An ubiquitous claim in mathematics is that such-and-such mathematical entity has a rich structure or more structure than another one. Most oftenly the entity is a structure - a set explicitly equipped with a structure, i.e. a graph, a group, etc. - but often enough it's just a set not explicitly equipped with a structure.

Examples:

1. Ash/Gross in Fearless Symmetry claim that the solution set [sic!] of a polynomial equation has "more structure" than just its cardinality (p.66).

2. They state on the other hand that the absolute Galois group $G_{\mathbb Q}$ "has a rich structure - much of it still unknown" (p. 87).

Two questions arise (for the beginner):

Question #1: If one considers the structure-richness of a genuine structure - as a set equipped with a structure (as in Example #2) -, what is a possible quantitative measure for richness?

The only answer that comes to my mind is something like the diversity of non-isomorphic (induced?) sub-structures. Is this formalizable?

Question #2: If only a set is mentioned - not explicitly equipped with a structure (as in Example #1) -, what sense does it make to talk of the richness of its structure?

I guess I have to presume a structure imposed on the set, and go back to Question #1. But often enough the structure to impose isn't obvious from the context (like in Example #1). Is this only because this is a book for the general audience? My impression is, that it's fairly often "left to the reader" to literally guess, which structure has to be imposed (to be a rich one).