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.

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.

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.

Mathematics is an art of human understanding.

Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It's just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.

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.