A mathematician is a machine for turning coffee into theorems.
We should make things as simple as possible, but not any simpler.
Computers are useless. They can only give you answers.
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.
, 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