Abstract
The post war years brought Brouwer back to topology and to intuitionism. Mostly ‘unfinished business’; mathematicians were catching up with Brouwer’s innovations, hence an exchange of ideas and problems. As the topology editor for the Mathematische Annalen Brouwer also got more papers to handle (e.g. Nielsen and Kerékjártó).
Most of Brouwer’s efforts, however, went into his intuitionism; from 1918 on he published substantial papers to put the subject on a firm footing. The first paper in the series introduced choice sequences and a constructive, but not finitistic, notion of set, now known as spread; furthermore the continuity principle—which was immediately applied to prove that the set of all number theoretic functions is not denumerable.
Brouwer got offers from Göttingen and Berlin, he remained however in Amsterdam on favorable conditions. One of those was that he could offer a position to Hermann Weyl, who in turn used the offer to improve his conditions in Zürich. The first international conference Brouwer attended after the war was the one in Nauheim, where he gave his first talk on intuitionistic mathematics, ‘Does every real number have a decimal expansion?’.
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Notes
- 1.
Brouwer to Klein, and Hilbert, 25 November 1918. ‘Möge das gesunde Herz Ihres Vaterlandes die heutige Krise überwinden, und mögen die deutsche Landen alsbald zu ungekanntes Blüte gedeihen in einer Welt der Gerechtigkeit! Das wünscht Ihnen Ihr Brouwer.’
- 2.
Diary of Van Eeden, 22 May 1918, 24 July 1918.
- 3.
A non-existent word, sounding like a pet name for a toddler, Wiessing (1960), p. 201.
- 4.
For a long time there were two quality newspapers that stood out in the Dutch newspaper world, the Algemeen Handelsblad, based at Amsterdam, and the Nieuwe Rotterdammer Courant. The two merged in the nineteen seventies.
- 5.
De ‘Mosgroene’.
- 6.
The Groene was later properly resurrected; it is still flourishing today in its own modest way.
- 7.
Wiessing to members of the KNAW, 21 October 1918.
- 8.
A former fellow member of CLIO, cf. p. 13.
- 9.
- 10.
Brouwer (1918c).
- 11.
Brouwer (1919i).
- 12.
Brouwer (1912d).
- 13.
- 14.
- 15.
Submitted, 29 March 1919. Brouwer (1919j).
- 16.
Brouwer to Riesz, 22 November 1921.
- 17.
Klein to Brouwer, 12 September 1919.
- 18.
Nielsen to Brouwer, 18 October 1919.
- 19.
Brouwer to Klein, 21 October 1919.
- 20.
Nielsen (1920).
- 21.
Nielsen’s paper was submitted on 15 January 1920 and Brouwer’s on 20 January 1920.
- 22.
Brouwer (1921b).
- 23.
Brouwer (1920).
- 24.
Brouwer to Klein, 19 September 1919.
- 25.
Akten der Phil. Fak. Az. II PH/36–d, Dozenten Generalia, 30 October 1919.
- 26.
- 27.
Carathódory left Göttingen in 1918 for Berlin. In 1920 he accepted a call from Athens, and subsequently went to Smyrna in order to organise the founding of a mathematics department in the new university.
- 28.
Bestand Philosophische Fakultät, Humboldt Universität, Nr.1468, Bl.313. The recommendation was signed by Carathéodory, Erhardt Schmidt, Planck, H.A. Schwarz, Nernst, Cohn. The qualification ‘deutschfreundlich’ was taken quite seriously after the war. The German nation felt itself misunderstood and surrounded by hostile nations.
- 29.
Hilbert to Weyl, 16 May 1920. ETH Hs.91: 606.
- 30.
Borel to Brouwer, 20 December 1919.
- 31.
Lorentz to Brouwer, 25 January 1920.
- 32.
Van Eeden’s diary, 27 January 1920. Brouwer was indeed short tempered where such obstructions as closed doors or the like were concerned. Brouwer used to drop in occasionally to see a journalist of the Handelsblad, living in Blaricum, with whom he was on good terms. Once, when he did not find the man in, Brouwer got so angry that he kicked in a window (oral communication, Mr. Crèvecoeur).
- 33.
Von Mises to Brouwer, 28 January 1920.
- 34.
Brouwer to Tellegen, 4 February 1920.
- 35.
Although the galloping inflation was still two years away, the German Reichsmark was not as strong as it used to be, and certainly weaker than the guilder.
- 36.
Mayor to Brouwer, 12 February 1920.
- 37.
Schoenflies to Brouwer, 16 February 1920, Brouwer to Schoenflies, 22 February 1920.
- 38.
In his second recommendation of 1920.
- 39.
Denjoy to Brouwer, 29 April 1920.
- 40.
Mulder (1917).
- 41.
Die Idee der Riemannschen Flächen.
- 42.
The two other events were Koebe’s solution of the uniformisation problem and Hilbert’s treatment of the Dirichlet Problem.
- 43.
Weyl to Klein, 6 May 1912.
- 44.
Das Kontinuum; Raum, Zeit und Materie.
- 45.
Cf. Brouwer to Fraenkel, 28 January 1927.
- 46.
Wiessing (1960), p. 33.
- 47.
Inherited from her father, see p. 249.
- 48.
The villa actually is still to be found in the Krodothal.
- 49.
J.H. Schogt, later succeeded by Belinfante.
- 50.
Weyl to the President of the ETH, 27 September 1920.
- 51.
Cf. Frey and Stammbach (1992).
- 52.
Brouwer to Weyl, 1 January 1921. ETH. HS.91:493.
- 53.
In March Brouwer asked for and obtained a three month sick leave.
- 54.
Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Allgemeine Mengenlehre. Brouwer (1918a). We will refer to the papers of this series as the Begründungs-papers.
- 55.
Brouwer (1925a). In the first presentation the definition is given in one sentence running over eleven lines!
- 56.
Menger (1979).
- 57.
Eine Menge ist ein Gesetz, auf Grund dessen, wenn immer wieder ein willkürlicher Ziffernkomplex der Folge ζ gewählt wird, jede dieser Wahlen entweder ein bestimmtes Zeichen, oder nichts erzeugt, oder aber die Hemmung des Prozesses und die definitive Vernichtung seines Resultates herbeiführt, wobei für jedes n nach jeder ungehemmten Folge von n−1 Wahlen wenigstens ein Ziffern-komplex angegeben werden kann, der wenn er als n-ter Ziffernkomplex gewählt wird, nicht die Hemmung des Prozesses herbeiführt. Jede in dieser Weise von der Menge erzeugte Zeichenfolge (welche also im allgemeinen nicht fertig darstellbar ist) heisst ein Element der Menge. Die gemeinsame Entstehungsart der Elemente einer Menge M werden wir ebenfalls kurz als die Menge M bezeichnen.
- 58.
Brouwer (1918a), p. 13, see also p. 238.
- 59.
A sniff of logic shows that this can be rewritten as \(\forall a (a \not\in X \leftrightarrow a \not\in Y)\).
- 60.
Cf. Weyl (1954).
- 61.
Klein to Weyl, 19 April 1918.
- 62.
Weyl (1918), p. IV.
- 63.
9 February 1918, cf. Polya (1972).
- 64.
2, 9, 16 December in the seminar of Fueter.
- 65.
Polya had advanced the view that research in set theory should not be restricted.
- 66.
Weyl (1921).
- 67.
Brouwer (1913b), note the ‘Freudian’ permutation.
- 68.
Weyl (1921), p. 66.
- 69.
That is, the construction of a specific object versus its non-effective existence.
- 70.
Cf. p. 101.
- 71.
We will indiscriminately use the terms ‘principle of the excluded third’, ‘principle of the excluded middle’. Brouwer preferred ‘principium tertii exclusi’. The acronym used here is PEM.
- 72.
van Dalen (1995).
- 73.
The translation of the German werdende Folge is somewhat problematic. Various adjectives such as ‘becoming’, ‘developing’, ‘emerging’, and ‘in statu nascendi’ have been used in the literature.
- 74.
Besitzt jede reelle Zahl eine Dezimalbruchentwicklung? (Brouwer 1921a).
- 75.
Elektrizität und Gravitation.
- 76.
Brouwer (1921a).
- 77.
Fricke to Klein, 28 September 1920.
- 78.
Brouwer (1908b).
- 79.
Brouwer (1919h).
- 80.
Brouwer (1952a).
- 81.
N.B. Brouwer’s ordinals were constructive, and hence countable.
- 82.
- 83.
Some people worry about the moment this problem A will be solved. There is no need to: there is an inexhaustible stock of unsolved problems; one can always take another one.
- 84.
A is inhabited if it contains an element, the positive analogue of non-empty.
- 85.
Van Eeden diaries, 22 January 1919.
- 86.
The assassin was interviewed after 67 years in a Dutch television program on 29 October 1991.
- 87.
The Institute was the Foundation supporting the Academy. For the structure of the organisation, see p. 260 and Schmitz (1990a).
- 88.
Borel to Brouwer, 18 December 1919.
- 89.
Brouwer to Schoenflies, 29 December 1919.
- 90.
Gutkind to Van Eeden, 15 January 1920.
- 91.
Borel to Brouwer, 20 September 1921.
- 92.
Brouwer was in a boyish way rather proud of the epithet ‘Bolschevist’, but it would go too far to attach any political significance to this fact. Let us add that there were some unconfirmed rumours that Brouwer met Lenin in Zürich.
- 93.
Cf. Het roode lampje (the red lamp), van Eeden (1921).
- 94.
Still a visitor and not a full member.
- 95.
Although legally it was only ‘suspended’.
- 96.
Diary Van Eeden, 22 May 1922.
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van Dalen, D. (2013). Mathematics After the War. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_8
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