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 ...

Isham Jameson

Isham Jameson

Isham Jameson

Isham Jameson