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This is a book about mathematical ideas that have developed in cultures outside the tradition as taught in, say, western universities. There are examples from the Maya of South America; the Borano and Malagasy of Africa; the Basque of Europe; the Tamil of India; the Balinese and Kodi of Indonesia; and the Marshall Islanders, Tongans and Tobriand Islanders of Oceania. It examines how mathematical ideas have developed and shaped these societies, and vice versa, and demonstrates patterns of mathematical thought that are very different from our own. In particular, it draws attention to how mathematical meaning emerges from the cultural contexts from which the ideas originate. The result is a book that leads us to reassess the assumptions of mathematics as understood by the reader. Some of our ideas taken to be universal are not, and some ideas thought to be exclusively our own are shared by others. The thing that makes this book stand out for me, is that it encourages the reader to reassess not just our mathematical assumptions, but our assumptions for how we structure society as a whole.

The book is fascinating for its subject matter and its presentation is clear and accessible. There are chapters on divination; methods of measuring time, both by natural cycles of nature and the stars, and by the cosmological belief systems of the society; map making and navigation; the social arrangements of communities; and patterns found in art. I particularly enjoyed the chapter on how communities organise themselves - there's an example of how the Basque constructed dwellings in a circle where each house is responsible for the well being of their neighbours and next nearest neighbours in distinct ways. The circular nature defies the emergence of hierarchies. In contrast an example from Tonga describes the mathematics of a ranking system that relies on gender, generation and age. The entire book is full of illustrations of people who have chosen to arrange their affairs in a different way than our own.

It does go into mathematical detail. While this detail in never difficult or inaccessible in itself (which is not to say that the mathematics isn't sophisticated - it's just that it should be understandable to anyone with high school level mathematical exposure) I did find myself glossing over some of the lengthier calculations. By necessity, they were presented in the language that a western mathematician is accustomed to. They demonstrate the rigour of the mathematics that was described, but I would happily have taken that on trust.

In general though, this is a fascinating book. It offers a global and humanistic perspective of the history of mathematics, and for anyone interested in mathematics, anthropology, and the way people choose their social arrangements, I highly recommend it.