A new paper on “The concept of given in Greek mathematics” by Nathan Sidoli contains some remarks pertaining to the issue of “geometrical algebra”. Ostensibly, Sidoli challenges what I wrote on this subject. But in fact I believe our views are not far apart.

The geometrical algebra issue, at bottom, stems from this conundrum: How different are the Greeks from us? Can we use our “mathematician’s empathy” to make sense of their work, and view differences in style and expression as superficial? Or is this the cardinal error of historical scholarship? In order to understand past thinkers, does sound historiography require us to put our own conceptions aside and start from the assumption that the Greeks inhabit a conceptual universe completely different than our own? When the Greeks do things that feel like algebra to us, are we feeling the underlying gist of their thinking, or are we merely seeing a reflection of our own ways of thinking that were not in the sources until we projected them there?

In my paper I argued for a fundamental unity rather than a fundamental chasm between ancient and modern thought. Specifically I defended these two theses:

GA1. The Greeks possessed a mode of reasoning analogous to our algebra, in the sense of a standardised and abstract way of dealing with the kinds of relations we would express using high school algebra. By and large, whenever we find it natural to interpret Greek mathematics in algebraic terms, the Greeks were capable of a functionally equivalent line of reasoning. If an algebraic interpretation of a Greek mathematical work suggests to us certain connections, strategies of proof, etc., then the Greeks could reach the same insights in a similarly routine fashion. This was an abstract, quantitative-relational mode of thought that was not confined to concrete geometrical configurations and not dependent on geometrical visualisation or formulation; in particular, it was obvious to the Greeks that the exact same kind of reasoning could just as well be applied to numerical relations as geometrical ones.

GA2. The Greeks were well aware of methods for solving quadratic problems (such as those exhibited in the Babylonian tradition). Books II and VI of the Elements contain propositions intended as a formalisation of the theoretical foundations of such methods.

In perfect agreement with these theses, Sidoli observes that many mathematicians from late antiquity onwards indeed switched back and forth between numerical and geometric perspectives, treating them as trivially equivalent:

“At least by the classical Islamicate period, and probably from much earlier, this blending of the geometrical and arithmetical readings of geometrical books of Euclidean works was commonplace.” Heron and Ptolemy did things of this nature and “there is no indication that they thought of their use of metrical analysis as in any way new, or innovative.” (§5.3)

Nevertheless Sidoli suspects that this was a later appropriation of the Euclidean material and not Euclid’s original intention:

“On balance I think [the geometrical algebra reading of Euclid] is less likely than the claim the Data was originally composed to address the needs of geometrical problem-solving and was then later repurposed as a means to justify and generalize metrical arguments.” (§6)

Sidoli’s arguments for this are as follows:

1. The propositions in Euclid that can be read in a GA way have a geometrical character that does not readily match up step-by-step with the arithmetical-algorithmic procedures the are, according to the GA interpretation, supposed to formalise. E.g.: “The only way this theorem could have any use to us in metrical, or arithmetic, problem-solving, is if we already knew, through independent considerations, what sorts of arithmetic operations to carry out, but were interested in an unrelated geometric approach as a purely theoretical justification of these operations.” (§3.4)

My reply: Indeed, which is why, in GA2, I spoke of “a formalisation of the theoretical foundations of such methods,” as opposed to a practical recipe version of them. Similarly, no one would use Elements Definition V.5 to compare magnitudes or Proposition I.2 to transfer lengths in everyday situations. Of course Euclid’s treatment is a highly refined formal system, far removed from giving practical recipes and very much concerned with investigating subtleties involved in founding all of mathematics on a minimalistic set of principles. It is indeed a matter of “a purely theoretical justification” just as Sidoli says. This is not in conflict with the GA theses.

2. The Data is not a systematic exposition of GA-type material. “At the very least, we must accept that if, in fact, Euclid had devised [the Data] in order to justify arithmetic procedures, he did so in a rather disorganized way.” (§6)

My reply: The Data is not a systematic exposition of anything. It’s a weird hodgepodge of theorems by anyone’s reckoning. No one, as far as I am aware, has ever made sense of the Data as a cogent treatise leading in a coherent way to any kind of natural goals, or given a good account of why Euclid selected the particular propositions he did. So the fact that the Data is not a perfect fit for the GA hypothesis is neither here nor there.

3. The correspondence between the arithmetical operations and their geometric equivalents is not systematically expounded by Euclid. Thus: “While Data 3 and 4 could be taken to justify adding and subtracting for general quantities, for multiplying and squaring, taking square roots and dividing, we must turn to Data 52, 55 and 57—but, as we saw, the proof of these theorems rely on the geometric construction of a square and a similar triangle.” (§6) “Proponents of [GA2] must contend with the fact that this highly geometrical, and frankly rather peculiar, proposition [Data 52] is the only candidate in the text for propositions demonstrating that the product of two given numbers is given.” (§3.4)

My reply: Euclid does after all establish the result, even if Sidoli thinks his manner is “peculiar,” so there is perhaps not much to discuss here. Especially not since the Data is an enigma on any reading whatsoever. But let us put those points aside for the sake of argument. Even so I do not think this proves much.

GA2 has to do with the fact that the very same extremely basic ruler-and-compass methods that suffice to build up all the geometry of the Elements are also sufficient to solve any quadratic equation and carry out various other manipulations of quadratic expressions. This is a sophisticated, nontrivial point regarding the scope of ruler-and-compass constructions. I do not see why this would need to be bundled, as Sidoli’s argument assumes it should be, with a pedantic foundational account of spelling out the elementary idea that products of numbers corresponds to areas of rectangles and so on. As Sidoli himself says: “there is no clear evidence that Greek mathematicians thought that the basic arithmetical operations needed to be justified, so there is no reason for us to believe that Euclid felt the need to engage in such a project” (§6). Indeed, so why could Euclid not simply have taken this for granted? I do not see how this speaks against the claim that Euclid was also interested in proving the nontrivial result that having a ruler and a compass means being able to solve any quadratic equation.

This does not contradict my point that Euclid’s concerns are foundational, because his theory remains internally foundational through and through. The intuitive association between arithmetical operations and geometrical constructs is not needed internally but only externally to identify the formal theory with other things. Such associations take place in a different arena of reasoning, not subject to the internal stringency of the theory. Just as in modern real analysis, say, mathematicians reason with the utmost care when developing the theory internally, but then switch to much more intuitive and informal notions when they apply it to differential equations arising from a physics problem. While the theory of real analysis is built up from set theory in a stringent, formal way, its association with its physical applications is taken for granted in an intuitive, common-sense way that is not incorporated in the formal theory itself. So also Euclid can sensibly do his formalised version of the theory of quadratic equations without having to specify internally within his system how this maps to “real world” ways of doing these things.

For all of these reasons I am not convinced that any conclusions regarding GA can be drawn from Euclid’s Data.

Beyond his main focus on the Data, Sidoli makes a few very brief general points on GA that are very interesting. Basically he argues “against” GA by saying that GA1 is so great that GA2 is not needed:

“GA1 is simply another way talking about the theory of the application of areas, which is not in any dispute.” (§3.4) “Almost no one would argue that it is not possible to make a reading of Elements [II] and VI as motivated by and justifying computational problem-solving. The question is rather whether such a reading [=GA2], or that through the theory of the application of areas [=GA1], is more broadly successful in explicating the ancient sources.” (§6)

This novel take is news to me. Traditionally, GA1 and GA2 have been considered naturally and closely associated. People have either accepted them both or rejected them both. Sidoli, instead, seems to construe them as mutually exclusive hypotheses in direct competition with one another.

I’m not sure what Sidoli means when he says “GA1 is not in any dispute.” Does he mean that he does not dispute it? Or that no one disputes it? The latter is certainly false. People absolutely do dispute GA1. They give arguments of the form “if the Greeks had truly been able to think algebraically [=GA1] then they would have done so-and-so differently than they did.” Such arguments are clearly a direct denial of GA1. The bulk of my GA paper is devoted to rebutting arguments (by Unguru, Mueller, Saito, Grattan-Guinness) of exactly this type.

So it is strange for me to see Sidoli claiming to disagree with me when he happily accepts GA1. In my eyes, if everyone accepts GA1 this is basically a win for GA, regardless of whether they also agree on GA2.