Friday, October 14, 2016

On Existence and Uniqueness

In engineering, it isn't uncommon to approach problems with an implicit assumption that they can be solved. In fact, we take a lot of pride in this problem-solving mindset. We ask "how", not "if"?

In a non-traditional department like mine, I have colleagues and collaborators from applied math. Their rigor and discipline prods them to approach new problems differently. 

Instead of asking "how can I start solving this?", the first question they ask is usually, "does a solution exist?"

If there is no solution, there is no point in looking for one. You can't find a black cat in a dark room, if it isn't there. 

If there is a solution, the next question to ask is: "is there one unique solution?", or are there many, perhaps, even infinite possible answers.

If there is a unique solution, any path that takes us to Rome will do. In practice, there is a preference for a path that might get us there fastest. We can begin thinking about an optimal algorithm.

If there are many possible solutions, and we seek only one, perhaps we can add additional constraints on the problem to discard most of the candidates. We can try to seek a solution that is optimal in some way.

There might be lessons from this mindset that are applicable to life in general.

When faced with a new problem, we might want to triage it into one of the three buckets.

Does this problem have a solution? If there is no solution, or the solution is completely out of one's control, then there is no point in being miserable about it.

As John Tukey once remarked, "Most problems are complex; they have a real part and an imaginary part." It is best to see isolate the real part, and see if it exists.

If there is a unique solution, then one should spend time finding the best method to solve the problem.

If, like the majority of life problems (who should I marry? what career should I pick?), there are multiple solutions, then one has to spend time formulating the best constraints or optimal criteria - before looking for a method of solution.

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