formal mathematics, so that mathematicians get a similar level of access to the domain of discourse as the LLM?
winwang 1 days ago [-]
Or mathematicial progress helps advance society and AI mathematical velocity doesn't offset certain blows to human mathematical velocity yet, so we get to lose progress for the good of an AI company's advertisement.
auggierose 1 days ago [-]
We don't lose progress by AI solving Navier Stokes. Get a grip on yourselves. I don't think there is a "progress" argument here without tying mathematics to "usefulness", and AI makes mathematics dramatically more useful.
simianwords 1 days ago [-]
I agree. Its so strange to see an institution externalising their specific problems. If they have a problem, they should adapt and fix it amongst themselves.
mrheosuper 1 days ago [-]
Whatabout mathematical progress that ruins humanity even more ? e.g a much more addicted algorithm than tiktok/facebook reel.
avazhi 8 hours ago [-]
What about it? That’s not a mathematical issue.
That’s like saying ‘what about the fact that a man has to walk up to his ex wife to shoot her in the head’ as an attempted criticism of walking.
unddoch 1 days ago [-]
For every important match problem solved by AI, without mathematicians we wouldn't know about the existence and importance of the problem.
Famous mathematical conjectures are social constructs, formed by decades of even centuries of attention given to them by members of the math community. Without it, the danger is that future math "progress" will be reduced to generating tables of Lean statements and a probable/unprovable bit generated by AI.
jillesvangurp 7 hours ago [-]
You make a good point about LLMs being (so far) great at identifying solutions to complex puzzles but not yet coming up with their own theories, questions, etc. The business of finding interesting questions to answer rather than answering them is the essence of what scientists do. Good science identifies more questions than it answers.
With Fermat's theorem, the genius was in the original theorem. Which then caused generations of mathematicians to break their heads over trying to prove it correct. Fermat didn't write down a proof. His theorem was famously just a scribble in a side line of a book. Probably it was something that he had an hunch about that he couldn't quickly falsify.
I don't think AI is being used much for coming up with new problems yet. But I don't see why that would not be possible either. It's the obvious next frontier after AI clears the backlog of existing theorems. I imagine scientists are already using AI to find new interesting problems to work on and generally explore the problem space. But fundamentally, the reflex of asking or imagining "if this is true, what else could be true" is something that distinguishes people from AIs. For now at least.
Xirdus 1 days ago [-]
Could just be a sampling bias. Humanity had something like 3000 years to make famous conjectures, whereas AI mathematicians have been around for a month or so. Give them time, I'm sure they'll start formulating highly consequential unsolved problems soon enough.
amoss 1 days ago [-]
Somewhat tiring that as alway any criticism is reduced to "but have you tried this on the latest model".
Xirdus 23 hours ago [-]
Those are the two extremes, both are just as bad. Don't excuse shortcomings of the current models with promises of future improvements. But also don't demand literal miracles in 21 business days.
auggierose 1 days ago [-]
Maybe tiring, but that is the reality. Note that mathematicians are only upset now that "have you tried this on the latest model" works for so many of their problems now, but didn't for the model before that.
eru 1 days ago [-]
Not just sampling bias, but also human bias.
In a sense, how do you know whether the problem your AI has just solved is important? A simple proxy is to just check whether humans have thought it's important.
That's also why famous open problems are a good benchmark or proxy: you don't need to convince the rest of the world that the problem your lab's new AI just solved is actually useful or hard.
xanderlewis 3 hours ago [-]
Not all of the 'famous' open problems 'mathematicians have failed to solve for decades' are actually that famous though. In some cases they've remained open because no one cared or had even heard of them. Those outside the mathematics community seem to think that all mathematicians can, and do, work on essentially all problems (for example, most mathematicians have in mind the millennium problems as a goal), but this is very far from the case. Most serious problem statements are not even particularly understandable to most mathematicians, let alone workable on.
There's also a difference between important and hard. There are important problems that turn out to be easy, and hard problems that turn out to be useless.
As usual, I think everyone would agree that something like "curing cancer" would be both hard and important!
I think the AI labs' current obsession with showing off mathematical results to uninformed outsiders is a cheap trick. If they were really interested in 'enabling human flourishing' (rather than just wowing, by any means possible, investors with more money than sense), they'd be showing off cures to diseases rather than solving obscure problems in combinatorics only previously considered by three Russians fifty years ago and then declaring that The Singularity is here.
xanderlewis 1 days ago [-]
> AI mathematicians have been around for a month or so.
LLMs have been around for years, and they're explicitly trained on the entire history of human mathematics (without which they'd be unable to do anything).
Xirdus 23 hours ago [-]
Yes, and they've done absolutely fuck all with this knowledge until very, very, very recently.
xanderlewis 3 hours ago [-]
That is what I'm saying. There's a gold rush to extract what can be done with LLMs, but no indication that it'll continue. Maybe it will, but the evidence for its continuation seems to be entirely hype based on the fact that we've suddenly thought of new things to try. It's not because 'the capabilities are improving at a staggering rate omg the singularity is here guys'; it's mostly because no one bothered to spend enough time and money (and currently absurd amounts are required) to give it a serious shot until recently.
I like the test proposed by Demis Hassabis. Something like: train an LLM on pre-Einstein physics and see if it can rediscover what Einstein did. Despite the incessant noise online, we seem to be no closer to this. If there's material evidence that we are, I'd sincerely like to hear about it!
yuzuquat 4 hours ago [-]
I'm curious how this will ultimately pan out. Presently, we scrutinize frontier labs because they're the ones able to easily deploy large agent swarms. It's not hard to imagine that as costs come down and models become better, smaller and smaller labs/organizations will increasingly have this same capability.
As some back of the envelope math, Openai claimed to have used ~300b output tokens to solve navier-stokes. 4x$200 plans allows me to burn ~2-3b input/output tokens a day. What's to stop a few highly motivated individuals with a reasonable background in mathematics from spinning up their own massive agent swarms in 6-12 months?
I think this fragile truce between ai-powered mathematics and overarching mathematical community is just that - fragile. In the same way the stable diffusion broke the dam with the artist community, we're seeing the same here.
LelouBil 1 days ago [-]
I saw an interview (in French) of Cédric Villani, speaking on behalf of him and other Fiels medalists, who said that in order to advance mathematics we need three things:
- ideas
- students
- problems
Basically, students to bring original ideas to try and solve existing problems and this generates new ideas and possibly new problems for new students to try and solve with new ideas and so on.
And he continued to say that the issue with LLMs in mathematics, is that he's afraid they could run out of problems, and so students wouldn't bother trying, and this could hurt understanding of mathematics as a whole.
It's totally not my field so I'm not sure what to think of it but it seems important
mikestylz 11 hours ago [-]
In the past we might have said horses, stone, and bronze were three things necessary to advance civilization. In the future I'm sure the AIs will prompt themselves with nice problems. But for now I agree with AGMAI that we should be careful not to disrupt the field too severely. It's reasonable to keep it around even if it's just functioning as a temporary "strategic reserve of expertise"
fasterik 14 hours ago [-]
I'm not a mathematician either, but I find it hard to believe we could run out of problems in pure mathematics. Pure math is like a game where if you get bored you can invent new rules. Maybe eventually all fields related to applied problems will be settled, and the value of pure mathematics will be relegated to the status of other games like chess and go. But that's so far away from current reality that I can't imagine it happening in the next century at least.
avazhi 8 hours ago [-]
If you need people to invent problems to solve, you may not be working in a useful area.
Time to get training as a plumber or electrician, I’d say.
astaza123 1 days ago [-]
As an answer to AI companies, this is so bad: instead of trying to find a path to a win-win-ish solution with some trade-offs, this says: sorry, we cannot think of any, so just stop making money, will you? Math needs better crisis managers.
But as an idea, this is even worse: does it mean to stop potential research to cold fusion, cancer and anything as long as it may touch some mathematician's interests, or does it mean math is so hopelessly irrelevant that this cannot be the case... Again, as a crisis manager, this is not how you pose it.
Makes me wonder, were they hired by Sam to sabotage?
Animats 1 days ago [-]
This paper wants AI companies to pay human mathematicians to understand AI-generated stuff.
That's an unusual ask.
unddoch 1 days ago [-]
If youre going to spend 10 million dollars on 10000 agents trying to solve some important maths problem I think it's reasonable to ask for some grants to help digest whatever they came up with.
Or you could hire mathematicians and do it in-house, but I guarantee you grants to PhD students are cheaper than silicon value salaries.
ChickeNES 15 hours ago [-]
Why pay for grants when you can just spend the money on compute more efficiently?
eli_gottlieb 15 hours ago [-]
Because the compute isn't gonna turn AI slop into a clear, readable paper.
gus_massa 13 hours ago [-]
Not for now, but it's AI is getting better. A big step is "refactoring" a very long proof into a few intermediate lemas and theorems that are more inteligible and useful for other proof. It may take a few years or decades in some cases.
Anyway, I expect AI to be better at "refactoring", but for now a centaur is better.
hodgehog11 1 hours ago [-]
Actually, for what you are mentioning, it is getting worse. There was a sweet spot somewhere around the release of GPT-o3, and ever since, the LLMs have been getting more accurate at solving problems, but worse at explaining how, and to hone in on what is interesting. This isn't surprising, as RL strategies shifted from RLHF to RLVR, so priorities during learning changed. I don't expect AI labs to reverse course on this. We can expect AI proofs to become increasingly incomprehensible over time.
6510 14 hours ago [-]
They aren't buying tokens, more likely it costs them 1m to run the 10000 agents for 88 hours. Still leaves room for the PhD ofc.
I'm trying to map it on other fields and cant help but think it is absurd. If some company spends 1m developing a new alloy, should metallurgists be upset when they publish the recipe?
1 days ago [-]
fultonn 14 hours ago [-]
AI companies are already paying human mathematicians, to help train their models. Including to understand AI-generated stuff. The pay is kind of shit but still quite a lot better than what we were paid as phd students, especially if you do it 40 hours a week (which, tbf, was very short week in grad school).
I know this because I spent a lot of social time around a Math dept in grad school and many of them are moonlighting working for subcontractors on production and evaluation of training data [1].
Anyways, mathematics is a bit of a funny discipline... some buckets:
There are lots of obvious cases where progress has obvious applications for incremental progress, and where you can ask for novel math from the perspective of the application or ask for applications from the perspective of novel math. That seems like the sort of thing that you could throw $10M at get something valuable without much human input. I do this, at much much much smaller dollar amounts, pretty regularly, and with open models, so nothing secret/sota/etc.
But there are also lots of cases where some progress gets made on something very pure and esoteric and the implications aren't really clear. Those are cases where just asking a bunch of nerds to marinate on it while interacting with a messy world in the full social generality of a modern university could make a lot of sense. The connection between the four color theorem and register allocation, for example, feels very... hard to do without humans free-associating in a diverse academic environment.
And, at last, there are cases where there are subfields or problems that are big and important in the way that driving a rare car or wearing a particularly fascinating rolex is important. Some of modern mathematics feels like it has its cultural roots in intellectual gamesmanship amongst wealthy gentleman and/or those under their patronage. A lot of the... less well-argued... objections come from this set.
Like I said, math is kind of a uniquely weird discipline. It's got aspects of practical gritty science, noblesse oblige/high-culture, fashion/taste, craftsmanship, tutelage, and so on.
--
[1] Some enterprising journalist should track down all the subcontractors used by OpenAI for mathematics data, then talk to all of those people and figure out how much breakthrough-adjacent data work was happening... pure conspiracy but I'd be curious for someone to explore the hypothesis that there were maybe people helping find fragments here and there and it wasn't all purely mechanical monkeying.
1 days ago [-]
joeldw 10 hours ago [-]
Something I've been thinking about that I don't quite know how to formalize is the role of _understanding_ in the cycle of progress. The other component would be discovery - generating proofs, creating new algorithms, etc. Discovery pushes the boundaries of a field, but understanding disseminates that knowledge, allowing the field to stand on the shoulders of the new discovery and perhaps make yet another.
Understanding is really the measure of utility. I suspect that an average human's intelligence is similar to one 100 years or even a century ago, and thus our ability to discover new theorems and proofs. The difference is the understanding. We not only have the last thousand years of discovery, but also the compaction, categorization, education, tutorials, etc, that allowed us to learn it effectively in a few years or decades of academia.
AI is accelerating the discovery part, but the _understanding_ part is just as vital for true progress.
philipwhiuk 3 hours ago [-]
This is really the Grant Sanderson position - reframe mathematics from proof-search to understanding-distribution.
dsign 1 days ago [-]
> However, it is now the case that AI can output mathematical arguments in situations without the human who prompted it being able to understand the arguments, verify them, or take responsibility for them.
I read this as "anybody can prompt, few can understand". And "we need more who can understand". If we had more mathematicians (than we have today) all of them piloting advanced models, the pie would grow. The problem is that AI capabilities drain (by disincentivizing) the education pipeline that would get us those mathematicians, and if recent rumbles about what AI is doing to education are to be believed, it does so many years before students even get to grad school.
IMHO, it is not that bad. Not having any human who understands linear algebra after the Butlerian Jihad is a win :-) .
octoberfranklin 7 hours ago [-]
and if recent rumbles about what AI is doing to education are to be believed
They are.
__alexs 6 hours ago [-]
Do the labs have a genuine interest in mathematics beyond generating interesting marketing material?
13 hours ago [-]
1 days ago [-]
philipwhiuk 3 hours ago [-]
Some of this is minor reasonable adjustments, much of it will just be ignored.
charcircuit 7 hours ago [-]
>and we ask them to stop testing advanced mathematical problems on proprietary models.
Being able to move the frontier of maths as fast as possible goes against this. Seeing where the latest models can take us is a positive thing. It seems the desire is to make it possible for someone outside the lab able to write a prompt to get "credit" for it. This is a selfish way of thinking. Prioritizing the discoverer over the math itself.
>AI labs that release substantial mathematical output without immediate accompanying human understanding must take responsibility for ensuring that human understanding will follow. In particular, AI labs should provide significant support, including funding, to help develop this understanding.
This is just reads like asking for a handout. Beyond a Lean proof I don't think they are obligated for providing any more interoperability of it. You can already manually go through all the lean statements by hand if you want. Considering how powerful AI proofs are the value of "human understanding" does not seem to be that high. It seems much more valuable to improve AI's understanding of math than humans.
>The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline.
In such a world it may be worth funding another AI lab who would be willing to sell you access to a powerful model for mathematics. Just because mathematics did frontier math in the past, that doesn't mean they are entitled to be at the frontier forever. I would rather advocate for open (to the public) models based off the fact that it allows society to tackle more problems from more angles speeding up the progress of math over than trying to make it about enshrining the previous of class of people into the frontier forever.
>As for publicly available models, unequal access due to economic and other factors risks establishing a multi-tier hierarchy across the mathematical world and greatly exacerbating existing inequalities that arise from institutional wealth and technological privilege.
We should not slow the progress of Mathematics down because that might cause inequality. The speed of results coming out should be the number 1 priority here.
globalnode 12 hours ago [-]
So it seems mathematics is a perfect playground for LLM's to show off their abilities. Is this the end of mathematics done by humans? probably.. but that still wont stop some people from going into the field, just like whats happening to computer programming. I find it strange that the two fields LLM's are built on are the two its trying hardest to replace humans in. Why will we have to teach mathematics / computing to the majority of kids at school anymore if we don't need a workforce competent in those skills? Will we just teach kids how to type fast so they can get the prompt in asap?
edit: also ive noticed when i ask llms maths questions they tend to introduces overly complicated things that arent relevant or make the problem harder. so maybe we will teach kids the basics of maths and how to interpret results? i am unsure.
areoform 1 days ago [-]
I applied to the caltech mathaton.
While applying, I looked at the current SoTA, (briefly) read through some of the papers, and realized that I am very far away from understanding them.
Understanding one of these proofs is the work of several months, years or lifetimes depending on whether or not something clicks. It requires a kind of stamina that I quite frankly don't have, but I would like to develop.
If the mathaton's organizers accept my team, I realized that I would spend the next few years working through the result.
So why apply to the Mathaton?
"Many years ago the great British explorer George Mallory, who was to die on Mount Everest, was asked why did he want to climb it. He said, 'Because it is there.'
Well, [theoretical math] is there, and we're going to climb it, and [topology] and [number theory] are there, and new hopes for knowledge and peace are there. And, therefore, as we set sail we ask God's blessing on the [~~most hazardous and dangerous and greatest adventure~~] on which [we have] ever embarked."
More seriously, I applied because I was hoping to get access to the models without the veil. I don't think people realize just how big the gap is between what exists behind the scenes at these entities, and what we get out here.
And it's frustrating. Because I think it's within the rights of frontier labs to decide whether or not to sell access to a product, but the labs aren't just doing that. They're trying to thumb the scale to make sure that none of us ever get access to these models at peak performance. Ever.
And I think humanity is worse off for that. I am worse off for that.
I have studied the shape and structure of historical technological revolutions (and I've written about it), and usually the world doesn't realize how big of a big deal the big deal is because the big deal is often flawed, broken, and under-delivers. In the short term.
In the long term...? The world changed in the past few months. I think mathematics is one small part of that.
For most of human history, higher mathematics would have been inaccessible to me, and other outsiders, no matter how well heeled. Mathematics is, or rather was, a living discipline that existed piecemeal in a handful of minds across the world. These people's time was finite and valuable. To just meet them, you'd have to jump through hoops, and spend years proving yourself.
There is no price for an hour of tutoring from Terence Tao. But now, with AI? You can have an entity with the capabilities of Terry Tao help you understand the subtleties of math.
AI has changed what mathematics is. And every prominent mathematician seems to know it. They feel like mathematics has been devalued, and in some ways it has. Mathematics has gone from being a living discipline kept alive by a chosen few to a wellspring everyone can sip from. For the first time in human existence, learning and accessing higher mathematics doesn't involve jumping through hoops and knowing the right people. You can just ask.
I can just ask.
Except I can't. Because that capability is being gate kept. And I want to know. I want to climb the mountain.
wbl 1 days ago [-]
"There is no royal road to Geometry" - Euclid. Mathematical knowledge isn't from tutoring but doing. And no one is against the AI helping explain the tricky parts to help you practice. Rather it's about dumping a giant bunch of low quality text with some Lean claiming a big result is done.
areoform 1 days ago [-]
Yes, it's why I love math. You can't buy fluency.
There are subtleties to mathematics that aren't easy to understand from the written page alone. It's why it's a living medium.
For example, as we're talking about LLMs... why not, there are ways to reason about vector spaces that weren't intuitive for me to understand. It's something that required talking things out with a friend who is a practising mathematician (albeit in training).
I am not smart enough to reconstruct all of mathematics on my own from scratches on paper alone. That back and forth is necessary. And it's something that you couldn't have "bought" for cutting edge math at any price a few months before this point in time. Because it exists in the minds of people and it needs lots of back and forths with those people.
It's why LLM proofs can be slop on paper. A proof that no one can check or understand is not but scratches on paper. BUT LLMs are also the solution to the problem they create. The machines that can generate proofs are also machines that can help us understand them.
The living medium can now be represented and scaled inside of a machine. I can now sit down at an airport and have that discussion. I think that's transformative for our species.
echelon 1 days ago [-]
> You can't buy fluency.
Just wait until BCI and/or fast Pavlovian conditioning force fed by agents into human learners.
I've been vibe coding my own SRS software that is vastly superior to my learning style than Anki, and I know I'm just scratching the surface of accelerated learning. Who knows where this goes.
MRI and glucose injections with agent-tutor steering and millisecond feedback to learning?
We might be able to Matrix "I Know Kung-Fu" things into brains one day.
fragmede 1 days ago [-]
You might need to add in chemically altered states of consciousness and magnetic stimulation of the right areas, but yes.
omnicognate 1 days ago [-]
> Except I can't. Because that capability is being gate kept.
In what way are these "gatekeepers" stopping you asking an LLM questions about maths?
areoform 1 days ago [-]
I would like to ask you a question.
Can you, I, or any mathematician who isn't well connected (let's say someone who is a young Maryam Mirzakhani or just someone who is in grad school) learn from the system that produced the solution to the unit distance problem? https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
You will notice that it says on the first page,
"first mathematically generated in one shot by an internal model at OpenAI"
And I want to talk to models of similar aptitude and capability to help me understand nuances of the proof. Mathematicians will happy to pay for this. I've heard that people and non-profits are putting together $$$ for this to get access to these systems so that they can all interrogate them.
But the issue is that we can't. And I'm using the royal we here.
The paper says that the labs shouldn't release proofs from models that mathematicians can't interrogate. It's very clear that the models we get as users aren't the models used to produce the breakthroughs. And as LLMs display emergent capabilities, it's uncertain whether or not the model actually understands what it's explaining.
Because if I don't understand it. Professional mathematicians who are subject experts don't understand it. Then how do we know the model does? How do we know that it's correctly representing the proof produced by a more capable model? It's not logical to take any random model at its word, unless we can verify. Or, if it's the same model that produced the proof.
And that's what the mathematicians want. Access to the actual models.
magus_stoopr 14 hours ago [-]
basically: "pretty pweez don't forget us peeeeple"
point: imo, the authors have no clue just how psychopathic and broken the execs are for the inc's that make these models
raverbashing 1 days ago [-]
Honestly what is "AI generated math"?
Everything is WFFs all the way down
bbor 13 hours ago [-]
Not much to say on the dogfight, but I am yet again pleading for "artificial" as an adjective. Artificial mathematics!
simianwords 1 days ago [-]
I disagree with this.
OpenAI should be allowed to produce whatever it wants but it just can't claim that it has actually solved without the due process like peer review. If for example OpenAI solves a new conjecture, OpenAI should be free to publish it in their blog or arxiv in whatever way they desire. It can be slop, it can be non-slop. No one should police it.
Mathematicians are free to use it or discard it. They shouldn't externalise their concerns and restrict labs.
Mathematics is seen today as the noblest and most aristocratic of professions. Turns out, AI disrupts it because access to capital/compute now decides the results. Mathematicians don't like this corruption - understandable.
Its like guild of accountants opposing the calculator and require a responsible release. haha
tttr 1 days ago [-]
There’s a reason why many of your posts are downvoted.
13 hours ago [-]
ritualdevin 16 hours ago [-]
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aidiscoverywire 1 days ago [-]
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Der_Einzige 1 days ago [-]
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ted_dunning 15 hours ago [-]
In normal English usage, "with extreme prejudice" is commonly understood to mean that you advocate killing the person in question.
Either you don't know what you are saying and should improve your language skills
Or you do and should be dropped from this site for advocating violence.
avazhi 12 hours ago [-]
Mathematicians surely have copped the biggest L of the AI era - and their transparently self-serving attempts to gatekeep their ivory towery when we now have something that can do it better and faster is, frankly, pathetic.
Time to get some real jobs, guys. When you're asking other entities to slow down with the discoveries so 'real people' can have their fun, you might not actually be participating in a useful profession.
This same day will come for other professions like pilots, lawyers, doctors, and teachers, but it's funny that the lowest hanging fruit really was what we'd all most expect lol.
You have a very institutionalised idea of what publishing means.
I think the spicy take is definitely this stance that longstanding mathematical problems shouldn't be used as benchmarks for (specifically proprietary) models. Stated right at the very top.
The justification is pretty clear.
> The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline.
It is all fun and games (for non mathematicians) when mathematicians can't compete with AI labs but I think the more dangerous direction is when this starts being true for the rest of everything. For example cybersecurity or whatnot. Hence why I think Anthropics whole stance of being completely against open anything is actually *extremely* dangerous due to the centralization of power which they completely ignore as a risk factor.
The most discussable thing in this is certainly the idea that labs should fund mathematicians to do expositions.
> One of our principles is that AI labs have a responsibility to provide support, including funding, for the development of human understanding of the AI mathematical output that they release.
Obviously this directionally sounds like the role of mathematicians would be shifting towards interpreting AI results instead of making proofs. I'm not sure who would should really be billed for that. Plus how would you decide who gets the grant?
An interesting thing is that by stating that that the lab that dropped the result provide the funding, this is *directly* proposing an example of taxing AI labs for displacing knowledge workers.
If Ramanujan could freely disclose and publish results that were arrived at by literal divine inspiration (he certainly didn't bother to write down "proofs" in the modern sense!) why do we suddenly need all this "responsible release" process stuff when it comes to AI-derived results? This makes no sense when compared with existing norms within the mathematical community.
It's almost like a lot of non-mathematicians are commenting and have no idea about how the field really works. Or at least, how it works at the top level.
"The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline."
Are mathematicians' private brains unfair if they're way better at solving open math problems than anyone else? Are they required to share everything they're working on, as they're working on it?
Should Andrew Wiles have been condemned? Wouldn't these arguments all have applied to him? Besides him working at more human speeds.
The goal is to expand and extend our understanding.
When an LLM produces a proof, nobody (including the LLM) gained new understanding ... Unless, of course, a human being sits down and understands it, and communicates the new understanding to the broader community.
In a world, where expensive AI models running thousands of agents produce the most remarkable results, how should the discipline of mathematics look like, when money and access to compute plays such a pivotal role?
With the Navier-Stokes solutions being out in the wild, any mathematician at an impoverished university can make a stab at it and help our understanding. Assume the solutions being held back by those who produced them at Harvard & Co. (until they have their nice articles ready), what would the role of mathematicians be at less affluent universities?
I do not assume AI companies will be ready to provide equal access to their AIs to worldwide mathematics.
If they want to simply generate theorems and proofs and post them on the internet, why not? Does doing that really inflict "substantial negative externalities ... on the mathematical community"?
My point was about what is and isn't the responsibility of AI labs. Why should they be required to invest more (or fewer) resources into math than they otherwise want to?
(If you ask me, safety and security are what they need to invest more into.)
"Academics are paid paltry salaries, university departments are cash-strapped, why are they studying these abstruse math problems instead of building rockets?"
Successfully advancing fields of math and science is a good thing. Why would they not do it. Why would they not want to prove their increasingly intelligent computers are able to solve unsolved, complex math problems humans have failed to solve for centuries and then release the solutions to the public. What do you want them to do, not try to solve them?
The question is whether we want mathematics to be dominated by whoever has the most capital. This is perilous because mathematics doesn't produce profits like commerce does. The AI labs might end up killing something off that they are unable to sustain a replacement for.
This is literally the economic bet of the big labs in the broader economy. You use your relative advantage to front run or outcompete.
I'm not sure this argument will work given it is essentially an argument against the thesis the big labs use to justify their valuations.
All your jobs are belong to them.
OpenAI was started because Elon and Sam thought Demis was going to make himself dictator and they wanted a piece of that action (reading between the lines). All the labs, even Anthropic have the same idea dressed in different marketing gloss. The major exception (I imagine) are Chinese labs, since the Communist Party of China will demand control over these labs and their models. Good! Democratic human control is a good thing. Certainly, there is more democracy in China than in America. That's the idea of socialism, democracy for the workers, not for the capitalists (we do the inverse).
I really don’t see how that could be an oversight. It’s the goal.
Maybe I'm alone on this, but for some reason these sorts of requests strike me as akin to gatekeeping how someone should breathe air. It's math... the numbers and symbols are just out there in the platonic realm available for anyone to do as they like with them. It's patently absurd to request other people to stop.
Ensuring credit where credit is due? That's fine. If your model incorporates the efforts of many others, then it's reasonable to request acknowledgement of everyone who contributed (even indirectly). But that's not what the request states — presumably their ask subsumes any advanced ML model, including those that weren't trained on a giant corpus of text.
[1] https://x.com/polynoamial/status/2093451221273387477
AI labs don't even need models to out compete mathematicians, they can employ enough mathematicians to out compete those who are not affiliated.
The questions is whether if we want research to be held in a oligopoly.
that is the way of technology. If i want to be just in the middle of the pack, not even a cutting edge, i have to buy a car, i can't just walk everywhere in reasonable time. I have to use a phone, i go to the doctor for modern medicine, etc. - note that pretty much everything is private companies goods and private services.
>where the frontier is in the hands of a single company.
We did invent anti-monopoly laws though to address such major issues with the private origin of the goods and services, and of course AI companies should be subject to it too (if you noticed, by stocking the scare of AI the BigAI companies are actually trying to get the anti-monopoly laws relaxed for them - and this is where our attention should be, the rest is just red herring)
>It takes what was previously an incredibly open field
It was nice riding a horse among the open rolling hills.
Difficult math problems are useful benchmarks and milestones. It's well worth throwing money at solving a problem if it drives competition and improves models significantly. The benefits become available to everyone, including the thousands of professional mathematicians who can use them to become more productive.
reminds an old sci-fi story where top engineers were shown a video of a genius inventor who invented anti-gravity and unfortunately died while testing his apparatus which was clearly anti-gravitating in the video. Thus believing that it is a solved problem, just need to rediscover the lost solution, the engineers quickly developed anti-gravity. The original video happened to be a fake specially created for that purpose.
>thus themselves holding a monopoly on certain aspects of mathematics
Monopoly prevents others. In science me knowing something doesn't prevent others from obtaining the same knowledge, especially in math where any knowledge is just a result of thinking.
The math establishment is trying to bring into the math the rules and notions similar to those of the patent and trademark laws. Those attempts should be outright rejected.
This is exactly what I was thinking as I read it, along with the bit that AI labs should support human understanding. The whole thing smells as they're trying to place this burden on labs that are just offering a tokens service; them publishing about particular topics is essentially a side quest in the first place IMO. If a community wants to create Math labs dedicated to understanding AI discoveries in the field, then they're free. If they want to petition AI labs for financial support, they're also free. But this wording where they're trying to dictate what AI labs should do (outside of their primary business) just smells.
- releasing papers using the normal process to allow peer review
- giving talks etc to disseminate knowledge so humans understand the result
- writing papers in a way (standard terminology etc) that allows mathematicians to digest the result (some AI math papers comprise a huge verbose load of non-standard terminology and waffle and then a massive lean proof. This is very hard for humans to actually understand, and means it's hard for others to take the work forward.)
- giving appropriate credit to results that are used to derive the work
It includes some specific recommendations for situations where the person prompting the model is not in a position to understand the output, and frankly these are really welcome given situations like the recent case at Anthropic where a non-mathematician at Anthropic prompted claude to make a significant improvement to the bounds of a problem related to the Riemann Zeta function[1] which led to widespread misreporting and claims (not by Anthropic themselves notably) that the Riemann hypothesis itself had been proved, which is emphatically not the case.
Research mathematics is fundamentally a collaborative activity and the way in which some of these results are released is done to maximise PR but means a ton of the mathematical value is left on the table.
[1] https://www.anthropic.com/research/riemann-zeta. As I understand it, the Riemann Hypothesis says that all non-trivial zeroes of the zeta function lie on a line called the critical line. Two centuries of previous work had established that at least something like 40.9% of the zeroes lie on the line and noone has ever found a non-trivial zero that does not lie on that line. Claude (with prompting from a non-mathematician to "try harder" etc) improved this bound massively to 67%. Now a lot of people said things like "OK so all we've got to do is to improve that to 100% and we've proved the RH", which is definitely not true unfortunately, because you can say that in the limit the proportion of the zeroes on the line is 100% and still have infinitely many which are not.
we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models
From point 2 under section "1. Background":
AI labs should provide significant support, including funding, to help develop this understanding
Point 3 under the same section:
The development of human understanding must remain organic and community led. It should not be directed by AI labs, even when the labs have produced the results.
If, as you say, they are "asking AI labs to hold themselves to the standards of the mathematical community", I must conclude that the mathematical community
1. Ideally wants a monopoly on mathematics research.
2. Demands money from those who dare violate their ideal monopoly.
3. Insists that the ideal monopoly remain in charge.
The basic purpose of mathematics has always been human understanding (see for example https://mathoverflow.net/a/44213), and the working group's various recommendations simply aim to ensure that computer-generated mathematical activity aligns with that purpose.
The motive is obvious and all over the comments. For you convenience, it can be summarized as gatekeeping. Start unpacking it and you will find ego (these are people at the top of a cloistered social pecking order who suddenly see that order threatened), entitlement (their writing reflects a belief that they own mathematics and have the right to dictate how it's done by others) and economics (academic funding follows prestige; being outdone by machines undermines the prestige, hence the funding).
> and also ignores the fact that your quote specifically targets the frontier labs, not LLMs or computational tools in general
That's like saying "your opposition to autonomous weapons ignores the fact that they specifically target [INSERT FAVORITE TARGET]". I ignore what's irrelevant.
If Ken Griffin had not chosen to donate $3 billion to CMU [1] and instead had spent that money on tokens to solve a bunch of open problems [2][3] using publicly available LLMs, do you seriously believe their reaction would have been more positive?
> The group (https://agmai.org/) comprises of world-renowned mathematicians, including several Fields medalists,
I am well aware of who they are, thank you very much.
> who have nothing left to prove.
In the old order now under threat.
> If they felt like it they could quit mathematics today and take private sector jobs paying far more than their professor salaries.
A common delusion among students and maybe even some professors who've never set foot outside academia. The authors of TFA know better, of course. Their current conditions amount to enjoying a comfortable living while pursuing their favorite hobby full time. In the private sector they would have to justify their salary by (gasp) working on something (OMG!) applied with a reasonable prospect of (would you believe it?) meaningfully contributing to their employer's (pardon my French) bottom line. A revolting thought, and prospective employers know that too. Both parties know it wouldn't work.
> the working group's various recommendations simply aim to ensure that computer-generated mathematical activity aligns with that purpose
If so it completely misses the mark.
[1] https://abcnews.com/Business/wireStory/hedge-fund-ceo-ken-gr...
[2] https://www.unsolvedmath.com/
[3] https://www.openproblemgarden.org/
The mathematics community has found that the tools have value long after being invented if the inventor leaves his tools in a specific format. The community is doing its best to preserve that - not because they want less toolmakers, but rather because they want the tools to be useful when they become needed.
An (obviously not 1-to-1) analogy is Amazon uses their platform sales data to push out almost identical AmazonBasic branded products. Small businesses can't compete, because Amazon has the economy of scale.
I am not saying that what Amazon doing is "wrong" per se, but I bet if small businesses have a voice they'd also ask Amazon to stop blatantly coping their most popular products.
It'd be weird to say that "the small businesses wants a monopoly on [insert product]."
As well, if the open models were as good as the proprietary ones, do you think they wouldn’t still have complaints?
> - releasing papers using the normal process to allow peer review
I'm not an academic but I've heard enough stories about how this can very much act as gatekeeping that I don't think it is a good request.
Pretty sure AI will do better for that.. mathematicians need to be centaurs like the rest of us and stop rhetoric that is going to make existing math centaurs feel like they might get math-cancelled
Centaurs?
https://en.wikipedia.org/wiki/Advanced_chess
Yes, and professions that do that - that vocally insist there is an essential human element to the craft - will likely fare better than the ones that say "whatever, code is code" or "whatever, math is math".
This is smart. We might not like it, but I don't know what's the end game for SWEs with their utilitarian attitudes to AI. We're digging our own grave. Meanwhile, professions such as writers or musicians are positioning themselves better by shunning "artists" that simply pull the lever. If you post gen AI poems or gen AI drawings on any artist forum, you're going to be eaten alive.
LLM generated results might benefit mathematical understanding if people can inspect their intermediate reasoning traces to discover erroneous human biases or patterns that they might have previously overlooked. Otherwise, the results might as well be produced by oracles.
"Intermediate reasoning traces" might be interesting but they are not by any means requited.
This letter has nothing to do with reasoning traces anyway: they just don't want math research to be front-run by internal models, that's all.
Uh, a hypothetical fully open source model would have the exact same problem, because LLMs rely on emergent phenomena and no one understands why they work.
no. The math result is a result only when it includes proof. The proof is the value here. The way somebody came to it isn't really important - we don't know how Newton came to his results, whether it was apple or pear, and it isn't really important. Or how Einstein was walking the city streets looking at the tower watches - it is just historic curiosity having no real value for science.
That has been one of the greatest thing about math departments - smooth talkers were always clearly visible as smooth talkers. You're either producing proofs, or you're anything but a mathematician.
I feel for mathematicians. They have similar situation like we have in programming. Well, we all just have to evolve and adjust (in particular reign in our pride as just in a few years - i think once LLMs start hitting 100T+ - we may loose our "top of God's creation" position). Any attempts at gatekeeping, ludditing, organizing in quasi observational/advisory boards really intended to protect their tenures, etc. ... - well, you just can't stop the wave.
It all reminds how Catholic Church insisted on responsible release of the Bible in German. The Church even unleashed the devastating 30 Years War trying to protect its monopoly on religion including the right to sell indulgences, etc.
>AI labs should provide significant support, including funding
And now all those "responsible math" and advisory boards would like to preserve their monopoly on math and would like to sell the indulgences to the AI labs. As usually it is all about money and power, not about science. As a Math PhD dropout myself i feel a bit of a shame and disappointment for that undignified scramble by the mathematics establishment. Being smart they should have led the way and show an example to the rest of humanity ...
This mischaracterizes the role of rigourous proofs in mathematical understanding. While undergrads and early grad students focus primarily on proofs, formalism recedes into the secondary role of honing intuition as mathematicians transition to their "post-rigourous" stage of development[1].
The importance of intuition in mathematics cannot be overstated. That's why people go to math talks even when the actual results are codified in papers. In a less formal setting, they get to pick the author's brain to learn their mental pictures and heuristics that don't make their way into papers. Those would be analogous to chain-of-thought traces and agent-to-agent messages for a computer generated result.
[1]: https://terrytao.wordpress.com/career-advice/theres-more-to-...
From the supervisor's perspective, people coming from pure mathematics were good at thinking about definitions. Coming up with useful definitions was the primary value they created, while theorems and proofs were just technical stuff they did to evaluate the value of proposed definitions.
My own background was in theoretical computer science, specifically algorithms. When you do algorithms without any qualifiers, you are studing them as mathematical objects in a simplified model of computation. The process often starts with a promising algorithmic idea. But if it looks like you can't prove anything nontrivial about the idea, you often stop studying it, regardless of the actual value of the idea. And if you manage to prove something, you start optimizing the algorithm for your theoretical model in order to prove better results. That usually makes it worse in practice.
The end result is that algorithms papers, both good and bad, typically contain theorems and proofs about algorithms nobody cares about. If you are a practicioner, you need to dig through all that noise to find the core algorithmic ideas, so that you can evaluate them in a more realistic setting. And if you are a theoretician, you are probably more interested in the techniques used in the proofs (which may also inspire future algorithmic ideas) than in the actual results.
You can find plenty of other similar situations. The value mathematicians create is rarely in the theorems and the proofs.
The law is also a sequence of letters available to anyone. Yet you cannot practice without passing the bar exam. And that is not the only example; many professions can only be practiced by licensed individuals: medical doctors, journalists, electricians...
You still can be punished for violating a law even if you hadn't passed the bar exam. So even without bar exam, you're supposed to know that sequence of letters and apply in your life exactly because it is available to anyone (until a law is officially published, it usually has no force)
> you cannot practice without passing the bar exam
you cannot represent other people in a court of law without passing the bar. you can absolutely read the law and represent yourself pro se.
> licensed individuals: medical doctors, journalists, electricians
first of all you don't need a license to practice journalism, the press credentials you're thinking are for the news agency itself. second of all doctors and electricians are licensed because there are material liabilities. you should read some more of that law that you're gatekeeping...
EDIT:
> News agencies and media organizations issue internal press credentials to verify a reporter's affiliation and identity, while official access credentials are provided by specific host institutions, government bodies, or event organizers.
https://2021-2025.state.gov/media-credentials-and-visa-infor...
EDIT2: it's hilarious to me that people here (again because of pure contrarianism and maybe anti-AI sentiment) are really arguing for a return to mathematical guilds. what's next? destroying all gutenberg presses?
EDIT3: freedom of the press is literally in the first amendment
Air is there for all to breathe. It's a more-or-less fungible, free resource for all to use and the consumption of it is a basic requirement for life.
The AI companies, in contrast are using vast financial, human and compute resources to train and operate specialised models that are not available to the public. The advisory group's job is to give non-binding advice on how they can use this privately owned technology in a responsible manner that avoids doing unnecessary harm to the mathematical community. To call that gatekeeping misses the point entirely.
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind.
The Navier Stokes proof came from an internal model that AFAIK still has not been released even in a limited way to scientists, let alone to the general public. Publicly available models are not what these mathematicians are talking about.
I understand the instinct here, I just disagree with the idea that we should limit mechanical automation to the rate of human understanding. That’s a very low ceiling
The chemical compounds of all kinds are also out there, available for anyone to do as they like with them. Say, mixing ammonium chlorate with peroxide, why not? Or potassium permanganate with powdered aluminum. It's patently absurd to request other people to stop.
Mix together whatever you want, and then you can fight the feds when they show up to enforce their monopoly.
1. Lol but they're literally not
2. A sample of chemical compound and a piece of math don't share literally any ontological qualities - you might as well have tried to make a comparison between math and nude pictures of the President
> you might as well have tried to make a comparison between math and nude pictures of the President.
The latter is but a very large number, interpreted in a particular way. Have you heard of "illegal numbers"? Yeah, apparently they exist. That damn state, treading on literally everything that humans might do as if it's any of its business.
https://en.wikipedia.org/wiki/Illegal_number
Or mathematics is more like a hobby, and while ai may spoil their fun, they need to move on like chess and go players.
Mathematical progress does (quite obviously, on the whole) help advance humanity, so progress is a good thing. The problem is that defunding mathematicians and handing over control to AI and the companies that create them will cause the subject to stagnate. Sure, for a while we might get progress on existing questions using (perhaps quite novel) combinations of existing techniques, but, so far, given the character of the results we’ve seen, there’s no indication that it will continue indefinitely. Even if it did, what would be the point? Huge textbooks full of work no one can understand or benefit from?
One possible analogy is that humans work to add new points to the space of mathematical knowledge, and AI then fleshes this out to attain the ‘convex hull’ of these points. Essentially, humans ‘invent’ the definitions and pose the questions and AI does the grunt work as well as some creative exploitation of known results and tools to bring down all the low-hanging fruit that follows (important note: what appears to be non-low-hanging fruit to us may in fact be technically low hanging once AI is involved; we saw this for example with the Jacobian conjecture). This seems to be the current situation, and to argue that humans are fully replaced it is necessary to argue that AI is adding points outside the convex hull of human mathematics. A sufficient example would be a first-principles AI proof using alien techniques, and this we haven’t seen so far.
The mathematics-chess comparison is, to put it bluntly, nonsense. I see where it comes from, but, as absolutely anyone with any research experience will tell you, mathematics is orders of magnitude (and this really isn’t strong enough) more open-ended, and doesn’t consist of a game one is seeking to ‘win’. The goal is understanding itself.
That’s like saying ‘what about the fact that a man has to walk up to his ex wife to shoot her in the head’ as an attempted criticism of walking.
Famous mathematical conjectures are social constructs, formed by decades of even centuries of attention given to them by members of the math community. Without it, the danger is that future math "progress" will be reduced to generating tables of Lean statements and a probable/unprovable bit generated by AI.
With Fermat's theorem, the genius was in the original theorem. Which then caused generations of mathematicians to break their heads over trying to prove it correct. Fermat didn't write down a proof. His theorem was famously just a scribble in a side line of a book. Probably it was something that he had an hunch about that he couldn't quickly falsify.
I don't think AI is being used much for coming up with new problems yet. But I don't see why that would not be possible either. It's the obvious next frontier after AI clears the backlog of existing theorems. I imagine scientists are already using AI to find new interesting problems to work on and generally explore the problem space. But fundamentally, the reflex of asking or imagining "if this is true, what else could be true" is something that distinguishes people from AIs. For now at least.
In a sense, how do you know whether the problem your AI has just solved is important? A simple proxy is to just check whether humans have thought it's important.
That's also why famous open problems are a good benchmark or proxy: you don't need to convince the rest of the world that the problem your lab's new AI just solved is actually useful or hard.
There's also a difference between important and hard. There are important problems that turn out to be easy, and hard problems that turn out to be useless.
As usual, I think everyone would agree that something like "curing cancer" would be both hard and important!
I think the AI labs' current obsession with showing off mathematical results to uninformed outsiders is a cheap trick. If they were really interested in 'enabling human flourishing' (rather than just wowing, by any means possible, investors with more money than sense), they'd be showing off cures to diseases rather than solving obscure problems in combinatorics only previously considered by three Russians fifty years ago and then declaring that The Singularity is here.
LLMs have been around for years, and they're explicitly trained on the entire history of human mathematics (without which they'd be unable to do anything).
I like the test proposed by Demis Hassabis. Something like: train an LLM on pre-Einstein physics and see if it can rediscover what Einstein did. Despite the incessant noise online, we seem to be no closer to this. If there's material evidence that we are, I'd sincerely like to hear about it!
As some back of the envelope math, Openai claimed to have used ~300b output tokens to solve navier-stokes. 4x$200 plans allows me to burn ~2-3b input/output tokens a day. What's to stop a few highly motivated individuals with a reasonable background in mathematics from spinning up their own massive agent swarms in 6-12 months?
I think this fragile truce between ai-powered mathematics and overarching mathematical community is just that - fragile. In the same way the stable diffusion broke the dam with the artist community, we're seeing the same here.
- ideas
- students
- problems
Basically, students to bring original ideas to try and solve existing problems and this generates new ideas and possibly new problems for new students to try and solve with new ideas and so on.
And he continued to say that the issue with LLMs in mathematics, is that he's afraid they could run out of problems, and so students wouldn't bother trying, and this could hurt understanding of mathematics as a whole.
It's totally not my field so I'm not sure what to think of it but it seems important
Time to get training as a plumber or electrician, I’d say.
But as an idea, this is even worse: does it mean to stop potential research to cold fusion, cancer and anything as long as it may touch some mathematician's interests, or does it mean math is so hopelessly irrelevant that this cannot be the case... Again, as a crisis manager, this is not how you pose it.
Makes me wonder, were they hired by Sam to sabotage?
Anyway, I expect AI to be better at "refactoring", but for now a centaur is better.
I'm trying to map it on other fields and cant help but think it is absurd. If some company spends 1m developing a new alloy, should metallurgists be upset when they publish the recipe?
I know this because I spent a lot of social time around a Math dept in grad school and many of them are moonlighting working for subcontractors on production and evaluation of training data [1].
Anyways, mathematics is a bit of a funny discipline... some buckets:
There are lots of obvious cases where progress has obvious applications for incremental progress, and where you can ask for novel math from the perspective of the application or ask for applications from the perspective of novel math. That seems like the sort of thing that you could throw $10M at get something valuable without much human input. I do this, at much much much smaller dollar amounts, pretty regularly, and with open models, so nothing secret/sota/etc.
But there are also lots of cases where some progress gets made on something very pure and esoteric and the implications aren't really clear. Those are cases where just asking a bunch of nerds to marinate on it while interacting with a messy world in the full social generality of a modern university could make a lot of sense. The connection between the four color theorem and register allocation, for example, feels very... hard to do without humans free-associating in a diverse academic environment.
And, at last, there are cases where there are subfields or problems that are big and important in the way that driving a rare car or wearing a particularly fascinating rolex is important. Some of modern mathematics feels like it has its cultural roots in intellectual gamesmanship amongst wealthy gentleman and/or those under their patronage. A lot of the... less well-argued... objections come from this set.
Like I said, math is kind of a uniquely weird discipline. It's got aspects of practical gritty science, noblesse oblige/high-culture, fashion/taste, craftsmanship, tutelage, and so on.
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[1] Some enterprising journalist should track down all the subcontractors used by OpenAI for mathematics data, then talk to all of those people and figure out how much breakthrough-adjacent data work was happening... pure conspiracy but I'd be curious for someone to explore the hypothesis that there were maybe people helping find fragments here and there and it wasn't all purely mechanical monkeying.
Understanding is really the measure of utility. I suspect that an average human's intelligence is similar to one 100 years or even a century ago, and thus our ability to discover new theorems and proofs. The difference is the understanding. We not only have the last thousand years of discovery, but also the compaction, categorization, education, tutorials, etc, that allowed us to learn it effectively in a few years or decades of academia.
AI is accelerating the discovery part, but the _understanding_ part is just as vital for true progress.
I read this as "anybody can prompt, few can understand". And "we need more who can understand". If we had more mathematicians (than we have today) all of them piloting advanced models, the pie would grow. The problem is that AI capabilities drain (by disincentivizing) the education pipeline that would get us those mathematicians, and if recent rumbles about what AI is doing to education are to be believed, it does so many years before students even get to grad school.
IMHO, it is not that bad. Not having any human who understands linear algebra after the Butlerian Jihad is a win :-) .
They are.
Being able to move the frontier of maths as fast as possible goes against this. Seeing where the latest models can take us is a positive thing. It seems the desire is to make it possible for someone outside the lab able to write a prompt to get "credit" for it. This is a selfish way of thinking. Prioritizing the discoverer over the math itself.
>AI labs that release substantial mathematical output without immediate accompanying human understanding must take responsibility for ensuring that human understanding will follow. In particular, AI labs should provide significant support, including funding, to help develop this understanding.
This is just reads like asking for a handout. Beyond a Lean proof I don't think they are obligated for providing any more interoperability of it. You can already manually go through all the lean statements by hand if you want. Considering how powerful AI proofs are the value of "human understanding" does not seem to be that high. It seems much more valuable to improve AI's understanding of math than humans.
>The use of proprietary internal models by AI labs to do mathematical research risks creating a two-tier system where labs outrun the rest of the field, effectively alienating the mathematical community from its own discipline.
In such a world it may be worth funding another AI lab who would be willing to sell you access to a powerful model for mathematics. Just because mathematics did frontier math in the past, that doesn't mean they are entitled to be at the frontier forever. I would rather advocate for open (to the public) models based off the fact that it allows society to tackle more problems from more angles speeding up the progress of math over than trying to make it about enshrining the previous of class of people into the frontier forever.
>As for publicly available models, unequal access due to economic and other factors risks establishing a multi-tier hierarchy across the mathematical world and greatly exacerbating existing inequalities that arise from institutional wealth and technological privilege.
We should not slow the progress of Mathematics down because that might cause inequality. The speed of results coming out should be the number 1 priority here.
edit: also ive noticed when i ask llms maths questions they tend to introduces overly complicated things that arent relevant or make the problem harder. so maybe we will teach kids the basics of maths and how to interpret results? i am unsure.
While applying, I looked at the current SoTA, (briefly) read through some of the papers, and realized that I am very far away from understanding them.
Understanding one of these proofs is the work of several months, years or lifetimes depending on whether or not something clicks. It requires a kind of stamina that I quite frankly don't have, but I would like to develop.
If the mathaton's organizers accept my team, I realized that I would spend the next few years working through the result.
So why apply to the Mathaton?
"Many years ago the great British explorer George Mallory, who was to die on Mount Everest, was asked why did he want to climb it. He said, 'Because it is there.'
Well, [theoretical math] is there, and we're going to climb it, and [topology] and [number theory] are there, and new hopes for knowledge and peace are there. And, therefore, as we set sail we ask God's blessing on the [~~most hazardous and dangerous and greatest adventure~~] on which [we have] ever embarked."
More seriously, I applied because I was hoping to get access to the models without the veil. I don't think people realize just how big the gap is between what exists behind the scenes at these entities, and what we get out here.
And it's frustrating. Because I think it's within the rights of frontier labs to decide whether or not to sell access to a product, but the labs aren't just doing that. They're trying to thumb the scale to make sure that none of us ever get access to these models at peak performance. Ever.
And I think humanity is worse off for that. I am worse off for that.
I have studied the shape and structure of historical technological revolutions (and I've written about it), and usually the world doesn't realize how big of a big deal the big deal is because the big deal is often flawed, broken, and under-delivers. In the short term.
In the long term...? The world changed in the past few months. I think mathematics is one small part of that.
For most of human history, higher mathematics would have been inaccessible to me, and other outsiders, no matter how well heeled. Mathematics is, or rather was, a living discipline that existed piecemeal in a handful of minds across the world. These people's time was finite and valuable. To just meet them, you'd have to jump through hoops, and spend years proving yourself.
There is no price for an hour of tutoring from Terence Tao. But now, with AI? You can have an entity with the capabilities of Terry Tao help you understand the subtleties of math.
AI has changed what mathematics is. And every prominent mathematician seems to know it. They feel like mathematics has been devalued, and in some ways it has. Mathematics has gone from being a living discipline kept alive by a chosen few to a wellspring everyone can sip from. For the first time in human existence, learning and accessing higher mathematics doesn't involve jumping through hoops and knowing the right people. You can just ask.
I can just ask.
Except I can't. Because that capability is being gate kept. And I want to know. I want to climb the mountain.
There are subtleties to mathematics that aren't easy to understand from the written page alone. It's why it's a living medium.
For example, as we're talking about LLMs... why not, there are ways to reason about vector spaces that weren't intuitive for me to understand. It's something that required talking things out with a friend who is a practising mathematician (albeit in training).
I am not smart enough to reconstruct all of mathematics on my own from scratches on paper alone. That back and forth is necessary. And it's something that you couldn't have "bought" for cutting edge math at any price a few months before this point in time. Because it exists in the minds of people and it needs lots of back and forths with those people.
It's why LLM proofs can be slop on paper. A proof that no one can check or understand is not but scratches on paper. BUT LLMs are also the solution to the problem they create. The machines that can generate proofs are also machines that can help us understand them.
The living medium can now be represented and scaled inside of a machine. I can now sit down at an airport and have that discussion. I think that's transformative for our species.
Just wait until BCI and/or fast Pavlovian conditioning force fed by agents into human learners.
I've been vibe coding my own SRS software that is vastly superior to my learning style than Anki, and I know I'm just scratching the surface of accelerated learning. Who knows where this goes.
MRI and glucose injections with agent-tutor steering and millisecond feedback to learning?
We might be able to Matrix "I Know Kung-Fu" things into brains one day.
In what way are these "gatekeepers" stopping you asking an LLM questions about maths?
Can you, I, or any mathematician who isn't well connected (let's say someone who is a young Maryam Mirzakhani or just someone who is in grad school) learn from the system that produced the solution to the unit distance problem? https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
You will notice that it says on the first page,
Mathematicians want to talk to the exact model variant whose summarized chain of thought is, https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...And I want to talk to models of similar aptitude and capability to help me understand nuances of the proof. Mathematicians will happy to pay for this. I've heard that people and non-profits are putting together $$$ for this to get access to these systems so that they can all interrogate them.
But the issue is that we can't. And I'm using the royal we here.
The paper says that the labs shouldn't release proofs from models that mathematicians can't interrogate. It's very clear that the models we get as users aren't the models used to produce the breakthroughs. And as LLMs display emergent capabilities, it's uncertain whether or not the model actually understands what it's explaining.
Because if I don't understand it. Professional mathematicians who are subject experts don't understand it. Then how do we know the model does? How do we know that it's correctly representing the proof produced by a more capable model? It's not logical to take any random model at its word, unless we can verify. Or, if it's the same model that produced the proof.
And that's what the mathematicians want. Access to the actual models.
point: imo, the authors have no clue just how psychopathic and broken the execs are for the inc's that make these models
Everything is WFFs all the way down
OpenAI should be allowed to produce whatever it wants but it just can't claim that it has actually solved without the due process like peer review. If for example OpenAI solves a new conjecture, OpenAI should be free to publish it in their blog or arxiv in whatever way they desire. It can be slop, it can be non-slop. No one should police it.
Mathematicians are free to use it or discard it. They shouldn't externalise their concerns and restrict labs.
Mathematics is seen today as the noblest and most aristocratic of professions. Turns out, AI disrupts it because access to capital/compute now decides the results. Mathematicians don't like this corruption - understandable.
Its like guild of accountants opposing the calculator and require a responsible release. haha
Either you don't know what you are saying and should improve your language skills
Or you do and should be dropped from this site for advocating violence.
Time to get some real jobs, guys. When you're asking other entities to slow down with the discoveries so 'real people' can have their fun, you might not actually be participating in a useful profession.
This same day will come for other professions like pilots, lawyers, doctors, and teachers, but it's funny that the lowest hanging fruit really was what we'd all most expect lol.