Towards an Arithmetical Logic The Arithmetical Foundations of Logic [Paperback]

This book offers an original contribution to the foundations of logic and mathematics and focuses on the internal logic of mathematical theories, from arithmetic or number theory to algebraic geometry. Arithmetical logic is the term used to refer to the internal logic of classical arithmetic, here called Fermat-Kronecker arithmetic and combines Fermats method of infinite descent with Kroneckers general arithmetic of homogeneous polynomials. The book also includes a treatment of theories in physics and mathematical physics to underscore the role of arithmetic from a constructivist viewpoint. The scope of the work intertwines historical, mathematical, logical and philosophical dimensions in a unified critical perspective; as such, it will appeal to a broad readership from mathematicians to logicians, to philosophers interested in foundational questions. Researchers and graduate students in the fields of philosophy and mathematics will benefit from the authors critical approach to the foundations of logic and mathematics.

Foreword.- 1.Introduction.- 2.Arithmetization of Analysis and Algebra.- 3.Arithmetization of Logic.- 4.Kronecker's Foundational Programme in Contemporary Mathematics.- 5.Arithmetical Foundations for Physical Theories.- 6.The Internal Logic of Constructive Mathematics.- 7.The Internal Consistency of Arithmetic with Infinite Descent. A Syntactical Proof.- 8 Conclusion. Arithmetism versus Logicism or Kronecker contra Frege.- References.

The present book consists of eight chapters devoted to the internal logic of arithmetic & . 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. (Roman Murawski, zbMATH 1339.03002, 2016)

Yvon Gauthier has taught formal logic and philosophy of science at the Universities of Sudbury, Toronto and Montreal for fifty years. He has l3)