Henri Poincaré

Logic, therefore, remains barren unless fertilized by intuition.

The true method of foreseeing the future of mathematics is to study its history and its present state.

Invention consists in avoiding the constructing of useless contraptions and in constructing the useful ones which are in the minority.

Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.

Absolute space, that is to say, the mark to which we must refer the earth to know whether it really moves, has no objective existence... The two propositions: 'The earth turns round' and 'It is more convenient to suppose the earth turns round' have the same meaning.

The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law.

It is by logic we prove, it is by intuition we invent.

The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things.

If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.

To understand is to perceive patterns.

The genesis of the mathematical creation is a problem which should intensely interest the psychologist.

There are no solved problems; there are only problems that are more or less solved.

The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight.

A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom.

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.

Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need for reflection.

The mathematical laws are not, properly speaking, laws of nature, but are, rather, creations of the human mind.

The mind is not only a light but a force; it has not only to see but to act.

The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.