Cauchy, infinitesimals and ghosts of departed quantifiers

TL;DR

This paper explores how infinitesimal procedures from Leibniz, Euler, and Cauchy can be better understood through nonstandard analysis, contrasting it with traditional Weierstrassian approaches and highlighting formalizations that clarify historical mathematical concepts.

Contribution

It demonstrates that nonstandard analysis provides more faithful formalizations of classical infinitesimal procedures than traditional frameworks, especially for Cauchy's sum theorem.

Findings

Nonstandard analysis aligns closely with historical infinitesimal procedures.

Weierstrassian formalizations often require reinterpretations or 'ghosts' of quantifiers.

Robinson's framework clarifies Cauchy's concepts of continuity and convergence.

Abstract

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. Keywords: historiography; infinitesimal; Latin model; butterfly model