Timeline for Was the early calculus inconsistent?
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| Apr 13, 2017 at 12:51 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 11, 2016 at 16:53 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Aug 21, 2013 at 17:37 | comment | added | Mikhail Katz | @Yemon: this is a potentially valid criticism. The answer to the criticism is carefully distinguishing between syntactic and semantic issues, as has been discussed elsewhere. At some point one has to confront the question as to which point of you is more meaningful on Leibniz: the one that claims that his infinitesimals were logically inconsistent and were finally swept away by Weierstrass, or the one that finds modern proxies to his Law of Continuity and Law of Homogeneity in the context of modern infinitesimal theories. The editors of Erkenntnis did not find obvious Whig flaws with our piece | |
| Aug 21, 2013 at 17:26 | comment | added | Yemon Choi | I agree they are separate issues, which is why I get a bit frustrated when Connes's comments - which always sounded to me like he was not thinking seriously but aiming for aphorism - are brought up. I might even argue that people addressing the infinitesimals of Leibniz and Euler by bringing up NSA are themselves drifting into Whig history... | |
| Aug 21, 2013 at 17:20 | comment | added | Mikhail Katz | @Yemon: you seem to be referring rather to mount Bur-sur-Yvette :-) Speaking of which, I had some discussions with Pierre Cartier who is very favorably disposed to infinitesimals both historical and modern (and his written about this). But I am not sure why discussions of the shortcomings of the coverage of the early calculus tend inevitably to turn into a court case on nonstandard analysis, as if ackowledging the bias of post-Weierstrassian historians will necessarily prove A. Robinson's point. The two issues are separate issues. | |
| Aug 21, 2013 at 17:08 | comment | added | Yemon Choi | It was, I'm afraid, purely hypothetical. But I picked three people who have done concrete harmonic/Fourier analysis of a high standard, certainly fluent in epsilons and deltas, who I suspect actually think in a more intuitive way. (Bourgain is just in there in case we are for some reason concentrating on what Fields Medallists may have brought down from Mount Sinai) | |
| Aug 21, 2013 at 16:51 | comment | added | Mikhail Katz | @Yemon: Could you provide some links on what Korner, Bourgain, and Nazarov may have said about infinitesimals? | |
| Aug 21, 2013 at 16:43 | comment | added | Yemon Choi | I also would care rather more what e.g. Thomas Korner or Jean Bourgain or Fedja Nazarov say about infinitesimals than anything Connes may have said in an attempt to be aphoristic, but perhaps that's just me :) | |
| Aug 21, 2013 at 16:41 | comment | added | Yemon Choi | Mikhail, yes I was reading the wrong thread title in my browser. Actually I agree with your last sentence, I have just never come across such denigration throughout my education as an analyst - which was as classical as you could imagine - and my teaching of the subject. FWIW I think the early approaches were incomplete (you have argued well against inconsistency) and that W's approach provides a fix, not nec. the only fix. | |
| Aug 21, 2013 at 16:39 | comment | added | Mikhail Katz | @Yemon: While I clearly perceive the sarcasm, I am not actually sure what you mean. The title of this question does not contain the word "history". Are you referring to the SE thread on the "victors"? Just to be perfectly clear, I am a great fan of Weierstrass in general and epsilontics in particular. I just don't think a post-Weierstrassian spin accompanied by denigration of the early masters does justice to the history you find fascinating. | |
| Aug 21, 2013 at 16:35 | comment | added | Yemon Choi | [deleted comment made when reading the wrong question] | |
| Aug 21, 2013 at 16:28 | history | edited | Mikhail Katz |
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| Jul 23, 2013 at 10:10 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Apr 8, 2013 at 12:45 | vote | accept | Mikhail Katz | ||
| Apr 8, 2013 at 12:29 | answer | added | Vladimir Kanovei | timeline score: 10 | |
| Apr 2, 2013 at 18:24 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
response to a common misconception
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| Mar 26, 2013 at 21:17 | answer | added | Paolo Giordano | timeline score: 4 | |
| Mar 23, 2013 at 20:00 | comment | added | Mikhail Katz | @Lee Mosher: you speak of a "very ahistorical assumption that the word 'consistent' was used by Bishop Berkeley in the same sense that it is used by modern logicians". Actually, it is exactly the same sense. Berkeley claimed the calculus was based on the inconsistency $(dx\not=0)\wedge(dx=0)$. | |
| Mar 22, 2013 at 9:04 | answer | added | Mikhail Katz | timeline score: 7 | |
| Mar 20, 2013 at 17:42 | comment | added | Mikhail Katz | @Joël: You wrote that "Yet I am still not convinced of the premise: is the claim that the early calculus was inconsistent really "ubiquitous"? I don't know the authors that Katz is mentioning to substantiate his claim: Boyer, Grabiner?" Grabiner is a historian. She wrote: "As Berkeley put it, the quantity we have called h 'might have signified either an increment or nothing. But then, which of these soever you make it signify, you must argue CONSISTENTLY with such its signification'. See maa.org/pubs/Calc_articles/ma002.pdf The implication is the early practices were inconsistent. | |
| Mar 20, 2013 at 16:58 | comment | added | Mikhail Katz | @Ben Brown: You wrote above that "They also treated certain quantities as both zero and non-zero in proofs, which is inconsistent with modern proof practice...Thus, the early practitioners found their techniques consistent, and modern commentators find their use of zero/non-zero quantities in proofs inconsistent." Berkeley precisely alleged that authors like Leibniz treated certain quantities as both zero and nonzero. But is Berkeley's criticism of Leibniz accurate? | |
| Mar 20, 2013 at 15:40 | answer | added | Philip Ehrlich | timeline score: 17 | |
| Mar 20, 2013 at 14:45 | answer | added | Lee Mosher | timeline score: 15 | |
| Mar 20, 2013 at 14:16 | answer | added | Wouter Stekelenburg | timeline score: 1 | |
| Mar 20, 2013 at 13:27 | answer | added | Andrej Bauer | timeline score: 13 | |
| Mar 20, 2013 at 0:02 | answer | added | Joël | timeline score: 42 | |
| Mar 19, 2013 at 23:17 | comment | added | Joël | ... contains only true, if not rigorously proved at that time, results. And a set of true results can not be, by any sense of the term, "inconsistent". So really, I am not sure what the question is about... | |
| Mar 19, 2013 at 23:13 | comment | added | Joël | Yes, I think too that the question is much more interesting as the first one about the "victors". Yet I am still not convinced of the premise: is the claim that the early calculus was inconsistent really "ubiquitous"? I don't know the authors that Katz is mentioning to substantiate his claim: Boyer, Grabiner? I don't read either a lot of calculus textbook. It seems to me that the naive, uniformed, dominant view about mathematicians was that the corpus of results of the calculus pioneers (Descartes, Pascal, Fermat, Newton, Leibniz, and you can continue with the Bernoulli(s), perhaps Euler).. | |
| Mar 19, 2013 at 22:56 | answer | added | Alexandre Eremenko | timeline score: 6 | |
| Mar 19, 2013 at 22:20 | comment | added | Joel David Hamkins | I find the question very interesting, and I am looking forward to reading answers posted by those with expertise in mathematical history. | |
| Mar 19, 2013 at 22:00 | comment | added | Ben Braun | ... Thus, the early practitioners found their techniques consistent, and modern commentators find their use of zero/non-zero quantities in proofs inconsistent. | |
| Mar 19, 2013 at 21:59 | comment | added | Ben Braun | The question is not clear, as the word "consistent" has multiple meanings. One must separately consider consistency of techniques and consistency of the arguments used to justify techniques. The practitioners of the early calculus viewed it as a set of legitimate methods for obtaining results that agreed with other methods and physical observation. They also treated certain quantities as both zero and non-zero in proofs, which is inconsistent with modern proof practice... | |
| Mar 19, 2013 at 20:18 | comment | added | Paul Siegel | I've occasionally wondered about this too, and here is how I made it precise in my mind. Many arguments in calculus were rather vague before the advent of deltas and epsilons, so it seems possible that there was a computation for which two different vague arguments gave different answers, or a vague argument which gave the right answer for one computation but the wrong answer in another. Did this ever happen? | |
| Mar 19, 2013 at 19:33 | comment | added | Mikhail Katz | @Yemon Choi: Berkeley is certainly taken seriously by philosophers, and perhaps more so than by mathematicians. Still, reliance on Berkeley's analysis of the calculus to label the latter "inconsistent" is ubiquitous in the history of math literature. If you are serious about questioning this, I can try to provide further references in addition to Boyer and Grabiner. | |
| Mar 19, 2013 at 19:32 | comment | added | Lee Mosher | At the root of this question is an implicit and very ahistorical assumption that the word "consistent" was used by Bishop Berkeley in the same sense that it is used by modern logicians. | |
| Mar 19, 2013 at 19:30 | comment | added | Mikhail Katz | @Ryan Budney: Authors like Boyer, Grabiner, as well as authors of calculus textbooks routinely claim such "inconsistency" ($dx\not=0$, $dx=0$, Q.E.D.). It could be that the early calculus was not "formal enough" as you suggest, but what do we make of such repeated claims in the literature? To illustrate a concept from the early calculus that was clearly inconsistent, consider Nieuwentijt's idea of an infinitesimal of the form $\frac{1}{\infty}$ that was supposed to be nilpotent. This is inconsistent by any modern standard, unlike Leibnizian calculus where the question is debatable as per above | |
| Mar 19, 2013 at 19:26 | comment | added | Yemon Choi | Berkeley is of course famous to most people for his ontology, not his views on calculus/fluxions. | |
| Mar 19, 2013 at 19:24 | comment | added | Yemon Choi | I prefer this question to the previous one, although we are now firmly in the realm of HPS when we try to understand what Berkeley meant by "inconsistency" (or for that matter, in what venues Vickers finds this assertion "ubiquitous"). | |
| Mar 19, 2013 at 19:20 | comment | added | Ryan Budney | It's not clear to me "early calculus" was formal enough to talk about whether or not it was consistent. There were things people did, so you can talk about the actions of people being consistent or inconsistent makes sense on a behavioral level, but by that standard they probably were fairly consistent. | |
| Mar 19, 2013 at 18:41 | comment | added | Gerald Edgar | subjective and argumentative? | |
| Mar 19, 2013 at 18:33 | history | edited | Mikhail Katz |
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| Mar 19, 2013 at 18:23 | history | asked | Mikhail Katz | CC BY-SA 3.0 |