## Abstract

*Foundations of Science* recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, *Conflicts between generalization, rigor, and intuition*. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses *text* with *context* and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit *context* of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz.

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## Notes

On page 17, Schubring cites Proclus’ correct reading of Euclid V.4, but as soon as Schubring attempts to paraphrase this in his own terms, he immediately gets it wrong by describing infinitesimals in terms of

*incommensurability*. The term*incommensurability*is used to describe phenomena related to irrationality, including both previous occurrences of the term in Schubring (2005) on pages 12 and 16.Leibniz uses the term

*finite*in his paraphrase of Euclid’s definition V.5 (or V.4, as discussed in footnote 3), but here he is dealing with a finite*integer**n*(which, in modern terminology, is tending to infinity), so that \(n\epsilon \) always stays less than 1 thereby violating the Archimedean property, if \(\epsilon \) is infinitesimal.Leibniz lists number V.5 for Euclid’s definition instead of V.4. In some editions of the

*Elements*this definition does appear as V.5. Thus, Euclid (1660) as translated by Barrow in 1660 provides the following definition in V.V (the notation “V.V” is from Barrow’s translation):*Those numbers are said to have a ratio betwixt them, which being multiplied may exceed one the other*. For our interpretation of this, see Sect. 3, Axiom E1.Schubring repeats the performance in 2015 when he claims: “I am analysing at length the methodological approach of Laugwitz (and Spalt), which consists in attributing to Cauchy (his) own ‘universe of discourse.’” (Schubring 2015, Sect. 3). But Spalt’s approach is not identical to Laugwitz’s!

Translation: “A function

*u*of a real variable*x*will be*continuous*between two given bounds on*x*if this function, taking for each intermediate value of*x*a unique finite value, an infinitely small increment given to the variable always produces, between the bounds in question, an infinitely small increment of the function itself.”Here Laugwitz is referring to Cauchy’s motto to the effect that “Mon but principal a été de concilier la rigueur, dont je m’étais fait une loi dans mon

*Cours d’analyse*avec la simplicité que produit la consideration directe des quantités infiniment petites.”The fact that Laugwitz had published articles in leading periodicals does not mean that he could not have said something wrong. However, it does suggest the existence of a strawman aspect of Schubring’s claims against him.

Fraser repeats the performance in 2015 when he claims that “Laugwitz, ... some two decades following the publication by Schmieden and him of the \(\Omega \)-calculus commenced to publish a series of articles arguing that their non-Archimedean formulation of analysis is well suited to interpret Cauchy’s results on series and integrals.” (Fraser 2015, p. 27) What Fraser fails to mention is that Laugwitz specifically separated his analysis of Cauchy’s

*procedures*from attempts to account*ontologically*for Cauchy’s infinitesimals in modern terms.Note that the term

*infinitesimal*itself was not coined until the 1670s, by either Mercator or Leibniz; see (Leibniz 1699, p. 63).Moigno’s

*chimerical*anti-infinitesimal thread has not remained without modern French adherents; see Kanovei et al. (2013).Leibniz’s dichotomy between assignable and inassignable quantity, on which his concept of infinitesimal was based, finds a rigorous mathematical treatment in the hyperreal number system (where an assignable number is a standard real number). Yet the mathematics of earlier times allowed an adequate intuitive understanding of the issue, sufficient to effectively and fruitfully use infinitesimals in mathematical practice, even though a semantic base (accounting for the ontology of a number) acceptable by modern standards was as yet unavailable. For further details on Leibniz’s theoretical strategy in dealing with infinitesimals see Katz and Sherry (2012), Katz and Sherry (2013), Sherry and Katz (2014).

Translation: “Nothing indicates, in the documents used, any principled hostility on the part of Cauchy to the elimination of algebraic analysis as an autonomous part placed at the beginning of the analysis course. What was important to him, on the other hand, was the presence of several items from this part, and the methods used in presenting them. …Note two particular points important to Cauchy, namely that continuous functions be placed at the beginning of differential calculus, and that the study of the convergence of series should find its place in the vicinity of Taylor’s formula, in differential and integral calculus.”

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## Acknowledgments

The work of V. Kanovei was partially supported by RFBR Grant 13-01-00006. M. Katz was partially funded by the Israel Science Foundation Grant No. 1517/12. We are grateful to the anonymous referees and to A. Alexander, R. Ely, and S. Kutateladze for their helpful comments. The influence of Hilton Kramer (1928–2012) is obvious.

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Błaszczyk, P., Kanovei, V., Katz, M.G. *et al.* Controversies in the Foundations of Analysis: Comments on Schubring’s *Conflicts*
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*Found Sci* **22**, 125–140 (2017). https://doi.org/10.1007/s10699-015-9473-4

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DOI: https://doi.org/10.1007/s10699-015-9473-4