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Axiomathes

pp 1–21 | Cite as

The Role of Intuition in Gödel’s and Robinson’s Points of View

  • Talia LevenEmail author
Original Paper
  • 17 Downloads

Abstract

Before Abraham Robinson and Kurt Gödel became familiar with Paul Cohen’s Results, both logicians held a naïve Platonic approach to philosophy. In this paper I demonstrate how Cohen’s results influenced both of them. Robinson declared himself a Formalist, while Gödel basically continued to hold onto the old Platonic approach. Why were the reactions of Gödel and Robinson to Cohen’s results so drastically different in spite of the fact that their initial philosophical positions were remarkably similar? I claim that the key to these different responses stems from the meanings that Gödel and Robinson gave to the concept of intuition, as well as to the relationship between epistemology and ontology. I also illustrate that although it might initially appear that Gödel’s and Robinson’s positions after Cohen’s results were quite different, this was not necessarily the case.

Keywords

Formalism Platonism Intuition Ontology Epistemology Completeness Formal language 

Notes

References

  1. Benacerraf P, Putnam H (eds) (1964) Philosophy of mathematics. Prentice Hall, New JerseyGoogle Scholar
  2. Dauben JW (1995) Abraham Robinson. Princeton University Press, PrincetonGoogle Scholar
  3. Dawson JW (1997) Logical dilemmas. A. K. Peters, WellesleyGoogle Scholar
  4. Feferman S et al (eds) (1990) Kurt Gödel collection works, vol 2. Oxford University Press, New YorkGoogle Scholar
  5. Feferman S et al (eds) (1995) Kurt Gödel collection works, vol 3. Oxford University Press, New YorkGoogle Scholar
  6. Feferman S et al (eds) (2003a) Kurt Gödel collection works, vol 4. Oxford University Press, New YorkGoogle Scholar
  7. Feferman S et al (eds) (2003b) Kurt Gödel collection works, vol 5. Oxford University Press, New YorkGoogle Scholar
  8. Gödel K (1933) The present situation in the foundation of mathematics’ lecture at the meeting of the American association. In: Feferman S et al (eds) Kurt Gödel collection works, vol 2. Oxford University Press, New York, pp 45–53Google Scholar
  9. Gödel K (1938) The consistency of the axiom of choice and of the generalized continuum-hypothesis. In: Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences), vol 24, p 12Google Scholar
  10. Gödel K (1940) Lecture on the consistency of the continuum hypothesis. In: Feferman S et al (eds) Kurt Gödel collection works, vol 3. Oxford University Press, New York, pp 175–185Google Scholar
  11. Gödel K (1947) What is Cantor’s continuum problem. In: Benacerraf P, Putnam H (eds) Philosophy of mathematics. Prentice Hall, New Jersey, pp 176–187Google Scholar
  12. Gödel K (1961) The modern development of the foundations of mathematics in the light of philosophy. In: Feferman S et al (eds) Kurt Gödel collection works, vol 3. Oxford University Press, New York, pp 374–387Google Scholar
  13. Gödel K (1964) What is Cantor’s continuum problem. In: Feferman S et al (eds) Kurt Gödel collection works, 1990, vol 2. Prentice Hall, Upper Saddle River, pp 254–270Google Scholar
  14. Gödel K (1972) On an extension of finitary mathematics which has not yet been used. In: Feferman S et al (eds) Kurt Gödel collection works, vol 2. Oxford University Press, New York, pp 271–280Google Scholar
  15. Gödel K (1995) Some basic theorems on the foundations of mathematics and their implications. In: Feferman S et al (eds) Kurt Gödel collection works, vol 3. Oxford University Press, New York, pp 304–323Google Scholar
  16. Hauser K (2006) Gödel program revisited part 1: the turn to phenomenology. Bull Symb Log 12:529–590CrossRefGoogle Scholar
  17. Keisler H et al (eds) (1979) Selected papers of abraham robinson. Yale University Press, LondonGoogle Scholar
  18. Posy JC (1998) Brouwer versus Hilbert: 1907–1928. Sci Context 11(2):291–325CrossRefGoogle Scholar
  19. Posy JC (2000) Epistemology, ontology, and the continuum. In: Grosholz E, Berger H (eds) The growth of mathematical knowledge. Kluwer, Alphen aan den Rijn, pp 199–219CrossRefGoogle Scholar
  20. Posy JC (2007) Kantian and Brouwerian infinity (not published, given as lecture at Krakow)Google Scholar
  21. Robinson A (1956) Complete theories. North-Holland, AmsterdamGoogle Scholar
  22. Robinson A (1964) Formalism 64. In: Proceedings of the international congress for logic, methodology and philosophy of science, Jerusalem, 1964, pp 228–246. North-Holland, Amsterdam; also in Keisler H et al (eds) (1979) Selected papers of Abraham Robinson, vol 2, pp 505–523. Yale University Press, LondonGoogle Scholar
  23. Robinson A (1970a) Infinite forcing in model theory. In: Proceedings of the second scandinavian logic symposium, Oslo 1970, pp 317–340. North Holland, Amsterdam, 1971; also in Keisler H et al (eds) (1979) Selected papers of Abraham Robinson, vol 2, pp 243–266. Yale University Press, LondonGoogle Scholar
  24. Robinson A (1970b) Forcing in model theory. In: Proceeding of the international congress of mathematicians, Nice, 1970, vol 1, pp 245–250. Gauthier-Villars, Paris, 1971; also in Keisler H et al (eds) (1979) Selected papers of Abraham Robinson, vol 1, pp 205–218. Yale University Press, LondonGoogle Scholar
  25. Robinson A (1970c) Completing theories by forcing (with J. Barwise). Math Log 1970, 2:119–142; also in Keisler H et al (eds) (1979) Selected papers of Abraham Robinson, vol 1, pp 229–242. Yale University Press, LondonGoogle Scholar
  26. Robinson A (1970d) From a formalist point of view. Dialectica 23:45–49CrossRefGoogle Scholar
  27. Robinson A (1971) Forcing in model theory. In: Symposia mathematica, vol 5. Istituto Nazional de Alta Mathematica, Rome, 1969/70; reprint. Academic Press, London, pp 69–82; also in Keisler H et al (eds) (1979) Selected papers of Abraham Robinson, vol 1, pp 205–218. Yale University Press, LondonGoogle Scholar
  28. Robinson A (1973) Concerning progress in the philosophy of the mathematics. In: Proceedings of the logic colloquium at Bristol 1973. North Holland, Amsterdam, 1975, pp 41–52; also in Keisler H et al (eds) 1979, vol 2, pp 556–567Google Scholar
  29. Tieszen R (2002) Gödel and the intuition of concepts. Synthese 133(3):363–391CrossRefGoogle Scholar
  30. Tieszen R (2005) Free variation and the intuition of geometric essences: some reflections on phenomenology and modern geometry. Philos Phenomenol Res 70(1):153–173CrossRefGoogle Scholar
  31. Tieszen R (2011) After Gödel. Oxford University Press, OxfordCrossRefGoogle Scholar
  32. Tieszen R (2015) Arithmetic, mathematical intuition, and evidence. Inquiry 58(1):26–58CrossRefGoogle Scholar
  33. Van Atten M, Kennedy J (2003) On the philosophical development of Kurt Gödel. Bull Symb Log 9(4):425–476CrossRefGoogle Scholar
  34. Wang H (1987) Reflection on Kurt Gödel. London MIT Press, LondonGoogle Scholar
  35. Wang H (1996) A logical journey. Cambridge MIT Press, CambridgeGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Computer Science, Levinsky College of EducationThe Open University of IsraelRosh Ha-AyinIsrael

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