You might want to consider reading the n-category cafe even if you don’t know what an n-category is — even if, antique as this view may be, you don’t care what an n-category is!
For instance, it’s the best place to read about the curious case of M. El Naschie, who’s published 322 of his own papers in the journal he edits for Elsevier.
More substantively: Tom Leinster has a beautiful pair of posts (part I, part II) about varying notions of “diversity” in population biology, and a way to capture all these notions as special cases of a general mathematical construction.
Drastic oversimplification: you might start by defining the diversity of an island beetle population to be the number of different species of beetles living there. But that misses something — a population with three equinumerous beetle species is more diverse than one where a single dominant species accounts for 98% of the beetles, with the remainder split evenly between the other two species. Part I of Leinster’s post is devoted to various measures that capture this behavior. In particular, he’ll explain why on the former island the effective number of species is 3 (just as you’d expect) while on the latter the “number” of species is not 3, but about 1.12 — in other words, the second island is very close to having just one kind of beetle.
In part II, Leinster discusses what happens when you take into account that some pairs of species are more similar than others.