Mathematics is the art of giving the same name to different things.
Thought is only a flash between two long nights, but this flash is everything.
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.
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?
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.
Mathematicians are born, not made.
Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.
If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Geometry is not true, it is advantageous.
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.
Mathematicians do not study objects, but relations between objects.
Mathematical discoveries, small or great are never born of spontaneous generation.
To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.
A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance.
It has adopted the geometry most advantageous to the species or, in other words, the most convenient.
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.
Facts do not speak.
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.
Impossible Problems Deep Looks Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.
Need we add that mathematicians themselves are not infallible?
Hypotheses are what we lack the least.
Point set topology is a disease from which the human race will soon recover.
To invent is to discern, to choose.
The mind uses its faculty for creativity only when experience forces it to do so.
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.
Just as houses are made of stones, so is science made of facts.
Stones Science is facts.
In the old days when people invented a new function they had something useful in mind.