Henri Poincaré Quotes

If, then, a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others, that will render an account equally well of all the particulars revealed by experiment.

What we call geometry is nothing but the study of formal properties of a certain continuous group [...]. The notion of this continuous group exists in our mind prior to all experience; but the assertion is no less true of the notion of many other continuous groups; for example, that which corresponds to the geometry of Lobachevsky. There are, accordingly, several geometries possible, and it remains to be seen how a choice is made between them. Among the continuous mathematical groups which our mind can construct, we choose that which deviates the least from that rough group, analogous to the physical continuum, which experience has brought to our knowledge as the group of displacements. Our choice is therefore not imposed by experience. It is simply guided by experience. But it remains free; we choose this geometry, not because it is more true, but because it is the more convenient.

Sociology is the science which has the most methods and the least results.

When navigators... determine a ... they must... calculate Paris time...[with] a chronometer set for Paris. The qualitative problem of simultaneity is made to depend upon the quantitative problem of the measurement of time.

We have then many kinds of intuition; first, the appeal to the senses and the imagination; next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning.

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.

Scientists believe there is a hierarchy of facts and that among them may be made a judicious choice. They are right, since otherwise there would be no science...

Toute définition implique un axiome, puisqu'elle affirme l'existence de l'objet défini. La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu'elle n'entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises.

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so lone as the relations remain unchanged. Content to them is irrelevant; they are interested in form only.

Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes.

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.

[Y]et when we speak of time... do we not unconsciously adopt this hypothesis... [and] put ourselves in the place of this imperfect god... [D]o not even the atheists put themselves in the place where god would be..?

In other words, the axioms of geometry (I do not speak of those of arithmetic) are merely disguised definitions.

The notion of infinity had long since been introduced into mathematics, but this infinity was what philosophers call a becoming; Mathematical infinity was only a quantity susceptible of growing beyond all limit; it was a variable quantity of which it could not be said that it had passed, but only that it would pass, all limits.

... between the elements of the continuum [there is supposed to be] a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point.