Mathematicians do not study objects, but relations between objects.
Mathematics is the art of giving the same name to different things.
Geometry is not true, it is advantageous.
Sociology is the science with the greatest number of methods and the least results.
The subliminal self is in no way inferior to the conscious self. It knows how to choose and to divine.
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.
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.
Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose.
All that is not thought is pure nothingness; since we can think only thoughts, and all the words we use to speak of things can express only thoughts, to say there is something other than thought is therefore an affirmation which can have no meaning.
What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.
Most striking at first is the appearance of sudden illumination, a manifest sign of long unconscious prior work.
All of mathematics is a tale about groups.
But all of my efforts served only to make me better acquainted with the difficulty, which in itself was something.
Just as houses are made of stones, so is science made of facts.
It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
Every good mathematician should also be a good chess player and vice versa.
A cat is witty, he has nerve, he knows how to do precisely the right thing at the right moment.
All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.
Experiment is the sole source of truth.