Well-Posed Problems

August 23, 2009

To continue along with the mathematical analogies that I built in my previous essay on iterative processes, I thought I would examine and riff on the concept of ‘well-posed problems.’ To be honest, this isn’t something that I have much experience with in mathematics. Because, typically, in all the math classes I’ve had, the problems are ALWAYS well-posed. That is, they’re always something that can be solved.

But life isn’t like math class. Typically, the majority of problems in my life aren’t well-posed. Heck, half of them aren’t even POSED. And if that’s the case, how can I expect to get anywhere with them?

To go back to the math metaphor again, what does it even mean for a problem to be well-posed? It means that a solution exists. That you can get there from here. So, on the flip-side, what does it mean if a problem isn’t well-posed? What does it mean for a problem to be ill-posed? It means that there IS NO SOLUTION. No matter how hard you try, you can’t get there from here. You can scratch your head, jump on one foot while patting your belly, or even bang your head against the wall continuously. That won’t get you to the solution. Because quite simply, that solution doesn’t exist.

And in life? You can go about life half asleep. That’s analogous to always dealing with ill-posed problems. And yes, it has its advantages. Like the fact that you don’t have to think quite so much. Because thinking is, like, hard. And if all you ever do is try to solve one ill-posed problem after another, you’ll always feel like you’re doing something. And if just doing ‘something’ is what you’re after, then damn if you’re not doing a good job!

But if you’re interested in doing anything more than ‘something,’ say ‘something in particular,’ then it’s going to require that you find a well-posed problem. Although life has the slight advantage over math in that sometimes you might start with an ill-posed problem and still get an answer (even a ‘correct’ answer, if that terminology has any meaning in the real world). But then you’re just being lucky, you’re not being smart. And although those two domains are not strictly exclusive, one is well within my control while the other is, almost by definition, not.

So, to move things forward you need to pose the ‘thing’ well. I guess that’s analogous to setting up real, honest, concrete goals. And I would imagine that setting those goals must also involve writing the thing down. Because the mind has a silly way of making it seem like you ‘know’ and ‘remember’ things, just because you happen to be able to keep a fuzzy concept of those things in your mind’s eye for a few seconds. Like this whole whole ‘well-posed problems’ analogy. It was stuck in my mind when I first thought of it, but it certainly didn’t look anything like it does now.

And I certainly don’t really have any concrete details as to how to tell whether a ‘life’ problem is well-posed. That will require more thinking (eke!) on my part.

Time to pose some problems. And iterate.

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