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Brouwer and Fraenkel on Intuitionism

Published online by Cambridge University Press:  15 January 2014

Dirk Van Dalen
Dirk Van Dalen*
Affiliation:
Department of Philosophy, Utrecht University, P.O. Box 8016, 3508 Tc Utrecht, The NetherlandsE-mail:[email protected]
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Extract

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.

The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional man like Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's book Zehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.

With the Grundlagenstreit reaching (in print!) a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.

The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.

I am indebted to Andreas Blass for his valuable suggestions and corrections.

Abraham Fraenkel (then still called Adolf) was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 3 , September 2000 , pp. 284 - 310
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[Bockstaele, 1949] Bockstaele, P., Het intuïtionisme bij de Franse wiskundigen, Verhandelingen van de Koninklijke Vlaamse Academie van Wetenschappen, Letteren en Schone Kunsten van België, vol. XI (32).Google Scholar
[Brouwer, 1930] Brouwer, L.E.J., A. Fraenkel, Zehn Vorlesungen Über die Grundlegung der Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 39, pp. 10–11.Google Scholar
[Brouwer, 1919A] Brouwer, L.E.J., BegrÜndung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil, Theorie der Punktmengen, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam (Eerste Sectie), vol. XII, no. 7, pp. 1–33.Google Scholar
[Brouwer, 1952] Brouwer, L.E.J., Historical background, principles and methods of intuitionism, South African Journal of Science, vol. 49, pp. 139–146.Google Scholar
[Brouwer, 1992] Brouwer, L.E.J., Intuitionismus, Bibliographisches Institut, Wissenschaftsverlag, Mannheim, edited by Dalen, D. van.Google Scholar
[Brouwer, 1928] Brouwer, L.E.J., Intuitionistische Betrachtungen Über den Formalismus, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the section of Sciences, vol. 36, p. 1189.Google Scholar
[Brouwer, 1926] Brouwer, L.E.J., Intuitionistische EinfÜhrung des Dimensionsbegriffes, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the section of Sciences, vol. 29, pp. 855–873.Google Scholar
[Brouwer, 1910] Brouwer, L.E.J., On the structure of perfect sets of points, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the section of Sciences, vol. 12, pp. 785–794.Google Scholar
[Brouwer, 1907] Brouwer, L.E.J., Over de grondslagen der wiskunde, Ph.D. thesis , Amsterdam.Google Scholar
[Brouwer, 1919B] Brouwer, L.E.J., Intuitionistische Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 28 (1919), pp. 203–208.Google Scholar
[Fraenkel, 1919] Fraenkel, A., Einleitung in die Mengenlehre, Springer Verlag, Berlin.CrossRefGoogle Scholar
[Fraenkel, 1923] Fraenkel, A., Einleitung in die Mengenlehre, second ed., Springer Verlag, Berlin.Google Scholar
[Fraenkel, 1927] Fraenkel, A., Zehn Vorlesungen Über die Grundlegung der Mengenlehre, Teubner.Google Scholar
[Fraenkel, 1967] Fraenkel, A. A., Lebenskreise. Aus den Erinnerungen eines JÜdischen Mathematikers, Deutsche Verlags-Anstallt, Stuttgart.Google Scholar
[Hilbert, 1905] Hilbert, D., Über die Grundlagen der Logik und der Arithmetik, Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13 August 1904, Teubner, Leipzig, pp. 174–185.Google Scholar
[Molk, 1885] Molk, J., Sur une notion qui comprend celle de la divisibilité et sur la théorie géneérale de l'élimination, Acta Mathematica, vol. 6, pp. 1–166.Google Scholar
[Dalen, van 1995] Dalen, D. van, Hermann Weyl's intuitionistic mathematics, this Bulletin, vol. 1, pp. 145–169.Google Scholar
[Dalen, van 1999] Dalen, D. van, Mystic, geometer, and intuitionist: L.E.J. Brouwer. Volume 1 – The Dawning Revolution, Oxford University Press, Oxford.Google Scholar
[Dalen, van 1990] Dalen, D. van, The war of the frogs and the mice, or the crisis of the Mathematische Annalen, Mathematical Intelligencer, vol. 12, pp. 17–31.CrossRefGoogle Scholar
[Veldman, 1976] Veldman, W., An intuitionistic completeness theorem for intuitionistic predicate logic, The Journal of Symbolic Logic, vol. 41, pp. 159–166.Google Scholar
[Weyl, 1921] Weyl, H., Über die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift, vol. 10, pp. 39–79.Google Scholar