Often, the key to research is figuring out how to redefine failure as success. Some stories: when Alan Turing published his epochal 1936 paper on Turing machines, he did so with great disappointment: he had recently learned that Alonzo Church had independently arrived at similar results using lambda calculus, and he didn’t know whether anyone would still be interested in his alternative, machine-based approach. In the early 1970s, Leonid Levin delayed publishing about NP-completeness for several years: apparently, his “real” goal was to prove graph isomorphism was NP-complete (something we now know is almost certainly false), and in his mind, he had failed. Instead, he merely had a few “trivialities,” like the definitions of P, NP, and NP-completeness, and the proof that satisfiability was NP-complete. And Levin’s experience is far from unique: again and again in mathematical research, you’ll find yourself saying something like: “goddammit, I’ve been trying for six months to prove Y, but I can only prove the different/weaker statement X!

Scott Aaronson on Philosophical Progress | Machine Intelligence Research Institute Sunday, December 15, 2013 @ 8:17pm