3 ms·

I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".

by n4r9 2mo ago

antonvs 2mo ago
It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.
- n4r9 2mo ago
  Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely linked to ideas of beauty.
  - antonvs 2mo ago
    I was referring to the Gowers quote you provided: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. That last part, "new and surprising but that with hindsight come to seem beautiful and natural," is an almost perfect description of how processing fluency (https://en.wikipedia.org/wiki/Processing_fluency https://en.wikipedia.org/wiki/Processing_fluency) or the mere-exposure effect (https://en.wikipedia.org/wiki/Mere-exposure_effect https://en.wikipedia.org/wiki/Mere-exposure_effect) works. There's plenty of other work along these lines. Gian-Carlo Rota wrote about this more philosophically in "The Phenomenology of Mathematical Beauty" (1997). He argued (among other things) that the same proof can go from opaque to beautiful purely through familiarity. My understanding is that no single person has "learned" finite simple group classification from top to bottom, so it's easy to see why it wouldn't fit this description - there's no-one in a position to say "with hindsight, it seems simple and natural." That makes it a poor example for your/Gowers' position.
    - n4r9 2mo ago
      Gowers claims that some methods appear beautiful and natural in hindsight, whilst others do not. That suggests beauty is more than a simple matter of familiarity.

I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".

by n4r9 2mo ago

I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".

antonvs 2mo ago

It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.

n4r9 2mo ago
Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely linked to ideas of beauty.
- antonvs 2mo ago
  I was referring to the Gowers quote you provided: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. That last part, "new and surprising but that with hindsight come to seem beautiful and natural," is an almost perfect description of how processing fluency (https://en.wikipedia.org/wiki/Processing_fluency https://en.wikipedia.org/wiki/Processing_fluency) or the mere-exposure effect (https://en.wikipedia.org/wiki/Mere-exposure_effect https://en.wikipedia.org/wiki/Mere-exposure_effect) works. There's plenty of other work along these lines. Gian-Carlo Rota wrote about this more philosophically in "The Phenomenology of Mathematical Beauty" (1997). He argued (among other things) that the same proof can go from opaque to beautiful purely through familiarity. My understanding is that no single person has "learned" finite simple group classification from top to bottom, so it's easy to see why it wouldn't fit this description - there's no-one in a position to say "with hindsight, it seems simple and natural." That makes it a poor example for your/Gowers' position.
  - n4r9 2mo ago
    Gowers claims that some methods appear beautiful and natural in hindsight, whilst others do not. That suggests beauty is more than a simple matter of familiarity.