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On what has been called Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums. (English) Zbl 1370.01009

This paper concerns the sixth proposition of Leibniz’s manuscript De quadratura arithmetica circuli, ellipseos et hyperbolae, dating to 1676. The today most accredited edition of this work is that published by Eberhard Knobloch in 1993 [Zbl 0919.01016]. De quadratura arithmetica consists of 51 propositions – many of them having numerous corollaries – where Leibniz offers the integration of several curves (among them, i.e., the logarithmic curve). Proposition 6 sounds like this: “Theorem 6 is most thorny. Therein it is overly carefully demonstrated that the procedure of constructing certain rectilinear step spaces (spatia rectilinea gradiformia) and in equal fashion polygons can be continued to such a degree that they differ from each other or from curves by a quantity which is smaller than any given quantity […]. It serves, however, (servit tamen) to lay the foundations of the whole method of indivisibles in the soundest way possible (firmissime).” [E. Knobloch, Hist. Math. 44, No. 3, 280–282 (2017; Zbl 1372.01012), pp. 280–282]. E. Knobloch [Synthese 133, No. 1–2, 59–73 (2002; Zbl 1032.01011)] claimed that Proposition 6 offers a general and rigorous foundation to integrate a conspicuous series of curves by means of what, nowadays, we call Riemannian sums. Other scholars as Levey, Arthur and Rabouin agree with him. In synthesis, this was the situation before the paper under review.
The author offers a completely different interpretation of Proposition 6. His main theses can be summarized in two items: 1) Proposition 6 is not a rigorous foundation to integrate a large class of curves; 2) Leibniz did not think it was. Rather, he considered this proposition as a sort of specific lemma to use in the following Proposition 7: nothing more than a particular theorem proved in the tradition of the indivisibles geometry and, after all, nothing more than an application of the method of exhaustion (p. 137). In substance, the author criticizes the idea that, by means of Proposition 6, Leibniz offered most of the conditions, fulfilling which, a curved area can be approximated by means of a series of infinitesimal rectilinear areas with a negligible error. Most of the author’s assertions as well as his general theses seem unsound: let us start from what I have indicated as item 2): Leibniz’s conviction to have rigorously founded the integral calculus within infinitesimal geometry (something partially different from infinitesimal calculus, a difference which the author seems, at best, to underestimate, as he uses indifferently both expressions). This is, so to say, the subjective perception Leibniz had of his own work. The whole question turns around the interpretation of the sentence translated by Knobloch as “It serves, however, (servit tamen) to lay the foundations of the whole method of indivisibles in the soundest way possible (firmissime)” [Zbl 0919.01016, p. 24] and by the author 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” (p. 136). The two translations are not significantly different and the whole question concerns the interpretation of that “It” I have written in italics. The author interprets it as “the strategy of proving that the approximation involved can be made arbitrarily good” (p. 137). However, there are neither contextual nor grammatical reasons to offer such an interpretation. It is a clear over-reading of Leibniz’s words, which seems to be based on an a-priori conviction of the author rather than on textual evidences. If we interpret – as all the other scholars mentioned by the author do and as it seems compulsory given the context and the Latin grammar – “it” as “Proposition 6”, no doubt is possible on the importance ascribed by Leibniz to this proposition in regard to the foundation of areas calculus within infinitesimal geometry. The author (pp. 136–138) refers to few writings, where Leibniz seems to underestimate his entire De quadratura arithmetica and, specifically, Proposition 6. However, these very few references are referred to a late period of Leibniz’s activity, when he had improved his methods and elaborated new and quicker techniques to express – anyway – concepts, great parts of which were already present in De quadratura arithmetica. Only in this sense, Leibniz conceived a critic to himself. On the other hand, the long references to the De quadratura arithmetica inserted by Leibniz in his Historia et origo calculi differentialis [in: C. I. Gerhardt (ed.), Mathematische Schriften. Band 5 (1855); reprint: Hildesheim: Georg Olms Verlag (1971; Zbl 1372.01097), pp. 401–404] leave no doubt on the importance he ascribed to his early work on the quadratures. Let us now see what can be called “the objective question”: did Leibniz offer a foundation to the integral calculus? On p. 137 of his paper, the author claims: “It makes no sense to speak of a ‘proof’ establishing the rigour of the method of exhaustion in general. It only makes sense to speak of individual instances of his method devised for individual propositions.” This is perfectly true. It is enough to read the ingenious works by Archimedes: the exhaustion is a general, non-heuristic method of proof, to apply which a new idea is necessary for any single proposition because it is not possible to establish a priori the conditions under which the method is applicable in general. However, after his analyses of Propositions 6 and 7 by Leibniz (Sections 2 and 4, pp. 139–141 and 142–144, where, by the way, a debatable relation between the curves \(f(x)\) and \(d(x)\) is established), the author himself points out (Section 5, pp. 144–146) a series of general conditions introduced by Leibniz for a curve to be integrable. These conditions are rather specific, albeit many of them are formulated implicitly by Leibniz. Therefore, Leibniz’s Proposition 6 offers a general foundation to integral calculus and it is something completely different as well as from the Greek method of exhaustion and Cavalieri’s indivisibles method, no doubt about this. Of course, the question if this foundation is absolutely rigorous makes sense and new researches in this direction might be an interesting contribution to the history of mathematics. Nonetheless, the author’s approach 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. Therefore, 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. Important particulars can be discussed.

MSC:

01A45 History of mathematics in the 17th century

Biographic References:

Leibniz, Gottfried Wilhelm
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References:

[1] Arthur, Richard, Leery bedfellows: Newton and Leibniz on the status of infinitesimals, (Goldenbaum, Ursula; Jesseph, Douglas, Infinitesimal Differences: Controversies between Leibniz and his Contemporaries (2008), de Gruyter), 7-30
[2] Child, J. M., The Early Mathematical Manuscripts of Leibniz (1920), Open Court Publishing: Open Court Publishing Chicago and London · JFM 47.0035.09
[3] Hofmann, J. E., Die Entwicklungsgeschichte der Leibnizschen Mathematik während des Aufenthaltes in Paris (1672-1676) (1949), Leibniz Verlag: Leibniz Verlag München · Zbl 0032.19303
[4] Knobloch, Eberhard, Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums, Synthese, 133, 59-73 (2002) · Zbl 1032.01011
[5] Leibniz, Gottfried Wilhelm, Mathematische Schriften. Band II (1855), Verlag von H.W. Schmidt: Verlag von H.W. Schmidt Berlin, Edited by C.I. Gerhardt · Zbl 1372.01098
[6] Leibniz, Gottfried Wilhelm, De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis, kritisch herausgegeben und kommentiert von Eberhard Knobloch, Göttingen, 1993, Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-physikalische Klasse, 3, 43 (1993) · Zbl 0919.01016
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[8] Leibniz, Gottfried Wilhelm, Sämtliche Schriften und Briefe. Reihe VII: Mathematische Schriften. Band 6: Arithmetische Kreisquadratur (2012), Academie Verlag: Academie Verlag Hannover · Zbl 1326.01072
[9] Leibniz, Gottfried Wilhelm, De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis. Edition of the Latin text by Knobloch (2016), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, German translation by Otto Hamborg · Zbl 1353.01022
[10] Levey, Samuel, Archimedes, infinitesimals, and the law of continuity: on Leibniz’s fictionalism, (Goldenbaum, Ursula; Jesseph, Douglas, Infinitesimal Differences: Controversies between Leibniz and his Contemporaries (2008), de Gruyter), 107-133
[11] Rabouin, David, Leibniz’s rigorous foundations of the method of indivisibles, (Jullien, Vincent, Seventeenth-Century Indivisibles Revisited (2015), Birkhäuser), 347-364 · Zbl 1326.01030
[12] Scholtz, Lucie, Die exakte Grundlegung der Infinitesimalrechnung bei Leibniz (1933), Offprint by Verlagsanstalt Hans Kretschmer, Görlitz-Biesnitz, 1934 · Zbl 0009.09801
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