Henri Poincaré Quotes

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics.

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.

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

"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.

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

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable.

The aim of Mathematical Physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations. It is besides, it is above all, to reveal to him the hidden harmony of things in making him see them in a new way.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

Logic sometimes creates monsters.

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

Le cerveau du savant, qui n'est qu'un coin de l'Univers, ne pourra jamais contenir l'univers tout entier

It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.
[1913, p210]

C'est avec l'intuition qu'on trouve, c'est avec la logique qu'on prouve

Les hommes ne s’entendent pas, parce qu’ils ne parlent pas la même langue et qu’il y a des langues qui ne s’apprennent pas.
[Men do not get along because they do not speak the same language and there are languages ​​that cannot be learned.]

Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.