En un mot, pour tirer la loi de l'expérience, if faut généraliser; c'est une nécessité qui s'impose à l'observateur le plus circonspect. In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.
Le savant doit ordonner ; on fait la science avec des faits comme une maison avec des pierres ; mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison. The Scientist must set in order. Science is built up with 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.
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.
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.
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.
Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.
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.
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.
To invent is to discern, to choose.
It is through science that we prove, but through intuition that we discover.
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.
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 is far better to foresee even without certainty than not to foresee at all.
Geometry is not true, it is advantageous.
It has adopted the geometry most advantageous to the species or, in other words, the most convenient.
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.
Hypotheses are what we lack the least.
Need we add that mathematicians themselves are not infallible?