The Mathematical Philosophy of Bertrand Russell: Origins and Development

248 Pages·1991·6.442 MB·English

by  Prof. Dr. Francisco A. Rodríguez-Consuegra (auth.)

Francisco A. Rodriguez-Consuegra
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
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to «Verwertungsgesellschaft Wort», Munich.
© 1991 Birkhauser Verlag Basel
Softcover reprint of the hardcover 1s t 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 x
Acknowledgements xi
Preface by Ivor Grattan-Guinness Xlll
Introduction
1. 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
2. The unpublished mathematical philosophy: 1898-1900 44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 81
2.8.1. The first contacts and opinions 81
2.8.2. The reasons for the rejection 86
viii The mathematical philosophy of Bertrand Russell
3. The contribution of Peano and his school 91
3.1. Logic 92
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
3.2. Arithmetic 105
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
3.3. Geometry 113
3.3.1. The geometric calculus and the principles of geometry 113
3.3.2. The 'logicist' geometry 117
3.4. The method 119
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
3.5. Peano's followers and their contributions 127
3.5.1. The various improvements 127
3.5.2. The transformation of definitions by abstraction into nominal ones 131
4. The principles of mathematics 135
4.1. The reaction to the Congress of 1900 135
4.1.1. The notes to the manuscripts: from Moore to Peano 135
4.1.2. The first writings 138
4.1.3. The acceptance of Cantor 141
4.2. Logic 144
4.2.1. The indefmables and the propositional function 144
4.2.2. Relations 151
Contents IX
4.3. Arithmetic 155
4.3.1. The definition of cardinal number 155
4.3.2. Finite and infinite 162
4.3.3. Quantity 164
4.3.4. Order 166
4.3.5. Ordinal numbers 169
4.3.6. Real numbers 173
4.4. Geometry 175
4.5. What Russell learned from Peano 181
S. Philosophical and methodological problems 185
5.1. Origin and evolution of Russell's logicism 185
5.2. The principle of abstraction 189
5.2.1. Origin and evolution 189
5.2.2. Assumptions and implications 194
5.3. The constructive definition 205
5.3.1. Nominal definitions 205
5.3.2. Mathematical and philosophical definitions 208
5.3.3. Analysis and ordinary language 211
5.4. Relational logic and ontology 215
Bibliography 224
B.l. Works by Russell 224
B.l.1. Unpublished manuscripts 224
B.1.2. Published or unpublished correspondence 225
B.l.3. Published works 225
B.2. Works by other authors 227
Keys to Russell's works
I also include the bibliographical reference, the only key used for the rest of articles or books
mentioned or quoted (see Bibliography).
AB Autobiography. 1967a
AMR Analysis of mathematical reasoning. m1898
CP The Collected Papers of Bertrand Russell. 1983a, 1984a ...
EA Essays in analysis. 1973a
FO Foundations of geometry, 1897a
FlAM Fundamental ideas and axioms ofm athematics. m1899
IMP Introduction to mathematical philosophy. 1919a
LK Logic and knowledge. 1956a
ML Mysticism and logic. 1918a
MPD My philosophical development. 1959a
OKEW Our knowledge of the external world. 1914a
PE Philosophical essays. 1910b
PL A critical exposition of the philosophy ofL eibniz, 1900a
PM Principia mathematica. 191Oa, 1912a, 1913a
POM1 Principles of mathematics. m1900
POM The principles of mathematics. 1903a
PP The problems ofp hilosophy. 1912b
PRM Portraits from memory. 1956b
TK Theory of knowledge (1913),1984a
Cross references
All references, no matter whether denoting a chapter, a section or a sub-section, are always
indicated through their numbers, i.e. 4, 4.5, 4.5.3.
Acknowledgements
I am very especially grateful to Ivor Grattan-Guinness, who encouraged me during the past years,
helped me with several publications, read and improved the whole manuscript of this book
through an impressive number of remarks, and wrote the Preface. I would like to express my
thanks also to Jesus Mosterfn for supervising the Ph. D. thesis which served as a partial ground
for this book, and for supporting me during the composition. I am also grateful to Juan J. Acero,
Alejandro Garciadiego, Mario G6mez, Javier de Lorenzo, Gregory Landini, Willard V. Quine,
Josep Rifa and Roberto Torretti for reading some chapters (or related materials) and sending me
valuable comments.
Concerning the unpublished materials which has been necessary to study, and all the related
information during many years, the kind help of Kenneth Blackwell, the Russell Archivist, and the
staff of The Bertrand Russell Archives (especially Carl Spadoni), McMaster University,
Hamilton, Ontario, Canada, has been decisive, together with the permission of Christopher Farley
(The Bertrand Russell Estate). The copyright of all the quoted unpublished material (manuscripts
and correspondence) is held by McMaster University, so my thanks are also due to Louis
Greenspan, Chairman of The Bertrand Russell Archives Copyright Permissions Committee, for
his explicit permission.
Some parts of this book have been previously published. Thus, earlier Spanish versions of
chapters 2, 3 and 4 have appeared as my 1988a, 1988b, 1988c and 1988d. Likewise, I have
reproduced parts of my English paper 1987c, mainly in 5.2. My thanks are due to both publishers
(the Department of Mathematics of the National Autonomous University of Mexico -
Mathesis-and Taylor & Francis, Ltd. -History and Philosophy ofL ogic) for their permission.
However, the way in which these materials appear here is quite different, for I have taken
advantage to make many corrections and improvements.
I am grateful to Benno Zimmermann, from Birkhiiuser Verlag, for having accepted this book
for publication, as well as to Doris Womer for her efficient help with the details. Also, I would
like to say that, although I myself typed the book camera-ready, the task was made much easier
thanks to my Macintosh computer, as well as to the following software: Ms. Word, Expressionist
and -in a few places--Hyper Card.
I cannot forget here the person of Bertrand Russell, whose works constituted the main reason I
studied philosophy, after having finished another completely different degree. I still remember my
frustration on reading, during a journey to Buenos Aires in 1970, the news of his death in La
Nacion, once I had decided to try to meet him some day. Michael Scott wrote (in his contribution
xu The mathematical philosophy of Bertrand Russell
to Schoenman 1967a) that it is impossible to write about Russell without offending someone,
perhaps including Russell himself. This idea, which I share, may be used to express my intention
of regarding this book, no matter what its arguments or conclusions can state, as a little homage to
him.
Cambrils, Tarragona, December 1990
After finishing this book, I received volume 8 of Russell: the Journal of the Bertrand Russell
Archives (the proceedings of a conference on Russell's philosophy), and the books Rereading
Russell: Essays in Bertrand Russell's metaphysics and epistemology (vol. XII of Minnesota
Studies in the Philosophy of Science), and E.R. Eames, Bertrand Russell's dialogue with his
contemporaries. I do not think that they contain ideas which can be regarded as relevant to the
results of my investigations here; see my essays-review (l990b, 1990c and 1991 a in the
Bibliography.)