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Phenomenology and Mathematics
Published online by Cambridge University Press: 27 November 2023
Summary
This Element explores the relationship between phenomenology and mathematics. Its focus is the mathematical thought of Edmund Husserl, founder of phenomenology, but other phenomenologists and phenomenologically-oriented mathematicians, including Weyl, Becker, Gödel, and Rota, are also discussed. After outlining the basic notions of Husserl's phenomenology, the author traces Husserl's journey from his early mathematical studies. Phenomenology's core concepts, such as intention and intuition, each contributed to the emergence of a phenomenological approach to mathematics. This Element examines the phenomenological conceptions of natural number, the continuum, geometry, formal systems, and the applicability of mathematics. It also situates the phenomenological approach in relation to other schools in the philosophy of mathematics-logicism, formalism, intuitionism, Platonism, the French epistemological school, and the philosophy of mathematical practice.
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- Online ISBN: 9781108993913Publisher: Cambridge University PressPrint publication: 21 December 2023
References
van Atten, M. (2007). Brouwer Meets Husserl: On the Phenomenology of Choice Sequences. Dordrecht: Springer. https://doi.org/10.1007/978-1-4020-5087-9.CrossRefGoogle Scholar
van Atten, M. (2010). Construction and Constitution in Mathematics. New Yearbook for Phenomenology and Phenomenological Philosophy, 10: 43–90.Google Scholar
van Atten, M., van Dalen, D., and Tieszen, R. (2002). Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum. Philosophia Mathematica, 10: 203–26. https://doi.org/10.1093/philmat/10.2.203.CrossRefGoogle Scholar
van Atten, M., and Kennedy, J. (2003). On the Philosophical Development of Kurt Gödel. Bulletin of Symbolic Logic, 9: 425–76.Google Scholar
Bachelard, S. (1968). A Study of Husserl’s Formal and Transcendental Logic. Translated by L. E. Embree. Evanston, IL: Northwestern University Press.Google Scholar
Becker, O. (1923). Beiträge zur Phänomenologischen Begründung der Geometrie und ihrer physikalischen Anwendungen. Jahrbuch für Philosophie und phänomenologische Forschung, 6: 385–560.Google Scholar
Becker, O. (1927). Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene. Halle: Max Niemeyer.Google Scholar
Brouwer, L. E. J. (1975). Intuitionism and Formalism. Bulletin of the American Mathematical Society, 20: 81–96. https://doi.org/10.1090/S0002-9904-1913-02440-6.CrossRefGoogle Scholar
Cantor, G. (1915). Contributions to the Founding of the Theory of Transfinite Numbers. Translated by P. Jourdain. New York: Dover.Google Scholar
Carter, J. (2019). Philosophy of Mathematical Practice: Motivations, Themes and Prospects. Philosophia Mathematica, 27: 1–32.https://doi.org/10.1093/philmat/nkz002.CrossRefGoogle Scholar
Cavaillès, J. (2021). On Logic and the Theory of Science. Translated by R. Mackay and K. Peden. Falmouth: Urbanomic Media.Google Scholar
Centrone, S. (2010). Logic and Philosophy of Mathematics in the Early Husserl. Dordrecht: Springer.CrossRefGoogle Scholar
Cobb-Stevens, R. (2002). Aristotelian Themes in Husserl’s Logical Investigations. In Zahavi, D. and Stjernfelt, F. (eds.), One Hundred Years of Phenomenology: Husserl’s Logical Investigations Revisited. Dordrecht: Kluwer, 79–92.https://doi.org/10.1007/978-94-017-0093-1_6.CrossRefGoogle Scholar
Da Silva, J. J. (2000). Husserl’s Two Notions of Completeness: Husserl and Hilbert on Completeness and Imaginary Elements in Mathematics. Synthese, 125: 417–38.CrossRefGoogle Scholar
Da Silva, J. J. (2013). How Sets Came to Be: The Concept of Set from a Phenomenological Perspective. New Yearbook for Phenomenology and Phenomenological Philosophy, 13: 84–100.Google Scholar
Da Silva, J. J. (2016). Husserl and Hilbert on Completeness, Still. Synthese, 193: 1925–47. https://doi.org/10.1007/s11229-015-0821-2.CrossRefGoogle Scholar
Derrida, J. (1989). Edmund Husserl’s Origin of Geometry: An Introduction. Translated by J. P. Leavey Jr. Lincoln, NB: University of Nebraska Press.Google Scholar
Descartes, R. (1985). Rules for the Direction of the Mind. Translated by D. Murdoch. In Cottingham, J., Stoothoff, R., and Murdoch, D. (eds.), The Philosophical Writings of Descartes, vol. 1. Cambridge: Cambridge University Press, 7–78. https://doi.org/10.1017/CBO9780511805042.004.CrossRefGoogle Scholar
Drummond, J. J. (1990). Husserlian Intentionality and Non-Foundational Realism. Dordrecht: Kluwer.CrossRefGoogle Scholar
Drummond, J. J. (2009). Phénoménologie et ontologie. Translated by G. Fréchette. Philosophiques, 36: 593–607. https://doi.org/10.7202/039488ar.CrossRefGoogle Scholar
Fine, K. (1998). Cantorian Abstraction: A Reconstruction and Defense. Journal of Philosophy, 95 (12): 599–634.https://doi.org/10.2307/2564641.Google Scholar
Føllesdal, D. (1994). Husserl and Frege: A Contribution to Elucidating the Origins of Phenomenological Philosophy. Translated by C. O. Hill. In Haaparanta, L. (ed.), Mind, Meaning, and Mathematics: Essays on the Philosophical Views of Husserl and Frege. Dordrecht: Kluwer, 3–47. https://doi.org/10.1007/978-94-015-8334-3_1.CrossRefGoogle Scholar
Frege, G. (1972). Review of Husserl’s Philosophy of Arithmetic. Translated by E. W. Kluge. Mind, 81: 321–37.Google Scholar
Frege, G. (1980a). Begriffsschrift, a Formula Language, Modeled upon That of Arithmetic, for Pure Thought. In van Heijenoort, J. (ed.), Frege and Gödel: Two Fundamental Texts in Mathematical Logic. Cambridge, MA: Harvard University Press, 1–82.Google Scholar
Frege, G. (1980b). The Foundations of Arithmetic: A Logico-Mathematical Inquiry into the Concept of Number. Translated by J. L. Austin. Evanston, IL: Northwestern University Press.Google Scholar
Gödel, K. (1961). The Modern Development of the Foundations of Mathematics in the Light of Philosophy. In Gödel, K., Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. N. (eds.), Collected Works, vol. 3 (1995). Oxford: Oxford University Press, 374–87.Google Scholar
Gödel, K. (1964). What is Cantor’s Continuum Problem? In Gödel, K., Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. N. (eds.), Collected Works, vol. 2 (1990). Oxford: Oxford University Press, 254–70.Google Scholar
Hartimo, M. (2007). Towards Completeness: Husserl on Theories of Manifolds 1890–1901. Synthese, 156: 281–310. https://doi.org/10.1007/s11229-006-0008-y.CrossRefGoogle Scholar
Hartimo, M. (2018). Husserl on Completeness, Definitely. Synthese, 195: 1509–27. https://doi.org/10.1007/s11229-016-1278-7.CrossRefGoogle Scholar
Hartimo, M. (2021). Husserl and Mathematics. Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108990905.CrossRefGoogle Scholar
Hauser, K. (2006). Gödel’s Program Revisited Part I: The Turn to Phenomenology. Bulletin of Symbolic Logic, 12: 529–90.https://doi.org/10.2178/bsl/1164056807.CrossRefGoogle Scholar
Heidegger, M. (1962). Being and Time. Translated by J. Macquarrie and E. Robinson. Oxford: Blackwell.Google Scholar
Hilbert, D. (1964). On the Infinite. In Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics: Selected Readings. Englewood Cliffs, NJ: Prentice-Hall, 134–51.Google Scholar
Hilbert, D. (1996). On the Concept of Number. Translated by W. Ewald. In Ewald, W. (ed.), From Kant to Hilbert, vol. 2. Oxford: Oxford University Press, 1092–5.Google Scholar
Hill, C. O. (2000). Abstraction and Idealization in Georg Cantor and Edmund Husserl Prior to 1895. In Hill, C. O. and Rosado Haddock, G. E. (eds.), Husserl or Frege: Meaning, Objectivity, and Mathematics. Chicago: Open Court, 109–36.Google Scholar
Hill, C. O. (2010). Husserl on Axiomatization and Arithmetic. In Hartimo, M. (ed.), Phenomenology and Mathematics. Dordrecht: Springer, 47–71.https://doi.org/10.1007/978-90-481-3729-9_3.CrossRefGoogle Scholar
Hintikka, J. (2003). The Notion of Intuition in Husserl. Revue internationale de philosophie, 224: 169–91.Google Scholar
Hopkins, B. C. (2005). Klein and Derrida on the Historicity of Meaning and the Meaning of Historicity in Husserl’s Crisis-Texts. Journal of the British Society for Phenomenology, 36: 179–87. https://doi.org/10.1080/00071773.2005.11006541.CrossRefGoogle Scholar
Hopkins, B. C. (2011). The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein. Bloomington, IN: Indiana University Press.Google Scholar
Husserl, E. (1956). Erste Philosophie. Erster Teil: Kritische Ideengeschichte. The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1975). Introduction to the Logical Investigations: A Draft of a Preface to the Logical Investigations. Edited by Fink, E.. Translated by P. J. Bossert and C. H. Curtis. The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1980). Ideas III: Phenomenology and the Foundations of the Sciences. Translated by T. Klein and W. Pohl. The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1983). Studien zur Arithmetik und Geometrie. Texte aus dem Nachlass (1886–1901). Edited by Strohmeyer, I.. The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1991). On the Phenomenology of the Consciousness of Internal Time (1893–1917). Translated by J. B. Brough. Dordrecht: Kluwer.CrossRefGoogle Scholar
Husserl, E. (1994). Early Writings in the Philosophy of Logic and Mathematics. Translated by D. Willard. Dordrecht: Kluwer.CrossRefGoogle Scholar
Husserl, E. (1997). Thing and Space: Lectures from 1907. Translated by R. Rojcewicz. Dordrecht: Springer.Google Scholar
Ierna, C. (2017). The Brentanist Philosophy of Mathematics in Edmund Husserl’s Early Works. In Centrone, S. (ed.), Essays on Husserl’s Logic and Philosophy of Mathematics. Dordrecht: Springer, 147–68. https://doi.org/10.1007/978-94-024-1132-4_7.Google Scholar
Ierna, C., and Lohmar, D. (2016). Husserl’s Manuscript A I 35. In Haddock, G. E. Rosado (ed.), Husserl and Analytic Philosophy. Berlin: De Gruyter, 289–320. https://doi.org/10.1515/9783110497373-011.CrossRefGoogle Scholar
Kant, E. (1998). Critique of Pure Reason. Translated by P. Guyer and A. W. Wood. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Klein, J. (1940). Phenomenology and the History of Science. In Farber, M. (ed.), Philosophical Essays in Memory of Edmund Husserl. Cambridge, MA: Harvard University Press, 143–63. https://doi.org/10.4159/harvard.9780674333512.c8.Google Scholar
Leng, M. (2002). Phenomenology and Mathematical Practice. Philosophia Mathematica, 10: 3–25.CrossRefGoogle Scholar
Linnebo, Ø. (2018). Platonism in the Philosophy of Mathematics. In E. N. Zalta (ed.), Stanford Encyclopedia of Philosophy (Spring 2018 Edition). https://plato.stanford.edu/archives/spr2018/entries/platonism-mathematics/.Google Scholar
Lohmar, D. (1990). Wo lag der Fehler der kategorialen Repräsentation? Zu Sinn und Reichweite einer Selbstkritik Husserls. Husserl Studies, 7: 179–97. https://doi.org/10.1007/BF00347584.CrossRefGoogle Scholar
Lohmar, D. (1993). On the Relation of Mathematical Objects to Time: Are Mathematical Objects Timeless, Overtemporal or Omnitemporal? Journal of the Indian Council of Philosophical Research, 10: 73–87.Google Scholar
Lohmar, D. (2000). Edmund Husserls “Formale und transzendentale Logik.” Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
Lohmar, D. (2004). The Transition of the Principle of Excluded Middle from a Principle of Logic to an Axiom: Husserl’s Hesitant Revisionism in Logic. New Yearbook for Phenomenology and Phenomenological Philosophy, 4: 53–68.Google Scholar
Maddy, P. (1980). Perception and Mathematical Intuition. Philosophical Review, 89: 163–96. https://doi.org/10.2307/2184647.CrossRefGoogle Scholar
Mahnke, D. (1966). From Hilbert to Husserl: First Introduction to Phenomenology, Especially that of Formal Mathematics. Translated by D. L. Boyer. Studies in History and Philosophy of Science, 8: 75–84. https://doi.org/10.1016/0039-3681(77)90020-6.Google Scholar
Majer, U. (1997). Husserl and Hilbert on Completeness: A Neglected Chapter in Early Twentieth Century Foundations of Mathematics. Synthese, 110: 37–56.CrossRefGoogle Scholar
Mancosu, P. (ed.). (2008). The Philosophy of Mathematical Practice. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199296453.001.0001.CrossRefGoogle Scholar
Mancosu, P. (2018). Explanation in Mathematics. In E. N. Zalta (ed.), Stanford Encyclopedia of Philosophy (Summer 2018 Edition). https://plato.stanford.edu/archives/sum2018/entries/mathematics-explanation/.Google Scholar
Mancosu, P., and Ryckman, T. (2002). Mathematics and Phenomenology: The Correspondence between O. Becker and H. Weyl. Philosophia Mathematica, 10: 130–202.CrossRefGoogle Scholar
Miller, J. P. (1982). Numbers in Presence and Absence: A Study of Husserl’s Philosophy of Mathematics. The Hague: Martinus Nijhoff. https://doi.org/10.1007/978-94-009-7624-5.CrossRefGoogle Scholar
Mohanty, J. N. (1977). Husserl and Frege: A New Look at Their Relationship. In Mohanty, J. N. (ed.), Readings on Edmund Husserl’s Logical Investigations. The Hague: Martinus Nijhoff, 22–32. https://doi.org/10.1007/978-94-010-1055-9_3.CrossRefGoogle Scholar
Mohanty, J. N. (1991). Husserl’s Formalism. In Seebohm, T. M., Føllesdal, D., and Mohanty, J. N. (eds.), Phenomenology and the Formal Sciences. Dordrecht: Kluwer, 93–105. https://doi.org/10.1007/978-94-011-2580-2_7.CrossRefGoogle Scholar
Nenon, T. (1997). Two Models of Foundation in the Logical Investigations. In Hopkins, B. C. (ed.), Husserl in Contemporary Context. Dordrecht: Kluwer, 97–114. https://doi.org/10.1007/978-94-017-1804-2_6.CrossRefGoogle Scholar
Parsons, C. (1980). Mathematical Intuition. Proceedings of the Aristotelian Society, 80: 145–68. https://doi.org/10.1093/aristotelian/80.1.145.CrossRefGoogle Scholar
Parsons, C. (2012). Husserl and the Linguistic Turn. In Parsons, C., From Kant to Husserl: Selected Essays. Cambridge, MA: Harvard University Press, 190–214. https://doi.org/10.4159/harvard.9780674065420.c13.CrossRefGoogle Scholar
Posy, C. J. (2020). Mathematical Intuitionism. Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108674485.CrossRefGoogle Scholar
Pradelle, D. (2000). L’archéologie du monde. Constitution de l‘espace, idéalisme et intuitionnisme chez Husserl. Dordrecht: Springer. https://doi.org/10.1007/978-94-024-1586-5.CrossRefGoogle Scholar
Pradelle, D. (2012). Par-delà la révolution copernicienne. Sujet transcendental et facultés chez Kant et Husserl. Paris: Presses universitaires de France.CrossRefGoogle Scholar
Pradelle, D. (2020). Intuition et idéalités. Phénoménologie des objets mathématiques. Paris: Presses universitaires de France.Google Scholar
Putnam, H. (1975). The Meaning of “Meaning.” In Putnam, H. (ed.), Mind, Language and Reality: Philosophical Papers, vol. 2. Cambridge: Cambridge University Press, 215–71. https://doi.org/10.1017/CBO9780511625251.CrossRefGoogle Scholar
Reck, E. H. (2013). Frege, Dedekind, and the Origins of Logicism. History and Philosophy of Logic, 34: 242–65. https://doi.org/10.1080/01445340.2013.806397.CrossRefGoogle Scholar
Reid, C. (1970). Hilbert. Berlin: Springer. https://doi.org/10.1007/978-3-662-28615-9.CrossRefGoogle Scholar
Reinach, A. (1989). Über den Begriff der Zahl. In Schuhmann, K. and Smith, B. (eds.), Sämtliche Werke, vol. 1. Munich: Philosophia Verlag, 515–29.Google Scholar
Rosado Haddock, G. E. (2010). Platonism, Phenomenology, and Interderivability. In Hartimo, M. (ed.), Phenomenology and Mathematics. Dordrecht: Springer, 23–46. https://doi.org/10.1007/978-90-481-3729-9_2.CrossRefGoogle Scholar
Rota, G.-C. (1989). Fundierung as a Logical Concept. The Monist, 72: 70–7. https://doi.org/10.5840/monist19897218.CrossRefGoogle Scholar
Rota, G.-C. (1990). Mathematics and Philosophy: The Story of a Misunderstanding. Review of Metaphysics, 44: 259–71.Google Scholar
Rota, G.-C. (1991). Mathematics and the Task of Phenomenology. In Seebohm, T. M., Føllesdal, D., and Mohanty, J. N. (eds.), Phenomenology and the Formal Sciences. Dordrecht: Kluwer, 133–8. https://doi.org/10.1007/978-94-011-2580-2_9.Google Scholar
Rota, G.-C. (1997b). Indiscrete Thoughts. Edited by Palombi, F.. Boston: Birkhäuser. https://doi.org/10.1007/978-0-8176-4781-0.CrossRefGoogle Scholar
Roubach, M. (2008). Being and Number in Heidegger’s Thought. London: Continuum. https://doi.org/10.5040/9781472546166.Google Scholar
Roubach, M. (2021). Numbers as Ideal Species: Husserlian and Contemporary Perspectives. New Yearbook for Phenomenology and Phenomenological Philosophy, 18: 537–45.Google Scholar
Roubach, M. (2022). Mathesis Universalis and Husserl’s Phenomenology. Axiomathes, 32: 627–37. https://doi.org/10.1007/s10516-021-09544-9.CrossRefGoogle Scholar
Smith, D. W. (2003). “Pure” Logic, Ontology, and Phenomenology. Revue internationale de philosophie, 57 (224/2): 133–56.Google Scholar
Smith, D. W. (2013). Husserl, 2nd ed. London: Routledge. https://doi.org/10.4324/9780203742952.CrossRefGoogle Scholar
Spiegelberg, H. (ed.). (1971). From Husserl to Heidegger: Excerpts from a 1928 Freiburg Diary by W. R. Boyce Gibson. Journal of the British Society for Phenomenology, 2: 58–83. https://doi.org/10.1080/00071773.1971.11006166.CrossRefGoogle Scholar
Steiner, M. (1978). Mathematical Explanation. Philosophical Studies, 34: 135–51. https://doi.org/10.1007/BF00354494.CrossRefGoogle Scholar
Thomasson, A. (2017). Husserl on Essences: A Reconstruction and Rehabilitation. Grazer Philosophische Studien, 94: 436–59. https://doi.org/10.1163/18756735-09403008.CrossRefGoogle Scholar
Tieszen, R. (1989). Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dordrecht: Kluwer. https://doi.org/10.1007/978-94-009-2293-8.CrossRefGoogle Scholar
Tieszen, R. (2005). Free Variation and the Intuition of Geometric Essences: Some Reflections on Phenomenology and Modern Geometry. Philosophy and Phenomenological Research, 70: 153–73. https://doi.org/10.1111/j.1933-1592.2005.tb00509.x.CrossRefGoogle Scholar
Tieszen, R. (2010). Mathematical Realism and Transcendental Phenomenological Idealism. In Hartimo, M. (ed.), Phenomenology and Mathematics. Dordrecht: Springer, 1–22. https://doi.org/10.1007/978-90-481-3729-9_1.Google Scholar
Tieszen, R. (2011). After Gödel: Platonism and Rationalism in Mathematics and Logic. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199606207.001.0001.CrossRefGoogle Scholar
Tieszen, R. (2012). Monads and Mathematics: Gödel and Husserl. Axiomathes, 22: 31–52. https://doi.org/10.1007/s10516-011-9162-z.CrossRefGoogle Scholar
Tragesser, R. S. (1984). Husserl and Realism in Logic and Mathematics. Cambridge: Cambridge University Press.Google Scholar
Tragesser, R. S. (1989). Sense Perceptual Intuition, Mathematical Existence, and Logical Imagination. Philosophia Mathematica, 2: 154–94. https://doi.org/10.1093/philmat/s2-4.2.154.Google Scholar
Webb, J. (2017). Paradox, Crisis, and Harmony in Phenomenology. In Centrone, S. (ed.), Essays on Husserl’s Logic and Philosophy of Mathematics. Dordrecht: Springer, 353–408. https://doi.org/10.1007/978-94-024-1132-4_14.CrossRefGoogle Scholar
Weyl, H. (1928). Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6: 86–8. https://doi.org/10.1007/978-3-663-16102-8_2.CrossRefGoogle Scholar
Weyl, H. (1949). Philosophy of Mathematics and Natural Science. Translated by O. Helmer. Princeton, NJ: Princeton University Press.Google Scholar
Weyl, H. (1987). The Continuum: A Critical Examination of the Foundation of Analysis. Translated by S. Pollard and T. Bole. Kirksville, MO: Thomas Jefferson University Press.Google Scholar
Weyl, H. (1998). On the New Foundational Crisis of Mathematics. Translated by B. Müller. In Mancosu, P. (ed.), From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press, 86–122.Google Scholar
Willard, D. (1980). Husserl on a Logic That Failed. Philosophical Review, 89: 46–64. https://doi.org/10.2307/2184863.CrossRefGoogle Scholar
Zahavi, D. (2017). Husserl’s Legacy: Phenomenology, Metaphysics, and Transcendental Philosophy. Oxford: Oxford University Press. https://doi.org/10.1093/oso/9780199684830.001.0001.Google Scholar
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