Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.

The real satisfaction from mathematics is in learning from others and sharing with others.

Mathematics is much less formally complete and precise than computer programs.

The product of mathematics is clarity and understanding. Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new.

Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide.

Mathematics is an art of human understanding.

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.

You don't see what you're seeing until you see it but when you do see it, it lets you see many other things.

As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.