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).