I disagree. Any given arbitrary problem is an instance of some broader set of problems, such that knowing how to solve that arbitrary problem allows you to trivially solve any other problem in that broader set. Conversely, if you can’t trivially solve it, that means you don’t understand the entirety of some fairly broad segment of concept-space.
In that sense, there’s no problems “without any interesting special structure” that are hard to solve relative to some conceptual toolbox. If you can’t trivially solve it, there are interesting structures to be uncovered on the way to solving it. (“Triviality” is a variable here, too. The more non-trivial solving a problem is, the higher the distance between your tools and the tools needed to solve it; the more ignorant you are, and so the more interesting structures you can uncover on the way to solving it.)
NP-complete problems are… I wanted to say “the exception”, but they’re not, really. They’re not conceptually hard to solve (unless it turns out P=NP after all, but surely not), just computationally so.
Edit: Again, Fermat’s Last Theorem is a good example. It’s the most random-ass problem among random-ass problems, but the solution to it required developing some broadly applicable tools.
I disagree. Any given arbitrary problem is an instance of some broader set of problems, such that knowing how to solve that arbitrary problem allows you to trivially solve any other problem in that broader set. Conversely, if you can’t trivially solve it, that means you don’t understand the entirety of some fairly broad segment of concept-space.
In that sense, there’s no problems “without any interesting special structure” that are hard to solve relative to some conceptual toolbox. If you can’t trivially solve it, there are interesting structures to be uncovered on the way to solving it. (“Triviality” is a variable here, too. The more non-trivial solving a problem is, the higher the distance between your tools and the tools needed to solve it; the more ignorant you are, and so the more interesting structures you can uncover on the way to solving it.)
NP-complete problems are… I wanted to say “the exception”, but they’re not, really. They’re not conceptually hard to solve (unless it turns out P=NP after all, but surely not), just computationally so.
Edit: Again, Fermat’s Last Theorem is a good example. It’s the most random-ass problem among random-ass problems, but the solution to it required developing some broadly applicable tools.