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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 intell

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.

x______________ 7d ago
I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.
Paracompact 7d ago
> 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.
- D-Machine 7d 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.
  - magicalist 7d ago
    > 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.
    - D-Machine 7d ago
      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.

x______________ 7d ago

I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.

Paracompact 7d ago

> 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.

D-Machine 7d 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.
- magicalist 7d ago
  > 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.
  - D-Machine 7d ago
    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.