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.
A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.
Geometry is not true, it is advantageous.
The mind uses its faculty for creativity only when experience forces it to do so.
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?
To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.
It has adopted the geometry most advantageous to the species or, in other words, the most convenient.
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.
Just as houses are made of stones, so is science made of facts.
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.
Science is facts.
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.
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.
No more than these machines need the mathematician know what he does.
Need we add that mathematicians themselves are not infallible?