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Was Frege a Logicist for Arithmetic?

  • Marco PanzaEmail author
Chapter

Abstract

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

References

  1. Benis-Sinaceur, H., M. Panza, and G. Sandu. 2015. Functions and Generality of Logic. Reflections on Frege’s and Dedekind’s Logicism. Cham, Heidelberg, New York, Dordrecht and London: Springer.CrossRefGoogle Scholar
  2. Boolos, G. 1998. Logic, Logic and Logic. With introductions and afterword by J.P. Burgess, ed. R. Jeffrey. Cambridge, MA and London: Harvard University Press.Google Scholar
  3. Boolos, G., and R.G. Heck Jr. Die Grundlagen der Arithmetik, §§ 82–3. In M. Schirn 1998, 407–428. Also in Boolos, 1998, 315–338.Google Scholar
  4. Carnap, R. 1929. Abriss der Logistik […]. Wien: J. Springer.Google Scholar
  5. Carnap, R. 1931. Die logizistische Grundlegung der Mathematik. Erkenntnis 2: 91–105.CrossRefGoogle Scholar
  6. Carnap, R. PMBP. The Logicist Foundation of Mathematics. In Philosophy of Mathematics, ed. P. Benacerraf and H. Putnam, 31–41. Englewood Cliffs: Prentice-Hall, 1964. English translation of Carnap 1931.Google Scholar
  7. Ferreira, F. (Forthcoming). Zigzag, and Fregean Arithmetic. Presented at the Workshop Ontological Commitment in Mathematics. In Memoriam of Aldo Antonelli, Paris, IHPST, December 14–15, 2015.Google Scholar
  8. Fraenkel, A. 1928. Einleitung in die Mengenlehre. Berlin Heidelberg: Springer Verlag. Third edition reworked and largely extended.Google Scholar
  9. Frege, G. 1884. Die Grundlagen der Arithmetik. Breslau: W. Köbner.Google Scholar
  10. Frege, G. 1893–1903. Die Grundgesetze der Arithmetick, vol. I–II. Jena: H. Pohle.Google Scholar
  11. Frege, G. GLAA. The Foundations of Arithmetic, trans J.L. Austin. Oxford: Blackwell, 1953.Google Scholar
  12. Frege, G. GGAER. Basic Laws of Arithmetic, trans. and ed. P.A. Ebert and M. Rossberg, with C. Wright. Oxford: Oxford University Press, 2013.Google Scholar
  13. Grattan-Guinness, I. 2000. The Search for Mathematical Roots. 1870–1940. Logic, Set Theory and the Foundation of Mathematics from Cantor Through Russell to Gödel. Princeton, NJ and Oxford: Princeton University Press.Google Scholar
  14. Heying, A. 1931. Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2: 106–115.CrossRefGoogle Scholar
  15. Kant, I. WL. Wiener Logik. In Gesammelte Schriften, vol. XXIV.2, 785–940. Berlin: Walter de Gruyter, 1966.Google Scholar
  16. Kant, I. LLY. Lectures on Logic, trans. and ed. J.M. Young. Cambridge: Cambridge University Press, 1992.Google Scholar
  17. Panza, M., and A. Sereni. Forthcoming. Frege’s Constraint and the Nature of Frege’s logicism. To appear on Revue of Symbolic Logic.Google Scholar
  18. Schirn, M. 1998. Philosophy of Mathematics Today. Oxford: Clarendon Press.Google Scholar
  19. von Neumann, J. 1931. Die formalistische Grundlegung der Mathematik. Erkenntnis 2: 116–121.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.CNRS, IHPST (CNRS and University of Paris 1, Panthéon-Sorbonne)ParisFrance
  2. 2.Chapman UniversityOrangeUSA

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