Journal for General Philosophy of Science

, Volume 39, Issue 2, pp 245–271 | Cite as

Kant’s Theory of Arithmetic: A Constructive Approach?

  • Kristina EngelhardEmail author
  • Peter Mittelstaedt


Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation.


Kant Arithmetic Construction Numbers Time 


  1. Allison, H. E. (2004). Kant’s transcendental idealism. An interpretation and defense (revised and enlarged edition. New Haven, London: Yale University Press.Google Scholar
  2. Barker, S. (1992). Kant's view of geometry: a partial defense. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 221–243). Dordrecht: Kluwer.Google Scholar
  3. Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, LXX(19), 661–679.CrossRefGoogle Scholar
  4. Bigelow, J. (2001). The reality of numbers. A physicalist’s philosophy of mathematics. Oxford: Oxford University Press (reprint of 1988).Google Scholar
  5. Brittan, G. (1992). Algebra and intuition. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 315–340). Dordrecht: Kluwer.Google Scholar
  6. Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Ed. E. Zermelo. Berlin: Springer.Google Scholar
  7. Carson, E. (1999). Kant on the method of mathematics. Journal of the History of Philosophy, XXXVII(4), 629–652.Google Scholar
  8. Falkenburg, B. (2000). Kants Kosmologie. Frankfurt: Klostermann.Google Scholar
  9. Frege, G. (1988). Grundlagen der Arithmetik. Ed. Ch. Thiel. Hamburg: Meiner.Google Scholar
  10. Friedman, M. (1992). Kant’s theory of geometry. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 177–220). Dordrecht: Kluwer.Google Scholar
  11. Hanna, R. (2002). Mathematics for humans. European Journal of Philosophy, 10(3), 328–352.CrossRefGoogle Scholar
  12. Hintikka, J. (1992a). Kant on the mathematical method. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 21–42). Dordrecht: Kluwer.Google Scholar
  13. Hintikka, J. (1992b). Kant’s transcendental method and his theory of mathematics. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 341–360). Dordrecht: Kluwer.Google Scholar
  14. Kamlah, W., & Lorenzen, P. (1967). Logische Propädeutik. Mannheim: Bibliographisches Institut.Google Scholar
  15. Kant, I. (1998). Critique of pure reason (P. Guyer & A. W. Wood, Trans.). Cambridge: Cambridge University Press. Cited as CpR.Google Scholar
  16. Kant, I. (1900 ff.). Gesammelte Schriften. Ed. by Deutsche Akademie der Wissenschaften (Cited as AA with roman numerals for volume and arabic numerals for pages, and n for note).Google Scholar
  17. Leibniz, G. W. (1882). Nouveaux essays. Ed. C. J. Gerhardt, Philosophische Schriften (Vol. V). Berlin (Reprint Hildesheim 1960).Google Scholar
  18. Longuenesse, B. (1998). Kant and the capacity to judge. Sensibility and discursivity in the transcendental analytic of the critique of pure reason. Princeton: Princeton University Press.Google Scholar
  19. Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Heidelberg: Springer.Google Scholar
  20. Melnick, A. (1992). The geometry of a form of intuition. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 245–256). Dordrecht: Kluwer.Google Scholar
  21. Parsons, C. (1992a). Kant’s philosophy of arithmetic (and postscript). In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 43–80). Dordrecht: Kluwer.Google Scholar
  22. Parsons, C. (1992b). Arithmetic and the categories. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 135–158). Dordrecht: Kluwer.Google Scholar
  23. Putnam, H. (1971). Philosophy of logic. New York: Harper.Google Scholar
  24. Quine, W. V. O. (1960). Word and object. Cambridge, MA: MIT Press.Google Scholar
  25. Shabel, L. (1998). Kant on the ‘symbolic construction’ of mathematical concepts. Studies in the History and Philosophy of Science, 29, 589–621.CrossRefGoogle Scholar
  26. Shabel, L. (2006). Kant’s philosophy of mathematics. In P. Guyer (Ed.), The Cambridge companion to Kant (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  27. Sutherland, D. (2004a). The role of magnitude in Kant’s critical philosophy. Canadian Journal of Philosophy, 34(4), 411–442.Google Scholar
  28. Sutherland, D. (2004b). Kant’s philosophy of mathematics and the Greek mathematical tradition. Philosophical Review, 113(2), 157–201.CrossRefGoogle Scholar
  29. Sutherland, D. (2005). The point of Kant’s axioms of intuition. Pacific Philosophical Quarterly, 86, 135–159.CrossRefGoogle Scholar
  30. Sutherland, D. (2006). Kant on arithmetic, algebra, and the theory of proportions. Journal of the History of Philosophy, 44(4), 533–558.CrossRefGoogle Scholar
  31. Tetens, H. (1994). Arithmetik: ein Apriori der Erfahrung? Dialektik, 3, 125–146.Google Scholar
  32. Thiel, C. (1995). Philosophie und Mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
  33. Weyl, H. (1927). Philosophie der Mathematik und Naturwissenschaft. In A. Baeumler & M. Schröter (Eds.), Handbuch der Philosophie, Abteilung II, Natur, Geist, Gott (pp. 1–162). München: Oldenburg.Google Scholar
  34. Weyl, H. (1966). Philosophie der Mathematik und Naturwissenschaft. (3rd edition). München: Oldenburg.Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Philosophisches Seminar der Universität zu KölnKölnGermany
  2. 2.Institut für Theoretische Physik der Universität zu KölnKölnGermany

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