Henri Poincaré: Other

“Now what is science? …it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship. Science, in other words, is a system of relations. …it is in relations alone that objectivity must be sought. …it is relations alone which can be regarded as objective.
External objects… are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation.”

Henri Poincaré book The Value of Science

Source: The Value of Science (1905), Ch. 11: Science and Reality

Nature , Appearance , Science

“Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.”

Henri Poincaré book Science and Hypothesis

Les mathématiciens n'étudient pas des objets, mais des relations entre les objets ; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matière ne leur importe pas, la forme seule les intéresse.
Source: Science and Hypothesis (1901), Ch. II: Dover abridged edition (1952), p. 20

Study , Change , Indifference

“We can not… escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. …Neither can this rule come to us from experience… This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. …it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead

Thinking

“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.”

Henri Poincaré book Science and Hypothesis

Si donc un phénomène comporte une explication mécanique complète, il en comportera une infinité d’autres qui rendront également bien compte de toutes les particularités révélées par l’expérience.
Source: Science and Hypothesis (1901), Ch. XII: Optics and Electricity, as translated by George Bruce Halsted (1913)

“Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?
This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.”

Henri Poincaré book The Value of Science

Source: The Value of Science (1905), Ch. 5: Analysis and Physics

Art , Music , Feeling , Mathematics

“We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them.
…We …choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: "The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions, are only the fruit of an unconscious opportunism."”

Henri Poincaré book The Value of Science

Source: The Value of Science (1905), Ch. 2: The Measure of Time

Law , Nature , Thinking , Success

“What is objective must be common to many minds and consequently transmissible from one to the other, and as this transmission can only come about by… discourse… we are even forced to conclude: no discourse no objectivity.”

Henri Poincaré book The Value of Science

Source: The Value of Science (1905), Ch. 11: Science and Reality

“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… One need only open the eyes to see that the conquests of industry which have enriched so many practical men would never have seen the light, if these practical men alone had existed and if they had not been preceded by unselfish devotees who died poor, who never thought of utility, and yet had a guide far other than caprice.
As Mach says, these devotees have spared their successors the trouble of thinking.”

Henri Poincaré book The Value of Science

Author's Essay Prefatory to the Translation: "The Choice of Facts," p.4
The Value of Science (1905)

Men , Thinking , Light , Science