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.

What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a mere association of ideas. It is to this association that what we call 'the sense of direction' is reducible. This feeling therefore can not be found in a single sensation

John Stuart Mill understood the word existence in a material and empirical sense; he meant that in defining a circle we assert that there are round things in nature.
In this form his opinion is inadmissible. Mathematics is independent of the existence of material objects. In mathematics the word exist can only have one meaning; it signifies exemption from contradiction. Thus rectified, Mill's thought becomes accurate. In defining an object, we assert that the definition involves no contradiction.

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.

only the relations between things. Outside of these relations there is no knowable reality.

Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only...

Logic remains barren, unless it is fertilized by intuition.

The mind has the faculty of creating 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

Il n’y pas d’infini actuel; les Cantoriens l’ont oublié, et ils sont tombés dans la contradiction. Il est vrai que le Cantorisme a rendu des services, mais c’était quand on l’appliquait à un vrai problème, dont les termes étaient nettement définis, et alors on pouvait marcher sans crainte.

... 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.

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.

. . . by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.

in every limited frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders.

The axioms of geometry, therefore, are neither synthetic a priori judgments nor experimental facts.
They are conventions; our choice among all possible conventions is guided by experimental facts; but it remains free and is limited only by the necessity of avoiding all contradiction. . . .
In other words, the axioms of geometry (I do not speak of those of arithmetic) are merely disguised definitions.
Then what are we to think of that question: Is the Euclidean geometry true?
It has no meaning.
As well ask whether the metric system is true and the old measures false.

When a body changes its place and its shape, we can no longer, by appropriate movements, bring back our sense-organs into the same relative situation with regard to this body; consequently we can no longer reestablish the primitive totality of impressions.