Well said. Advanced math is mostly about working with properties higher up the chain of abstraction, and then seeing what happens when you bring the insights learned up there back down to more concrete examples.

From an OO point of view, the real numbers inherit almost every useful trait: they're a field, they have a topology, they have a measure. Studying the parent classes, so to speak, gives you abstract algebra, topology, and analysis, respectively.

Once you get the basics of each, you can study how they interact. Then, once that stuff is clear, they can be recombined in beautiful ways to give you new objects to study.