Abstract
Cut off from his international contacts, in particular Göttingen, Brouwer returned to his foundational research. He lectured on The theory of point sets, a course on constructive set theory. In the 1916/17 course he introduced choice sequences and the continuity principle. Now that he saw how to exploit the apparent weakness of his intuitionistic mathematics, he decided to build his new intuitionism in a systematic way. There were some signs during the war years that something was brewing, but the first papers appeared after the war. Part of his efforts were directed at a project called Significs, a study of language and meaning following Lady Welby and Frederik van Eeden, the author and first psychiatrist in Holland. A considerable effort was made to create a group of people with a common interest in the subject, including an International Academy, etc. In the later years of the war Brouwer proposed, together with colleagues, to found a section of the Dutch Airforce (which was hardly existing at the time) for the study and application of air reconnaissance (photogrammetry). Finally, from now on Brouwer was spending more time on university/faculty/Academy matters.
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Notes
- 1.
Brouwer to Scheltema, 15 August 1914.
- 2.
An analysis of the ideas of the French constructivists can be found in the excellent and instructive survey (Bockstaele 1949).
- 3.
Die Mathematische Methode, Hölder (1924), p. 98. He refers to an earlier statement of the same tenor in 1892.
- 4.
Brouwer (1913b), p. 91.
- 5.
Die allgemeine Functionentheorie, 1882.
- 6.
Zermelo (1904).
- 7.
- 8.
We will from now on call the French intuitionists, Borel, Lebesgue, Baire, Poincaré, Hadamard, and their associates semi-intuitionists in accordance with present usage. Brouwer used in his later publications the name old-intuitionists for his predecessors.
- 9.
Richard (1905).
- 10.
Brouwer (1914), p. 79.
- 11.
ρ is the sequence of natural numbers; by ‘governed by induction’ Brouwer means ‘given by a law’.
- 12.
Brouwer was not quite consistent in his use of ‘fundamental sequence’. Usually (and certainly in the early foundational papers) this was a lawlike sequence, but from time-to-time Brouwer also speaks of fundamental sequences of arbitrarily chosen elements. One has to be careful in reading his papers, usually the meaning is clear from the context, but not always!
- 13.
f 1 and ρ stand for the sets of all choice sequences of natural numbers and the set of the natural numbers.
- 14.
Cantor (1892).
- 15.
Brouwer had seen that the diagonal method is perfectly acceptable to a constructivist; this is implicit in the inaugural lecture of 1912; it is explicit in the set-theory lectures, which were constructive (in the spirit of the dissertation); Brouwer provided the non-constructive arguments with the caveat ‘this is not constructive’.
- 16.
‘Functions from Baire space to the natural numbers are continuous’. In traditional mathematics this is, of course, not the case.
- 17.
Note, however, that Bishop managed to regain a good deal of mathematics in the Kronecker–Borel tradition. The price to be paid was a general strengthening of assumptions.
- 18.
Brouwer (1917a).
- 19.
Among friends he was called ‘Jaap’ or ‘Joop’.
- 20.
The Dutch sinologist and author. This Henri Borel will frequently be mentioned in the history of Significs. The reader should not confuse him with the French mathematician.
- 21.
- 22.
Little John.
- 23.
Redekunstige grondslag van verstandhouding, published in Studies 1–3, 1897.
- 24.
Victoria Alexandrina, the Hon. Lady Welby, wife of Sir William Welby-Gregory, 4th Baronet (1837–1912). I am indebted to Adrian Mathias for the information on Lady Welby.
- 25.
Cf. p. 57.
- 26.
Pseudonym Volker.
- 27.
Siderische Geburt—Seraphische Wanderungen vom Tode der Welt zu Taufe der Tat, Volker (1910).
- 28.
Gutkind (1930).
- 29.
Van Eeden to Lady Welby, 7 October 1910.
- 30.
Van Eeden to Lady Welby, 16 November 1910.
- 31.
Mauthner (1906).
- 32.
Welt-Eroberung durch Helden-Liebe, van Eeden and Gutkind (1911).
- 33.
Aufruf an die Kulturwelt, Kellermann (1915).
- 34.
August 1914.
- 35.
Verslag van de gewone vergadering der wis- en natuurkundige afdeling XXIII (1e gedeelte) p. 828.
- 36.
Published in de Nieuwe Amsterdammer of 3 February 1917.
- 37.
Bij ‘t licht van de oorlogsvlam.
- 38.
Vaderlandsliefde, menschenliefde en opvoeding.
- 39.
Cf. p. 4.
- 40.
Huishoudschool.
- 41.
The house is still there; an old man who had lived in Harzburg his whole life, vividly remembered Professor Brouwer mit seiner Kusine. He recalled that Brouwer offered his father to invest the family capital of only 18 Marks into Dutch guilders, which was strictly forbidden by the currency laws in this period of galloping inflation after World War I, just to help them out. The father refused the offer.
- 42.
Brouwer to Schoenflies, 28 March 1915, 10 June 1915.
- 43.
Lorentz to Brouwer, 11 June 1915.
- 44.
As reported in a letter from Brouwer to Zeeman, 19 June 1915.
- 45.
Also materiell wird von Ihnen nichts weiteres verlangt als dass Sie da sind.
- 46.
Brouwer to Zeeman, 19 June 1915.
- 47.
Blaschke to Brouwer, 4 November 1915.
- 48.
Brouwer to Blaschke, draft 19 November 1915, Cf. CW II, p. 410.
- 49.
- 50.
van Eeden (1971), Diary 11 January 1916.
- 51.
Cf. Fontijn (1990).
- 52.
Ibid. 28 February 1916.
- 53.
Diary Van Eeden, 11 August 1916.
- 54.
Cf. p. 14.
- 55.
Een Machtig Brouwsel; ‘brouwer’ is Dutch for ‘brewer’.
- 56.
Schmitz (1990b).
- 57.
- 58.
- 59.
Internationaal Instituut voor Wijsbegeerte.
- 60.
Manifesto of March–April 1916, cf. Schmitz (1990a).
- 61.
Cf. Schmitz (1990a), p. 220 ff.
- 62.
begrippen en begripsverhoudingen.
- 63.
Mannoury Nu en morgen. Signifische varia. (unpublished) 1939. Cf. Schmitz (1990a), p. 226.
- 64.
Borel to Buber, 7 December 1916.
- 65.
Buber to Brouwer, 1918, Gutkind (1919).
- 66.
Buber to Borel, 17 March 1917.
- 67.
Brouwer to Buber, 4 February 1918.
- 68.
Gutkind (1919).
- 69.
Propria Cures 9 September 1918.
- 70.
Brouwer to Zeeman, 8 May 1916.
- 71.
On the significance of mathematical logic for philosophy, 1903.
- 72.
Faculty to Curators, 9 January 1917.
- 73.
Over de Sociale Betekenis van de Wiskundige Denkvorm.
- 74.
Ueber Stabilität periodischer ebener Bahnen, Wiener Berichte 1886. Brouwer to Korteweg 15 October 1915.
- 75.
Korteweg to Brouwer, 17 October 1915, 18 October 1915, Brouwer to Korteweg 19 October 1915.
- 76.
Brouwer to Korteweg, 29 November 1915, 1 December 1915, 2 December 1915.
- 77.
The reference to Brouwer’s card is erroneous. There must have been a verbal exchange.
- 78.
Mannoury (1937).
- 79.
- 80.
Brouwer to Schoenflies 5 August 1916.
- 81.
Brouwer to Schoenflies 27 August 1916.
- 82.
De Nieuwe Amsterdammer 19 June 1916.
- 83.
Brouwer to Minister of Defence, 18 September 1915.
- 84.
Snijders to Brouwer, 12 October 1915.
- 85.
- 86.
In 1915 the so-called landstorm was created, an organisation comparable to the National Guard. It was intended for volunteers who had completed their first military training, but who were not called up in the mobilisation.
- 87.
Brouwer to Zeeman, 17 December 1917.
- 88.
Cf. p. 48. The passage occurs in the draft but not in the final letter (Brouwer to Lorentz 16 January 1918).
- 89.
It is a fact that Lorentz could not say ‘no’ to requests from the government. Another instance is his involvement in the ‘Zuiderzee project’, which took up much of his valuable time.
- 90.
A military airfield of long standing. After the second world war an American squadron was based at Soesterberg.
- 91.
As already suggested in Brouwer to Lorentz, 19 February 1918.
- 92.
Brouwer to Lorentz, 8 September 1918.
- 93.
First having obtained Zeeman’s advice. Brouwer to Zeeman, 12 July 1918.
- 94.
Fokker to Brouwer, 24 March 1919.
- 95.
Oral communication C. Emmer, Brouwer’s friend and family doctor.
- 96.
The gentleman-burglar Arsène Lupin, was the hero of detective novels of M. Leblanc.
- 97.
A fictitious village.
- 98.
Het Naakt Model, Grote dagen in Knollenbroek, De vergulde Krakeling.
- 99.
Brouwer to Scheltema, 27 September 1917 and 8 September 1918.
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van Dalen, D. (2013). The War Years. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_7
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