The aim of mathematics is to explain as much as possible in simple terms.

I dislike frontiers, political or intellectual, and I find that ignoring them is an essential catalyst for creative thought. Ideas should flow without hindrance in their natural course.

Would you rather be deaf or blind?

There is no clear-cut distinction between example and theory

Any good theorem should have several proofs, the more the better.

Everything useful in mathematics has been devised for a purpose. Even if you don't know it, the guy who did it first, he knew what he was doing. Banach didn't just develop Banach spaces for the sake of it. He wanted to put many spaces under one heading. Without knowing the examples, the whole thing is pointless.

If theory is the role of the architect, then such beautiful proofs are the role of the craftsman. Of course, as with the great renaissance artists, such roles are not mutually exclusive. A great cathedral has both structural impressiveness and delicate detail. A great mathematical theory should similarly be beautiful on both large and small scales.