If you don't know how to make a decision, then don't make it

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If you don't know how to make a decision, then don't make it

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[QUICK DESCRIPTION]
Very often a proof requires one to choose some object that will make the rest of the proof work. And very often it is far from obvious how to make the choice. Often a good way of getting round the problem is to take an ''arbitrary'' object of the given type, give it a name $X$, and continue with the proof as if you ''had'' chosen $X$. Along the way, you will find that you need to assume certain properties $P_1,\dots,P_k$ of $X$. Then your original problem is reduced to the question "Does there exist an object of the given type with properties $P_1,\dots,P_k$?" Often, this is a much more straightforward question than the main problem you were trying to solve.

A very simple example would be doing an analysis argument where a certain parameter has to be small enough. But it would be nice to have some much more structural examples. An example already appears in the article [[How to solve cubic and quartic equations]] in the course of the discussion of the general solution of a cubic.

See also "[[Keep parameters unspecified until it is clear how to optimize them]]".

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If you don't know how to make a decision, then don't make it

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