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