There is a widespread misconception among educated laypeople that mathematics is primarily about logic and proof. No serious mathematician would deny that logical rigor has played an essential role in the development of mathematics. But the essential role that intuition plays in mathematical progress is little known. The asymmetry between the high level of awareness of the importance of logical rigor and the low level of awareness of the importance of intuition has led to educated laypeople to have a heavily distorted view of mathematics. Below I've collected some quotes from great mathematicians about the value of intuition in mathematics. I welcome any references to counterbalancing quotes from people of similar caliber

In Why Johnny Can't Add, mathematician and historian Morris Kline quotes Felix Klein saying

You can often hear from non‐mathematicians, especially from philosophers, that mathematics consists exclusively in drawing conclusions from clearly stated premises; and that, in this process, it makes no difference what these premises signify, whether they are true or false, provided only that they do not contradict one another. But a person who has done productive mathematical work will talk quite differently. In fact those persons are thinking only of the crystallized form into which finished mathematical theories are finally cast. The investigator himself, however, in mathematics, as in every other science, does not work in this rigorous deductive fashion. On the contrary, he makes essential use of his fantasy and proceeds inductively, aided by heuristic expedients. One can give numerous examples of mathematicians who have discovered theorems of the greatest importance which they were unable to prove. Should one, then, refuse to recognize this as a great accomplishment and, in deference to the above definition, insist that this is not mathematics, and that only the successors who supply polished proofs are doing real mathematics? After all, it is an arbitrary thing how the word is to be used, but no judgment of value can deny that the inductive work of the person who first announces the theorem is at least as valuable as the deductive work of the one who first proves it. For both are equally necessary, and the discovery is the presupposition of the later conclusion.

According to an obituary by Robert Langlands, mathematician Harish-Chandra said

In mathematics we agree that clear thinking is very important, but fuzzy thinking is just as important as clear thinking

As reported by Hermann Weyl, while lecturing on Bernhard Riemann Felix Klein said

Undoubtedly, the capstone of every mathematical theory is a convincing proof of all its assertions. Undoubtedly, mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who have distinguished themselves by intuition than by rigorous proofs.

In his essay on Mathematical Creation Henri Poincare wrote

Numerous mathematicians