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26 October 2001
An Amateur Looks at the History of Numbers
Georges Ifrah, The Universal History of Numbers. From Prehistory to the Invention of the Computer. Translated by David Bellos et al. John Wiley and Sons, 1998. 633 pages.
ISBN 0-471-39340-1Reviewed by Sheldon Greaves
Amateur scientists around the world can take heart at what one of their own has accomplished with the publication of this book. It is, quite simply, the most complete study of the invention and evolution of numbers ever published in a single volume. Spanning a wide array of cultures and vast stretches of history, it contains an astonishing breadth and depth of scholarship.
This book began, as do most great intellectual achievements, with a deceptively simple question. The author, Georges Ifrah, was working as a math teacher when one of his young students asked, "where did numbers come from?" Ifrah realized that he did not have a good explanation at hand, and so went looking for one in the library. To his dismay, he found that while there was a great deal to be had on the history of math, virtually nothing could be found on the history and development of numbers and counting. Apparently it was too banal a topic to grab the interest of the experts.
Gradually, Ifrah's interest in the question became an obsession. He eventually was able to secure enough support to devote himself to his work on a nearly full-time basis. His study of the history of numbers and counting and their influence on the development of humanity was eventually published (in French) as Histoire universelle des chiffres.
One of the more striking insights of this book is that counting and numbering is not the obvious concept we assume it is. Ifrah demonstrates how primitive cultures past and present came to counting slowly, haltingly. He also shows how some cultures have managed to obtain "numerical" information without the benefit of counting or arithmetic. For example, he explains how some archaic tribes sending warriors off to fight give each warrior a small pebble. The returning survivors deposit their pebbles in a basket or other container as a way of gauging the number of casualties.
Ifrah has done his homework. His writing is dense, yet clear. He takes very little for granted on the part of the reader and unlike many professional experts, he takes the time to explain things in detail for the non-specialist. And his coverage is positively intimidating. Stone age, Mesopotamian, Chinese, Sanskrit, western Europe, Oceania, Mesoamerica... it's all here and in remarkable detail.
For example, his explanation of the Mesopotamian sexagesimal (base 60) system, which forms the basis of our 360° circle and our timekeeping notation, is one of the best I have ever seen. The other material in the section on Mesopotamian numbers shows an excellent grasp of the material and even the original language. Although I'm certain he does not know Akkadian or Sumerian, he clearly worked with someone who did. His other sections display similar levels of intellectual craftsmanship, with odd exceptions. While his text and prose is outstanding, his bibliography contains some unexpected mistakes. For example, he cites Otto Neugebauer's classic The Exact Sciences in Antiquity, but lists the author's first name as "Oscar". Multiple works by assyriologist Pierre Amiet are listed under "A. Amiet." But the titles seem to be correct, so readers wishing to explore further shouldn't have too much trouble following Ifrah's research in the literature.
Bibliographic issues notwithstanding, this is a marvelous book. Counting systems and numeric history from around the globe, and from preliterate time to the first calculating machines are described in enthralling detail. I don't care how much you know about math or history, you will learn a lot just from leafing through this seemingly endless stream of numeric delights.
