Interesting list. I’d be curious to see other people’s.
My experience with (3) is the opposite, in three respects.
First, I often encounter problems in my research that I understand completely. No magic or dark corners remain. Nonetheless, finding a solution remains challenging. When I do find the solution, I don’t feel as if the problem has been further illuminated.
Second, the various lines of attack I develop with an impartial understanding of a problem are usually much more creative and varied than after my understanding fully solidifies. When I first encounter a problem I usually take at least a day or two to enumerate possible solutions before actually studying the problem.
Three, I’ve successfully solved complex problems I didn’t understand at the time by blindly trying approaches my intuition threw at me.
First, I often encounter problems in my research that I understand completely. No magic or dark corners remain. Nonetheless, finding a solution remains challenging. When I do find the solution, I don’t feel as if the problem has been further illuminated.
I’d really like to see an example. For me, understanding the solution is part of the problem. I read “understand completely” as “can characterize the solution to the problem as a mapping from inputs to outputs”, and such a characterization should itself be a solution.
I work in applied math. The bulk of my work is aimed at doing something efficiently. Given a problem (a physical model, plus some discretization), I’ll have to find suitable algorithms to solve the resulting equations. I usually understand enough about the physics (and artificial physics introduced via the discrete setting) to have a complete understanding of the limits of the efficiency of any implementation. Nonetheless, actually creating the implementation which maximizes efficiency is not straightforward (otherwise I’d have no job!).
I guess you could classify this as two different problems: the base problem of understanding, in an abstract sense, the bounds on efficiency, and the actual problem of constructing algorithms which will touch those bounds. In my mind though, they feel intimately related in a way that I can’t unravel.
Interesting list. I’d be curious to see other people’s.
My experience with (3) is the opposite, in three respects.
First, I often encounter problems in my research that I understand completely. No magic or dark corners remain. Nonetheless, finding a solution remains challenging. When I do find the solution, I don’t feel as if the problem has been further illuminated.
Second, the various lines of attack I develop with an impartial understanding of a problem are usually much more creative and varied than after my understanding fully solidifies. When I first encounter a problem I usually take at least a day or two to enumerate possible solutions before actually studying the problem.
Three, I’ve successfully solved complex problems I didn’t understand at the time by blindly trying approaches my intuition threw at me.
I’d really like to see an example. For me, understanding the solution is part of the problem. I read “understand completely” as “can characterize the solution to the problem as a mapping from inputs to outputs”, and such a characterization should itself be a solution.
I work in applied math. The bulk of my work is aimed at doing something efficiently. Given a problem (a physical model, plus some discretization), I’ll have to find suitable algorithms to solve the resulting equations. I usually understand enough about the physics (and artificial physics introduced via the discrete setting) to have a complete understanding of the limits of the efficiency of any implementation. Nonetheless, actually creating the implementation which maximizes efficiency is not straightforward (otherwise I’d have no job!).
I guess you could classify this as two different problems: the base problem of understanding, in an abstract sense, the bounds on efficiency, and the actual problem of constructing algorithms which will touch those bounds. In my mind though, they feel intimately related in a way that I can’t unravel.
You are disputing definitions of the word “understand”.
ETA: Wrong.
Explicitly so, and asking for an example to be sure.
Actually, no, you are not disputing definitions, you are asking for a clarification. My bad.