Mathematicians do not study objects, but relations between objects.

Mathematicians are born, not made.

A small error in the former will produce an enormous error in the latter.

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

Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.

Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.

How is an error possible in mathematics?

Thought is only a flash between two long nights, but this flash is everything.

In the old days when people invented a new function they had something useful in mind.

Mathematics is the art of giving the same name to different things.

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

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.

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.

To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.

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.

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.

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.

Science is facts.

Mathematical discoveries, small or great are never born of spontaneous generation.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.

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

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.

Need we add that mathematicians themselves are not infallible?

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.

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.

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

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.

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.

Hypotheses are what we lack the least.

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

It has adopted the geometry most advantageous to the species or, in other words, the most convenient.

Facts do not speak.

Just as houses are made of stones, so is science made of facts.

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.

Geometry is not true, it is advantageous.

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

Mathematicians do not study objects, but relations between objects.

To invent is to discern, to choose.

Exactness cannot be established in the arguments unless it is first introduced into the definitions.

Astronomy is useful because it raises us above ourselves; it is useful because it is grand. It shows us how small is man's body, how great his mind, since his intelligence can embrace the whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony.

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.

Geometry is the art of correct reasoning from incorrectly drawn figures.

Sociology is the science which has the most methods and the least results.

As I say, there was this movement to try to bring philosophers and mathematicians together into an organization where they would talk to each other. An organization wasn't effective unless you had a journal. That's about all I know.

Mathematicians are born, not made.

As far as I know, only a small minority of mathematicians, even of those with Platonist views, accept the idea that there may be mathematical facts which are true but unknowable.

Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof.