Towards an Arithmetical Logic

The idea of an internal logic of arithmetic or arithmetical logic is inspired by a variety of motives in the foundations of mathematics. The development of mathematical logic in the twentieth century, from Hilbert to the contemporary scene, could be interpreted as a continuous tread leading to arithmetical logic. Arithmetization of analysis with Cauchy, Weierstrass and Dedekind and arithmetization of algebra with Kronecker have lead to the foundational inquiries initiated by Hilbert. Frege’s logical foundations of mathematics, mainly arithmetic, have contributed to clarify philosophical motives and although Frege’s logicism has not achieved its goals, it has given birth to Russell’s theory of types and to some extent to Zermelo’s set-theoretic cumulative hierarchy while launching philosophical logic and philosophy of language. But the “arithmetism” I have in mind here is mostly anti-Fregean in that in turns logicism upside down and asks the question: “how far can we go into logic with arithmetic alone” rather than the Fregean question: “how far can we go into arithmetic with deductive logic alone?” It is Kronecker’s polynomial arithmetic that guides here and the purpose of this book is to see how far a Kroneckian constructivist program can go in the arithmetization (and algebraization) of logic in the twenty-first century. The present work has been conceived has a sequel to my 2002 book Internal Logic, Foundations of Mathematics from Kronecker to Hilbert (Kluwer) and as a continuation of my efforts towards an arithmetical logic.

Hilbert is not the originator of the expression “metamathematics”, but he is the first to define it as the theory of formal systems designed to capture the internal logic (“inhaltliche Logik”) of mathematics and it is the internal logic of arithmetic, what I call arithmetical logic, that was to be his primary concern.

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).