The global biodiversity crisis is one of humanity's most urgent problems, but even quantifying biological diversity is a difficult mathematical and conceptual challenge. This book brings new mathematical rigour to the ongoing debate. It was born of research in category theory, is given strength by information theory, and is fed by the ancient field of functional equations. It applies the power of the axiomatic method to a biological problem of pressing concern, but it also presents new theorems that stand up as mathematics in their own right, independently of any application. The question 'what is diversity?' has surprising mathematical depth, and this book covers a wide breadth of mathematics, from functional equations to geometric measure theory, from probability theory to number theory. Despite this range, the mathematical prerequisites are few: the main narrative thread of this book requires no more than an undergraduate course in analysis.

'The book shows that the theory of diversity measurement is fertile soil for new mathematics, just as much as the neighboring but far more thoroughly worked field of information theory.' Hirokazu Nishimura, ZB Math Reviews
'Each new mathematical concept is introduced from fundamental principles, in a lucid, engaging style, keeping the core biological application in the foreground … Leinster provides us with the tools to clearly assess what we are losing, so that we may focus better on what we can save.' Benjamin Allen, The Quarterly Review of Biology
'The book is a welcome addition to the theory of biodiversity in that it covers the topics of entropy and diversity in a mathematically rigorous yet fairly accessible way … Recommended.' M. Bona, Choice Magazine

Tom Leinster is Professor of Category Theory at the University of Edinburgh, a member of the University of Glasgow's Boyd Orr Centre for Population and Ecosystem Health, and co-author of a highly-cited Ecology article on measuring biodiversity. He was awarded the 2019 Chauvenet Prize for mathematical writing.