The Renaissance was the time when Western Europeans lost their awe of the Ancients and realized that they had as much to contribute to civilization and society as the Greeks and Romans had contributed. To modern eyes, the puzzle is not that this should have occurred, but that it should have taken so long for people to lose their inferiority complex.
–J. Gribbin

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.
—R. Feynman

The fastest computation is one you don't do.

Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.
–D. Hilbert

An idea which can be used once is a trick. If it can be used more than once it becomes a method.
–G. Pólya

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.
–I.N. Herstein

Nothing can stop an idea whose time has come.
–V. Hugo

... before going farther I need to mention a few things in my life that have shaped my opinions. The first occurred at Los Alamos during WWII when we were designing atomic bombs. Shortly before the first field test (you realize no small scale experiment can be done―either you have critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen―after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, "What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?" I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, "Never mind, Hamming, no one will ever blame you." Yes, we risked all the life we knew of in the known universe on some mathematics. Mathematics is not merely an idle art form, it is an essential part of our society.
—R. Hamming

[On Alonzo Church] The person lecturing us was logic incarnate.
–G.C. Rota