Yet the secret of the productivity of genius will always lie in posing new questions and divining new theorems that shall disclose valuable results and connections. Without the creation of new points of view, without new goals being set up, mathematics , for all the rigor of its proofs , would soon exhaust itself and begin to stagnate. Thus, mathematics is advanced , in a certain sense, most by those who are more distinguished by intuition than by rigorous proofs. 255p
316p on axioms of groups Felix Klein
It is a peculiar fact about the genesis and growth of new disciplines that too much rigor too early imposed stifles the imagination and stultifies invention. A certain freedom from the strictures of sustained formality tends to promote the development of a subject in its early stages, even if this means the risk of a certain amount of error. Nonetheless, there comes a time in the development of any field when standards of rigor have to be tightened.
Wisdom of the West, p. 280
In fact the opposition of instinct and reason is mainly illusory. Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new.
-Our Knowledge of the External World,
The intuitive notion of a circle, for instance, is made precise when we define it as the set of points equidistant from a fixed point (the center); thus a definition can transform figures into mathematical language. For some people, a good definition is even a work of art. The great mathematician Alexander Grothendieck once wrote, "Around the age of twelve...I learned the definition of a circle. It impressed me by its simplicity and its evident truth, whereas previously, the property of 'perfect rotundity' of the circle had seemed to me a reality mysterious beyond words. It was at that moment... that I glimpsed for the first time the creative power of a' good' mathematical definition... still, even today, it seems that the fascination this power exercises over me has lost nothing of its force."
D'Alembert, Jean Le Rond (1717-1783) Just go on..and faith will soon return. [To a friend hesitant with respect to infinitesimals.] In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhauser, 1981
C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente.
"It is by logic that we prove, but by intuition that we discover."
-- Henri Poincaré http://en.wikiquote.org/wiki/Henri_Poincar%C3%A9
- Henri Lebesgue
No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary.