Reviel Netz's views on geometric algebra

Question

There have been some recent developments in reevaluation of the geometric algebra interpretation of ancient greek mathematics, where Reviel Netz is an expert. I don't have enough background in the field but would like to know what Reviel Netz's position is.

Answer 1

The issue certainly still inspires passions, for context see our previous discussion in Current ways of thinking in the History of Mathematics.

Netz's position is expressed in his panoramic book Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations. It is not so much a return to the geometric algebra debate as such, as to the conflict of philosophies behind it. Netz favors a compromise between the liberties of rational reconstruction and the strictures of context of the times, somewhat reminiscent of Knorr's views, for the sake of evolutionary view of history. In particular, he criticizes Unguru for neglecting "the dynamics of the transformation from the ancient to the modern. Unguru’s premise was that of a great divide separating ancient from modern thinking. The assumption of a great divide, in itself, is not conducive to the study of the dynamics leading from one side of the divide to the other". Netz tries to identify the trends behind historical dynamics rather than simply present snapshots of separate epochs in their contexts:"...Late Antiquity and the Middle Ages were characterized by a culture of books-referring-to-other-books (what I call a deuteronomic culture). This emphasized ordering and arranging previously given science: that is, it emphasized the systematic features of science. Early Greek mathematics,on the other hand, was more interested in the unique properties of isolated problems. The emphasis on the systematic led to an emphasis on the relations between concepts, giving rise to the features we associate with 'algebra'".

As expected, Netz's historiography is controversial. Acerbi's very long review of the book disputes Netz's translations and even parsing of the sources, and criticizes him for "unsatisfactory command and highly tendentious use of primary sources and of secondary literature, unwarranted resort to overgeneralization and to modern mathematical concepts... and treating the received texts and diagrams as if they were what their alleged author wrote in the first place". Despite praising the book for "fascinating and deep theses, brilliant argumentation, an unmistakably flamboyant style, lucid and rhetorically very effective expositions of the difficult proofs presented, wide-ranging interpretative perspectives, and refined tools of analysis" he calls it "utterly disappointing" because in his opinion Netz adjusts facts to fit his picture of historical development. Acerbi's conclusion is:"The main thesis itself is a true masterpiece of interpretative insight; yet, I believe, it is simply not supported by the textual evidence adduced. The only conclusion one draws after reading Netz’ book is that the Greek tradition is a dead end, and that Arabic mathematicians reconsidered the whole issue on entirely new grounds. Despite the author’s efforts, there is no continuous trajectory from problems to equations..."