Timeline for A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
Current License: CC BY-SA 4.0
41 events
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| Oct 17, 2020 at 18:44 | history | edited | Manfred Weis | CC BY-SA 4.0 |
deleted a duplicate "have"
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| Nov 28, 2019 at 17:25 | comment | added | R W | Thank you for the reference - and sorry for the silence :) | |
| Nov 28, 2019 at 14:31 | comment | added | Fabrice Pautot | FYI, very interesting French paper here | |
| Nov 20, 2019 at 10:06 | vote | accept | Fabrice Pautot | ||
| Nov 20, 2019 at 10:06 | comment | added | Fabrice Pautot | Anyway, thanks a lot again, your answer is, by far, the best one from my own point of view. Therefore, I'm happy to validate it. | |
| Nov 14, 2019 at 16:58 | comment | added | Fabrice Pautot | As French people say: snif snif. :( I would be happy to continue the discussion with you. | |
| Nov 12, 2019 at 4:29 | comment | added | Fabrice Pautot | Regarding Poincaré's "infiniment petite", should I/we understand that Poincaré does not accept the tertium non datur, that is the sum rule of probability theory? | |
| Nov 10, 2019 at 22:41 | comment | added | Fabrice Pautot | Thanks for Pier paper, quite interesting. Did Poincaré believe in probability theory? Certainly not! Only in very special circumstances, e.g. his marvelous method of arbitrary functions, improved by Borel and Fréchet (missing in this paper). But he's right: my own starting point was precisely this question "Can we postulate a probabilistic, statistical model?". It took me 4 years, 1999-2002, to answer this single question and the answer is NO, probability theory is not hypothetico-deductive. Poincaré could not see this because he was not reasoning in terms of information... | |
| Nov 10, 2019 at 7:28 | comment | added | Fabrice Pautot | I would say that sometimes the job can be very easy (e.g. for the Behrens-Fisher problem, obtain the limit Bayes factors) but absolutely NECESSARY, for what is required is not the ratio of the limits, which is undefined 0/0, but the limit of the ratios, which is not obtained by Lebesgue integrals. | |
| Nov 10, 2019 at 7:21 | comment | added | Fabrice Pautot | Translated from the French version of Cartier Le calcul des probabilités de Poincaré, I don't have the English one, p.10, 14.9: "After that, it only remains (!) to do some combinatorics in order to compute complex finite sums, and then some asymptotic calculus to obtain the limit probabilities. The job may be difficult, but the framework is basic and without a trap. A good part of the calculus of probabilities, and statistical physics, can be played on this stage." | |
| Nov 9, 2019 at 20:52 | comment | added | Fabrice Pautot | ... In order to get the correct solution, we have to forget actual infinity and go back to potential infinity in order to take the limit of the solutions to the discrete problems. By construction, they are obtained as Riemann or Henstock-Kurzweil integrals, not Lebesgue's. Measure theory itself disqualifies the Lebesgue integral in some problems of probability theory. Poincaré could have stated something like this. | |
| Nov 9, 2019 at 20:48 | comment | added | Fabrice Pautot | I recap: the Behrens-Fisher problem intuitively makes sense, it has vital applications such as clinical trials (placebo vs treatment). However, from the point of view of Bayesian probability theory + measure theory it does not make sense, for the probability that the numerical values of two continuous parameters are equal to each other is equal to zero, a priori and a posteriori. That's the reason why the standard "solution" found in any textbook is completely wrong, in particular it violates measure theory... | |
| Nov 9, 2019 at 9:05 | comment | added | Fabrice Pautot | ... For then one is forbidden to take into account both the sensible intuition and the intuition a priori that preside over the mathematical conception of the continuum and whose importance is only understood once the positive virtue of the potential infinity is recognized." | |
| Nov 9, 2019 at 9:05 | comment | added | Fabrice Pautot | p. 208: "If, indeed, Poincaré defends the philosophical thesis according to which we must understand the continuum as a potential infinity, it is not for lack of questioning it, but because it finds in potential infinity a virtue that actual infinity does not possess. It is therefore not only because of the paradoxes that it engenders that Poincaré rejects actual infinity, it is because by considering the infinity of the continuum as actual and not as potential, we do not see how it gives mathematics the specificity that makes it the privileged language of physics... | |
| Nov 9, 2019 at 9:03 | comment | added | Fabrice Pautot | Translated from paper [7], p.207: "Thus, Poincaré's solution consists in saying that if we consider the "set" of the rational numbers as a potential infinity (...) and not as an actual infinity, the idea which governs the resolution of the contradiction of the physical continuum is present without the cloudiness of the physical continuum being lost." | |
| Nov 9, 2019 at 8:22 | comment | added | Fabrice Pautot | Borel shows Poincaré that they can solve new mathematical probabilistic problems thanks to the Lebesgue integral. Poincaré could have shown Borel that some physical, statistical probabilistic problems cannot be solved with the Lebesgue integral. That's what I believe in. Please tell me if this makes sense or not. | |
| Nov 9, 2019 at 8:05 | comment | added | Fabrice Pautot | The solution to this problem, which I owe to Poincaré himself, is what motivated the present question. I'm convinced Poincaré would have found it by himself... instantaneously because it is perfectly in line with what he used to say about the physical continuum and the mathematical one, see reference [7] in my draft paper/note: the suitable infinity for physics is the potential one, not the actual one, therefore we should pass to the limit at the end, not at the beginning, otherwise the problem is degenerate, and this is why it took such as long time to solve the Behrens-Fisher problem........ | |
| Nov 9, 2019 at 2:14 | comment | added | Fabrice Pautot | This problem for instance. I would be more than happy to get your feedback on it.................. | |
| Nov 9, 2019 at 2:07 | comment | added | Fabrice Pautot | To come back to my main question please: what if we can exhibit a probabilistic problem whose solution is given ONLY by a limit of a sequence of Riemann sums? | |
| Nov 9, 2019 at 2:03 | comment | added | Fabrice Pautot | ... but Poincaré says just above that the probability that x is 'incommensurable", irrational is equal to 1. Therefore the probability that x is rational should be exactly equal to 0, isn't it? What do I miss? | |
| Nov 9, 2019 at 2:03 | comment | added | R W | Yes - I agree - just wanted to emphasize that the reality is somewhat different from what it is supposed to be :) Have a look at Doob's articles - they are quite entertaining. | |
| Nov 9, 2019 at 1:59 | comment | added | Fabrice Pautot | Thanks a lot. I mean, Kolmogorov Grundbegriffe is (supposed to be) the standard axiomatic, measure-theoretic system for probability theory even if there exists many alternative systems (Cox, Nelson, Henstock, Renyi, Bernstein...). | |
| Nov 9, 2019 at 1:57 | comment | added | R W | Concerning the reference I just wanted to make sure I don't miss anything. I think the reason why Poincaré says "infiniment petite" is exactly that he wanted to avoid giving any precise definition of this quantity. | |
| Nov 9, 2019 at 1:54 | comment | added | R W | Concerning the statement that "since Kolmogorov ... probability theory ... relies on the Lebesgue integral". Alas, this is quite far from the reality. Most probabilists don't know measure theory, are afraid of using it, and, in particular, avoid dealing with probability measures (!). There are several very instructive articles by Doob about it written in the 90s (for instance, Probability vs. Measure which he concludes by saying that "adherents of each aspect [of probability] are human and therefore scorn adherents of the other"). I could give more recent examples as well... | |
| Nov 9, 2019 at 1:53 | comment | added | Fabrice Pautot | Sorry, I don't get your point about the precise reference. I just mean that i) Borel says "La réponse évidente (1, p. 148) aux deux questions précédentes est zéro" ii) Poincaré actually does not take the trouble to prove that it is "infiniment petite". Of course, that the integral vanishes is quite an intuitive result. I don't really understand why Poincaré says "infiniment petite" instead of 0? | |
| Nov 9, 2019 at 1:42 | comment | added | R W | Could you give a precise reference for "so obvious that they don't deserve to be proved by Poincaré"? Actually, Borel in his 1905 paper quotes the first 1896 edition of Poincaré's book (p.126). The passage from p.126 of the first edition is without any changes reproduced in the 1912 edition as well (on p.148 which you mention). However, most likely Poincaré did not need any theory to conclude that any "common sense" integral of a function which is non-zero only at rational points vanishes (moreover, literally he just says that the corresponding probability is "infiniment petite"). | |
| Nov 9, 2019 at 1:05 | vote | accept | Fabrice Pautot | ||
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| Nov 9, 2019 at 1:02 | vote | accept | Fabrice Pautot | ||
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| Nov 9, 2019 at 1:02 | comment | added | Fabrice Pautot | Borel says that, thanks to the new Lebesgue integral, we can finally prove results that are so obvious that they don't deserve to be proved by Poincaré! :) Who first proved that the "integral" of the Dirichlet function over [0,1] is 0? | |
| Nov 9, 2019 at 0:47 | comment | added | Fabrice Pautot | Another less important but interesting question please: in the 1905 paper, Borel says that the integral of the Dirichlet function over [0,1] is obviously equal to zero, citing Poincaré who actually gives this result without a proof, on page 148 of Le calcul des probabilités, second edition, 1912. Then, Borel explains that this result cannot be derived with the classical Darboux integral, only with the new Lebesgue integral. So, how did Poincaré derive/prove/guess this result??? Did Poincaré use the Lebesgue integral, after all??? | |
| Nov 9, 2019 at 0:36 | comment | added | Fabrice Pautot | Very interesting, isn't it? Now, please let me ask you one question. Of course, any Riemann-integrable function is Lebesgue-integrable. But can the limit of a sequence of Riemann sums be interpreted as a Lebesgue integral??? I ask this question because if we can ever exhibit a probabilistic problem whose solution is given by a limit of a sequence of Riemann sums and if such limit is, by definition, a Riemann or an Henstock-Kurzweil integral but not a Lebesgue integral, then we could prove that Poincaré was right in being reluctant to rely on the Lebesgue integral within probabilty theory. | |
| Nov 9, 2019 at 0:20 | comment | added | Fabrice Pautot | Your input is CRUCIAL. I'd like to discuss with you more extensively if ever possible please. May I recap the situation like this: for some reasons to be understood, Poincaré could ESPECIALLY not welcome the Lebesgue integral because he was also a physicist and, in particular (he used to conceive le calcul des probabilités as a branch of mathematical physics), a probabilist... and a philosopher of science. But since Kolmogorov Grundbegriffe (and even Borel 1905 paper as you pointed it out), probability theory, for mathematicians, physicists and statisticians, relies on the Lebesgue integral. | |
| S Nov 7, 2019 at 15:44 | history | edited | R W | CC BY-SA 4.0 |
Added translation of quote. Could probably be done better by someone who actually knows French.
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| S Nov 7, 2019 at 15:44 | history | suggested | eirikdaude | CC BY-SA 4.0 |
Added translation of quote. Could probably be done better by someone who actually knows French.
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| Nov 7, 2019 at 13:19 | review | Suggested edits | |||
| S Nov 7, 2019 at 15:44 | |||||
| Nov 6, 2019 at 21:56 | comment | added | Fabrice Pautot | Therefore, I will answer my own question. Hope you will appreciate... | |
| Nov 6, 2019 at 21:54 | comment | added | Fabrice Pautot | Very nice answer. Are you French RW? I've already read Poincaré, Borel, Cartier and Mazliak (who's doing a great job) but not Pier and Bru (who's doing a great job too and I should read his book). In fact, by a probabilistic reasoning (probability theory belonging to mathematical physics according to Poincaré), I guessed that Poincaré the physicist and philosopher of science could not welcome the Lebesgue integral and measure theory, and I was not surprised at all to check a posteriori that he never said anything about them, probably in order to preserve his friendship with Borel... | |
| Nov 6, 2019 at 19:12 | history | edited | R W | CC BY-SA 4.0 |
added 18 characters in body
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| Nov 6, 2019 at 18:43 | history | edited | R W | CC BY-SA 4.0 |
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| Nov 5, 2019 at 22:24 | history | edited | user44143 | CC BY-SA 4.0 |
Added translation, made references less of an interruption
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| Nov 5, 2019 at 20:25 | history | answered | R W | CC BY-SA 4.0 |