Advertisement

Abstract

Brouwer’s Thesis is a somewhat ambiguous book, it contains a purely mathematical part, dealing with ‘Hilbert 5’, i.e. the elimination of differentiability conditions in the theory of Lie groups, and a number of geometrical investigations. But the larger part was the presentation of a personal approach to the foundations of mathematics together with well-argued criticism of contemporary schools. The chapter makes extensive use of archive material, that allows us to follow how Brouwer’s ideas evolved. It contains the fundamental material on Brouwer’s ur-intuition, the genesis of the natural numbers and the continuum. Furthermore Brouwer’s views and first steps in intuitionistic logic are discussed. The dissertation and the archive material shows that Brouwer’s philosophical principles went beyond just mathematics.

Keywords

Number Class Mathematical System Continuum Problem Continuum Hypothesis Outer World 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Bellaar-Spruyt, C.: Leerboek der Formeele Logica. Bewerkt naar de dictaten van wijlen Prof. Dr. C.B. Spruyt door M. Honigh. Vincent Loosjes, Haarlem (1903) Google Scholar
  2. Brouwer, L.E.J.: Polydimensional vectordistributions. K. Ned. Akad. Wet. Proc. Sect. Sci. 9, 66–78 (1906a) Google Scholar
  3. Brouwer, L.E.J.: The force field of the non-Euclidean spaces with negative curvature. K. Ned. Akad. Wet. Proc. Sect. Sci. 9, 116–133 (1906b) Google Scholar
  4. Brouwer, L.E.J.: The force field of the non-Euclidean spaces with positive curvature. K. Ned. Akad. Wet. Proc. Sect. Sci. 9, 250–266 (1906c). Corr. in Brouwer (1909g) Google Scholar
  5. Brouwer, L.E.J.: Die mögliche Mächtigkeiten. In: Castelnuovo, G. (ed.) Atti IV Congr. Intern. Mat. Roma, vol. 3, pp. 569–571. Accad. Naz. Lincei, Roma (1908a) Google Scholar
  6. Brouwer, L.E.J.: De onbetrouwbaarheid der logische principes. Tijdschr. Wijsb. 2, 152–158 (1908b) Google Scholar
  7. Brouwer, L.E.J.: Remark on multiple integrals. K. Ned. Akad. Wet. Proc. Sect. Sci. 22, 150–154 (1919k) Google Scholar
  8. Brouwer, L.E.J.: Mathematik, Wissenschaft und Sprache. Monatshefte Math. Phys. 36, 153–164 (1929a) MathSciNetMATHCrossRefGoogle Scholar
  9. Brouwer, L.E.J.: Die Struktur des Kontinuums [Sonderabdruck] (1930) MATHGoogle Scholar
  10. Brouwer, L.E.J.: Willen, Weten, Spreken. Euclides 9, 177–193 (1933a) Google Scholar
  11. 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) Google Scholar
  12. Browder, F.I. (ed.): Hilbert’s Problems. Proc. of Symposia in Pure Mathematics. Am. Math. Soc., Providence (1976). 2 vols. Google Scholar
  13. Freudenthal, H.: Zur Geschichte der Grundlagen der Geometrie. Zugleich eine Besprechung der 8. Aufl. von Hilberts “Grundlagen der Geometrie”. Nieuw Arch. Wiskd. 5, 105–142 (1957) MathSciNetMATHGoogle Scholar
  14. Gray, J.J.: The Hilbert Challenge. Oxford University Press, Oxford (2000) MATHGoogle Scholar
  15. Hilbert, D.: Mathematische Probleme. Nachr. Ges. Wiss. Gött., Math.-Phys. Kl. 999, 253–297 (1900) Google Scholar
  16. Hilbert, D.: Über die Grundlagen der Logik und der Arithmetik. In: Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13 August 1904, pp. 174–185. Teubner, Leipzig (1905) Google Scholar
  17. Mannoury, G.: Review. Over de Grondslagen van de Wiskunde. De Beweging 3, 241–249 (1907) Google Scholar
  18. Poincaré, H.: Sur les résidus des integrales doubles. Acta Math. 9, 321–380 (1887) MathSciNetMATHCrossRefGoogle Scholar
  19. Poincaré, H.: Analysis situs. J. Éc. Polytech. 1, 1–123 (1895) Google Scholar
  20. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. III. Gauthier-Villars, Paris (1899) Google Scholar
  21. Poincaré, H.: Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathematische Physik. Teubner, Leipzig (1910) Google Scholar
  22. Rogers, H. Jr.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967) MATHGoogle Scholar
  23. Ular, A.: Le Livre de La Voie et la ligne-droite de LAO-TSË. Éditions de la Revue Blanche, Paris (1902) Google Scholar
  24. van Atten, M.: The hypothetical judgement in the history of intuitionistic logic. In: Glymour, C., Wang, W., Westerståhl, D. (eds.) Logic, Methodology, and Philosophy of Science XIII: Proceedings of the 2007 International Congress in Beijing, vol. 13. King’s College Publications, London (2008) Google Scholar
  25. van Dalen, D.: L.E.J. Brouwer en De Grondslagen van de wiskunde. Epsilon, Utrecht (2001b) MATHGoogle Scholar
  26. van Dalen, D.: Kolmogorov and Brouwer on constructive implication and the Ex Falso rule. Russ. Math. Surv. 59, 247–257 (2004) CrossRefGoogle Scholar
  27. van Stigt, W.P.: The rejected parts of Brouwer’s dissertation on the foundations of mathematics. Hist. Math. 6, 385–404 (1979) MATHCrossRefGoogle Scholar
  28. van Stigt, W.P.: Brouwer’s Intuitionism. North-Holland, Amsterdam (1990) MATHGoogle Scholar
  29. Zermelo, E.: Beweis dasz jede Mengen wohlgeordnet werden kann. Math. Ann. 59, 514–516 (1904) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Dirk van Dalen
    • 1
  1. 1.Department of PhilosophyUtrecht UniversityUtrechtNetherlands

Personalised recommendations