Navigating Difficult Math: Intuition vs Proofs

[JanClaesen]

So how should I approach more difficult math?
On a lower math level it is possible to really understand a certain property, relation, formula, ... you can see where it's coming from: intuitive and/or by quickly derivating it in your head.

But as the math becomes more difficult, this approach becomes of course impossible, the proofs become too hard to 'quickly derivate them in your head', so do I just have to accept certain properties, trusting on a proof I made in the past, without 'really' understanding what I'm doing?
And what about certain intuitive concepts rigorously defined, like limits. Sometimes the 'symbols and calculations' make sense in a certain proof, but I don't really have the feeling I really understand what I just did, it doesn't seem reducible to fundamental logic/axioma's, there's no intuitive way to understand it.

I hope this post makes a bit sense.

 

[Tac-Tics]

When you approach an unfamiliar problem, you start taking tools out of your tool box until you've solved it.

If you're doing a geometric problem, try using your intuition to find obvious solutions. If you're working with something very algebraic, start throwing down equations. If you are working with objects with simple properties (sets, algebras, topologies), focus on their definitions and the basic theorems behind them.

There's no one way to do anything. But every technique you learn has some applicability to new problems, so look at each new problem from as many angles as you can.