The twentieth century was a time of big changes in our understanding of Greek mathematics. Many parts of what David Fowler has called "the standard story" have been challenged, and the scholarly consensus has certainly changed as a result. Three examples will have to suffice:
- The standard story suggested that the discovery of incommensurability represented a kind of "foundational crisis" for Greek mathematics, requiring a rethinking of, for example, the concept of ratio. This consensus has been challenged, and many scholars have come to doubt its correctness.
- In the early 20th century, the consensus was that texts such as Book II of Euclid's Elements were best understood as "geometric algebra". This idea was sharply criticized by Sabetai Unguru in the 1960s and has largely been abandoned by historians (who are now more interested in understanding how the Greek approach differs from what later became known as "algebra").
- The standard story paid little attention to social context. Historians today are interested in finding out how Greek mathematicians lived, what they did from day to day, and how the very sophisticated pure mathematics of the great texts got transmitted (or not) to scientists and practitioners.
In Classics in the History of Greek Mathematics, Christianidis has provided a very useful collection of papers that encompass many of these changes. Here one can find Unguru's original papers (and the angry responses they elicited), discussions of the supposed "crisis of foundations", papers on the origin of Greek axiomatics, on geometry and "algebra". The papers, alas, are in their original languages (German, French, and English), but anglophone readers can be reassured that most of the articles are in English, especially the more recent ones. In any case, this is an excellent place to go to learn what the historians have been up to, and the give-and-take of debate makes it fascinating. The book is probably too expensive for non-fanatic individual readers, but it is definitely a "must buy" for libraries.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of MAA Reviews.