Cauchy's continuum

Details Cauchy's continuum Cauchy's continuum

2011-08-21

arxiv.org/abs/1108.4201v2

Perspectives on Science 19 (2011), no. 4, 426-452

Mathematics

History and Overview

29 pages, 5 figures. arXiv admin note: text overlap with arXiv:some 1104.0375

Scientific paper

10.1162/POSC_a_00047

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.

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