Reply to Paolo Bussotti’s review of my Leibniz-Riemann paper

According to several leading Leibniz scholars, one of Leibniz’s unpublished manuscripts, written in his youth, contains a theorem establishing a perfectly rigorous foundation for the calculus, equivalent to Riemann sum integration theory. In a recent paper, I criticised this view and gave a much more modest interpretation of Leibniz’s supposed masterpiece (which for some mysterious reason he didn’t get around to publishing in the remaining 40 years of his life, even though it supposedly prefigured major ideas of 19th-century mathematics by 200 years).

One of the authors I challenged was Eberhard Knobloch (who, incidentally, was just awarded the most prestigious prize in the field). He replied with indignation in a letter to the editors, calling my paper “completely unacceptable.” I have submitted a reply which will hopefully appear soon (update: here it is).

In the meantime, a review of my paper just appeared in the Zentralblatt. The review is written by Paolo Bussotti, who spent three months as a guest researcher hosted by Knobloch in 2014. This so-called review is not really a review of my paper at all, but rather a regurgitation of Knobloch’s letter to the editors, which Bussotti follows slavishly.

Bussotti cites Knobloch’s letter parenthetically, but in no way indicates that “his” critique is in fact nothing but a point-by-point regurgitation of everything Knobloch said in his letter. Bussotti’s phraseology will lead readers to think that he is offering an independent judgement, when in reality he is parroting Knobloch’s letter. For instance, Bussotti writes:

> [Blåsjö’s] main theses can be summarized in two items: 1) … 2) … let us start from what I have indicated as item 2)

This summary of my view in terms of these two theses is due to Knobloch, who even explicitly labelled them (1) and (2). But those who do not have Knobloch’s letter in front of them will surely be mislead by this kind of phrasing into believing that Bussotti has carried out his own independent analysis, rather than simply transcribed almost literally the exact view of his friend, who is one of the parties in the conflict.

Although Bussotti obediently follows Knobloch on every single point of substance, he does manage to introduce some absurd misunderstandings of his own. For instance, he writes:

> The whole question turns around the interpretation of the sentence translated by Knobloch as “It serves, however, to lay the foundations of the whole method of indivisibles in the soundest way possible” and by Blåsjö as “Whence it will be permissible to use the method of indivisibles proceeding by spaces formed by steps or by sums of ordinates as strictly demonstrated”. The two translations are not significantly different and the whole question concerns the interpretation of that “it”.

It is difficult to fathom how Bussotti could have gotten it into his head that these two quotes are “two translations” of the same passage and even that they are “not significantly different.” They are of course completely different quotes and obviously do not refer to the same passage in Leibniz. The relevant quote in my paper is on a different page altogether (137), with a translation that follows Knobloch virtually verbatim.

In any case, the notion that “the whole question” comes down to this one sentence (as Bussotti claims twice) is absurd. It does, however, square well with Knobloch’s letter, which opens with a critique of my reading of this passage.

As for the substantive point at stake, it concerns whether the “it” in question refers to Proposition 6 (as the standard view has it) or to the idea of its proof (as I claim). Bussotti regurgitates (without saying so) Knobloch’s argument that it must be the former, for reasons of Latin grammar. I do not deny that the “it” is Proposition 6 grammatically speaking. But this proves nothing. As seen in my paper, in the very same passage Leibniz uses the very same “it” as follows: “In it, it is demonstrated in fastidious detail that …” Thus Leibniz is obviously using “it” (i.e., “Proposition 6”) quite loosely as a way of referring to the whole passage of text (somewhat like a chapter heading, say), rather than to the propositional statement per se (which is what Knobloch’s interpretation needs). Thus my interpretation is not at all inconsistent with the text.

Knobloch also raised a quibble about whether Leibniz’s proposition should be called a foundation of infinitesimal geometry or of infinitesimal calculus. Knobloch tried to allege that he spoke only of the former and that the latter is an anachronistic misnomer introduced by me. Bussotti duly parrots the same point:

> infinitesimal geometry [is] partially different from infinitesimal calculus, a difference which [Blåsjö] seems, at best, to underestimate, as he uses indifferently both expressions.

But the insinuation that I somehow introduced this false equivocation is absurd. The notion that Leibniz’s proposition provides a foundation for the calculus is clearly and explicitly present in the works I criticise. In fact, later, when it suits his purposes, Bussotti himself goes on to reaffirm exactly this:

> Leibniz’s proposition 6 offers a general foundation to integral calculus …, no doubt about this.

Why, then, is he bitching that I spoke of calculus instead of infinitesimal geometry, if he himself uses the same terminology and thinks there is “no doubt” that it is accurate? Bussotti’s critique is not even coherent, let alone sound.

Bussotti’s review ends with an accurate and revealing observation:

> The approach of [Blåsjö] does not seem favourable to edify new and collaborative researches in the line traced by Knobloch and by the other scholars who have studied Leibniz [for] many years. … My conviction is that new insights as to the concept of rigour in Leibniz can be achieved taking into account that the general picture traced by these authors is basically correct.

This seems to me an accurate description of a kind of implicit axiom of modern historiography, namely that scholarship should be collaborative rather than critical. You should pat your friends on the back, not question them. This may be a sound policy if we want academia to be a feel-good social club. But as a recipe for intellectual progress I think it is fundamentally misconceived.