Comment by D-Machine
1 day ago
> 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.
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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.
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> 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.