Are the basic axioms in mathematics empirical

Philosophy of Science of Mathematics

Theory of Science pp 189-208 | Cite as

  • Stephan Kornmesser
  • Wilhelm Buettemeyer
First Online:


Mathematics has been a subject of epistemological reflections since ancient times. The indistinguishable status of their objects of investigation, the exemplary precision of their conceptualization and argumentation, the unequivocal validity of their theorems and their remarkable arrangement in axiom systems as well as the versatile applicability of their quite abstract results required an explanation. At the same time, the positive assessment just described resulted in a yardstick against which mathematical theories had to be measured from an epistemological point of view.

This is a preview of subscription content, log in to check access.


Unable to display preview. Download preview PDF.


  1. Aristotle: Second analytics [about 350 BC Chr.]. Einl., Übers. And Komm. Horst Seidl, gr.-dt. Würzburg / Amsterdam 21987. Google Scholar
  2. Euclid: The Elements [about 300 BC Chr.]. Trans. V. Clemens Thaer. 4th, exp. Edition Frankfurt a. M. 2003. Google Scholar
  3. Blaise Pascal's reflections on geometry in general: "De l’esprit géométrique" and "De l’art de persuader" [around 1655]. With German translation and comm. By Jean-Pierre Schobinger. Basel 1974.Google Scholar
  4. Bourbaki, Nicolas: "L’architecture des mathématiques". In: François Le Lionnais (ed.): The grands courants de la pensée mathématique. Paris 1962, 35-47; German in: Michael Otte (ed.): Mathematician on math. Berlin 1974, 140–159.Google Scholar
  5. Brouwer, Luitzen Egbertus Jan: Collected Works. Vol. I: Philosophy and Foundations of Mathematics [1905-55]. Amsterdam 1975.Google Scholar
  6. Cohen, Paul J .: Set Theory and the Continuum Hypothesis. Reading, Mass. 1966; repr. Mineola, N.Y. 2008. Google Scholar
  7. Frege, thank God: Basic laws of arithmetic. Derived in terms of terms [1893-1903]. 2 vols. Ed. Thomas Muller et al. Paderborn 2009.Google Scholar
  8. Frege, thank God: The basics of arithmetic. A logical mathematical investigation into the concept of number [1884]. Stuttgart 2011. Google Scholar
  9. Gentzen, Gerhard: "The consistency of pure number theory". In: Mathematical annals 112: 493-565 (1936); Reprint Darmstadt 1967.Google Scholar
  10. Gödel, Kurt: "About formally undecidable propositions of Principia Mathematica and related systems I". In: Monthly booklets of mathematics and physics 38: 173-198 (1931). Google Scholar
  11. Hilbert, David: Hilbertiana. Five essays [1918-25]. Darmstadt 1964.Google Scholar
  12. Hilbert, David: Basics of geometry [1899]. Stuttgart 141999. Google Scholar
  13. Lorenzen, Paul: "The consistency of classical analysis". In: Mathematical journal 54: 1-24 (1951). Google Scholar
  14. Lorenzen, Paul: Differential and integral. A constructive introduction to classical analysis. Frankfurt a. M. 1965. Google Scholar
  15. Lorenzen, Paul: Textbook of constructive philosophy of science [1987]. Stuttgart 2000. Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, a part of Springer Nature 2020

Authors and Affiliations

  1. 1. Institute for PhilosophyCarl von Ossietzky UniversityOldenburgGermany