Timeline for Did Cauchy think that uniform and pointwise convergence were equivalent?
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10 events
| when toggle format | what | by | license | comment | |
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| May 4, 2017 at 13:48 | comment | added | Mikhail Katz | @AndrásBátkai, the book by Grabiner is not "wonderful" but on the contrary profoundly flawed. See this post and the article linked there. | |
| Jul 30, 2014 at 11:13 | comment | added | Mikhail Katz | ...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. | |
| Jul 30, 2014 at 11:12 | comment | added | Mikhail Katz | @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... | |
| Jul 30, 2014 at 10:23 | comment | added | Yemon Choi | @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. | |
| Jul 30, 2014 at 10:15 | comment | added | Mikhail Katz | @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. | |
| Jul 30, 2014 at 10:07 | comment | added | Yemon Choi | @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. | |
| Jul 30, 2014 at 9:51 | comment | added | Mikhail Katz | @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. | |
| Jul 21, 2014 at 18:39 | comment | added | arjun | 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. | |
| S Jul 21, 2014 at 15:50 | history | answered | András Bátkai | CC BY-SA 3.0 | |
| S Jul 21, 2014 at 15:50 | history | made wiki | Post Made Community Wiki by András Bátkai |