No more than these machines need the mathematician know what he does.

Science is facts.

The mind uses its faculty for creativity only when experience forces it to do so.

Thus, they are free to replace some objects by others so long as the relations remain unchanged.

To invent is to discern, to choose.

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment.

A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.

Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence.

If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing.

If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws.

It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.

One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.

The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law.

It is through science that we prove, but through intuition that we discover.

A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.

Point set topology is a disease from which the human race will soon recover.

Science is built up of facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

It is far better to foresee even without certainty than not to foresee at all.