Abstract math and making proofs

Question

What is abstract math about? I think we can not visualise probably what we read. Or can we? I am talking about the theorems, definitions and proofs in areas of math like Riemannian geometry, differentiable manifolds, group theory, functional analysis with vector spaces, spaces with inner product,.

So, what remains to do? Think of the logic behind math, remember what you read and when given the chance to prove an exercise or problem, use logic? But what can give me the directions and instructions to prove what i want to prove? Thank you.

Answer 1

When learning advanced (or simple, too) mathematics what I see is a process of sense-making. This is a combination of visualization, thinking in terms of processes, building intuition by calculating a lot, knowing key examples, etc.

When learning one develops these thinking skills that allow them, personally, to process mathematics. Some of them are learned from others, others built by oneself.

When asked to prove something, there are several strategies. There is a famous book of Pólya György (in English: George Pólya) about this, for example. One key step I find myself taking is using the representations I have at hand to figure out if the claim is true or what happens if I try to contradict it, and by this mean understand the problem better.