Thread

If math is more than proof, we need to better celebrate the rest of it

433 points · 285 comments · num42

ForgotMyUUID · 1789802300 · focus · HN ↗
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
1. Geof25 · 1789804528 · focus · HN ↗
  People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
  It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
  1. Paracompact · 1789808204 · focus · HN ↗
    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.
    D-Machine · 1789811001 · focus · HN ↗
    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.
    Paracompact · 1789812927 · focus · HN ↗
    > 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 · 1789817165 · focus · HN ↗
    > 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.
    catgary · 1789842655 · focus · HN ↗
    > 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.
    D-Machine · 1789843854 · focus · HN ↗
    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.
    magicalist · 1789844525 · focus · HN ↗
    > 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.
    D-Machine · 1789846963 · focus · HN ↗
    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.

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