Who Gave You the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus

Alexandre Borovik, Mikhail G. Katz

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered. © 2011 Springer Science+Business Media B.V.
    Original languageEnglish
    Pages (from-to)245-276
    Number of pages31
    JournalFoundations of Science
    Volume17
    Issue number3
    DOIs
    Publication statusPublished - Aug 2012

    Keywords

    • Archimedean axiom
    • Bernoulli
    • Cauchy
    • Continuity
    • Continuum
    • du Bois-Reymond
    • Epsilontics
    • Felix Klein
    • Hyperreals
    • Infinitesimal
    • Stolz
    • Sum theorem
    • Transfer principle
    • Ultraproduct
    • Weierstrass

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