Felix Klein once said,
Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof.
Till now I thought the opposite. I thought that it is the rigorous methods of proof that requires more ingenuity because in my opinion is is 'quite easy' to make conjectures but in many cases unimaginably difficult to prove it.
Consider the following viewpoints,
View 1: Every conjecture is intuitive at the beginning. But it is useless unless we prove it because simply deducing consequences if it had been true isn't worth a mathematician's time, perhaps. What if any of the deduced consequences doesn't contradict any of the established theorems?. So what in the end in essential, the rigorous proof of course.
View 2: It's not true that simply deducing consequences if it had been true isn't worth a mathematician's time. What if in course of this study (though logical but perhaps not always practical) the mathematician has derived a contradiction with the established theorems? And also, let's grant for some time that rigorous methods of proof is mostly important but you must have something to prove and that something must come from intuition.
Probably this question is not best suited for this site, but I am eager to know the thoughts of my fellow MSE users on this subject.