Terence Tao

Mathematical progress is often a dialogue between the general and the specific.

There is no permanent place in the world for ugly mathematics.

Use the wastebasket. Not every calculation needs to be preserved.

The best way to learn is to do; the worst way to teach is to talk.

In mathematics, you don't understand things. You just get used to them.

A mathematical theory is not considered complete until you can explain it to the first person you meet on the street.

The difference between a good mathematician and a great one is often the ability to ask the right questions.

Rigour is to the mathematician what morality is to the philosopher.

The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' but 'That's funny...'

If a problem is too hard, try a simpler one.

The map is not the territory, but a good map is incredibly useful for navigating the territory.

An argument that is conceptually clear is often more valuable than a technically correct but opaque one.

The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise.

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs.

A mathematical model should be as simple as possible, but no simpler.

The deepest theorems often have the simplest, most elegant statements.

Mathematics is the art of giving the same name to different things.

Progress in mathematics often comes from finding the right compromise between generality and specificity.

The only way to learn mathematics is to do mathematics.

A good notation can be worth a thousand theorems.