A mathematician is a machine for turning coffee into theorems.
Paul Erdős

We should make things as simple as possible, but not any simpler.
Albert Einstein

Computers are useless. They can only give you answers.
Pablo Picasso

My impression is that I have never obtained anything at low cost. All my results have been preceded by preparatory ones – setting up work, a very long experimentation – hoping that at the end of this experimentation, an incredibly simple idea occurs which comes and solves the problem. And then you need to go through the checking period, almost intolerable because of the fear you have of being mistaken. I will never let anyone believe that you can wait just like this until results come all by themselves.
Alain Connes, EMS Newsletter, March 2007.

Proof is the end product of a long interaction between creative imagination and critical reasoning. Without proof the program remains incomplete, but without the imaginative input it never gets started.
Sir Michael Atiyah, "Advice to a Young Mathematician", The Princeton Companion to Mathematics.

On a typical day, I get no ideas, but on a good day, I get a wrong idea.
Wrong ideas are ingredients in the pot.
Add enough ingredients and the stew cooks. With luck, it tastes good.

Charles Fefferman in Mathematicians: an outer view of the inner world.

The musical notes are only five in number
but their melodies are so numerous that one cannot hear them all.
The primary colors are only five in number
but their combinations are so infinite that one cannot visualize them all.
The flavors are only five in number
but their blends are so various that one cannot taste them all.
In battle there are only the normal and extraordinary forces,
but their combinations are limitless; none can comprehend them all.

Sun Tzu, The Art of War.