As a research mathematician, working in number theory, who is interested in the history of his own field, I have done some reading in the History of Mathematics, particularly that of Ancient Greek and Hellenistic mathematics. When I compare the way history of mathematics is done now with the way it was done, say 50 years ago, I am struck by certain distinct changes.
To be specific, I have noticed a certain trend, among historians today, to attack "naive" or "Whiggist" approaches to their subject, which they perceive as having dominated the field for too long. Put roughly, at least as far as I understand it, a naive way of doing historiography would be to force a "modern" reading upon historical mathematical texts. In contrast, a more refined, say "postmodern", approach would be to deliberately refrain from forcing ancient terminology in (what would be perceived to be) a modern straitjacket.
An infamous example that is central to this debate is the so-called "Geometrical Algebra," which is an interpretation of Book II of Euclid's Elements that originated in the work of Zeuthen in the 19th century. Indeed, the critique of "Geometrical Algebra" seems to be the flagship of the postmodern movement (as I will call it for ease of reference). This discussion is already quite old (1970s I think), but it is still quite often referred to in order to make a point. Here is the debate in a nutshell: in Book II of the Elements, Euclid discusses propositions such as
II.1. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.
II.4. If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments.
Now the proponents of "Geometrical Algebra" would say that II.1 is just Euclid's way of stating that $a(b_1+\ldots+b_n)=ab_1+\ldots+ab_n$, in other words the distributive law, and II.4 is just Euclid's way of stating that $(a+b)^2=a^2+b^2+2ab$. Personally, I find these readings quite convincing. On the other hand, postmodern historians of mathematics would label this line of reasoning "anachronistic", and that is of course to some extent simply true. At the same time, there seems to be a certain vacuity about just labeling a certain reading anachronistic, without substituting a new reading in its place.
Without getting into this whole discussion any further, I was wondering the following things:
1. What is the current status of this debate?I presume that it has died down somewhat, since the 70s are a while back. So I would also like to know:
2. How are opinions divided nowadays? (For example, on the Geometrical Algebra issue.) Has one camp "won" the debate, or are there several "camps" still carrying on independently from each other?
I look forward to reading your comments.