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.

Derrière la série de Fourier, d'autres séries analogues sont entrées dans la domaine de l'analyse; elles y sont entrees par la même porte; elles ont été imaginées en vue des applications.After the Fourier series, other series have entered the domain of analysis; they entered by the same door; they have been imagined in view of applications.

Deviner avant de démontrer! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes.Guessing before proving! Need I remind you that it is so that all important discoveries have been made?

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.

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.

La pensée n'est qu'un écliar au milieu d'une longue nuit. Mais c'est cet éclair qui est tout.Thought is only a flash in the middle of a long night. But this flash means everything.

How is it that there are so many minds that are incapable of understanding mathematics? ... the skeleton of our understanding, ... and actually they are the majority. ... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education.

If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.