Abstract
The main purpose of this essay is to show that Kant’s account of the nature of mathematical thinking is still highly relevant to the philosophy of mathematics. For Kant has raised important questions which, whether or not one agrees with his answers, must be answered if the nature of pure mathematics and of its application to empirical phenomena is to be understood. The essay begins with a brief summary of Kant’s position, with special emphasis on those of his theses which are still regarded as central by contemporary mathematicians and philosophers (§1). There follows a discussion of some postKantian developments and reactions to Kant’s philosophy of mathematics (§2). The essay ends with an outline of an alternative view of the structure and function of pure and applied mathematics (§3).
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Notes
All references to Kant’s works refer to the Akademie edition, briefly AK. The Roman letters refer to the volumes of the edition. The following further abbreviations are used: C pure R for the second edition of the Critique of Pure Reason,C pr R for the Critique of Practical Reason, Proleg for Prolegomena to any Future Metaphysics.
For a more detailed discussion of Kant’s account of the Categories and Ideas of pure reason see my “On Kant’s Theory of Concepts” in Proc. of the Sixth International Kant Congress, ed. by G. Funke and Th. M. Seebohm, Washington, 1990, pp. 55–70.
See “The Independence of the Continuum Hypothesis” in Proc. Nat. Ac. Sc. vol. 50, 51, 1963–64.
See Subtle is the Lord by A. Pais, O.U.P., 1982, ch. 12.
For a discussion of the justifiable predictive use of incompatible scientific theories see my “On Scientific Information, Explanation and Progress” in Proceedings of the Seventh International Congress of Logic, Methodology and Philosophy of Science, edited by Ruth Barcan Marcus et al., North Holland, 1986, pp. 1–15.
See Annals of Mathematics, vol. 37, No. 4, 1936, pp. 823–843.
For a more detailed discussion see chapters IV to VII of my Philosophy of Mathematics, London, 1960, also Dover Publications, New York, 1986.
See volume 2 of Grundlagen der Mathematik by D. Hilbert and P. Bernays, pp. 416–42.
“Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” in Monatschefte far Mathematik und Physik, vol. 38, 1931, §1.
See e.g. “What is Cantor’s Continuum Problem?” in American Monthly, Vol. 54, 1947, pp. 515–525.
See “Historical Background, Principles and Methods of Intuitionism” in South African Journal of Science,vol. 49, 1952, pp. 132–146.
For a more detailed discussion, see my Experience and Theory, London, 1966, and “Über Sprachspiele und rechtliche Institutionen” in Proceedings of the 5th International Wittgenstein Symposium, Vienna, 1981, pp. 480–491.
See Theory of Games and Economic Behaviour, 2nd edition, Princeton, 1947, p. 21.
See e.g. Paul Bernays, “Mathematics as a Domain of Theoretical Science and of Mental Experience” in Logic Colloquium 1973, ed. by H. E. Rose and J. C. Shepherdson, Amsterdam, 1975.
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Körner, S. (1994). On Kant’s Philosophy of Mathematics from a Present-Day Perspective. In: Parrini, P. (eds) Kant and Contemporary Epistemology. The University of Western Ontario Series in Philosophy of Science, vol 54. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0834-8_4
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DOI: https://doi.org/10.1007/978-94-011-0834-8_4
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