My instinctive answer is that no, there’s no special property, no neat compact reason. It just so happens that these specific pairings of (conceptual toolbox,research problem) have a very large L. If we were a different or alien culture, that has historically pursued different branches of math and started from different native concepts, these problems might’ve been trivial, and some of our trivial problems might’ve been tremendously difficult.
I guess “what native concepts we start with” and therefore “what math branches we historically pursue” might not be entirely arbitrary, conditioning on the fact that we’re evolved life. There might even be some generalities across most probable instances of naturally-evolved minds. But that’s the only source of non-arbitrariness I see here.
A starker example is the Fermat’s Last Theorem. There’s really no reason I can see why it “had” to have a proof over 100 pages long. (Also, this is why I think Fermat was full of it when he claimed to have solved it. I’m sure mathematicians following him have cycled through all possible math-tools of his time and their neighborhoods, so unless he built a really long chain of math results in his private time and then published none of it, he couldn’t have done it.)
My instinctive answer is that no, there’s no special property, no neat compact reason. It just so happens that these specific pairings of (conceptual toolbox,research problem) have a very large L. If we were a different or alien culture, that has historically pursued different branches of math and started from different native concepts, these problems might’ve been trivial, and some of our trivial problems might’ve been tremendously difficult.
I guess “what native concepts we start with” and therefore “what math branches we historically pursue” might not be entirely arbitrary, conditioning on the fact that we’re evolved life. There might even be some generalities across most probable instances of naturally-evolved minds. But that’s the only source of non-arbitrariness I see here.
A starker example is the Fermat’s Last Theorem. There’s really no reason I can see why it “had” to have a proof over 100 pages long. (Also, this is why I think Fermat was full of it when he claimed to have solved it. I’m sure mathematicians following him have cycled through all possible math-tools of his time and their neighborhoods, so unless he built a really long chain of math results in his private time and then published none of it, he couldn’t have done it.)