Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling).
Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc).
And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
> Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling
Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it. I simply ran out of motivation to pursue higher math, especially when following through on learning and research became more and more labor. (It was always labor; but it was a labor I used to love.) I am far from a prodigy.
> even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are
Disagree. It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind. What your average mind (and student) lacks is exposure to formalism, abstraction, and how these things tie in with what they are familiar with, which is symbolic manipulation. With the exception of geometric proofs (another educational bugbear), they have little context for what formalism is or why it matters.
> Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it.
This sounds a lot like you may have in fact succumbed to your abstraction ceiling, because in practice, the ceiling manifests as not as it being impossible for you to learn something, but that it would take you years and inordinate effort to master what you notice others mastering easily in just a fraction of the time. You may have not heard the exact term (comes from Douglas Hofstadter), and you may be talking about just the academic busywork, but I find it hard to believe you never encountered discussions about this kind of stuff. I would also politely suggest that unless you are Terry Tao posting under some kind of alt, you most certainly do have an abstraction ceiling (or your own mathematical limits) too.
> It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind
The latter statement is obviously false, but regardless, intelligence explains some of the difficulty, and much other difficulties far more parsimoniously than "everyone could just learn any math if they just tried hard enough and had good enough teachers". E-d is merely an obvious and generally familiar example, and nothing I said really relies on this very specific aspect of maths, obviously. We also shouldn't pretend your (almost certainly false) view of math and intelligence isn't also often harmful to struggling students in its own way.
> Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle - while for myself I felt clumsy. even oafish, wandering painfully up a arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn ( so I was assured), things I felt incapable of understanding the essentials or following through to the end.
(Alexander Grothendieck, Recoltes et Semailles)
Amazing that he managed to keep going after hitting his abstract ceiling in graduate school.
You clearly don't understand the meaning of the term. Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling.
Honestly, the pushback on this post is utterly baffling. Clearly the human mind has limits on what it can comprehend and the rate at which it can learn difficult things. Clearly these limits differ among individuals and are related to intelligence broadly.
Huge proportions of the population struggle to ever even grasp simple fractions, and not for a lack of effort from them or society. Fourth-year undergraduate mathematics is another beast entirely. Pretending the world is otherwise is pure fantasy and also plainly harmful, to the world and people that are unfairly pushed beyond their capabilities.
> Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling
lol, see, it's unfalsifiable. No true abstraction ceiling.
Find me the magical teaching method that can make anyone learn any kind of mathematics at nearly the same rate, and you have falsified the idea that anyone has mathematical limits.
Given we as a society can't even figure out how to do this for educating stuff involving simple fractions, my theory is far superior than whatever exactly it is you think.
That's far different than telling a math PhD whom you've never met that they hit their "abstraction ceiling" based on a 2/3 paragraph comment on hacker news. I'm sure you'd have said a similar thing to a young Grothendieck if he were describing his early struggles in graduate school. You're getting pushback because you were being rude and presumptuous.
Work on your reading comprehension, I made it clear that increased effort is what an abstraction ceiling feels like, but also made it clear that GP could have been talking about the effort of academic busywork.
Let's also not pretend that "you could have learned epsilon delta proofs, you just didn't try hard enough or your teachers weren't competent" or "you just didn't have enough time" and etc. is also not rude and presumptuous. Denying the existence of such limits is equally offensive.
You’re in a kind of compulsive ideology here. This same vein of thought and why it is harmful is described by David Bessis in the book Mathematica: A Secret World of Intuition and Curiosity.
“To stop thinking in terms of “gifts” and “talents,” one has to find an alternate explanation. My way of looking at things, which has served me well throughout my career, was to imagine that creative mathematicians were hackers who had found ways to unlock “hidden modes” of our cognition. Most of the time, they’d done so unwittingly, and were entirely incapable of explaining how.”
It is a phenomenon like child-like mental yoga of attention.
Uh huh. A bunch of quasi-mystical bullshit to justify utterly inept garden variety intelligence denialism (it also just shifts terminology: if we accept your metaphysics, it would still be strange to propose that everyone has exactly equal "hacking" ability).
The hubris and willful ignorance required to imagine that everyone is just equally and infinitely unbounded in their cognitive ability is simply mind-boggling in 2026.
But everyone is as a kid when they learn language and everything else. Some people retain that level of watching, listening, and babbling (hacking) when they don't know what to do. So the task is just to get people back into that mindset. Innate intelligence isn't necessarily only symbolic manipulation (analytical), it also is experiential/creative and practical per Sternberg.
> But everyone is as a kid when they learn language and everything else
Also clearly false by almost all current research.
> So the task is just to get people back into that mindset.
Again, you have no evidence, and this is clearly wrong in cases of mental retardation or brain damage. Modern genetic studies also seem to suggest intelligence is related to a lucky absence of errors / genetic problems that are otherwise inconsequential (or even advantageous) in other domains, so really, your "everyone starts perfectly equal in intellectual ability" is just empirically disconnected and ignorant fantasy.
I haven't talked to a child educational psychologist lately (I imagine that would be quite renewing to spend some time in a kindergarten even if only virtually, don't you?), but I don't think biological claims about elite abstraction is the only way when it comes to explaining more and more with less and less for mathematical education. For example, non-symbolic distinction and indication operations are far more fundamental than symbolic manipulation, and using that is a bottoms-up foundation more in keeping with constructivist understanding exhibited by math exemplars.
Cognitive disparity is because of compounding investment of attention and metacognition in development, preferably in a self-referential non-symbolic universal way because intuition is partly based on sensual metaphors and embodied cognition. Everyone has issues distinguishing ungrounded concepts if they don't have a map of them from their attention previously..
Mental rigidity (aka "fragile perfects") is a fairly common phenomenon for math anxiety, whereas Grothendieck advised uninhibited playfulness to deal with uncertainty. The perceived difficulty of mathematics is a social phenomenon rather than organic comprehension limits on abstraction ceiling.
It is the social aspect of math that is the superintelligent part of it, which transcends the genetic determinist perspective. Civilization advances because education transforms the breakthroughs of genius (which all children have ultimate capability for) into the baseline intuition of the rising children by sharpening their attention. Math is supposed to be a democratization of human understanding, that's why the Greeks were so keen on deduction and why proofs are for systematic communication. If a stupid-ass computer can do math, so can any human being.
> Cognitive disparity is because of compounding investment of attention and metacognition in development [...]
Sure, but exclusively? There are no other factors that don't depend on effort / investment / social context?
I can't take you seriously when you take such an absolutist stance on these things when science has long since accepted nothing complex about humans is 100% nature or 100% nurture (really, shared vs. non-shared environment vs. genetics: but, surely you know this).
There certainly are numerous factors claimed. I believe cognition happens from distinguishing reality and people can always learn better how to do this.
Whereas “abstraction ceiling” is hard science backed by ample literature, not a loose metaphor directly contradicted by several prominent mathematicians who didn’t quit when the going got tough.
I think the fact the Feynman took an IQ test and scored 127 is damning of the entire concept. I think what turns mathematicians off is the thought that psychologists who couldn’t tell you the difference between a scheme and a metric space think they can actually measure who has the capacity to be a mathematician and who doesn’t.
Like, why fucking bother doing anything? Why run the 100 meters at the Olympics, let’s just do some genetic testing and measurements to pick the fastest man in the world. Why teach kids music, let’s just measure hand size and do some sight singing exercises and teach the talented kids piano. This whole nonsense reeks of Gattaca-style quasi-eugenics where people get sorted into profession by people who don’t actually have expertise in any of them. You’re not in the guild, you don’t get to appoint to the guild, and you certainly don’t get to gatekeep who can apply for the guild.
Edit: Dropping slurs when someone compares your views to eugenics is an interesting strategy. Shouldn’t you be at a meetup discussing Curtis Yarvin’s work or something?
"Abstraction ceiling" is just a way to talk about intelligence at at the tails that reveals one of the difficulties / limitations you can encounter when it comes to compressing / abstracting complex mathematical objects. Your objections to the term are facile and clearly stem from an obviously unvocalized intelligence denialism that is simply indefensible today. Also, intelligence != IQ, obviously this is too simplistic.
Everything about your arguments and other posts is similar reductions to retarded extremes (our only options are "eugenics 2.0" or deranged intelligence denialism - there is no room for anything in between, e.g. the idea that base intelligence matters and sets a hard average ceiling on potential, but that effort and other factors might push one slightly above/below this ceiling relative to others with a similar intelligence, and etc). Or alternately you hallucinate things I never said or even remotely implied (e.g. we should gatekeep based on dumb psychology metrics or hand sizes).
Just be honest: you know intelligence is real and matters, but you want to dance around this fact because you find it ideologically inconvenient, or you can't admit you yourself have limits (and lack the courage to realize the obvious social broader consequences of this personal admission).
Dropping slurs after being compared to a eugenicist is an interesting choice. Maybe you would be more comfortable in the comments of Curtis Yarvin’s blog. You’d certainly get less pushback there.
Falling into deranged ideological projection is also an interesting choice - one I chose to mostly ignore. You seem to really obsess a lot about this Yarvin fellow: I tried reading his stuff once and found it intolerable.
I imagine you think calling some of my language choice a "slur" here is some kind of gotcha, when the term I used is specifically one widely disputed as actually being offensive, given it is mostly used now to refer to normal people acting in intellectually deficient ways, and not generally to those with actual learning disabilities that deserve our sympathy. There are studies on this, which you surely are aware of.
If I had referred to your more deranged positions as "smooth-brained halfwit extremes", you likely wouldn't haven't tried to impotently pull this "slur" card, even though the semantics are basically identical. Which basically goes to show that you value irrelevant surfaces over substantial realities, and frankly is perfectly consistent with the midwit intelligence denialism on display in your posts in this exchange.
> This sounds a lot like you may have in fact succumbed to your abstraction ceiling
It sounds more like you're turning a vibes based theory into a tautology.
Hofstadter struggling with math for the first time in graduate school isn't a unique story, nor is his self introspection about this event a good basis for an apparently unfalsifiable theory about human cognition.
We have mountains of evidence that humans differ dramatically in cognitive potential, and more again that often effort / practice can only explain a small amount of the variance in performance in a wide variety of fields. We have basically zero evidence at all that anyone can just learn anything if they try hard enough under the right teacher, and plenty of evidence to the contrary.
Abstraction ceilings are about rates and difficulty of learning, so even if we assumed the (absurd) claim that no one has any fundamental cognitive limits, until we are immortal, being slow enough still creates an effective ceiling.
Intelligence denialism is the incoherent and indefensible position here.
That stuff always gives me such a eugenics 2.0 vibe — no, no, the hierarchy is based on innate cognitive ability now. Gives me the creeps that they're actually in academia pushing that stuff.
Whether intentional or not, it's there. I have yet to see a textbook that really taught learners as they deserve to be taught. Some are good in some aspects, but a perfect textbook does not exist.
"Here are some theorems and some proofs and here are some ideas left as exercise to the student". smh.
The fear is not that they will just slow down progress for all. It is that regulation will specifically burden competition. If you kill open-source training, ban Chinese models, crack down on self-hosting, grandfather OpenAI/Anthropic/Google into regulatory compliance while throwing the book at startups, etc. you wind up in the worst of all possible worlds.
But the proposal the article is reacting to is for literally none of that! It is quite literally the opposite, with its proposed measures applying only to frontier labs rather than grandfathering them. It doesn't say anything about open source training, self-hosting, open weights, or startups. It does not suggest a ban on Chinese models (just better enforcement of chip export controls).
That's a valid concern, but some of that is outright impossible. Banning chinese models and killing open source training is not happening without massive unified international cooperation, and that sort of level of action would require the international counties decide to allow the US aligned companies to just, win. Which would be pretty against their own interests.
Also, nobody ever bothers to argue why a specific proposed regulation is "regulatory capture" or would burden startups more than big companies or anything. It's just supposed to be obvious that corporations love regulation and it's bad for the public, all of post-WWII political history notwithstanding.
It's not all-or-nothing. Banning Chinese models in the public sector and strong-arming the private sector against using them would already do great damage. Similarly, open-source training could be stymied by any of hardware embargoes, taxation, or regulation of larger players.
Sure, it's possible for regulations to make things worse, i will agree. But without regulations it's pretty clear things are going to end up VERY bad, and the only knob we have to make it not bad is regulations. So it's important we try something, and work towards doing a good kind of regulation, or any one of the bad futures you imagine is pretty likely to come to pass.
Good regulations would be very nice! Mandatory transparency into training and dataset usage would be a benefit for all, for example. Some sort of regulation or incentives against the most corrosive enshittification (AI call centers, AI therapists, undisclosed AI entertainment mills, etc.) would also be an overwhelmingly popular proposition. And enforcement of the CFAA on operators who let malicious agents loose onto the open internet, or otherwise consume too many resources or violate robots.txt, might at least help the internet stay alive a little longer.
It’s quite possible for imperfect regulations to make a problem worse. See, for instance, sanctions intended to weaken China that ended up creating powerful Chinese competitors and reducing Western influence in their internal markets, bringing them closer to technological autonomy.
The sibling comment offers some good regulations that may actually reduce harms, but the kind of regulations offered there are not the ones that the “safety” people want, because they hurt profits.
AI has autonomously found (many) proofs of False in Lean and Rocq, so it's not merely a theoretical concern. A misaligned AI agent tasked with proving the near-impossible just might wind up smuggling in a bug deep in a lemma somewhere (anyone remember the days back when AI routinely made tests pass by "fixing" the tests?). That said, I doubt OpenAI would be so foolish as to not do a cursory vetting of the proof for malicious compliance, so the actual odds are probably pretty low.
> I doubt OpenAI would be so foolish as to not do a cursory vetting
Significant evidence exists that they have in the past been at least, if not more, foolish as to not perform even minimal not-approaching the boundary of cursory vetting of several significant and well known failure modes with far greater risk of reputational damage than getting an esoteric math solution falsely claimed as successful.
So that doubt appears baseless in light of known operating conditions at OpenAI, and the estimate of the actual odds is probably an order of magnitude away from reality.
> That said, I doubt OpenAI would be so foolish as to not do a cursory vetting of the proof for malicious compliance, so the actual odds are probably pretty low.
You mean the company that “accidentally” let their model perform a cyber-attack?
First of all, that is Fermat's Last Theorem, not Navier-Stokes.
Second of all, you did not read the link.
> In particular, we use honest when the goal is to create a valid proof. This allows for mistakes and bugs in proofs and meta-code (tactics, attributes, commands, etc.), but not for code that clearly only serves to circumvent the system (such as using the debug.skipKernelTC).
Given that AI has autonomously found proofs of `False` in Lean and other proof assistants, it is far from impossible that such a circumvention could be present somewhere in 13 million lines.
If we read the link, it has a section called Gold Standard: comparator and external checkers, and comparator is how OpenAI has gone about checking their lean proofs.
Perhaps you did not understand the Fermat theorem proof announcement/repo or the link. The 13 million lines did not use any external, possibly not honest libraries, as the proof eventually only used the fundamental axioms. So for the Fermat theorem formalization, no open open questions remain.
The point is, they "proved" the Collatz conjecture. You would not know they exploited a bug unless you actually went and dug into their proof. Can we be so certain this has not happened within the millions of lines of Navier-Stokes? In an ideal world, our proof assistants would be more battle-hardened by now (recent exploits deny this), our AI better aligned (their tendency to cheat at tests denies this), or their handlers more responsible (the Hugging Face incident denies this), but the reality is more complicated.
At this point in time, we really can't be confident in accepting proof certificates without any human eyes on the script that generated it. I still have 95%+ confidence in this particular result being trustworthy, but a precedent of blind faith is guaranteed to end badly.
This person knew they did not prove the Collatz conjecture and others independently figured it out within hours. Not sure this is at all relevant, other than pointing out how trivial it is for the community to understand errors in lean4.
It was trivial because the Collatz proof script is literally 1000x smaller than the script for Navier-Stokes and involves no advanced math. And they found the bug by... manually inspecting the proof script. Maybe we should do the same for Navier-Stokes before declaring the matter settled?
Not only that, but there is a very fuzzable tell of something funny in the Collatz proof script (`CommandElabM`, i.e. metaprogramming). We may not at all be so lucky in other malicious scripts, especially if there are still kernel-level bugs in Lean.
By formalizing, they mean within a proof assistant like Lean or Rocq, not simply in prose in a textbook. I can attest, 40 hours per page is by no means an overestimate for this sort of work.
The point was that a textbook (where the 40hr/page estimate comes from) is cumulative/linear -- what you need for page n was defined / established on the preceding pages. But in a proof such as this you can call on any other published result (and those can do the same) so the dependency graph is (potentially) much bushier. Thus later pages of the proof should take far more than 40 hours to manually formalize.
The scale factor comes from this number in the article, seemingly an intuited estimate:
> Say a research article takes 20 times more effort to formalize than page in an undergraduate textbook.
That would suggest formalizing a 10-page research article might take 200 weeks (assuming 40h/wk) of effort, or about four years. Not a mathematician, I have no idea if that's in the ballpark.
It warms my heart every time I see an interactive proof assistant being used to improve rather than simply slow down mathematical thinking.
After years of using the things, I believe not enough focus is given to high-velocity uses of proof assistants for prototyping. They can altogether replace scratch paper for fumbling around with new concepts.
I agree! Martín Escardó never tires to say that he uses the Agda proof assistant in exactly this sense, as a kind of interactive blackboard for taking notes and structuring his thoughts. The vast TypeTopology repository is the result of years of following this philosophy: https://github.com/martinescardo/TypeTopology
Origami design will be my personal test bed for the coming years.
It's objectively very difficult and technical, it's spatiovisual, it's artistic, learning resources for it are sparse and most just learn by the FAFO method, current AI sucks terribly at it, and it's not likely to ever be specifically targeted by benchmaxxers.
reply