Was Cauchy prescient?

Question

Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853.

Michael Segre, writing in Archive for History of Exact Sciences, claimed concerning Cauchy's sum theorem:

What is amazing here is Cauchy's attitude. He totally disregarded Fourier's counterexample and did not admit having made a mistake: not only did he "prove" his theorem, but he repeated it in a paper read to the Academie des Sciences as late as 1853. (page 233 in Segre, Michael. Peano's axioms in their historical context. Arch. Hist. Exact Sci. 48 (1994), no. 3-4, 201-342)

For his part, Umberto Bottazzini wrote:

The language of infinites and infinitesimals that Cauchy used here seemed ever more inadequate to treat the sophisticated and complex questions then being posed by analysis... The problems posed by the study of nature, such as those Fourier had faced, now reappeared everywhere in the most delicate questions of "pure" analysis and necessarily led to the elaboration of techniques of inquiry considerably more refined than those that had served French mathematicians at the beginning of the century. Infinitesimals were to disappear from mathematical practice in the face of Weierstrass' epsilon and delta notation (p. 208 in Bottazzini, Umberto. The higher calculus: a history of real and complex analysis from Euler to Weierstrass. Translated from the Italian by Warren Van Egmond. Springer-Verlag, New York, 1986)

These authors make Cauchy appear rather obstinate with regard to what is described by some historians as his famous "mistake". To a number of mathematicians who have studied Cauchy's work, such claims by historians seem surprising. Are we to accept them at face value? Is there more to the story than meets the eye?

An analysis of this question by my coauthors and myself is presented in this 2017 publication in Foundations of Science. Additional relevant material is referenced at this regularly updated site. What I am seeking are other possible responses to this question from people who have examined Cauchy's writings.

Note 1. I included in the article (on page 6) an extensive quotation from Cauchy that includes in particular his improbable substitution of $x=\frac{1}{n}$ in the remainder term $r_n$; see (new version of) article linked above. To a mathematician trained in the Weierstrassian framework this looks like a freshman calculus error. However, Robinson's framework enables an interpretation of this as evaluation at an infinitesimal point. Recall that the salient mathematical point here is that uniform convergence is expressible by a pointwise condition in the extended continuum. This is analogous to uniform continuity being expressible by a pointwise condition, namely S-continuity or microcontinuity (this last point is not strictly speaking related to the sum theorem but may help sort this out for those not closely familiar with the framework).

Note 2. For a related discussion of Cauchy see this MSE post.

Note 3. A detailed response to objections by Jesper Luetzen, Craig Fraser, and others appears in this 2017 publication in Mat. Stud.

Answer 1

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1971) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent study in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

Answer 2

After having read Katz' article, I must say I am not convinced and find that the standard interpretation, namely that of Cauchy making a mistake in 1821 and failing to acknowledging it or correcting it properly in 1853 is closer to the truth. In other words, even after reading your paper, I see nothing more than meets the eye.

One of your main point is the word "toujours" (always) which appears in the 1853 version of Cauchy's theorem, but not in the 1821 version. Quoting your paper, the 1821 version says

When the various terms of series $u_0 +u_1 +u_2 + \dots +u_n + u_{n+1} + \dots$ are functions of the same variable $x$, continuous with respect to this variable in the neighborhood of a particular value for which the series converges, the sum s of the series is also a continuous function of $x$ in the neighborhood of this particular value.

(I would have liked to see the French version, by the way).

The 1853 version is:

Théorème 1. Si les différents termes de la série $$u_0,u_1,u_2,\dots,u_n,u_{n+1},\dots \ \ (1)$$ sont des fonctions de la variable réelle $x$, continues, par rapport à cette variable, entre des limites données; si, d’ailleurs, la somme $u_n +u_{n+1} + \dots + u_{n′−1}$ devient toujours infiniment petite pour des valeurs infiniment grandes des nombres entiers $n$ et $n′ > n$, la série (1) sera convergente et la somme $s$ de la série sera, entre les limites données, fonction continue de la variable $x$.

You interpret "toujours" as meaning "for real (archimedean) and for infinitesimal values of the variable $x$". But I note that it is more natural to interpret it simply as meaning "for all real (archimedean) values of $x$". This interpretation would be enough to make the 1853 statement different, precisely with a stronger hypothesis, than the 1821 statement, for plainly the 1821 statement requires only the convergence of the series for a particular value $x_0$ (and the continuity of the $u_n$ on a neighborhood of $x_0$) to conclude the continuity of the sum $s$ at $x_0$. Thus we would have two statements of Cauchy's theorem, which both happen to be false.

The second important point of your argument is the discussion of Cauchy's treatment of a potential counter-example related lo Abel's objection in section 2.3. Cauchy claims that this is not a counter-example to his 1853 theorem because it fails some hypothesis. But here, since you give no quotation of Cauchy, it is impossible to know if Cauchy's arguments support your interpretation or are simply mistaken.

Answer 3

For convenience of our readers I provide a summary of the article linked in the question:

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... Interpretation of texts written in the nineteenth century, and the meaning we give to technical terms, procedures, theories, and the like are closely related to what we already know as well as our expectations and assumptions. This paper provides evidence that a change in the cultural-technical framework of a historian provides new explanations, which are arguably more natural, and new insights into Cauchy’s work.

Any serious interpretation of Cauchy's proof of his sum theorem has to take into account his argument involving the point generated by the sequence $(\frac{1}{n})$. I am not aware of any reasonable interpretation of such a point as a nonzero point of a standard Archimedean continuum.