Towards an Arithmetical Logic: The Arithmetical Foundations of Logic

✍ Scribed by Yvon Gauthier (auth.)

• Publisher: Birkhäuser Basel
• Year: 2015
• Tongue: English
• Leaves: 193
• Series: Studies in Universal Logic
• Edition: 1
• Category: Library

✦ Synopsis

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 Fermat’s method of infinite descent with Kronecker’s 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 author’s critical approach to the foundations of logic and mathematics.

✦ Table of Contents

Front Matter....Pages i-xi
Introduction: The Internal Logic of Arithmetic....Pages 1-3
Arithmetization of Analysis and Algebra....Pages 5-24
Arithmetization of Logic....Pages 25-53
Kronecker’s Foundational Programme in Contemporary Mathematics....Pages 55-70
Arithmetical Foundations for Physical Theories....Pages 71-116
The Internal Logic of Constructive Mathematics....Pages 117-134
The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof....Pages 135-162
Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege....Pages 163-177
Back Matter....Pages 179-184

✦ Subjects

Mathematical Logic and Foundations