Leibniz on The Elimination of Infinitesimals

Abstract

There can scarcely be any question that the most important of Leibniz's many mathematical contributions was his development of the calculus differentialis. Although his work in such fields as algebra, series summations, combinatorics, determinant theory, and other areas can be seen (retrospectively at least) as original and even groundbreaking, his efforts in these disciplines were not generally made public his day. In contrast, historians of mathematics routinely speak of a "Leibnizian tradition" in analysis that traces back to a series of published papers from the 1680s and constitutes a clear advance in European mathematics. Yet, although the infinitesimal methods Leibniz introduced with his calculus were powerful new techniques opening a vast field of new results, these methods raised significant conceptual and methodological issues. The specific worry voiced by several of Leibniz's mathematical contemporaries was that these new methods were fundamentally unrigorous, so that they not only offended against criteria of mathematical intelligibility, but were also liable to lead to error.

From the Intractable to the Undetermined : Between Calculus and Geometry, Leibnizian Thoughts on ⁰⁄₀ (1700-1706)

“…Cependant, l'un et l'autre textes restent inédits du vivant de Leibniz. Leurs éditions de nos jours ont fait l'objet d'analyses [Jesseph 2015], [Pasini 1993]. Les deux textes sont proches en partie car ils contiennent un développement qui s'appuie sur une configuration géométrique identique 10 .…”

“…Ces critiques portent sur le statut existentiel des différentielles et sur leur mode de fonctionnement, sujets sur lesquels Leibniz est conduit à apporter des éclaircissements. Ces derniers ont fait l'objet de nombreuses études philosophiques et historiques [Beeley 2008], [Jesseph 2015], [Mancosu 1996], [Pasini 1985a[Pasini ,b, 1988[Pasini , 1993, [Vermij 1989]. Dans le contexte de la querelle des infiniment petits à l'Académie royale des sciences, qui débute en juillet 1700, Michel Rolle (1652Rolle ( -1719 reprend et reformule ces critiques.…”

Leibniz�s well-founded fictions and their interpetations

“…While the "common method of indivisibles" 40 Jesseph argues in [52] that the techniques in DQA were limited by their reliance on the knowledge of the tangent lines to the curve (and/or the corresponding differential). Therefore the applicability of the techniques depended on the availability of such data.…”

“…Jesseph's translation: " 'Nor does it matter whether there are such quantities in nature, for it suffices that they are introduced as fictions, since they allow the abbreviations of speech and thought in the discovery as well as demonstration'(Leibniz 1993, p. 69)" (Jesseph[52], 2015, p. 198). A longer passage including this one was quoted in Parmentier's French translation in Section 4.9.…”

Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania

“…This indicates that Leibniz embraces what we refer to as a Bernoullian continuum (though certainly not a non-Archimedean continuum in a modern settheoretic sense), contrary to Ishiguro's Chapter 5. [Jesseph 2015] shows that strategies Leibniz employed in the attempt to show that such fictions are acceptable because the use of infinitesimals can ultimately be eliminated have to presume the correctness of an infinitesimal inference (i.e., inference exploiting infinitesimals), namely identifying the tangent line to a curve as part of the construction. In the case of conic sections this strategy succeeds because the tangents are already known from Apollonius.…”