In *What is a number?* *Mathematical concepts and their origins,* mathematics professor Robert Tubbs takes a look at the role mathematical concepts like number and infinity have played in humanistic endeavors in human history, from the 5^{th} century B.C.E. to the present day. It’s a math book without equations which focuses on ideas which any interested reader with at least a high school education can understand.

Tubbs begins with a discussion of Pythagoras (6^{th} century B.C.E.) and his discovery of the correspondence between harmonious sounds and mathematical ratios, reportedly sparked by his observation of the different pitched produces by blacksmiths’ hammers striking anvils: the sound of two hammers is only harmonious when the weights of the hammers can be expressed as a ratio of two whole numbers. Or that’s the version we get from the Roman philosopher Macrobius (395–423 A.D.) who admittedly was writing centuries after the fact, but it’s a reasonable illustration of the point. Pythagoras further inferred that the physical world itself was mathematical (rather than the more typical modern view that it can be understood or analyzed through mathematics) and his ideas proved influential in creative endeavors as well as (or perhaps more so than) in scientific fields: Tubbs traces the influence of Pythagoras on philosophy and poetry from St. Augustine to Allen Ginsburg, exemplifying this book’s far-reaching and eclectic approach.

Other topics covered in *What is a number?* include the concept of infinity (which had no place in Pythagoras’ orderly and finite universe), application of axiomatic reasoning to Christian theology (a useful discussion in case you’ve always wondered, as I often have, why anyone would be concerned with trying to “prove” the existence of God), theories of time and its measurement from the ancient Mayans forward, and theories of vision and their influence on art. If there is a single salient theme in this book it is the differing attempts man has made to understand his world and the role mathematics has played in it, and the corresponding influence of mathematically-derived views of the world on creative works such as visual art and poetry.

*What is a number?* is very clearly written and would be a good choice for undergraduate humanities courses which include a non-technical treatment of mathematics history and philosophy. One of Tubbs’ stated purposes is to refute the commonly-held view that mathematics is an esoteric subject removed from ordinary human endeavors and this volume, which is also enjoyable enough to interest the general reader, should go a long way toward achieving that goal. A number of well-chosen illustrations and diagrams help clarify the ideas presented and the volume also includes a 10-page bibliography and detailed index.

Robert Tubbs is a professor in the Department of Mathematics at the University of Colorado at Boulder where his research includes Number Theory and the Intellectual History of Mathematical Ideas. He is co-author of *Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory* with Edward B. Burger (Springer, 2004).

Sarah Boslaugh is a freelance writer, statistical consultant and Instructor in the Washington University School of Medicine in St. Louis, MO. Her books include *An Intermediate Guide to SPSS Programming: Using Syntax for Data Management* (Sage, 2004), *Secondary Data Sources for Public Health: A Practical Guide* (Cambridge, 2007), and *Statistics in a Nutshell* (O'Reilly, 2008), and she served as Editor-in-Chief for *The Encyclopedia of Epidemiology* (Sage, 2008).