Michael N. Fried’s obituary of Sabetai Unguru has just appeared. There is no love lost between my historiographical convictions and theirs. Indeed, Fried reiterates as much in the obituary with a thinly veiled insult: Unguru’s works “evoke strong reactions and inspire probing historical research (see Blåsjö 2016 for the former and Corry 2013 for the latter).”

In the obituary, Fried summarises Unguru’s view as follows:

“Sabetai’s stubborn refusal to use modern symbolism or modern mathematical frameworks in interpreting the past … arose from an earnest and profound respect for thinkers of the past; they deserved a sympathetic, responsive and sensitive hearing. To use modern symbolism was not merely to translate a text but more pointedly to dismiss or ignore the ancient mathematicians’ own geometrical ways of thinking: it was in effect to put words in their mouths, to listen to oneself, not to them. The discipline of history, in his view, required listening deeply and making one’s best effort to understand the other.”

Of course I could not agree more with these goals, but Unguru’s ban on (modern) paraphrase is a terrible way of trying to achieve them. The dogma that absence of paraphrase means maximal fidelity to historical thought is a fallacy that is bound to prevent sympathetic historical understanding rather than facilitate it.

For example, consider how Grassmann proved the dimension theorem for linear subspaces. Grassmann proved this theorem twice: in §126 of the 1844 edition of his book *Die Ausdehnungslehre*, and again in §25 of the 1862 edition.

Grassmann’s 1862 proof would fit essentially word for word in a modern textbook in linear algebra. The modern reader can see right away that he is (in his own words) talking about spans, bases, linear independence, etc. exactly as we would today.

Grassmann’s 1844 proof looks wildly different. First of all it is almost purely verbal. Unlike in the 1862 proof, there is no notation for the basis vectors of the various subspaces, for example. But more fundamentally different still is the terminology, which is very strange and alien to the modern mind. There are no spaces or subspaces, but rather “magnitudes.” There are no intersections but “outer factors”; no spans but “outer products.” There are no linearly independent sets of vectors but rather “magnitudes” that have no common factors.

Consider how we would read Grassmann’s 1844 proof if we were to adopt Unguru’s principle (quoted by Fried in the obituary):

“[The historian’s] main effort should be in the direction of showing the extent to which past ideas were unlike modern ones, irrespective of the fact that they might (or might not) have led to the modern ideas. This is a wise methodological tack, since it enables the historian to avoid reductive anachronism while channeling his historical empathy toward an understanding of the past in its own right.”

This is not wise. It is not wise to assume that the reading that makes the text as alien to modern thought as possible is necessarily the same thing as the the reading that is maximally historically sensitive. This is a fallacious dogma—a preconceived notion that frequently directly forbids the historian from pursuing the interpretation actually intended by the historical author.

If we applied Unguru’s logic to Grassmann’s 1844 proof, we would conclude that the apparent resemblance to the modern version of the theorem was completely alien to Grassmann’s way of thinking. Grassmann evidently used a strongly arithmetical paradigm that didn’t have any of the standard conceptual machinery of modern linear algebra: a radical difference in expression that must reflect a radical difference in thought.

But, in fact, when Grassmann gives his 1862 proof, he adds the remark that this is the same theorem as that of his 1844 edition, and that the proof idea is the same in both cases.

So, according to Grassmann’s own testimony, it would have been correct to read the 1844 proof in a modernised manner. That would have been the reading that would have been faithful to the historical actor’s own ideas, as he explicitly says.

In other words, Unguru’s dogma would have actively prevented us from understanding the text as intended by the author. The dogmatic historian following in Unguru’s footsteps would by no means be maximally empathetic and sensitive to historical thought in its own right. Rather, such a historian would have presumptively ruled out the correct historical interpretation before even trying to comprehend the proof, based on a small-minded historiographical precept.

Let us consider a second example, from Fried & Unguru’s book *Apollonius of Perga’s Conica: Text, Context, Subtext*. On page 96 they discuss Serenus’s work on the section of a cylinder. What kind of curve results when you intersect a cylinder with a plane? Serenus found the “symptoma” of the section, that is to say, basically its (Cartesian) equation. In modern terms, we can see from the equation that the curve is an ellipse. But Fried & Unguru refuse to accept this, since that is an algebraic way of thinking that, according to them, is alien to Greek mathematics. Hence they claim that Serenus didn’t conclude from the equation, but only from a subsequent cone construction, that the curve was an ellipse:

“Serenus, significantly, includes a proposition, the very next one [Prop. 20], showing that ‘it is possible to exhibit a cone and a cylinder which are alike cut in one and the same ellipse’, as if to show that one can conclude only now that the section having the ellipse's symptoma is indeed an ellipse. Like Apollonius, then, Serenus thought that a curve having the symptoma of an ellipse still could not simply be called an ellipse until he could find a cone of which it was a section.”

But the exact opposite is the case. Serenus literally and explicitly does the exact opposite what Fried & Unguru say. He literally explicitly concludes from the symptoma that it is “evident” that the section is an ellipse (well before introducing the cone that Fried & Unguru are talking about).

Fried & Unguru arrived at their erroneous interpretation because they decided, as a matter of preconceived historiographical orthodoxy, that the “algebraic” interpretation must be impossible. Because of their blind adherence to this fallacious historiographical dogma, Fried & Unguru are committing precisely the sin they have sworn to avoid: namely, looking at history and seeing nothing but a reflection of their own preconceived notions.