The Nonlinear Library LW - A simple model of math skill by Alex Altair
Jul 21, 2024
13:38
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: A simple model of math skill, published by Alex Altair on July 21, 2024 on LessWrong.
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 the generating skills go in the opposite order.
This feels like an elegant mnemonic to me, although of course it's a very simplified model. Treat every statement below as a description of the model, and not a claim about the totality of doing mathematics.
Understanding
Understanding these more or less has to go in the above order, because proofs are of theorems, and theorems are about defined objects. Let's look at each level.
Definitions
You might think that definitions are relatively easy to understand. That's usually true in natural languages; you often already have the concept, and you just don't happen to know that there's already a word for that.
Math definitions are sometimes immediately understandable. Everyone knows what a natural number is, and even the concept of a prime number isn't very hard to understand. I get the impression that in number theory, the proofs are often the hard part, where you have to come up with some very clever techniques to prove theorems that high schoolers can understand (Fermat's last theorem, the Collatz conjecture, the twin primes conjecture).
In contrast, in category theory, the definitions are often hard to understand. (Not because they're complicated per se, but because they're abstract.) Once you understand the definitions, then understanding proofs and theorems can be relatively immediate in category theory.
Sometimes the definitions have an immediate intuitive understanding, and the hard part is understanding exactly how the formal definition is a formalization of your intuition. In a calculus class, you'll spend quite a long time understanding the derivative and integral, even though they're just the slope of the tangent and the area under the curve, respectively.
You also might think that definitions were mostly in textbooks, laid down by Euclid or Euler or something. At least in the fields that I'm reading papers from, it seems like most papers have definitions (usually multiple). This is probably especially true for papers that are trying to help form a paradigm. In those cases, the essential purpose of the paper is to propose the definitions as the new paradigm, and the theorems are set forth as arguments that those definitions are useful.
Theorems
Theorems are in some sense the meat of mathematics. They tell you what you can do with the objects you've formalized. If you can't do anything meaty with an object, then you're probably holding the wrong object.
Once you understand the objects of discussion, you have to understand what the theorem statement is even saying. I think this tends to be more immediate, especially because often, all the content has been pushed into the definitions, and the theorem will be a simpler linking statement, like "all As are Bs" or "All As can be decomposed into a B and a C".
For example, the fundamental theorem of calculus...
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 the generating skills go in the opposite order.
This feels like an elegant mnemonic to me, although of course it's a very simplified model. Treat every statement below as a description of the model, and not a claim about the totality of doing mathematics.
Understanding
Understanding these more or less has to go in the above order, because proofs are of theorems, and theorems are about defined objects. Let's look at each level.
Definitions
You might think that definitions are relatively easy to understand. That's usually true in natural languages; you often already have the concept, and you just don't happen to know that there's already a word for that.
Math definitions are sometimes immediately understandable. Everyone knows what a natural number is, and even the concept of a prime number isn't very hard to understand. I get the impression that in number theory, the proofs are often the hard part, where you have to come up with some very clever techniques to prove theorems that high schoolers can understand (Fermat's last theorem, the Collatz conjecture, the twin primes conjecture).
In contrast, in category theory, the definitions are often hard to understand. (Not because they're complicated per se, but because they're abstract.) Once you understand the definitions, then understanding proofs and theorems can be relatively immediate in category theory.
Sometimes the definitions have an immediate intuitive understanding, and the hard part is understanding exactly how the formal definition is a formalization of your intuition. In a calculus class, you'll spend quite a long time understanding the derivative and integral, even though they're just the slope of the tangent and the area under the curve, respectively.
You also might think that definitions were mostly in textbooks, laid down by Euclid or Euler or something. At least in the fields that I'm reading papers from, it seems like most papers have definitions (usually multiple). This is probably especially true for papers that are trying to help form a paradigm. In those cases, the essential purpose of the paper is to propose the definitions as the new paradigm, and the theorems are set forth as arguments that those definitions are useful.
Theorems
Theorems are in some sense the meat of mathematics. They tell you what you can do with the objects you've formalized. If you can't do anything meaty with an object, then you're probably holding the wrong object.
Once you understand the objects of discussion, you have to understand what the theorem statement is even saying. I think this tends to be more immediate, especially because often, all the content has been pushed into the definitions, and the theorem will be a simpler linking statement, like "all As are Bs" or "All As can be decomposed into a B and a C".
For example, the fundamental theorem of calculus...