Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by other… - Henri Poincaré

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so lone as the relations remain unchanged. Content to them is irrelevant; they are interested in form only.

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 very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all the theorems with which so many volumes are filled, are only indirect ways of saying that A is A?

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. ...we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it.