are multi-stage (sequentially dependant?) breakthroughs more impressive than a similar number of breakthroughs that aren’t sequentially dependant or that happen far apart in time from each other?
Yes, because… it means they couldn’t have been finding low-hanging fruit. When one problem leads to another, you don’t get to wander off and look for easier ones, you have to keep going down one of these few avenues of this particular cave system. So if someone solved a contiguous chain of problems you can be sure that some of those were probably genuinely really hard. It also requires them to develop their own understanding of something that nobody could help them with, and to internalize that deeply enough to keep going.
Sequences like this occur naturally in real-world projects, so if they’re avoiding them it’s kinda telling.
more are needed until something valuable can be produced produced?
?. More are needed before we can make a judgement. I’d believe that lots of value can be produced without any of these big leaps.
But you can also just judge each breakthrough separately, conditional on what they had access to. If they’re deep into a problem past where anyone has been and then go further, that might be more impressive, but it may not be, in case it’s easy to identify the next (possibly hard) subproblem after solving the last subproblem. So you can approach it locally/greedily, without thinking ahead much to where you need to go, only about where you are now and the next step. I think upper-year and grad-level pure math and theoretical computer science problems can be like this, although maybe not as hard as you’re asking for.
Something harder I have in mind would be something like having a non-local/non-greedy approach to solving a problem, where you have a major breakthrough just to get a sketch of a proof or to come up with possible lemmas, and then it takes further breakthroughs to close things up. If your sketch is wrong, then all the work can become basically useless, and you don’t progress things for the next people to try (except by ruling out a dead end).
It’s like thinking more moves ahead in chess with multiple hard moves to identify, compared to just making the same number of hard moves to identify in the same game, but never part of the same sequence simulation in your head. Both are multi-stage, but the second one is local/greedy and isn’t more impressive than making the same number of hard to identify moves across games, fixing the total number of games played.
Also, breakthroughs across very different areas rather than all concentrated in the same area demonstrates greater flexibility and generalizability of their strengths.
Yes, because… it means they couldn’t have been finding low-hanging fruit. When one problem leads to another, you don’t get to wander off and look for easier ones, you have to keep going down one of these few avenues of this particular cave system. So if someone solved a contiguous chain of problems you can be sure that some of those were probably genuinely really hard. It also requires them to develop their own understanding of something that nobody could help them with, and to internalize that deeply enough to keep going.
Sequences like this occur naturally in real-world projects, so if they’re avoiding them it’s kinda telling.
?. More are needed before we can make a judgement. I’d believe that lots of value can be produced without any of these big leaps.
But you can also just judge each breakthrough separately, conditional on what they had access to. If they’re deep into a problem past where anyone has been and then go further, that might be more impressive, but it may not be, in case it’s easy to identify the next (possibly hard) subproblem after solving the last subproblem. So you can approach it locally/greedily, without thinking ahead much to where you need to go, only about where you are now and the next step. I think upper-year and grad-level pure math and theoretical computer science problems can be like this, although maybe not as hard as you’re asking for.
Something harder I have in mind would be something like having a non-local/non-greedy approach to solving a problem, where you have a major breakthrough just to get a sketch of a proof or to come up with possible lemmas, and then it takes further breakthroughs to close things up. If your sketch is wrong, then all the work can become basically useless, and you don’t progress things for the next people to try (except by ruling out a dead end).
It’s like thinking more moves ahead in chess with multiple hard moves to identify, compared to just making the same number of hard moves to identify in the same game, but never part of the same sequence simulation in your head. Both are multi-stage, but the second one is local/greedy and isn’t more impressive than making the same number of hard to identify moves across games, fixing the total number of games played.
Also, breakthroughs across very different areas rather than all concentrated in the same area demonstrates greater flexibility and generalizability of their strengths.