Axiomathes

, Volume 23, Issue 1, pp 109–135 | Cite as

Are Mathematical Theories Reducible to Non-analytic Foundations?

Original Paper

Abstract

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.

Keywords

Axiom of Choice Finitistic First-level idealization Gödel’s incompleteness theorems Individual-substrate Infinite Intentionality Second-level idealization Skolem-Löwenheim Theorem 

References

  1. Adams R (1979) Primitive thisness and primitive identity. J Philos 76(1):5–26CrossRefGoogle Scholar
  2. Bell LJ, Slomson BA (2006) Models and ultraproducts. Dover Publications, New YorkGoogle Scholar
  3. Cohen P (1966) Set theory and the continuum hypothesis. W.A. Benjamin, New York, MAGoogle Scholar
  4. French S (1989) Identity and individuality in classical and quantum physics. Australas J Philos 67(4):432–446CrossRefGoogle Scholar
  5. Føllesdal D (1969) Husserl’s notion of noema. J Philos 66:680–687CrossRefGoogle Scholar
  6. Goldblatt R (1985) On the role of the Baire category theorem and dependent choice in the foundation of logic. J Symbol Logic 50(2):412–422CrossRefGoogle Scholar
  7. Halpern DJ, Levy A (1971) The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic set theory. AMS Proceedings, pp 83–134Google Scholar
  8. Hill CO (2010) Husserl on axiomatization and arithmetic. In: Hartimo M (ed) Phenomenology and mathematics. Springer, BerlinGoogle Scholar
  9. Hill CO, Rosado Haddock GE (2000) Husserl or Frege? Meaning, objectivity, and mathematics. Open Court, La SalleGoogle Scholar
  10. Husserl E (1962) Die Krisis der Europäischen Wissenschaften und die Transzendentale Phänomenologie. Hua, Band VI, hgb. W. Biemel. M. Nijhoff, Den HaagGoogle Scholar
  11. Husserl E (1966) Zur Phänomenologie des Inneren Zeibewusstseins. Hua, Band X, hgb. R. Boehm. M. Nijhoff, Den HaagGoogle Scholar
  12. Husserl E (1974) Formale und Transzendentale Logik. Hua, Band XVII, hgb. P. Janssen. M. Nijhoff, Den HaagGoogle Scholar
  13. Husserl E (1975) Logische Untersuchungen. (Prolegomena zur Reinen Logik), Hua, Band XVIII, hgb. E. Holenstein. M. Nijhoff, Den HaagGoogle Scholar
  14. Husserl E (1976) Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Erstes Buch, Hua, Band III/I, hgb. K. Schuhmann. M. Nijhoff, Den HaagGoogle Scholar
  15. Husserl E (1984) Logische Untersuchungen. Hua, Band XIX1, (zweiter Band, erster Teil), hgb. U. Panzer. M. Nijhoff, Den HaagGoogle Scholar
  16. Husserl E (2002) Logische Untersuchungen, Ergänzungsband, Erster Teil, Hua, Band XX/I, hgb. U. Melle. Kluwer, DordrectGoogle Scholar
  17. Kanamori A (2008) Cohen and set theory. Bull Symbolic Logic 14(3):351–378CrossRefGoogle Scholar
  18. Kleene SC (1980) Introduction to metamathematics. North-Holland Publication Co, New YorkGoogle Scholar
  19. Krause D, Coelho AMN (2005) Identity, indiscernibility, and philosophical claims. Axiomathes 15:191–210CrossRefGoogle Scholar
  20. Kunen K (1982) Set theory. An introduction to independence proofs. Elsevier, AmsterdamGoogle Scholar
  21. Lavine S (1994) Understanding the infinite. Harvard University Press, CambridgeGoogle Scholar
  22. Livadas S (2010) Impredicativity of continuum in phenomenology and in non-cantorian theories. In: Carsetti A (ed) Causality, meaningful complexity and embodied cognition. Springer, Berlin, pp 185–199Google Scholar
  23. Livadas S (2011) The expressional limits of formal language in the notion of quantum observation. Axiomathes, 1–25. doi:10.1007/s10516-011-9168-6. (Online ISSN 1122-1151)
  24. Lohmar D (2002) Elements of a phenomenological justification of logical principles, including an appendix […] on the transfiniteness of the set of real numbers. Philos Math 10(3):227–250CrossRefGoogle Scholar
  25. Popper K (1934) Logik der Forschung. Springer, WienGoogle Scholar
  26. Rosado Haddock GE (1987) Husserl’s epistemology of mathematics and the foundation of platonism in mathematics. Husserl Stud 4(2):81–102CrossRefGoogle Scholar
  27. Schoenfield J (1967) Mathematical logic. Addison Wesley Publication, ReadingGoogle Scholar
  28. Simmons FG (1963) Introduction to topology and modern analysis. McGraw-Hill Kogakusha Ltd, TokyoGoogle Scholar
  29. Sokolowski R (1974) Husserlian meditations. Northwestern University Press, EvanstonGoogle Scholar
  30. Tieszen R (2005) Phenomenology, logic, and the philosophy of mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  31. van Atten M (2006) Brouwer meets Husserl: on the phenomenology of choice sequences, synthese library, vol 335, Springer, DordrechtGoogle Scholar
  32. van Atten M, Van Dalen D, Tieszen R (2002) Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum. Philos Math 10(3):203–226CrossRefGoogle Scholar
  33. van Dalen D (2004) Logic and structure. Springer, BerlinGoogle Scholar
  34. van Fraasen B (1991) Quantum mechanics: an empiricist view. Clarendon Press, OxfordCrossRefGoogle Scholar
  35. Woodin HW (2001) The continuum hypothesis I & II. N Am Math Soc 48–6 (resp. 567-576 & 681-690)Google Scholar
  36. Woodin HW (2011) The realm of the infinite. In: Heller M, Woodin WH (eds) Infinity: new research frontiers. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.PatrasGreece

Personalised recommendations