Ten Misconceptions from the History of Analysis and Their Debunking

Abstract

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.

46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.5456

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• Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
• Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics
• Almost equal: Method of adequality from Diophantus to Fermat and beyond
• Leibniz's Laws of Continuity and Homogeneity
• A Cauchy-Dirac delta function
• Interpreting the infinitesimal mathematics of Leibniz and Euler
• 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
• Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners
• Assigning probabilities to non-Lipschitz mechanical systems
• Analyzing Benardete's comment on decimal notation
• Hyperspace fermions,Möbius transformations, Krein space, fermion doubling, dark matter
• Infinitesimals as an issue in neo-Kantian philosophy of science
• A cognitive analysis of Cauchy's conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus
• Nonstandard techniques and nowhere differentiable functions I: A dense family of generalized blancmange functions
• Is mathematical history written by the victors?