  This book brings new mathematical rigour to the ongoing vigorous debate on
how to quantify biological diversity. The question "what is diversity?" has
surprising mathematical depth, and breadth too: this book involves parts of
mathematics ranging from information theory, functional equations and
probability theory to category theory, geometric measure theory and number
theory. It applies the power of the axiomatic method to a biological problem of
pressing concern, but the new concepts and theorems are also motivated from a
purely mathematical perspective.
  The main narrative thread requires no more than an undergraduate course in
analysis. No familiarity with entropy or diversity is assumed.
