In this companion website-assisted textbook, the author develops the claim that if you can define clear axioms for what a good diversity measure should do—symmetry, continuity, sensitivity to evenness, and sensible behaviour under pooling—and you’re led to an entropy-based family, not an arbitrary menu. From these first principles, Fleisch derives Shannon and Rényi measures and their Hill-number equivalents, which count effective species, weight rare types transparently, and partition diversity into alpha, beta, and gamma without sleight of hand.

Fleisch tackles familiar puzzles—how adding a rare category can change little, why two communities can be different yet equally diverse—and grounds the argument with compact proofs and crisp examples from ecology, markets, and language.

Metrics quietly steer conservation targets, inequality dashboards, and learning algorithms. Choosing them by axioms, not habit, improves comparability, avoids perverse incentives, and makes arguments about “diversity” precise enough to act on.