One conclusion I drew from this: In order to solve a difficult problem, reframe it in as many different ways as possible since each framing has a different attack surface, i.e., each framing suggests different potential solutions.
Well, suggests. Problems are problems and solutions are solutions. I find it more helpful to think of everything at once, rather than focusing on ideas one at a time. Like how in a math problem, when you’re stuck you should start writing down all the related equations you can think of.
Like how in a math problem, when you’re stuck you should start writing down all the related equations you can think of.
This strikes me as doing something similar, or maybe the same thing. For example, in probability theory, often we care about whether a sequence of random variables converges in distribution to another random variable. If you work straight from the definition of convergence in distribution, you’ll write down the CDF of a variable in the sequence and then try to show that it converges to the other random variable (except at discontinuity points). This can often be messy, but thinking in terms of characteristic functions suggests a different way of approaching the problem: show that the characteristic function of the sequence converges to the characteristic function of the other random variable. This method is often easier.
Writing down all related equations and theorems would allow you do see this other
possible avenue of attack (there’s a theorem: X_n converges in dist to X iff the characteristic function of X_n converges to that of X). As soon as you start trying to solve the problem in terms of characteristic functions, you have reframed the problem. Writing down all those related equations just allowed you to see how you could reframe it.
I was a little over-general. But basically, you can try a solution without only thinking about the problem in that one way. If the problem is complicated, yeah, one might need to simplify it in order to work on it, and if that’s the case you can get better results by comparing multiple simplifications. But it’s better not to simplify at all, when you can.
For example, I don’t need to simplify the problem of global warming. But it’s reasonable that other people might need to, so yeah, in that sort of case go ahead and reframe.
One conclusion I drew from this: In order to solve a difficult problem, reframe it in as many different ways as possible since each framing has a different attack surface, i.e., each framing suggests different potential solutions.
Well, suggests. Problems are problems and solutions are solutions. I find it more helpful to think of everything at once, rather than focusing on ideas one at a time. Like how in a math problem, when you’re stuck you should start writing down all the related equations you can think of.
This strikes me as doing something similar, or maybe the same thing. For example, in probability theory, often we care about whether a sequence of random variables converges in distribution to another random variable. If you work straight from the definition of convergence in distribution, you’ll write down the CDF of a variable in the sequence and then try to show that it converges to the other random variable (except at discontinuity points). This can often be messy, but thinking in terms of characteristic functions suggests a different way of approaching the problem: show that the characteristic function of the sequence converges to the characteristic function of the other random variable. This method is often easier.
Writing down all related equations and theorems would allow you do see this other possible avenue of attack (there’s a theorem: X_n converges in dist to X iff the characteristic function of X_n converges to that of X). As soon as you start trying to solve the problem in terms of characteristic functions, you have reframed the problem. Writing down all those related equations just allowed you to see how you could reframe it.
I was a little over-general. But basically, you can try a solution without only thinking about the problem in that one way. If the problem is complicated, yeah, one might need to simplify it in order to work on it, and if that’s the case you can get better results by comparing multiple simplifications. But it’s better not to simplify at all, when you can.
For example, I don’t need to simplify the problem of global warming. But it’s reasonable that other people might need to, so yeah, in that sort of case go ahead and reframe.