Henri Poincare Quotes

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.

The essence of mathematics lies precisely in its freedom.

Modern mathematics, that most astounding of intellectual creations, has projected the mind's eye through infinite time and the mind's hand into boundless space.

Humanity has known for a long time what fractals are. It is a very strange situation in which an idea which each time I look at all documents have deeper and deeper roots, never (how to say it), jelled.

A mathematician ... has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.

The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience

Now the word-symbols of conceptual ideas have passed so long from hand to hand in the service of the understanding, that they have gradually lost all such fanciful reference.

When a man knows that the abstraction ten exists - nothing on earth can stop him from looking for the fact of eleven.

Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.

A mind that is very selective to forms... is apt to use its images metaphorically, to exploit their possible significance for the conception of remote or intangible ideas.

One should not be too distracted by definitions. Ideas transcend definitions.

Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.

To a mathematician the eleventh means only a single unit: to the bushman who cannot count further than his ten fingers it is an incalculable myriad.

In mathematics ... we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

Strange! I don't understand how it is that we can write mathematical expressions and calculate what the thing is going to do without being able to picture it.

The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.

Mathematics deals exclusively with the relations of concepts to each other without consideration of their relation to experience.

I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.

Mathematicians do not deal in objects, but in 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.

Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

Mathematics doesn’t care about those beyond the numbers.

Time is but the measure of the difficulty of a conception. Pure thought has scarcely any need of time, since it perceives the two ends of an idea almost at the same moment.

When an idea is not robust enough to stand expression in simple terms, it is a sign that it should be rejected.