Relativity. The Theory and its Philosophy

265 Pages·1980·17.361 MB·English

by  Roger B. Angel and Mario Bunge (Auth.)

FOUNDATIONS & PHILOSOPHY OF SCIENCE & TECHNOLOGY
General Editor. MARIO BUNGE, McGill University. Montreal. Canada
This series has three goals. The editor formulates them as follows:
(1) To encourage the systematic exploration of the foundations of science and
technology
(2) To foster research into the epistemological, semantical, ontological and ethical
dimensions of scientific and technological research
(3) To keep scientists, technologists, scientific and technological administrators and
policy makers informed about progress in the foundations and philosophy of
science and technology
Some Titles in the Series
AGASSI, J.
The Philosophy of Technology
BUCHTEL, H.
The Conceptual Nervous System
BUNGE, M.
The Mind-Body Problem
HATCHER, W.
The Logical Foundations of Mathematics
SIMPSON, G.
Why and How. Some Problems and Methods in Historical Biology
WILDER, R.
Mathematics as a Cultural System
Pergamon Journals of Related Interest
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Editor:
Prof. Gerd Buchdahl, Department of History and Philosophy of Science,
University of Cambridge, England
This journal is designed to encourage complementary approaches to history of science
and philosophy of science. Developments in history and philosophy of science have
amply illustrated that philosophical discussion requires reference to its historical
dimensions and relevant discussions of historical issues can obviously not proceed
very far without consideration of critical problems in philosophy. Studies publishes
detailed philosophical analyses of material in history of the philosophy of science, in
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* Free specimen copies available on request.
Relativity:
The Theory and its Philosophy
by
ROGER B. ANGEL
Concordia University, Montreal, Canada
PERGAMON PRESS
OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT
UK Pergamon Press Ltd., Headington Hill Hall,
Oxford OX3 OBW, England
USA Pergamon Press Inc., Maxwell House, Fairview
Park, Elmsford, New York 10523, USA
CANADA Pergamon of Canada, Suite 104, 150 Consumers
Road, Willowdale, Ontario M2J 1P9, Canada
AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544,
Potts Point, NSW 2011, Australia
FRANCE Pergamon press SARL, 24 rue des Ecoles,
75240 Paris, Cedex 05, France
FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus,
OF GERMANY Hammerweg 6, Federal Republic of Germany
Copyright © 1980 Roger B. Angel
All Rights Reserved. No part of this publication may
be reproduced, stored in a retrieval system or
transmitted in any form or by any means: electronic,
electrostatic, magnetic tape, mechanical,
photocopying, recording or otherwise, without
permission in writing from the publishers
First edition 1980
British Library Cataloguing in Publication Data
Angel, Roger B
Relativity. -(Foundations & philosophy of science
& technology). - (Pergamon international library).
1. Relativity (Physics)
I. Title II. Series
530.1ΊΌ1 QC173.585 79-^1551
ISBN 0-08-025197-8 (Hardcover)
ISBN 0-08-025196- X (Flexicover)
Printed and bound in Great Britain by
William Clowes (Beccles) Limited, Beccles and London
FOR
UNNI AND VICKI
Preface
MY principal aim in writing this book was to help to remedy the situation which has long
been of concern to university teachers of the philosophy of physics—namely, that few
students are sufficiently prepared in both philosophy and physics either to have access to
the serious literature or to undertake serious work of their own in this field. Too often, the
student of philosophy must avail himself of the popular literature. Consequently, he is
unable to distinguish the genuine scientific content of a physical theory from the
particular philosophical axe which its expositor happens to be grinding. I, too, have axes
to grind. However, I have made an effort to keep the two as distinct as one may reasonably
expect.
I attempt to provide a completely self-contained treatment of the philosophical
foundations of the theory of relativity. By that, I naturally do not intend that the book
contains everything that needs to be known, or even a significant fraction thereof. I merely
mean that it should not be necessary for the average reader to master other works in order
to attain a reasonable understanding of everything in this one. I have assumed that the
reader dimly recalls the rudiments of elementary algebra, including the use of exponents,
and those of elementary geometry. I have also taken for granted a passing familiarity with
the notation of first-order logic. On this meagre basis, I attempt to provide a survey of the
most essential mathematical techniques and concepts which seem to me to be
indispensable to an understanding of the foundations of both the special and general
theories of relativity. In short, the book includes a crash course in applied mathematics,
ranging from elementary trigonometry to the classical tensor calculus. Needless to say,
much more attention has been given to mathematical intuition than to rigorous
presentation. This mathematical treatment has been divided between Chapter 1 and
Chapter 7, so that the interest and patience of the reader would not be too severely tried.
At the same time, I employ these mathematical tools in an exposition of the two theories,
which I hope is an adequate basis for serious, independent philosophical reflection.
Finally, I devote several chapters to the exposition and exploration of what I take to be
among the more central and interesting philosophical problems and points of view which
arise from these theories.
The level of treatment is between the merely popular and that of the high-grade treatise,
whether scientific or philosophical. However, I hope that in bridging the gap between the
two, I succeed in providing the industrious reader with direct access to the latter. That has
been my goal, and the several peculiarities which characterize this book derive from that
intention. I am firmly convinced that anyone who has serious aspirations in the field of
philosophy, particularly those parts which touch on epistemology or metaphysics, should
have a reasonably detailed knowledge of at least one scientific theory. The purpose of a
textbook is to open intellectual doors. I shall be content if this book succeeds in so doing.
ix
x Preface
The preface of a first book provides its author with a unique opportunity to
acknowledge his intellectual debts. Mine are too numerous to be mentioned in their
entirety. In matters mathematical, I have found Wrede's Introduction to Vector and
Tensor Analysis to strike an ideal middle ground between rigour and readability. Of the
many books on relativity which I have studied, I have profited most from Bergmann's
Introduction to the Theory of Relativity, Anderson's Principles of Relativity Physics,
Adler, Bazin and Schiffer's Introduction to General Relativity, and Ohanian's Gravitation
and Spacetime. But these are just a few of the books which are mentioned in the
bibliographies which are appended to each chapter.
No writer in this field can fail to be aware of a debt to Professor Adolf Grünbaum, who
is largely responsible for the current high level of interest in, and quality of, the philosophy
of space and time. Even those who express disagreement with various of his conclusions,
including myself, must allow that he has led the way to many of the more interesting
highways and byways of this branch of philosophy. I owe a personal debt to many
colleagues including Dr. Barry Frank of the department of physics and Dr. Harold
Proppe of the department of mathematics, both of Concordia University, for their patient
response to my many requests for advice and information.
My greatest intellectual debt is to my teacher, Professor Mario Bunge and to his
copious writings and especially his Foundations of Physics. Although our styles are very
different, those readers who are familiar with his work will recognize his influence on
scores of pages throughout this book, including, I venture to say, those in which I disagree
with him in matters of detail. It was he who first taught me how to understand science and
how to philosophize about it. Indeed, it was he who made me fully aware that the former is
a necessary condition of the latter. Whatever may be of merit in these pages is due to his
profound influence on my philosophic outlook. The mistakes, of course, are of my own
invention.
Finally, I express my gratitude to my wife Unni and my daughter Vicki, to both of
whom this book is dedicated, for remaining cheerful through all of the author's ups and
downs.
Montreal, June 1979
List of Logical Symbols
Λ = conjunction meaning "and".
V = disjunction meaning "either-or".
-> = material implication meaning "is a sufficient condition for".
<-► = biconditional meaning "is a necessary and sufficient condition for".
~ = negation meaning "not".
(x) = universal quantifier meaning "for every x".
( 3x) = existential quantifier meaning "there is an x such that".
Px = monadic predicate meaning "x is P".
Pxy = dyadic predicate meaning "x is P to >'".
Pxyz = triadic predicate meaning, for example, "x is P to y and z".
XI
1
Mathematical Preliminaries
FOR the reader with little or no mathematical background, this introductory chapter may
well prove to be the hardest in the book. In effect, he will be entering an entirely new
conceptual world. Moreover, the subject-matter encompasses a great many fundamental
mathematical concepts which are normally acquired, step by step, over a period of years. I
strongly advise you to read this material quite slowly. The worked ex#ririples should be
studied until you are able to reproduce them with only a minimum of reference tojhe text.
In this way, you will find that you make rapid gains in both confidence and facility. Do not
be discouraged in finding that you are unable to recall various details. They will gradually
become fixed in your memory as you continue to refer back to them, as needed, in the
course of reading the later chapters. At all times, you should be sufficiently motivated by
the thought of acquiring one of the most valuable, and ultimately fascinating, keys to
knowledge. Once you have obtained it, you will find that it opens incredibly many doors in
both science and philosophy. For so high a return, the investment will prove relatively
slight.
Sets, Relations, Functions
The most fundamental concept of classical mathematics is that of a set. The term set is
roughly synonymous with those of class and collection. It refers to any arbitrary grouping
of entities of any kind. Conventionally, sets are abstractly symbolized by upper-case Latin
letters, e.g. A, B and C. When a set is finite and sufficiently small, it may also be displayed by
enclosing the names of the objects which it comprises in braces. Thus, we may write: A
= {1, 2, 3}, meaning that A is to symbolize the set consisting of the numbers 1,2 and 3. The
objects which are the members of a given set are called its elements. The relation of "be-
longing to" or "being a member of a given set which holds between a set and any one of its
elements is symbolized by the lower-case Greek epsilon (ε). Thus, the following expression
is obviously true: 3ε{1, 2, 3}. There is a close connection between being an element or
member of a given set and having a property of a given kind. For example, to have the
property yellow is to be an element of the set of all yellow objects. Thus, when
the necessary and sufficient condition for membership in the set A is the possession
of the property P, we may symbolize the set A by the expression {x|Px}, which simply
means the set of all objects x having the property P. Finally, it should be noted that a set is
not an ordered collection. That is to say, that the order in which the elements are taken has
;
2 Relativity: The Theory and its Philosophy
no effect on the set as such. Hence, {1,2,3} = {3,1,2}. By the same token, the repetition of
one or more elements has no significance. Thus, {1, 2, 3} = {1, 2, 3, 2}.
The most important relation which may hold between two sets is the subset relation.
Abstractly, A is a subset of B if and only if every element of A is an element of B. This
relation is symbolized by A ^ B. Employing standard logical notation, we may write the
defining expression for such a relation as
A Ç B <_► (x) (χεΑ -► χεΒ)
Thus, we see that {1, 2} ç {1, 2, 3}. Similarly, the set of ants is a subset of the set of insects,
which is in turn a subset of the set of animals. The last example reveals an important
property of the subset relation—namely, that it is transitive. In general, if A ç B and
ßcC, then A ^C. The transitivity of the subset relation is sufficient to distinguish it from
the membership relation, which lacks this property. For example, Plato is an element of
the set of philosophers. The set of philosophers is an element of the set of learned
professions. However, Plato is clearly not an element of the set of learned professions.
That is simply to say that Plato is not a learned profession. Thus, 1ε{ 1, 2, 3} but it is not the
case that 1 ç {1,2, 3}, although it is the case that {1} ^ {1, 2, 3}. The foregoing reveals the
necessity for drawing a sharp distinction between 1 and {1J. The former is the number one
while the latter is the set whose sole member is the number one. Clearly, such a set is not a
number. Sets with but one member are called unit sets or singletons. Finally, it is to be
noted that every set is a subset of itself, e.g. {1, 2, 3} £ {1, 2, 3}. That is to say, that the
subset relation is not asymmetric. However, we may define the asymmetric relation of
proper subset symbolized by c. A c B if and only if A ç B and A Φ B.
There are various operations which may be performed on sets. The reader will be
familiar with the notion of operation in terms of the well-known operations of arithmetic
such as addition and multiplication. Intuitively, these are operations on pairs of numbers
such that the result of a given operation is another number. Similarly, operations on sets
are operations performed on pairs of sets such that the result will be another set. The most
elementary operation on sets simply consists in combining all of the elements of two sets to
form a third set. Such an operation is called union and is symbolized by u. The defining
expression for u is (x) (x ε A u B «-► x ε A V χ ε B). The intersection of A and B,
symbolized A n B consists in forming the set consisting of all elements which belong to
both A and B. (x) (χεΑ ηΒ^χεΑΛχεΒ). The relative difference of two sets, which is
obtained by forming the set whose elements are all and only those elements which belong
to the first set but not the second, is defined by (x)(xε A — B ^χεΑ A x φΒ)
Examples:
{1,2,3} u{3,4,5} = {1,2,3,4,5}
{1,2,3} n{3,4,5} = {3}
{1,2,3} ^{3,4,5} = {1,2}
In order that these operations be defined for all arbitrary sets, it is necessary to introduce a
distinguished set which is called the empty set. This is literally the set which has no
members. Since a set is determined or defined simply in terms of its members, it follows
that the empty set is unique. If two sets are empty, then they have precisely the same
membership and are to be regarded as identical. While there may be philosophical
objections to identifying the set of unicorns with the set of ten-mile-high buildings, such
a treatment is adequate to mathematical contexts. The empty set is conventionally
Mathematical Preliminaries 3
symbolized by 0. Thus, {1, 2} n {3,4} = {1, 2} n {Plato} = 0. Sets whose intersection
is 0 are said to be disjoint.
So far we have discussed the very general notion of set. While most of our examples have
referred to sets of numbers, this was simply for notational convenience. A set may consist
of any sorts of objects whatsoever, including other sets. We now turn to the topic of
relations, which will be treated as sets consisting of objects or elements of a more restricted
kind. Specifically, we define a relation as a set of «-tuples (sometimes spelled "entuple"). An
«-tuple is a sequence of n objects taken in a fixed order. Thus, we may speak of the first
element, the second element and more generally of the ith element of the «-tuple, where i
= 1, 2, . . . , n. An «-tuple may be displayed in a manner similar to that in which one
displays a set. However, the convention in this case is to enclose the elements (or
coordinates) in pointed brackets. For example, < 1, 2, 3 > is the triple whose first element is
one, whose second element is two, and so forth. Since an «-tuple is ordered, the order in
which the elements are taken is, of course, significant. This is the principal difference
between a set and an «-tuple. Thus, while {1, 2, 3} = {1, 3, 2}, < 1, 2, 3 > φ < 1, 3, 2 >. We
shall adopt the convention of using a lower-case indexed letter to symbolize an «-tuple.
Thus, a is the «-tuple, where i ranges over all the positive integers from 1 to «. For example,
{
if a = < 1, 2, 3 >, then a = 1, a = 2, a — 3. On the other hand, if/), = < 1, 3, 2 >, then b
x x 2 3 x
= 1, b = 3, b = 2. In general a = b only if a = b a = b , . . . , a = b . It is obvious
2 3 x x x u 2 2 n n
that, unlike the case of a set, the repetition of elements is of significance. Thus,
< 1, 2, 3 > Φ < 1, 2, 3, 2 >. It should perhaps be added that in a more elegant mathematical
treatment of the subject, the notion of «-tuple is definable in terms of set and element-
hood.
We are particularly interested in the notion of a binary relation, which will be treated as a
set of couples. For example, suppose we have a set A = {1, 2, 3}. Then we may define on A
a relation of 'less than' or " < " as follows:
<={<1,2>,<1,3>,<2,3>}.
Conventionally, relations are abstractly symbolized by upper-case R with a numerical
subscript, e.g. R R , etc. The use of the upper-case is wholly appropriate since, on this
l9 2
interpretation, a relation is merely a special type of set. For example, 'husband of is the set
of all couples such that the first element or coordinate of each couple is the husband of the
corresponding second element. It is interesting to note that just as the notion of set was
seen to correspond to the logical notion of property or monadic predicate, so the notion of
binary relation corresponds in precisely the same way to that of dyadic predicate. Thus, we
may represent a binary relation—for example, 'brother of—by the formal expression:
{ <x, y > | Bxy}. That is the set of all couples, <x, y >, such that x is the brother of y.
In so far as a binary relation is a bonafide set, all of the operations which are defined for
sets in general are applicable to binary relations. Thus, we may speak of the intersection,
union, etc., of two or more relations. However, in what follows we shall be more interested
in those operations which are defined specifically for binary relations. Firstly, there is the
operation of forming the converse of a relation. The converse of a relation R is denoted by
R. It may be defined as follows:
(x,y}eR <-> <y, x}eR.
In effect, then, the converse o(R is formed by reversing the order of elements in each of the
couples belongmg to R. Thus, the converse of "less than" ( < ) is "greater than" ( > ).