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Leibniz on the requisites of an exact arithmetical quadrature

Federico Raffo Quintana

Studies in History and Philosophy of Science Part A 67 (C):65-73 (2018)

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Leibniz's rigorous foundation of infinitesimal geometry by means of riemannian sums.Eberhard Knobloch - 2002 - Synthese 133 (1-2):59 - 73.details In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) Download Export citation Bookmark 17 citations
Leibniz’s syncategorematic infinitesimals.Richard T. W. Arthur - 2013 - Archive for History of Exact Sciences 67 (5):553-593.details In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, (...) Download Export citation Bookmark 8 citations
Detleff Clüver: An Early Opponent of the Leibnizian Differential Calculus.Paolo Mancosu & Ezio Vailati - 1990 - Centaurus 33 (3):325-344.details Download Export citation Bookmark 5 citations
Nova methodus investigandi. On the Concept of Analysis in John Wallis’s Mathematical Writings.Philip Beeley - 2013 - Studia Leibnitiana 45 (1):42-58.details Download Export citation Bookmark 1 citation
Leery Bedfellows: Newton and Leibniz on the Status of Infinitesimals.Richard Arthur - 2008 - In Douglas Jesseph & Ursula Goldenbaum (eds.), Infinitesimal Differences: Controversies Between Leibniz and His Contemporaries. Walter de Gruyter.details Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) Download Export citation Bookmark 11 citations
Leibniz' Einführung des Transzendenten.Herbert Breger - forthcoming - Studia Leibnitiana.details Download Export citation Bookmark 6 citations
Actual Infinitesimals in Leibniz's Early Thought.Richard T. W. Arthur - unknowndetails Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no (...) Download Export citation Bookmark 4 citations
Leibniz und Wallis.Joseph Ehrenfried Hofmann - 1973 - Studia Leibnitiana 5 (2):245 - 281.details Download Export citation Bookmark 1 citation

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