Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege

Abstract

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).