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  1. Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav Sergeyev - 2019 - Foundations of Science 24 (1):153-170.
    This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...)
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  • 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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  • Gregory’s Sixth Operation.Tiziana Bascelli, Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Tahl Nowik, David M. Schaps & David Sherry - 2018 - Foundations of Science 23 (1):133-144.
    In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...)
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