I was always delighted by the way in which Thom discussed mathematics, using sentences obviously having no strict logical meaning at all. While I was never able to completely free myself from the straitjacket of logic, I was forever poisoned by the dream of the irresponsible mathematical speculation with no exact meaning. “One can always find imbeciles to prove theorems” was, according to Thom's students, his principle.
Vladimir Arnol'd
I don't see how anybody understands what is happening in physics today. Even I don't understand much which I would like to learn from physics books. But with me, if I don't understand something, then I go to the telephone and call up Debye or Born, and they come and explain it to me. And then I understand it -- but what do other people do?
David Hilbert
Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of, and couldn't exist without, the many months of stumbling around in the dark that precede them.
Andrew Wiles