Problems are often stated in vague terms... because it is quite uncertain what the problems really are.

The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work-that is, correctly to describe phenomena from a reasonably wide area.

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.

Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number - there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.

Technological possibilities are irresistible to man. If man can go to the moon, he will. If he can control the climate, he will.

If you tell me precisely what it is a machine cannot do, then I can always make a machine which will do just that.