Erkenntnis 78 (3):571-625 (2013)

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Abstract
Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity
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DOI 10.1007/s10670-012-9370-y
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References found in this work BETA

Philosophical Investigations.Ludwig Josef Johann Wittgenstein - 1953 - New York, NY, USA: Wiley-Blackwell.
The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.
The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
The Principles of Mathematics.Bertrand Arthur William Russell - 1903 - Cambridge, England: Allen & Unwin.

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Citations of this work BETA

Infinitesimals as an Issue of Neo-Kantian Philosophy of Science.Thomas Mormann & Mikhail Katz - 2013 - Hopos: The Journal of the International Society for the History of Philosophy of Science (2):236-280.

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