This fascinating collection of essays is a must-have for those who are interested in the history and philosophy of mathematics but are tired of the usual "foundational" discussion of formalism versus platonism. As the editors point out in their introduction, there are other ways to think about what mathematics is. In this case, the focus is on understanding what mathematicians actually do (or have done).

Part I of the book focuses on the role of visualization in mathematics, and especially on how visualization interacts with mathematical reasoning. Anyone who is familiar with "Proofs without Words" knows that there is something here to study. The essays in this section are not light reading, but they raise and explore interesting issues. Mancosu's essay, for example, discusses the "return of the visual" in recent mathematics, contrasting 19th century attitudes ("see, no pictures!") with recent work (for example, on fractals).

The essays in part II are about mathematical explanation and styles of reasoning. They include an essay by Reviel Netz on the aesthetics of mathematics, which tries to think through what mathematicians mean when they say an argument is beautiful, two essays on styles of reasoning in other mathematical cultures (Høyrup on ancient Mesopotamia, Chemla on China), and two more straightforwardly philosophical essays.

All in all, this is a book that libraries will want to have, particularly if they strive to have good collections on the history and philosophy of mathematics.

Fernando Q. Gouvêa is Professor of Mathematics and Colby College and the co-author of *Math through the Ages: A Gentle History of Mathematics for Teachers and Others*.