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Selected Essays on Pre- and Early Modern Mathematical Practice

Jens Høyrup
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Gouvêa
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Jens Høyrup is a well-known and influential historian of premodern mathematics. He is most famous for his work on Old Babylonian “algebra,” culminating in the monograph Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin. But Høyrup’s interests range widely, including most premodern mathematics, sociological and philosophical issues, and, more recently, early modern mathematics.

The massive volume under review collects in its almost 1000 pages 31 recent articles by Høyrup, presented with an introductory overview. The articles have been reformatted but not rewritten. The introductory overview explains the origins of each of the essays and occasionally comments on further developments. All of the essays have been published previously, but several of them appeared in edited volumes and may not be easy to access, so it is good to have them collected here.

The book is divided in two parts. In the first part, the essays are mostly on the history (narrowly understood) of mathematics. The second part ranges more widely into historiography, sociology, and philosophy.

There are a lot of interesting essays here. Høyrup is very interested in mathematical problems, including “recreational” problems. He pays attention to material that is sometimes considered too elementary to be of interest, often with illuminating results.

Also interesting is the focus on mathematical practice: how mathematicians actually operated, the methods they used, and their reasoning styles. One of the articles, for example,

… grew out of [his] uneasiness with explanations of ancient (or otherwise unfamiliar) ways of thinking or expressing thought through postulated cognitive inability or postulated incompatible modes of thought.

In other words, the goal is to understand how people thought and why they thought that way, not to simply assume an unpassable gulf between us and them.

Høyrup is not always easy to read, but he is always interesting and not afraid of breaking with “what everyone knows.” (He says that “taboos when respected have consequences; outlawed concepts cannot contribute to the creation of new knowledge.“) Historians of mathematics will find much food for thought in this book.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.