I was a graph theory junkie long ago and even moved to Budapest for awhile to study among the greats. While I was there I started working on Barnette's Conjecture which came to occupy my thoughts over the next 24 years of my life, on and off as I worked in many different fields. Last summer I even thought for a few days that I had actually solved it.
But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.
There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.
I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.
> Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.
This is the part that gives me the strangest feeling about it all, because you're not the only one with this experience. I've experienced this too on different problems, as have many researchers across many fields.
I disagree with the Fields Medalists on the majority of their complaints. AI math is happening and there's no going back. However, on one point I increasingly agree: virtually none of this stuff is possible with technology any normal citizen has access to. I have no problem with AI models making revolutionary advances in math or science. Where I start to have a problem is when the AI models making these advances are tightly withheld, proprietary, and seemingly never released with these capabilities intact. This has been the case for all of 2026 so far.
I suspect that this is in fact the source of much of the angst. None of this progress is reproducible outside of one or two teams inside OpenAI and Anthropic. It's becoming an incredible concentration of power that I don't know that we've ever quite seen before. Right now, it feels harmless because it's being used for wonky math problems that aren't (yet) practical for anything. But great power never stays harmless. History has taught us that countless times, in countless different forms.
> I suspect that this is in fact the source of much of the angst.
Why do you "suspect" this as if it's some hidden motivation when the very first paragraph of the advisory group's statement (linked from the OpenAI post) says:
> 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. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
Tao and others in that group have been strongly and publicly pro AI from the start. They are not advocating "going back". They're objecting to the strip mining of open problems using proprietary technology.
I don't think the strip mining metaphor is appropriate. Mining is a zero-sum game; if I mine something, nobody else can go and mine the same resources I did. Mathematical problems don't go away when AI finds a Lean proof. They create new opportunities for humans to study the solutions, learn new techniques from them, identify promising directions for future research, discover alternative/more beautiful proofs, and write expositions for other humans.
Strip mining is very apt if you view the economics of the present system as "effort -> recognition -> career advancement". Even in strip mining, the resources that had been buried are now available for use in the broader economy. What's no longer available is the living that was to be had digging them out.
The problem isn't effort, though. All of the things I mentioned constitute effort and could be rewarded. The job economy was created by mathematicians incentivizing the proof of difficult theorems above all else and valuing all other work at approximately zero as far as career advancement was concerned. Now they're pulling a 180 and claiming that math was never really about proving theorems, but that's contradicted by their revealed preferences. The strip-mining problem only exists if they continue with the status quo ante.
Mathematicians aren't homogeneous. There are mathematicians valuing pedagogy, collaboration, bridge-building, theory building, along with those that chase the 'difficult theorems', to name a few, and there are lots of flavors within each class, with lots of blending and blurring. You infer that mathematicians prefer the status quo simply because it is the status quo -- with a little thought, you'll recognize that this is a fairly silly notion.
There are myriad circumstances where the values of most practitioners differ from the status quo, which is nevertheless well-entrenched. This can arise from inertia, or from outside forces, such as broader cultural milieu, integration with larger institutions, or contending with economic realities. If you think that these do not and haven't historically played a role in determining the job economy and that math is a pure field where mathematicians could comfortably shape it according solely to their own ideals then you are naive
And, in addition, many mathematicians are graduate students or postdocs hoping to line up a permanent job soon.
For example, if you look at Terry Tao's blog, he has a tremendous amount of first-class expository writing. So, too (to some extent) do junior mathematicians -- but, unfortunately, this tends to not be highly valued by the job market. Grad students and postdocs have learned that to succeed they need to play by the existing rules of the game.
Well, the board has just been yanked from underneath them. People like me can afford the sort of idealism and soul-searching that the parent comment describes, but junior mathematicians face a very unenviable set of circumstances.
1) I never said the problem was effort; I was trying to explain the strip mining analogy, and it's one of the two anchors that make the analogy work.
2) Mathematicians didn't create this economy; it was foisted upon them by the same managerial mentality that brought us "publish or perish" and "the monthly sales quota".
3) I can't tell if you honestly don't get why the strip-mining analogy resonates, or...?
Here's another analogy: if we suddenly discovered personal teleportation, and marathon runners were complaining that it was ruining the sport, would you say "they're pulling a 180 and claiming that marathon running was never really about getting to a point 26 miles away as fast as possible, but that's contradicted by their revealed preferences"?
The strip mining analogy is better though, because it captures the sense of irreversible goal-loss when a problem goes from being "unsolved" to "solved".
As far as I understand, even with "publish or perish", peer reviewers decide what counts as an important enough paper to be published in a prestigous journal, and committees of peers decide whether or not, say, an expository article on arXiv or a textbook counts toward hiring or tenure. Again, as far as I understand, those things have largely not been rewarded in the past.
I like your marathon example, but maybe not for the reasons you intended. The community of marathoners decides the rules of a marathon. You don't need a hypothetical teleporter; you're already not allowed to use a bicycle, performance-enhancing drugs, or shoes that don't fit the specifications. The rules are updated to adapt to changing technology. Yes, I'm arguing that the strip-mining analogy doesn't make sense because mathematics is in the same situation. There's nothing stopping peer reviewers and hiring/tenure committees from changing the rules about which kinds of effort confer recognition and career advancement.
NO. Most definitely not. Nor am I advocating any sort of gatekeeping.
My entire point is that there is a legitimate perspective, held by real people, from which the "strip mining" analogy is perfectly apt. There were fishermen working these waters for many generations before the trawlers came, and they see the situation differently than the people who've got a vested interest in the company running the factory fishing operations.
Strip mining is an extraordinarily appropriate metaphor.
Imagine a mine has an unknown number of rare materials. And you know the general location of a few of the most valuable spots. But you don't know what may be valuable right next to it. If the pieces that we know are valuable are suddenly gone, the incentive to mine that particular area drops considerably, dropping the chance to discover potentially brand new materials that would have been found the normal way.
That's an empirical claim. I could equally well say that doing an automated search of the problem space and having a database of results and open problems will identify vastly more interesting and valuable areas. Again, the idea that math is some exhaustible material is a metaphor, not an established fact. I'm willing to change my view as new evidence comes in, but I think we're going to have to wait and see what the landscape looks like in a few years.
> the incentive to mine that particular area drops considerably, dropping the chance to discover potentially brand new materials that would have been found the normal way.
FWIW, I think the metaphor breaks down with this framing. This isn't really a problem associated with strip mining, what's left behind is generally low or negative value (toxic). I'd suggest a different metaphor, from Wikipedia:
> This process involves the removal of all ground vegetation in the area, which is a detriment to the environment.[19] Topsoil may be placed over the tailing along with planting trees and other vegetation. Another reclamation method involves filling in the hole with water to create an artificial lake. Large tailing piles left behind may contain heavy metals which can leach out acids such as lead and copper and enter into water systems.
This feels very similar to the issues with algorithmic problem "mining". It has the potential to destroy the human ecosystems surrounding these problems, leaving barren wasteland behind where nothing can grow or flourish.
I hope sincerely hope they don't currently use "proprietary technology" like:
Wolfram Mathematica ($890/yr)
Magma ($2500/yr)
Maple ($680/yr)
COMSOL ($1500,yr)
Matlab ($500+/yr)
Seems like a very strange position to take, in my opinion.
Why does the field of mathematics suddenly now need to be "fair" and give everyone access to the same tools? Has that ever been the case in academics? It's always been a competition for name-recognition, grants, institutions, etc.
Macsyma / Maxima was an MIT developed CAS system back in the 60's that was proprietery until they sold it off to IBM for a tidy sum. Magma actually has free access if you're in the US, otherwise you pay. That's not to mention proprietary MATLAB toolboxes or specialized Stata modules.
Likewise, a lot of the above packages have pretty sweet site-wide deals with R1 universities. If you're at a smaller, foreign one, you're out of luck.
You can't compare that to a model that cost tens of millions to train, millions more to run, and if you don't work at OpenAI you can't get it at all no matter how much you're willing to pay.
Everything about it is secret. There is no API access or official model card that could tell you how it works. You don't know how it came up with the results, how much time it had to spend, if it's a pure LLM or a neuro symbolic AI or a human in the loop system that still needs hundreds of trained mathematicians. Math always cared about transparency (maybe except Stephen Wolfram) and this is the complete opposite.
And because they don't share reasoning traces or anything at all other than finished results, no one knows how much credit really belongs to them. It's perfectly possible that their system is searching the internet for promising leads, connecting them all together to build a coherent proof and then completely throwing away the sources.
I don't think it works like that but no one can know for sure because they never show us what it's really doing.
Regarding the advisory group, OpenAI claims to “have drawn on their advice”, which would include not dumping a bunch of AI slop, with the footnote that if they do do that, at least fund the process of digesting it.
At the same time, there's a new note at the bottom of agmai.org stating how they've been in contact with OpenAI about this particular release, and they say that “we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully”.
So, what's going on there; is this British English for “they didn't follow anything at all”? Because from my perspective, it looks like they doubled down on the Navier–Stokes approach of trying to maximize PR gain while being as lazy as possible about actually contributing anything back to science, releasing only slop that may or may not be correct and may or may not be straight up plagiarism, as has been the case earlier.
If I were on the AGMAI board, I'd feel terribly exploited when reading that press release, yet their response is modest.
Hairer, if you're reading this: is there any indication whatsoever that AGMAI was anything but a cheap way for OpenAI to science-wash their press release?
Now I know there are issues with the field and how just answering these questions may cause broader problems, but I feel like the posted results is far from slop. We can't just call any output slop, or it loses all meaning.
If it was slop, it'd not be causing the issues the group are concerned about - they're not saying "the problem is we're getting loads of incorrect proofs thrown about that are nonsense".
When you blanket a set of things with a pejorative, and it turns out that some of the members of that set are demonstrably and definitively NOT covered by that pejorative, and that all the pejorative means at bottom is "I don't like", all you've accomplished in the long run is to call into question any future legitimate use of that pejorative. It is tempting, especially when heated, to stretch an invective, but it will ironically only lead to the death of its utility over time.
So the fact that the Library of Babel (i.e. all possible books) contains occasional gems means that you can't object to using it on principle? That would seem to follow from your logic.
What about a filtered set "all well formed books"? Or "all well formed books that are plausible enough that they could convince a reasonable person, regardless of their accuracy"?
It's generally taken that a cup of sewage in a barrel of wine makes a barrel of sewage. Surely a reasonable person could claim that a barrel of sewage was still sewage, even if it contained several cups of wine?
I find it hilarious how people seem to think spending enormous effort to fix obvious failures, only to immediately uncover a new round of obvious failures is a recipe for exponential takeoff. So I guess we're both going to be pretty amused over the next few years, huh?
They're not calling any output slop, they're calling indecipherable output slop. The management class responsible for hiring, firing, and paying people doesn't possess the domain knowledge to say for certain whether or not LLM output is optimal (which, in this context, means correct), but they will trust that it's good enough to justify further automation / fewer grant approvals / etc. So in that sense, slop can and will cause the economic issues people are concerned about.
University boards want the prestige of successful research programs. Doing the hard work to get something demonstrably true is going to lose out economically in this paradigm, where we are all being conditioned to uncritically ooh and aah at the incantations being elicited from these magic boxes. The oracles even have legions of zealots who will berate you for not being sufficiently deferential and reverent, or worse, accuse you of blasphemy. If for no other reason, I agree with using the term to express all of the above succinctly, even if LLMs can be helpful tools generally.
<< The management class responsible for hiring, firing, and paying people doesn't possess the domain knowledge to say for certain whether or not LLM output is optimal
To be fair, management rarely possessed domain knowledge in rudimentary management.
In the context of maths papers, the term has come to refer to papers having the shortcomings that are, for whatever reason, typical of LLM out, including things like using non-standard terminology all over the place, emphasizing easy steps while leaping over harder ones, having bizarre organisation, and, importantly, failing to properly cover existing work and as a result being hard to tell from plagiarism.
The degree to which these issues feature will differ, but it is generally the case that converting the output to proper research requires significant effort, hence the AGMAI recommendations being what they are, and not performing that effort tends to come off as laziness or incompetence, so I can see how slop has become the popular term.
> We can't just call any output slop, or it loses all meaning.
The term “ai slop” is not supposed to discriminate good ai output from bad, the entire purpose of the phrase is a blanket term that delegitimizes all ai output.
You are missing the biggest benefit of a self storage business - the appreciation of the underlying real estate. When you dig into the financials of the major self storage businesses you'll see they are essentially REITs that have better cashflow. They can pick an up-and-coming area, do a minimal build-out with low annual overhead, and then down the road when the facility would be needing overhauls and maintenance the underlying property has typically appreciated so much that it dwarfs all other associated revenue streams and makes sense to sell and raze the existing structure. Great business model if you have a long enough timeline.
This is also why some startups trying to revolutionize the self storage model had extreme headwinds - if you are renting the underlying properties and trying to make the storage business profitable you are at an extreme disadvantage to the larger players who can subsidize operating costs with portfolio appreciation.
This same concept is also true of McDonalds historically - they own the land, lease it to franchisees, and realize the appreciation themselves. McDonalds are fairly economically valuable, so the capital gains by the time they're cleared and resold as development often dwarfs the value of the franchise fees.
The whole reason why the Catholic Church doesn't allow priests to be married anymore was because it used to be that the individual churches were mostly the property of the priest, and would be passed down generationally (with some restrictions). The church phased in the ban on marriage so that over time, the churches would all in up in the estate of a priest with no heirs, where they'd become the property more formally of the Catholic Church proper. Today the Catholic Church is the single largest landowner in the world, with about 170 million acres.
More recent religious organizations have taken note of that, with LDS owning about 2.4M acres.
Your comment dragged me into a rabbit hole, which is always a welcome surprise.
From my readings, one can always debate whether this reason for celibacy is whole or not (of course, more holy reasons were given, like purity of the ritual among others), but avoiding inheritance from father to sons was certainly a big factor. By barring heirs from the equation, it solved the whole dispute over inheritance. Yet, when it comes to clerical office and priesthood, "property" is not quite right, since it's mostly about usufruct and benefices, moreso. Which can be significant assets, that goes without saying, hence the Church did everything it could, for holy and not-so-holy-reasons, that traditional generational transmission does not apply. No next generation, no problem.
Quite importantly to the Roman Catholic Church, they wanted to avoid dynasties of priests that would grow in importance and out of their control, so they did whatever they could (ban marriage and fatherhood) so that a mis-aligned priest would only be a temporary nuisance to the Institution.
A lot went into this short burst of investigation, like the fact that early Churches were private property of land-owners, building churches for indulgences and power, reinforcing their ties with the Church, and I may dig deeper. Thanks for your comment again, if you have some references you recommend, I'd appreciate you share them!
I'd like to point out that the CA assembly has passed AB-1542 and it is likely going to be signed by the governor this week. This would make the sale and sharing of "sensitive" personal information illegal, and one of the sensitive personal categories is geolocation data that can map an individual to within a 1850-ft radius.
In my understanding, this pretty much makes this type of driver data illegal to sell or share. CalPrivacy's enforcement division has their eye on connected car manufacturers already, so we'll see what they do with that.
Not true. If they do that they are violating the law and that will show up in the audits. They have to delete the data, all records associated with that identifier, now including any 3rd party data that they bought from other brokers and did not collect or buy originally. There have been many loopholes for years but they are all being closed.
You can invoke CCPA (if applicable) and directly ask a covered business or registered data broker for an exact output of your existing information on their systems. This is honestly rare and most companies will find ways to deny you - asking for various documents to prove your identity and hoping you quit responding.
If you are a CA resident you can also get on the state's DROP platform to request that all registered data brokers delete your data, but it's only for deletion and not the other 4 types of data requests allowed under the CCPA (right to know, right to collect, right to opt out, and right to limit use).
Reminder that smart TV ids are part of California's DROP platform, and that you can (and should) request that all registered data brokers delete data associated with your television.
I grew up playing X-Wing and TIE Fighter, editing missions was my first exposure to "programming" and the notion that you could change the bits inside the computer to make different outcomes occur.
I also fondly remember my cousins and I taking turns playing missions, and the two that were not playing would swivel and tilt our big office chair to give the sensation of motion, punching the chair with our fists when taking heavy fire.
I've been building the infra for data brokers to connect to DROP (easy), actually effect deletions (hard), and make sure the data stays deleted (harder): forgetmenaut.com
DROP is pretty significant considering that it's the first compliance system meant to have an immediate effect (delete the data), backward-looking effect (forward a legally-binding deletion request to everyone that data was sold to or shared with), and a forward-looking effect (never let that record re-enter your system, in perpetuity). This is significantly more tracking and auditing infrastructure than anyone in the industry has ever normally run, not to mention that the request volume is 100-10000x what most of these brokers would process in previous years.
We'll see how well companies actually managed to comply when audits are performed for every registered broker in 24 months. I also think the impending prosecutions (and likely bankruptcies) of several unregistered data brokers will encourage the others to take it more seriously.
Well the CPPA (state regulator) just hit General Motors with a $12.75M fine for selling data to two registered data brokers, and made the brokers who received the data delete it all: https://ccpa.world/enforcement/gm-onstar-smart-driver
Does it apply to Google? Well that's an interesting question. I think the answer is yes but the practical matter is that the CPPA is going to get some legal precedent and some more lawyers on staff before they take on Google. At the current number of requests in the DROP platform they could determine Google is an unregistered data broker and fine them $25B+, but I don't think they are going to do that this year.
I think within 36 months they will take the legal victories from prosecuting the first set of unregistered data brokers and apply it to the real players in the data ecosystem. At least, that's what I would do if I were Michael Macko.
But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.
There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.
I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.