> 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.
It’s like someone had Claude binge on Galton, Murray and Yarvin then let it loose on Hacker News.
Well you're just a generally rude person, aren't you.