Henri Poincaré - Intuition Quotes

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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
It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.
In French: C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente.

Henri Poincaré
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
Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.

Henri Poincaré
The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.

Henri Poincaré
It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent.

Henri Poincaré
Logic and intuition have each their necessary role. Each is, indispensable. Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.

Henri Poincaré
This intuition of mathematical order, which enables us to perceive harmonies and hidden relations, cannot be present in everyone.

Henri Poincaré