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 p… - Henri Poincaré

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

Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet, and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?

The syllogism cannot teach us anything essentially new. It must be conceded that mathematical reasoning has of itself a sort of creative virtue, and consequently differs from the syllogism.

Cette harmonie que l’intelligence humaine croit découvrir dans la nature, existe-t-elle en dehors de cette intelligence ? Non, sans doute, une réalité complètement indépendante de l’esprit qui la conçoit, la voit ou la sent, c’est une impossibilité. Un monde si extérieur que cela, si même il existait, nous serait à jamais inaccessible. Un monde si extérieur que cela, si même il existait, nous serait à jamais inaccessible.