The art of doing mathematics is finding that special case that contains all the germs of generality.
—David Hilbert

A proof by contradiction is dangerous because if you make a mistake, it helps you. Because you make a mistake and you go on, and then something wrong appears and you say ‘I have it proven’.

I didn't make a decision to pursue astronomy. Rather, it just grabbed me, and I had no thought of escaping.
—Carl Sagan

A problem is grand in science if it lies before us unsolved and we see some way for us to make some headway into it.
—Richard Feynman

Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can.
—John Von Neumann's housekeeper, describing her employer

A trick used twice is a method.

The fastest computation is the one you don't do.

If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.

It would be an unfair exaggeration to say that mathematicians rig their questions so that they can answer them, but there is undoubtedly a grain of truth in the statement. The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable.
—Michael Atiyah