Higher level mathematics is extremely proof based!! Everything that one does in mathematics relies on rigorous proofs. Proofs are sometimes difficult for people, but proving can be quite fun
Higher mathematics basically has the following steps in its development:
- Finding the right tools to describe something
- Proving that the tools work under certain circumstances.
- Extending the tools so that they work under more general circumstances
- Proving that they work there also.
This is a very broad description, but let me give you an example of what pure mathematics does:
Problem: finding the area under a curve.
Solution: the integral
So, in the first step, we prove that the integral actually does the right job and behaves exactly how we want to.
However, we cannot integrate every function! So, one tries to extend the notion of an integral and this gives us the Lebesgue integral. We prove that the Lebesgue integral corresponds to the original integral in the usual cases and we prove that the Lebesgue integral works how we expect it to work.
All that pure math is about it to rigorize intuition and to abstract the familiar.