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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.
Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Logic, Symbolic and mathematical. --- Philosophy. --- Logic of mathematics --- Mathematics, Logic of --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics. --- Mathematical logic. --- Mathematical Logic and Foundations. --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism
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Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.
Arithmétique --- Logique mathématique --- Mathématiques --- Logique --- Fondements --- Philosophie --- Logic, Symbolic and mathematical. --- Mathematics --- Philosophy. --- Logic, Symbolic and mathematical --- Logic of mathematics --- Mathematics, Logic of --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Philosophy --- Arithmétique. --- Logique mathématique. --- Fondements. --- Philosophie. --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Mathematical logic. --- Mathematics. --- Logic. --- Philosophy and science. --- Mathematical Logic and Foundations. --- Mathematics, general. --- Philosophy of Science. --- Science and philosophy --- Science --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Intellect --- Psychology --- Reasoning --- Thought and thinking --- Math --- Methodology
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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.
Mathematical logic --- wiskunde --- logica
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