Meaning and Existence in Mathematics

167 Pages·1972·7.215 MB·English

by  Charles Castonguay (auth.)

Library of
Exact Philosophy
Editor:
Mario Bunge, Montreal
Co-editors:
Sir Alfred Jules Ayer, Oxford
Rudolf Carnap t, Los Angeles, Calif.
Herbert Feigl, Minneapolis, Minn.
Victor Kraft, Wien
Sir Karl Popper, Penn
Springer -Verlag Wien New York
Library of Exact Philosophy 9
Charles Castonguay
Meaning and
Existence
in Mathematics
Springer-Verlag Wien New York 1972
Printing type: Sabon Roman
Composed and printed by Herbert Hiessberger, Pottenstein
Binding work: Karl Scheibe, Wien
Design: Hans Joachim Boning, Wien
ISBN-13:978-3-7091-7115-8 e-ISBN-13:978-3-7091-7113-4
DOl: 10.1007/978-3-7091-7113-4
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.
© 1972 by Springer-Verlag/Wien
Softcover reprint of the hardcover 1st edition 1972
General Preface to the LEP
The aim of the Library of Exact Philosophy is to keep alive the
spirit, if not the letter, of the Vienna Circle. It will consequently
adopt high standards of rigor: it will demand the clear statement
of problems, their careful handling with the relevant logical or
mathematical tools, and a critical analysis of the assumptions and
results of every piece of philosophical research.
Like the Vienna Circle, the Library of Exact Philosophy sees in
mathematics and science the wellsprings of contemporary intellectual
culture as well as sources of inspiration for some of the problems
and methods of philosophy. The library of Exact Philosophy will
also stress the desirability of regarding philosophical research as
a cooperative enterprise carried out with exact tools and with the
purpose of extending, deepening, and systematizing our knowledge
about human knowledge.
But, unlike the Vienna Circle, the Library of Exact Philosophy
will not adopt a school attitude. It will encourage constructive work
done across school frontiers and it will attempt to minimize sterile
quarrels. And it will not restrict the kinds of philosophical problem:
the Library of Exact Philosophy will welcome not only logic,
semantics and epistemology, but also metaphysics, value theory
and ethics as long as they are conceived in a clear and cogent way,
and are in agreement with contemporary science.
Montreal, January 1970
Mario Bunge
Preface
The take-over of the philosophy of mathematics by mathematical
logic is not complete. The central problems examined in this book
lie in the fringe area between the two, and by their very nature
will no doubt continue to fall partly within the philosophical re
mainder. In seeking to treat these problems with a properly sober
mixture of rhyme and reason, I have tried to keep philosophical
jargon to a minimum and to avoid excessive mathematical compli
cation. The reader with a philosophical background should be
familiar with the formal syntactico-semantical explications of proof
and truth, especially if he wishes to linger on Chapter 1, after which
it is easier philosophical sailing; while the mathematician need only
know that to "explicate" a concept consists in clarifying a heretofore
vague notion by proposing a clearer (sometimes formal) definition
or formulation for it. More seriously, the interested mathematician
will find occasional recourse to EDWARD'S Encyclopedia of Philos
ophy (cf. bibliography) highly rewarding. Sections 2.5 and 2.7 are
of interest mainly to philosophers.
The bibliography only contains works referred to in the text.
References are made by giving the author's surname followed by the
year of publication, the latter enclosed in parentheses. When the
author referred to is obvious from the context, the surname is
dropped, and even the year of publication or "ibid." may be
dropped when the same publication is referred to exclusively over
the course of several paragraphs. In some quotations I have con
verted the notation of the original to suit my own.
This book developed from a thesis presented to the Faculty of
Graduate Studies of McGill University in 1971. Both thesis and
x Preface
book owe much to the encouragement of MARIO BUNGE, who deftly
and patiently stimulated my growing interest in these philosophical
problems. The content and form of the book have also benefited
from suggestions of PHILIP OLIN and JOHN TRENTMAN, and, espe
cially, WALTER BURGESS and FRANZ OPPACHER. Other intellectual
debts are acknowledged through my frequent references to the
bibliography.
I hope that the questions posed and the views advanced here
may provoke the reader to further improve upon them.
Ottawa, November 1972
Charles Castonguay
Contents
Introduction 1
1. Extension and Intension 9
1.1 The Basic Doctrine 9
1.2 A Set-theoretic Formulation 12
1.3 Extension and Intension in Formalized Theories 16
1.4 Intension as Comprehension 22
1.5 Calculi of Extensions and Intensions 26
1.6 Extension and Intension of Theories 30
1.7 Intension as Connotation: Core Intension 33
1.8 Vagueness 34
1.9 Intensional Autonomy 38
2. Meaning 41
2.1 Correspondence and Coherence Views 41
2.2 Meaning as Intension/Extension 46
2.3 Meaning of Constructs in Mathematical Theories 49
2.4 Meaning in Formal Theories 54
2.5 C. I. Lewis on Meaning 58
2.6 Truth in Theory and Truth in Practice 64
2.7 Nonexistent Possibles 68
3. Existence 74
3.1 The Thesis that Existence is Consistency 74
3.2 Empiricist Notions of Existence 78
3.3 Objectivity and Evidence 83
3.4 A Seasoned Constructivism: Piaget's Genetic Epistemology 87
3.5 Heuristics and Mathematical Existence 91
3.6 Style 94
3.7 Sets and the Semantics of Mathematics 99
3.8 Categories and the De-ontologization of Mathematics 102
XII Contents
4. Reduction 111
4.1 Reduction in Mathematics 111
4.2 Meaning-preserving Correspondences 118
4.3 Explanation v. Reduction 124
4.4 Ontological Commitment 131
4.5 Ontological Reduction 138
Bibliography 144
Index of Names 153
Subject Index 156
Partial List of Symbols 159
Introduction
The referential view of meaning surely offers the most transparent
conception of the meaning of an expression in a language, identi
fying as it does such meaning with the relation between the expres
sion and the extralinguistic entities to which the expression is taken
to refer. Whatever limited success this primitive view may have
achieved in clarifying the meaning of concepts in factual theories
dealing with the physical world, its application to mathematics im
mediately prompts delicate questions concerning the ontological and
epistemological status of the presumed non-linguistic mathematical
referents: What is the mode of existence of these entities. and how
does the mathematician have access to them? In the present work
we adopt an active mode of approach to these questions, and ask
what sense can be made of mathematical existence, and how mean
ing and knowledge are derived in the course of mathematical
activity.
To this end we will strive in what follows to draw close to
what mathematics is, and attend to mathematics as process, rather
than speculate on what it might be. And though a definite construc
tivist, kinetic view of mathematical certainty will emanate from our
deliberations, we will not directly concern ourselves with the prob
lem of the foundation of mathematical knowledge. Not that we feel
that speculative philosophy lays any stronger claim on the latter
problem; on the contrary, this is one philosophical domain where
mathematics has seen to its own arms, and where significant dis
cussion requires an especially developed mathematical competence,
which we will not pretend to assume.
In spite of the silence of the various forms of mathematical
Platonism, realism, and empiricism before the two questions which
1 LEP 9