Cauchy, infinitesimals and ghosts of departed quantifiers

Abstract

. Cauchy, infinitesimals and ghosts of departed quantifiers, Mat. Stud. 47 (2017), 115-144.Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence.As case studies, we analyze the approaches of Craig Fraser and Jesper Lützen to Cauchy's contributions to infinitesimal analysis, as well as Fraser's approach toward Leibniz's theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser's interpretive framework.

Cauchy's Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals

“…F (a) (2.1) (Cauchy [14], 1827, p. 289; counter (2.1) added) A formula equivalent to (2.1) was proposed by Dirac a century later. 2 The expression α α 2 + (µ − a) 2 (2.2) occurring in Cauchy's formula is known as the Cauchy distribution in probability theory. Here Cauchy specifies a function which meets the criteria as set forth by Dirac a century later.…”

“….+u n ′ −1 should be 1 As discussed in (Laugwitz [33], 1989), a further condition needs to be imposed on α and ǫ in modern mathematics to ensure the correctness of the formula. 2 The key property of the Dirac delta "function" δ(x) is exemplified by the defining formula toujours infiniment petite pour des valeurs infiniment grandes des nombres entiers n et n ′ > n . .…”

Analytic functions in a unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties

“…. [1] the cultivated intuition just discussed, which has been developed under the influence of logical deduction and might almost be called a form of memory; but rather of [2] the naïve intuition, largely a natural gift, which is unconsciously increased by minute study of one branch or other of the science. (ibid., p. 245-246)…”

“…(ibid.) Thus Klein's intuition [1] is cultivated (under the influence of logical deduction), while intuition [2] is the most basic in his vision of mathematics, namely prelogical:…”

Metric completions, the Heine-Borel property, and approachability

“…Additional material on Robinson's framework can be found in [6]. The historical significance of Robinson's framework for infinitesimal analysis in relation to the work of Fermat, Gregory, Leibniz, Euler, and Cauchy has been analyzed respectively in [7], [8], [9], [10], [11], and elsewhere. The approach is not without its critics; see e.g., [12].…”