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A History of Mathematics: From Mesopotamia to Modernity

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A slightly more descriptive title for this book would be On the History of Mathematics, because the book is not a chronology and detailed narrative of the development of mathematics over the course of human history, but rather a careful, questioning look at selected past moments in mathematics. It does not attempt to tell a comprehensive story of its subject, and in fact ponders at times how such a story should be told. The writing style is polished and reflective. The author often compares the methods, notation, meanings, and possible intentions of earlier mathematicians to those of our own, and contemplates what the differences might imply for our understanding of the texts. The book is a scholarly, thoughtful overview, and would work well as an introductory supplement to more comprehensive general histories of mathematics.

Hodgkin refers often throughout the text to Fauvel and Gray's The History of Mathematics: A Reader.

Brief Contents
Introduction
1. Babylonian mathematics
2. Greeks and 'origins'
3. Greeks, practical and theoretical
4. Chinese mathematics
5. Islam, neglect and discovery
6. Understanding the scientific revolution
7. The calculus
8. Geometry and space
9. Modernity and its anxieties
10. A chaotic end?
Conclusion
Bibliography
Index

"We have not, unfortunately, resisted the temptation to cover too wide a sweep, from Babylon in 2000 BCE to Princeton 10 years ago. We have, however, selected, leaving out (for example) Egypt, the Indian contribution aside from Kerala, and most of the European eighteenth and nineteenth centuries. Sometimes a chapter focuses on a culture, sometimes on a historical period, sometimes (the calculus) on a specific event or turning-point. At each stage our concern will be to raise questions, to consider how the various authorities address them, perhaps to give an opinion of our own, and certainly to prompt you for one.

"Accordingly, the emphasis falls sometimes on history itself, and sometimes on historiography: the study of what historians are doing." (4)

Mr. Hodgkin gives a great overview of the history of mathematics, the current state of historical arguments, and all the references (including websites) for further study. At 262 pages it is very readable - I was not looking for a ponderous work with every possible fact catalogued. His approach is refreshingly irreverent and even funny:
"10th century Damascus must surely have been unique as a place where copying the text of Euclid could earn you a living." and
"Perhaps rather than decrying the 'low level' of geometry present in Vitruvius's architecture, we should think about the fact that it was a Roman, rather than a Greek, who bothered to write such a treatise....We have different cultures (cohabiting in the same empire) with different ideas of what a book is for."
I have slogged my way through many math histories without learning half as much, and to be entertained as well is more than one hopes for in such a book.

Hodgkin is a historiography geek with no interest in writing a history of mathematics other than to nitpick about details. Basically, each chapter summarises the conventional story---usually rather scornfully, and too briefly for anyone to gain from it---and then dwells on a myriad of minuscule objections to this version raised by highly specialised historians and published (for a reason, I would say) in highly specialised journals. This piling up of obscure historiographical hypotheses rarely makes a coherent point, let alone does it contribute to any substantial understanding of the history of mathematics.

Although the chapter topics follow the current model of history of mathematics text books (compare the table of contents Victor J. Katz's history of mathematics; notably similar), the text has a strength, depth, and honesty found all too seldom in a text book mathematical history. This is not the typical text-book on technical history that can be dismissed (as Victor J. Katz's should be) as "a pack of lies" with only "slight exageration" (to quote William Berkson's Fields of Force).

Also, the text is bold enough to quote and translate the actual and typical style of presentation used in Bourbaki meetings: "tu es demembere foutu Bourbaki" ("you are dismmembered [..]) [a telegram sent by Bourbaki group to Cartan, informing him that his book was accepted and would be published]. Luke Hodgkin's text dispenses with the asterisk (see p.241).