Some propositions, Dr. Godel showed, can be neither proved nor disproved within the confines of mathematical logic. Uncle Petros, after devoting his life to Goldbach's conjecture, finally lost hope when he believed that Goldbach's conjecture might be undecidable. Perhaps he gave up too soon. Working mathematicians generally ignore the possibility that proofs they are pursuing are impossible. For them, the techniques invented and principles illuminated by pursuing the proof of a hard problem are the real rewards.
Fame is nice too.