6 found
Alexandre Borovik [5]Alexandre V. Borovik [2]
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Alexandre Borovik
University of Manchester
  1.  65
    Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus.Alexandre Borovik & Mikhail G. Katz - 2012 - Foundations of Science 17 (3):245-276.
    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, (...)
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  2.  41
    An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals.Alexandre Borovik, Renling Jin & Mikhail G. Katz - 2012 - Notre Dame Journal of Formal Logic 53 (4):557-570.
    A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...)
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  3.  26
    A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos.Piotr Błaszczyk, Alexandre Borovik, Vladimir Kanovei, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze & David Sherry - 2016 - Logica Universalis 10 (4):393-405.
    We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...)
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  4.  17
    Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms.Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps & David Sherry - 2018 - Foundations of Science 23 (2):267-296.
    Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.
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  5.  28
    On the Schur-Zassenhaus Theorem for Groups of Finite Morley Rank.Alexandre V. Borovik & Ali Nesin - 1992 - Journal of Symbolic Logic 57 (4):1469-1477.
  6.  22
    Schur-Zassenhaus Theorem Revisited.Alexandre V. Borovik & Ali Nesin - 1994 - Journal of Symbolic Logic 59 (1):283-291.