Abstract
After the War of the Frogs and the Mice Brouwer more or less retired from the scene. Freudenthal, Hopf’s student, was appointed as his assistant in Amsterdam. Brouwer fought an investment scandal involving a health spa in Budapest, and founded a new mathematics journal, Compositio Mathematica. Heyting introduced his formal system for intuitionistic logic and arithmetic. The foundational atmosphere clearly was improving. The rise of the nazi regime is discussed, including an attempt of the new authorities to lure Brouwer to Göttingen; which predictably failed. At the end of the thirties Brouwer briefly returned to topology with a proof of the triangulation property for differentiable manifolds, only to find out that an American mathematician, Cairns, had already solved that case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Brouwer to Wirtinger, 11.I.1929.
- 2.
Brouwer to Ehrenhaft, 1.IV.1930.
- 3.
Brouwer to Ehrenhaft, 19.IV.1930.
- 4.
Part of the file was taken to Moscow after Brouwer’s death by Alexandrov, and the remains that survived two fires and several transfers is in the Brouwer Archive.
- 5.
For more on Freudenthal’s Berlin years, see Freudenthal (1987a).
- 6.
Freudenthal may have been a bit prejudiced; Rutherford mentions another experience: ‘In the winter of 1929 Professor Weitzenböck pointed out to me that there was no complete account of the theory of modular invariants embodying the work of Dickson, Glenn and Hazlett. … The substance of Part II is largely taken from a course of lectures entitled “Algebraïsche theorie der lichamen” which Professor Weitzenböck delivered in Amsterdam University during the session 1929–30.’ Rutherford (1932).
- 7.
According to Freudenthal, Hurewicz presented Gödel’s incompleteness theorem in a seminar. He also refereed foundational papers (including Heyting’s big logic papers).
- 8.
Freudenthal to Hopf, 22.XII.1930.
- 9.
First semester of 1930/31.
- 10.
De Donder to Brouwer, 26.X.1929.
- 11.
Brouwer to De Donder, 13.VI.1930.
- 12.
Brouwer to De Donder, 9.X.1930.
- 13.
Heyting (1931b).
- 14.
Brouwer to Heyting, 20.IX.1930.
- 15.
Heyting (1931a).
- 16.
- 17.
Cf. Herbrand (1971), pp. 273, 282 ff.
- 18.
Freudenthal to Hopf, 2.IX.1931. The reference to bishops in jail is probably a bit of irreverent gossip, the notes in the archive do not mention such a thing.
- 19.
He visited Budapest at least 37 times between 1930 and 1939!
- 20.
However, the last correspondence concerning Sodalitas was dated 1951, and the last remittance took place in 1971!
- 21.
Departementshoofd.
- 22.
Assistants earned at that time 1000 guilders. The proposed raise was 200 guilders.
- 23.
The so-called ‘openbare les’, i.e. ‘public lecture’.
- 24.
Speakers: Alexandrov, Borsuk, Hurewicz, Kaufmann, Knaster.
- 25.
Mathematische Grundlagenforschung. Intuitionismus, Beweistheorie.
- 26.
Heyting, oral communication.
- 27.
Brouwer to Korteweg, 23.I.1907.
- 28.
See e.g. van Dalen (2004).
- 29.
- 30.
The student who took the notes was David van Dantzig, who was to contribute to the foundations himself much later.
- 31.
Most of the historical information on Van Dantzig is taken from Alberts’ biography, Alberts (2000).
- 32.
See p. 4.
- 33.
Van der Waerden to Schouten, 22.IV.1932.
- 34.
van Dantzig (1931).
- 35.
Ehrenfest to Van der Waerden, 6.II.1930.
- 36.
Courant to Ehrenfest, 9.II.1930. Courant informed Ehrenfest that Van der Waerden was number 3 on the list for Hilbert’s succession.
- 37.
Einfachste Grundbegriffe der Topologie, Alexandrov (1932).
- 38.
Combinatorial and set theoretic topology.
- 39.
Alexandroff (1932), p. 26.
- 40.
Brouwer to Alexandrov, 20.X.1932.
- 41.
Lize to Brouwer, 7.IX.1932.
- 42.
It should be pointed out that a knighthood in the Netherlands carried no side benefits; there is no title attached to it and it does not lend social or legal status to a person. It is no more and no less than a sign of royal appreciation.
- 43.
Brouwer to Van der Corput, 23.XI.1933. Cf. p. 600.
- 44.
Brouwer to Van der Corput, 5.X.1935.
- 45.
Vollenhove (1918).
- 46.
Herman Dooyeweerd (1884–1977) was a law professor at the Calvinist Vrije Universiteit. His fame rests on his philosophy, as presented in De wijsbegeerte van de wetsidee (1935–36).
- 47.
For a discussion of the role of intuitionism in Van Vollenhoven’s thinking, see Blauwendraat (2004).
- 48.
- 49.
Revész (1933).
- 50.
Oral communication, Mrs. C. Vuijsje.
- 51.
- 52.
- 53.
Pinl and Furtmüller (1973).
- 54.
Schappacher and Kneser (1990), p. 27. Schappacher pointed out that the authorities might have been in a hurry to handle the Göttingen mathematics department, in order to forestall possible student actions against the institute or individual mathematicians. The institute was viewed as a ‘bastion of Marxism’. For background information on the political landscape and the ensuring developments, see also the above mentioned exposition.
- 55.
See Reid (1986), Chap. 15.
- 56.
Teichmüller would have become one of the top mathematicians of his generation, had he survived the war. He volunteered in 1939 for military service and fell in 1943 at the eastern front. For mathematical and historical information, see Schappacher and Scholz (1992).
- 57.
Mannoury to Brouwer, 17.VI.1933.
- 58.
The National Socialist regime carried the political use of language to unknown heights; such terms as ‘Gleichschalten’ had a normal everyday meaning, but under the regime it acquired a new one: ‘following the Nazidoctrine’, or even cruder, ‘eliminating opposition and deviant ideas and practice’.
- 59.
- 60.
Abstrakte Geometrie, Vahlen (1905b).
- 61.
- 62.
Vahlen (1911).
- 63.
- 64.
Einstein to Hedwig Born, Einstein and Born (1969).
- 65.
Communication of H. Freudenthal.
- 66.
Mehrtens (1987). Our presentation makes substantial use of this paper.
- 67.
Bieberbach’s inaugural lecture, Bieberbach (1914), the translation is Mehrtens’.
- 68.
Bieberbach (1924).
- 69.
Boutroux (1920). Bieberbach read the German translation.
- 70.
A society of patrons and supporters of education in the exact sciences.
- 71.
Vom Wissenschaftsideal der Mathematiker, 15.II.1926.
- 72.
Schütz Abteilung, the storm troopers of the Party.
- 73.
Mehrtens (1987).
- 74.
Ibid. p. 227.
- 75.
- 76.
Lindner (1980), p. 95.
- 77.
Mehrtens (1987), p. 228.
- 78.
Mehrtens (1987), p. 228.
- 79.
See Menzler-Trott (2001), Chap. 4, and the literature cited in that book.
- 80.
Hardy (1934). Also in Math. Intelligencer, 6 (1984).
- 81.
Cf. p. 562.
- 82.
Brouwer (1909a, 1912a, 1919b). Noordhoff listed Brouwer’s publications regularly in his catalogue. In 1922, 1926, 1928, 1933: De onbetrouwbaarheid der logische principes, Het Wezen der Meetkunde, Intuïtionisme en Formalisme (collected in Wiskunde, Waarheid, Werkelijkheid), Over de Grondslagen der Wiskunde, Luchtvaart en Photogrammetrie. In 1938, 1940,1942, 1948, De onbetrouwbaarheid der logische principes, Het Wezen der Meetkunde, Intuïtionisme en Formalisme. In 1949 only De Uitdrukkingswijze der Wetenschap (containing Brouwer 1933b), this item appeared for the last time in the catalogue of 1958.
- 83.
Brouwer to Noordhoff, 10.X.1929, cf. also p. 504.
- 84.
From Brouwer’s letter to Veblen, 11.X.1930.
- 85.
Hadamard to Einstein, 16.X.1930.
- 86.
Einstein to Hadamard, 15.XI.1930.
- 87.
This is a somewhat free translation of the German text. There is probably an English version somewhere in some archive, but I have not found any.
- 88.
Freudenthal to Hopf, 22.XII.1930.
- 89.
Brouwer schimpft jetzt auf Alexandrov.
- 90.
Bieberbach to Brouwer, 21.VI.1934.
- 91.
- 92.
Cf. Remmert (1999), p. 18.
- 93.
Bieberbach to Brouwer, 8.I.1935.
- 94.
Brouwer to Bieberbach, 15.I.1935.
- 95.
Bieberbach to Doetsch, 19.I.1935.
- 96.
Brouwer to Doetsch, 20.III.1935.
- 97.
Doetsch to Feigl, 16.VII.1934.
- 98.
- 99.
Alexandroff–Hopf, Topologie. Dedicated to Brouwer.
- 100.
‘… aus der Redaktion vom Compositio Mathematica sämtliche reichsdeutsche Mitglieder ausgeschieden sind, …’
- 101.
Blumenthal to Hilbert, 11.XI.1933.
- 102.
Schappacher (1987), p. 354.
- 103.
- 104.
Brouwer was in German circles described as deutschfreundlich. This term acquired after 1933 a very specific meaning: pro Nazi. But before 1933 it just meant what it said: sympathetic towards Germany and Germans. It is not unusual for commentators to confuse the two meanings.
- 105.
Tornier to Brouwer, 19.VI.1934.
- 106.
Ich erlaube mir nun die Anfrage, ob Sie, den viele deutsche Mathematiker mit mir schon lange für einen der grössten Forscher von typische germanischer Prägung halten, bereit wären, den alten Ruf der Göttinger Mathematik neu begründen zu helfen.
- 107.
Bohr to Veblen, 11.VIII.1934, cf. Segal (1986).
- 108.
Siegmund-Schultze (2001), p. 191.
- 109.
Brouwer to Veblen, 20.X.1934.
- 110.
Sister society of the AA and AAA.
- 111.
Interview, H.P. de Klerk, junior.
- 112.
gemeenteraden.
- 113.
Private communication, J.J. Oversteegen.
- 114.
Weitzenböck (1923).
- 115.
Balke (1973), p. 101.
- 116.
Helen Ernst, an artist.
- 117.
Hübner (2002).
- 118.
eine Hütte “mit bohème-artiger Verpflegung”.
- 119.
There are some post 1945 documents in the archive that refer to land property in Poland. No details are given.
- 120.
- 121.
- 122.
The fact that Heyting was an associate editor may be explained by the distribution of the specialisms. There was already ample topological expertise in the board, but Heyting was the only foundationalist.
- 123.
Inaugural address, 28.V.1931.
- 124.
Brouwer to Freudenthal, 20.VIII.1935.
- 125.
Freudenthal to Brouwer, 12.VIII.1936.
- 126.
In Dutch, a ‘juffrouw’. Today it would be insulting, it was not so in the old days.
- 127.
Brouwer to Freudenthal, 17.VIII.1936.
- 128.
Freudenthal, oral communication.
- 129.
Ibid.
- 130.
To commemorate the centenary of Van der Waals’ birth.
- 131.
- 132.
Oral communication, Freudenthal.
- 133.
Freudenthal slightly extended the results by showing that C q-manifolds allowed C q-triangulations.
- 134.
Brouwer to Freudenthal, 8.VII.1939.
- 135.
Brouwer to Rosenthal, 4.X.1939.
- 136.
Brouwer to Freudenthal, 4.X.1939.
- 137.
Brouwer to Freudenthal, 25.X.1939.
- 138.
Freudenthal to Brouwer, 27.X.1939.
- 139.
Freudenthal to Brouwer, 18.III.1940; Cairns to Freudenthal, 14.I.1940; Cairns (1935).
- 140.
Brouwer to Freudenthal, 30.IV.1940.
- 141.
Brouwer (1939).
- 142.
Freudenthal (1939).
- 143.
Oral communication Mrs. J.F. Heyting-van Anrooy, the later wife of Arend Heyting.
- 144.
Published in De Tribune, a communist journal, 16.I.1937. See also Fasseur (2001), p. 144 ff.
- 145.
Bonger took his life when the Germans invaded Holland.
- 146.
Pontryagin to Lefschetz, 29.XII.1939.
- 147.
Brouwer to Freudenthal, 2.III.1940.
- 148.
See Behnke (1978).
- 149.
Behnke to Van Dalen, 27.XI.1976.
- 150.
Blumenthal (1935).
- 151.
Blumenthal’s 1939–1943 diaries have been published by Volkmar Felsch, Felsch (2011).
- 152.
A transit camp from where Jews were sent to the camps in Germany.
- 153.
Cf. p. 623, see also Siegmund-Schultze (1984).
- 154.
In Brouwer’s words ‘sein tiefsinniges und suggestives, viel zu wenig gewürdigtes Werk über geometrische Grundlagenfragen; “Abstrakte Geometrie”’.
- 155.
Wim Bierens de Haan.
- 156.
Most of the information on this episode was provided by Mrs. Van Wering.
- 157.
Mayor of Hilversum to Brouwer, 15.VI.1938.
- 158.
Trustees to council, 13.VII.1939.
- 159.
A semi-official institution providing (mostly evening) courses in various areas.
- 160.
repetitor.
References
Alberts, G.: Twee geesten van de wiskunde. Biografie van David van Dantzig. CWI, Amsterdam (2000)
Alexandroff, P.: Einfachste Grundbegriffe der Topologie. Springer, Berlin (1932)
Alexandrov, P.S.: Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen. Math. Ann. 106, 161–238 (1932)
Balke, E.: Chroniknotizen. Aus den Errinnerungen des Freundes (1902–1989). L. Mundschenk KG, Soltau (1973)
Behnke, H.: Semesterberichte. VandenHoeck und Ruprecht, Göttingen (1978)
Bieberbach, L.: Über die Grundlagen der modernen Mathematik. Die Geisteswissenschaften 1, 896–901 (1914)
Bieberbach, L.: Ganesh Prasad. Mathematical research in the last 20 years (review). Dtsch. Literaturztg. 45, 725–727 (1924)
Bieberbach, L.: Persöhnlichkeitsstruktur und Mathematisches Schaffen. Unterrichtsblätter Math. Naturwiss. 40, 236–243 (1934)
Blauwendraat, H.: Worsteling naar waarheid. De opkomst van Wiskunde en Informatica aan de VU. Meinema, Zoetermeer (2004)
Blumenthal, O.: Lebensgeschichte von David Hilbert. In: David Hilbert, Gesammelte Abhandlungen. III, vol. 3, pp. 388–429. Springer, Berlin (1935)
Bohr, H.: Ny Matematik i Tyskland. Berlinske Aften (1934)
Boutroux, P.: L’idéal scientifique des Mathématiciens dans l’antiquité et dans les temps modernes. Felix Alcan, Paris (1920)
Brouwer, L.E.J.: Leven, Kunst en Mystiek. Waltman, Delft (1905). Translation by W.P. van Stigt in Notre Dame J. Form. Log. 37, 381–429 (1966)
Brouwer, L.E.J.: Het wezen der meetkunde. Clausen, Amsterdam (1909a). Inaugural address privaat docent, 12.10.1909. Also in Brouwer (1919b)
Brouwer, L.E.J.: Intuïtionisme en Formalisme. Clausen, Amsterdam (1912a). Inaugural address, professor
Brouwer, L.E.J.: Wiskunde, Waarheid, Werkelijkheid. Noordhoff, Groningen (1919b)
Brouwer, L.E.J.: Intuïtionistische splitsing van mathematische grondbegrippen. K. Ned. Akad. Wet. Versl. Gewone Vergad. Afd. Natuurkd. 32, 877–880 (1923d)
Brouwer, L.E.J.: Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresber. Dtsch. Math.-Ver. 33, 251–256 (1925d)
Brouwer, L.E.J.: Mathematik, Wissenschaft und Sprache. Monatshefte Math. Phys. 36, 153–164 (1929a)
Brouwer, L.E.J.: Willen, Weten, Spreken. Euclides 9, 177–193 (1933a)
Brouwer, L.E.J.: Willen, Weten, Spreken. In: Brouwer, L.E.J., Clay, J., et al. (eds.) De uitdrukkingswijze der wetenschap, pp. 43–63. Noordhoff, Groningen (1933b)
Brouwer, L.E.J.: Zum Triangulationsproblem. Indag. Math. 1, 248–253 (1939)
Brouwer, L.E.J.: Consciousness, philosophy and mathematics. In: Proceedings of the 10th International Congress of Philosophy, Amsterdam, 1948, vol. 3, pp. 1235–1249 (1949c)
Cairns, S.S.: Triangulation of the manifold of class one. Bull. Am. Math. Soc. 41, 549–552 (1935)
Craig, G.A.: Germany 1866–1945. Oxford University Press, Oxford (1981)
Dehn, M.: K.Th. Vahlen, Abstrakte Geometrie (review). Jahresber. Dtsch. Math.-Ver. 14, 535–537 (1905)
Einstein, A., Born, M.: Briefwechsel 1916–1955. Nymphenburger Verlagshandlung, Munich (1969)
Fasseur, C.: Wilhelmina. Krijgshaftig in een vormeloze jas. Balans, Amsterdam (2001)
Felsch, V.: Otto Blumenthals Tagebücher. Ein Aachener Mathematik Professor erleidet die NS-Diktatur in Deutschland, den Niederlanden und Theresienstadt. Hartung Gorre Verlag, Konstanz (2011)
Freudenthal, H.: Zur intuitionistischen Deutung logischer Formeln. Compos. Math. 4, 112–116 (1936)
Freudenthal, H.: Die Triangulation der differentierbaren Mannigfaltigkeiten. Indag. Math. 1, 311–332 (1939). Nachtrag, Ind. Math. 2, 249 (1940)
Freudenthal, H.: Berlin 1923–1930. Studienerinnerungen von Hans Freudenthal. de Gruyter, Berlin (1987a)
Hardy, G.H.: The J-type and the S-type among mathematicians. Nature 134, 250 (1934)
Herbrand, J.: Logical Writings. Goldfarb, W. (ed.). Harvard University Press, Cambridge (1971)
Hesseling, D.E.: Gnomes in the Fog. The Reception of Brouwer’s Intuitionism in the 1920s. Birkhäuser, Basel (2002)
Heyting, A.: Die Intuitionistische Mathematik. Forsch. Fortschr. 7, 38–39 (1931a)
Heyting, A.: Philosphische Grundlegung der Mathematik. Blätter für Deutsche Philosophie (4), 1930 (review). Jahresber. Dtsch. Math.-Ver. 40, 50–52 (1931b)
Heyting, A.: Sur la logique intuitionniste. A propos d’un article de MM. Barzin et Errera. Enseign. Math. 31, 121–122 (1932)
Heyting, A.: Bemerkung zu dem Aufsatz von Hern Freudenthal “Zur intuitionistischen Deutung logischer Formeln”. Compos. Math. 4, 117–118 (1936a)
Hübner, H.: Ein zerbrechliches Menschenskind – Helen Ernst (1904–1948). Biographie einer antifaschistischen Künstlerin zwischen Athen, Zürich, Berlin, Amsterdam, Ravensbrück und Schwerin. Trafo Verlag, Berlin (2002)
James, I.M. (ed.): The History of Topology. Elsevier, Amsterdam (1999)
Kuiper, N.H.: A short history of triangulation and related matters. In: Baayen, P.C., van Dulst, D., Oosterhoff, J. (eds.) Proceedings Bicentennial Congress Wiskundig Genootschap, vol. 1, pp. 61–79. Mathematisch Centrum, Amsterdam (1979)
Kuiper, J.J.C.: Ideas and explorations. Brouwer’s road to intuitionism. Ph.D. thesis, Utrecht University (2004)
Lindner, H.: Deutsche und gegentypische Mathematik. Zur Begründung einer arteigenen Mathematik im Dritten Reich durch Ludwig Bieberbach. Suhrkamp, Frankfurt am Main (1980)
Mehrtens, H.: Das “Dritte Reich” in der Naturwissenschaftsgeschichte: Literaturbericht und Problemskizze. In: Mehrtens, H., Richter, S. (eds.) Naturwissenschaft, Technik und NS-Ideologie. Suhrkamp, Frankfurt (1980)
Mehrtens, H.: Ludwig Bieberbach and Deutsche Mathematik. In: Philips, E.R. (ed.) Studies in History of Mathematics, pp. 195–241. Math. Assoc. of America, Washington (1987)
Menzler-Trott, E.: Gentzens Problem. Mathematische Logik im nationalsozialistischen Deutschland. Birkhäuser, Basel (2001)
Pinl, M., Furtmüller, L.: Mathematicians under Hitler. In: Year Book XVIII. Publications of the Leo Baeck Institute, vol. 18, pp. 129–184. Secker & Warburg, London (1973)
Reid, C.: Hilbert–Courant. Springer, Berlin (1986)
Remmert, V.R.: Mathematicians at war. Power struggles in Nazi Germany’s mathematical community: Gustav Doetsch and Wilhelm Süss. Rev. Hist. Math. 5, 7–59 (1999)
Remmert, V.R.: Die Deutsche Mathematiker-Vereinigung im “Dritten Reich”: Fach- und Parteipolitik. DMV-Mitt. 12, 223–245 (2004a)
Remmert, V.R.: Die Deutsche Mathematiker-Vereinigung im “Dritten Reich”: Krisenjahre und Konsolidierung. DMV-Mitt. 12, 159–177 (2004b)
Revész, G.: Das Schöpferisch-Persönliche und das Kollektive in ihrem kulturhistorischen Zusammenhang. J.C.B. Mohr (Paul Siebeck), Tübingen (1933)
Rutherford, D.: Modular Invariants. Cambridge University Press, Cambridge (1932)
Schappacher, N.: Das Mathematische Institut der Universität Göttingen 1929–1950. In: Becker, H., Dahms, H.-J., Wegeler, C. (eds.) Die Universität Göttingen unter dem National Sozialismus, pp. 345–373. K.G. Saur, München (1987)
Schappacher, N., Kneser, M.: Fachverband – Institut – Staat. Streiflichter auf das Verhältnis von Mathematik zu Gesellswchaft und Politik in Deutschland sei 1890 unter besonderer Berücksichtigung der Zeit des Nationalsozialismus. In: Fischer, G., Hirzebruch, F., Scharlau, W., Törnig, W. (eds.) Festschrift zur Jubiläum der DMV. Vieweg, Braunschweig (1990)
Schappacher, N., Scholz, E.: Oswald Teichmüller – Leben und Werk. Jahresber. Dtsch. Math.-Ver. 94, 1–39 (1992)
Segal, S.L.: Mathematics and German politics: the National Socialist experience. Hist. Math. 13, 118–135 (1986)
Siegmund-Schultze, R.: Theodor Vahlen zum Schuldanteil eines deutschen Mathematikers am faschistischen Miszbrauch der Wissenschaft. NTM Schriftenr. Gesch. Naturwiss. Tech. Med. 21, 17–32 (1984)
Siegmund-Schultze, R.: Rockefeller and the Internationalization of Mathematics Between the Two World Wars. Birkhäuser, Basel (2001)
Vahlen, K.Th.: Max Dehns Besprechung meiner “Abstrakte Geometrie”. Jahresber. Dtsch. Math.-Ver. 14, 591–595 (1905a)
Vahlen, Th.: Abstrakte Geometrie. Untersuchungen über die Grundlagen der Euklischen und nicht-Euklidischen Geometrie. Teubner, Leipzig (1905b). Second edition appeared as supplement to Deutsche Mathematik. Hirzel, Leipzig (1940)
Vahlen, Th.: Konstruktionen und Approximationen in systematischer Darstellung: eine Vorstufe zur höheren Geometrie. Teubner, Leipzig (1911)
Vahlen, Th.: Ballistik. de Gruyter, Berlin (1922)
Vahlen, Th.: Ballistik, 2nd edn. de Gruyter, Berlin (1942)
van Dalen, D.: Kolmogorov and Brouwer on constructive implication and the Ex Falso rule. Russ. Math. Surv. 59, 247–257 (2004)
van Dantzig, D.: Studien over topologische algebra. Ph.D. thesis, Rijksuniversiteit Groningen (1931)
Vollenhove, D.H.T.: De wijsbegeerte van de wiskunde van theïstisch standpunt. Ph.D. thesis, Vrije Universiteit, Amsterdam (1918)
Weitzenböck, R.: Invarianten-Theorie. Noordhoff, Groningen (1923)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
van Dalen, D. (2013). The Thirties. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_15
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4616-2_15
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4615-5
Online ISBN: 978-1-4471-4616-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)