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I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.

arjun
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    See answers or comments by me and Greg Graviton at http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong – KConrad Jul 21 '14 at 14:41
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    I have heard something along the lines of the following: Cauchy's conception of real numbers included infinitesimals. So for him pointwise convergence included infinitesimal pointwise convergence. In some rigorous sense, this does end up being uniform convergence. I believe this may be true in synthetic differential geometry, for instance. I probably read this in a MO comment somewhere, but I cannot find it. Maybe someone reading this could back up this perspective? – Steven Gubkin Jul 21 '14 at 18:02
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    @StevenGubkin, indeed uniform convergence is equivalent to a certain pointwise condition (in the extended domain) for the natural extensions of the functions. – Mikhail Katz Jun 16 '16 at 14:14

3 Answers3

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The issue of Cauchy's understanding of continuity is a subject of lively historical debate. Grabiner represents only one view in this debate. Laugwitz has published a series of scholarly articles studying the issue, including this 1987 publication in Historia Mathematica.

No discussion of this issue is complete without mentioning Cauchy's article dating from 1853 where he deals with convergence of series of functions and acknowledges that the condition as stated in 1821 needs to be modified.

Here Cauchy seems to introduce a condition close to uniform convergence which is stronger than one used in his earlier works. See for instance this 2013 publication in Foundations of Science and this 2018 publication in Mat.Stud..

Mikhail Katz
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See the wonderful book by Judith Grabiner, The Origins of Cauchy's Rigorous Calculus (1981), page 140 : "Actually, his [Cauchy's] proof implicitly assumed the function to be uniformly continuous, though he did not distinguish between continuity and uniform continuity, just as he had not distinguished between convergence and uniform convergence."

András Bátkai
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    Thanks for the reference to the book. In the link above, @GregGraviton argues that Cauchy knew the difference and this is because he includes infinitesimals in the definition of pointwise convergence. So perhaps it would be unfair to say, "Cauchy implicitly assumes uniform continuity". Instead, perhaps one should say, "infinitesimals are included in Cauchy's definition of pointwise convergence, and this can be shown to imply uniform convergence". Greg Graviton's argument appears compelling, and here is a paper about this. – arjun Jul 21 '14 at 18:39
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    @AndrásBátkai, it seems to me that Grabiner's comment about "uniform continuity" is not "unfair" but rather mathematically incorrect. She seems to confuse uniform convergence and uniform continuity. Every continuous function on a compact interval is also uniformly continuous, so introducing this distincyion here is irrelevant. – Mikhail Katz Jul 30 '14 at 09:51
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    @katz What evidence is there that Grabiner does not know the difference between unif convergence and unif cty? It makes perfect sense to say: Cauchy was using the fact that in setting X, Property Y is equivalent to Property Z, but did not justify this. – Yemon Choi Jul 30 '14 at 10:07
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    @YemonChoi, I am basing myself on the quotation provided by Andras. The difference between uniform continuity and continuity is irrelevant here, as I explained. Furthermore you are misrepresenting my comment. I did not say that she "does not know the difference" but rathern that her comment here is irrelevant. It is mathematically incorrect to insist on the parallel between convergence and continuity since, unlike the case of convergence, there is no distinction between the two notions of continuity that's relevant to this case. – Mikhail Katz Jul 30 '14 at 10:15
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    @katz although the two notions of continuity coincide in this case, it is not automatic, and I interpret Grabiner's comment as suggesting: "since Cauchy did not yet comment on the semantic distinction between these notions, he might not have realized at first that in general one needs to explicitly state uniformity of the delta to make various plausible statements correct, and this is the same kind of oversight when he defines convergence of functions." I'm not claiming her assertion is historically correct, but it makes mathematical sense to me. – Yemon Choi Jul 30 '14 at 10:23
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    @Yemon, thanks for your comment. I think you mean "syntactic distinction" rather than "semantic distinction" since semantically there is no difference between continuity and uniform continuity on a compact, at least in classical logic. But what's the meaning of your comment on "uniformity of the delta"? Grabiner explicitly acknowledges in the introduction to her book that Cauchy never gave an epsilon delta definition of continuity. He consistently gave an infinitesimal definition of continuity in all his publications... – Mikhail Katz Jul 30 '14 at 11:12
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    ...Furthermore in 1853 Cauchy seems to extend his requirement to a larger domain by using the term "toujours", so if anything the relevant modern analogy would be between S-continuity at real points only versus at all points of the extended domain. – Mikhail Katz Jul 30 '14 at 11:13
  • @AndrásBátkai, the book by Grabiner is not "wonderful" but on the contrary profoundly flawed. See this post and the article linked there. – Mikhail Katz May 04 '17 at 13:48
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I recommend the reading of a wonderful 15-page text of Imre Lakatos, "Another case study of the method of proofs and refutations", which discuss precisely what Cauchy knew and didn't know and did and didn't do about this. It is published as an appendix in "Proofs and Refutations".

Google Book page of "Proofs and Refutations":

Joël
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    Could you please provide a link? – Mikhail Katz Mar 16 '17 at 16:02
  • It is a book, so I'm not sure it is available freely on the web. But I had a link to the google book page. – Joël Mar 16 '17 at 16:05
  • I am not saying that Lakatos' vision of the issue is compatible with the others expressed here. I remember that Lakatos is quite dismissive against the history pages in Bourbaki's element devoted to the question, but in my memory (which is faint) he doesn't mention infinitesimals. But anyway, if only for the style, it is a good read. – Joël Mar 16 '17 at 16:08
  • Oh, I thought you were referring to his essay that was published posthumously in the mathematical intelligencer in 1978 (the first volume of the journal I believe). That's an interesting text and I think quite different from what he wrote in proofs and refutations. – Mikhail Katz Mar 16 '17 at 16:23
  • Interesting, I'll read it. – Joël Mar 16 '17 at 19:31
  • So what did you find out by reading it? – Mikhail Katz May 09 '17 at 12:30
  • I do not have access to it (except by paying $35, with as a question of principle I do not want to do). My university ("maudit soit son nom!") gives me access to the Math Intel. Archive after 1993 only. If you know an open source access, I would be vert thankful. – Joël May 09 '17 at 18:51
  • Joël, I sent you a pdf via email. – Mikhail Katz May 10 '17 at 11:37