Abstract
Brouwer had to defend himself on three fronts: the Formalism-Intuitionism conflict, the War of the frogs and the mice, The Menger dimension conflict. The War of the frogs and the mice is extensively discussed, since it brought in fact the end of the foundational conflict. The Menger conflict deals with the priority for the dimension definition.
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Notes
- 1.
Cf. Lietzmann (1942).
- 2.
Alexandrov (1969).
- 3.
Überall und scheinbar überall definierte Funktionen.
- 4.
Alexandrov to Hopf, 12.VIII.1926.
- 5.
Brouwer to B. Adama van Scheltema, 22.VIII.1926.
- 6.
Über Definitionsbereiche von Funktionen.
- 7.
Cf. p. 352 ff.
- 8.
Courant to Springer, 2.VIII.1927.
- 9.
See Reid (1970), Chap. 21.
- 10.
Die Grundlagen der Mathematik, Hilbert (1928).
- 11.
- 12.
Cf. Brouwer (1913b), p. 88.
- 13.
Ibid, p. 77.
- 14.
The reference to ‘worthless scrip’ suggests that Hilbert has Weyl’s New Crisis-paper in mind. But it covers Brouwer’s views equally well.
- 15.
It is simple to prove ∃x(x+2=7).
- 16.
A most interesting claim. Given a number of extra specifications, there are good arguments for Hilbert’s claim when made for the system of Gentzen.
- 17.
Weyl (1928).
- 18.
Cf. van Dalen (1999c).
- 19.
- 20.
‘Die Eisdecke war in Schollen zerborsten, und jetzt ward das Element des Flieszenden bald vollendes Herr über das Feste. L.E.J. Brouwer entwirft—und das ist eine Leistung von der gröszten Erkenntnistheoretischen Tragweite—eine strenge mathematische Theorie des Kontinuums, die es nicht als starres Sein, sondern als Medium freien Werdens faszt.’ Weyl (1928).
- 21.
Weyl (1946), p. 275.
- 22.
Weyl (1932).
- 23.
Weyl (1944), p. 157.
- 24.
Von Neumann to Carnap, 7.VI.1931.
- 25.
- 26.
Courant (1981), p. 161.
- 27.
The note is not dated, but it is almost certainly written in his later years. It also mentions Russell, Kant and Mannoury.
- 28.
Courant (1981), p. 160.
- 29.
Courant (1981).
- 30.
Cf. p. 304 ff.
- 31.
Cf. p. 371 ff.
- 32.
Cf. Sect. 9.1.
- 33.
Brouwer to Bieberbach, 23.I.1928.
- 34.
Dated March 1923 and addressed to the members of the DMV.
- 35.
Cf. p. 329.
- 36.
Cf. Schroeder-Gudehus (1966), pp. 248, 250.
- 37.
Sommerfeld to Brouwer, 24.III.1928.
- 38.
Perron to Bieberbach, 6.V.1928. Perron inquired if Bieberbach would go to Bologna and give a talk.
- 39.
Bieberbach to Ziehen, 18.VI.1928.
- 40.
Cf. Bieberbach to Courant, 15.VII.1928.
- 41.
Hilbert to rectors of German universities, 29.VI.1928.
- 42.
Perron to Landau, 2.VII.1928.
- 43.
…Dieses Schreiben nimmt in schroffster Weise Stellung gegen den diesen Herbst in Bologna stattfindenden Internationalen Mathematiker-Kongress, denunziert den Kongress als eine Veranstaltung des conseil internationale de recherches und rät daher dringend an zu einer Ablehnung der an die Universitäten usw. ergangener Einladung.
- 44.
Von Mises to Courant, 7.VII.1928.
- 45.
He referred tongue in cheek to the Hilbert–Bieberbach discussion, noting that ‘it is a good illustration of your remark that inside the mathematics society no controversies exist’.
- 46.
Bieberbach to Courant, 15.VII.1928.
- 47.
Apparently a meeting of the editors of the M.A. editors at the time of the annual meeting of the DMV.
- 48.
Bohr to Brouwer, 3.VII.1928.
- 49.
Härlen to Brouwer, 6.IX.1928.
- 50.
Reid (1970), p. 188.
- 51.
Courant to Springer, 3.X.1928.
- 52.
A wealth of information on Carathéodory can found in the recent biography (Georgiadou 2004).
- 53.
Gawehn to Von Mises, 9.XI.1928.
- 54.
Carathéodory had also received a copy, and probably Blumenthal as well.
- 55.
Einstein to Carathéodory, 19.X.1928.
- 56.
Einstein to Hilbert, 19.X.1928.
- 57.
Carathéodory to Einstein, 20.X.1928.
- 58.
An enigmatic remark. It is far from clear what Cara (as Carathéodory was called among friends) had in mind. The only possible point of friction around 1918 could have been the offer of a chair in Göttingen to Brouwer, who subsequently turned it down. But is hard to see how that could have bothered Hilbert.
- 59.
Einstein to Carathéodory, 23.X.1928.
- 60.
This probably refers to the troubles in Schleswig-Holstein during roughly the same period when farmers resisted the tax policies of the government. Hans Fallada has sketched the episode in his Bauern, Bonzen und Bomben.
- 61.
Oral communication, Mrs. F. Heyting-van Anrooy.
- 62.
Carathéodory to Courant, 3.XI.1928.
- 63.
Blumenthal to Courant, 4.XI.1928.
- 64.
Brouwer to Courant, 6.XI.1928.
- 65.
Courant to Brouwer, 10.XI.1928.
- 66.
Cf. Freudenthal to Hopf, 22.I.1932. Freudenthal wrote that in Brouwer’s opinion Hilbert was not pleased at all, feeling that Brouwer was trying to do him out of a valuable collaborator.
- 67.
Cf. p. 558.
- 68.
Cf. p. 528. The last preserved written communication was dated 20.VIII.1919.
- 69.
Blumenthal to the editors, 16.XI.1928.
- 70.
Blumenthal to Courant, 28.XI.1928.
- 71.
The contract had indeed a clause to the effect that changes in the editorial board required the assent of the publisher. The problem was of course how to interpret ‘editor’.
- 72.
Cf. Mehrtens (1987).
- 73.
This suggestion of the publisher encouraged the impression that the conflict had a political origin. Blumenthal complained to Courant (letter of 18 November 1928) ‘… the bad thing is, that Brouwer managed to move everything on to the political plane, just what Carathéodory thought he had prevented’. The idea of mediation was not pursued.
- 74.
Brouwer indeed founded a new journal, the Compositio Mathematica, with the Dutch publisher Noordhoff.
- 75.
Bohr and Courant to Blumenthal, 14.XI.1928.
- 76.
The above mentioned Aktennotiz.
- 77.
Ibid.
- 78.
Blumenthal is here less than truthful. Cara had not authorised Hilbert. He had probably tried to pacify Hilbert, who took this for the desired approval. Cara wrote, referring to Blumenthal’s draft, ‘In fact I have not given an authorisation in the legal sense in my letter to Hilbert.’ (Carathéodory to Courant, 14.XI.1928.)
- 79.
If anything, Brouwer’s circular letter could been condemned rather as divisive, than as insulting.
- 80.
Cf. Sect. 9.3.
- 81.
Bieberbach to Blumenthal, 24.XI.1928.
- 82.
25.II.1920.
- 83.
War of the frogs and the mice—a Greek play of unknown author; a late medieval German version, Froschmeuseler, is from the hand of Rollenhagen.
- 84.
Springer to Courant, 17.I.1928.
- 85.
Cara to Blumenthal, 27.XI.1928.
- 86.
Courant to Carathéodory, 30.XI.1928.
- 87.
Blumenthal to Courant and Bohr, 4.XII.1928.
- 88.
Blumenthal to Bohr, 5.XII.1928.
- 89.
Blumenthal to Courant, 16.XII.1928.
- 90.
Carathéodory to Courant, 19.XII.1928.
- 91.
Courant to Carathéodory, 23.XII.1928.
- 92.
Carathéodory to Courant, 12.XII.1928.
- 93.
Courant to Carathéodory, 15.XII.1928.
- 94.
Carathéodory to Courant, 19.XII.1928.
- 95.
Courant to Carathéodory, 23.XII.1928.
- 96.
Cf. letter to Einstein, October 15.
- 97.
Blumenthal to Courant, 31.XII.1928.
- 98.
Courant to Brouwer, 23.XII.1928.
- 99.
The German term is ‘Übergriff’, which could also be translated as ‘impertinence’. In view of Blumenthal’s reputation, that version seems implausible.
- 100.
[Brouwer’s footnote] One might think for a moment that the communication of such utterances carried something incorrect, because one assumes with respect to these naturally an atmosphere of confidentiality. This assumption of confidence, however, and the solidarity which it presumes, in as far as it is not invalidated by the end of the conversation, is certainly not compatible with Carathéodory’s later silence in the face of Blumenthal’s later false representations. Moreover, even justifiable scruples must take second place in the present case, where a discussion of wicked defamation and isolation of status is concerned—just as in the case of the question of a witness in a court of law.
- 101.
Only a thoroughly confused person would thus run the risk of a court case for falsification. If true, this supports Carathéodory’s and Brouwer’s view.
- 102.
The available evidence points at Mohrmann.
- 103.
Ehrenfest to Van der Waerden, 8.X.1928.
- 104.
Alexandrov to Hopf, 6.XII.1928.
- 105.
Brouwer to Heyting, 17.VII.1928.
- 106.
Brouwer to Heyting, 28.IX.1929.
- 107.
Brouwer to Weyl, 16.II.1928.
- 108.
Euwe (1929).
- 109.
Er hat die Sachen überhaupt nicht gelesen. Interview, Bernays 18.VII.1977.
- 110.
To be precise, the 6th Deutsche Physiker- und Mathematikertagung, and the 91st Versammlung der Gesellschaft Deutscher Naturforscher und Ärzte. September 5–7, 1930 Königsberg. Proceedings published in Erkenntnis (23), 1931.
- 111.
Weyl (1946), p. 275.
- 112.
Erkenntniss, 1931, pp. 147, 148.
- 113.
Bernays to Gödel, 10.XII.1930, cf. Dawson (1997), p. 282.
- 114.
Hilbert (1930).
- 115.
Gödel called this the Von Neumann conjecture: ‘If there is a finitary consistency proof at all, then it can be formalised. Then Gödel’s theorem yields the impossibility of a consistency proof at all’ (15.1.1931). See also Dawson (1997), p. 68 ff.
- 116.
8.X.1929.
- 117.
Fraenkel (1927), p. 154.
- 118.
Bernays to Hilbert, 25.X.1925.
- 119.
Hilbert (1930).
- 120.
For an instructive discussion of a particular instance of perpetual criticism, see Hodges (1998).
- 121.
Hilbert (1931).
- 122.
Zehn Vorlesungen über die Grundlegung der Mengenlehre, 1927.
- 123.
Hausdorff to Fraenkel, 20.II.1927.
- 124.
Study to Fraenkel 5.III.1927.
- 125.
Einleitung in die Mengenlehre.
- 126.
Hausdorff to Fraenkel, 19.XI.1928.
- 127.
Finsler (1925).
- 128.
About the scientific ordeal of mathematicians (Vom Wissenschafts ideal der Mathematiker), 1926.
- 129.
Pierpont (1928), p. 37.
- 130.
Grelling (1928).
- 131.
Cf. p. 499.
- 132.
Weil to Fréchet, 31.I.1927.
- 133.
Menger (1928b).
- 134.
Brouwer (1924e).
- 135.
Hurewicz to Menger, 10.X.1928.
- 136.
Brouwer (1928f).
- 137.
Brouwer to Hahn, 4.VIII.1929.
- 138.
Moreover, a recent preliminary analysis by the Dutch forensic institute showed no deviation among the corresponding handwritten corrections.
- 139.
Brouwer uses Brieftasche; it is more likely that he was carrying a small type of briefcase that was very common at the time, than a wallet.
- 140.
Brouwer to Hahn, 9.VIII.1929.
- 141.
Brouwer to Hahn, 17.VIII.1929.
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van Dalen, D. (2013). The Three Battles. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_14
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