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Husserl's Philosophy of Mathematical Practice
Published online by Cambridge University Press: 04 December 2024
Summary
Husserl's Philosophy of Mathematical Practice explores the applicability of the phenomenological method to philosophy of mathematical practice. The first section elaborates on Husserl's own understanding of the method of radical sense-investigation (Besinnung), with which he thought the mathematics of his time should be approached. The second section shows how Husserl himself practiced it, tracking both constructive and platonistic features in mathematical practice. Finally, the third section situates Husserlian phenomenology within the contemporary philosophy of mathematical practice, where the examined styles are more diverse. Husserl's phenomenology is presented as a method, not a fixed doctrine, applicable to study and unify philosophy of mathematical practice and the metaphysics implied in it. In so doing, this Element develops Husserl's philosophy of mathematical practice as a species of Kantian critical philosophy and asks after the conditions of possibility of social and self-critical mathematical practices.
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- Online ISBN: 9781009165709Publisher: Cambridge University PressPrint publication: 16 January 2025
References
Antonutti Marfori, M. & Quinon, P. (2021). Intensionality in Mathematics: Problems and Prospects. Special Issue on Intensionality in Mathematics. Synthese, 198 (5), 995–999.CrossRefGoogle Scholar
Avigad, J. (2008). Computers in Mathematical Inquiry. In Mancosu, P., ed., Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 302–316.CrossRefGoogle Scholar
Bachelard, S. (1968). A Study of Husserl’s Formal and Transcendental Logic, Evanston: Northwestern University Press.Google Scholar
Baldwin, J. T. (2018). Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism. New York: Cambridge University Press.CrossRefGoogle 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
Bernays, P. (1935). On Platonism in Mathematics. In Benacerraf, P. & Putnam, H., eds., Philosophy of Mathematics, Selected Readings, 2nd ed. Cambridge: Cambridge University Press, pp. 258–271.Google Scholar
Button, T. & Walsh, S. (2018). Philosophy and Model Theory, Oxford: Oxford University Press.CrossRefGoogle Scholar
Carnap, R. (1936). Logical Syntax of Language, Smeaton, A., transl., London: Routledge & Kegan Paul.Google Scholar
Carnap, R. (1950). Empiricism, Semantics, and Ontology. Revue Internationale de Philosophie, 4 (11), 20–40.Google Scholar
Carr, D. (1999). The Paradox of Subjectivity: The Self in the Transcendental Tradition, Oxford: Oxford University Press.CrossRefGoogle Scholar
Carr, D. (2022). Phenomenology as Critical Method. In Aldea, S., Carr, D., & Heinämaa, S., eds., Phenomenology as Critique: Why Method Matters. New York: Routledge, pp. 9–24.CrossRefGoogle Scholar
Carter, J. (2019). Philosophy of Mathematical Practice – Motivations, Themes and Prospects. Philosophia Mathematica, 27 (1), 1–32.CrossRefGoogle Scholar
Corfield, D. (2003). Towards a Philosophy of Real Mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Corry, L. (2004). Introduction: The History of Modern Mathematics – Writing and Rewriting. Science in Context, 17 (1/2), 1–21.CrossRefGoogle Scholar
Crosilla, L. (2022). The Entanglement of Logic and Set Theory, Constructively. Inquiry, 65 (6), 638–659.CrossRefGoogle Scholar
Crowell, S. (2001). Husserl, Heidegger and the Space of Meaning, Evanston: Northwestern University Press.CrossRefGoogle Scholar
Dreben, B. and Kanamori, A. (1997). Hilbert and Set Theory. Synthese, 110, 77–125.CrossRefGoogle Scholar
Dybjer, P. & Palmgren, E. (2020). Intuitionistic Type Theory. The Stanford Encyclopedia of Philosophy (Summer 2020 Edition), Edward, N. Zalta, ed., https://plato.stanford.edu/archives/sum2020/entries/type-theory-intuitionistic/.Google Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. II, Oxford: Clarendon Press.Google Scholar
Feferman, S. (1985). Intensionality in Mathematics. Journal of Philosophical Logic, 14, 41–55.CrossRefGoogle Scholar
Feferman, S. (1999). Does Mathematics Need New Axioms? American Mathematical Monthly, 106, 99–111.CrossRefGoogle Scholar
Ferreirós, J. (2016). Mathematical Knowledge and the Interplay of Practices, Princeton: Princeton University Press.CrossRefGoogle Scholar
Ferreirós, J. & Gray, J. eds. (2006). Architecture of Modern Mathematics: Essays in History and Philosophy, Oxford: Oxford University Press.CrossRefGoogle Scholar
Føllesdal, D. (1958). Husserl und Frege, Ein Beitrag zur Beleuchtung der Entstehung der Phänomenologischen Philosophie, Oslo: H. Aschehoug.Google Scholar
Giardino, V. (2017). The Practical Turn in Philosophy of Mathematics: A Portrait of a Young Discipline. Phenomenology and Mind, 12, 18–28.Google Scholar
Gödel, K. (1961). The Modern Development of the Foundations of Mathematics in the Light of Philosophy (*1961/?). In Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., & Solovay, R. N., eds., Kurt Gödel, Collected Works 3, Unpublished Essays and Lectures. Oxford: Oxford University Press, pp. 374–387.Google Scholar
Griffith, A. M. (2018). Social Construction as Grounding. Philosophy and Phenomenological Research, 97 (2), 393–409.CrossRefGoogle Scholar
Hamkins, J. D. (2024). How the Continuum Hypothesis Could Have Been a Fundamental Axiom. Journal for the Philosophy of Mathematics, arxiv: https://arxiv.org/abs/2407.02463.Google Scholar
Hartimo, M. (2006). Mathematical Roots of Phenomenology: Husserl and the Concept of Number. Journal of History and Philosophy of Logic, 27 (4), 319–337.CrossRefGoogle Scholar
Hartimo, M. (2010). The Development of Mathematics and the Birth of Phenomenology. In Hartimo, M., ed., Phenomenology and Mathematics. Dordrecht: Springer, pp. 107–121.CrossRefGoogle Scholar
Hartimo, M. (2017a). Husserl and Hilbert. In Centrone, S., ed., Essays on Husserl’s Logic and Philosophy of Mathematics. Dordrecht: Springer, pp. 245–263.CrossRefGoogle Scholar
Hartimo, M. (2017b). Husserl and Gödel’s Incompleteness Theorems. The Review of Symbolic Logic, 10 (4), 638–650.CrossRefGoogle Scholar
Hartimo, M. (2018). Radical Besinnung in Formale und transzendentale Logik (1929). Husserl Studies, 34, 247–266.CrossRefGoogle Scholar
Hartimo, M. (2019). Husserl on Kant, and the Critical View of Logic. Inquiry: An Interdisciplinary Journal of Philosophy, 65 (6), 707–724.CrossRefGoogle Scholar
Hartimo, M. (2020a). Husserl’s Transcendentalization of Mathematical Naturalism. Journal of Transcendental Philosophy, 1 (3), 289–306.CrossRefGoogle Scholar
Hartimo, M. (2020b). Husserl on “Besinnung” and Formal Ontology. In Kjosavik, F. & Serck-Hanssen, C., eds., Metametaphysics and the Sciences: Historical and Philosophical Perspectives. New York: Routledge, pp. 200–215.Google Scholar
Hartimo, M. (2021a). Husserl and Mathematics, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hartimo, M. (2021b). Formal and Transcendental Logic – Husserl’s Most Mature Reflection on Mathematics and Logic. In Jacobs, H., ed., The Husserlian Mind. London: Routledge, pp. 50–59.CrossRefGoogle Scholar
Hartimo, M. (2021c). The Chimera of Logicism: Husserl’s Criticism of Frege. In Boccuni, F. & Sereni, A., eds., Origins and Varieties of Logicism: A Foundational Journey in the Philosophy of Mathematics. New York: Routledge, pp. 197–214.CrossRefGoogle Scholar
Hartimo, M. (2022a). Epistemic Values and Their Phenomenological Critique. In Heinämaa, S., Hartimo, M., & Hirvonen, I., eds., Contemporary Phenomenologies of Normativity: Norms, Goals, and Values. New York: Routledge, pp. 234–251.Google Scholar
Hartimo, M. (2022b). Radical Besinnung as a Method for Critique. In Aldea, A., Carr, D., & Heinämaa, S., eds., Phenomenology as Critique: Why Method Matters. New York: Routledge, pp. 80–94.CrossRefGoogle Scholar
Hartimo, M. & Okada, M. (2018). Syntactic Reduction in Husserl’s Early Phenomenology of Arithmetic. Synthese, 193 (3), 937–969. https://doi.org/10.1007/s11229-015-0779-0.CrossRefGoogle Scholar
Hartimo, M. & Rytilä, J. (2023). No Magic: From Phenomenology of Practice to Social Ontology of Mathematics. Topoi, 42, 283–295. https://doi.org/10.1007/s11245-022-09859-1.CrossRefGoogle Scholar
Heinämaa, S., Hartimo, M., & Hirvonen, I. (2022). Introduction: Phenomenological Approaches to Normativity. In Heinämaa, S., Hartimo, M., & Hirvonen, I., eds., Contemporary Phenomenologies of Normativity: Norms, Goals, and Values. New York: Routledge, pp. 1–16.CrossRefGoogle Scholar
Hilbert, D. (1996). Die logischen Grundlagen der Mathematik. Mathematische Annalen, 88, 151–165.CrossRefGoogle Scholar
Hilbert, D. (1922). Logical Foundations of Mathematics. Lecture Given at the Deutsche Naturforscher-Gesellschaft, September 1922. In Ewald, W. B., transl., and ed., From Kant to Hilbert, a Source Book in the Foundations of Mathematics. Oxford: Clarendon Press, pp. 1134–1148.Google Scholar
Hill, C. O. (2019). Translator’s Introduction. In Husserl, E., ed., Logic and General Theory of Science Lectures 1917/18, with Supplementary Texts from the First Version of 1910/11. Cham: Springer, pp. XXIII–L.Google Scholar
Hill, C. O. & Rosado Haddock, G. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago: Open Court.Google Scholar
Hirvonen, I. (2022). Reconciling the Noema Debate. Axiomathes, 32 (Suppl 3), 901–929. https://doi.org/10.1007/s10516-022-09643-1.CrossRefGoogle Scholar
Husserl, E. (1969). Formal and Transcendental Logic, Cairns, D., transl., The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1970a). Philosophie der Arithmetik, Eley, L., ed., Husserliana XII, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1970b). The Crisis of European Sciences and Transcendental Phenomenology, an Introduction to Phenomenological Philosophy, Carr, D., transl., Evanston: Northwestern University Press.Google Scholar
Husserl, E. (1973a). Cartesianische Meditationen und Pariser Vorträge, Strasser, S., ed., Husserliana, I, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1973b). Zur Phänomenologie der Intersubjektivität. Texte aus dem Nachlaß. Erster Teil: 1905–1920, Kern, I., ed., Husserliana XIII, The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1973c). Experience and Judgment: Investigations in a Genealogy of Logic, Landgrebe, L., ed., Churchill, J. S. Churchill & Ameriks, K., transl., Evanston: Northwestern University Press.Google Scholar
Husserl, E. (1974). Formale und transzendentale Logik: Versuch einer Kritik der logischen Vernunft, Janssen, P., ed., Husserliana XVII, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1975). Prolegomena zur reinen Logik, Holenstein, E., ed., Husserliana XVIII, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1976a). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie, Schuhmann, K., ed., Husserliana III/1, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1976b). Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie, Biemel, W., ed., Husserliana VI, The Hague: Martinus Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1981). Philosophy as Rigorous Science. In Lauer, Q., transl., McCormick, P. & Elliston, F. A., eds., Husserl, Shorter Works. South Bend: University of Notre Dame Press, pp. 166–197.Google Scholar
Husserl, E. (1984). Logische Untersuchungen. Zweiter Band, Erster Teil. Untersuchungen zur Phänomenologie und Theorie der Erkenntnis, Panzer, U., ed., Husserliana XIX/I, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1985). Erfahrung und Urteil: Untersuchungen zur Genealogie der Logik, 6th ed., Landgrebe, L., ed., Hamburg: Felix Meiner Verlag.Google Scholar
Husserl, E. (1987). Vorlesungen über Bedeutungslehre Sommersemester 1908, Panzer, U., ed., Husserliana XXVI, The Hague: Martinus Nijhoff.Google Scholar
Husserl, E. (1996). Logik und allgemeine Wissenschaftstheorie: Vorlesungen 1917/18, Panzer, U., ed., Husserliana XXX, Dordrecht: Kluwer.CrossRefGoogle Scholar
Husserl, E. (1999). Cartesian Meditations: An Introduction to Phenomenology, Cairns, D., transl., Dordrecht: Kluwer.Google Scholar
Husserl, E. (2001a). Logical Investigations, Vol. I, Findlay, J. N., transl., London: Routledge.Google Scholar
Husserl, E. (2001b). Logical Investigations, Vol. II, Findlay, J. N., transl., London: Routledge.Google Scholar
Husserl, E. (2001c). Natur und Geist: Vorlesungen Sommersemester 1927, Weiler, M., ed., Husserliana XXXII. Dordrecht: Kluwer.CrossRefGoogle Scholar
Husserl, E. (2001d). Husserls Manuskripte zu seinem Göttinger Doppelvortrag von 1901. Husserl Studies, 17, 87–123.Google Scholar
Husserl, E. (2003). Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887-1901, Willard, D., transl., Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Husserl, E. (2012). Zur Lehre vom Wesen und zur Methode der Eidetischen Variation: Texte aus dem Nachlass 1891–1935, Fonfara, D., ed., Husserliana XLI. Dordrecht: Springer.Google Scholar
Husserl, E. (2014). Ideas for a Pure Phenomenology and Phenomenological Philosophy. First Book: General Introduction to Pure Phenomenology, Dahlstrom, D. O., transl., Indianapolis: Hackett.Google Scholar
Husserl, E. (2019). Logic and General Theory of Science, Hill, C. O., transl., Cham: Springer.CrossRefGoogle Scholar
Iemhoff, R. (2020). Intuitionism in the Philosophy of Mathematics. The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), Zalta, Edward N., ed., https://plato.stanford.edu/archives/fall2020/entries/intuitionism/.Google Scholar
Kant, I. (1998). Critique of Pure Reason, Guyer, P. & Wood, A. W. transl. and eds., Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kisiel, T. (1970). Phenomenology as the Science of Science. In Kockelmans, J. & Kisiel, T., eds., Phenomenology and the Natural Sciences. Evanston: Northwestern University Press, pp. 5–44.Google Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. Oxford: Oxford University Press.Google Scholar
Klev, A. (2017). Husserl’s Logical Grammar. History and Philosophy of Logic, 39 (3), 232–269.CrossRefGoogle Scholar
Kouri Kissel, T. & Shapiro, S. (2020). Logical Pluralism and Normativity. Inquiry, 63 (3–4), 389–410.CrossRefGoogle Scholar
Leng, M. (2002). Phenomenology and Mathematical Practice. Philosophia Mathematica, 10 (3), 3–25.CrossRefGoogle Scholar
Ierna, C. & Lohmar, D. (2016). Husserl’s Manuscript A I 35. In Rosado Haddock, G. E., ed., Husserl and Analytic Philosophy. Berlin: De Gruyter, pp. 289–320.CrossRefGoogle Scholar
Maddy, P. (2007). Second Philosophy: A Naturalistic Method, Oxford: Oxford University Press.CrossRefGoogle Scholar
Maddy, P. (2008). How Applied Mathematics Became Pure. The Review of Symbolic Logic, 1 (1), 16–41.CrossRefGoogle Scholar
Maddy, P. (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford: Oxford University Press.CrossRefGoogle Scholar
Maddy, P. & Väänänen, J. (2023). Philosophical Uses of Categoricity Arguments. Cambridge Elements in the Philosophy of Mathematics. Cambridge: Cambridge University Press.Google Scholar
Mancosu, P. ed. (2008). Philosophy of Mathematical Practice, Oxford: Oxford University Press.CrossRefGoogle Scholar
Moon, S. (2023). Husserlian Philosophy of Mathematical Practice: An Empathy-First Approach. Dissertation. University of California, Irvine.Google Scholar
Okada, M. (2013). Husserl and Hilbert on Completeness and Husserl’s Term Rewrite-Based Theory of Multiplicity. 24th International Conference on Rewriting Techniques and Applications, RTA, 4-19.Google Scholar
von Plato, J. (2017). The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age, Princeton: Princeton University Press.Google Scholar
Putnam, H. (1980). Models and Reality. Journal of Symbolic Logic, 45 (3), 464–482.CrossRefGoogle Scholar
Queloz, M. (2021). The Practical Origins of Ideas: Genealogy as Conceptual Reverse-Engineering, Oxford: Oxford University Press.CrossRefGoogle Scholar
Rosado Haddock, G. (2006). Husserl’s Philosophy of Mathematics: Its Origin and Relevance. Husserl Studies, 22, 193–222.CrossRefGoogle Scholar
Shapiro, S. (1997). Philosophy of Mathematics, Structure and Ontology, Oxford: Oxford University Press.Google Scholar
Tieszen, R. (2011). After Gödel: Platonism and Rationalism in Mathematics and Logic, Oxford: Oxford University Press.CrossRefGoogle Scholar
Wachtel, A. (2024). The Phenomenological Concept of Definiteness: Husserl vs. His Interpreters, and Tertium non Datur. The New Yearbook for Phenomenology and Phenomenological Philosophy, 22, 188–208.CrossRefGoogle Scholar
Weyl, H. (1985). Axiomatic versus Constructive Procedures in Mathematics. The Mathematical Intelligencer, 7 (4), 12–17, 38.CrossRefGoogle Scholar
Zahavi, D. (2002). Transcendental Subjectivity and Metaphysics: A Discussion of David Carr’s Paradox of Subjectivity. Human Studies, 25, 103–116.CrossRefGoogle Scholar
Zahavi, D. (2017). Husserl’s Legacy: Phenomenology, Metaphysics, and Transcendental Philosophy, Oxford: Oxford University Press.Google Scholar
Zermelo, E. (1908). Investigations in the Foundations of Set Theory I. In van Heijenoort, J., ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge: Harvard University Press [1967], pp. 201–215.Google Scholar
Zermelo, E. (1930). On Boundary Numbers and Domains of Sets: New Investigations in the Foundations of Set Theory. In Ewald, W. B., ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 2, Oxford: Clarendon Press [1996], pp. 1219–1233.Google Scholar
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