Proof from Intuition and Failed Attempts to Prove Formally
In fact, I think that sometimes repeated failure is evidence for the insight when it is repeated failure by multiple people. Think of the history of failure to prove Fermat's last theorem. Personally, I never doubted the theorem for a second and I doubt I am alone in believing that the repeated failure to provide a proof did not provide much if any evidence that it was false. Or consider what a history to prove Goldbach's conjecture would look like (I haven't looked to see if there is an actual history of attempts to do so). The very fact that so many people have the insight that it is true is what is guiding all these (sadly failed) attempts, and the (partial) independence of the testimony can be surprisingly strong evidence when modeled probabilistically. And it helps when there is considerable conceptual similarity among the attempts, for the insights are often of the form "considerations pertaining to X support Y" (and we just can't get the bridge in formal logic yet).
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