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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”
4. Cum Prodiisset
4.2 Law of Continuity, with Examples
Brief Summary:
In Cum Prodiisset Leibniz discusses his Law of Continuity. According to one formulation, in a continuous transition, the final ending (the terminus) of the transition may be included with that transition.
Summary
The basis of the calculus that Leibniz formulates in Cum prodiisset is his Law of Continuity (LC). It takes a variety of forms. Here is one formulation:
In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included.9 [[ft 9: Boyer claims that Leibniz used this formulation of LC in ‘‘a letter to [Pierre] Bayle in 1687’’ (Boyer 1959, p. 217). Boyer’s claim contains two errors. First, the work in question is not a letter to Bayle but | rather the Letter of Mr. Leibniz on a general principle useful in explaining the laws of nature, etc. (Leibniz 1687). Second, while this letter does deal with Leibniz’ continuity principle, it does not contain the formulation In any supposed continuous transition, ending in any terminus, etc.; instead, it postulates that an infinitesimal change of input should result in an infinitesimal change in the output (this principle was popularized by Cauchy in 1821 as the definition of continuity in Cauchy 1821, p. 34). Boyer’s erroneous claims have been reproduced by numerous authors, including Kline (1972, p. 385).]]
[[KS 577. footnote, 577-578]]
The final terminus in this explanation is an ending of a transition. KS then give five reasons that the terminus “encompasses inassignable quantities” [[infinitesimal quantities, see KS ‘Leibniz’s Laws of Continuity and Homogeneity” http://arxiv.org/pdf/1211.7188.pdf. See Page 578 of KS ‘Leibniz’s Infinitesimal’ for the five reasons.]]
In Cum Prodiisset, Leibniz offers a number of examples for how the Law of Continuity can be applied. KS will focus on three of them [[quoting]]:
(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).
(2) Invoking the idea that the term equality may refer to equality up to an infinitesimal error, Leibniz writes: when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small (Child 1920, p. 148).
(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).
[[KS 579, for more on the ellipse example and on the law of continuity, see this entry on Leibniz’ letter to Malebranche.]
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.
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