Internal Logic : Foundations of Mathematics from Kronecker to Hilbert - Y. Gauthier

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Internal Logic

Foundations of Mathematics from Kronecker to Hilbert

By: Y. Gauthier

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Hardcover | 30 June 2002

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268 Pages

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24.77 x 16.51 x 1.91

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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.

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Preface	p. vii
Introduction: Foundations of Mathematics
Introduction	p. 1
Philosophy of mathematics and foundational research	p. 3
Mother-theories	p. 4
Mathematical logic	p. 10
The theory of mathematical practice	p. 14
Indefinite and indeterminate as foundational concepts	p. 15
Conclusion: The internal logic of polynomial arithmetic	p. 20
From Hilbert to Kronecker
Introduction: Metamathematics	p. 22
Arithmetic	p. 24
Algebra	p. 26
Logic	p. 30
Elliptic functions	p. 33
Kronecker's programme	p. 38
Conclusion: The arithmetization programme	p. 48
The Consistency of Arithmetic Revisited
Introduction	p. 50
Finitism	p. 52
Fermat's descent	p. 54
Logic	p. 58
The polynomial setting	p. 60
Internal and external consistency of arithmetic	p. 65
Conclusion	p. 76
The Internal Consistency of Arithmetic with Infinite Descent
Introduction	p. 81
Logic	p. 84
Semantics	p. 87
Arithmetic	p. 90
The polynomial translation	p. 96
The consistency proof	p. 101
Concluding remarks	p. 114
From Kronecker to Brouwer
Introduction: Cantor	p. 118
Frege's foundations of arithmetic	p. 124
Husserl's theory of multiplicites	p. 128
Weyl's constructivism	p. 132
Brouwer's intuitionism and local negation	p. 135
Negation by default	p. 144
Conclusion	p. 147
Hilbert and the Foundations of Physics
Introduction: The consistency of physical theories	p. 148
Quantum Mechanics and the local observer	p. 149
Locality and complementation in Quantum Mechanics	p. 159
Renormalization procedures in quantum field theories	p. 173
The construction and the dynamical structure of physical theories	p. 181
The notion of physical theory	p. 183
Conclusion	p. 185
Conclusion. Internal Logic: From Kronecker to Hilbert and Beyond
Introduction: Constructivism	p. 186
Logic	p. 187
Finitism	p. 191
Modular logic	p. 193
Arithmetic	p. 201
Complexity	p. 208
Conclusion	p. 212
References	p. 215
Subject Index	p. 231
Name Index	p. 235
Table of Contents provided by Ingram. All Rights Reserved.

Internal Logic

Foundations of Mathematics from Kronecker to Hilbert

At a Glance

Hardcover

Shipping

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Defective items

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Paperback

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