When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound?
How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them?
Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? How do we tell truth from fiction?
I get the impression that the value of unproven conjectures is more in the new math and techniques that may be discovered - by humans - trying to prove/disprove them, rather than much utility in any eventual result.
Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not.
These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new.
However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go.
Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.
It just occurred to me that this "it's not the destination, it's the journey", what you learn and contribute by trying to solve a problem, may point to what I'm guessing is an unpopular conclusion...
It may actually be a negative, rather than a positive, for AI to have solved these problems.
If humans had continued to work on these problems, then its quite possible they may have invented new math along the way and benefited the field. Now that the problems are solved, in one manner, I would assume that the level of human interest in them is much diminished, and the likelihood of these problems generating the same benefit for mathematics has diminished. Erdos chose his problems precisely because he thought they could advance the field.
AI itself does not appear to have benefited from solving these problems - they are hard for a human but apparently fairly easy for an AI. We don't just have one result here, proven after great labor, but 10 (with more results rumored to be withheld), with the effort/cost to generate them reported to be low.
The temptation to announce AI solutions to problems that are hard for humans, easy for AI, may be hard to resist, but perhaps there would be more benefit to be had to leave the hard for human problems for humans to solve, and be more impressed when AI solves problems that are hard for AI.
As Hans Moravec noted "What's easy for humans is hard for computers, and what's hard for humans is easy for computers", and while AI is trying to make inroads into that, it still remains fundamentally true.
I'm not a mathematician, so I may be talking nonsense. Would it not be equally possible that AI solutions to these problems have, or start revealing new maths, or new techniques. I doubt that is the case for these type of problems, but it may well be in the future. And maybe these techniques are seen by the right people, who have the correct intuition of how they may be a bridge in other problems.
Or another angle. Given that, from my amateur understanding, a lot of these problems are proof by counter-example, could you not argue that at least AI eliminates low hanging fruit and more important problems can gain focus.
Of course, I could equally argue that the low hanging fruit is needed to train up mathematicians.
So yeah, in the end, I don't know. But am pretty certain that AI is not going to be limited by these sort of proofs for much longer.
My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy.
An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?
You could regard existing mathematical techniques as a set of Lego pieces. If you build a proof just using those pre-existing Lego pieces then it may be a new proof (a new Lego model), but it's not a new Lego piece - a new piece of mathematical machinery.
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.
No. Mathematicians who considered themselves experts on exactly those conjectures were bewildered.
>And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems
...
>Is there some broader context or theory within which this would've been the obvious thing to do? What other results can be proven using these techniques? What is it telling us about quantum information or operator theory? I have no idea
I saw another expert on Ehrhart express their bafflement (but I can't find that link now)
PP (actually the "broad career mathematicians" themselves) were being slightly disingenuous... But the specific domain experts seem to be understating how much over their head it actually was. We're in totally unknown territory here..
we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done!
(Steel Manning:
1. maybe any chatbot would need 1000x more tokens than were used to arrive at the result to understand it to its own satisfaction.
2. It's possible that Einstein did not fully understand General Relativity
> we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done!
Why would that be optimistic? Seems like the pessimistic reading to me, if anything. Or are you having a bout of Schadenfreude?
I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.
Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?
The number of geniuses in every field that congregate in HN is incredible, too bad they're here responding to comments instead of revolutionizing nuclear energy, math or medicine.
I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it.
I might have overstated a bit, but by 9th grade (15 year old) this is what was taught to us back then.
1. "You need to slow down neutrons so they can react"
or 2. "Here's the graphs of how the neutron absorption and scattering cross sections vary with neutron temperature for H-1, H-2, H-3, Be-9, C-12, O-16, Fe-54, Fe-56, Fe-57, U-233, U-235, U-238, Pu-239, …"
If it was the former, you didn't learn "nuclear engineering".
ofc, you can't even start talking about moderation and not mention cross-sections
(which is a simplification in itself, but that's best left until 2nd-3rd year in uni)
But for general understanding, .. there is stuff that slows neutrons. some is more effective, some less. There is also activation. It is why tanks and ifvs were lined with polyethylene or similar on the inside back in cold war - it had lots of hydrogen. But for controlling a power plant that is not enough - why?
having a lab assignment detecting beta-emission from 40-K (potassium-40 salt) is of course unthinkable nowadays. or idk, assembling a interferometer out of a HeNe laser for example. Whoa, no! Kids can get hurt!
I went to a decent high school, this was not something covered in 7th grade. It isn't covered in my daughter's school either.
For the 2026 version of me out there, please ignore. It is nerd posturing, and as real as the boomers at your gym claiming to have benched 225/315/405 in high school, despite having terrible form while doing 185.
This seems like an extreme exaggeration from a few people claiming to not understand a very recent result. This cycle of a new result being discovered, and reaearchers needing some time to truly digest and disassemble it, is normal.
I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)".
Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?
Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.
Yes, although a counterpoint to that is that if these weren't long-standing published problems that have stumped humans, would anyone care? What is important here, the problem/proof, or the fact that they are famous problems that have stumped humans (and therefore make good bragging material).
I'm not sure that a problem being famous, or hard for a human, is the same as it being important.
Take Fermat's last theorem as an example - was a proof of it ever considered to be, then or now, consequential, or was it the Paris Hilton of problems - famous for being famous? A problem whose legend grew because it it so simple to state, with the challenge that there might be a simple proof, and that turned out to be so difficult to solve that it therefore became prestigious to do so?
Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
No. More like "I am mathematician and I have digested this sloppy writing and understand this to be an application of Foo theory to the Bar problem with a Baz twist. The prior publication omits crucial references to..."
The field has been dealing with this for a long time.
Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0].
The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B.
Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new.
This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research.
Mathematics is an unusually dense (if not the most dense...by a few large steps) field. So lots of areas of mathematics are extremely deep and narrow without any real shortcuts, even for seasoned mathematicians.
Physics has bounds of concern, where as mathematics would encompass all physics, as well as all possible alternative physics, in all possible forms. Maybe not unbounded, but close to it.
The field of mathematics is smart about this and knows the difference between a pile of Lean code and understanding, and mathematicians try to get from the unintuitive explanations to something that makes more sense, e.g. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the... (where incidentally Tao used a chatbot to help take apart the problem, but with a lot of interaction and work from his side).
Making things understandable is mathematics, and more generally a kind of intelligence, and is crucial to continued progress. You couldn't use algebraic geometry to disprove a conjecture if people hadn't organized (what could have been just) a pile of random observations into something called algebraic geometry.
Historically LLMs have done best where it's possible to train using an objectively verifiable reward function. Computer programs are pretty good on this front and so are Lean proofs. (Of course, they don't only do things you can RLVR heavily, but those have progressed fastest.) Not sure where 'making mathematical knowledge more understandable' falls on that spectrum.
Understandability isn't only important for advanced math. Keeping computer programs from becoming a mess is a challenge in high-level organization too, and the chat with the user is an explanation task. If you look online at what people say about large LLM-built codebases (SlopCodeBench is a neat effort to make make it concrete, but common wisdom seems to mostly agree on the general problem) and chatbot prose, I don't think everyone considers those solved problems!
It's hard to tell how thoroughly the labs grasp and care about this at an organization-wide level. I'm sure at least some maybe-results exist inside labs but haven't been published because the humans couldn't verify them and didn't want to be embarrassed with a false result. (Maybe also why counterexamples are a lot of the first results published: often simple to verify, even if hard to obtain.) A good sign would be if results in a few months come out more like what mathematicians consider well-written papers explaining results in a more intuitive way, fewer shocking announcements of bare counterexamples in tweets. It's probably a slow climb to get there.
I think this is where Terry Tao hit the nail on the head. OpenAI has made claims, but the mathematics community does not accept garbage. It's less that they've done a disservice to the community; more that they've done a disservice to their own reputation.
I agree. Prompting 5.6 Sol to "Review these writeups and include appropriate citations and credit where appropriate" probably would have been a step in the right direction.
Then again, this would require them to actually care about the mathematics.
So a good fraction of famous old math puzzles with no practical use are famous because they are in some way similar to problems people actually care about. If you can solve the toy problem that is in some way simpler, maybe you can use the same methods to solve the "real" problem.
One open question is whether these machine solutions to these problems will act as springboards to future research, either when given to human mathematicians, or when used to train future machine models.
One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.
And then there's G. H. Hardy, famous number theorist, who wrote an "Apology" for mathematics in the sense of defending the study of number theory for its own sake because it's beautiful.
It is one in the same! The apology is, of course, defending the beauty of the pursuit but at the same time the writing acknowledges the (alleged) utter lack of practicality.
It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.
The problem is more that the puzzles had an original context and reasons why they arose from "real" problems directly or mathematical problems trying to solve "real" problems etc. but the presentation and abstraction hides this pretty well for people not "near" the problems.
Yes, that's correct. I've done research in pure math, but left after the postdoc level. I enjoyed it for its artistic sake but it's not something that gives anything practical. Most pure mathematicians don't ever work on practical problems either. They just teach and in return get to do their hyperspecialized research. It's done for its own sake. (Some pure math eventually ends up having applications but it's rarer and rarer and it wasn't very common to begin with. Like applications to crypto or whatever, but that was always a tiny slice to begin with.)
The meaning of "practical use" is all about context - who/when/where/why/what - and so it would be kind of hard to definitely claim, in an intellectually honest way, that a piece of math is an old puzzle with "no practical use whatsoever".
Unless you want to be the guy in the 19th century making fun of Boole algebra for having no practical use. You might be right, but not for long.
Thanks to Martin Gardner's books back in the day we had genious hackers doing just that, recreative programming... until it guided our lifes: GNU/Linux, GNU itself, GNU Emacs, the BSD's, Rogue with the Curses library (and tons of Unix serious stuff), Unix itself to play SpaceWar in a PDP10...
even the ZMachine it's an astounding example of running virtual machines on 8 bit machines 20 years earlier than Java.
Most modern pure math has no practical use whatsoever. It's amazing how few people understand this. Mathematics is a service department and maybe one in a hundred thousand results have any practical use. (Speaking as someone with a PhD and over a decade in pure math with several published papers...)
Your observation is perfectly on point, I think the season of companies announcing breakthroughs might be over soon (unless they somehow manage a major achievement, P vs NP or similar). At the same time, mathematicians will be left with superintelligent machines solving the actual math for them, much like software engineers nowadays. This was unexpected, and unexpected at this scale up to a couple of months ago.
My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.
That axiom is clearly false. AI can interact with external systems, which means it is not just compressed knowledge - it has the ability to access new information.
I find some interesting parallels in chess, which often has lots of analogs with math to begin with. But chess went from a pure human endeavor to one where supercomputers aided by world class players finally managed to eek out a slightly suspicious win against a world champion (approximately where we are now in math) and to now a days - where your phone could easily crush the world's strongest player, who is also probably the strongest player of all time.
The way the chess world adapted this was initially to try to understand the machine. After all chess, like math, is complete information - so you can easily see the computers 'thoughts' in terms of the exact moves its saying are best in a variation and how it might respond to any other idea. But it quickly became clear that this wasn't working so well.
Players would regularly get positions that the computer says 'and black wins' and then proceed to lose it convincingly, simply because the positions were so extremely weird and difficult to play that even if it might be technically winning, it's the sort of position where you're walking a fine line with lots of complex moves to find. Humans aren't computers and even the best of us can't play like one in weird positions.
Now a days they're taken more in balance. The computer's evaluation of a position is probably about as good as you can get, but playability matters much more in practical terms. Knowing the eval of a position doesn't really matter if you don't understand the position. Knowing the answer can help with understanding (for instance computers have radically reshaped and improved human understanding of space in chess as we noticed computers obsessing over it) but I think the days of 'oh the computer says it's winning, so I should be able to take it from here' are near to gone.
It's more like FOSS project using and you ask a developer who works at a different company how to build this project and he says "I don't know automake, perl and there are lots of macros"
What are the odds that this random FOSS project has solved the problem of building software and nobody noticed. Close to 0. Don't confuse accidental complexity for transcendental depth.
In college the punchline for all the engineer, physicist and mathematician jokes were something like "The mathematician says: Yes, there is a solution."
In all serious "I don't understand any of this it's way over my head."
> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them?
Theoretical science isn't about usefulness per se, but knowledge and understanding for its own sake, so a better way to say this is to point out that you only benefit in such cases if you understand things yourself. If understanding is the goal - which it is in the case of true theory - then the only way to attain that goal is to actually attain understanding. What good is it if an LLM produces a valid proof, but no one grasps it? This isn't like digging a ditch where it doesn't matter who does it or how he does it as long as you have a ditch. Here, the ditch is knowledge, as it were.
i doubt anyone really wants, or cares, to understand what the AI comes up with. If it's not related to your work or your name isn't associated with the discovery then I would think time is better spent on things that are.
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Huh?
Mathematics is an extremely wide subject, it's perfectly normal even for two professional mathematicians not to understand each other's work. Have you considered that maybe they just don't work in that area?
Are you implying OpenAI's paper (which was, by the way, edited by humans and provided Lean certificates for most of the proofs) is actually gibberish? That's flat-earth levels of conspiracy.
There's no honor in being stupid, yet I must admit I am stupid, as there's even less honor in being stupid and pretending otherwise.
A lot of people were super hyped about OpenAI's 10 discoveries, but I still don't understand what they mean, and even if I did, what are the implications.
Like, what are non-sofic groups, and what follows from the conclusion that they exist?
I mean in the sense that quantum mechanics might make my head spin, but it's because of that that we have stuff like semiconductors, which have been one of the most significant discoveries.
The Fourier transform is one of the reasons we have fast telecommunications and radars.
What practical things are possible or might be possible due to these results?
> What practical things are possible or might be possible due to these results?
Not much, they're all fairly minor problems that have been solved, of ~entirely niche academic interest. It's so far more just that AI _can_ solve novel math problems, ones that humans didn't accidentally train the answer in and it just spit it back out.
> Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands
An unsettling number of people do seem to be more than ready for this. In fact I'd say they seem almost gleeful about it. Thinking is hard and they don't seem to like doing it!
>If a math problem falls in the forest but nobody is around to understand it does it make a sound?
This is not new nor unique. There is plenty of research, especially in math, which can really only be understood by a few people in the entire world. It is not uncommon for a proof to be presented by a mathematician which, initially, is only understood to that mathematician, and it can take a long time for even another mathematician who is an expert in the same field to be able to confidentially say they understood it.
>I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
This is meaningless in a vacuum. If you give a novel proof in some niche subfield of topology to a competent mathematics researcher who focuses in number theory, they'd say the same thing regardless of if a human or machine wrote the proof. A bunch of "career mathematicians" on Twitter proclaiming this doesn't mean anything other than these people aren't currently equipped to understand the contents of the proofs. That's fine and normal, but the idea that anybody with a PhD in Math should be able to pick up one of these proofs and give it a skim and be able to say, "ahh, yes, quite clever, it seems so obvious in retrospect," is absurd. That's not how this kind of research works.
>Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands?
Nobody is blindly accepting these proofs as valid. ChatGPT isn't spitting out a wall of text and proclaiming that they've solved a previously unsolved math problem while everyone is saying, "well if an LLM says it, it must be true!" lol
These proofs are being checked by automated systems (which have been in-use well before LLMs have existed) as well as being checked over by actual experts who are actually capable of (and motivated to) verifying these proofs. But that work still isn't done. There's enough evidence that these companies are confident in saying these proofs are correct, but there's going to be a lot of ongoing work from people to continue to verify and, more importantly, understand these proofs. It's literally some of these people's full-time jobs to do this.
>How do we tell truth from fiction?
When was the last time you verified even a classical, relatively simple mathematical assertion? How often are you just relying on a larger system of experts to ensure that we're not just blindly accepting fiction as truth?
That's not to try to stick it to you personally, but it's just highlight that there's an entire system in-place here that you're not aware of and that you don't have an understanding of that is working just fine including in this context. Real mathematicians aren't going to lazily start letting OpenAI assert whatever they want about their products solving these kinds of problems without heavy scrutiny.
Maybe the "beautiful, elegant" math is really just accidentally that way, just the tiny cross section our dumb human brains can understand. The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.
Generating a value of type "Either (Int, String) Bool" is proving that there's at least one integer and at least one string, or there's at least one valid boolean value. Except in Haskell, where it could also be an infinite loop.
I have the same questions about human mathematicians!
I can't tell if xkcd #435 is still true, or if math is just as mushy as everything else seems to be. When a math proof can only be understood by a handful of people, what does that mean about that proof? I think the LLMs are pushing a problem that existed already and pushing it further.
The AIs seem to have some combination of very broad familiarity with math (enabling relevant things from other subfields to be brought in to the proof) as well as patience and "sitzfleisch" (stamina in working through details even if they aren't immediately obviously promising.)
An obvious area for improvement would be automated generation of new conjectures and attempts to prove (or disprove) them, with the discovered arguments then being used as training for refined models. This will require autoformalization to check the results as there will be too many for manual verification.
The mathematics are above my intellectual capacities, but I still find the man and his lifestyle fascinating. Though I admit his heart attack probably wasn't a coincidence, sadly.
I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS.
I often see comments that some of the users here long to "make a computer do neat tricks all day", and I can't help but think meager sponsorship could go a long way in this area. Existing grant structures, being much more traditional, are constrained by their cost.
> I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS.
The problem is that Paul Erdos was eccentric but he was also Paul Erdos. I'm not saying you're implying that, but I feel it's a similar line of thinking to how popular culture often romanticizes autism and Asperger's because some very smart people are (allegedly) affected. The same group includes people who need 24/7 care.
Computer science at the frontier is as specialized and hard as mathematics, you need years of study to truly understand a field well enough to make meaningful contributions. You can try sponsoring me if you really want, but I don't think you'd be spending your money wisely in expectation.
FWIW I've worked in two different academic labs, both of which would have benefited from a runaway worker barreling down random paths, as long as they were being productive in some capacity.
When Gwern needed a fellowship to come to SF he got one. When he had his “great idea” it was funded. The truly unique people out there find a way. Any government bureaucrat will just find the Jason Ardays of the world.
That's survivorship bias. If there are truly unique people who never get a chance to do what they would be good at, we don't hear about them.
Private sector bureaucrats are no different from government ones. The fundamental issue is that if you have a system, it's going to fail outliers. Almost by definition. And the situation is even worse if the system uses metrics for making decisions.
Well, if the statement is "we need some eccentric thinkers" then survivors suffice. We have some eccentric thinkers. If the statement is "we need more eccentric thinkers" then I think I'm going to demand the rigor of "explain why we need to increase the quantity and why your proposed intervention is the best way to get there with evidence behind it".
Honestly, I think our extremely liberal disability benefits scheme achieves this to a large extent (albeit possibly with downsides for both the recipient and society).
Look at all of the crazy stuff made by unemployed trans people/autists over the past decade, disproportionately from countries with strong welfare states.
something i've just realized : today long-standing maths problems are falling. It's great intellectually but won't probably have an immediate impact on our lives.
Now, what will happen once long-standing physics ( and chemistry and biology) problems will start to fall and at the same rate ?
Then we're going to enter a totally different world.
It's hard to see problems in those fields falling at anywhere the same rate as math, because they are all experimental fields.
There may be some problems of type type "why does X happen?" that appear answerable in terms of known science, but even these would need verification. If you want to make advances in fundamental physics, then a promising AI-generated theory might take a decade and billions of dollars to prove or disprove.
Math is a rather unique field in being entirely theoretical, axiomatic and self-referential. It is basically the best possible case not just for AI to advance without needing experimental verification, but also specifically for today's AI technology of auto-regressive LLMs and RL training, whereby valid reasoning steps learnt in one context will also be valid in another context (i.e. there is some generalizability of learnt reasoning) as long as you have learnt the pertinent aspects of that context that the validity depends on.
For example, protein folding has been figured out by AI. This was a very big moment for science and yielded a nobel prize. Alphafold 1 happened 4 years before the first version of ChatGPT and the LLM craze we see today.
But yeah, most problems in physics, chemistry or biology require labs on top of actual hard thinking. You need to be able to design experiments in a certain way. Once you have the funding, the right tools, the right people to use those tools, then you can use LLMs to increase the speed of the calculations and so on.
Not really. These same techniques fall flat on their face when applied to most physics and chemistry problems. All of academia has already been doing ML4Science for the last 8 years. God knows how many billions have been spent.
The only two major highlights are weather modeling and folded protein backbone prediction.
Mostly everything else, either lacks enough data, or there are contraits on the size of the foundational models that render them impractical or they just fail to generalize.
Connect the AI to a fully automated lab and it may be able to generate and harvest those necessary data.
It may also produce Supercovid in the process, so...
We don't have to worry about new math as much as we do about new biology. Although "breaking all current encryption" would have some nasty consequences as well.
It seems that AIs are really good at finding counterexamples now.
Even if progress by AIs in proving conjectures lags, it seems likely that AIs collectively will, in the next few years, find counterexamples to nearly all the Erdős (and other) conjectures that are actually false and also provably false.
That means we will able to assume that nearly all the remaining conjectures are either true or undecidable.
Surely, that's good for folks who just want to know where the truth boundaries in mathematics lie.
It's obviously causing a lot of soul-searching amongst professional mathematicians.
Arguably, they should have given less weight for the last 100 years to Hardy's view in 'A Mathematician's Apology' [0]:
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.
Rota takes a much more balanced view in 'Indiscrete Thoughts' [1].
"Problem Solvers" take Hardy's view:
> ... The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. Mathematical exposition is regarded as an inferior undertaking. ...
While for "theorizers":
> Mathematical exposition is considered a more difficult undertaking than mathematical research.
If professional mathematicians can reinvent themselves, there will be plenty of work left to do to explain the results of AIs to other humans.
There probably needs to be a new career path into professional pure mathematics other than doing novel research in a PhD.
Disappointing lack of "Why" in an article that starts with it.
Are they actually doing something new and novel, or are they just absorbing that "a=b as was proven in transcendental hyper-circular group theory; and b=c was proven in universal quantum superposition"; and they're the first to find the connection that a=c? And several of the problems are counterexamples, not novel proofs of correctness?
Its fascinating either way, but it'd be nice to actually understand more of what is happening.
One of the recent problems was actually proven with elementary results. It did use a technique not usually applied to the problem space, but it wasn’t a hyper specialized result.
We have now passed the point where the AI bots were annoying. Now it is the reactionary “everything even possibly slightly remotely tangentially connected to AI must be deplatformed” rants everywhere that are even worse.
It would be tempting to assume the commenter here has no idea what Renaissance is or how they made so much money.
The "commentator"'s account also has a karma of negative 3 ! I've never even seen a negative karma before !
There is a pattern where the most divisive comments, that one might suspect of being from bots, do tend to come from accounts with single or double digit karma. Maybe HN tries to identify and remove these, which is why they tend to be new ?
There is no such thing as the Renaissance Fund. Quanta receives funds from the Simons Foundation... I also don't know where you get it from that Renaissance Technologies is heavily invested in AI. Any sources to back up your claims because I suspect you made all that up?
> The company is widely considered one of the most profitable hedge funds in history, generating an estimated $7 billion to $8 billion in annual revenue solely from management and performance fees
When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound?
How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them?
Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? How do we tell truth from fiction?
I get the impression that the value of unproven conjectures is more in the new math and techniques that may be discovered - by humans - trying to prove/disprove them, rather than much utility in any eventual result.
Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not.
These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new.
However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go.
Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.
It just occurred to me that this "it's not the destination, it's the journey", what you learn and contribute by trying to solve a problem, may point to what I'm guessing is an unpopular conclusion...
It may actually be a negative, rather than a positive, for AI to have solved these problems.
If humans had continued to work on these problems, then its quite possible they may have invented new math along the way and benefited the field. Now that the problems are solved, in one manner, I would assume that the level of human interest in them is much diminished, and the likelihood of these problems generating the same benefit for mathematics has diminished. Erdos chose his problems precisely because he thought they could advance the field.
AI itself does not appear to have benefited from solving these problems - they are hard for a human but apparently fairly easy for an AI. We don't just have one result here, proven after great labor, but 10 (with more results rumored to be withheld), with the effort/cost to generate them reported to be low.
The temptation to announce AI solutions to problems that are hard for humans, easy for AI, may be hard to resist, but perhaps there would be more benefit to be had to leave the hard for human problems for humans to solve, and be more impressed when AI solves problems that are hard for AI.
As Hans Moravec noted "What's easy for humans is hard for computers, and what's hard for humans is easy for computers", and while AI is trying to make inroads into that, it still remains fundamentally true.
I'm not a mathematician, so I may be talking nonsense. Would it not be equally possible that AI solutions to these problems have, or start revealing new maths, or new techniques. I doubt that is the case for these type of problems, but it may well be in the future. And maybe these techniques are seen by the right people, who have the correct intuition of how they may be a bridge in other problems.
Or another angle. Given that, from my amateur understanding, a lot of these problems are proof by counter-example, could you not argue that at least AI eliminates low hanging fruit and more important problems can gain focus.
Of course, I could equally argue that the low hanging fruit is needed to train up mathematicians.
So yeah, in the end, I don't know. But am pretty certain that AI is not going to be limited by these sort of proofs for much longer.
> are using known math to solve them rather than inventing anything new.
Isn't a new proof new math? If not, what qualifies as new math?
My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy.
An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?
You could regard existing mathematical techniques as a set of Lego pieces. If you build a proof just using those pre-existing Lego pieces then it may be a new proof (a new Lego model), but it's not a new Lego piece - a new piece of mathematical machinery.
To be fair, very few proofs involve the creation of new machinery wholesale.
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.
No. Mathematicians who considered themselves experts on exactly those conjectures were bewildered.
>And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems
...
>Is there some broader context or theory within which this would've been the obvious thing to do? What other results can be proven using these techniques? What is it telling us about quantum information or operator theory? I have no idea
https://bsky.app/profile/henryyuen.bsky.social/post/3ms2jpch...
That guy has been featured in quantamag too, btw
I saw another expert on Ehrhart express their bafflement (but I can't find that link now)
PP (actually the "broad career mathematicians" themselves) were being slightly disingenuous... But the specific domain experts seem to be understating how much over their head it actually was. We're in totally unknown territory here..
we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done!
(Steel Manning:
1. maybe any chatbot would need 1000x more tokens than were used to arrive at the result to understand it to its own satisfaction.
2. It's possible that Einstein did not fully understand General Relativity
> we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done!
Why would that be optimistic? Seems like the pessimistic reading to me, if anything. Or are you having a bout of Schadenfreude?
I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.
Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?
The number of geniuses in every field that congregate in HN is incredible, too bad they're here responding to comments instead of revolutionizing nuclear energy, math or medicine.
I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it.
I might have overstated a bit, but by 9th grade (15 year old) this is what was taught to us back then.
> why neutron moderation is needed
What you learned, was it more like:
1. "You need to slow down neutrons so they can react"
or 2. "Here's the graphs of how the neutron absorption and scattering cross sections vary with neutron temperature for H-1, H-2, H-3, Be-9, C-12, O-16, Fe-54, Fe-56, Fe-57, U-233, U-235, U-238, Pu-239, …"
If it was the former, you didn't learn "nuclear engineering".
ofc, you can't even start talking about moderation and not mention cross-sections
(which is a simplification in itself, but that's best left until 2nd-3rd year in uni)
But for general understanding, .. there is stuff that slows neutrons. some is more effective, some less. There is also activation. It is why tanks and ifvs were lined with polyethylene or similar on the inside back in cold war - it had lots of hydrogen. But for controlling a power plant that is not enough - why?
and then we answer why.
I'm pretty confident that no 15 year old would have learned about void coefficients if Chernobyl hadn't happened.
Man you went to a way cooler high school than me. I definitely was not taught anything that would help me understand how a nuclear reactor work.
Not even basic thermodynamics?
having a lab assignment detecting beta-emission from 40-K (potassium-40 salt) is of course unthinkable nowadays. or idk, assembling a interferometer out of a HeNe laser for example. Whoa, no! Kids can get hurt!
Being familiar with the basics of fission power generation does not put the design of a competitive plant within your wheelhouse.
Nor understanding and judging a proposed design, yes.
I went to a decent high school, this was not something covered in 7th grade. It isn't covered in my daughter's school either.
For the 2026 version of me out there, please ignore. It is nerd posturing, and as real as the boomers at your gym claiming to have benched 225/315/405 in high school, despite having terrible form while doing 185.
By that standard, I could've been an astronaut at that age.
Newton's laws of motions are not hard. Making a rocket that doesn't kill the occupant, is.
This seems like an extreme exaggeration from a few people claiming to not understand a very recent result. This cycle of a new result being discovered, and reaearchers needing some time to truly digest and disassemble it, is normal.
I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)".
Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?
Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.
Yes, although a counterpoint to that is that if these weren't long-standing published problems that have stumped humans, would anyone care? What is important here, the problem/proof, or the fact that they are famous problems that have stumped humans (and therefore make good bragging material).
I'm not sure that a problem being famous, or hard for a human, is the same as it being important.
Take Fermat's last theorem as an example - was a proof of it ever considered to be, then or now, consequential, or was it the Paris Hilton of problems - famous for being famous? A problem whose legend grew because it it so simple to state, with the challenge that there might be a simple proof, and that turned out to be so difficult to solve that it therefore became prestigious to do so?
To continue the analogy, it’s like writing answers that are so convoluted that no one can understand them.
Latest presentation of Terence Tao on what current advancements in AI mean for math discusses (among other things) those issues: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...
Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
No. More like "I am mathematician and I have digested this sloppy writing and understand this to be an application of Foo theory to the Bar problem with a Baz twist. The prior publication omits crucial references to..."
The writing's already pretty good, even Terrence Tao calls it flawless, if you read the slides. That's not where the LLM requires help.
The field has been dealing with this for a long time.
Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0].
The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B.
Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new.
This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research.
[0] https://www.newscientist.com/article/1997488-wikipedia-size-...
Mathematics is an unusually dense (if not the most dense...by a few large steps) field. So lots of areas of mathematics are extremely deep and narrow without any real shortcuts, even for seasoned mathematicians.
Awesome observation. What would you consider denser?
Physics possibly.
Physics has bounds of concern, where as mathematics would encompass all physics, as well as all possible alternative physics, in all possible forms. Maybe not unbounded, but close to it.
QWEN-3.6-27B
P.S. Couldn't resist :p
Unlike 3.6 35B.
Is it dense or MoE?
It's a good model Sir!
The field of mathematics is smart about this and knows the difference between a pile of Lean code and understanding, and mathematicians try to get from the unintuitive explanations to something that makes more sense, e.g. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the... (where incidentally Tao used a chatbot to help take apart the problem, but with a lot of interaction and work from his side).
Making things understandable is mathematics, and more generally a kind of intelligence, and is crucial to continued progress. You couldn't use algebraic geometry to disprove a conjecture if people hadn't organized (what could have been just) a pile of random observations into something called algebraic geometry.
Historically LLMs have done best where it's possible to train using an objectively verifiable reward function. Computer programs are pretty good on this front and so are Lean proofs. (Of course, they don't only do things you can RLVR heavily, but those have progressed fastest.) Not sure where 'making mathematical knowledge more understandable' falls on that spectrum.
Understandability isn't only important for advanced math. Keeping computer programs from becoming a mess is a challenge in high-level organization too, and the chat with the user is an explanation task. If you look online at what people say about large LLM-built codebases (SlopCodeBench is a neat effort to make make it concrete, but common wisdom seems to mostly agree on the general problem) and chatbot prose, I don't think everyone considers those solved problems!
It's hard to tell how thoroughly the labs grasp and care about this at an organization-wide level. I'm sure at least some maybe-results exist inside labs but haven't been published because the humans couldn't verify them and didn't want to be embarrassed with a false result. (Maybe also why counterexamples are a lot of the first results published: often simple to verify, even if hard to obtain.) A good sign would be if results in a few months come out more like what mathematicians consider well-written papers explaining results in a more intuitive way, fewer shocking announcements of bare counterexamples in tweets. It's probably a slow climb to get there.
Even the experts on the problem being solved find the writeups nearly impossible to read.
Example: https://nitter.poast.org/henryquantum/status/208362369543662...
Seems like a disservice to the community that openai put so little effort into producing good writeups...
I think this is where Terry Tao hit the nail on the head. OpenAI has made claims, but the mathematics community does not accept garbage. It's less that they've done a disservice to the community; more that they've done a disservice to their own reputation.
I agree. Prompting 5.6 Sol to "Review these writeups and include appropriate citations and credit where appropriate" probably would have been a step in the right direction.
Then again, this would require them to actually care about the mathematics.
I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?
So a good fraction of famous old math puzzles with no practical use are famous because they are in some way similar to problems people actually care about. If you can solve the toy problem that is in some way simpler, maybe you can use the same methods to solve the "real" problem.
One open question is whether these machine solutions to these problems will act as springboards to future research, either when given to human mathematicians, or when used to train future machine models.
One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.
And then there's G. H. Hardy, famous number theorist, who wrote an "Apology" for mathematics in the sense of defending the study of number theory for its own sake because it's beautiful.
Which one were you referring to?
It is one in the same! The apology is, of course, defending the beauty of the pursuit but at the same time the writing acknowledges the (alleged) utter lack of practicality.
It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.
True. Though ~99% is still useless in the end.
We are not at the end, how could you possibly be confident in this claim?
As they say: 80% of marketing is useless, you just don't know in advance which 80%. Same with math
The problem is more that the puzzles had an original context and reasons why they arose from "real" problems directly or mathematical problems trying to solve "real" problems etc. but the presentation and abstraction hides this pretty well for people not "near" the problems.
Isn't most of math this way? Occasionally, we find a practical use for it, but that's usually not the point.
Yes, that's correct. I've done research in pure math, but left after the postdoc level. I enjoyed it for its artistic sake but it's not something that gives anything practical. Most pure mathematicians don't ever work on practical problems either. They just teach and in return get to do their hyperspecialized research. It's done for its own sake. (Some pure math eventually ends up having applications but it's rarer and rarer and it wasn't very common to begin with. Like applications to crypto or whatever, but that was always a tiny slice to begin with.)
The meaning of "practical use" is all about context - who/when/where/why/what - and so it would be kind of hard to definitely claim, in an intellectually honest way, that a piece of math is an old puzzle with "no practical use whatsoever".
Unless you want to be the guy in the 19th century making fun of Boole algebra for having no practical use. You might be right, but not for long.
Thanks to Martin Gardner's books back in the day we had genious hackers doing just that, recreative programming... until it guided our lifes: GNU/Linux, GNU itself, GNU Emacs, the BSD's, Rogue with the Curses library (and tons of Unix serious stuff), Unix itself to play SpaceWar in a PDP10...
even the ZMachine it's an astounding example of running virtual machines on 8 bit machines 20 years earlier than Java.
Most modern pure math has no practical use whatsoever. It's amazing how few people understand this. Mathematics is a service department and maybe one in a hundred thousand results have any practical use. (Speaking as someone with a PhD and over a decade in pure math with several published papers...)
Your observation is perfectly on point, I think the season of companies announcing breakthroughs might be over soon (unless they somehow manage a major achievement, P vs NP or similar). At the same time, mathematicians will be left with superintelligent machines solving the actual math for them, much like software engineers nowadays. This was unexpected, and unexpected at this scale up to a couple of months ago.
My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.
That axiom is clearly false. AI can interact with external systems, which means it is not just compressed knowledge - it has the ability to access new information.
Why does being a compressed corpus of knowledge mean it can't discover things? What if it has knowledge of techniques to discover proofs?
It does and that's what's happening.
Pattern matching and replacing can be Turing Complete, from sed to rewriting languages.
So it's just a matter of combinatorics.
https://wiki.xxiivv.com/site/rewriting.html
I find some interesting parallels in chess, which often has lots of analogs with math to begin with. But chess went from a pure human endeavor to one where supercomputers aided by world class players finally managed to eek out a slightly suspicious win against a world champion (approximately where we are now in math) and to now a days - where your phone could easily crush the world's strongest player, who is also probably the strongest player of all time.
The way the chess world adapted this was initially to try to understand the machine. After all chess, like math, is complete information - so you can easily see the computers 'thoughts' in terms of the exact moves its saying are best in a variation and how it might respond to any other idea. But it quickly became clear that this wasn't working so well.
Players would regularly get positions that the computer says 'and black wins' and then proceed to lose it convincingly, simply because the positions were so extremely weird and difficult to play that even if it might be technically winning, it's the sort of position where you're walking a fine line with lots of complex moves to find. Humans aren't computers and even the best of us can't play like one in weird positions.
Now a days they're taken more in balance. The computer's evaluation of a position is probably about as good as you can get, but playability matters much more in practical terms. Knowing the eval of a position doesn't really matter if you don't understand the position. Knowing the answer can help with understanding (for instance computers have radically reshaped and improved human understanding of space in chess as we noticed computers obsessing over it) but I think the days of 'oh the computer says it's winning, so I should be able to take it from here' are near to gone.
It's more like FOSS project using and you ask a developer who works at a different company how to build this project and he says "I don't know automake, perl and there are lots of macros"
What are the odds that this random FOSS project has solved the problem of building software and nobody noticed. Close to 0. Don't confuse accidental complexity for transcendental depth.
In college the punchline for all the engineer, physicist and mathematician jokes were something like "The mathematician says: Yes, there is a solution."
In all serious "I don't understand any of this it's way over my head."
> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them?
Theoretical science isn't about usefulness per se, but knowledge and understanding for its own sake, so a better way to say this is to point out that you only benefit in such cases if you understand things yourself. If understanding is the goal - which it is in the case of true theory - then the only way to attain that goal is to actually attain understanding. What good is it if an LLM produces a valid proof, but no one grasps it? This isn't like digging a ditch where it doesn't matter who does it or how he does it as long as you have a ditch. Here, the ditch is knowledge, as it were.
This has been an ongoing debate since the computerized proof of the four color theorem fifty years ago.
i doubt anyone really wants, or cares, to understand what the AI comes up with. If it's not related to your work or your name isn't associated with the discovery then I would think time is better spent on things that are.
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
Huh?
Mathematics is an extremely wide subject, it's perfectly normal even for two professional mathematicians not to understand each other's work. Have you considered that maybe they just don't work in that area?
Are you implying OpenAI's paper (which was, by the way, edited by humans and provided Lean certificates for most of the proofs) is actually gibberish? That's flat-earth levels of conspiracy.
They claimed them as advances, not breakthroughs.
There's no honor in being stupid, yet I must admit I am stupid, as there's even less honor in being stupid and pretending otherwise.
A lot of people were super hyped about OpenAI's 10 discoveries, but I still don't understand what they mean, and even if I did, what are the implications.
Like, what are non-sofic groups, and what follows from the conclusion that they exist?
I mean in the sense that quantum mechanics might make my head spin, but it's because of that that we have stuff like semiconductors, which have been one of the most significant discoveries.
The Fourier transform is one of the reasons we have fast telecommunications and radars.
What practical things are possible or might be possible due to these results?
> What practical things are possible or might be possible due to these results?
Not much, they're all fairly minor problems that have been solved, of ~entirely niche academic interest. It's so far more just that AI _can_ solve novel math problems, ones that humans didn't accidentally train the answer in and it just spit it back out.
> Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands
An unsettling number of people do seem to be more than ready for this. In fact I'd say they seem almost gleeful about it. Thinking is hard and they don't seem to like doing it!
>If a math problem falls in the forest but nobody is around to understand it does it make a sound?
This is not new nor unique. There is plenty of research, especially in math, which can really only be understood by a few people in the entire world. It is not uncommon for a proof to be presented by a mathematician which, initially, is only understood to that mathematician, and it can take a long time for even another mathematician who is an expert in the same field to be able to confidentially say they understood it.
>I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”.
This is meaningless in a vacuum. If you give a novel proof in some niche subfield of topology to a competent mathematics researcher who focuses in number theory, they'd say the same thing regardless of if a human or machine wrote the proof. A bunch of "career mathematicians" on Twitter proclaiming this doesn't mean anything other than these people aren't currently equipped to understand the contents of the proofs. That's fine and normal, but the idea that anybody with a PhD in Math should be able to pick up one of these proofs and give it a skim and be able to say, "ahh, yes, quite clever, it seems so obvious in retrospect," is absurd. That's not how this kind of research works.
>Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands?
Nobody is blindly accepting these proofs as valid. ChatGPT isn't spitting out a wall of text and proclaiming that they've solved a previously unsolved math problem while everyone is saying, "well if an LLM says it, it must be true!" lol
These proofs are being checked by automated systems (which have been in-use well before LLMs have existed) as well as being checked over by actual experts who are actually capable of (and motivated to) verifying these proofs. But that work still isn't done. There's enough evidence that these companies are confident in saying these proofs are correct, but there's going to be a lot of ongoing work from people to continue to verify and, more importantly, understand these proofs. It's literally some of these people's full-time jobs to do this.
>How do we tell truth from fiction?
When was the last time you verified even a classical, relatively simple mathematical assertion? How often are you just relying on a larger system of experts to ensure that we're not just blindly accepting fiction as truth?
That's not to try to stick it to you personally, but it's just highlight that there's an entire system in-place here that you're not aware of and that you don't have an understanding of that is working just fine including in this context. Real mathematicians aren't going to lazily start letting OpenAI assert whatever they want about their products solving these kinds of problems without heavy scrutiny.
No offense taken, I appreciate the time you took to respond and clarify these things.
> How can we possibly make use of these breakthroughs if we don’t understand them?
I mean, arguably the development of LLMs matches that description.
Maybe the "beautiful, elegant" math is really just accidentally that way, just the tiny cross section our dumb human brains can understand. The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.
> The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.
It is, provably: per Curry–Howard correspondence, any program you write is a proof of a theorem, and it is indeed mathematically meaningless.
Generating a value of type "Either (Int, String) Bool" is proving that there's at least one integer and at least one string, or there's at least one valid boolean value. Except in Haskell, where it could also be an infinite loop.
I, for one, welcome our giant supercomputer overlords.
I have the same questions about human mathematicians!
I can't tell if xkcd #435 is still true, or if math is just as mushy as everything else seems to be. When a math proof can only be understood by a handful of people, what does that mean about that proof? I think the LLMs are pushing a problem that existed already and pushing it further.
[0] https://xkcd.com/435/
That's why OpenAI also published machine-checkable proofs.
The process is very, very faintly similar to running a typechecker over your software sources.
They're publishing machine checkable proofs because that's the only way they, themselves, can check them.
They don't understand the math either.
For 3.5 days we had a machine checkable proof the Collatz conjecture was false. There turned out to be a bug in the machine checker.
Yes, however any LLM agent you asked about that proof could tell you that. (And even most humans who know a little bit about the bug.)
You are right that Lean isn't great in this respect, and people are working on proof formalisations that are less prone to bugs.
Non-Erdős problems are also falling.
The AIs seem to have some combination of very broad familiarity with math (enabling relevant things from other subfields to be brought in to the proof) as well as patience and "sitzfleisch" (stamina in working through details even if they aren't immediately obviously promising.)
An obvious area for improvement would be automated generation of new conjectures and attempts to prove (or disprove) them, with the discovered arguments then being used as training for refined models. This will require autoformalization to check the results as there will be too many for manual verification.
Generating new conjectures is easy. It's generating novel conjectures that's hard.
On the other hand, quantity has a quality all its own.
Conjectures that survive the gauntlet of immediate proof/disproof could also be interesting.
I applaud how you articulated this.
The mathematics are above my intellectual capacities, but I still find the man and his lifestyle fascinating. Though I admit his heart attack probably wasn't a coincidence, sadly.
I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS.
I often see comments that some of the users here long to "make a computer do neat tricks all day", and I can't help but think meager sponsorship could go a long way in this area. Existing grant structures, being much more traditional, are constrained by their cost.
> I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS.
The problem is that Paul Erdos was eccentric but he was also Paul Erdos. I'm not saying you're implying that, but I feel it's a similar line of thinking to how popular culture often romanticizes autism and Asperger's because some very smart people are (allegedly) affected. The same group includes people who need 24/7 care.
Computer science at the frontier is as specialized and hard as mathematics, you need years of study to truly understand a field well enough to make meaningful contributions. You can try sponsoring me if you really want, but I don't think you'd be spending your money wisely in expectation.
FWIW I've worked in two different academic labs, both of which would have benefited from a runaway worker barreling down random paths, as long as they were being productive in some capacity.
> [...] as long as they were being productive in some capacity.
I'm afraid you only know that after the fact.
You might also like TempleOS, or at least the context and history behind it.
I guess that's the crux of the issue
> Though I admit his heart attack probably wasn't a coincidence, sadly.
He died aged 83...
I didn't know that. A poor assumption I made that his amohetamine use shortened his life significantly.
Most drugs aren't as unhealthy as the government wants you to believe.
When Gwern needed a fellowship to come to SF he got one. When he had his “great idea” it was funded. The truly unique people out there find a way. Any government bureaucrat will just find the Jason Ardays of the world.
That's survivorship bias. If there are truly unique people who never get a chance to do what they would be good at, we don't hear about them.
Private sector bureaucrats are no different from government ones. The fundamental issue is that if you have a system, it's going to fail outliers. Almost by definition. And the situation is even worse if the system uses metrics for making decisions.
Well, if the statement is "we need some eccentric thinkers" then survivors suffice. We have some eccentric thinkers. If the statement is "we need more eccentric thinkers" then I think I'm going to demand the rigor of "explain why we need to increase the quantity and why your proposed intervention is the best way to get there with evidence behind it".
Honestly, I think our extremely liberal disability benefits scheme achieves this to a large extent (albeit possibly with downsides for both the recipient and society).
Look at all of the crazy stuff made by unemployed trans people/autists over the past decade, disproportionately from countries with strong welfare states.
something i've just realized : today long-standing maths problems are falling. It's great intellectually but won't probably have an immediate impact on our lives.
Now, what will happen once long-standing physics ( and chemistry and biology) problems will start to fall and at the same rate ?
Then we're going to enter a totally different world.
It's hard to see problems in those fields falling at anywhere the same rate as math, because they are all experimental fields.
There may be some problems of type type "why does X happen?" that appear answerable in terms of known science, but even these would need verification. If you want to make advances in fundamental physics, then a promising AI-generated theory might take a decade and billions of dollars to prove or disprove.
Math is a rather unique field in being entirely theoretical, axiomatic and self-referential. It is basically the best possible case not just for AI to advance without needing experimental verification, but also specifically for today's AI technology of auto-regressive LLMs and RL training, whereby valid reasoning steps learnt in one context will also be valid in another context (i.e. there is some generalizability of learnt reasoning) as long as you have learnt the pertinent aspects of that context that the validity depends on.
For example, protein folding has been figured out by AI. This was a very big moment for science and yielded a nobel prize. Alphafold 1 happened 4 years before the first version of ChatGPT and the LLM craze we see today.
But yeah, most problems in physics, chemistry or biology require labs on top of actual hard thinking. You need to be able to design experiments in a certain way. Once you have the funding, the right tools, the right people to use those tools, then you can use LLMs to increase the speed of the calculations and so on.
There have been other discoveries though by deepmind: https://deepmind.google/blog/millions-of-new-materials-disco...
Those long standing math problems solution may serve as building block for solving the experimental science ones.
Not really. These same techniques fall flat on their face when applied to most physics and chemistry problems. All of academia has already been doing ML4Science for the last 8 years. God knows how many billions have been spent.
The only two major highlights are weather modeling and folded protein backbone prediction.
Mostly everything else, either lacks enough data, or there are contraits on the size of the foundational models that render them impractical or they just fail to generalize.
Connect the AI to a fully automated lab and it may be able to generate and harvest those necessary data.
It may also produce Supercovid in the process, so...
We don't have to worry about new math as much as we do about new biology. Although "breaking all current encryption" would have some nasty consequences as well.
It seems that AIs are really good at finding counterexamples now.
Even if progress by AIs in proving conjectures lags, it seems likely that AIs collectively will, in the next few years, find counterexamples to nearly all the Erdős (and other) conjectures that are actually false and also provably false.
That means we will able to assume that nearly all the remaining conjectures are either true or undecidable.
Surely, that's good for folks who just want to know where the truth boundaries in mathematics lie.
It's obviously causing a lot of soul-searching amongst professional mathematicians.
Arguably, they should have given less weight for the last 100 years to Hardy's view in 'A Mathematician's Apology' [0]:
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.
Rota takes a much more balanced view in 'Indiscrete Thoughts' [1].
"Problem Solvers" take Hardy's view:
> ... The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. Mathematical exposition is regarded as an inferior undertaking. ...
While for "theorizers":
> Mathematical exposition is considered a more difficult undertaking than mathematical research.
If professional mathematicians can reinvent themselves, there will be plenty of work left to do to explain the results of AIs to other humans.
There probably needs to be a new career path into professional pure mathematics other than doing novel research in a PhD.
[0] https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
[1] https://ncatlab.org/nlab/show/Gian-Carlo+Rota
https://ncatlab.org/nlab/show/Gian-Carlo+Rota
I'm not sure we should take it as a given that AI will not also become better than humans at mathematical exposition.
Disappointing lack of "Why" in an article that starts with it.
Are they actually doing something new and novel, or are they just absorbing that "a=b as was proven in transcendental hyper-circular group theory; and b=c was proven in universal quantum superposition"; and they're the first to find the connection that a=c? And several of the problems are counterexamples, not novel proofs of correctness?
Its fascinating either way, but it'd be nice to actually understand more of what is happening.
One of the recent problems was actually proven with elementary results. It did use a technique not usually applied to the problem space, but it wasn’t a hyper specialized result.
Because pattern recognition and deduction is automatable and LLMs are superhuman at low depth high depth problems. Next
is probably a mix of other math papers that combined can solve this, the ingredients were already out there and they got trained with it.
could be, I guess we'll see.
but even that, if done correctly, is quite impressive. It sure sounds very useful given the number of papers out there.
Quantamagazine is owned by the heavily AI invested Renaissance Fund.
The whole article is an ad that covertly or overtly inserts how websites are built with ChatGPT, how humans say that AI is better than them etc.
This is incidentally the future of chatbots. I could not have written this comment without Illy Espresso. Would you like to find a cafe near you?
We have now passed the point where the AI bots were annoying. Now it is the reactionary “everything even possibly slightly remotely tangentially connected to AI must be deplatformed” rants everywhere that are even worse.
It would be tempting to assume the commenter here has no idea what Renaissance is or how they made so much money.
The "commentator"'s account also has a karma of negative 3 ! I've never even seen a negative karma before !
There is a pattern where the most divisive comments, that one might suspect of being from bots, do tend to come from accounts with single or double digit karma. Maybe HN tries to identify and remove these, which is why they tend to be new ?
There is no such thing as the Renaissance Fund. Quanta receives funds from the Simons Foundation... I also don't know where you get it from that Renaissance Technologies is heavily invested in AI. Any sources to back up your claims because I suspect you made all that up?
There is. As for their investments, I don't know:
https://www.simonsfoundation.org/about/our-history/
>Quantamagazine is owned by the heavily AI invested Renaissance Fund.
Except it's not? It's owned by a private foundation where the only link is both were founded by the same dude who hasn't run either in like 15 years.
That seems like a pretty big thing to just gloss over like it's just set-dressing.
Which fund would that be?
https://www.rentec.com/Home.action?index=true
> The company is widely considered one of the most profitable hedge funds in history, generating an estimated $7 billion to $8 billion in annual revenue solely from management and performance fees
If you don't know Ren you should!
I meant which fund is supposed to be AI heavy? RIEF? RIDA? Medallion? Aren't they all quant?