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.
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2$\begingroup$ See answers or comments by me and Greg Graviton at mathoverflow.net/questions/35468/… $\endgroup$– KConradCommented Jul 21, 2014 at 14:41
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8$\begingroup$ 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? $\endgroup$– Steven GubkinCommented Jul 21, 2014 at 18:02
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2$\begingroup$ @StevenGubkin, indeed uniform convergence is equivalent to a certain pointwise condition (in the extended domain) for the natural extensions of the functions. $\endgroup$– Mikhail KatzCommented Jun 16, 2016 at 14:14
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3 Answers
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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..
answered Jul 30, 2014 at 9:33
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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."
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3$\begingroup$ 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. $\endgroup$– arjunCommented Jul 21, 2014 at 18:39
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3$\begingroup$ @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. $\endgroup$ Commented Jul 30, 2014 at 9:51
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1$\begingroup$ @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. $\endgroup$ Commented Jul 30, 2014 at 10:07
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4$\begingroup$ @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. $\endgroup$ Commented Jul 30, 2014 at 10:15
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1$\begingroup$ @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. $\endgroup$ Commented Jul 30, 2014 at 10:23
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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":
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1$\begingroup$ Could you please provide a link? $\endgroup$ Commented Mar 16, 2017 at 16:02
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$\begingroup$ 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. $\endgroup$– JoëlCommented Mar 16, 2017 at 16:05
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$\begingroup$ 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. $\endgroup$– JoëlCommented Mar 16, 2017 at 16:08
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$\begingroup$ 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. $\endgroup$ Commented Mar 16, 2017 at 16:23
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