Continuity between Cauchy and Bolzano: issues of antecedents and priority

Abstract

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 Cours d'Analyse, may have plagiarized Bolzano's Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by Kästner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially.

Acknowledgments

The authors thank the anonymous referees for numerous suggestions that helped improve the article. We are grateful to Olivier Azzola, archivist at the Ecole Polytechnique, for granting access to Cauchy's handwritten course summaries reproduced in Figure 1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 In the original German: ‘Stetigkeit definiert Cauchy inhaltlich so wie Bolzano’ (Heuser 1991, 691). Heuser goes on to present Cauchy's first 1821 definition in terms of $f(x+\alpha )-f\left(x\right)$ (see Section 2.2) but fails to mention the fact that Cauchy describes α as an infinitely small increment.

2 Note, however, that Bolzano did exploit infinitesimals in his later writings; see for example, Grattan-Guinness (1970, note 29, 379), Trlifajová (2018) and Fila (2020).

3 The equivalence of such a definition with the $ϵ,\delta$ one requires the axiom of choice.

4 Translation: ‘The limit of a continuous function of several variables is [equal to] the same function of their limit. Consequences of this Theorem with regard to the continuity of composite functions dependent on a single variable.’ The reference for this particular lesson in the Archives of the Ecole Polytechnique is as follows: Le 4 Mars 1817, la leçon 20. Archives E. P., X II C7, Registre d'instruction 1816–1817.

5 Belhoste places it even earlier, in 1816: ‘according to the Registres, Cauchy knew the modern concept of continuity as far back as March 1817, but the “invention” was anterior, as shown by the instructional program of December 1816’ Belhoste (1991, 255, note 6).

6 Siegmund-Schultze (2009) writes: ‘By and large, with few exceptions to be noted below, the translation is fine’.

7 In the original: ‘En d'autres termes, la fonction $f\left(x\right)$ restera continue par rapport à x entre les limites données, si, entre ces limites, un accroissement infiniment petit de la variable produit toujours un accroissement infiniment petit de la fonction elle-même’ (Cauchy 1821, 34–35).

8 Meaning dissolution, that is, absence (of continuity).

9 Similarly, in a recently published book, Rusnock and Šebestík mention that ‘there has been speculation that Cauchy may have learned a thing or two from Bolzano’ (Rusnock and Šebestík 2019, 49); see also note 3 there.

10 Grattan-Guinness apparently means ‘source.’

11 Cauchy had discussed continuity even earlier, in an 1814 article on complex functions (see Freudenthal 1971, 380). However, that discussion stayed at the intuitive level and cannot be described as reasonably precise.

12 Note that we take no position with regard to which definition was closer to a modern one, Bolzano's or Cauchy's (Bolzano's was arguably closer to the modern Epsilontik standard). The point we are arguing is that both were reasonably precise in the sense specified.

13 In the original: ‘Stetig heißt nähmlich eine Function, wenn die Veränderung, die sie bey einer gewissen Veränderung ihrer Wurzel erfährt, kleiner als jede gegebene Größe zu werden vermag, wenn man nur jene klein genug nimmt’ Bolzano (1816, 34). Note that Bolzano repeatedly uses Wurzel in the sense of ‘input to a function’; see for example, footnote on page 11 of Bolzano (1817). The issue is discussed in Russ (2004, 256, note f).

14 We translate Reihe as ‘sequence’, even though it is often translated as ‘series’, since ‘series’ nowadays is a standard technical term which is not appropriate here, and moreover the German term Reihe can mean either ‘sequence’ or ‘series’.

15 The German conjunction so dass, especially in Kästner's (now obsolete) spelling as two separate words, resembles the English ‘such that’; in the present case, however, this is a false friend. In fact ‘as a consequence’ is one of several standard translations of the German conjunction sodass.

16 Kästner's phrasing nach einander folgender could possibly be interpreted as the statement that the terms mentioned here are immediate successor elements, in particular since the standard technical translation for ‘immediate sucessor element’ is Nachfolger. This, however, could not be what Kästner meant to say. Kästner's phrasing (note that he does not say Nachfolger outright) is sufficiently vague to allow for an interpretation where he means to speak of two terms which follow shortly one after another, though there are other terms in between.

17 In the original: ‘In einer Reihe von Grössen, erfolgt das Wachsthum oder das Abnehmen derselben, nach dem Gesetze der Stetigkeit (lege continui) wenn nach jedem Gliede der Reihe eines folget, oder vor ihm vorhergehen kann, das so wenig als man nur will von dem angenommenen Gliede unterschieden ist, so daß der Unterschied zweyer nach einander folgender Glieder, weniger als jede gegebene Grösse betragen kann.’ This was quoted in Spalt (2015, 283). In our translation, we try to strike a balance between literalness and readability in line with the approach taken in Blåsjö and Hogendijk (2018).

18 Perhaps a better translation is ‘too broad’.

19 According to Kröger (2014, Abbildung 10), there were two edititions of this treatise. These are Kästner (1766, 1793). In the 1793 edition of Kästner's treatise referred to by Bolzano as Auflage 2, Section 183 appears on page 543.

20 Sebestik also points out that Bolzano and Cauchy's definitions of continuity could have been ‘the result of a critical reflection on the texts by Euler and Lagrange’ (Sebestik (1992), 110, 81, 83).

21 Not to be confused with his law of continuity. For a detailed discussion see Katz and Sherry (2013), Sherry and Katz (2012), Bascelli et al. (2016), Bair et al. (2017b), and Bair et al. (2018).

22 In modern analysis, the sequence-condition is equivalent to continuity for first-countable spaces.

23 Including Cauchy's own definition in 1814, in an article on complex functions quoted by Freudenthal; cf. note 11.

Additional information

Funding

E. Fuentes Guillén was supported by the Postdoctoral Scholarship Program of the Dirección General de Asuntos del Personal Académico (DGAPA-UNAM). V Kanovei was supported by RFBR [grant no. 18-29-13037].