Skip to main content
SpringerLink
Log in

Gödel, Mathematics, and Possible Worlds

  • Chapter
  • First Online:
Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 35))

  • 1003 Accesses

Abstract

Hintikka has claimed that Gödel did not believe in possible worlds and that the actualism this induces is the motivation behind his Platonism. I argue that Hintikka is wrong about what Gödel believed, and that, moreover, there exists a phenomenological unification of Gödel’s Platonism and possible worlds theory. This text was written for a special issue of Axiomathes on the philosophy of Nicolai Hartmann, which explains the two introductory paragraphs.

Originally published as van Atten 2001. Copyright © 2001 Springer Science+Business Media.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    [​​[This book was published in 1937; I will be citing the second edition, Hartmann 1949.]​​]

  2. 2.

    The term ‘actualism’ is ambiguous, depending on whether one makes a distinction between the existence of a world and its obtaining. If one does, one could be an actualist and still believe in possible worlds by holding that possible worlds exist but only the actual world obtains. Evidently, this is not Hintikka’s use here.

  3. 3.

    Yourgrau (1999, 44) writes that ‘for a mathematical Platonist like Gödel, the mere possibility of a formal structure would, I take it, imply its actual (mathematical) existence’. I am in broad agreement with this statement but think ‘possibility’ needs qualification; moreover, if we take Gödel’s turn to phenomenology into account, there is room for a substantial argument for the position thus qualified.

  4. 4.

    This brief discussion of notions of possibility is adapted from van Atten (2002).

  5. 5.

    Sehr vieles ist logisch möglich, was nicht wesensmöglich ist. Die Logik muß mit ‘unmöglichen Gegenständen’ rechnen, d.h. mit wesensunmöglichen (viereckiger Kreis); je nachdem, was sie an vorgegebenen Merkmalen (etwa des Kreises) anerkennt, sind solche Gegenstände für sie möglich oder nicht. Die ‘Unmöglichkeit’ solcher Gegenstände ist eben eine Seinsunmöglichkeit (etwa eine geometrische), nicht eine logische. (Hartmann 1949, 323)

  6. 6.

    [Wirklichkeit und Unwirklichkeit] spielen keine eigene Rolle im idealen Sein neben der Möglichkeit des Seins und Nichtseins. Sie sind mit ihr gesetzt, sind ein Selbstverständliches, besagen nicht ‘mehr’ als das Seinkönnen und Nichtseinkönnen. (Hartmann 1949, 318)

  7. 7.

    Denn Erkenntnis kann vomSein ihres Gegenstandes aus überhaupt nicht impliziert werden. Das ideale Sein ist an sich nicht weniger indifferent gegen die Idealerkenntnis, als das reale Sein gegen die Realerkenntnis. (Hartmann 1949, 464)

  8. 8.

    Prinzipiell entspricht (im Apriori der unbedingten Wesensallgemeinheit) jedem ‘wahrhaft seienden’ Gegenstand die Idee eines möglichen Bewußtseins, in welchem der Gegenstand selbst originär und dabei volkommen adäquat erfaßbar ist. Umgekehrt, wenn diese Möglichkeit gewährleistet ist, ist eo ipso der Gegenstand wahrhaft seiend. (Husserl 1976a, 329)

  9. 9.

    The Paris Lectures and the Cartesian Meditations are not the same work, but the latter is an elaboration of the former.

  10. 10.

    dann ergibt sich das Korrelat unserer faktischen Erfahrung, genannt ‘die wirkliche Welt’, als Spezialfall mannigfaltiger möglicher Welten und Unwelten, die ihrerseits nicht anderes sind als Korrelate wesensmöglicher Abwandlungen der Idee ‘erfahrendes Bewußtsein’ mit mehr oder minder geordneten Erfahrungzusammenhängen. (Husserl 1976a, 100)

  11. 11.

    Die Idee dieser Transzendenz ist also das eidetische Korrelat der reinen Idee dieser ausweisenden Erfahrung. Das gilt für jede erdenkliche Art von Transzendenz, die als Wirklichkeit oder Möglichkeit soll behandelt werden können. (Husserl 1976a, 101)

References

  • van Atten, M. (2001). Gödel, mathematics, and possible worlds. Axiomathes, 12(3–4), 355–363. Included in this volume as Chap. 7.

    Google Scholar 

  • van Atten, M. (2002). Why Husserl should have been a strong revisionist in mathematics. Husserl Studies, 18(1), 1–18.

    Article  Google Scholar 

  • Gödel, K. (1949a). A remark about the relationship between relativity theory and idealistic philosophy. In Schilpp (1949, pp. 447–450). Reprinted, with original page numbers in the margin, in Gödel (1990, pp. 202–207).

    Google Scholar 

  • Gödel, K. (*1951). Some basic theorems on the foundations of mathematics and their implications. Lecture, published in Gödel (1995, pp. 304–323).

    Google Scholar 

  • Gödel, K. (*1961/?). The modern development of the foundations of mathematics in the light of philosophy. Lecture draft in German, published, with an English translation, in Gödel (1995, pp. 374–387). The English title is Gödel’s.

    Google Scholar 

  • Gödel, K. (1990). Publications 1938–1974 (Collected works, Vol. 2; S. Feferman, J. Dawson, Jr., S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort, Eds.). Oxford: Oxford University Press.

    Google Scholar 

  • Gödel, K. (1995). Unpublished essays and lectures (Collected works, Vol. 3; S. Feferman, J. Dawson, Jr., S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort, Eds.). Oxford: Oxford University Press.

    Google Scholar 

  • Hartmann, N. (1949). Möglichkeit und Wirklichkeit (2nd ed.). Meisenheim am Glan: Westkulturverlag Anton Hain.

    Google Scholar 

  • Harvey, C., & Hintikka, J. (1991). Modalization and modalities. In Seebohm et al. (1991, pp. 59–77).

    Google Scholar 

  • Hintikka, J. (1975). The intentions of intentionality. Boston: Reidel.

    Book  Google Scholar 

  • Hintikka, J. (1998). On Gödel’s philosophical assumptions. Synthese, 114, 13–23.

    Article  Google Scholar 

  • Hintikka, J. (2000). On Gödel. Belmont: Wadsworth.

    Google Scholar 

  • Husserl, E. (1976a). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie: Erstes Buch. 1. Halbband: Text der 1.–3. Auflage (Husserliana, Vol. 3/1; K. Schuhmann, Ed.). Den Haag: Martinus Nijhoff.

    Google Scholar 

  • Husserl, E. (1983). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy: First book: General introduction to phenomenology (Edmund Husserl collected works, Vol. 2; F. Kersten, Trans.). Dordrecht: Kluwer.

    Google Scholar 

  • Husserl, E. (1984a). Logische Untersuchungen: Zweiter Band, 1. Teil (Husserliana, Vol. 19/1; U. Panzer, Ed.). Den Haag: Martinus Nijhoff.

    Book  Google Scholar 

  • Mohanty, J. N. (1981). Intentionality and ‘possible worlds’. Revue Internationale de Philosophie, 35, 91–112. (Reprint in Mohanty 1985, pp. 25–44).

    Google Scholar 

  • Mohanty, J. N. (1982). Husserl and Frege. Bloomington: Indiana University Press.

    Google Scholar 

  • Mohanty, J. N. (1984). Husserl on ‘possibility’. Husserl Studies, 1, 13–29. (Reprint in Mohanty 1999, pp. 152–167).

    Google Scholar 

  • Mohanty, J. N. (1985). The possibility of transcendental philosophy. Dordrecht: Martinus Nijhoff.

    Book  Google Scholar 

  • Mohanty, J. N. (1990). Phenomenology and the modalities. Acta Philosophica Fennica, 49, 110–122. (Reprint in Mohanty 1999, pp. 168–179).

    Google Scholar 

  • Mohanty, J. N. (1999). Logic, truth and the modalities: From a phenomenological perspective. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Russell, B. (1919). Introduction to mathematical philosophy. London: Allen/Unwin.

    Google Scholar 

  • van der Schaar, M. (2001). Hartmann’s rejection of the notion of evidence. Axiomathes, 12, 285–297.

    Article  Google Scholar 

  • Schilpp, P. A. (Ed.). (1949). Albert Einstein: Philospher-scientist (The Library of Living Philosophers, Vol. 7). Evanston: Library of Living Philosophers.

    Google Scholar 

  • Seebohm, T., Føllesdal, D., & Mohanty, J. N. (Eds.). (1991). Phenomenology and the formal sciences. Dordrecht: Kluwer.

    Google Scholar 

  • Wang, H. (1987). Reflections on Kurt Gödel. Cambridge, MA: MIT.

    Google Scholar 

  • Wang, H. (1996). A logical journey: From Gödel to philosophy. Cambridge, MA: MIT.

    Google Scholar 

  • Yourgrau, P. (1999). Gödel meets Einstein. Chicago: Open Court.

    Google Scholar 

Download references

Acknowledgements

I am thankful to Igor Douven, Leon Horsten, and Palle Yourgrau for their comments on the draft. The work described in this paper was done under a Postdoctoral Fellowship from the Fund for Scientific Research-Flanders (Belgium), which is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Sciences, Normes, Décision (CNRS/Paris IV), CNRS, Paris, France

    Mark van Atten

Authors
  1. Mark van Atten
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

van Atten, M. (2015). Gödel, Mathematics, and Possible Worlds. In: Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Logic, Epistemology, and the Unity of Science, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-10031-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation