Towards an arithmetical logic. The arithmetical foundations of logic
The book under review is a monograph that offers a contribution to the foundations of logic and mathematics. The focus is put on the internal logic of mathematical theories, from arithmetic and number theory to algebraic geometry. The title term ``arithmetical logic'' refers to internal logic of classical arithmetic -- in the book it is called Fermat-Kronecker arithmetic and constantly opposed to Peano arithmetic. It combines Fermat's method of infinite descent with Kronecker's general arithmetic of homogeneous polynomials. It is claimed that there is no set-theoretic element in pure arithmetic while Peano and Dedekind formalized arithmetic using the transfinite set-theoretic framework. Also theories in physics and mathematical physics are considered in the book with the aim to underscore the role of arithmetic from a constructivist point of view. The book is a continuation of the author's earlier book [Internal logic. Foundations of mathematics from Kronecker to Hilbert. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1019.03001)]. The present book consists of eight chapters devoted to the internal logic of arithmetic (Introduction), arithmetization of analysis and algebra (Chapter 1), arithmetization of logic (Chapter 2), Kronecker's foundational program in contemporary mathematics (Chapter 4), arithmetical foundations for physical theories (Chapter 5), the internal logic of constructive mathematics (Chapter 6), the internal consistency of arithmetic with infinite descent (Chapter 7) and to arithmetism versus logicism (Chapter 8). Historical, mathematical, logical and philosophical aspects are considered and intertwined in the book. It can be interesting for logicians and mathematicians as well as for philosophers of mathematics.
scientific article