Cauchy's continuum

TL;DR

This paper reexamines Cauchy's 1821 sum theorem, clarifying his original hypothesis involving a single variable and the conditions under which the theorem holds, contrasting it with later interpretations involving uniform convergence.

Contribution

It clarifies Cauchy's original formulation of the sum theorem and distinguishes it from the later uniform convergence interpretation, emphasizing the role of single-variable conditions.

Findings

Cauchy's hypothesis involves a single variable x, not a pair of variables.

Cauchy's error term r_n(x) tends to zero uniformly over x, including infinitesimal values.

The traditional uniform convergence interpretation differs from Cauchy's original formulation.

Abstract

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy's hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term r_n = r_n(x) to go to zero at all values of x, including the infinitesimal value generated by 1/n, explicitly specified by Cauchy. If one wishes to understand Cauchy's modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits.