There is literally not one aspect of the universe that is provably finite. Not in any direction, not at any scale, not in any way you can think to look at it.
Maybe I misunderstand you, but I don’t think I agree with that.
E.g. the observable universe is finite. Its observable age is finite. The number of known elementary particle types is finite. The number of particles within the observable universe is finite (according to the current understanding). Etc.
There are many examples.
Do we disagree on this, or am I perhaps misunderstanding your position here?
Key word is provably. We can assume the universe is finite because we don't have evidence against it, but no one knows if it's possible to prove or reject that for good.
Infinity is an abstraction. The universe cannot be "infinite" in the mathematical sense, as the mathematical infinity is an abstraction. The only meaningful question is "is mathematical infinity required to produce good models of the universe/physics?" and the answer so far is "no". We use infinity because it is convenient, but it is not really needed.
There is no reason to think that "infinity" will be essential to describing the universe any more than any other possible abstraction that we have not so far constructed.
> The universe cannot be "infinite" in the mathematical sense
Why not? An infinitely large universe goes against our intuitions, but our intuitions were formed by the experience of living on a finite planet and have no bearing on what's true on a galactic scale – just like they have no bearing on what's true at the quantum scale. There is no objective reason to assume that the universe isn't infinite.
I would say that there is nothing in intuition that goes against infinities, really. Quite the opposite, eg most ancient greek philosophy assumed matter could be divided indefinitely while retaining its properties. It was scientific and mathematical development that actually showed that finite/discrete models are enough to describe physics.
Moreover, even if finitism is not enough, as I said in mathematics infinity is specific concepts/abstractions, and there are even infinities with different cardinalities, not to mention the axiomatic system within which it is defined produces different properties of infinity. Which infinity is the universe of all these? I am not even sure if this question can be really meaningful, unless we get to a completely different description of the universe than the ones we have where such statements can actually make sense. The best we can get imo is to show that some accurate physical model requires infinity, and that seems already quite hard thing to show.
Observable universe is just the part of the universe from where light had time to hit the Earth already. It is growing pretty much at the speed of light, not because of physical changes to the space-time, but because the light from farther away finally arrived. So while it's definitely finite, I wouldn't say it disproves the claim that nothing about the universe is known to be finite.
Because of inflation, there is a distance at which the objects not already in our light cone are moving away faster than the speed of light. So the amount of stuff that we will be able to observe is limited.
besides the other points raised, even within the observable universe there are some assumed infinities as well as things that might be infinite, might not be.
e.g. time and space likely aren't quantized, you have an infinite amount of points in between any two arbitrary points in spacetime. don't mention planck scale black holes, it's a meme and even most phycisists who like to cite it haven't actually thought about it for more than 5 minutes.
I have the exact opposite experience. There nothing I can think of that is infinite in nature. Not only is it impossible to prove infinity empirically ("in the universe"; the only thing that actually matters) but there are exactly zero actual things with an infinite nature of any kind that I can think of. It's a useful imaginary construct that makes fancy math work, that's all.
For example, the special theory of relativity doesn't work if space-time has any "lowest scale" (i.e. if the distance between any two events can be expressed as n * min_dist, where n is an integer and min_dist is a constant that has the same value for all observers), it you can't infinitely sub-divide it, since then lengths couldn't contract the way that the Lorrentz transforms predicts. Lots of math in QM and QFT also requires space and time to be continuous for other reasons as well.
Also, none of our current laws put any kind of limit on the life span of the Universe as a whole, so space-time is also infinitely increasing, even if heat death or big crunch models are correct.
Of course, none of this is provable per se - it is fundamentally impossible to run an experiment in finite time that proves whether something is finite or infinite. It could be that SR and QFT are wrong in their continuous space assumptions, but only in ways that are too subtle to measure. Still, as it stands, we don't have any theory of how the world works which allows for a discrete space-time, even as an approximation - we need space-time to be continuous for our current theories to work at all.
Well, our physics is famously incomplete and even if it weren't: math needing infinity is in no way shape or form proof of anything but the nature of math which itself could be nothing but a highly particular construct.
I always find it interesting nobody seriously questions why something so extraordinarily basic as geometry pervades "the universe". I'd say it's much more likely we are mathy creatures and cannot do anything but perceive logical building blocks somewhat reminding me of a guy using a flashlight proclaiming the universe comprises various configurations of spherical lights.
I can imagine infinity just fine and I'm sure many useful things can be inferred from it, but that doesn't mean it exists. Lots of math is possible but does not exist in reality and that's fine, I'm just being honest about the very real boundaries here. I get the feeling many people confuse the map for the territory.
> there are exactly zero actual things with an infinite nature of any kind that I can think of
That's because your imagination is bounded by being a monkey that has spent its entire life in a finite environment — no offense. The universe, however, isn't bounded the limits of our imagination or intuition.
Yes, that's all very interesting (truly) but where does your actual lived experience come into play? That's the part I am talking about, the "in the universe" part, not the "could be" or "in imaginary parts of the possible universe" but the actual, experience-able universe. That's what separates math from physics and IMO math is the one that has to prove its worth not the other way around. I can construct endless internally consistent imaginary worlds with zero bearing on (any) reality.
And not one aspect that’s provably infinite, either, sl I don’t think your argument is interesting.
There are many things about the universe, though, that are finite to the best of our understanding. As in, any infinities that crop up in our models are properly understood to be cases where the models are wrong or incomplete.
On the other hand, every single observation anyone has ever made was a finite value of something.
Even when things are potentially infinite, there seems to be a conspiracy of the universe to make it finite. The size of space may be infinite, but due to a finite speed of light and the finite age of the universe we can only ever observe a finite slice. When attempting to zoom in to an infinite degree onto supposedly point-like particles, quantum mechanics makes everything fuzzy and finite again. Black holes appear to have infinite curvature at the center, but it's unobservable due to the event horizon.
This principle seems so natural that we even apply it in advance: In quantum field theory, there are infinities in the equations when done naively, which we deem so unnatural that we apply funny tricks which make mathematicians cringe, but they miraculously work out so well that we get theory and experiments to agree to an astonishing degree.
And yet the vast majority of physicists treat infinity in the equations as a sign of them being incomplete, and dismiss the possibility of the universe being infinite in any way
I would like to point out that there is also no proof of God not existing either
Who needs calculus anyways? I'd rather focus learning linear algebra especially after realizing a derivative is just a linear transformation and integration problems share a lot with determinants (areas).
Maybe so many more people would engage with mathemathics when they realize this.
I can't really think of anyone. Maybe just mathematicians, physicists, economists, computer scientists, chemists and engineers. Apparently they even use something called back propagation in AI training, which is nothing but chain rule from ... calculus.
But yeah, I can't see it being relevant since we have linear algebra already.
Maybe you missed my point but I was talking about calculus in the theoretical sense i.e study of functions in the reals.
In every way you are referring to people using calculus in an applied sense always been reliant on linear algebra.
Python libraries for doing backpropagation is built on numerical techniques that computes and is definitely not just manipulating symbols and calling it chain rule.
No, back propagation is just the chain rule from calculus. d(f(g)) = df(g)dg. It is applied to linear algebra, but with linear algebra alone you will never arrive to it.
You can just admit you don't understand calculus (and go and learn it if you care). No reason to go around doing mental gymnastics about it.
Good luck optimizing without being able to use the condition f'(x) = 0. You might think you can do everything numerically, but when your function has parameters you're in trouble.
Geez, I don't know why I keep arguing with you people. I've seen several examples where just integrating or derivating something gets you an analytic solution that you can compute in an instant, rather than having to start doing some 3 dimension integrals numerically.
I had rejection sensitivity growing up. It was a rather negative experience. At the time, I would have traded it away for pretty much anything. As an adult however, I wonder if whatever aspect of my personality led to the rejection sensitivity also led to certain “benefits” in terms of the ability to achieve various things I wanted to (via correlated traits).
If I could trade away just the rejection sensitivity, I would. If I had to trade both the rejection sensitivity along with the correlated traits, I’m not so sure that I would in fact. This sort of decision is obviously very individual however. The tradeoffs may not be worth it for many.
> 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.
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?
>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.
The only reason the world has mathematics at all is because of mathematicians. Most of the value in these results is to the people who understand them, not to some random fantasy science fiction project you’re imagining is going to happen because navier stokes is solved.
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.
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?
>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.
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.
> 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."
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.
To an ordinary person this sounds like, mathematicians were doing something, a team of scientists at OpenAI beat them fair and square in honest conditions and now mathematicians are whining and requesting artificial protections. I am not saying if this is what I believe, I meant this is how it will look from outside.
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.
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
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.
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.
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.
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
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.
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.
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]."
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.
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.
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?
Is “gate keeping” a new toy for ai bros or something? Peer review has a purpose… You guys should try to understand systems before you celebrate their death.
I freely admit I am not an academic. I just note that there is a lot of negative sentiment about people trying to get stuff published, and a lot of it looks from an external perspective to be elitist gatekeeping.
Please read that again, I am not on the inside here, so anything I say is just based on reading stuff from aspiring academics and articles in that mode. To my amateur eyes, there seems to be a problem in the academic world, of some sort, but how big it is, or isn't, I don't know.
Is that ok?
So, assume I, a complete amateur, no PhD, no connections or academic record to speak of (I have an MSc only, from the 90s, in a third world country) thinks I've discovered something, how do I go about "publishing"?
To me, I could easily see value where I self publish somewhere, where other similar non-connected people can see, crowd source the verification (basically peer review) and if it turns out to be useful, it'll be used.
Maybe it would be easy for me to get it formally published, but I doubt it.
First of all if you think you had discovered something in a pre-AI world, there’s a 99% chance it’s some triviality or wrong. Mathematics is extremely rich and people spend their entire lives thinking about it. Undergrads can’t comprehend what the grad students are studying. Grad students can’t comprehend what the post docs are doing. And it just goes and goes.
In a post-AI world there’s a mountain of high quality work being published by people who actually know what’s going on. You can probably make a discovery with chatgpt right now, but why should someone who’s got a ton of things to do already read your article? You’re free to put it online, but why should you be free to force someone to read something?
Second of all, it’s extremely aggravating to read all these software people criticizing mathematicians, who almost universally have low salaries and who chose to go into this profession because they care about contributing to human knowledge.
>In a post-AI world there’s a mountain of high quality work being published by people who actually know what’s going on. You can probably make a discovery with chatgpt right now, but why should someone who’s got a ton of things to do already read your article? You’re free to put it online, but why should you be free to force someone to read something?
But aren't you literally just proving what I'm saying? The original argument was that everyone should have to post through peer reviewed journals, I said that is probably gatekeeping too far, and your response above just confirms my suspicions.
> Second of all, it’s extremely aggravating to read all these software people criticizing mathematicians, who almost universally have low salaries and who chose to go into this profession because they care about contributing to human knowledge.
I probably agree with you, except I have never and am not criticising mathematicians. I am only saying that there are quite a lot of reports around that the academic world is fierce and that sometimes there are high barriers to getting something recognised.
No idea if it is true or not, but my personal opinion is that it isn't going to matter much in the future. If it exists, then people will work around any gatekeeping.
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.
“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
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.
>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
(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).
Next time you make any argument to protect something you find meaningful and important, and something you believe you add value to, just remember it’s pure gate keeping and everyone can see it.
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.
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.
>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 ...
> 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.
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.
> 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.
> 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.
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.
The other side of this is that there actually are utilitarian arguments for human involvement. Unfortunately SE is dominated by misanthropic types who dismiss these out of hand.
I doubt it. You have to make these 1000 times cheaper at the same level as the internal model at OpenAI for random (wealthy) people to start knocking down major conjectures.
> 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...
>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)
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
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
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.
If your life's work were to get outsourced to AI and you lost your job/the opportunity to do it, I doubt you would make this argument.
I would rather live in a world where mathematicians get the opportunity to do math with AI, rather than AI companies doing the most important mathematical work and leaving them behind. I hope this for all professions that people want to do.
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.
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.
Why are you confusing the substance/field with the tools to use it? He said air is free which is correct. You and parent are complaining that air compressors cost money and are proprietary.
> 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.
> 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
> 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.
When you make a sacrifice, it has to make sense, and must have some chance for success. In this case it needs to be recognized by others, and a contemporary person can hardly understand any other sacrifice than monetary.
I think the perspective from inside vs outside the buildings is important. Consider Meta’s classic campus vs their newer buildings designed by Frank Gehry. I like the exterior appearance of the classic campus much better as it feels “friendlier”, but from the inside, it flips drastically. The newer buildings are far superior in every way compared to the dimly lit, fluorescent feeling you get inside the older buildings with the small windows.
Contrarian take: I think the ability of AI to produce valid mathematical proofs (even inscrutable ones) is absolutely fantastic. Mathematics as a profession does not have a monopoly over math itself any more than professional pianists have a monopoly on who plays piano, when they play, and how.
I have sympathy for any jobs that might be affected (much as my own job has become more tenuous in software engineering). And if the field is disrupted by chaos that makes the research process unproductive, that's bad too and should of course be handled by applying better organization within the institutions that tend to perform mathematical research.
But to a large degree, the notion that "sloppy AI proofs are bad for mathematics research" seems like a total failure of the imagination to me. Attempting to find shorter proofs or more elegant proofs can be turned back in on itself via proof theory. There are proofs in Presburger arithmetic that are doubly exponential in the length of the sentence. Yet a more powerful theory like PA makes quick work of such theorems. The explainability or "subjective beauty" of a proof can be quantified and optimized against. Optimization itself can be optimized against. I really don't understand how this magical ability to know the truth of more theorems much more quickly—even via an "ugly" route—is anything but a net positive.
Am I missing something obvious? Isn’t this just a simple DB query to see the state history of the “Data Controls” → “Improve model for everyone” toggle in the settings? Just report whether that was ever on and over what time period.
Multiple OpenAI staff have publicly said they cannot do that as accessing specific user settings without their consent (or legal requirement) violates their internal privacy policy.
However, the mathematicians could easily declare whether they had the toggle on or off. Yet curiously, they will not say!
Zero reason to believe open AI employees, big tech has history of employees/subcontractors creeping on exes and minors, someone with keys to creep on customer for valuations is more parsimonious.
It’s not about what they say but what they do and right now no frontier AI company deserves anything but skepticism based on how they act verse what they say.
So if the scientist(s) will reports that they had it enabled, it would mean that OpenAI lied, when they made a statement about not using their chats for training, right?:)
Yes you're missing several obvious things. Even saving the last changed date (nevermind every change date or what the change was) for every setting for every user would be earth crushingly wasteful. The value by itself isn't even worth including in backups.
Storing the last changed date for every single person on earth (even though not every person is an OpenAI customer) is something you could easily do on a laptop. It would be a rounding error for OpenAI.
I don't know what format they use for storage, but Iceberg would be a reasonable choice. A date in iceberg format is 4 bytes[1]. I checked postgres as well as a reference point. It also uses 4 bytes for a date, so whatever they use it's going to be about that.
Current world population is just shy of 8.3 Billion people [2].
4 bytes times 8.3 billion people gives 30.92 GiB. [3] OpenAI's training data will be in the petabyte range at least.
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