Some things are formally defined terms of art and some things aren't. There are some people here who don't merely want to learn a model for the naturals, they want some kind of spiritual essence of a number. It's useful to say that number is an informal term, and that the fuzzy boundaries you think with are adequate for communication.

Similarly, "crazy" is not a term of art for psychiatry. That doesn't mean a lack of definition, and lacking formalism does not mean inadequate for communication. By using a fuzzy term for communication, you imply a taste in curation and abstraction. It means you don't want to get into the details of what crazy means.

> That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc

A metric is a term of art, and if that's the level of specificity you wanted, then it's adequate.