2011/08/31 by Karin U. Katz, Mikhail G. Katz
Mathematics · #math.HO #math.CA #math.LO #msc:01A85 #msc:26E35 #msc:03A05 #msc:97A20 #msc:97C30
paper · pdf · doi:10.1162/posc_a_00047
published as Perspectives on Science 19 (2011), no. 4, 426-452 · 29 pages, 5 figures. arXiv admin note: text overlap with arXiv:some 1104.0375
arxiv created 2011/10/17 · arxiv updated 2011/10/18
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 rn = rn(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.