Henri Poincaré Quotes

if the contrary takes place, if these two muscular sensations vary independently of one another, we shall have to take account of one more independent variable, and 'complete visual space' will appear to us as a physical continuum of four dimensions.

Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.

Formerly, when a new function was invented, it was in view of some practical end. To-day they are invented on purpose to show our ancestors' reasonings at fault, and we shall never get anything more than that out of them.

If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages.

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

He has begun by supposing that light has a constant velocity... the same in all directions. This... could never be verified directly by experiment... The postulate... resembling the ... furnishes us with a new rule for the investigation of simultaneity.

Tout le monde y croit cependant, me disait un jour M. Lippmann, car les expérimentateurs s'imaginent que c'est un théorème de mathématiques, et les mathématiciens que c'est un fait expérimental.

The essential characteristic of reasoning by recurrence [induction] is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.

L’état actuel et l’avenir de la physique mathematique (1904)

But these labels can only be finite in number. On that score, psychologic time should be discontinuous.

If we study the history of science we see happen two inverse phenomena... Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities.
...No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations.

only reach him after having traversed reflecting media of complicated form. The two indications which serve us in judging distances would cease to be connected by a constant relation. A being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual space.

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.]

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 but principal de l'enseignement mathématique est de développer certaines facultés de l'esprit et parmi elles l'intuition n'est pas la moins précieuse. C'est par elle que le monde mathématique reste en contact avec le monde réel et quand les mathématiques pures pourraient s'en passer, il faudrait toujours y avoir recours pour combler l'abîme qui sépare le symbole de la réalité.

Naukę buduje się z faktów tak jak dom buduje się z cegieł, ale samo nagromadzenie faktów nie jest jeszcze nauką, podobnie jak kupa cegieł nie jest domem.

The ones who are preoccupied by logic are above all; 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 others are guided by intuition and, at the first stroke, make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.