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A Dialogical Account of the Intersubjectivity of Intuitionism

  • Clément LionEmail author
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Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 44)

Abstract

The present paper aims at integrating the phenomenological reading of Brouwerian intuitionism into the domain of semantics, by challenging the claim that the very meaning of mathematical expressions—expressions of free choice sequences included—is invariable and objectively determinable and that, accordingly, any deictic expression should be removed from mathematics. By introducing constructability into the constitution of meaning itself and by considering meaning as a “social act”, we try to map another route into intersubjectivity, based on the distinction between the play-level and the strategic level, which has been further developed in the dialogical framework, following the work of Paul Lorenzen. It is suggested that the steps towards such a route can be retraced from Oskar Becker’s original “Cartesian” approach to intersubjectivity, which facilitates a new reading of Brouwer’s own way of conceptualizing “mutual understanding”. In doing so, our general purpose is therefore to promote an insertion of dialogical constructivism into Mark van Atten’s take on the intuitionist Creating Subject.

Keywords

Actual/potential Creating subject Deictics Dialogical logic Free choice sequences Intersubjectivity Intuitionism Occasional expressions Phenomenological reduction Transcendental subject Turing machine 

Notes

Acknowledgements

Thanks to Shahid Rahman for his endless support and to Claudio Majolino for suggesting that I read and study Oskar Becker in the first place. To Mark van Atten, who patiently answered my innumerable questions, whose writings introduced me to Brouwer’s thought (making the present study possible), and who read and provided invaluable clarifications for the first version of this paper, I owe an enormous debt of gratitude. Finally, I would like to thank Christina Weiss, for having given me the opportunity to write it.

References

  1. Bassler, O. B. (2006). Book review: Mark van Atten, On Brouwer. Notre Dame Journal of Formal Logic, 47(4), 581–599.CrossRefGoogle Scholar
  2. Becker, O. (1927). Mathematische Existenz: Untersuchungen zur Logik und Ontologie mathematischer Phänomene. Halle a. d. S.: Niemeyer.Google Scholar
  3. Becker, O. (1929). Von der Hinfälligkeit des Schönen und der Abenteuerlichkeit des Künstlers. I: Jahrbuch für Philosophie und phänomenologische Forschung. Ergänzungsband. Halle: Husserl-Festschrift (pp. 27–52).Google Scholar
  4. Becker, O. (1936). Husserl und Descartes. Archiv für Rechts- und Sozialphilosophie, 30, 616–621.Google Scholar
  5. Becker, O. (1962). Zwei phänomenologische Betrachtungen zum Realismusproblem. In K. Hartmann (Ed.), Lebendiger Realismus : Festschrift für Johannes Thyssen (pp. 1–26). Bonn: H. Bouvier.Google Scholar
  6. Brandom, R. (2000). Articulating reasons: An introduction to inferentialism. Cambridge, London: Harvard University Press.Google Scholar
  7. Broch, H. (1933). Die unbekannte Größe. Berlin: Fischer Verlag (2. Auflage 2016. Suhrkamp Taschenuch).Google Scholar
  8. Brouwer, L. E. J. (1949). Consciousness, philosophy, and mathematics. In Proceedings of the Tenth International Congress of Philosophy, Amsterdam, August 11–18, 1948. North-Holland Publishing Company, Amsterdam 1949, pp. 1235–1249. The Journal of Symbolic Logic, 14(2), 1235–1249.Google Scholar
  9. Brouwer, L. E. (1975). Collected works. In A. Heyting (Ed.),North-Holland, Amsterdam.Google Scholar
  10. Bühler, K. (2011). Theory of language: The representational function of language (D. Fraser Goodwin, Trans.). Amsterdam, Philadelphia: John Benjamins Publishing Company.Google Scholar
  11. Cavell, S. (1979). The claim of reason. Oxford, New York: Oxford University Press.Google Scholar
  12. Dango, A. B. (2016). Approche dialogique de la révision des croyances dans le contexte de la théorie constructive des types de Per Martin-Löf. London: College Publication.Google Scholar
  13. Descartes, R., (1999). Oeuvres philosophiques. (F. Alquié, Ed.) II: 1638–1642. Paris: Garnier.Google Scholar
  14. Dummett, M. (1977). Elements of Intuitionism. Oxford: Clarendon Press.Google Scholar
  15. Gethmann, C. F. (2002). Hermeneutische Philosophie und Logischer Intuitionismus. In J. Mittelstrass & A. Gethmann, Die Philosophie und die Wissenschaften (Fink). München (pp. 109–128).Google Scholar
  16. Heidegger, M. (1979). Sein und Zeit. Tübingen: M. Niemeyer.Google Scholar
  17. Heyting, A. (1931). Die intuistonnische Grundlegung der Mathematik. Erkenntnis, 2, 106–115.CrossRefGoogle Scholar
  18. Husserl, E. (1962). L’origine de la géométrie (J. Derrida, Trans.). Paris: Presses Universitaires de France.Google Scholar
  19. Husserl, E. (1969). Formal and transcendental logic (D. Cairns, Trans.). The Hague: Martinus Nijhoff.Google Scholar
  20. Husserl, E. (1973a). Experience and judgment: Investigations in a genealogy of logic (L. Landgrebe, Ed., S. Churchill & K. Ameriks, Trans.). London: Routledge and K. Paul.Google Scholar
  21. Husserl, E. (1973b). Zur Phänomenologie der Intersubjektivität: Texte aus dem Nachlass. Zweiter Teil: 1921–1928. In Kern, I., & Breda, H. L. van (Ed.). The Hague: Martinus Nijhoff.Google Scholar
  22. Husserl, E. (1983). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy: First book: General introduction to a pure phenomenology (F. Kersten, Trans.). The Hague, Boston; Lancaster: Kluwer Academic Publishers.Google Scholar
  23. Husserl, E. (1995). Cartesianische Meditationen: Eine Einleitung in die Phänomenologie (Vol. 3, duchgesehen Auflage). Hamburg: Meiner.Google Scholar
  24. Husserl, E. (2001a). Logical investigations. Vol. 1 (J. N. Findlay, Trans.). London: Routledge.Google Scholar
  25. Husserl, E. (2001b). Logical investigations. Vol. 2 (J. N. Findlay, Trans.). London: Routledge.Google Scholar
  26. Kern, I. (1964). Husserl und Kant. Den Haag: Nijhoff.CrossRefGoogle Scholar
  27. Kreisel, G. (1967). Informal rigour and completeness proofs. In I. Lakatos (Ed.). Studies in logic and the foundations of mathematics (Vol. 47, pp. 138–186). Elsevier, Amsterdam.CrossRefGoogle Scholar
  28. Lorenz, K. (1972). Der dialogische Wahrheitsbegriff, Neue Hefte für Philosophie. H 213, pp. 111–123.Google Scholar
  29. Lorenz, K. (2009). Dialogischer Konstruktivismus. Berlin, New York: De Gruyter.Google Scholar
  30. Lorenz, K. (2010). Logic, language and method: On polarities in human experience. Berlin, New-York: De Gruyter.Google Scholar
  31. Lorenzen, P. (1960). Logik und Agon, Atti. Congr. Internat. De Filosofia, vol. 4. Sansoni, Firenze, pp. 187–194. Reprinted in Lorenzen, P., & Lorenz K. (1978) Dialogische Logik, Darmstadt: wissenschaftl. Buchgesellschaft.Google Scholar
  32. Lorenzen, P., & Schwemmer, O. (1975). Konstruktive Logik, Ethik und Wissenschaftstheorie. Meisenheim: Anton Hein.Google Scholar
  33. Marion, M. (1998). Wittgenstein, finitism, and the foundations of mathematics. New York: Oxford University Press.Google Scholar
  34. Martin-Löf, P. (1990). Mathematics of infinity. In Proceedings of the International Conference on Computer Logic (pp. 146–197). London, UK, UK: Springer-Verlag.CrossRefGoogle Scholar
  35. Martin-Löf, P. (2017). Assertion and request. Lecture held at Oslo, 2017. Transcription by A. Klev.Google Scholar
  36. Mittelstrass, J. (2002). Oskar Becker und Paul Lorenzen oder: die Begegnung zwischen Phänomenologie und Konstrüktivismus. In J. Mittelstrass & A. Gethmann, Die Philosophie und die Wissenschaften (Fink). München (pp. 65–83).Google Scholar
  37. Nancy, J.-L. (2000). La pensée dérobée. Paris: Galilée.CrossRefGoogle Scholar
  38. Narasina, R. (2009). The chequered history of epistemology and science: The intuitionist interlude. In B. Ray (Éd.), Different types of history (pp. 106–112). Pearson Education India.Google Scholar
  39. Prabhavananda, & Isherwood, C. (Trans.). (1956). Bhagavad-Gita: The song of God. London: Phoenix House.Google Scholar
  40. Rahman, S., Redmond, J., & Clerbout, N. (2016) N. Objective Knowledge and the not Dispensability of Epistemic Subjects. Some remarks on Popper’s notion of objective knowledge. Cahiers d’Epistémologie (Vol. 5, pp. 25–53). L’Harmattan.Google Scholar
  41. Rahman, S., McConaughey, Z., Klev, A., & Clerbout, N. (2018). Immanent reasoning or equality in action: A plaidoyer for the play level. Dordrecht: Springer.CrossRefGoogle Scholar
  42. Reinach, A. (2012). The Apriori foundations of the civil law: Along with the lecture. In J. F. Crosby (Ed.). «Concerning phenomenology». Frankfurt: Ontos.Google Scholar
  43. Ricoeur, P. (2004). À l’école de la phénoménologie. Paris: J. Vrin.Google Scholar
  44. Shafiei, M. (2018). Meaning and Intentionality: A dialogical approach. London: College Publications.Google Scholar
  45. Sundholm, G. (1984). Brouwer’s anticipation of the principles of charity. Proceedings of the Aristotelian Society, 85, 263–276.CrossRefGoogle Scholar
  46. Sundholm, G. (2014). Constructive recursive functions, church’s thesis, and Brouwer’s theory of the creating subject: Afterthoughts on a Parisian joint session. Constructivity and computability in historical and philosophical perspective (pp. 1–35). Dordrecht: Springer.Google Scholar
  47. Troelstra, A. (1977). Choice sequences: A chapter of Intuitionist Mathematics. Oxford: Clarendon Press.Google Scholar
  48. Troelstra, A., & van Dalen, D. (1988). Constructivism in mathematics: An introduction (Vol. 1). Amsterdam: Elsevier.Google Scholar
  49. van Atten, M. (2002a). On Brouwer. Belmont, Calif.: Wadsworth Publishing Co Inc.Google Scholar
  50. van Atten, M. (2002b). Phenomenology’s reception of Brouwer’s choice sequences. In V. Peckhaus (Ed.), Oskar Becker und die Philosophie der Mathematik (pp. 101–107). München: Fink Verlag.Google Scholar
  51. van Atten, M. (2007). Brouwer meets Husserl: On the phenomenology of choice sequences. Netherlands: Springer.CrossRefGoogle Scholar
  52. van Atten, M. (2015). Troelstra’s Paradox and Markov’s Principle. In G. Alberts, L. Bergmans, & P. Muller (Eds.), Dutch significs and early criticism of the vienna circle. Dordrecht: Springer. (Forthcoming, Preprint on Hal, archives ouvertes).Google Scholar
  53. van Atten, M. (2018). The creating subject, the Brouwer-Kripke Schema, and infinite Proofs. Forthcoming in Indigationes Mathematica.Google Scholar
  54. van Atten, M., & van Dalen, D. (2002). Arguments for the continuity principle. The Bulletin of Symbolic Logic, 8(3), 329–347.CrossRefGoogle Scholar
  55. Van der Schaar, M. (2011). The cognitive act and the first-person perspective: An epistemology for constructive type theory. Synthese, 180, 391–417.CrossRefGoogle Scholar
  56. Wittgenstein, L. (1989). Philosophische Bemerkungen. In R. Rhees (Ed.). Frankfurt am Main: Suhrkamp.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de LilleLilleFrance

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