Abstract
Standard wisdom has it that mathematical progress has eclipsed Kant’s view of mathematics on three fronts: intuition, infinity and the continuum. Not surprisingly, these very areas define Brouwer’s own relation to Kant, for Brouwer attempted to recreate the Kantian picture of the continuum by updating Kant’s notions of infinity and intuition in a set theoretic context. I will show that when we look carefully at how Brouwer does this, we will find a certain internal tension (a “disequilibrium”) between his epistemology of intuition and the ontology of infinite objects that he must adopt. However, I will also show that when we search the corresponding Kantian notions for that same disequilibrium, then — despite first impressions — we will find equilibrium instead. Because of this, I will suggest at the end that the basic components of Kant’s eighteenth century view provide a foundation for important parts of classical rather than intuitionistic mathematics. This, in turn, will lead to a reassessment of Brouwer’s Kantianism, and the way that mathematics has progressed from Kant’s time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bottazzini, U. (1986). The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass.Stuttgart: Springer-Verlag.
Brouwer, L. E. J. (1907). Over de Grondslagen der Wiskunde. Dissertation, University of Amsterdam, translated and reprinted in Brouwer (1975, 11–101).
Brouwer, L. E. J. (1912). “Intuitionism and Formalism.” Bulletin of the.American Mathematical.Society. Volume 20: 81–96.
Brouwer, L. E. J. (1919). Begründung der Mengenlehre unabhägig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil, Theorie der Punkmengen. Amsterdam: Verh. Koninklijke Akademie van Wetenschappen te Amsterdam. Pages 3–33.
Brouwer, L. E. J. (1927). “Über Definitionsbereiche von Funktionen.” Mathematische Annalen. Volume 97: 60–7 5.
Brouwer, L. E. J. (1948a). “Essentieel Negatieve Eigenschappen.” Koninklijke Akademie van Wetenschappen te Amsterdam, Proc. Volume 51. Translated and reprinted in Brouwer (1975, 478–9).
Brouwer, L. E. J. (1948b). “Consciousness, Philosophy and Mathematics.” Proceedings of the Tenth International Congress of Philosophy. Amsterdam. Pages 1235–49.
Brouwer, L. E. J. (1952). “Historical Background, Principles and Methods of Intuitionism.” South African Journal ofScience. Volume 49: 139–46.
Brouwer, L. E. J. (1975). Collected Works. A. Heyting. (Ed.). Volume I. Amsterdam: North Holland.
Cellucci, Carlo. (1999). “The Growth of Mathematical Knowledge: An Open World View.” this volume.
Euler, L. (1755). “Remarques sur les Mémoires Précédens de M. Bernoulli.” in Euler (1952, 233–54).
Euler, L. (1952). Opera Omnia. Series 2. Volume 10.
Dummett, Michael. (1977). Elements ofIntuitionism. Oxford: Oxford University Press.
Dummett, Michael. (1978a). Truth and Other Enigm’s. Cambridge: Harvard University Press.
Dummett, Michael. (1978b). “The Philosophical Basis of Intuitionistic Logic.” in Dummett (1978a).
Friedman, Michael. (1992). Kant and the Exact Sciences. Cambridge: Harvard University Press.
Gardner, Martin (1991). The Unexpected Hanging and Other Mathematical Diversions. Chicago: University of Chicago Press.
Gödel, Kurt. (1940a). The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Annals of Mathematics Studies. Volume 3. Princeton: Princeton University Press.
Gödel, Kurt. (1940b). Lecture on the Consistency of the Continuum Hypothesis. Brown University. Reprinted in Gödel (1995, 175–85).
Gödel, Kurt. (1951). “Some Basic Theorems on the Foundations of Mathematics and Their Implications.” Josiah Willard Gibbs Lecture, delivered at a meeting of the American Mathematical Society on December 26, 1951. Reprinted in Gödel (1995, 290–323).
Gödel, Kurt. (1995). Collected Works. Volume III: Unpublished Essays and Lectures. S. Fefferman et. al. (Eds.) Oxford: Oxford University Press.
Kant, Immanuel. (1781/87). Critique of Pure Reason. Translated by N. K. Smith. New York: St. Martin’s Press.
Kant, Immanuel (1992). Lectures on Logic. J. M. Young. (Ed.). Cambridge: Cambridge University Press.
Kaplan, D. and Montague, R. (1960). “A Paradox Regained.” Notre Dame Journal of Formal Logic. Volume 1: 79–90.
Leibniz, G. W. (1686). Discourse on Metaphysics. Translated and reprinted in Loemker (1956).
Loemker, L. E. (Ed.). (1956). Gottfried Wilhelm Leibniz: Philosophical Papers and Letters. Reidel.
Posy, Carl. J. (1984). “Kant’s Mathematical Realism.” The Monist. Volume 67: 115–34. Reprinted in Posy (1992).
Posy, Carl J. (1991). “Kant and Conceptual Semantics,” Topoi. Volume 10: 67–78.
Posy, Carl J. (Ed.). (1992). Kant’s Philosophy of Mathematics: Modern Essays. Dordrecht: Kluwer Academic Publishers.
Posy, Carl J. (1995). “Unity, Identity, Infinity: Leibnizian Themes in Kant’s Philosophy of Mathematics.” Proceedings of the Eighth International Kant Congress. Marquette University Press.
Russell, Bertrand. (1897). An Essay on the Foundations of Geometry. Cambridge: Cambridge University Press.
Russell, Bertrand (1903). The Principles of Mathematics. Cambridge: Cambridge University Press.
Troelstra, A. S. (1977). Choice Sequences: A Chapter of Intuitionistic Mathematics. Oxford: Oxford University Press.
Yushkevich, A. P. (1976). “The Concept of Function up to the Middle of the Nineteenth Century.” Archive for the History of the Exact Sciences. Volume 16: 37–85.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Posy, C.J. (2000). Epistemology, Ontology and the Continuum. In: Grosholz, E., Breger, H. (eds) The Growth of Mathematical Knowledge. Synthese Library, vol 289. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9558-2_14
Download citation
DOI: https://doi.org/10.1007/978-94-015-9558-2_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5391-6
Online ISBN: 978-94-015-9558-2
eBook Packages: Springer Book Archive