How did Leibniz view the foundations of infinitesimals? A new paper by Rabouin & Arthur gives one answer. Among other things, they address some critiques I raised. My reflections on their new paper are as follows.

First some context. There were a range of options available for justifying infinitesimal methods in the 17th century. One could argue that:

1. Infinitesimals can be completely avoided in the manner of the method of exhaustion of the Greeks, which is functionally equivalent to infinitesimal reasoning yet is impeccably rigorous since it is based entirely on finitistic reasoning. Anytime one uses infinitesimal language, this is to be understood as a mere shorthand that could always in principle be translated into a proof “in the manner of the Ancients” if needed.

2. Infinitesimals can be thought of as very small numbers (such as dx=0.00001), in which case infinitesimal results are strictly speaking only approximations, but approximations that can be made arbitrarily accurate (that is to say, the error can be made less than any assignable magnitude) by making the dx smaller and smaller.

3. Infinitesimals can be thought of as new entities that enlarge the universe of real numbers, analogous to complex numbers or points at infinity in projective geometry.

(4. More exotically: Nature is inherently infinitesimal. The calculus does not approximate curves by polygons; rather, all curves genuinely are actually straight on the micro level.)

(5. Perhaps not an independent strategy but a shorthand manner of speaking that ultimately reduces to one of the above: Infinitesimals are “useful fictions” — heuristic devices that work regardless of their existential or foundational status.)

Leibniz often alluded to these ways of justifying infinitesimals. He seems to have been quite happy to take a pluralistic approach: not only is each of these ways of justifying infinitesimals quite convincing separately, but the sheer multitude of plausible approaches in itself adds further credibility.

Indeed, it is not necessary for the working mathematician to insist on one of these approaches and exclude the others; on the contrary, the flexibility afforded by the multiplicity of lenses is creatively useful. It is “the business of the metaphysician,” as Leibniz says, to worry about which is the “real” or “true” foundation. Since this is an issue of little consequence to actual mathematical practice, it is wiser for the mathematician to remain neutral and agnostic and not waste time with it.

Many things Leibniz says fits this picture well. But Rabouin & Arthur argue against it. The say that, on the contrary, Leibniz had “very definite views” on what the right foundations of infinitesimals were, “from which he never wavered.” According to this interpretation, the pluralistic tendencies of Leibniz’s public statements on the foundations of infinitesimals are to be seen more as argumentative strategies adopted to the context and audience than genuine expressions of Leibniz’s own views.

For my part, I think it makes a lot of sense to take Leibniz’s pluralistic statements at face value. Moreover, I don’t think the difference between this and the view of Rabouin & Arthur amounts to all that much, for many purposes. The fact remains, any way you slice it, that Leibniz argued skilfully for the pluralistic approach in many writings, especially in direct reply to criticism of the foundations of the calculus. This effectively makes him a pluralist for most intents and purposes, I would say. I find this more interesting and important for the development of mathematics than esoteric debates about what Leibniz allegedly “really” thought privately, based on unpublished manuscripts.

In any case, that is the general background. Now let’s look at the specific points of contention between my paper and that of Rabouin & Arthur.

Some scholars (including Rabouin and Arthur in previous publications) have claimed 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. That’s what I argued in my paper mentioned above.

Against this, 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.

I say: No, that’s not what those documents show. 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 of the treatise — “curious particulars” indeed, rather than unprecedented foundational material. The allegedly innovative foundational parts are not partially published and abstracted in the Actes but the geometrical results are, such as the series for pi/4. Nor do the foundational parts have any evident connection to Descartes. So foundations of infinitesimals 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 explains why he never did such things again, and why he would only use it to give to those who were 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. They could have made it a lot clearer that 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 obviously says absolutely nothing about the alleged significance of the DQA as foundational for 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.

In sum, I do not see any compelling evidence that Leibniz thought of the DQA as foundationally important.

In my paper I also argued the same point internalistically: the mathematical details of the DQA are nothing special as far as foundations are concerned, and certainly not as remarkably rigorous and general as has been claimed.

Rabouin & Arthur claim to address this: in their introduction they promise to “show how Leibniz’s method in the DQA builds on and improves upon the extant method of indivisibles.” But in fact they say hardly anything about this. The only explicit claims about how Leibniz allegedly goes beyond his predecessors in this way that I could find are a few sentences around notes 30-32, 39-41. I very much doubt that the rather incidental things pointed to there were either new or significant. In any case, one could certainly not conclude that they were from these vague allusions.

A high burden of proof falls on those who make extravagant claims about the allegedly profound and proto-modern insights in the unpublished DQA. The new paper by Rabouin & Arthur is interesting and useful in many respects, but it certainly falls short of meeting that burden.