The earth's biodiversity is now declining so rapidly that many believe this to be the start of one of the greatest mass extinctions in the history of life on earth. Taking action requires not only political will and cultural change, but also logically correct methods for quantifying and reasoning about diversity - because without them, we risk channelling conservation funds in the wrong direction. The mathematics of biodiversity turns out to be surprisingly deep. Even making the concept of diversity precise is a major challenge. It also involves branches of mathematics that most people think of as highly "pure" rather than "applied" (an increasingly unhelpful distinction). I will tell a small part of this story, touching parts of mathematics from geometry to information theory to category theory.