How is an error possible in mathematics?
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.
Mathematicians are born, not made.
If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.
A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.
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 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.
Point set topology is a disease from which the human race will soon recover.
It is far better to foresee even without certainty than not to foresee at all.
Facts do not speak.
Just as houses are made of stones, so is science made of facts.
Hypotheses are what we lack the least.
It is through science that we prove, but through intuition that we discover.
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.
Geometry is not true, it is advantageous.
Mathematicians do not study objects, but relations between objects.
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 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.
Mathematics is the art of giving the same name to different things.
Need we add that mathematicians themselves are not infallible?
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 to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living.
In the old days when people invented a new function they had something useful in mind.
Science is facts.
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?
A small error in the former will produce an enormous error in the latter.
Thought is only a flash between two long nights, but this flash is everything.
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.
Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.
The mind uses its faculty for creativity only when experience forces it to do so.
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.
No more than these machines need the mathematician know what he does.
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.
To invent is to discern, to choose.
Thus, they are free to replace some objects by others so long as the relations remain unchanged.
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.
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.
Mathematical discoveries, small or great are never born of spontaneous generation.
