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

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.

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

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.

We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.

Mathematics is an art of human understanding.

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