Comment on Rabouin & Arthur (2020)
Some scholars claim that Leibniz had a very sophisticated understanding of the foundations of infinitesimal methods that was far ahead of his time. But he didn’t actually publish these ideas. On the contrary, in public communication he was often laid back about the foundations of the calculus. He seemed uninterested in addressing the issue, and only made some rather laconic remarks about it when pressed by others. Nevertheless, some scholars maintain that the unpublished De quadratura arithmetica (DQA), written in his youth, a decade before his first calculus publication, contains these brilliant foundational insights.
I disagree. In my view, as far as the foundations of infinitesimals is concerned, the DQA is quite unremarkable and basically just rehashes ideas that were commonplace among leading mathematicians at the time. I also believe that this was Leibniz’s own view. I argued this in a 2017 paper.
A new paper by key advocates of the opposite view has just appeared. The authors, Rabouin & Arthur claim that:
> there are in fact many documents in which Leibniz refers to the DQA, most of the time very explicitly, as the place to go to find a justification for the use of infinitesimals.
No. That’s not what those documents say. Leibniz references the DQA but hardly any of these references even concern the justification of infinitesimals at all, let alone explicitly say that the DQA is “the [!] place to go” for such justifications.
Let’s go through them all one at a time.
> Gerhardt published a Compendium of the DQA, which Leibniz prepared for publication.
In this Compendium, Leibniz explicates at length the specific geometrical results of the treatise, and gives very short shrift to the allegedly so insightful parts on the foundations of infinitesimal methods.
> Moreover, Leibniz certainly sought to publish the treatise itself. In an exchange of letters in 1682, he discusses the project of publishing the DQA.
1682 is before any calculus publication, and hence irrelevant for our purposes.
> More interestingly, though, ten years later Leibniz reopened the possibility of publishing the DQA. In a letter …, he wrote: “… the distractions that I then had did not permit me to lay it out in full, and I contented myself by giving certain abstracts in the Actes of Leipzig. … One could add a preamble containing some curious particulars on what Mons. Descartes invented or took from elsewhere.”
Again it seems that Leibniz does not have in mind the foundational parts, but the particular geometrical results. That is what he published in the Actes, and that is what is related to what Descartes said. Foundations of infinitesimals does not fit these descriptions, so that does not appear to be the aspect of the DQA that Leibniz has in mind.
> Bodenhausen signalled to Leibniz that it would be very useful to have at one’s disposal a gentle introduction to shut the mouths of the Euclidean “Pied Pipers” (Rattenfänger), who were hostile to the new method [i.e., the calculus]. Leibniz responded positively to the demand and sent … a presentation of the calculus for those who were trained in the “manner of the Ancients”. And what did he provide on this occasion? A presentation of Prop. 6 of the DQA accompanied by a translation into the differential calculus corresponding to Prop. 8. To be sure, all of these results are at the time superseded by the many researches in which Leibniz had been engaged since then, and he does warn his correspondent that the results from the DQA are almost immediate with the new calculus. But precisely, it is all the more striking that when coming to a translation of this calculus into the language of the Ancients, the only example he has to provide in 1690 is still prop. 6 of the DQA.
None of this contradicts my interpretation. This is all consistent with Leibniz regarding the DQA as a tedious explication of standard material as far as foundations is concerned. This explain why he never did such things again, and why he would only use it to give to idiots who are completely out of touch with the mathematics of the time.
> When Leibniz is pressed to explain his method of quadrature to someone having difficulty with it in 1695, the “most elegant” way he can conceive of demonstrating it is not in terms of the more powerful methods he has developed since his youth, but by reference to the very presentation in the DQA which those methods had, according to Jesseph and Blåsjö, rendered inadequate and obsolete.
I think Rabouin & Arthur are quite deceptive here. Note well: Leibniz’s “method of quadrature” here means his method of quadrature of the circle, i.e., his series for pi/4. That is to say, what is at stake is one particular result, not the general method of quadrature employed in the calculus. The fact that certain very specific results can be proved “elegantly” by DQA methods says absolutely nothing about the alleged significance of the DQA as foundational for the infinitesimal methods overall.
> Moreover, contrary to Blåsjö’s claim that he never quoted any of its results in foundational discussions, we have … also what Leibniz wrote to Johann Bernoulli in 1698.
Here indeed Leibniz mentions that a specific and not particularly important point Bernoulli made is similar to one he made in an obscure part of DQA. This obviously has nothing to do with Leibniz claiming that the DQA was anything like “the place to go to find a justification for the use of infinitesimals” in any way, shape, or form. On the contrary, Rabouin & Arthur themselves quote Leibniz as immediately saying in the same letter that “it is always the case that what is concluded by means of the infinite and infinitely small can be evinced by a reductio ad absurdum by my method of incomparables (the Lemmas for which I gave in the Acta)” — in other words, Leibniz is referring to his published works for the foundations of his calculus, with no indication whatsoever that the earlier DQA that he just mentioned has a more profound treatment of those very issues.