# Current ways of thinking in the History of Mathematics

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.

and

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?

• You may be interested in this article... – Marius Kempe Sep 29 '15 at 20:14
• @MariusKempe: thank you, yes I had read that one. I thought it was forceful and well-written. It received a (somewhat condescending) reply in the same issue of the same magazine (J. of Hum. Math., Vol. 4(2), 2014, pp. 124-136, to be found online at scholarship.claremont.edu/cgi/…), which at least signals that the debate isn't completely dead yet. But it does seem quite close to being dead, if the only attacks on the modern consensus are coming from a PhD student. – RP_ Oct 1 '15 at 17:53
• There is now a very good article summarizing the debate by Jens Høyrup, 'What is “geometric algebra”, and what has it been in historiography?' – Marius Kempe Jan 25 '16 at 4:23

I just wrote a paper going through the case against the geometrical algebra and arguing that the modern consensus against it is ill-founded and an associated blog post. But my opinion is a minority one. The geometrical algebra hypothesis has indeed been declared dead with considerable confidence by the modern generation of historians. In my introduction I quote these statements from establishment historians, which are a very accurate description of the state of the matter in recent years: “Unguru’s position [against geometrical algebra] could now be regarded as the accepted orthodoxy.” “It is clear that the old historiography has been overcome. … There are very few who still believe in such historiographical artefacts as … geometric algebra.”

As mentioned in the comments, see also my critique the modern consensus in the historiography of mathematics in general, as well as its reply from a representative of the modern establishment opinion.

Histories of science and mathematics are very young, only emerging as disciplines in the early 20th century. This means that most of the received view (tradition) was created by non-historians, who often had expository, didactic and human interest purposes as their primary motivation. This encouraged a view of history as a sequence of bursts of genius, and rational reconstruction as the method of analyzing it, reading past results as imperfect prototypes of modern concepts, perhaps in a different guise, towards which they develop. Grattan-Guinness coined the term "heritage", as opposed to "history", for this kind of approach in History or Heritage? An Important Distinction in Mathematics and for Mathematics Education. It often produces conceptually cleaned up and notationally modernized sketches, and for many purposes this may well be desired. Both tendencies were taken to the extreme however in E.T. Bell's controversial Men of Mathematics, and they largely dominate historical parts of many mathematical textbooks.

The main changes occurred in two stages, neither exclusive to the history of mathematics. In 1930s the traditional historical internalism was challenged by demands for attention to social and cultural context, the so-called Hessen thesis. The second stage was a turn from rational reconstruction to detailed analysis of contemporary sources, and more sceptical view of mathematical speculations. For the history of mathematics specifically the watershed was Burkert's Lore and Science: Studies of Pythagoras, Philolaus and Plato, that exposed almost all of traditional story about Pythagoras as a myth fabricated by Plato's successors at the Academy. "Mystic, yes... but a mathematician, no". Burkert's is an impeccable work of classical scholarship with exhaustive analysis of multiple original sources, many of them obscure, that left little room for doubt and set a standard for subsequent work.

Geometric algebra was a natural hypothesis under the rational reconstruction methodology. It is very appealing if modern algebra is taken as the final cause, and gained wide popularity after van der Waerden's eloquent popularization. But its speculative nature and conflation of Babylonian arithmetic with algebra led Szabo to question it in late 60s, and Unguru to reject it forcefully in 1975, more on the debate here. A modern view can be found in Fowler's Mathematics of Plato's Academy, which is directly indebted to Burkert for its methodology and many insights:

"The idea that a length is a number is so deeply ingrained in our thought that it takes a conscious effort to conceive of an approach to geometry that does not make such an assumption. It is such an arithmetized interpretation that led historians to describe Book II of the elements as "geometric algebra". Fowler argues that Greek geometry was completely non-arithmetized. The strongest evidence comes from his analysis of the very difficult Book X, where he shows, I think successfully, that the way Euclid (or Theaetetus?) structures the argument precludes an arithmetical approach. Most historians would agree with this description of Greek geometry, I think. Many, however, see this characteristic as something to be explained... Fowler turns this on its head, and argues that in fact the non-arithmetized nature of Greek geometry should be read as evidence that Greek geometry does not derive from Babylonian sources. Essentially, Fowler argues that this approach to geometry is just as "natural" as the arithmetized one that we now use, and that it therefore is not necessary to explain why the Greeks used it".

While the geometric algebra debate may have been the loudest the points made apply more broadly. This is what Unguru argued, for example:

"To read ancient mathematical texts with modern mathematics in mind is the safest method for misunderstanding the character of ancient mathematics, in which philosophical presuppositions and metaphysical commitments played a much more fundamental and decisive role than they play in modern mathematics".

For example, one finds Euclid interpreted by Hilbert and Klein as a founder of axiomatic method in the popular textbooks of Greenberg and Hartshorne. But Euclid's Elements read in the context of the times (known from multiple mathematical and non-mathematical sources), turn out to have more in common with modern constructivism and intuitionism than with Hilbert's axiomatic formalization. See the historical part of Friedman's Kant's Theory of Geometry and section 3 of Acerbi's Euclid's Pseudaria.

• Thank you; this is an intriguing perspective. I've read bits of Burkert and a little more of Fowler. I was already impressed, and because of what you write I think I'll read more. But, what you seem to say is: there isn't any real debate about these issues anymore; historians of science have all grown up now; and all have embraced the new methods. This then is bad news for "realists" such as me, who are inclined to credit arguments like "while the Greeks phrased their arguments geometrically, they meant them algebraically". Even a modern historian such Wilbur Knorr acknowledges as much. – RP_ Sep 29 '15 at 22:43
• I think it is fair to say that historians mostly embraced the new view, but what people meant is a philosophical question, it will always be open to interpretation. Rovelli writes "History of science may have two distinct objectives. The first is to reconstruct the historical complexity of an author or a period. The second is to understand how we got to know what we know... As a scientist of today, I respect the historians working within the first perspective (without which there would be no history at all), but I regret a trend that undervalues the second". arxiv.org/abs/1312.4057 – Conifold Sep 30 '15 at 1:52
• Blasjo even says "alleged superiority of modern historiographical standards ultimately rests on a dubious redefinition of the purpose of history". On Knorr see Unguru's (mostly positive) review jstor.org/stable/230094?seq=1#page_scan_tab_contents except "W.K. opts for the preservation of the concept "geometric algebra" as a useful interpretive term... seems to confuse and conflate anachronistically arithmetic and algebra... Ahistorical asides of this kind undermine W.K.'s own methodological approach (pp. 5-14) and diminish its effectiveness", etc. – Conifold Sep 30 '15 at 1:56
• @René I agree that Knorr's compromise on interpretational debate is promising. Geometric algebra may not be a good example for it because the work of Mueller, Berggren and Fowler removed Waerden's main factual arguments: that book II is used "algebraically" by Euclid, that there was Babylonian transmission, and that it was not useful geometrically. Knorr wrote "while I shall continue to use the term geometric algebra I will never mean by it anything more than Book II type geometry", but he does see Greek diagrams as vehicles for structural relations, like modern symbols, so keeps the term. – Conifold Oct 1 '15 at 21:28
• @René You may also like Netz's position on historical dynamics of mathematical ideas, which is in the spirit of Knorr's compromise. katz recently asked a question about him, and his example shows that at least in the more general sense the debate is still very much alive hsm.stackexchange.com/questions/3349/… – Conifold Jan 21 '16 at 18:58
1. Historians of mathematics currently are trying, IMHO, to be overly exact; I've seen some of them criticize the use of the word "triangle" in connection with Babylonian mathematics since Babylonians did not have the concept of angle. To me, this is ridiculous, not only because in German triangle is Dreieck (3-gon). I do regret the current trade of making history of mathematics a playground for four of five specialists.

2. As for the geometric algebra debate: the original idea was that there was some sort of algebra before the Greeks, i.e. Euclid, translated it into geometry. This position, as far as I can see, is dead. I still like to think of Euclid II as geometric algebra because these results are used by Euclid not in geometric contexts, but in algebraic ones (e.g. in the construction of Pythagorean triples).

3. Let me add something I just have seen in an article by Jens Hoyrup (Les Lais: or, or, What Ever Became of Mesopotamian Mathematics?):

"In the main, Neugebauer was thus fully right; but what went into Greek "geometrical algebra" was not the sophisticated scribal algebra of the Old Babylonian era but the anonymous riddle tradition of Mesopotamian origin."

• Thank you for your reply. I do sympathize with your point 1. On the other hand, I do think that the "postmodern" historiographers raise interesting epistemological concerns (how far should we go in identifying ancient and modern concepts, what role did ancient mathematics play within the wider whole of ancient thought, etc.). But I guess I'm too much of a Wittgensteinian (i.e. I think that philosophical questions need to be cured or dissolved rather than answered) to like the answers they've come up with. – RP_ Oct 1 '15 at 19:49

You wrote "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." You mentioned geometric algebra as an example, but did not limit your question to this context.

The received views in current historiography tend to (1) accept dogmas that have been developed when the state of mathematics was very different from what it is today, and strongly affected by what mathematicians believed then; and (2) be suspicious of relying on procedures in modern mathematics to shed light on procedures of historical mathematics. In this way, received historiography has developed its own set of what you refer to as "Whiggist" (or to be more polite, presentist) positions.

Aside from the issue of geometrical algebra, we have challenged such received views on the history of analysis in a number of recent publications, including Is mathematical history written by the victors? and some follow-up articles.

• That looks like a very interesting article, but could you say a little more on how this fits in with my question? – RP_ Jan 21 '16 at 1:18
• @René, thanks for your comment. I tried to respond in the body of my answer. – Mikhail Katz Jan 21 '16 at 8:06