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Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.

Of course, Lebesgue and Poincaré knew each other, they even met on several occasions and shared a common close friend, Émile Borel.

However, it seems Lebesgue never wrote to Poincaré and, according to Lettres d’Henri Lebesgue à Émile Borel, note 321, p. 370

… la seule information, de seconde main, que nous avons sur l’intérêt de Poincaré pour la « nouvelle analyse » de Borel, Baire et Lebesgue

the only second-hand information we have on Poincaré's interest in the "new analysis" of Borel, Baire and Lebesgue

is this, Lebesgue to Borel, 1904, p. 84:

J’ai appris que Poincaré trouve mon livre bien ; je ne sais pas jusqu’à quel point cela est exact, mais j’en ai été tout de même très flatté ; je ne croyais pas que Poincaré sût mon existence.

I learned that Poincaré finds my book good; I do not know to what extent that is accurate, but I nevertheless was very flattered; I did not believe that Poincaré knew of my existence.

See also note 197, p. 359

Nous ne connaissons aucune réaction de Poincaré aux travaux de Borel, Baire et Lebesgue.

We do not know any reaction of Poincaré to the works of Borel, Baire and Lebesgue.

To my mind this situation is totally unexpected, almost incredible: the Lebesgue integral and measure theory are major mathematical achievements but Poincaré, the ultimate mathematical authority at this time, does not say anything??? What does it mean?

So, please, are you aware of any explicit or implicit statement by Poincaré on the Lebesgue integral or measure theory?

If you are not, how would you interpret Poincaré’s silence?

Pure disinterest? Why? Discomfort? Why? Something else?

This question is somewhat opinion-based, but

The true method of forecasting the future of mathematics is the study of its history and current state.

according to Poincaré and his silence is a complete historical mystery, at least to me.

ThiKu
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Fabrice Pautot
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    Well, at that time Poincaré was working on many subjects different from integration theory (relativity, analysis situs, dynamical systems, math foundations), so maybe he did not have the time to write extensively about Lebesgue's new theory. – Francesco Polizzi Nov 05 '19 at 06:33
  • As an aside, the first non-Lebesgue-misurable set (Vitali example) was constructed in 1905. I do not know any reaction of Poincaré to this, but I bet he was not happy, given his aversion to the Axiom of Choice. – Francesco Polizzi Nov 05 '19 at 06:39
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    @FrancescoPolizzi Thanks. Yes, Poincaré was quite busy :) but not a single statement between 1904 and 1912 at least about such important works is really hard to understand, precisely because when Poincaré is not happy (e.g. Cantor set theory), he's not reluctant to tell it. Moreover, see e.g. p. 260, Lebesgue to Borel, 1910: "I met Poincaré only to talk about Drach.". It seems like Lebesgue even never discussed his works privately with Poincaré! Same for Borel, apparently. Crazy story. – Fabrice Pautot Nov 05 '19 at 08:47
  • For comparison it might be useful to know whether Poincaré reacted to other developments he might have been concerned with, e.g. Lie groups. BTW one option is that he reacted in some way but there is no known written trace of it. – YCor Nov 05 '19 at 15:19
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    @ Polizzi "first non-Lebesgue-misurable set" do you mean miserable or measurable ? :) – meh Nov 05 '19 at 21:45
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    @aginensky misurabile :) – R W Nov 05 '19 at 21:49
  • @YCor That's my own option, see my comment below to Alexandre Eremenko. I fear I even guess his reaction. Please see my post https://mathoverflow.net/questions/339918/has-the-behrens-fisher-problem-been-solved ... that you voted for deletion! Physical continuum vs mathematical continuum, potential infinity vs actual infinity, Lebesgue vs Henstock-Kurzweil integrals, Kolmogorov vs Henstock probability theories are the keys. – Fabrice Pautot Nov 06 '19 at 08:49
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    It seems to me that we can only speculate to answer the question. I, on the contrary, think that there's nothing unusual in what you describe. Even nowadays not all the big names working in a field talk actively to each other. – user347489 Nov 07 '19 at 01:51
  • @user347489 I'm gonna answer my own question because I actually deduce that Poincaré the physicist could not welcome the Lebesgue integral and measure theory so that it is not surprising to see that he never said anything about them. More precisely, I believe Poincaré could have said something like this: "Measure theory disqualifies the Lebesgue integral."....................................... – Fabrice Pautot Nov 07 '19 at 06:26
  • this and the answers were a good read! – AlexArvanitakis Nov 07 '19 at 19:57

4 Answers4

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It has nothing to do with the conflict with Borel which developed later, and one can find a pretty explicit answer in the aforementioned letters of Lebesgue to Borel.

(These letters were first published in 1991 in Cahiers du séminaire d’histoire des mathématiques; selected letters with updated commentaries were also published later by Bru and Dugac in an extremely interesting separate book.)

In letter CL (May 30, 1910) Lebesgue clearly states:

Poincaré m'ignore; ce que j'ai fait ne s'écrit pas en formules.

Poincaré ignores me, [because] what I have done can not be written in formulas.

EDIT In interpreting this statement of Lebesgue I trust the authority of Bru and Dugac who in "Les lendemains de l'intégrale" accompany this passage with a footnote (missing in the 1991 publication) stating that

Dans [the 1908 ICM address] Poincaré ne semble pas considérer l'intégrale de Lebesgue comme faisant partie de "l'avenir des mathématiques", puisqu'il ne mentionne pas du tout la théorie des fonctions de variable réelle de Borel, Baire et Lebesgue.

In [the 1908 ICM address] Poincaré does not seem to consider the Lebesgue integral as a part of the "future of mathematics", as he does not mention at all the theory of functions of a real variable of Borel, Baire and Lebesgue.

I would rather interpret the meaning of "formulas" in the words of Lebesgue in a more straighforward and naive way. It seems to me that he was referring to the opposition which was more recently so vividly revoked by Arnold in the form of "mathematics as an experimental science" vs "destructive bourbakism".

By the way, it is interesting to mention that the first applications of the Lebesgue theory were - may be surprisingly - not to analysis, but to probability (and the departure point of Borel's Remarques sur certaines questions de probabilité, 1905 is clearly and explicitly the first edition of Poincaré's "Calcul des probabilités"). Poincaré had taught probability for 10 years and remained active in this area (let me just mention "Le hasard" that appeared first in 1907 and then was included as a chapter in "Science et méthode", 1908 and the second revised edition of "Calcul des probabilités", 1912), and still he makes no mention of Lebesgue's theory. This issue has been addressed, and there are excellent articles by Pier (Henri Poincaré croyait-il au calcul des probabilités?, 1996), Cartier (Le Calcul des Probabilités de Poincaré, 2006, the English version is a bit more detailed) and Mazliak (Poincaré et le hasard, 2012 or the English version). To sum them up,

[Poincaré's] seemingly limited taste for new mathematical techniques, in particular measure theory and Lebesgue’s integration, though they could have provided decisive tools to tackle numerous problems (Mazliak)

is explained by his approach of

a physicist and not of a mathematican (Cartier)

to these problems.

Manfred Weis
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R W
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    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... – Fabrice Pautot Nov 06 '19 at 21:54
  • Therefore, I will answer my own question. Hope you will appreciate... – Fabrice Pautot Nov 06 '19 at 21:56
  • 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. – Fabrice Pautot Nov 09 '19 at 00:20
  • 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. – Fabrice Pautot Nov 09 '19 at 00:36
  • 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??? – Fabrice Pautot Nov 09 '19 at 00:47
  • 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? – Fabrice Pautot Nov 09 '19 at 01:02
  • 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"). – R W Nov 09 '19 at 01:42
  • 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? – Fabrice Pautot Nov 09 '19 at 01:53
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    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... – R W Nov 09 '19 at 01:54
  • 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. – R W Nov 09 '19 at 01:57
  • 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...). – Fabrice Pautot Nov 09 '19 at 01:59
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    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. – R W Nov 09 '19 at 02:03
  • ... 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? – Fabrice Pautot Nov 09 '19 at 02:03
  • 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? – Fabrice Pautot Nov 09 '19 at 02:07
  • This problem for instance. I would be more than happy to get your feedback on it.................. – Fabrice Pautot Nov 09 '19 at 02:14
  • 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........ – Fabrice Pautot Nov 09 '19 at 08:05
  • 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. – Fabrice Pautot Nov 09 '19 at 08:22
  • 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." – Fabrice Pautot Nov 09 '19 at 09:03
  • 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... – Fabrice Pautot Nov 09 '19 at 09:05
  • ... 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." – Fabrice Pautot Nov 09 '19 at 09:05
  • 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... – Fabrice Pautot Nov 09 '19 at 20:48
  • ... 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. – Fabrice Pautot Nov 09 '19 at 20:52
  • 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." – Fabrice Pautot Nov 10 '19 at 07:21
  • 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. – Fabrice Pautot Nov 10 '19 at 07:28
  • 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... – Fabrice Pautot Nov 10 '19 at 22:41
  • 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? – Fabrice Pautot Nov 12 '19 at 04:29
  • As French people say: snif snif. :( I would be happy to continue the discussion with you. – Fabrice Pautot Nov 14 '19 at 16:58
  • 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. – Fabrice Pautot Nov 20 '19 at 10:06
  • FYI, very interesting French paper here – Fabrice Pautot Nov 28 '19 at 14:31
  • Thank you for the reference - and sorry for the silence :) – R W Nov 28 '19 at 17:25
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The main applications of Lebesgue integral to concrete problems of analysis found before Poincare's death are the Riesz-Fischer theorem (1907) and Fatou's work (1906). All this is somewhat remote from the main interests of Poincare. Applications of measure theory to mechanics (ergodic theory) were found later, after his death.

You cannot expect even the greatest mathematician to react quickly to ALL important discoveries.

EDIT. Recently I read an old survey of harmonic analysis written by N. Wiener, "Historical background of harmonic analysis". I cite:

"...The notion of "almost all" has became an accepted part of the equipment of every physicist. It was under the influence of ideas belonging to this domain that Poincare at the end of the last century developed the philosophy in questions of the theory of probability which marked the first really great progress in that theory since the days of Laplace."

"The ideas of statistical randomness and phenomena of zero probability were current among the physicists and mathematicians in Paris around 1900, and it was in a medium heavily ionized by these ideas that Borel and Lebesgue solved the mathematical problem of measure."

I discussed with colleagues what does he exactly mean by "first really great progress since the days of Laplace", and we decided that this was his famous "return theorem". And certainly this was very long before the measure theoretic foundations of probability "were laid down by Kolmogorov".

Alexandre Eremenko
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  • Of course, that's a possibility even if i) 8 years (1904-1912) is a very long time for Poincaré ii) a non-reactive Poincaré to a major next-door, French discovery is simply not Poincaré iii) if Poincaré the mathematician might not have reacted "quickly", there are good reasons to think that Poincaré the physicist might have reacted immediately and very strongly (e.g. Borel never understood Poincaré about the physical continuum vs the mathematical one). My own guess is that Poincaré reacted, perhaps very strongly, in 1904, but that he decided to keep his reaction for himself. – Fabrice Pautot Nov 06 '19 at 08:13
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Poincaré studied with Hermite, who famously in a letter 1893 to Stieltjes wrote „I turn with terror and horror from this lamentable scourge of continuous functions with no derivative.“ Poincaré himself is often quoted „Heretofore when a new function was invented it was for some practical end; today they are invented expressly to put at fault the reasoning of our fathers; and one will never get more from them than that.“ Of course these quotes are older than the Lebesgue integral, yet they may explain why integration of pathological functions was not considered to be important by Poincaré and other French mathematicians.

ThiKu
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  • The quote you attribute to Poincare is actually due to Andre Bloch. This is what he wrote in connection with Fatou's example, which is known nowadays as the "Fatou-Bieberbach example". – Alexandre Eremenko Nov 08 '19 at 12:50
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    @AlexandreEremenko No, Thiku is right, this is Poincaré's. Original quote: "Autrefois, quand on inventait une fonction nouvelle, c'était en vue de quelque but pratique ; aujourd'hui, on les invente tout exprès pour mettre en défaut les raisonnements de nos pères, et on n'en tirera jamais que cela.". Science et méthode, bottom of page 132 http://jubilotheque.upmc.fr/fonds-physchim/PC_000305_001/document.pdf?name=PC_000305_001_pdf.pdf – Fabrice Pautot Nov 08 '19 at 13:56
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    @Fabrice Pautot: Thank you very much. I clearly remember seeing this in Bloch. Probably Bloch CITES Poincare. – Alexandre Eremenko Nov 08 '19 at 14:06
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    @AlexandreEremenko My pleasure. – Fabrice Pautot Nov 08 '19 at 14:09
  • I believe we can take this statement by Poincaré as an implicit criticism of the Lebesgue integral. Poincaré might have something like Dirichlet function in mind, which is Lebesgue-integrable but not Riemann-integrable, but not very "practical". Therefore, I'll be happy to validate your answer if you basically agree with this interpretation. – Fabrice Pautot Nov 08 '19 at 17:09
  • Don‘t ask me, ask Poincaré :-) . You’re right that the quote is from „Science et Méthode“ which appeared in 1908, so certainly Poincaré knew Lebesgue‘s 1902 thesis at the time. Looking at the context of the quote in „Science et Méthode“ it seems to refer to many things. The Lebesgue-integrable non-Riemann-integrable functions might have been one of them, but certainly not the only one. – ThiKu Nov 08 '19 at 22:15
  • @ThiKu Please, any other suggestion for the pathological functions? – Fabrice Pautot Nov 08 '19 at 23:19
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    Many Fourier series yield functions which are Lebegue-integrable but not Riemann-integrable. The Riesz-Fischer theorem $L^2(\left[-\pi,\pi\right])=l^2({\mathbf N})$ wouldn‘t be true with Riemann‘s notion of integral. – ThiKu Nov 08 '19 at 23:35
  • @ThiKu Ok, thanks. R W input below is crucial............... The Lebesgue was not very exciting for Poincaré the mathematician, but for Poincaré the physicist, the probabilist (he used to conceive probability theory not as a branch of mathematics but as a branch of mathematical physics) and the philosopher of sicence, that's another story....... – Fabrice Pautot Nov 08 '19 at 23:41
  • @FabricePautot I can understand that enlarging the class of integrable functions to not very practical functions, if not pathological, may not look like an exiting achievement for a mathematical physicist. But the great improvement of Lebesgue theory is not in “quantity” (enlarging the class), but in “quality”: completeness methods, limit theorems, etc, which are of great practical interest. So I agree with you, the story has a mysterious side – Pietro Majer May 29 '22 at 21:26
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The Lebesgue-Borel conflict may provide a hint why Poincaré was, like Borel, not impressed by Lebesgue's contribution:

As we hinted earlier, all was not well between Borel and Lebesgue and their long-standing friendship deteriorated until it finally collapsed, at Lebesgue's instigation, in 1917. The evidence we have is provided by letters preserved at the Institute Poincaré, which Lebesgue wrote to Borel starting in 1901. The reasons, both psychological and scientific, are complex. To begin with, Borel, along with such luminaries as Kronecker and Poincaré, was a constructivist, so he rejected Lebesgue's generalisation of his measure concept as having no meaning since it was non-constructive.

—G. T. Q. Hoare and N. J. Lord, 'Intégrale, longueur, aire' the Centenary of the Lebesgue Integral, The Mathematical Gazette Vol. 86, No. 505 (2002) pp. 3–27, doi:10.2307/3621569.

kodlu
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Carlo Beenakker
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    Thanks Carlo, I'm aware of the deterioration of the friendship between Borel and Lebesgue. But according to Lebesgue quotation above, Poincaré was rather happy with his works at least in 1904! Unfortunately, we don't know how Lebesgue learned that "Poincaré finds my book good." Another main issue for understanding this strange story is that the letters from Borel to Lebesgue are lost, unfortunately. – Fabrice Pautot Nov 05 '19 at 09:34
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    Are there any evidences that Poincaré was aware of Vitali's counterexample? – Francesco Polizzi Nov 05 '19 at 09:40
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    @FrancescoPolizzi I'd like to know! – Fabrice Pautot Nov 05 '19 at 14:38