Abstract
My aim in this paper is to consider Leibniz’s response to concerns raised about the foundations of his differential calculus, and specifically with his doctrine that infinitesimals are “fictions,” albeit fictions so well-founded that their use will never lead to error. I begin with a very brief sketch of the traditional conception of rigorous demonstration and the methodological disputes engendered by the advent of the Leibnizian calculus differentialis. I then examine Leibniz’s claim that infinitesimal magnitudes are fictions, and consider two strategies he employed in the attempt to show that such fictions are acceptable.
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Notes
- 1.
The Aristotelian account of demonstrative knowledge is summarized and analyzed in Hankinson (1995). Aristotle’s treatment of first principles is aptly summarized in Heath’s commentary to the Euclidean Elements, (Euclid [1925] 1956, 1: 117–123).
- 2.
On Descartes’ conception of rigor, see Bos (2001, Chap. 15); Pascal’s conception of rigorous proof is articulated in his essay “L’esprit géométrique” in Pascal (1963, pp. 348–359); Hobbes’s account of demonstration is summarized in the first of his Six Lessons (1656); Wallis’s conception of rigor is outlined in the first three chapters of his Mathesis Universalis in Wallis (1693-1699, 1: 17–24); Barrow’s approach is summarized in the fourth of his Lectiones Mathematicae (Barrow 1860, 1: 63–76).
- 3.
On the Archimedean method of exhaustion, see Dijksterhuis (1987, pp. 130−134).
- 4.
On Descartes and the question of what counts as properly geometrical, see Bos (2001).
- 5.
- 6.
The locus classicus for such characterizations of the infinitesimal is L’Hôpital (1696).
- 7.
- 8.
See Mancosu (1996, pp. 168–176) for more on Rolle’s objections.
- 9.
The literature on the Leibnizian understanding of infinitesimals is extensive. The most complete study is Bos (1974), which distinguishes two Leibnizian approaches to the infinitesimal: one relying upon exhaustion, the other involving the Leibnizian “law of continuity”, intended to provide a theoretical basis for Leibniz’s fictionalism. Ishiguro (1990, Chap. 5) reads Leibnizian fictionalism as adopting a “syncategorematic” view in which infinitesimal terms do not denote and play the role of logical fictions. This interpretation has been endorsed and extended by Arthur (2008), Knobloch (1994, 2002), and Levey (2008). Precisely when Leibniz committed to the fictionalist approach is a subject of some scholarly disagreement. Levey (2008, p. 107) declares that Leibniz “abandoned any ontology of actual infinitesimals” in the 1670s, and both Arthur and Knobloch accept variants of this timeline. I have argued elsewhere (Jesseph 1998) that Leibniz’s correspondence with Wallis and Jean Bernoulli in the 1690s betrays serious reservations about the reality of infinitesimals (in opposition to Bernoulli) but also a disinclination to take them as “nothing” (contrary to Wallis). Nothing I say here depends upon settling this specific issue.
- 10.
This remark appears in Leibniz’s response to the criticisms of the calculus voiced by Bernard Nieuwentijt (GM V, 322).
- 11.
Hofmann (1974, p. 2) aptly characterizes the state of Leibniz’s mathematical knowledge upon his arrival in Paris as “deplorable.”
- 12.
- 13.
One might note, in passing, that this is precisely the kind of argument that Newton attempts when he bases his method of fluxions on “ultimate ratios of evanescent increments.” The connection between Leibniz and Newton on this issue is explored in Arthur (2008).
- 14.
This is but one of several statements of the law. The original appearance to which Leibniz refers is in the Nouvelles de la république des lettres in July of 1687 (GP III, 53–54). In a manuscript known as Cum prodiisset… written around 1701, Leibniz stated the law as holding “In any proposed continual transition ending in any terminus, it is permitted to formulate a general reasoning in which the last terminus may be contained” (Leibniz 1846, p. 40). See Schubring (2005, pp. 174–186) for an interpretation of the role of the law in Leibniz’s mathematics .
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Jesseph, D. (2015). Leibniz on The Elimination of Infinitesimals. In: Goethe, N., Beeley, P., Rabouin, D. (eds) G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes, vol 41. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9664-4_9
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