I've noticed that when trying to understand a math paper, there are a few different ways my skill level can be the blocker. Some of these ways line up with some typical levels of organization in math papers:

Definitions: a formalization of the kind of objects we're even talking about.
Theorems: propositions on what properties are true of these objects.
Proofs: demonstrations that the theorems are true of the objects, using known and accepted previous theorems and methods of inference.

Understanding a piece of math will require understanding each of these things in order. It can be very useful to identify which of type of thing I'm stuck on, because the different types can require totally different strategies.

Beyond reading papers, I'm also trying to produce new and useful mathematics. Each of these three levels has another associated skill of generating them. But it seems to me that [...]

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Outline:

(01:29) Understanding

(01:42) Definitions

(03:28) Theorems

(04:55) Proofs

(07:29) Generating

(07:52) Proofs

(08:58) Theorems

(10:08) Definitions

(12:50) Skill improvement

(14:24) Paradigm formation starts at the end

The original text contained 2 footnotes which were omitted from this narration.

The original text contained 1 image which was described by AI.

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First published:
July 21st, 2024

Source:
https://www.lesswrong.com/posts/EF8tvShQJ5cbdZzTb/a-simple-model-of-math-skill

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Narrated by TYPE III AUDIO.

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“A simple model of math skill” by Alex_Altair

LessWrong (30+ Karma)