Reply to Pambuccian on Greek constructivism

A strangely petty review by Victor Pambuccian of my article on Greek operationalism has appeared in zbMATH. Its supposed critiques are as follows.

“Although the reader is told that the concept [of operationalism] will be defined in 2.2.1, one doesn’t find a definition there (except for a quotation from P. W. Bridgman [The logic of modern physics. New York: Macmillan (1927)], which is supposed to explain the ‘core principle of operationalism’), and needs to read the entire paper for bits and pieces about what operationalism is.”

In 2.2.1 I cite and discuss in some detail the definition of operationalism given by Nobel Prize-winning Harvard physicist Percy Bridgman. And later I flesh out this definition by showing how it plays out in many concrete examples. What’s wrong with explaining important concepts both in general terms and through examples? Isn’t it a good thing that the key concept of my interpretative lens is consistently applied and developed throughout my argument? Why the reviewer thinks this is something to be snarky about I do not understand.

“The reviewer found several shortcomings. There is … an absence of a genuine dialogue with the views of other authors in the history and philosophy pf [sic] Greek mathematics.”

Ironic that the reviewer styles himself a champion of “genuine dialogue” while writing a bitter and whiny review that ignores 99% of what I wrote.

“The author asks various questions that appear to make other views incoherent, but that cannot be called a genuine dialogue. … Those the author disagrees with are: W. R. Knorr, S. Menn, K. Saito and N. Sidoli, W. Burkert, F. Acerbi, and S. Cuomo.”

In other words, I engage directly with the main experts in the field, and I explain specifically why I disagree with their views, and I pose specific challenges to those maintaining such views. Which is obviously a dialogue, as well as normal scholarly practice and a good recipe for critically assessing competing historical interpretations. But somehow this is bad because it’s not “genuine” enough? What does that even mean?

“Most problematic is a dismissal of a purported standard Platonic interpretation of ancient Greek geometry, without a clear exposition of what such a view would be.”

I do not understand what part of my argument the reviewer thinks he is disagreeing with. Occasionally I refer to various standard views, but I do not merely “purport” that these are standard; rather, I give exact citations to leading scholars unequivocally expressing those views. For instance, I say that “a standard view is that Euclid studiously avoided the use of superposition whenever he could”, for which I give the references Heath, 1956, I.249, I.225; Knorr, 1978, 161; Mancosu, 1996, 29; Mueller 1981, 22.

My article is not about Plato or Platonism. I do occasionally refer to Platonic views in phrases such as: “Plato’s opinion was reportedly that mathematicians who ‘descended to the things of sense’ were ‘corrupters and destroyers of the pure excellence of geometry’ (Plutarch)”; “Platonic emphasis on ‘pure thought’ and detachment from physicality”; “the Platonic belief that the objects of mathematics are eternal and independent of human actions situated in space and time”. Seems straightforward to me. Why is this not “clear” enough for the reviewer?

“The reader is under the impression that Plato presented some sort of otherworldly view of geometry, and had a connection to the mathematics of his day similar to that of Spinoza, Hegel, or Wittgenstein. This is certainly not the standard view.”

Of course the idea that Plato viewed geometry as “otherworldly” is completely standard. Although my article is not about Plato, I do refer to this view in passing. For instance I cite Jesseph, 2015, 205, who wrote: “Dimensionless points, breadthless lines, and depthless surfaces of Euclidean geometry were not traditionally taken to be the sort of thing one might encounter while walking down the street. Whether such items were characterized as Platonic objects inhabiting a separate realm of geometric forms, or as abstractions arising from experience, it was generally agreed that the objects of geometry and the space in which they are located could not be identified with material objects or the space of everyday experience.”

Although, again, my paper is not about Plato, it is true that I suggest in passing (in order to explain why my paper is NOT about Plato) that perhaps Plato might have “had a connection to the mathematics of his day similar to that of Spinoza, Hegel, or Wittgenstein”. What is the reviewer’s objection? That this is not standard? Of course I never claimed that this was a standard view.

“Already A. D. Steele had clarified Plato’s position with respect to drawing instruments and his displeasure not with motion or mechanics, but with the abandoning of conceptual geometry.”

Again, my paper is not about Plato. My paper argues that Greek geometers embraced instruments, motion, and mechanics. If Plato did so too, great. That only strengthens my thesis.

My paper is not making any argument about what Plato’s opinion was. I cite some famous passages in which ideas that are blatantly anti-instruments/motion/mechanics are associated with Plato. I then give counterarguments to the views expressed in those passages. Whether these passages are truly representative of Plato’s opinion is immaterial to my argument.

“That he never defended what passes today as Platonism in mathematics was explained at length in P. Pritchard, as well as earlier, by A. M. Frenkian.”

Fine with me. Nowhere in my paper do I claim that Plato “defended what passes today as Platonism”. Why is the reviewer bringing this up as if it were a “shortcoming” of my article? That makes no sense.

“Portraying Plato as a matematical [sic] simpleton also runs counter to the conclusion reached by C. Lattmann, that it was Plato who was responsible for the transformation of Greek mathematics into the abstract, general, and deductive form encountered in Euclid’s Elements.”

This is a wildly speculative view that is not shared by any established historian of Greek geometry as far as I am aware. Why is the mere fact that one person believes this brought up in a paragraph of “shortcomings” of my article (which, again, is not even remotely an article about Plato, even though that is evidently the only hobby horse the reviewer cares about)?

“C. Lattmann [Mathematische Modellierung bei Platon zwischen Thales und Euklid. Science, Technology, and Medicine in Ancient Cultures 9. Berlin: De Gruyter (2019)] (a title not found among the references, although it deals at length with the duplication of the square)”

Indeed, my article, which was submitted in 2020 and published online at the journal website in April 2021, does not cite this obscure monograph in German which was published in 2019. The reviewer is in all seriousness bringing this up as evidence of “shortcomings” of my article. I do not understand what possessed the reviewer to think that such malicious pettiness was appropriate for a zbMATH review.

“The reviewer never thought that ‘modern mathematicians can hardly wrap their heads around how [angle trisection or cube duplication] could even be considered a research problem at all’. … The thought that the obsession with geometric construction problems would be incomprehensible to a modern mathematician is not something the reviewer has encountered before in conversation or in writing, nor has he ever found it in the least surprising or in need of justification in the context of a general, exact, deductive enterprise of the kind Greek geometry is.”

Greek geometers devised at least 13 different cube duplications and at least 7 different angle trisections. Name any other research problem that they solved in anywhere near as many different ways. There are none. The obsession with these problems is unique and singular.

For example, in surviving sources, Greek mathematicians give exactly one construction of an ellipse through five given points, and exactly one proof of the isoperimetric property of the circle. Why not 13 times each? Indeed, subsequent history shows that these problems can very fruitfully be tackled in a multitude of interesting ways that were readily within the reach of Greek mathematicians. But instead they preferred to double a cube for the 13th time. And so it goes throughout their entire tradition.

Why were the three classical construction problems virtually the ONLY problems the Greeks were obsessed with proving over and over and over again? Why was a 13th way of doubling a cube a higher priority than a second way of drawing an ellipse through five points, let alone a first way of doing something new, such as, say, drawing a tangent to a cycloid? Because Greek geometry was a “general, exact, deductive enterprise”? That is a nonsensical answer that does not understand the question.

Or to put it another way: Show me a modern geometry textbook that treats cube duplication as an important problem for reasons other historical interest. There is no such book, even though modern geometry is no less a “general, exact, deductive enterprise” than Greek geometry. So obviously the mere preoccupation with “general, exact, deductive” geometry is not enough to explain interest in cube duplication.