Henri Poincaré Quotes

Hilbert justifies vicious circles by defining a ’proof’ only by postulates, thus turning it into a new dead mathematical element. But should not an existence proof or the absence of possible contradictions be given for this new symbol? And is this not just moving the difficulty?
(...)
From our point of view Hilbert’s ‘replâtrage’ is superfluous.

The true geometrician makes this selection judiciously, because he is guided by a sure instinct, or by some vague consciousness of I know not what profounder and more hidden geometry, which alone gives a value to the constructed edifice.
To seek the origin of this instinct, and to study the laws of this profound geometry which can be felt but not expressed, would be a noble task for the philosophers who will not allow that logic is all. But this is not the point of view I wish to take, and this is not the way I wish to state the question. This instinct I have been speaking of is necessary to the discoverer, but it seems at first as if we could do without it for the study of the science once created. Well, what I want to find out is, whether it is true that once the principles of logic are admitted we can, I will not say discover, but demonstrate all mathematical truths without making a fresh appeal to intuition.

Que l'on cherche à se représenter la figure formée par ces deux courbes et leurs intersections en nombre infini dont chacune correspond à une solution doublement asymplotique. ces intersections forment une sorte de treillis, de tissu, de réseau à maille infiniment serrées ; chacune de ces deux courbes ne doit jamais se recouper elle-même, mais elle doit se replier sur elle même de manière infiniment complexe pour venir recouper une infinité de fois toutes les mailles du réseau. On sera frappé de la complexité de cette figure, que je ne cherche même pas à tracer. Rien de plus propre à nous donner une idée de la complication du problème des trois corps et en général de tous les problèmes de la Dynamique où il n'y a pas d'intégrale uniforme et où les séries de Bohlin sont divergentes.

Il ne faut pas comparer la marche de la science aux transformations d’une ville, où les édifices vieillis sont impitoyablement jetés à bas pour faire place aux constructions nouvelles, mais à l’évolution continue des types zoologiques qui se développent sans cesse et finissent par devenir méconnaissables aux regards vulgaires, mais où un œil exercé retrouve toujours les traces du travail antérieur des siècles passés. Il ne faut donc pas croire que les théories démodées ont été stériles et vaines.

(1) The rules applied are exceedingly various. (2) It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of the measurement of time; no matter whether a chronometer is used, or whether account must be taken of a velocity of transmission, as... of light, because such a velocity could not be measured without measuring a time.

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.

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.

The mind uses its creative faculty only when experience requires it.

Is not my present nearer my past of yesterday than the present of Sirius?

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.

Si toutes les parties de l’univers sont solidaires dans une certaine mesure, un phénomène quelconque ne sera pas l’effet d’une cause unique, mais la résultante de causes infiniment nombreuses ; il est, dit-on souvent, la conséquence de l’état de l’univers un instant auparavant.

Let us try to represent the figure formed by these two curves and their intersections in infinite number, each corresponding to a doubly asymptotic solution, these intersections form a kind of mesh, of fabric, of infinitely tight network; each of the two curves must never intersect itself, but it must fold back on itself in a very complex way in order to cross an infinite number of times all the meshes of the network. On will be struck by the complexity of this figure, which I do not even try to draw. Nothing is more likely to give us an idea of the complexity of the three-body problem and in general of all the problems of dynamics where there is no uniform integral and where the Bohlin series are divergent.

The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all the theorems with which so many volumes are filled, are only indirect ways of saying that A is A?

In recapitulation, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. Its power is limited only by the necessity of avoiding all contradiction; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto.

[I]t is the sidereal day, that is, the duration of the rotation of the earth, which is the constant unit of time. ...However ...[many] astronomers ...think that the tides act as a check on our globe, and that the rotation of the earth is becoming slower and slower. Thus would be explained the apparent acceleration of the motion of the moon...