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Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms

Abstract

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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Notes

  1. 1.

    A debate of long standing concerns the issue of whether Cauchy modified or clarified the hypothesis of the sum theorem of 1853 as compared to 1821. Our analysis of the 1853 text is independent of this debate and our main conclusions are compatible with either view.

  2. 2.

    Some historians are fond of recycling the claim that Abraham Robinson used model theory to develop his system with infinitesimals. What they tend to overlook is not merely the fact that an alternative construction of the hyperreals via an ultrapower requires nothing more than a serious undergraduate course in algebra (covering the existence of a maximal ideal), but more significantly the distinction between procedures and foundations (see Sect. 1.2) which highlights the point that whether one uses Weierstrass’s foundations or Robinson’s is of little import, procedurally speaking.

  3. 3.

    See http://u.cs.biu.ac.il/~katzmik/infinitesimals.html for a more detailed list.

  4. 4.

    Here the equation numbers (1) and (3) are in Cauchy’s text as reprinted in Cauchy (1900).

  5. 5.

    The equation number (2) is in Cauchy’s original text.

  6. 6.

    The equation number (4) is in Cauchy’s original text.

  7. 7.

    Historians often use the term punctiform continuum to refer to a continuum made out of points (as for example the traditional set-theoretic \({{\mathrm{{{\mathbb {R}}}}}}\)). Earlier notions of continuum are generally thought to be non-punctiform (i.e., not made out of points). The term punctiform also has a technical meaning in topology unrelated to the above distinction.

  8. 8.

    Nelson’s syntactic recasting of Robinson’s framework is a good illustration, in that the logical procedures in Nelson’s framework are certainly up to modern standards of rigor and are expressed in the classical Zermelo–Fraenkel set theory (ZFC). With respect to an enriched set-theoretic language, infinitesimals in Nelson’s Internal Set Theory (IST) can be found within the real number system \({{\mathrm{{{\mathbb {R}}}}}}\) itself. The semantic/ontological issues are handled in an appendix to Nelson (1977), showing Nelson’s IST to be a conservative extension of ZFC.

  9. 9.

    A similar criterion was formulated in Fraenkel (1928, pp.  116–117). For a discussion of the Klein–Fraenkel criterion see Kanovei et al. (2013, Section  6.1).

  10. 10.

    For the role of these in a possible construction of the hyperreals see Sect. 5.2.

  11. 11.

    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.”

  12. 12.

    The fact that Laugwitz had published articles in leading periodicals does not mean that he couldn’t have said something wrong. However, it does suggest the existence of a strawman aspect of Schubring’s sweeping claims against him.

  13. 13.

    The transfer principle is a type of theorem that, depending on the context, asserts that rules, laws or procedures valid for a certain number system (or more general mathematical structure), still apply (i.e., are transfered) to an extended number system (or more general mathematical structure). Thus, the familiar extension \({{\mathrm{{\mathbb {Q}}}}}\,\hookrightarrow\, {{\mathrm{{{\mathbb {R}}}}}}\) preserves the properties of an ordered field. To give a negative example, the extension \({{\mathrm{{{\mathbb {R}}}}}}\,\hookrightarrow \,{{\mathrm{{{\mathbb {R}}}}}}\cup \{\pm \infty \}\) of the real numbers to the so-called extended reals does not preserve the properties of an ordered field. The hyperreal extension \({{\mathrm{{{\mathbb {R}}}}}}\,\hookrightarrow \,{{}^{*}{{\mathrm{{{\mathbb {R}}}}}}}\) preserves all first-order properties. For example, the identity \(\sin ^2 x+\cos ^2 x=1\) remains valid for all hyperreal x, including infinitesimal and infinite values of \(x\in {{}^{*}{{\mathrm{{{\mathbb {R}}}}}}}\). In particular, the properties of the reciprocal function remain the same after it is extended to the hyperreal domain.

    Arsac tries to explain nonstandard analysis but he seems to be as unaware of the transfer principle as Viertel: “Une fonction continue au sens habituel est une fonction continue aux points standards, mais elle ne l’est pas obligatoirement aux points non standard” (Arsac 2013, p. 133). Contrary to his claim, the natural extension of a continuous function f will be continuous at all hyperreal points c in the sense of the standard definition \(\forall \epsilon>0\;\exists \delta >0:|x-c|<\delta\,{\implies}\, |f(x)-f(c)|<\epsilon \), by the transfer principle. Indeed, Arsac confused S-continuity and continuity...

  14. 14.

    The symbol \(\delta \) is not used in reference to an integer tending to infinity.

  15. 15.

    Grattan-Guinness acknowledges this point GG4 implicitly in his summary of Cauchy’s proof in (Grattan-Guinness 1970, p. 123). Meanwhile, an \(\varepsilon \) does occur in Cauchy on p. 32; see Sect. 2.2.

  16. 16.

    This is a reference to Cauchy’s sum (3) namely \(u_n+u_{n+1} \ldots u_{n^{\prime }-1}\) discussed in Sect. 2.

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Acknowledgements

V. Kanovei was supported in part by the RFBR Grant Number 17-01-00705. M.  Katz was partially funded by the Israel Science Foundation Grant Number 1517/12. We are grateful to Dave L.  Renfro for helpful suggestions.

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Bascelli, T., Błaszczyk, P., Borovik, A. et al. Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms. Found Sci 23, 267–296 (2018). https://doi.org/10.1007/s10699-017-9534-y

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Keywords

  • Cauchy’s infinitesimal
  • Sum theorem
  • Quantifier alternation
  • Uniform convergence
  • Foundational paradigms