Cauchy, infinitesimals and ghosts of departed quantifiers

J. Bair, P. Blaszczyk, R. Ely, V. Henry, V. Kanovei, K. U. Katz, M. G. Katz, T. Kudryk, S. S. Kutateladze, T. McGaffey, T. Mormann, D. M. Schaps, D. Sherry
HEC-ULG, University of Liege, 4000 Belgium; Institute of Mathematics, Pedagogical University of Cracow, Poland; Department of Mathematics, University of Idaho, Moscow, Russia; Department of Mathematics, University of Namur, Belgium; IPPI, Moscow, and MIIT, Moscow, Russia; Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; Department of Mathematics, Lviv National University, Ukraine; Sobolev Institute of Mathematics, Novosibirsk State University, Russia; Rice University, US; Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, Donostia San Sebastian, Spain; Department of Classical Studies, Bar Ilan University, Ramat Gan, Israel; Department of Philosophy, Northern Arizona University, Flagstaff, US
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson’s framework, while Leibniz’s law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson’s framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy’s procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson’s framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. As case studies, we analyze the approaches of Craig Fraser and Jesper L¨utzen to Cauchy’s contributions to infinitesimal analysis, as well as Fraser’s approach toward Leibniz’s theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser’s interpretive framework.
historiography; infinitesimal; Latin model; butterfly model; law of continuity; ontology; practice; Cauchy; Leibniz
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