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This question has been discussed on several sites including MathOverflow but with not definite result. Presumely HSE is best suited.

Jeremy Gray denies that Poincare said, "Later generations will regard Mengenlehre as a disease from which one has recovered." [1]

Skolem however notes as a rumor "he is reported to have said in a talk at the international congress of mathematicians at Rome in 1908 that in future set theory would be considered as a disease from which one has recovered." [2]

Is there further evidence in support of Skolem?

1) Jeremy Gray, "Did Poincare say 'set theory is a disease'?", Math. Intelligencer 13 (1991) 19-22. 2) Thoralf Skolem: "Über die Grundlagendiskussionen in der Mathematik", Den Syvende Skandinav. Matematikerkongr. Oslo (1929).

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  • $\begingroup$ Could you please provide links to the earlier discussions, at least the one on MO you mention? $\endgroup$ – Ben May 19 '17 at 10:54
  • $\begingroup$ See Poincaré and set theory $\endgroup$ – Mauro ALLEGRANZA May 19 '17 at 11:25
  • $\begingroup$ See also What was the motivation behind Poincaré saying that topology was a disease $\endgroup$ – Mauro ALLEGRANZA May 19 '17 at 11:30
  • $\begingroup$ "Skolem notes as a rumor"... as you can see from the related material, the quote is very "indirect: Gray says that Pierpont says taht Holder said, but it seems that we have no direct source. But Poincaré approach against Cantorism was very critical, and thus the quote is for sure "in the spirit" of Poincaré. $\endgroup$ – Mauro ALLEGRANZA May 19 '17 at 11:44
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    $\begingroup$ @Andrés E. Caicedo I hope that all posters are aware of the background of their questions. $\endgroup$ – Otto May 23 '17 at 12:19
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I would say that the result is pretty definite. Gray does not simply deny that the quote is genuine, he quotes the scholarship of Moore (Zermelo' s Axiom of Choice), Cassinet-Guillemot (L'Axiome du choix) and Dauben (Georg Cantor), all of whom neglect to mention it. Moreover, the sources that do mention it, like Skolem and Kline, trace it to Poincare's remarks at the Fourth International Congress of Mathematicians in Rome, 1908, but Poincare's essay circulated there, The Future of Mathematics, has nothing of the sort. The closest he comes to curing "disease" is:

"I think, and I am not the only one who does, that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case [beau cas pathologique]."

The "not be completely defined by a finite number of words" is the phrasing that French proto-constructivists (Borel, Baire, Lebesgue) and Poincare used since 1905 to criticize Zermelo's proof of the well-ordering theorem from the axiom of choice, and similar arguments, see Where did Borel stress that $\mathbb{Q}$ being effectively enumerable by $\mathbb{N}$ is not about its size? It was motivated by the Richard's paradox (of the smallest number not definable in finitely many words), and led to Poincare's predicativist vicious circle principle. Thus, the beau cas pathologique referred specifically to Zermelo-type "constructions":

"Thus, although I am favorably disposed to accept Zermelo's Axiom, I reject his proof, which for an instant had made me believe that aleph-one could actually exist."

Better yet, Skolem's sources are reported rumours, and Kline's is... E.T. Bell, whose reputation for making up and embellishing stories is legendary, for some of his handiwork see What resources are available for lives of recent mathematicians besides E.T. Bell's Men of Mathematics? What's the famous story about a mathematician who gave a talk without saying a word? and Did Cauchy forget or lose mathematical papers aside from Abel's and Galois's? (hint: Cauchy did not even lose Galois', but why let facts get in the way of a good story). Bell quotes the above passage before adding the "later generations will regard Mengenlehre as a disease from which one has recovered", dates it to 1905, and Kline copies him despite knowing that the essay is from 1908.

But this time the juicy story fell into Bell's lap. The "quote" first appears in Pierpont's address of 1928 (Pierpont at least got the date right), and his source was likely Hölder's 1924 book. One or both of them were likely the sources of Skolem's "reportage" too. Except Hölder offers the "quote" as a summary, and he was not at the Congress himself. There is no trace of the "quote" before 1924, so Gray's reconstruction of events seems very reasonable:

"Hölder, who was not at the Rome Congress but knew that Poincare's address had caused a stir, offered at the end of his book a summary of what Poincare had said. The beau cas pathologique became a disease, a not unfair translation possibly coloured by the gossip that would also have spread news of Poincare's opinions. The optimism that Poincare conveyed became the suggestion that one will look back on the disease and see that one has recovered from it."

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I cannot answer the question but only quote some circumstancial evidence.

Firstly, I have seen related quotes which are with certainty by Poincaré:

For my part I think, and I am not alone in so thinking, that the important thing is never to introduce any entities but such as can be completely defined in a finite number of words. Whatever be the remedy adopted, we can promise ourselves the joy of the doctor called in to follow a fine pathological case. (Henri Poincaré: "Science and Method", Nelson, London, 1914 p. 44)

There is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. (Henri Poincaré: "Science and Method", Nelson, London, 1914 p. 195)

Secondly, Cantor was well aware of Poincaré's opposition:

So I understand quite well the opposition of Mons. Poincaré, by which I felt myself honoured, so he never had in his mind to honour me, as I am sure. (Cantor in a letter to Russell, 19 Sept 1911)

Here is a reference to Poincaré's talk in a letter of Cantor to Grace Chisholm Young of June 1908:

Erstaunt war ich über das, was Herr Henri Poincaré im April d. J. in seiner römischen Vorlesung über den „Cantorisme" sagte, doch kann ich nicht behaupten, dass es mich aus meinem Gleichmuth gebracht hat. Seit vielen Jahren sehe ich es kommen, dass sich die französische Akademie (aufgehetzt von der Berliner Akademie) in ihrem Zorn wider das unerhörte Transfinite resp. „Actuale Unendliche", das von mir ausgegangen ist, einmal fürchterlich blamiren würde. Dies scheint in Rom zur Thatsache geworden zu sein. Wie bisher, rühre ich auch in Zukunft keinen Finger um, was ich brachte gegen den verblendeten Hochmut der Akademiker diesseits und jenseits des Rheins zu vertheidigen.

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  • $\begingroup$ @Francios I have reversed one edit in order to restore Cantor's original English. $\endgroup$ – Otto May 19 '17 at 20:58
  • $\begingroup$ Ah, sorry. You seem right, though maybe not. So... $\endgroup$ – Francois Ziegler May 20 '17 at 16:01

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