We define diversity measures that take account of the varying similarities between species, and show how they can be used. We state an unexpected theorem on maximizing diversity: there is a single abundance distribution that maximizes diversity from all viewpoints simultaneously. There follows a broad-brush survey of magnitude, which is closely related to maximum diversity and is defined in the very wide generality of enriched categories. In the case of metric spaces, magnitude encodes fundamental geometric invariants of size (such as volume, surface area and dimension) and is related to the concept of capacity in potential theory.

We introduce two families of deformations of Shannon entropy: the q-logarithmic entropies (also called “Tsallis entropies”) and the Rényi entropies. We explain how the exponentials of the Rényi entropies, called the Hill numbers, convey information about the diversity and structure of an ecological community. We introduce the power means, which lie at the technical heart of this book. We give functional equations characterizing the q-logarithmic entropies on the one hand, and the Hill numbers on the other.