Abstract
Brouwer was extremely critical of Hilbert’s formalism, which was completely opposed to the intuitionistic philosophy of mathematics. Brouwer trod softly in his publications in order to avoid conflict. However, when Hermann Weyl joined Brouwer’s intuitionism, his ‘New crisis-paper’ the tone changed. Hilbert told Brouwer and Weyl in unmistakable terms how wrong they were. The Grundlagenstreit was born. The conflict dragged on for years.
From time to time the scientific discussion was replaced by external causes. One such was the planned publication of the Riemann memorial volume by the Mathematische Annalen. A conflict arose when the participation of French mathematicians could in this project of their arch enemy was questioned. One section of the editorial board objected to the French participation, which was advocated by Hilbert. The conflict left bad feelings.
Inspired by the contributions of Alexandrov, Menger, and Urysohn, Brouwer collected in 1925 a small group of topologists in Amsterdam, including Vietoris and Wilson. The brief concentration of topologists was later called the Dutch topological school.
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Notes
- 1.
Weyl to Mulder, 24.VII.1910.
- 2.
See Sieg (1999).
- 3.
Hilbert claimed that Kronecker rejected set theory on the grounds of the paradox of the set of all sets. This is rather a misjudgement of Kronecker’s motives, moreover Kronecker died before the paradoxes surfaced.
- 4.
- 5.
Brouwer (1919a).
- 6.
There are certain indications that Hilbert’s independent position during the war had made him no friends among his colleagues. Born wrote in a letter to Hilbert (27.VIII.1919) that he guessed that Hilbert’s action was largely determined by his relationship with the ‘Göttingen’ colleagues, who mostly subscribed to the Alldeutschtum. In view of the treatment by this group during the war, Hilbert’s threat to leave Göttingen might not have been empty talk.
- 7.
- 8.
See Sect. 8.8.
- 9.
Since there is no elaboration of this remark, one can only make an educated guess as to what these similarities were. A possible candidate could be Weyl’s treatment of quantified statements as ‘judgement abstracts’. In Hilbert’s writings a similar phenomenon surfaces in the distinction between finitary, meaningful statements and quantified, ideal statements, cf. Hilbert (1926), p. 174. In Hilbert’s papers no reference to Weyl’s ideas is to be found. There is, however, a reference in Hilbert and Bernays (1934, 1939).
- 10.
Festschrift udgivet af Københavns Universitet. November 1921, p. 114.
- 11.
Natur und mathematisches Erkennen (March 14), Axiomenlehre und Widerspruchsfreiheit (15 and 17 March). I am indebted to Sigurd Elkjaer for the information and documentation on Hilbert’s Copenhagen lectures and doctorate.
- 12.
See Hesseling (2002) for details.
- 13.
Hilbert (1922).
- 14.
Cf. p. 313.
- 15.
Hilbert (1922), p. 159.
- 16.
Cf. p. 314 ff.
- 17.
Cf. pp. 313, 313.
- 18.
- 19.
Kneser (1925).
- 20.
- 21.
Abbreviated as PEM, principle of the excluded middle.
- 22.
Brouwer (1923f), p. 2.
- 23.
Brouwer (1919h).
- 24.
In Fraenkel (1923), p. 239, Brouwer is quoted slightly differently: ‘For intuitionists, however, ‘consistency’ has by no means the same meaning as ‘existence’, no more than ‘a crime which’ cannot be detected by prescribed means of investigation stops being a crime.’
- 25.
Brouwer (1907), p. 138 (footnote), pp. 141, 142.
- 26.
Cf. I.10.4, p. 392. Brouwer had picked a hard, although elementary, problem. Only in 1998 was the sequence 0123456789 found in the decimal expansion of π, Borwein (1998). Needless to say that the demise of a single counterexample does not weaken the intuitionist cause—there are enough new counterexamples.
- 27.
- 28.
Cf. p. 386; see van Dalen (2000) for more on Fraenkel and intuitionism.
- 29.
Einleitung in die Mengenlehre, Fraenkel (1919).
- 30.
Knopp to Fraenkel, 2.I.1924.
- 31.
Fraenkel (1923), p. 164.
- 32.
Fraenkel (1923), p. 173.
- 33.
See p. 386.
- 34.
Hilbert (1923).
- 35.
Fraenkel (1923), p. 239.
- 36.
Belinfante (1923).
- 37.
Belinfante had not made use of Brouwer’s new results, such as the fan theorem and the continuity theorem. From a modern point of view, one could say that he belonged to the Bishop school avant la lettre.
- 38.
- 39.
Weitzenböck to Weyl, 16.IV.1923.
- 40.
Brouwer to Bieberbach, 1924, undated. See p. 406.
- 41.
Konsequenzen des intuitionistischen Standpunktes in der Mathematik.
- 42.
Reid (1970), p. 184.
- 43.
Since Blumenthal, Brouwer and Klein were at the time entangled in the unpleasant Mohrmann affair, there was no shortage of topics, see also p. 583.
- 44.
In the same letter he told of his strongly favourable impression of Kneser and Neugebauer.
- 45.
Reid (1970), p. 184.
- 46.
Drost (1952), p. 26.
- 47.
Brouwer (1929b).
- 48.
1921—1; 1923—4; 1924—7, 1925—2; 1926—2; 1927—3; 1928—2; 1929—1 (not counting the multiple versions).
- 49.
Interview, A. Heyting.
- 50.
Ziffernkomplex. In Brouwer (1925a) Brouwer used ‘Nummer’.
- 51.
e.g. Brouwer (1954a).
- 52.
Brouwer (1925a).
- 53.
Strictly speaking one particular second-order restriction had been implicitly recognised all along, namely the restriction of all future choices to a law.
- 54.
- 55.
Brouwer (1942a).
- 56.
Cambridge Lectures 1946–1950, Brouwer (1981).
- 57.
Of higher-order restriction.
- 58.
Brouwer (1981), p. 13.
- 59.
- 60.
See Hesseling (1999).
- 61.
Hilbert 1904, reprinted in the later versions of Grundlagen der Geometrie.
- 62.
See p. 125
- 63.
Brouwer (1928b).
- 64.
- 65.
Bernstein (1919), pp. 63–78.
- 66.
- 67.
Bernays (1922), p. 11.
- 68.
Ibid. p. 10.
- 69.
Ibid. p. 15.
- 70.
That is to say, Bernays pointed out in his writings, cf. Bernays (1922), that Hilbert did not lack in appreciation for constructive metamathematics. For some reason Hilbert just did not see, or did not wish to see, his intellectual debt to Brouwer and Weyl.
- 71.
The terminology is anachronistic; Brouwer started to use the name ‘intuitionism’ for his program after World War I.
- 72.
Brouwer (1907), Chap. 3.
- 73.
Hilbert (1922).
- 74.
Cf. Rowe (1989).
- 75.
Grundlagen der Mathematik I (1934), II (1939).
- 76.
See p. 442. Brouwer’s Nauheim lecture predates the Grundlagenstreit; it is by all standards a totally innocuous academic treatise on effective procedures, without programmatic claims. Nobody could possibly take exception to it. Therefore I will not view it as part of the foundational conflict.
- 77.
See p. 445.
- 78.
Curiously enough Brouwer was in Münster on June 1 and 2. At least that was what Brouwer announced in a letter to Gerda Holdert, 14.V.1925: ‘I have to be in Münster on June 1, 2.’ He may or may not have gone to Münster after all, but it remains a curious coincidence.
- 79.
Über das Unendliche und die Begründung der Mathematik.
- 80.
Hilbert (1926), p. 163.
- 81.
Ibid. p. 170.
- 82.
The fact that Hilbert accepted the modest constructive approach: potential infinite for cardinalities in the style of Brouwer 1907, of course does not imply that he would accept, or even contemplate, choice sequences.
- 83.
Hilbert (1923), p. 160.
- 84.
Hilbert used the German ‘finit’ instead of ‘endlich’. Following Kleene, the term finitary has been adopted in the foundations of mathematics.
- 85.
Hilbert (1926), p. 179. A more modern version would run roughly as follows: if a finitary statement is provable in the extended system, then it is already provable in the finitary system (the extension is conservative).
- 86.
Courant to Hilbert, 10.IX.1925.
- 87.
- 88.
See p. 332.
- 89.
Brouwer to Blumenthal, 1.XI.1924.
- 90.
Cf. p. 329.
- 91.
The editorial board contained two kinds of editors: the proper editors, listed on the cover as ‘Herausgeber’ (publishers), and associated editors, listed under the heading ‘unter Mitwirkung von’ (with the co-operation of). When relevant, we will call the first kind ‘managing editors’ and the second kind ‘associate editors’.
- 92.
Carathéodory to Brouwer, 6.XI.1924.
- 93.
‘ohne Gefahr zu laufen, von der ganzen Meute der Banausen angebellt zu werden.’
- 94.
Blumenthal to Einstein, 15.XII.1924.
- 95.
Einstein to Blumenthal, 16.XII.1924.
- 96.
Bieberbach to Blumenthal, 12.I.1925.
- 97.
Hölder to Blumenthal, 13.I.1925.
- 98.
Einstein to Blumenthal, 20.I.1925.
- 99.
‘wenn absolute Beweise deutschfreundlicher Gesinnung und Betätigung vorliegen’.
- 100.
Carathéodory to Blumenthal, 20.I.1925.
- 101.
Cf. p. 627.
- 102.
Bieberbach to Blumenthal, 23.I.1925.
- 103.
Bieberbach to Hilbert, 25.II.1925.
- 104.
Cf. p. 551.
- 105.
Minutes of the meeting of the Wiskundig Genootschap of 24.II.1923.
- 106.
In 1924 Einstein joined the C.C.I., an organisation of the League of Nations, after all. The larger part of my information on the scientific organisations after the war is drawn from Schröder-Gudehus’ books, Schroeder-Gudehus (1966, 1978), which are highly recommended for their treatment and the use of sources.
- 107.
Karo (1926). It is worth noting that Karo was a Jew, and that he emigrated to the USA during the Nazi period. The protest against the boycott of German science and scientists was supported by a large section of the German academic community.
- 108.
The motto is: Je maintiendrai.
- 109.
Schroeder-Gudehus (1966), p. 248.
- 110.
Kerkhof to Brouwer, 4.V.1925.
- 111.
Brouwer to Kerkhof, 11.V.1925.
- 112.
Witness the remark of Sommerfeld in a letter to Blumenthal (26.I.1925): ‘I can but experience it as humiliating that Brouwer feels more German than we do.’
- 113.
Cf. Schroeder-Gudehus (1966), p. 207.
- 114.
Cf. p. 329.
- 115.
Der geistige Krieg gegen Deutschland, Karo (1926).
- 116.
As testified by letters in the Alexandrov archive.
- 117.
- 118.
Smid became a specialist in insurance mathematics. Max Euwe won in 1935 the world title in chess. Later he became a professor in computer science.
- 119.
Brouwer actually lived most of the time in Blaricum, but right at the border with Laren. This border has moved back and forth in the course of time. Sometimes he rented a house in Laren or in Blaricum. A large part of his letters bears the name of Laren, later letters show Blaricum.
- 120.
Interview Van der Waerden.
- 121.
Oral communication F. Kuiper.
- 122.
Max Euwe told a different story. When Van der Waerden asked his question, Brouwer replied sternly and then concluded: ‘Mr. Van der Waerden, I advise you to study the material completely before you start a discussion.’ The story went from student generation to student generation. Freudenthal’s version was that ‘Van der Waerden thought to have found a mistake. But Brouwer was completely right.’
- 123.
Emmy Noether to Brouwer, 14.XI.1925.
- 124.
‘The algebraic foundations of the geometry of number’, [i.e. enumerative geometry], van der Waerden (1926).
- 125.
de Vries (1936).
- 126.
For biographical information on Van der Waerden, see Springer (1997) and the papers in Nieuw Archief voor Wiskunde, 1994(12) no. 3.
- 127.
Oral communication Gerda Holdert.
- 128.
Menger to Brouwer, 3.VII.1925.
- 129.
L.E.J. Brouwer in tiefer Verehrung gewidmet.
- 130.
Menger (1979).
- 131.
Here and in the following pages I quote from Menger (1979).
- 132.
Alexandrov (1969), p. 117.
- 133.
Die Topologie in und um Holland in den Jahren 1920–1930, Alexandrov (1969).
- 134.
Cf. MacLane (1981).
- 135.
E. Noether to Brouwer, 7.1X.1919. For the Karlsruhe meeting, see p. 175 ff.
- 136.
Brouwer to E. Noether, 21.XI.1925.
- 137.
The University of Göttingen.
- 138.
Alexandrov to E. Noether, 11.XI.1925.
- 139.
Rectal, as the hospital tradition was.
- 140.
Oral communication, Mrs. J.F. Heyting-van Anrooy.
- 141.
Alexandrov to Hopf, 10.IV.1927.
- 142.
We recall that Urysohn discussed his dimension theory for the first time (with Alexandrov) in August 1921 (cf. p. 397 and Johnson 1981, p. 228). He lectured on the topic in Moscow in the academic year 1921/22, and submitted three printed notes on the subject to the Moscow Mathematical Society. The first internationally accessible version appeared in the Comptes Rendus in 1922. Menger developed his ideas on dimension theory between April 1921 ands February 1922. An account was submitted to Hahn in November 1922, and was withdrawn after Hahn discovered a mistake in it. Menger’s first publication followed in December 1923, Menger (1923).
- 143.
Menger to Brouwer, 10.IV.1926.
- 144.
Schreier to Menger, 7.IV.1926.
- 145.
Hahn to Brouwer, 10.IV.1926. It is not clear whether Menger or Brouwer asked for this information.
- 146.
Menger (1926).
- 147.
Brouwer to Hahn, 22.X.1929.
- 148.
Brouwer (1976), p. 567.
- 149.
Brouwer was off to Batz, to meet Alexandrov.
- 150.
Menger to Brouwer, 19.VIII.1926.
- 151.
In oprechte waardering van Uw werk en in dankbaarheid U opgedragen van den schrijver. Menger had for all practical purposes mastered the Dutch language.
- 152.
Brouwer to Hahn, 27.VIII.1929.
- 153.
Brouwer to Hahn, 22.X.1929.
- 154.
Menger (1979), p. 246.
- 155.
One can sympathise with Menger, although my experiences were the other way round. When I first learned about analytic sets, I thought, ‘Why, these are spreads!’
- 156.
Menger (1979), pp. 86, 246.
- 157.
- 158.
See Menger (2003), p. 4.
- 159.
Heyting (1931b).
- 160.
Brouwer to Hahn, 22.X.1929.
- 161.
For comparison, here are the two formulations: Menger—‘in einem allerdings weniger bekannten kurzen Aufsatz (Crelle Journ. 142, S.146–152) eine Definition n-dimensionaler Kontinua gegeben, die nach Korrektur (Amsterdamer Akademieber. XXVI, 1923) mit unserer Definition des n-dimensionalen Kontinuums äquivalent ist’; Brouwer—‘in einem allerdings weniger bekannten kurzen Aufsatz (Journ.f.Math. 142, S.146–152; vgl. auch die Korrektur eines daselbst befindlichen Schreibfehlers in den Amsterdamer Proceedings 26, S.796) eine Definition n-dimensionaler Kontinua gegeben, die mit unserem Definition des n-dimensionalen Kontinuums äquivalent ist.’ The reader will appreciate the difference in suggestive force.
- 162.
Menger (1979), p. 247.
- 163.
i.e. the theorem that ℝn has natural dimension n.
- 164.
Cf. van Dalen (1999a), in which it is shown that, e.g., the irrationals (the complement of the rationals) are indecomposable, and hence 1-dimensional.
- 165.
Alexandrov to Menger, 1925 (no precise date known).
- 166.
Cf. Menger (1994), p. 200.
- 167.
Tumarkin (1926), Menger to Brouwer, 8.IV.1927.
- 168.
CW II, p. 564.
- 169.
Enzyklopädie der mathematischen Wissenschaften, Dritter Band, Geometrie, Ch. 13. Enzyklopädie der mathematischen Wissenschaften (submitted 15.X.1929). In fact Klein had asked Brouwer to engage Vietoris to write the chapter. Alexandrov and Menger had opposed the idea (interview Vietoris).
- 170.
Menger to Brouwer, 17.I.1928.
- 171.
Geschichtliche Entwicklung der Topologie, Feigl (1928).
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van Dalen, D. (2013). Progress, Recognition, and Frictions. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_12
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