The descriptive math part was very good, thanks - and that's why I resisted downvoting the post. My problem is that the conclusion omits the hugely important factor that categories are useful for specific goals, and the kind of techniques you are suggesting (essentially unsupervised techniques) are context-free.

E.g. is a dead cow more similar to a dead (fixed from 'live') horse or to a live cow? (It clearly depends what you want to do with it)