A better approach to answering why a specific problem is super-hard given a certain level of knowledge is doing a post-mortem on hard-but-solved problems. A few random examples:
Fermat’s Last Theorem
Poincare Conjecture
Trisecting an angle
Non-Euclidean geometry (search for a proof of Euclid’s 5th postulate)
Calculation of planetary motion (from epicycles to Newton’s laws)
One could ask the following questions, one easy and one hard:
What made these problems easy to state by hard to solve?
Given the level of knowledge at the time the problem was stated, what contemporary signs of “hardness” one could have noticed?
Once/if you answer these, and figure out some patterns of hardness, you can then think about yet-unsolved problems and see if you can spot the same signs. If you are lucky, it might also give you promising directions for fruitful research.
Edit: it may well be that there is no recognizable pattern, some peaks are taller than others but all are hidden in the clouds and the only way to find out is through the hard work of climbing them.
A nice way to do such a post-mortem would be to actually ask the people who were there if they thought the problem was Super Hard, why so, and how they did update after the solution was found.
A better approach to answering why a specific problem is super-hard given a certain level of knowledge is doing a post-mortem on hard-but-solved problems. A few random examples:
Fermat’s Last Theorem
Poincare Conjecture
Trisecting an angle
Non-Euclidean geometry (search for a proof of Euclid’s 5th postulate)
Calculation of planetary motion (from epicycles to Newton’s laws)
One could ask the following questions, one easy and one hard:
What made these problems easy to state by hard to solve?
Given the level of knowledge at the time the problem was stated, what contemporary signs of “hardness” one could have noticed?
Once/if you answer these, and figure out some patterns of hardness, you can then think about yet-unsolved problems and see if you can spot the same signs. If you are lucky, it might also give you promising directions for fruitful research.
Edit: it may well be that there is no recognizable pattern, some peaks are taller than others but all are hidden in the clouds and the only way to find out is through the hard work of climbing them.
Interesting.
A nice way to do such a post-mortem would be to actually ask the people who were there if they thought the problem was Super Hard, why so, and how they did update after the solution was found.
Thanks!