I liked this post. It partially answers the question “why is it useful for selection theorems like VNM, Cox to take in several intuitions and prove that something is uniquely specified?” It seems that
Ad-hoc mathematical definitions result from underconstrained sets of intuitions, and are not useful because they are but one in a large space of possible definitions
Good definitions result from perfectly constrained sets of intuitions. In addition to the uniqueness, you have all intuition required to uniquely specify a definition, and so you probably have captured every important intuition
Good definitions/theorems can also tell us about our intuitions:
Impossibility theorems result from overconstrained sets of intuitions, and are often required to tell you that your intuitions are wrong
Very good definitions (e.g. those in fundamental theorems) result from consistent overconstrained sets of intuitions, and give additional evidence that our intuitions are consistent
I liked this post. It partially answers the question “why is it useful for selection theorems like VNM, Cox to take in several intuitions and prove that something is uniquely specified?” It seems that
Ad-hoc mathematical definitions result from underconstrained sets of intuitions, and are not useful because they are but one in a large space of possible definitions
Good definitions result from perfectly constrained sets of intuitions. In addition to the uniqueness, you have all intuition required to uniquely specify a definition, and so you probably have captured every important intuition
Good definitions/theorems can also tell us about our intuitions:
Impossibility theorems result from overconstrained sets of intuitions, and are often required to tell you that your intuitions are wrong
Very good definitions (e.g. those in fundamental theorems) result from consistent overconstrained sets of intuitions, and give additional evidence that our intuitions are consistent