Katz and Sherry’s [Pt.3] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 3 ‘A Pair of Leibnizian Methodologies, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

3. A Pair of Leibnizian Methodologies

Brief Summary:

Leibniz had two methodologies, which the authors call the A-methodology (using Archimedes’ exhaustion) and the B-methodology (using infinitesimals). Recent Leibniz scholars either acknowledge both of Leibniz’ methodologies or just the first type. The authors believe those in the second camp are misreading Leibniz’ notion of the infinitesimal’s fictionality.

Summary

Leibniz had two infinitesimal calculus methodologies: one by exhaustion and one using the law of continuity. (KS 575) The first relies on Archimedes’ exhaustion method, and the authors call it the ‘A-methodology’. The second uses infinitesimals and is called the ‘B-methodology’.

Leibniz considered infinitesimals as fictions. In his time, this was a controversial position, especially for some of his disciples, like Bernoulli, l’Hôpital, and Varignon. Accordiing to Ferraro, “Leibniz’s infinitesimals enjoy an ideal ontological status similar to that of the complex numbers, surd (irrational) exponents, and other ideal quantities.” (576)

The authors will now examine how commentators attribute either both A and B methodologies or just the A-methodology. They first quote from Leibniz’ 1702 letter to Varignon.

Here Leibniz outlines a geometrical argument involving quantities c and e described as ‘‘not absolutely nothing’’, and goes on to comment that c and e [KS quoting Leibniz:]

are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements (Leibniz et al. 1702, pp. 104–105). [KS 576]

Jesseph argues that Leibniz proposes both A and B methodologies. Like Bos, Jesseph emphasizes Leibniz’ law of continuity and regards it not as a mathematical principle but rather as a “a general methodological rule with applications in mathematics, physics, metaphysics, and other sciences’’ (KS 576 quoting Jesseph ibid p.21).

Recent work on Leibniz’ calculus is divided into two camps: 1) Those who recognize both methodologies (Bos, Ferraro, Horváth, Jesseph, and Laugwitz), and 2) those who have a syncategorematic interpretation that only recognizes the A-methodology. The authors believe that the second reading “is due to an incorrect analysis of Leibniz’s fictionalism.” (KS 577)

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html