The Mathematical Philosophy of Bertrand Russell: Origins and Development
by Ivor Grattan-Guinness Until twenty years ago the outline history of logicism was well known. Frege had had the important ideas, until he was eclipsed by Wittgenstein. Russell was important in publicising the former and tutoring the latter, and also for
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The Mathematical Philosophy of Bertrand Russell: Origins and Development
1991
Birkhauser Verlag
Basel· Boston· Berlin
Author's address Prof. Dr. Francisco A. Rodriguez-Consuegra Pere Martell, 7, 2° 0 E-43001 Tarragona, Spain
Deutsche Bibliothek Cataloging-in-Publication Data Rodrlguez-Consuegra, Francisco A.: The mathematical philosophy of Bertrand Russell: origins and development / Francisco A. Rodriguez-Consuegra. - Basel; Boston; Berlin: Birkhauser, 1991 ISBN 978-3-0348-7535-6 ISBN 978-3-0348-7533-2 (eBook) 00110.1007/978-3-0348-7533-2
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use a fee is payable to «Verwertungsgesellschaft Wort», Munich.
© 1991 Birkhauser Verlag Basel Softcover reprint of the hardcover 1st edition 1991 Printed from the author's camera-ready manuscripts on acid-free paper in Germany ISBN 978-3-0348-7535-6
To Ana, who made my return to work possible
Contents
Keys to Russell's works and Cross references Acknowledgements Preface by Ivor Grattan-Guinness
x
xi Xlll
Introduction
1.
2.
Methodological and logicist background
5
1.1.
Boole and Peirce
5
1.2.
Dedekind and Cantor
13
1.3.
Coutu rat and Whitehead
19
1.4.
Bradley and Moore
27
1.5.
Foundations of geometry
36
The unpublished mathematical philosophy: 1898-1900
44
2.1.
The genesis of the 1898-1900 manuscripts
44
2.2.
Logic, mathematics and ontology
49
2.3.
The evolution of the main concepts
57
2.4.
Concepts, axioms, presupposition and implication
62
2.5.
The contradiction and the infinite
69
2.6.
Relations and the 'principle of abstraction'
72
2.7.
The method of definition
77
2.8.
The gradual approach to Cantor 2.8.1. The first contacts and opinions 81 2.8.2. The reasons for the rejection 86
81
viii
3.
The mathematical philosophy of Bertrand Russell
The contribution of Peano and his school
91
3.1.
92
3.2.
3.3.
3.4.
3.5.
4.
Logic 3.1.1. Objective and stages 92 3.1.2. Primitives, logical order and interdefinability 93 3.1.3. Implication, inclusion and membership 97 3.1.4. Classes, propositions and individuals 99 3.1.5. Mathematical propositions and quantification 101 3.1.6. Relations, functions, classes, properties and propositions 103 Arithmetic 3.2.1. The axioms and their interpretation: Dedekind 105 3.2.2. The definability of number 107 3.2.3. Real numbers: construction and definition 108 3.2.4. The 'logicist' arithmetic: Cantor 110 Geometry 3.3.1. The geometric calculus and the principles of geometry 113 3.3.2. The 'logicist' geometry 117 The method 3.4.1. Axiomatics 119 3.4.2. Definitions 121 3.4.3. The defmition by abstraction 124 3.4.4. Simplicity, analysis and intuition 125 Peano's followers and their contributions 3.5.1. The various improvements
