Cauchy's continuum

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.

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

Cited by in corpus (16)

• Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
• Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
• Ten Misconceptions from the History of Analysis and Their Debunking
• Leibniz's Laws of Continuity and Homogeneity
• Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics
• Almost equal: Method of adequality from Diophantus to Fermat and beyond
• A Cauchy-Dirac delta function
• Interpreting the infinitesimal mathematics of Leibniz and Euler
• Cauchy's infinitesimals, his sum theorem, and foundational paradigms
• Toward a history of mathematics focused on procedures
• Cauchy, infinitesimals and ghosts of departed quantifiers
• An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals
• Controversies in the foundations of analysis: Comments on Schubring's Conflicts
• Small oscillations of the pendulum, Euler's method, and adequality
• Historical infinitesimalists and modern historiography of infinitesimals
• Is Leibnizian calculus embeddable in first order logic?