Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege

  • Yvon Gauthier
Part of the Studies in Universal Logic book series (SUL)


I understand arithmetical philosophy on the model of Russell’s mathematical philosophy as an internal examination of arithmetical concepts—in the case of Russell, the internal examination of logical and general mathematical concepts (Russell 1919). From the mathematical point of view, Kronecker’s general arithmetic could be seen as a successor to Newton’s Arithmetica Universalis in the sense that both Newton and Kronecker wanted to integrate algebra into a general or universal arithmetic. The two texts “On the concept of number” (Kronecker 1887a,b) and his last lectures in Berlin “On the concept of number in mathematics” (see the German text edited by Boniface and Schappacher 2001) summarize Kronecker’s conception of number or whole number (integer).


Algebraic Variety Homotopy Theory Internal Logic General Arithmetic Peano Arithmetic 
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  1. M. Agrawal, N. Kayal and N. Saxena (2004): PRIMES is in P. Annals of Mathematics, 160:781–793.zbMATHMathSciNetCrossRefGoogle Scholar
  2. J. Boniface and N. Schappacher (2001): Sur le concept de nombre en mathématique. Cours inédit de Leopold Kronecker à Berlin en 1891. Revue d’histoire des mathématiques, 7:207–275.Google Scholar
  3. Y. Gauthier (2002): Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. Kluwer, Synthese Library, Dordrecht/Boston/London.zbMATHGoogle Scholar
  4. A. Granville and T.J. Tucker (2002): It’s as Easy as ABC. Notices of the AMS, 1224–1231.Google Scholar
  5. D. Hilbert (1935): Gesammelte Abhandlungen 3 vols. Chelsea, New York.Google Scholar
  6. L. Kronecker (1886): Zur Theorie der elliptischen Funktionen. Werke, IV:309–318.Google Scholar
  7. L. Kronecker (1887a): Ein Fundamentalsatz der allgemeinen Arithmetik. Werke, II:211–241.Google Scholar
  8. L. Kronecker (1887b): Über den Zahlbegriff. Werke, III:251–274.Google Scholar
  9. L. Kronecker (1901): Vorlesungen über Zahlentheorie I. K. Hensel, ed. Teubner, Leipzig.Google Scholar
  10. B. Russell (1919): Introduction to Mathematical Philosophy. Allen and Unwin, London, UK.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontrealMontrealCanada

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