Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

Mikhail G. Katz, David M Sherry

Research output: Contribution to journalArticle

26 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)571-625
Number of pages55
JournalErkenntnis
Volume78
Issue number3
DOIs
StatePublished - 2013

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Calculus
Resources
Differential Calculus
Fictionality
Gottfried Wilhelm Leibniz
Transcendental
Inconsistency
Homogeneity
Exception
Reasoning
Logic
Calculi
Criticism
Fiction
Continuity
History
Strategy
Modern Logic
Fallacies
Paraphrase

ASJC Scopus subject areas

  • Philosophy
  • Logic

Cite this

Leibniz's Infinitesimals : Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond. / Katz, Mikhail G.; Sherry, David M.

In: Erkenntnis, Vol. 78, No. 3, 2013, p. 571-625.

Research output: Contribution to journalArticle

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