[if φ(n) ≈ Kn2] then this would have consequences of the greatest magnitude. That is to say, it would clearly indicate that, despite the unsolvability of the [Hilbert] Entscheidungsproblem, the mental effort of the mathematician in the case of the yes-or-no questions would be completely replaced by machines. … [this] seems to me, however, within the realm of possibility.
I conjecture that there is no good algorithm for the traveling salesman problem. My reasons are the same as for any mathematical conjecture: (1) It is a legitimate mathematical possibility, and (2) I do not know.
In this paper we give theorems that suggest, but do not imply, that these problems, as well as many others, will remain intractable perpetually.
If you have ever attempted a crossword puzzle, you know that it is much harder to solve it from scratch than to verify a solution provided by someone else. Likewise, solving a math homework problem by yourself is usually much harder than reading and understanding a solution provided by your instructor. The usual explanation for this difference of effort is that finding a solution to a crossword puzzle, or a math problem, requires creative effort. Verifying a solution is much easier since somebody else has already done the creative part.