Source: AMS, Nov 2009
Gelfand: You mentioned computers. What has changed in mathematics since their appearance?
Manin: What has changed in pure mathematics? The unique possibility of doing large-scale physical experiments in mental reality arose. We can try the most improbable things. More exactly, not the most improbable things, but things that Euler could do even without a computer. Gauss could also do them. But now, what Euler and Gauss could do, any mathematician can do, sitting at his desk.
The story of the development of the general theory of relativity is a striking example. Not only did Einstein not know the mathematics he needed, but he didn’t even know that such mathematics existed when he started understanding the general theory of relativity in 1907 in his own brilliantly intuitive language. After several years dedicated to the study of quanta, he returned to gravitation and in 1912 wrote to his friend Marcel Grossmann: “You’ve got to help me, or I will go out of my mind!”
A program arises when a great mathematical mind sees something as a whole, or not as a whole, but as something more than a single detail. But it is seen at first only vaguely.