Richard Swinburne - Space and Time (1981)
Richard Swinburne - Space and Time (1981)
Richard Swinburne - Space and Time (1981)
Richard Swinburne
Second Edition
:0 Richard Swinburne 1968, 1981
Published by
tHE MACMILLAN PRESS LTD
London and Basingstoke
Companies and representatIVes
throughout the world
Swinburne, Richard
Space and time - 2nd ed.
I. Space and ttme
l. Title
114 BN632
ISBN 0-333-29072-0
8765432
02 01 00 99 98 97 96
Contents
Preface to the Second Edition vii
Preface to the First Edition viii
btroduction: N.:cessity and Contingency
Place and Matter 10
2 Spaces 26
3 Absolute Space 42
4 Distance and Direction -(i) Primary Tests 61
5 Distance and Direction -(ii) Secondary Tests 84
6 The Geometry of Space 98
7 The Dimensions of Space 114
8 Past and Future 131
9 Logical Limits to Spatio-Temporal Knowledge 144
10 Times and the Topology of Time 165
II Time Measurement and Absolute Time 177
12 Physical Limits to Knowledge of the Universe-
(i) Horizons 206
13 Physical Limits to Knowledge of the Universe-
(ii) Past/Future Sign Asymmetry 215
14 The Size and Geometry of the Universe 233
15 The Beginning and End of the Universe 247
Index 263
Preface to the Second Edition
The first edition of Space and Time was published in 1968. It was
intended both as a guide for students to the scientific and philosophical
literature of the subject and as a vehicle of my own philosophical views.
In the twelve years which have now passed since I finished writing the
first edition, there have been very considerable advances in astronomy,
physical cosmology, and particle physics, as well as a great number of
philosophical books and articles on space and time. As I find myself still
in agreement with most of the detailed arguments of the first
edition, I thought it worthwhile to produce a new edition, revised to take
account of these scientific and philosophical developments.
On one crucial philosophical issue, however, I have come to hold a
very different view from the unargued view implicit in the first edition. In
1968 I was a verificationist; I assumed that if no evidence of observation
could in any way count for or against some statement, then necessarily
that statement was empty of factual meaning. I now think that this is not
so; a statement may make a meaningful factual claim even if evidence of
observation can give no grounds for believing it to be true or for
believing it to be false. In the new Introduction I give bnef reasons for
my change of view. It has necessitated a redescription of a few of my
results at vanous places in the book.
I am most grateful to Mrs Yvonne Quirke for her patient typing of
sections of the second edition. '
Preface to the First Edition
I am most grateful to all who read or heard earlier versions of various
chapters and produced helpful criticisms of them, and especially to
Dr Elizabeth Harte, Dr Mary Hesse, Mr J. L. Mackie, Mr Roger
Montague, Professor Alan White, and the late Mr Edward Whitley.
They saved me from many silly mistakes and provided me with useful
distinctions and arguments. I am also most grateful to my wife for
.reading the proofs. .
Some of the material has previously appeared in articles in journals,
although some of the conclusions which I reach in this book are
substantially different from conclusions reached in the articles. I am
grateful to the editors of the following journals for permission to use
material from the articles cited: Analysis ('Times', 1965, 25, 185-91,
'Conditions for Bitemporality', 1965, 26, 47-50, 'Knowledge of Past
and Future', 1966, 26, 166-72); Philosophical Quarterly (' Affecting the
Past', 1966, 16, 341-47); Philosophy of Science ('Cosmological
Horizons', 1966, 33, 210-14); and Proceedings of The Aristotelian
Society (,The Beginning of the Universe', Supplementary Volume, 1966,
40, 124- 38). In the references to these articles and to articles throughout
the book, the first number indicates the year of publication of the
journal, the second number the volume of the journal, and the third
number gives the pages ofthejournal in which the article is to be found.
I have placed at the end of each chapter a bibliography of the most
important works on the main topics of the chapter, from which the
argument is developed. The bibliography also includes elementary
works expounding the scientific theories or mathematical systems
discussed in the chapter. Numbers in square brackets refer to the
bibliography at the end of the chapter. Works referred to in the text
which are important only for minor points in my argument are cited in
footnotes but not included in the bibliography.
Introduction:
Necessity and Contingency
This book has two aims. The first and major aim is to analyse the
meaning of various claims about Space and Time, to show which
properties Space and Time must have as a matter of logical necessity,
and to show what are the logically necessary limits to our knowledge of
events in Space and Time. It thus considers such questions as whether
Space must of logical necessity hav~ three dImensions, what it means to
say that an object is at rest, or whether it is logically possible that we
should know as much about the future as we do about the past. The
second and subsidiary aim is to describe, where a definite conclusion can
be reached, the most general contlOgent properties of the Space and
Time with which we are familiar, and the physical limits to our
knowledge of events in Space and Time; or, when there is still scientific
dispute, to set forward the rival scientific theories on the subject. It is
thus for example concerned to describe theories about the age of
Universe, and the limits to our knowledge of the Universe caused by the
finite velocity of light.
The plan of the book is roughly as follows. In Chapters 1-7, I analyse
the meaning of spatial terms and the status of propositions about Space,
leaving unanalysed the meaning of temporal terms. In Chapters 8-11, I
then conduct the analysis of temporal terms and the propositions about
Time. In Chapters 12-15, I examine what are the physical limits to our
knowledge of the Universe and what sort of conclusions we can reach
about its general spatio-temporal character.
Considerations of what would be a proper length for the book
uDfortunately preclude discussing at any great length problems of space
and time on the very small scale, such as whether distances and temporal
intervals are divisible without limit. Such problems are much fewer and
to my mind of much less philosophical interest than problems of space
and time on the very large scale, and adequate discussion of the former
would necessitate full treatment of Quantum Theory, just as adequate
discussion of large-scale problems necessitates full treatment of the
2 Space and Time
Theory of Relativity. For the latter but not for the former I have found
room.
For the benefit of any unfamiliar with the terms I must give a brief and
necessarily very superficial exposition of the distinction between the
logically impossible, the logically contingent and the logically necessary.
To the initiates I apologise.
The distinction between the logically contingent and the logically
necessary is an ancient one but it was first formalised explicitly by
Kant [I] as his distinction between the synthetic and the analytic. Kant's
formulation of this distinction suffers from various deficiencies and I
will give a more modern one.
All statements are either logically contingent (synthetic or factual),
logically necessary (analytic), or logically impossible. Any statement
which is either logically necessary or logically contingent is logically
possible. A logically impossible statement is one which entails a
contradiction. For example, 'There are squares which do not have four
sides' is logically impossible; for we mean by 'square' 'figure with four
equal sides and for equal angles', and so the statement entails the claim
that there are figures which both do and do not have four sides. An
analytic or logically necessary statement is one whose negation is
logically impossible. The negation of a statement is the statement which
says that things are not as the first statement said. Thus the negation of
There are men on the moon' is There are no men on the moon'. Hence
'All squares have four sides' is a logically necessary statement. So too is
'All bachelors are unmarried', 'Acceleration is the rate at which velocity
increases', '2+2 =4', and 'If there is a God, he is omnipotent'. We
mean by 'God' a being who is omnipotent, omniscient, perfectly good
etc. Hence 'it is not the case that if there is a God, he is omnipotent' is
logically impossible, for it entails the claim that 'It is not the case that if
there is an omnipotent being, there is an omnipotent being'. Logically
impossible statements must be false, whatever the world is like; logically
necessary statements must be true, whatever the world is like.
A logically contingent or factual or synthetic statement is one which
does not entail a contradiction, and whose negation does not entail a
contradiction. Among synthetic statements are 'Kangaroos live wild
only in Australia', 'Julius Caesar crossed the Rubicon in 49 D.C.', and
'Silver dissolves in nitric acid'. There is no contradiction involved in any
of these statements, nor in their negations. It may be false to claim that
kangaroos live wild in parts of the world other than Australia, but I do
not contradict myself in claiming this. Synthetic statements may be true
or false; which they are depends on how the world is. Philosophers
Introduction: Necessity and Contingency 3
distinguish between sentences on the one hand and statements or
propositions on the other. A sentence is a grammatically weJl-formed
group of words. A statement is what is expressed by a sentence when the
utterer of the sentence thereby asserts that something is the case (as
opposed to, for example, issuing a command or asking a question).
'Socrates is a man' expresses the same statement as any sentence which
has the same meaning as it. Thus it expresses the same statement as the
Latin 'Socrates est homo' or the French 'Socrate est un homme'.
Different sentences may thus express the same statement. The same
words may, however, in the course of time acquire different meanings
and so the same sentence may express different statements. 'Judge Smith
is an indifferent judge' used to mean that Judge Smith is an impartial
judge, but now it means that he is ajudge who is not concerned about the
issue of this case.
Given their current meanings, the words in a sentence determine
which statement that sentence currently makes. If the words of a.
sentence with their current meanings are alone sufficient to ensure that
that sentence makes a true statement, then that statement will be
analytic. That the sentence 'all bachelors are unmarried' makes a true
statement is guaranteed by the meanings of the words which it contains.
If the words of a sentence with their current meanings are not sufficient
to guarantee either the truth or falsity of the statement which it makes
that statement will be synthetic. My inquiry is into the nature of
statements about Space and Time as made by sentences uttered during
the period of history in which I am writing by normal users of language
or, in the case of sentences using technical scientific terms, by normal
users of the scientific language.
We do not need to observe the world to find out whether or not a
statement is analytically true. We need only reflect on the concepts
denoted by the terms in the sentence making the statement; on the
meaning of those -terms. We can ascertain that the statements are
analytically true independently of other experience of the world, that is
a priori. Statements, the truth or falsity of which we can ascertain in this
way, are termed a priori statements. It would seem also natural to
suppose that we must use our experience of the world to find out whether
or not synthetic statements are true. Only a study of animal ecology will
tell us whether or not kangaroos live wild only in Australia; only a study
of documents and monuments will tell us in which year Caesar crossed
, the Rubicon. Statements, the truth or falsity of which we can find out
only by experience, are termed a posteriori. So it is natural to maintain
that while all analytic statements are a priori, all synthetic statements are
4 Space and Time
a posteriori. Several philosophers, and c1assicaJly Kant, have, however,
claimed that there are synthetic a priori statements - that is, statements
informative about the world, the truth of which can be ascertained
independently of experience of the world. Kant claimed that the
statements of geometry and 'every event has a cause' were such
statements. Most philosophers of the British empirical tradition have,
however, denied that there are any synthetic a priori statements.
Following the hints of Hume, 1 they have claimed that the distinction.
between the synthetic and the analytic coincides with that between the
a posteriori and the a priori. There is not space to argue this issue, and so
I will take for granted this coincidence, as seems initially reasonable.
When discussing the views of Kant or others in his tradition on some
statement which they claim to be a synthetic a priori truth, having stated
their claim, I shall subseq uently represent the claim as the claim that the
statement was analytic or 10gical1y necessary.
A number of recent writers, and most famously Quine [2], have
denied the applicability to our discourse of this distinction between the
synthetic and the analytic. One reason which they have had for denying
this is that in their view there is no clear criterion for when sentences do or
do not have the same meaning and so for whether they make the same
statement. A similar reason isihat any precise definition of 'analytic' and
'synthetic' brings in notions like 'containing' or 'entailing' a 'con-
tradiction', and that in their view there is no clear criterion for when one
statement 'entails' another, or when we have a 'contradiction'. The
Quinean attack however seems to me to fail because all these words
either have an established use in ordinary language or can be defined by
words which do have such a use. We often say of two sentences that they
have the same meaning or that they say the same. If we could not
distinguish whether two sentences did or did not have the same meaning,
we could never translate from one language to another; and, despite the
difficulties of translating some sentences into some languages, we often
make perfect translations. True, whether or not two sentences mean the
same may sometimes depend on the context in which these sentences are
uttered, but that does not affect the main point. Likewise we can give
clear examples of statements which 'entail' other statements or 'con-
tradict' other statements, and these words can be defined by such
expressions as 'denies what the first statement says' or 'would contradict
yourself if you said this and denied that', which have established uses in
'David Hume, Enquiry Concerning Human Understanding (1st edn 1748) ed.
L. A. Selby-Bigge; 2nd edn (Oxford, 1902) p. 25.
Introduction: Necessity and Contingency 5
ordinary language. There is unfortunately no space in this book for a full
discussion of Quine's attack on the distinction, and the book must
assume that the Quinean attack can be dealt with adequately on the
above lines. (For further discussion see [3] and [4].)
The examples which we have taken so far of analytic and logically
impossible statements are ones where it is easy to see the logical status of
the statements. In this book we shall be concerned with statements
whose logical status is in no way obvious-e.g. 'no agent can bring
about a past state of affairs' or 'there is only o,ne Space'. How are we to
show whether such a statement is analytic, synthetic or logically
impossible? By inquiring whether either the statement which the
sentence makes or its negation entails a contradiction. You can show that
a statement entails a contradiction by deducing that contradiction from
it. The only way to show that a statement does not entail a contradiction
is to show that it follows from some other statement which does not
entail a contradiction. Plenty of ordinary statements are to all ap-
pearances free of contradiction, and must be taken to be so in the absence
of proof to the contrary. Thus 'Today is Friday', 'This wall is green', 'All
swans are white', 'All students wear jeal1S' etc. are in this category. If you
can show that a doubtful statement q follows from a statement or
conjunction of statements p, which are to all appearances free of
contradiction, you have done all that can be done towards showing that q
is free of contradiction, i.e. describes a logically possible state of affairs.
In other words, to show that a statement of doubtful standing (e.g.
'There is more than one space') is free of contradiction you have to
describe in statements apparently free of contradiction a world in
which that statement is true. It is a matter of showing such a world to be
conceivable, by describing in detail what it would be like for it to be true
in intelligible statements. Thus initially There is a man who has two
bodies' looks as if it might be logically impossible. But we can go. on to
describe how it could be true- one subject of experience might be able
to control two sets of limbs directly in the way in which we control one
set of limbs directly; and acquire knowledge of the world as a result of
stimuli impinging on two sets of sense organs. Developing the picture we
show in statements apparently free of contradiction what it would ~like
for the doubtful statement to be true, and thus show it to be logically
possible. In this book I shall spell out what it would be like for there to be
two spaces, and for physical geometry to be non-Euclidean, and thereby
show these to be logical possibilities.
You can show that a statement q entails a contradiction, by deducing
that contradiction from it. But to do that, you need to know what a
6 Space and Time
statement entails. It is not always obvious what a statement entails. You
can show that a statement q entails. a statement p, by showing that the
only worlds in which q holds of which there seems to be an intelligible
description are also worlds in which p holds. Of course the fact that a
given individual cannot describe a world in which 'q but not p' holds in
ways which he finds intelligible does not conclusively prove that q entails
p; but the fact that no one can give such a description is evidence (and the
only evidence there can be) of what the speakers of the language are
committed to when they utter the sentence which makes the statement
that q. Thus to show that 'There is a surface which is red and green all
over' is logically impossible, you need to show that 'This is a surface
which is red all over' entails 'This is not green all over'. The only way to
show this entailment is to show that any attempt to describe a surface
which is red and green all over ends up with an unintelligible description.
In this book I attempt to show, for example, that 'There is a four-
dimensional world' is logica1ly impossible by showing that any attempt
. to describe such a world ends up unintelligibly.
Many writers in the tradition oflogical positivism have claimed that a
statement can only be a synthetic or factual statement jf there could
(logically) be some observable evidence which, if observed, could count
for or.against it. This doctrine is often called verificationism. 'Once upon
a time the Earth was red-hot' is, on this doctrine a synthetic statement
because it could be confirmed or disconfirmed by geological evidence
now available, or indeed because it could have been observed to be true at
an earlier time by rational beings situated on other planets. By contrast,
positivists claimed, a non-analytic statement such as 'Reality is one'
which for logical reasons could not be confirmed or disconfirmed by
observation was really logically impossible or in their terminology
'meaningless'. However I know of no good argument for accepting
verificationism. A sentence may make a perfectly meaningful
statement - the words which occur in it may be meaningful and be put
together in accord with normal grammatical rules; and there may be no
contradiction involved in the statement or its negation - and yet there
may be reason of logic why the statement is nQt in any way confirmable
or disconfirmable by observation. An apparent example is 'There is an
uninhabited planet on which a tree dances once a year without there
being any evidence of its activity'. (Under evidence I include both
physical traces left behind by this tree, and evidence of trees observed to
dance in similar circumstances elsewhere.) Clearly there can be no
evidence of observation that an event of a certain kind occurs in
circumstances where there is no evidence of its occurrence. Nor can there
Introduction: Necessity and Contingency 7
be evidence of observation against it; for it is very unlikely a priori, and
there is nothing particular which, if observed, would make it any less
likely. So the cited statement cannot in any way be confirmed or
disconfirmed by observation, and yet there seems no reason apart from
verificationist dogma for believing that it is not a meaningful synthetic
statement (although of course not one which we have any reason for
believing to be true). So mere unconfirmability by itself does not show
lack of factual meaning.
However the converse does appear to hold. If there can be evidence of
observation which confirms some statement, that statement must make a
meaningful factual claim; for if it contained a contradiction nothing
could add to the likelihood of its truth. There are certain normal
inductive standards according to which evidence would confirm a
statement, if that statement is not logically impossible- e.g. the
statement is confirm~d if it successfully predicts the evidence and is
simple (in the way which I spell out on pp. 43tT). If there could be such
evidence which apparently confirms a statement then (in the absence of
positive grounds for supposing the statement to contain a contradiction)
that suggests that it really does confirm the statement and so that the
statement is a meaningful factual statement. I shall therefore sometimes
argue that where some claim about space and time (e.g. the existence of
Absolute Space, which I discuss in Chapter 3) could apparently be
confirmed by evidence of observation, that provides reason for believing
it to be a meaningful factual statement.
As well as distinguishing between the logically necessary, possible, and
impossible, I need a further distinction between the physically necessary,
possible and impossible; and the merely practically necessary, possible
and impossible. The physically necessary is whatever is necessitated by
the laws of nature. The physically impossible is whatever is ruled out by
them; the physically possible is whatever is permitted by them. Thus
(given the truth of the current physical theory of matter) it is physically
necessary that the quantity of matter-and-energy be conserved in any
physical interaction. It is physically impossible (given the truth of
modern Quantum Theory) for an observer to determine the position (x)
and momentum (Px) of any particle more accurately than within the
ranges ru:, fl.px, where fl.x.fl.Px ~~, h being Planck's constant. Or, to take
411:
. an example which we shall have to consider later in much detail, it is
physically impossible to travel between two points faster than light
(c. 300,000 km per sec). It is on the other hand physically possible for an
observer to measure the position and momenta of all particles within a
8 Space and Time
room within the ranges ~x, ~Px ( ~x. ~Px ~ 4:). or to travel at more
than 200,000 km per sec over some surface.
What is physically impossible is what is ruled out by the permissible
ways in which physical objects change their positions and characteristics,
or by the permissible ways in which, from_whatever arrangement they
start, they can be arranged: What is practically impossible is what is ruled
out not merely by this but by the actual arrangements of the constituents
of the Universe at the time at issue. It is what cannot he done by a man
given the way in which the constituents of the Universe behave and the
way they are at present arranged. Thus, provided with a suitable
calculating device or the material, labour, and plans for constructing one,
I could find out within the above-mentioned range the positions and
momenta of all the particles in this room at some given temporal instant.
Provided with a suitable space rocket or the material, labour, and plans
for constructing one, I could travel across the Earth at 200,000 km per
sec. I have not got these devices nor the material, labour, and plans for
constructing them and so the doing of the aforementioned things is not
practically possible-at any rate, for the immediate future. What is not
practically possible now may become practically possible if, by human
effort or otherwise, the constituents of the Universe become differently
arranged. If, given their present arrangement and the laws of nature
governing their change of arrangement, they could never adopt a future
arrangement permitting the doing of some thing, the doing of that thing
will never be practically possible. Yet if, whatever the present arrange-
ment of the constituents of the Universe, a given effect could never be
produced, the producing of it is physically impossible. What a human
agent can now effect given the present arrangement of the constituents of
the Universe is now practically possible. What is practically necessary is
what, if anything, a human agent in these circumstances must effect.
These distinctions I have made only loosely, as my work is not devoted
primarily to making such distinctions but to using them to clarify the
characteristics of Space and Time. But I hope that I have made them with
sufficient clarity for my use of them to provide subsequent illumination.
BIBLIOGRAPHY
I Two- or one-dimensional entities like patches of light or lines are not normally
said to occupy places but to have positions. Such uses of 'place' as 'the place of
an -argument in the thesis' or 'the place of religion in the home' are clearly
metaphorical.
2 Aristotle ([ I], Chapter 2) regarded the enclosing surface as defining, the
primary place of a material object.
Place and Matter II
Alternatively where an object is may be described by identifying
smaller places or points enclosed within the object. My field may be said
to lie at the intersection of the 55th parallel oflatitude north and the first
meridian oflongitude west. What this means is that the place of my field
is a region of unspecified volume surrounding the point formed by the
intersection of those two lines. Both identifying a place which contains
among other distinct places the primary place, and identifying a smaller
place or point enclosed by the object, give less precise descriptions of
where an object is than the most precise description which can be given.
A place is identified by describing its spatial relations to material
objects forming a frame of reference, and which are for this purpose
regarded as fixed. Any set of material objects which over a period of time
retain the same spatial relations among themselves may form a frame of
reference and will form such a frame if they are used for locating places
and objects. Other material objects which over that period of time retain
the same spatial relations with certain material objects forming a frame
of reference form the same frame as they do.
An object (or place) is spatially related to another object (or place) if it
is in some direction at ~ome distance from it. The spatial relations
between objects (and places) are their distance and - given theapplica-
bility of the concept (see Chapter 4) - direction from each other. Two
objects retain the same spatial relations if they remain at the same
distance in the same direction from each other (it is logically necessary
that whatever spatial relations two places defined by reference to the
same frame have to each other at one temporal instant they have at all
instants). Thus a group of ships at sea keeping station may form a frame
of reference since they retain the same spatial relations to each other.
Any rigid bod'y may form a frame of reference since it ~onsists of parts
which are material objects, the spatial relations of which to each other
do not change. Thus the Earth or a ship can form a frame of reference.
One may say where a ship is by saying on what part of the Earth it is
situated, or where a wreck is by saying how far it is in what direction
from our ship. To say that an object is In the same place as, or is in a
different place from, what it was before is to say that its spatial relations
to the objects of the frame have not, or have, altered. To say that an
object has remained still over a period is usually to say that its parts have
not changed their places over that period. The parts change their places
either if the object as a whole changes its place or if they merely
interchange their places (in the case of a rigid body, if the parts
interchange their places, without the rigid body changing its place, the
body rotates).
Place and Matter 13
The standard examples of material objects are the solid things which
resist our pressure and which we can lift up and push around. Rocks,
tables and chairs, people and houses, clothes, cars and plates are all
standard examples of material objects. Anything which is composed of
such things, in the sense that its spatial parts are such things, is itself also
a material object. Thus the Earth and the other planets are said to be
material objects because they are composed of things like lumps of rock.
Likewise anything of which the standard examples of material objects
are composed, in the sense that it is one of the spatial parts of such an
object, is itself also a material object. Molecules of which solids are
composed are material objects because tables and chairs, etc. are formed
from molecules joined together. Further, anything which by occupying a
place thereby as a matter of logical necessity excludes other material
objects from that place is itself a material object. What is meant by this is
not that other material objects calTDot get into the place, but that they
cannot get into the place without at the same time excluding the other
object from it. Anthony Quinton has caned this property of material
objects the property of logical impenetrability, and has neatly dis-
tinguished it from hardness. He writes: 'Nothing can be where the
impenetrable is; less hard things cannot get into the place where the hard
is' ([5], p. 341). The impenetrability is said to be logical because if
anything did occupy the same place at the same instant of time as a
material object we would for that reason refuse it the title of material
object. A magnetic field may occupy the same place at the same instant
as a chair, a sound may Occupy the same place at the same instant as a
volume of air - but this means that magnetic fields and sounds cannot
be material objects. It is in virtue of this criterion of impenetrability that
the molecules which compose a gas or a liquid are said to be material
objects. If they occupy a place, no other material object can do so at the
same time. If there is a volume of air in a cylinder, you cannot keep a
solid iron piston there at the same time.
We have thus started with standard cases of material objects and
extended the scope of the term by means of the notions of composition
and logical impenetrability, thereby funy delineating the class of things
commonly called material objects. In so doing we have taken for granted
the often urged claim that the following proposition (A) is a logically
necessary truth:
(A) No two material objects can be in the same place at the same
temporal instant.
14 Space and Time
This claim and the remarks of the last page must however be qualified
in an important way. We are to understand by 'place' primary place. My
toothbrush and the shaving soap can both be in the bathroom at the
same instant of time, but the toothbrush cannot have the shaving-soap
case as its primary place if the shaving soap does. Clearly too a material
object can be in the primary place of another material object. A speck of
dust can be in the shaving-soap case as well as the shaving soap. But the
speck of dust does not have the shaving-soap case as its primary place.
The primary place of the speck of dust is a small region within the
shaving-soap case.
But could there not be two material objects which completely
interpenetrated each other and had the same outer surface? No. Because
if two things interpenetrated each other so thoroughly that they could
not be separated, they would not be tenned material objects. If you mix
some strawberries and a lump of cream, all material objects, to make
strawberry ice-cream, neither the strawberries nor the lump of cream
exist as such any longer. The small particles of strawberry and the small
particles of cream are material objects - but of them it is true that they
do not have the same primary place as other small particles. If two
objects can be separated from each other, then they will have different
enclosing surfaces. Anyone who marks off where they are will mark off
different regions of Space. I conclude that the following amended
version of proposition (A) is a logically necessary truth.
(A') No two material objects can have the same primary place at the
same temporal instant.
(8) No spatial thing can be in.two different places at the same temporal
instant.
(8') No spatial thing can have two different primary places at the same
temporal instant.
BIBLIOGRAPHY
Suppose that on going to bed at home and falling asleep you found
yourself to all appearances waking up in a hut raised on poles at the
edge of a lake. A dusky woman, whom you realise to be your wife, tells
you to go out and catch some fish. The aream continues with the
apparent length of an ordinary human day, replete with an appropri-
ate and causally coherent variety of tropical incident. At last you
climb up the rope ladder to your hut and fall asleep. At once you find
yourself awaking at home to the world of normal responsibilities and
expectations. The next night life by the side of the tropical lake
continues in a coherent and natural way from the point at which it left
off. And so it goes on. Injuries given in England leave scars in
England, insults given at the lakeside complicate lakeside personal
relations .... Now if this whole state of affairs came about it would
not be very unreasonable to say that we lived in two worlds ([2], pp.
141f).
In this myth, Quinton argues, our life in the one set of surroundings is
just as coherent as that in the other, and so there are no grounds for
saying that one of the lives is only a dream. There are other people by the
lake and in our ordinary surroundings, and to anyone who urges that
one group of people does not really exist, that we only dream about
them, the answer must be made that whatever grounds there are for
saying that, there are the same grounds for saying it about the other
people. There are, however, on Quinton's myth, no grounds for saying
that the objects of one set of surroundings are spatially related to those
of the other, for however much exploration we do in one life, we can
never find the objects of the other.
Now is Quinton's myth coherent? Does it describe, without con-
tradiction, a world in which there are two spaces, and a world in which
an inhabitant would have grounds for believing that there are two
spaces. I shall argue that, with certain amendments to the myth, Quinton
is successful in showing what he wishes to show. But, first, I must deal
with four objections to Quinton's argument. Some are objections to the
claim that Quinton has given a coherent description of a world of two
spaces, and some are objections to the claim that an inhabitant would
have grounds to believe that he lived in a two-space world.
The first objection is that there is an inconsistency in Quinton's
account as stated in that while a man is in England, living the life of the
30 Space and Time
English day, his body, according to Quinton, lies asleep by the
lakeside! -and presumably, conversely. The man is only awake in one
place at one temporal instant, but asleep and awake, he is to be found in
two places at one temporal instant. And this violates the principle which
we have noted (p. 18) to be a logically necessary truth, that no spatial
thing can have two different primary .places at the same temporal
instant. We might attempt to avoid this difficulty by urging that the true
man is the man's mind or soul and that that was only in one place at one
time, and that the man's mind was connected with two different bodies ..
But all this would need a lot of philosophical justification to make
coherent, and there is no need to take this route to meet the objection.
We need only amend the myth so that soon after people who have these
odd experiences go to sleep, they just vanish from their beds, and soon
after they reappear. they wake up. Then it will not be the case that they
are in two places at one temporal instant.
The second objection is that although a man who had the sort of
experiences which Quinton describes would have no grounds for saying
that life in one set of surroundings was real while life in the other set of
surroundings was only a dream, he might have good grounds for
doubting the reality of both lives and for believing that he was subject to
perpetual hallucinations. His experiences might seem so mysterious and
disordered that he reasonably doubted whether either sort of experience·
was properly described as genuine. This might, I think. be the case if only
one man had this sort of experience and all the other people of both his
environments reported to him (or appeared to report to him) that they
did not. For then in each set of surroundings when he told people of his
other life. they would not believe him - since (hey would have no other
evidence than his word for crediting what he said. Hence throughout his
life everybody he met would doubt the truth of half of his reports of
previous events. They would say that he 'only dreamt it'. And each of his
reports would be doubted on half the occasions on which it was made. In
such circumstances the man might reasonably hold that he had no
grounds for believing any of his own reports of past events, that his
whole life, and not merely part of it, was a dream. At that point he would
presumably believe himself to be unable to wake up, and So would
become demented.
The difficulty can be met by supposing that all or most of the people of
the man's two environments reported to him similar experiences of
I [2]. p. 141. 'Your wife says. "You were very restless last night. What were you
dreaming about?'"
Spaces 31
double lives to those which he experienced. One way is to suppose that
'everyone's dream-life was coherent but that no person's dream-life
corresponded with anyone else's' ([2], p. 143). This position formally
meets the objection which I have made, but one might still wonder
whether this description was a sufficiently coherent one. If everybody's
lives were so dilTerent from those of others one might well wonder if
there was sufficient coherence about the world for a public language
applied to it to make sense. However, a coherent situation will certainly
arise 'if we suppose that the dreams of everyone in England reveal a
coherent order of events in our mystical lake district and let everyone
have one and only one correlated lake-dweller whose waking ex-
periences are his dreams' ([2], p. 143). If one supposes in this way that
one meets in the new environment people of the old, and that the·
possibility of travel between the two environments is recognised in each,
this objection would seem to have been met. Quinton describes such a .
situation as a possible but inessential variant on his myth, but the
considerations which I have adduced would seem to suggest that it is a
highly desirable, if not essential, variant, if the myth is to do the job for
which he intends it. In Quinton's variant:
there are various ways in which we can suppose that people who know
each other in England could come to recognise one another at the
lakeside, for example, by drawing self-portraits from memory or by
agreeing, when in England, to meet at some lakeside landmark. The
injury which I do you at the lakeside may be revenged not there but in
England ([2), pp. 141 f).
in the most favourable possible way .... Not only do all Charles's
memory claims that can be checked fit the pattern of Fawkes' life as
known to historians; but others that cannot be checked are plausible,
provide explanations of unexplained facts and so on. Are we to say
that Charles is now Guy Fawkes, that Guy Fawkes has come to life
again in Charles's body, or some such thing? ([5], pp. 237f.).
What this shows is that in the simpler case where only Charles claimed to
Spaces 33
be Guy Fawkes, it would be 'vacuous' ([5], p. 240) to say that Charles
was Guy Fawkes, for there would be no difference between saying that
Charles was Guy Fawkes and saying that he was exactly similar to Guy
Fawkes.
The most general justification which Williams provides of the last
stage of this argument is that 'identity is a one-one relation, and that no
principle can be a criterion of identity for things of type T if it relies on
what is logically a one-many or many-many relation between things of
type T' ( [7], pp. 440. If we say in the simpler case (where only Charles,
but not Robert, claimed to be Guy Fawkes) that Charles is Guy Fawkes,
we are using a criterion which could lead us to contradict ourselves.
Williams demands that a criterion of identity should be such that
rigorous application of it should of logical necessity never lead to self-
contradiction; in particular, consistent application of it should oflogical
necessity never lead us to say A = B and A = C .but C i= B. If we say that
Charles is Guy Fawkes, we could on exactly similar grounds have to say
that Robert is Guy Fawkes, but since Charles is not Robert, we would
have contradicted ourselves. The criterion of bodily continuity,
Williams suggests, does ensure. that only one later person can be the
same person as an earlier person, for only one person can have his body;
whereas very many later persons could satisfy the memory and character
criterion.
In the past twenty years there has been a very <;onsiderable
philosophical literature on the subject of personal identity. Many
writers sympathetic to Williams' general approach have modified it in
one important respect. The continuity which guarantees personal
identity is not, they claim, continuity of the whole body but of that part
of the body which is causally responsible for person's memory and
character - viz. continuity of brain. So if A's brain is taken out of A's
~ody and put into B's body and B's brain is taken out of B's body and
put into A's body, and the two brains are connected up with their new
bodies so that we have fully functioning persons, then thereafter the
person who had B's body would be A, and the person who had A's body
would be B. There would be such continuity of memory and character
that we would let brain continuity dictate who we said was A, rather
than continuity of the rest of the body.
But a major difficulty with Williams' original view and this revised
view is that in fact, as expanded so far, neither of them guarantee the
uniqueness for which Williams sought. This is especially evident in
connection with the revised view that brain-continuity is a necessary
condition of personal identity. The human brain consists of two
34 Space and Time
hemispheres. It is known that if one or other hemisphere is damaged in
an accident, the person may continue to live and shpw much normal
human behaviour. So it is logically possible (and much more likely to
occur than the Guy Fawkes story related by Williams) that A's brain
could be taken out of A's body, and the right hemisphere be placed in
one empty skull, the left hemisphere be placed in another empty skull,
and both half-brains be connected up so that we have two fully
functioning persons. Let us call the new persons Band C. Both Band C
have brain-continuity with A (and so, we may reasonably suppose,
behave like A and make memory claims similar to those which A made).
By the criterion of brain continuity, ,as so far expanded, both would be
A. The duplication which Williams saw as damaging the memory and
character criterion is equally damaging to the brain continuity criterion.
In the face of this difficulty exponents of the latter view have usually
added a clause to the effect that if two later persons both satisfy the
criterion of brain continuity for being the same person as a previous
person. then neither are that person. For example, David Wiggins
suggested that we
That is, to be the same person, you have to have a bodily organ causally
necessary and sufficient for memory and character, i.e. some of the same
brain. and no part of that brain must animate another person.
However. as Wiggins 'appreciate~, there is in a definition of this kind
the following awkwardness. that a man's identity (i.e. whether or not he
is identical with a certain past person) could, as a matter of logical
necessity, depend on the success or failure of an operation to a brain
other than his own; For suppose, as in the previous story, that A's brain
is taken out of A's body and the right hemisphere placed in one empty
skull and the left in another. Suppose that the transplant of the right
hemisphere takes and that we have a fully functioning person, B. If the
other transplant fails, there will be only one part of the original brain
animating a person, and so 011 a Wiggins-type account that person will
be identical with the original person. B will be A. But if the other
transplant takes neither B nor anyone else will be A. But it seems
Spaces 35
logically absurd to suppose that who B is depends (as a matter of logic)
on what happens as the result of an operation in a body other than his
own, or that I can survive an operation only if some transplant fails.
Something has gone badly wrong in this kind of theory of personal
identity, and two different kinds of theory of personal identity have been
developed in recent years which do meet this kind of difficulty. One is a
theory which takes the approach of Williams and Wiggins much further
by stressing the similarity of personal identity to the identity of other
kinds of material objects. For desks and ships, for armies and countries,
our criteria of identity are not rigorous enough to give unique answers in
all cases. We saw in Chapter I that there was no unique true answer as to
which ship was the ship of Theseus. Sometimes the answer that B is the
same object as an original object A is as near to the truth as an answer
that a different object C is A. If we want uniqueness we may need
arbitrary decisions, especially in cases where there is any extensive
transplantation of parts ~ whether they be parts of ships or parts of
persons. A number of writers have stressed this, but one writer in this
tradition who has been much discussed is Derek Parfit [II]. He goes on
to claim that what is important is not so much personal identity, but
what he calls psychological continuity. Very roughly the latter is
similarity of memory and character causally linked. Thus a person P at
time t' is psychologically continuous with Pat t, if P' remembers what P
remembers and P's memory causes (e.g. via continuity of brain matter)
p"s memory. Now two later personscartnot both be identical with P, but
two later persons can both be psychologically continuous with P, and so
P can 'survive' as both, although not be identical with both.
Psychological continuity is a matter of degree, since similarity of
memory (and character) and extent of causal dependence can vary from
total to zero. Personal identity in general involves psychological
continuity (or, perhaps in its absence, bodily continuity) but psychologi-
cal continuity can hold where there is no identity. Two persons at
different times are the same person 'if they are psychologically
continuous and there is no person who is contemporary with either and
psychologically continuous with the other' ([II] p. 13).
On a theory ofpersonaJ identity such as that of Williams and Wiggins
some sort of spatio-temporal continuity relative to a previously
identifiable frame is necessary for personal identity, and hence move-
ment of a person from one sp:ice to another is not possible. On Parfit's
view the more important concept of psychological continuity involves
causality. I do not see any need for him to hold that causality involves
spatio-temporal continuity. But he might well hold this, and if he did
36 Space and Time
again motion between spaces would not be logically possible (since the
identity of persons involves psychological continuity, or perhaps in its
absence, bodily continuity).
It seems to me however that Partil's approach begs crucial factual
questions and is very difficult to make coherent. In our transplant story
look at the matter from A's point of view, before the operation. Suppose
the person in B's body is to be tortured and the person in C's body is to
be rewarded. A wishes to be rewarded and not to be tortured, but has he
cause for hope or fear? That depends whether he will experience the
feelings of the body into which his left brain hemisphere is transplanted,
or the feelings of the body into which his right brain hemisphere is
transplanted, or of neither, or of both. Suppose that both transplants
take. In that case Parfit seems to suggest that the last suggestion is right;
that A will at any rate in some measure have the feelings both of B and of
C. But it is hard, to put it mildly, to make sense of how that can be, since
Band Cwill not experience each other's feelings. Each of the other three
suggestions seems coherent enough, seems to contain no self-
contradiction. In the first case, although both Band C would behave like
A and make somewhat similar memory claims to A, only C would be A.
A's hopes to be rewarded would be fulfilled. In the second case, although
both Band C would behave like A and make somewhat similar memory
claims to A, only B would be A. A's fears about being tortured would be
fulfilled. In the third case despite the similarity of memory and
character, neither B nor C would be A. Each of these seems to be a
factual possibility. Yet how can logic tell us which will happen? In the
circumstances described, surely one and only one suggestion w_ould
provide the true description of the situation, and yet we would not know
which was the true description, nor might we have any idea about how to
go about finding out which was the true description.
All of this suggests a very different theory of personal identity which
along with others I have myself advocated in recent years (e.g. in [12]).
On this view personal identity is something ultimate, not further
analysable. It is not constituted either by similarity of memory and
character or by bodily continuity, although it is a basic inductive
principle (analogous to the much wider criterion of similarity discussed
in the last chapter) that similarity of memory and character and
continuity of body are strong evidence that two persons are the same
and the lack of such similarities and continuity is strong evidence that
two persons are not the same. If P' at time t' does not have a body
continuous with that of Pat t, that is good grounds for supposing that P'
is not the same person as P. It is not however conclusive grounds. They
Spaces- 37
could be the same person, and the very good satisfaction of the other
criterion (similarity of memory and character) would be good grounds
for supposing that they are the same, especially if there was no rival
candidate P" at the same time as P' who satisfied the criterion of
similarity of memory and character equally well. If the criterion is
merely evidence of identity, and not constitutive of it, no logical
difficulties arise. The logical possibility of a Robert turning up does not
make the similarity of Charles to Guy Fawkes any less evidence of his
being Guy Fawkes. If Robert does in fact tum up, we do not know which
of Charles and Robert is Guy Fawkes. They cannot both be, but one of
them may be-despite our ignorance.
This theory of personal identity has the consequence that the survival
conditions for persons are very different from those for other material
objects. This is not surprising because cars and ships do not have feelings
or hopes to survive, and therefore there is nothing puzzling about there
being two later ships, each of which is as much the same ship as the
earlier ship as is the other ship. But this does seem an incoherent
suggestion when we come to talk about persons. On this theory of
personal identity, continuity of body and so spatio-temporal continuity
relative to a previously identifiable frame is not a necessary condition of
personal identity, and so persons may move between spaces. Evidence of
identity of persons in a new space with persons in an old space would be
provided by very strong satisfaction of the criterion of similarity and
memory by one and only one person P' in the new space for each person
P in the old space.
Finally, I consider a fourth objection which may be made to
Quinton'S myth. It claims that as Quinton describes the situation; we do
not have adequate evidence for believing that the two 'places: belong to
two different spaces, even though they may in fact do so. It might be
. granted that both 'places' were real places, and that men passed from the
one to the other discontinuously. But an objector could ask, what real
reason have we for supposing that the second place is not spatially
related to the first? QUinton's answer here is brief. 'Suppose that I am in
a position to institute the most thorough geographical-investigations
and however protractedly and can!fully these are pursued they fail to
reveal anywhere on earth like my lake. But could we not then say that it
must be on some other planet? We could, but it would be gratuitous to
do so. There could well be no positive reason whatever. beyond our
fondness for the Kantian thesis, for saying that the lake is located
somewhere in ordinary physical space, and there are in the circum-
stances envisaged, good reasons for denying its location there' ([2],
38 Space and Time
p. 143). Quinton's 'good reasons' are presumably that we have in the one
environment looked for the place of our other. life and not found it. But
having looked for a thing and not found it is fairly weak reason for
believing it not to exist unless we have looked at most of the places where
it could possibly be. An objector might well urge in such a case -as
Quinton has described'that the place of the other life is just as likely to be
beyond the furthest observed galaxy as to be not spatially related to
ourselves, and hence that it is not reasonable to postulate the odder
alternative. There could be two kinds of further good reason which
might make Quinton'.s alternative the more reasonable one to adopt.
The first kind of good reason would be that the laws of nature
observed to hold in one life were markedly different from those observed
to hold in the other. Thus, suppose the men transported from one
environment to the other include scientists. They observe that in one life
an inverse square law of gravitational attraction holds, while in the other
life an inverse cube law holds, without there being any obvious en-
vironmental differences (e.g. vast density of matter in one life) to which
the difference of law could be attributed. This difference holds in all of
many observed regions in each life. The scientists are now faced with two
alternatives. If they claim that the two environments are both in the
same space, they have to admit a discontinuity of scientific laws, that one
law holds in one environment and in the other environment a different
law holds; at the boundary between the two regions matter becomes
subject to a discontinuous change of behaviour (there being no evidence
within each region of any gradual change of law). To adopt this
hypothesis would thus be to introduce an awkward complexity into
science, and the scientist followi~ the principle of scientific metho-
dology that he ought to adopt the simplest hypothesis consistent with
observations might well prefer the (admittedly provisional) hypothesis
that the two places had no spatial relations to each other. Spatial
discontinuity of law operation is something that the scientist has never
since the seventeenth century allowed into science (on Aristotelian
physics there was just such a discontinuity: one set of laws governed
terrestrial phenomena, and another set celestial phenomena).
The second kind of good reason" for adopting Quinton's alternative
would, if it applied, be much stronger than the first. It might turn out
that the space of at any rate one of the lives was a closed or finite space; 1
that is, contained only a finite number of places of finite volume, so that
1 For more adequate characterisation of a closed space and of the grounds for
claiming that a space was a closed space, see Chapter 6.
Spaces 39
in whatever direction one set off from a given place, one would
eventually arrive back at where one started. Now in that case one could
in principle fully explore all the places in the space of one of the lives and
failure to find among them the places of the other life would be very good
evidence indeed for the former not being spatially related to the latter.
Even if the space was too big to be fully explored in practice, we could
explore some of it and our evidence of failure to find the other
environment would be much better evidence of its non-existence in that
space if the space were a finite one than if the space were an infinite one.
Further, although we might not be able fully to explore all the regions of
the finite space, we might be able to Jearn quite a lot about them by
telescopes and similar devices so as to have quite reasonable evidence
that the other environment was not in that space.
I conclude that we could have good evidence that the two environ-
ments were not spatially related and hence that they belonged to two
different spaces.
Thus the four possible objections to Quinton's original thesis can be
rebutted and the thesis stands. It is not a necessary truth that there is
only one space; and that there is more than one space is something of
which men could have evidence.
In order for a man to pass from one space to another, he must have a
means of travel other than the sort of way by which we normally get to
places, which, to give it a name, I propose to call local motion. In local
motion between two places A and B I can travel by many routes, one of
them being a shortest line. If I travel by local motion on a shortest line
from A to B, then as I move Iget further away from A and nearer to B, in
the sense that a man would have to lay more and more measuring rods to
cover the distance along the line between myself and A and fewer and
fewer to cover the distance along the line between myself and B. All of
this holds if I travel on foot, by bus or by rocket or-mutatis
mutandis- ifit is not I that is travelling but a marble or a bullet. If travel
by local motion is along any other line than a shortest line then distance
from any place on the line to A or B can be measured, and in my travel
from A after I have reached a certain place on the line I get nearer and
nearer to B.
To get to another space we would have to use a means other than local
motion, one lacking the characteristics described above. Quinton's
method of getting there by falling asleep dearly satisfies this demand,
but there are many other methods which would do so. You might only
need to recite the opening words of the Koran and in a flash your
surroundings are entirely new. To get back again you recite the closing
~o Space and Time
words and in a flash you are in the old surroundings. Here again there is
no question of travelling along a line, distance from places along which
to the starting or finishing points could be measured.
Other spaces are not merely a matter of philosophical thought-
experiment. The traditional Christian doctrine of Heaven may most
plausibly be regarded as a doctrine about another space. Christians have
always maintained that in Heaven after the General Resurrection men
will be embodied (though these bodies will lack many of their earthly
limitations). Bodies must be located at places. So Heaven must be a
place. The medievals in general believed that the Heaven of the blessed
was sit~ated just beyond the sphere of the fixed stars. But seventeenth-
century astronomy showed that view to be mistaken. There is no 'sphere'
of fixed stars. The telescopes that pry daily ever deeper into the Universe
have revealed nothing resembling Heaven. Even if the Christian
maintains that Heaven lies far beyond the furthest observed galaxy,
there is still the difficulty that different laws of nature are said to hold in
Heaven, different from those which hold within the observable Universe
(bodies are freed from their 'earthly limitations'), and this, as we have
noted, makes awkward the assertion that both belong to the same space.
If the Christian wishes to maintain the doctrine that Heaven is a place,
he does much better to claim that it is a place not spatially related to
Earth. Such a doctrine is, we have urged in this chapter, a logically
possible one; and indeed, although this is not my thesis here,.one much
more consistent than the medieval view with much traditional Christian
theology.
The claim of this chapter has been that it is not a logically necessary
truth that there is only one space. However, unless we accept the
Christian or some other such metaphysic, we have no reliable infor-
mation about any space except our own, and the remainder of the book
will be concerned with the actual and possible properties of this space.
BIBLIOGRAPHY
1 I have omitted rrom this place the words 'similar and'. I ignore them ror the
I [1]296a -b. For interpretation see the note on p. 242 of the edition in The Loeb
and ed. Ed~ard Rosen), Dover edition (New York, 1959), p. 57.
48 Space and Time
moving or not. Galileo backed up this view by urging that a stone
dropped from the top of the mast of a moving ship has as well as its
downward motion, the ship's motion, which is why - approximately-
it falls to the bottom of the mast. 1 A simple explanation of the latter
phenomenon followed from Galileo's mechanics, whereas you would
have to complicate the old system to account for it. The principle of
Galileo's mechanics used above was generalised by Huygens and
Newton as the principle of mechanical relativity -that the same laws of
mechanics hold in any frame of reference moved with uniform
rectilinear motion relative to a given frame. A frame of reference relative
to which Newton's laws of motion hold is known as an inertial frame.
This is - to a high degree of approximation -:- a frame stationary or in
uniform motion relative to the 'fixed stars'2 or-less approximately-
relative to the Sun, or - to a yet smaller degree of approximation - for
purposes of small-scale mechanical experiments, the Earth (since its
motion is during periods of time a few seconds long approximately
rectilinear), Newton's laws of motion were, it was held, the simplest
which apply to any frames. Consequently if we consider only mechanics
there will be an infinite number offrames relative to which scientific laws
are highly simple, and none relative to which they are yet more simple.
With whatever uniform velocity the ship moves relative to the Earth, the
same laws of mechanics will hold on it. Of two bodies moving uniformly
relatively to each other, mechanics cannot decide which is 'really
moving'.
This showed that the simple conclusions of Copernicus, Kepler, and
Galileo himself were unjustified. One could not conclude that the Sun
was absolutely still, only that it was either absolutely still or in absolute
uniform motion (viz. in uniform motion relative to Absolute Space). Yet
for two bodies moving non-uniformly relative to each other, Huygens
and Newton showed that the laws of mechanics would differ, dependent
on which one took as the frame of reference, and the laws of mechanics
relative to an inertial frame would be simpler than laws relative to a non-
inertial frame (viz. one with non-uniform motion relative to an inertial
~1
be,lieve that they will succeed we should adopt the
theory. If we have no reason either way, we may, if we
~
choose, adopt the theory as a provisional hypothesis,
and such a theory will be significant because capable of
being further tested. Mach's arguments show exactly
the reverse of what he thinks they show,2 viz. they
show that if Absolute Space did exist, one could
demonstrate the existence of rotation relative to it.
(;;\2 Reichenbach [9] discusses these issues with an
artificial example. He imagines two world systems far
~
apart from each other, each of an Earth (E) sur-
rounded by a shell of stars (F)-see Figure I.E! and
F2 are stationary relative to each other, and so are E2
Figure I and F!; but E] and F] are in relative rotation, and so
are E2 and F2 • Now, he argues, if centrifugal effects,
1 See, e.g. K. R. Popper, The Logic of Scientific Discovery (London, 1959),
passim.
2 One could just interpret Mach in the cited passage as claiming that it was rash
Absolute Space 55
such as water stationary relative to the Earth having a deformed surface,
are found in both EI and E 2, that shows that they are produced by mere
relative rotation of earth and shell, and hence we do not have to, indeed
cannot, attribute centrifugal etTects to absolute rotation; in our terms,
EI and E2 will be frames of reference relative to which the laws of nature
are equally simple. But if centrifugal etTects are found in only one Earth,
say E I ' clearly no mere relative rotation of earth and shell Can be
responsible for them. Hence he concludes, as we have done, that Mach's
principle that centrifugal forces are connected only with relative
rotation of masses 'is undoubtedly empirical' ([9], p. 217). For
observation can settle this issue.
However, Reichenbach claims that, although we must postulate
absolute rotation if centrifugal etTects are found only in Ep we still
cannot tell whether it is EI or E2 that is rotating absolutely. For, he
claims, we do not know whether centrifugal forces are produced by the
absolute rotation of a body or the absolute rotation round it of a shell. If
the former, then EI is rotating absolutely; if the latter, E2 is rotating
absolutely, but we do not know which.
But this is not so. We can perfectly well determine by other
experiments whether centrifugal forces are produced by the absolute
rotation of a body or by the absolute rotation of a surrounding shell. We
have only to rotate a body on EI in the opposite direction to that in
which FI is rotating relative to EI and see if greater centrifugal etTects
appear in it. If they do, it is the absolute rotation of a body which
produces in it centrifugal etTects. If they do not, then it is the rotation of
the surrounding shell which produces centrifugal etTects. Hence we can
determine whether EI or E2 is rotating absolutely. Reichenbach's
example shows that if Absolute Space did exist, one could detect by the
issue of experiments which bodies were rotating relative to it, and not
merely that some bodies were rotating relative to it but not which bodies
those were.
However, we have urged, the claim that there is Absolute Space is the
claim that there is a single most basic frame and not a class of equally
basic frames. A nd for scientific, not philosophical, reasons, such a claim
is not justified. Still, if the empirical data were such that a justifiable
to adopt a theory that the bucket was rotating relative to Absolute Space when it
rotated relative to the fixed stars and showed deformation, until we knew the
result of experiments with a thickened bucket. But Mach seems to be claiming
not this, but that until such experiments had been performed, the theory was not
significant.
56 Space and Time
scientific distinction (in the sense earlier described) could be made
between absolute velocity and mere relative velocity (and not, as
discussed above, merely between absolute rotation and mere relative
rotation) a Machian could always raise a similar objection to that cited
earlier against the interpretation of such data. Certainly, he might
admit, it is simpler today to refer the laws of nature to a frame K than to
a frame K moving in uniform motion relative to it, and to attribute
certain effects found on K but not on K to the l11JJtion of K' relative to K.
But if you say on the basis of this that K is absolutely at rest, you commit
yourself to the doctrine that the following experiment cou~d be
performed: stop K so that the simple laws which held on K relative to K
now also hold on K' relative to K, and then give K a uniform motion
relative to K in such a way that the same laws continue to hold on K
relative to K, and the effects will appear on K. 'But no one is competent
to say how the experiment would turn out.' But in those circumstances
the supporters of the doctrine of Absolute Space are predicting that the
experiment could be performed and have the result described, and claim
that they have some grounds for making this prediction. The opponents
of the doctrine of Absolute Space must claim that at some stage the
experiment will break down: either K and K cannot be held stationary in
such a way that the laws originally holding on K hold on both, or K
cannot be given a uniformhlotion relative to K' so that the simple laws
continue to hold on K, or, if such motion can be given, the effects in
question will not appear on K. They must claim this for otherwise the
effects in question cannot be produced by the mere relative motion of K
and K. The fact that a doctrine of Absolute Space does have such
predictive consequences and so can be confirmed or disconfirmed by
observation shows, even for the verificationist, that talk about Absolute
Space is significant, and, on present scientific data, that the assertion of
its existence is unjustified.
There is another thesis which is sometimes confused with the thesis of
the existence of Absolute Space, as I have presented that thesis and as-
historically - I claim that it has on the whole been discussed. On this
second thesis to say that Space is absolute is to say that its properties are
independent of the physical objects which it contains, whereas to say
that it is relative is to say that those properties are so dependent. The
properties of Space are its geometrical properties, being Euclidean or
hyperbolic and such-like. I will call these theses the thesis that Space is
M-absolute and the thesis that it is M-relative. Einstein certainly held
the latter thesis: 'According to the general theory of relativity, the
geometrical properties of space are not independent, but they are
Absolute Space 57
determined by matter' ([7], p. 113). And Newton in claiming that
Absolute Space 'remains always similar' (words which I omitted from
the definition of Absolute Space cited on p. 50 while considering the
main issue) was committed to the view that Space was M-absolute, for
since the density of matter and so physical objects in general varies from
place to place (however they be identified) his claim implies that the
geometry of Space does not depend on the density of physical objects.
Which of these two theses is correct and how one could prove which was
correct are questions which we must postpone until Chapter 6 where we
will consider how it can be shown what is the geometry of Space. What is
important here is that we should realise that the two theses are different
theses from the ones with which I have been concerned for most of this
chapter. Any confusion between them arises from the fact that Newton
was both an absolutist and an M-absolutist and Einstein was both a
relativist and an M-relativist. I
Clearly a man could consistently be an absolutist and an M-relativist.
Aristotle could perfectly well have held that the geometry of Space
varied from place to place with the density of the matter which it
contained, while holding to the view that the laws of nature were simpler
if referred to the Earth than if referred to any other frame. And, even
more obviously, a man could be, as Leibniz clearly was, a relativist and
an M-absolutist. A man might hold that there were many equibasic
frames of reference while asserting that geometry remained Euclidean
whatever objects occupied Space. Matter might deform measuring rods,
he could argue, so that they were inaccurate measures of Space - but
Space itself remained Euclidean. The two sets of theses are completely
independent of each other.
A question closely connected historically with the questions whether
Space is absolute or whether it is M-absolute is the question whether
Space would exist if there were no physical objects occupying Space.
This question is sometimes raised by asking whether, if the Universe
came into existence at a certain temporal instant, there was empty Space
before that instant existing from all eternity; or whether, if the Universe
were destroyed, Space would continue to exist.
The connection of this question with the preceding ones is this. The
absolutist and the M-absolutist tend to think of Space as an entity, not
very dissimilar from the physical objects which occupy it. For the
absolutist it is something relative to which physical objects may move,
I For examples of the interweaving of the different doctrines see [II) and the
BIBLIOGRAPHY
1 These tests for a line being a straight line do not presuppose the truth of any one
type of geometry. They are compatible with Euclidean or hyperbolic geometry or
geometry of any other type. Sometimes geometers only use the term 'straight line'
where the geometry is assumed to be Euclidean. Ordinarily however, even if we
knew nothing about geometry, we would seem to be able to ascertain that a line is
a straight line. Hence I make no assumption about geometry in using the term
'straight line'.
64 Space and Time
The other primary test for a line being a straight line we may term the
three-surface test, since it begins with a three-surface test for a plane. If a
surface A fits snugly on another surface B, and B fits snugly on a third
surface C, and C fits snugly on A (such that for all points IX on A touching
a point f3 on B, the point f3 touching a point y on C, then when A fits
snugly on C the points a and y fit on each other) then A, Band C are all
planes. Then if two planes have a common edge, that edge is a straight
line. For practical reasons the second test can only be used over very
,short distances, but where both tests can be used, we expect the tests to
give the same results as each other. If they do, then the line that by both is
a straight line is really a straight line and the surface that by both is a
plane is really a plane. The description of the tests elucidates our
ordinary concepts of 'straight line' and 'plane'. If in one instance the two
tests gave different results, we whould not know in that instance how to
apply our concepts. If frequently whatever satisfied one test did not
satisfy the other and conversely, then our ordinary concepts would have
proved useless. We would have to modify them, so that perhaps
satisfaction of one test alone sufficed for a line to be termed a straight
line. There is no logical necessity that the tests should give the same
results, but empirically we have always found, and all modern theories
of mechanics predict that we will always find, that when the one test is
satisfied, the other is too. Similar considerations would apply if there are
further primary tests for a line being a straight line.
To measure the direction of B from A we have to measure the angles
which a straight line from A to B makes with other straight lines which
pass through A. Normally measuring the angles made by the straight line
from A to B with any two straight lines will suffice, with measurement of
the distance along the former line, uniquely to identify B (with peculiar
geometries or in spaces of dimensions greater than three, if such there
could be, more angular measurements would be needed - for this see
Chapter 7). Measurement of the necessary number of angles gives the
direction of B. To measure the angles, we construct a protractor. We
mark a straight line on the plane surface of a rigid body. Along this line
we lay a thin rigid rod rigidly connected at one end to another thin rigid
straight rod. The test that the rods are straight is that they fit along
straight lines. The test of rigid connection is that the distance between
any two points on the two rods remains constant as the rod is moved.
The rods are then rotated in the plane of the surface about the point
where they are joined. The angle between the two rods in their original
position is given some arbitrary value. The angle between the rods in
Distance and Direction - (i) Primary Tests 65
each subsequent position which they take up is then said to have the
same value. There are 360 degrees in a complete circle. So when the rigid
rods are so joined that 360 of the angles between them make the whole
circle, then the angle between them will be one degree. We mark ofT the
divisions by straight lines meeting at a centre point on the plane surface
which we now call a protractor. We can then measure the angle between
any two intersecting straight lines by fitting the protractor on to them so
that one lies along its base line and the other coincides with another line
. of it. The number of unit angles marked on the protractor which fit
between the two straight lines giv~ the angle between the two lines. It is
conceivable that there may be more than one direction of B from A.
This will arise if more than one straight line goes from A to B - which
will not however occur if the geometry of Space is Euclidean or of many
other types. The applicability of the concept of direction pressupposes .
the applicability of the concept of a straight line. Given the latter, then
any two points which can be connected by a line (and so are at some
distance from each other) can be connected by a straight line, for the
shortest line will be the straight line. Hence any two spatially related
points will lie in some direction from each other.
The distance and direction established above is the distance and
direction from A to B at the temporal instant at which the rod placed on
the straight line stationary relative to the starting point touches B. (How
to ascertain which instant at A is simultaneous with thiS instant is a
question which we shall consider in Chapter 11.)
To apply the criteria described we need to be able to recognise
coincidences (of rod and mark) which we can do either visually or .
tactually. It should be noted that even if we rely on visual means to detect
coincidences, we do not thereby pre-suppose the rectilinear propagation
of light. Even if light travelled in circles, a coincidence would remain a
coincidence.
The methods for measuring distance and direction which I have
described are methods for measuring the distance between material
objects (e.g. the distance from this wall to that wall). They may be used
for ascertaining the distance between other spatial things, in so far as the
boundaries of those things are distinct enough for distance to be
measured. But places as well as spatial things have from each other
distance and direction. To measure the distance between where A is and
where B was is to measure the distance between two places. In order to
identify the present place which is the same as the place previously
occupied by B, we have to specify or take for granted a frame of
66 Space and Time
reference.! The distance between two places P and Q is naturally
supposed to be the distance between two objects occupying those places.
At this point difficulties arise in applying the ordinary concept or
distance. It is a consequence or the SPecial Theory or Relativity, and or all
more general theories or mechanics which include it, that the distance so
measured between two places depends on the velocity or the objects
occupying those places - and this, whatever standard or simultaneity is
adopted. Thus suppose that two places on an inertial rrame P and Q are
occupied at the same instant, as judged by some standard or simultaneity,
respectively by two objects a and b stationary relative to each other and
that the distance rrom P to Qas measured by the distance rrom a to b is x.
Suppose another object a' to be at virtually the same place as a at that
instant and to be passing a with a velocity v away rrom b. The distance
rrom P to Qas measured by the distance rrom a' to b will then be different
rrom x - whatever our standard or simultaneity. Suppose that the
standard is the light signal method (to be described in Chapter 11) so that
clocks on a and b have been synchronised by light signals sent between
them. Then the distance rrom a' to b will be - according to special
1 For some of the difficulties in this definition and analysis of what it does not
include as a universal force see [2], ch. 3, section A, and Brian Ellis 'Universal
and Differential Forces', British Journal for the Philosophy of Science (1963) 14,
177 -94. .
2 Reichenbach himself made the point that his concept 'of universal force is more
general and contains the concept of the coincidence preserving force as a special
case'-[I] p. 27.
78 Space and Time
1 [2] ch. I, section D. Grunbaum argues this case against Eddington, who makes
my simple point.
82 Space and Time
would be of no use because all intervals would contain the same number
of points. See p. 26 for discrete, dense, and continuous spaces.)
Grunbaum calls such a way of measuring distance giving to space an
intrinsic metric; and he contrasts it with measuring by such means as
rods, which he calls giving to space an extrinsic metric. He claims that if
we can give to space an intrinsic metric, so that we measure distance by
counting points, we ought to do so because 'the object of a
metric ... is ... to tell the intrinsic story in so far as possible.' ([5]
p. 576.), However, to measure distance by counting points, would involve
making the assumption that each point took up an equal volume, and
this would seem an arbitrary assumption unless by our normal standards
of measurement (i.e. those involved in the extrinsic one) they did so.
Grunbaum claims that extrinsic metrics are 'convention-laden', whereas
I have suggested that it is part of the meaning of 'distance' that by and
large it is the kind of thing which is determined by rigid rods. (For further
detailed criticism of Grunbaum see [6].)
True, as I have admitted, criteria in this region may not,always be
completely clear. There will be obvious cases where the criteria yield one
unambiguous result The distance from a certain point on one wall of my
sitting-room to a certain other point on another wall is twenty feet. How
to show this is unambiguously clear. Yet what value we give to a distance
in the neighbourhood of a very massive body may depend on which of
two possible ways we correct our measurements, and the ordinary ways
of correcting measurements_extrapolated to this case may allow either
way of correcting measurements. But although there is a certain
vagueness about the correct application of the concept of distance, as of
all concepts, in extraordinary circumstances, to the difficulties of which I
have drawn attention, there are perfectly clear tests for distance in
ordinary circumstances. If the philosopher allows us to introduce
completely different tests for 'distance', as Reichenbach and Grunbaum
do, he is analysing a concept of his own invention.
In analysing the meaning of 'distance' and 'direction' I have been
concerned to describe the primary tests for distance and direction. We do
not often make measurements by the methods which I have described.
Even on a small scale we do not normaIly find the straight line between
two objects by doing a complicated series of manoeuvres with a ruler,
trying to fit it the least possible number of times between the two objects,
let alone by finding a common edge to two surfaces shown to be planes
by the three-surface test. Rather, we stretch a string between the objects
and measure along it.Jn the next chapter I propose to show that other
tests for distance and direction in normal use are secondary tests, and
Distance and Direction --(i) Primary Tests 83
also to describe in a little detail the kind of secondary tests which we use
on the cosmological scale. The latter will be a useful preliminary to
subsequent analysis of the conclusions of cosmology about the size and
geometry of the Universe.
BIBLIOGRAPHY
I The same is true of distance and direction on the very small scale, but as stated
in the Introduction, this work would become too bulky were it to provide
adequate treatment of problems of the very small scale.
Distance and Direction-(ii) Secondary Tests 87
thermometer near the centre of Jupiter to test the latter assertion; but the
fact remains that what the assertion means is that, if we could take such a
thermometer there, it would record a certain reading. It is often
practically impossible to test the truth of statements by using primary
tests. But this practical impossibility does not affect the meaning of a
statement. My claim is that the methods which we use for measuring
directions and large distances in astronomy we use because we believe
that the results attained by using them are the same as the results which
would be attained by using the primary tests, if we could use these. We
must now investigate the methods which we use in practice on the large
scale.
In measuring distance and direction on a large scale we assume that
relations between bodies found locally hold outside the range for which
they have been tested. Clearly in general we are justified in doing so. It is
characteristic of science to work on the assumption that the unknown
resembles the known, that characteristics of the behaviour of matter
found locally hold beyond the ranges for which they have been tested.
The kind of circumstances under which we are not justified in
extrapolating properties will be considered on pp. 93 fT.
The normal method for measuring the direction of a distant body is
from the angle made by light rays emanating from it. The body is stated
to be (or rather, to have been at the instant at which the light rays left it)
in the direction in which arrive light rays emanating from it. The use of
this method assumes that light travels in straight lines in vacuo (except
when near massive bodies) and that the space between Earth and the
stars is very nearly a vacuum. The first assumption has been tested on
Earth. The second assumption is justified by considerations of
simplicity - the planets obey Newton's laws only if we assume that the
space in which they move is very nearly a vacuum. Unless we make this
assumption the explanation of planetary motion would become very
complicated. Hence we may say that the assumption has been tested
within the solar system. We now assume that these assumptions hold for
Space far beyond the solar system, and hence can estimate the direction
of distant bodies. -
The most important method for measuring the distance of near-by
stars is the method of parallax and is a more complicated form of the
kind of method described on p. 85. This assumes, as well as that
directions can be estimated in the way just described, that the geometry
of Space. is Euclidean. This has been found to be so on Earth, and it is
now assumed that it holds for Space far beyond the Earth.
The method of parallax uses a short distance measured by other
88 Space and Time
methods, that is ultimately by rigid body or taut string methods, this
distance being the base line. The angles at each end of the base line made
with the base line by light rays impinging from the distant object are
mea~ured and the distance of the object calculated by the formulae on
p. 85. This, basically, is the method by which the distance from the Earth
to the Sun and the Moon and the planets was established (the grounds
for using light rays to ascertain direction being, when these distances
were originally established, weaker than those stated above). The first
stage was to ascertain the radius of the Earth. This was done in the fourth
century s.c. by measuring the different angles subtended at different
places on the Earth by light rays from the Sun. But it could be done by
rigid body methods, ifmen were prepared to take the necessary trouble.
The circumference of the Earth could be measured and t~e radius
calculated from it by the use of the formula of Euclidean geometry that
the circumference of a sphere is 2n times its radius. That the Earth is
approximately a sphere can be shown by measuring many great circles on
it (that is, lines on the surface of the Earth, marking the shortest distance
along the surface between any two points on the line) and showing that
these are of approximately equal length. We then obtain a base line for
our astronomical measurements by measuring the distance along the
surface of the Earth between two places on it A and B and then from the
known radius of the Earth calculating, using Euclidean geometry, the
. distance along the straight line through the Earth between A and B. This
gives the length of the base line b. The use of this method informs us
where the distant object was when the light left it. (For a base line long
relative to the distances being measured, we have to add a correction to
allow for the different periods of time taken by the light to come to the
two points on the base line from the distant object. For a base line like
one on the surface of the Earth, this factor may be ignored.) We can thus
measure the distance from the Earth to the Sun, Moon, and planets.
We can then infer, as Kepler did,from the change of the Sun's position
relative to the Earth the path of the ellipse traced out annually by the
Earth round the Sun. The Earth moves some 184 million miles from one
side of the Sun to the other during the half-year. We can now use a
straight line from one side of the ellipse to the other as base line for the
calculation of much larger distances by the method of parallax. This is
normally done as follows.
We measure the angles of a star at different instants during the year
and find two equal and opposite beafings of the star (ex = )' in Figure 4).
The Earth's path being nearly circular, the base line b is at all periods of
the year approximately 184 million miles. The angle fJ/2 (90 0
ex) is then
-
Distance and Direction - (ii) Secondary Tests 89
known as the annual parallax of the star. a = c = __ b_. The first star
. 2cosa
parallax was calculated in 1838 by
B F. W. Bessel, who found that the annual·
parallax of the star 61 Cygni was 0.31", and
hence that the star was at a distance of some
54 million million miles from the Earth.
Astronomical distances are normally mea-
sured in parsecs. A star with an annual
parallax of 1" is said to be at a distance of 1
parsec from the Earth. Hence 61 Cygni lies at
a distance of some three parsecs. The nearest
star to the Earth, Proxima Centauri, lies at a
distance of some one-and-a-half parsecs.
The calculation of distance by parallax
A'----..J....Lb..l-----:-l C thus works on the assumption that certain
Figure 4 locally tested regularities (Euclidean geo-
metry, the rectilinear propagation of light)
hold far outside the range for which they
have been tested. Thereby the range within which distances can be
measured is enormously extended. Using the Earth's orbit as base line
astronomers can calculate distances up to 100 parsecs. A more
complicated method known as the method of statistical parallax is used
to calculate distances up to 500 parsecs. I shall not describe this method
as itis somewhat complicated (for a brief description see [I] pp. 29- 34).
Hence I shall state without proof my thesis about it that it works by
assuming that certain regularities observed to hold locally hold far
outside the range for which they have been tested.
A variety of further methods are used to calculate greater distances,
the most important for medium distances being the 'headlights' method.
Certain stars, especially the Cepheids, are variable stars whose lumin-
osity increases and decreases in a characteristic way, the period of
oscillation being constant for each Cepheid but varying for different
Cepheids from one day to many years. The distance 0f some of these
stars near to us can be calculated by the method of statistical parallax. We
assume that the locally tested law that the luminosity impinging on an
observer from a distant object varies inversely as the square of the
distance of the observer from it holds far outside the range for which it
has been tested (a correction has to be added when the distant object is in
motion relative to the observer. We have evidence from the absence of
significant Doppler shift-see p. 91-that such corrections would be
90 Space and Time
insignificant for stars of our own or near galaxies). Hence, knowing the
distance from ourselves of near-by Cepheids we can calculate how bright
they would appear to an observer at a distance of ten parsecs. The
luminosity impinging from a star on an observer on the Earth is ·known
as its apparent luminosity. The luminosity from a star which would
impinge on an observer at a distance from it often parsecs. is known as its
absolute luminosity.
From its apparent luminosity (l) and known distance from the Earth
in parsecs (d) we can thus calculate for any star within 500 parsecs from
. L d2
the Earth the absolute luminosity (L) of the star by the formula T = 10 2
Jd 2
orL=-
100·
Instead of dealing in apparent and absolute luminosity, astronomers
normally deal in apparent and absolute magnitude. Apparent magnitude
is related to apparent luminosity logarithmically by the equation f2
= 1001'"2 - '"' )/5, where II and.1 2 are the apparent luminosities of two stars
and m] and m2 their apparent magnitudes, the apparent magnitude of
certain stars being fixed by ancient convention. Absolute magnitude is.
the apparent magnitude which would be calculated from luminosity
impinging on an observer at a distance of 10 parsecs. Absolute
magnitude· (M), apparent magnitude (m) and distance in parsecs
and radially close. Given the locally tested law that light travels through
space in straight lines, stars close to others on a photograph of the sky
will be in fact tangentially close to them. The evidence that a tangentially
close star is radially close to another is that it is of similar apparent
magnitude and has a similar velocity of approach or recession. For we
know from observations within the region where distances can be
calculated by the method of statistical parallax that tangentially close
stars of similar apparent magnitude which have a similar velocity of
approach or recession are normally also radially close to each other. We
assume that this locally tested characteristic of stars also holds outside
the range of observation. We ascertain a star's velocity of approach to or
recession from the Earth by studying its Doppler shift. We know from
observations on the Earth that the spectrum of light from a body B as
observed at A is shifted towards the violet end of the spectrum if B is
approaching A, and towards the red if B is receding from A. We assume
this to hold for stars and galaxies far outside the region for which it has
been tested and know of no other explanation of a red-shift (apart from
the gravitational red-shift predicted by General Relativity which would
not normally be relevant). We. also assume the locally tested law that the
elements Of matter have the same spectra in all places, and so that there
are only certain possible emitted spectra. Hence from a received
spectrum, given that all lines have been shifted together, we can infer
what was the emitted spectrum and so infer velocity of recesion or
approach. That such an inference can always be made confirms the law
that the elements are the same at all places and have there the same
spectra. The proportion by which the wave-lengths of received light
exceed those of emitted light due to a velocity of mutual recession of
source and observer is known as the Doppler shift of the source.
Using all these tools we can infer which stars are near to a given
Cepheid and hence their approximate distance from the Earth. Once we
know which stars are near to each other, we realise that the Universe is
populated by large groups of stars, called galaxies, to one of which our
solar system belongs; and that the galaxies keep together in larger units
called clusters. We can infer the distances of near-by galaxies from the
Cepheids in them.
There are many similar methods of calculating distances of stars in our
own or near-by galaxies which lie outside the range of calculation by
statistical parallax. They all depend for their operation, like the two
methods so far described, on assuming some locally tested physical
regularity to hold outside the region for which its operation has been
tested.
92 Space and Time
Having by such means found the distance of near-by galaxies, we can
calculate from their apparent magnitude and diameter, their absolute
magnitude and actual diameter. We then notice that the absolute
magnitude and the diameter, of galaxies varies only within a narrow
range. Assuming once again that what holds locally holds more widely,
we are now in a position to judge the distance of a distant galaxy from its
apparent magnitude. For we know within a range its absolute magnitude
and given the inverse square law of diminution of luminosity can infer
thence its distance. (Evidence from experiments on Earth shows that a
correction has to be added to the formula for a light source with .a
velocity of recession from the observer, S9 that a source with a velocity
yielding a Doppler shift of z lies at a distance _1_ times that which
1 +z
would otherwise have been estimated.) A similar inference can also be
made from the apparent diameter of a galaxy indicated by the angle
which it subtends for an observer on Earth (see Figure 5). If we know
within a range the actual diameter of a galaxy, we can then infer, within a
range, its distance from the Earth,
given the rectilinear propagation of
light and Euclidean geometry. This
method as stated is not of much use
for other than near-by galaxies since
the angle subtended by others is too
small to be determined accurately.
But a similar method can be applied
to clusters of galaxies. We observe
:% + - angle subtcmded that galaxies, of similar magnitudes
at earth . with similar recessional velocities ap-
pear close together on photographic
Figure 5 plates. We infer that they form a
cI uster of galaxies close to each other.
We find the actual diameters of near-by clusters, infer that what holds
locally holds outside the tested range, and thus infer from the angle
subtended by them at the Earth the distances of clusters of galaxies.
Distances of individual galaxies or clusters calculated by these methods
are obviously very rough, but the methods are very useful for calculating
the average distance oflarge numbers of galaxies or clusters, when errors
made in calculating the distance of individual galaxies cancel out.
The most important method for ascertaining the distance of the most
distant galaxies is from their red-shift. Assuming on the grounds stated
earlier that red-shift indicates recession, we note a functional relation
Distance and Direction - (ii) Secondary Tests 93
between the distance of a galaxy and the velocity of its recession
indicated by its red-shift. The astronomer E. P. Hubble who first noted
this relation claimed that this was a linear relationship between velocity
(V)and distance (d)V = Hd (H being 'Hubble's constant'), but some later
astrQnomers have claimed that the relation is non-linear. But granted
that observations on galaxies whose distances can be ascertained by such
other means as the method of apparent magnitude supports Hubble's
law, we can go on to use it to establish distances of the most distant
galaxies observed, i.e. distances up to 1,000 m. parsecs from ourselves.
(Although there are the good grounds described earlier for holding that
red-shift shows velocity of recession in the case of stars and galaxies,
there are grounds for suspecting that the red-shift of quasars may have a
different cause. Quasars are strange star-like objects first discovered in
the mid-sixties about whose nature scientists are still very puzzled.)
So by continual extrapolations from local observations, the distances
of more and more distant objects have been ascertained. I have sketched
only in the briefest outline a few of the methods used. I have given the
outline to illustrate the methods offinding out the distance and direction
of distant objects by further and further generalising from local patterns
of phonemena. Since we use these methods because we believe that
certain patterns of phenomena hold beyond the range fOJ which they
have been observed to hold, if we find reason for no longer holding this
belief about certain patterns of phenomena, we ought to abandon any
methods of estimating distance and direction which presuppose its truth.
There could be two distinct kinds of reason for believing that certain
patterns of phenomena do not hold beyond a region for which they had
originally been observed to hold. The first kind of reason would be that
subsequent observations did nQt confirm the original observations
that one property in question held in a local region. Thus using some
method I we may have noted that the Universe within the range of study
by that method had a property P. Assuming that P holds outside that
range of observation, we can then use a method II which embodies this
assumption to find further distances. Suppose now that by more careful,
precise and painstaking use of method I, we find that P does not in fact
hold of the Universe within the range of observation by method I and
hence there are no grounds for assuming it to hold outside that range. In
that case the use of method II to ascertain distance would not be justified.
Astronomical methods of ascertaining distance are continually being
modified in the light of such considerations. Thus all methods which
work from the luminosity of distant objects need an estimate of the
amount of light absorbed by interstellar and intergalactic dust. Given
94 Space and Time
absolute magnitude and apparent magnitude, we can only infer distance
if 'we judge insignificant or give a value to this factor. Initially the
astronomer may judge this factor to be insignificant; and by some
method II (e.g. inferring the distance of a galaxy from its apparent
magnitude as described on p. 92) may infer its distanee. However, more
careful estimates of the amount of intergalactic absorption may lead him
to revise his method II.
The second kind of reason for believing that certain patterns of
phenomena do not hold beyond the region for which they had originally
been observed to hold could arise even if subsequent observations
confirmed that the property did hold in the region studied by the original
method. This kind of reason would be that the assumption that it held
beyond the region for which it had been observed to hold proved
incompatible with the reasonable supposition that various other proper-
ties held beyond the range for which they had been observed to hold.
Thus suppose that by using some method I, we observe that properties P,
Q, R, and S hold of the regio!! of the Universe which can be studied by
that method. The assumption that P holds universally means that we can
use a method II for ascertaining further distances. The assumption that
Q, R, and S hold universally means that we can use respectively methods
III, IV, and V for ascertaining further distances. Suppc;>se now that the
use of method IV gave different estimates of distance from the other
methods. Then either R does not hold beyond the range studied by
method I, or p. Q, and S do not.
. If we have no general theory of physics on a cosmological scale, the
simplest assumption to make is that the odd property R does not hold
outside the range of observation by method I, and so that method IV is
not a justifiable method of ascertainjng distance. However, if we have a
cosmological theory, the situation is different. On the basis of obser-
vations that properties P, Q, R, and S hold locally a scientific theory may
be set up which gives for those observations a moresimp\e and coherent
explanation than any other theory compatible with them. It may,
however, be a consequence of the theory that while R holds universally,
P, Q, and S do not hold universally. Thus it may be a consequence of the
theory that P, Q, and S only hold in a small region and that in a wider
region a property different from P, Q, and S would hold, but that the
region studied was too small for a difference from P, Q, and S to be
noticed. Hence P, Q, and S would not be extrapolatable properties. The
use of methods II, III, and V which assume them to hold beyond the
original range would not be justified and would have to be corrected in
the light of the theory. What is happening here is that the theory reveals
Distance and Direction - (ii) Secondary Tests 95
that one or more local properties, in our example R, are more
fundamental than others and the assumption that these hold universally
will mean that others do not. Cosmological theories developed from the
General Theory of Relativity and rivals to it have involved such
modifications to methods of estimating distance. The General Theory
cast doubt on the general validity of Euclidean geometry. Since all the
secondary tests for distance assume that geometry is Euclidean, they
must be modified if that assumption turns out to be unjustified. Various
forms of General Theory and other cosmological theories lead to
different conclusions about the relation which they claim holds uni-
versally between the distribution of matter and the geometry of Space. In
so far as anyone theory is well substantiated, we must assume that the
relation which it shows to hold locally holds universally. Hence from our
knowledge of the density of matter on a large scale we can infer the
geometry of Space on a large scale, and hence ascertain thllt it is different
from one which to a high degree of approximation holds on the small
scale, viz. Euclidean geometry. The theory by extrapolating a certain
property and assuming it to hold universally showed that a different
property would not hold universally. Whether we are to adopt a theory
which assumes a property P to hold universally, or a theory which
assumes another property R to hold universally, depends on the relative
simplicity of the two theories. The character of evidence for any general
cosmological theory like General Relativity and the conclusions about
the geometry of the Universe which can be derived from it will be
examined in Chapter 14. Clearly there are v.arious difficulties in building
up such a theory - we shall have to show, for example, how we can reach
a conclusion about the density of matter without assuming a certain
geometry to hold. But assuming that we can by these means show that a
certain geometry, other than the Euclidean which to a high "degree of
approximation holds locally, holds universally, we should have to
modify such methods of calculating distance as the method of parallax
and the method of apparent magnitude which take for granted the
universal applicability of Euclidean geometry.
So in these ways the methods described of ascertaining distance and
direction can be and are being continually refined in the light of
experience. The fact that all the methods described of ascertaining
distance and direction beyond the Earth are open to review and
refinement indicates that the methods are secondary tests. We use
secondary tests because we believe them correlated with primary tests.
To the extent that we have evidence that this correlation does not hold
we amend the secondary tests. If empirical evidence could show that a
96 Space and Time
test for distance did not give good results, that shows that the test is a
secondary test. Empirical evidence could not show this for primary tests,
since empirical evidence cannot refute a definition.
It is a consequence of these points that various concepts used by
cosmologists related to the concept of distance are not in fact concepts of
distance. Cosmologists have defined several different 'distances' of
objects differing in respect of the method of estimating them. Thus there
are 'distance by apparent size', 'luminosity distance', and 'parallax
distance'. The distance by apparent size of a gala~y G is -~, where D
2 tan (X
is the average diameter of a local galaxy and (X the angle subtended at the
general do, give rise to different distances and there is no one of them which can
be labelled as the "correct" distance' ([3] p. 147). Different procedures give
different measurements, but they are not all measurements of distance.
Trautman may be right when he claims that 'proper distance' (viz. distance
relative to the frame in which the first object is stationary) 'is of no practical use',
in cosmology. But he is misguided to recommend the use in cosmology of
Distance and Direction - (ii) Secondary Tests 97
81 8L10GRAPH Y
J Strictly speaking, you are allowed to make a cut so long as you join the body up
For any line L and point P not on L, there is more than one line L'
through P parallel to L. (If there is more than one parallel line, it can
then be proved that there are infinitely many.)
Euclid's other axioms remain. But with this different axiom, many of
the theorems of hyperbolic geometry are radically different from those of
Euclidean geometry. For instance, the interior angles of a triangle sum to
less than 180°. The angle sum is smaller, the greater the length of sides of
the triangle. The ratio of the circumference to the diameter of a circle is
greater than 1£. There can be no similar figures which are not congruent.
This means that if you take some figure and compare it with another
figure with sides of different lengths but having the same ratios to each
The Geometry of Space 103
other-say a pentagon with each of its five sides of equal length
compared with another pentagon also having each of its five sides of
equal length to each other but longer or shorter than those of the other
pentagon, then the angles oLthe two figures and hence the shape of the
figures will be different. There are many other such differences from
Euclidean geometry.1
Elliptic geometry is obtained by amending the fifth postulate to:
For any line L and any point P not on L, there are no lines through P
parallel to L.
Euclid's first three postulates, if interpreted as they normally are, have
to be modified to make this new postulate consistent with the rest of the
Euclidean system. 2
The properties of elliptic geometry include the following. The interior
angles of a triangle sum to more than 180°. The angle sum is greater, the
greater the length of the sides of the triangle. The ratio of a circumference
to the diameter of a circle is less than n. There can be no similar figures
which are not congruent.
fJl
r
small area of the surface has the metric M I' it follows that the surface is
the surface of a sphere. But if we know that the surface has the topology
TI (say, the topology of a disc) in a small region and assume that it has
everywhere, we still do not know what is the topology of the surface as a
whole. The surface may be a plane or the surface of a sphere - in every
small region these have identical topologies. It should be added that
although many different topologies of Space are compatible with the
assumption that in every small region the topology is the same topology
Tl> this assumption does rule out many different topologies of Space.
But it does not enable us to choose, for example, between Euclidean and
elliptic geometries which have in every small region the same topology.
Wher~as the assumption that every small region of Space has a metric
M 1 does uniquely identify the metric of Space as a whole. Hence if we are
to extrapolate from properties of an explored region to find the topology
of Space as a whole, we must use its metrical properties.
In his discussion of the question of what we should say about the
topology of Space if the region explored appeared closed, Reichenbach
({7] § 12) urged that we could always consistently deny any conclusion
based on exploration with certain measuring rods about the topology of
Space by postulating causal anomalies. He considers a case where a man
leaves his study, passes through a number of spherical shells, each
completely enclosing the next one, and finds himself in a room exactly
like his study. There are then, Reichenbach urges, two possible
interpretations of what has happened. Either geometry is non-Euclidean
(e.g. the torus geometry postulated by Reichenbach), so that you can pass
through spherical surfaces enclosing each other and find yourselfback at
your starting point, or the place wh~re the man found himself was not
really his starting point. The room was merely one very like his study.
Reichenbach's explorer does an experiment to see if the new room is
really his old study. 'He writes down his thoughts on a sheet of paper,
. adds a code word, locks the paper in a drawer, puts the key in his pocket,
and leaves shell 5 . .- .. Arriving at 1 he finds his room, opens the drawer
with the key he put into his pocket, and recognises on the slip of paper
the same words which he had written down in shell 5' t [7] p. 64). Here,
claims Reichenbach, the observer can either take this evidence as
evidence that the two places are the same place or postulate action at a
distance between the two places, so that his writing something down in
one place produced a similar piece of writing in the other place without
intervening places being affected (viz. there being no known wireless
devices or such transmitting through the shells and thus producing the
effects). Now, claims Reichenbach, such action at a distance is to be
110 Space and Time
considered a 'causal anomaly' of the type that science attempts to avoid.
We can, he says, still claim that geometry is Euclidean but to do so we
would have to postulate causal anomalies, and this he advises us not to
do. .
Two points must however now be made against Reichenbach's claim
about his parable. The first is that, contrary to what Reichenbach seems
to suppose, there is a truth here. One of the above interpretations is true
and the other false, whether or not the observer could come to know
which is which. We can see this by reflecting on the fact that it is at any
rate logically possible (and perhaps physically possible) that the
investigator could grow in size until his arm reached right through the
shells. Let him then touch one study with one hand and reach through
the shells with the other arm. If the geometry was Euclidean, he would
touch a different study with the second arm. If it was non-Euclidean, he
would touch the same study and hold the first hand with the second. The
two situations are different. Secondly even if an observer could not grow
in size, he still has good reasons for a belief that the topology is non-
Euclidean. The first good reason is that as we have seen, simplicity is
evidence of truth, and the fact that it is necessary to postulate an extra
and strange force producing duplication of effects at a distance in order
to save Euclidean topology is good reason for holding that the topology
is non-Euclidean. The second good reason is that, as we saw in Chapter
I, the detailed similarity of a present object to a past object is evidence
that the two objects are the same object - in the absence of positive
evidence that they are not. The fact that both studies are qualitatively
similar and surrounded by qualitatively similar objects in each case, as
far as you go from the two studies, is strong evidence that the two studies
are the same material object, and so that the topology is non-Euclidean.
The general point which seems to lie behind Reichenbach's use of his
parable is that one can 'choose' whether to say that two objects are or are
not the same object. Once a decision has been made, with respect to all
objects linked by paths, which objects are identical to which other
objects, the topology of space is fixed. The metric is then detennined by
which paths are straight lines, and this, I urged earlier, is not in general
an arbitrary matter. I now urge that the topology is also no arbitrary
matter. There is a truth as to whether or not objects are identical, and
one about which we can have good evidence. That' evidence of
qualitative similarity is evidence of identity is, as we saw in Chapter 1, a
basic criterion for our interpretation of experience.
All of this provides a conclusive objection to the Kantian position that
Euclidean geometry is the only geometry which, it is logically possible,
The Geometry of Space III
can hold of physical space. For the topology of Space might be non-
Euclidean, and observers might have good reason for believing that,
which they could only deny by denying that evidence of qualitative
similarity is evidence of identity. And if we denied that, as we saW in
Chapter I, we would never be able to make any justified judgements at
all about the identity of material objects.
It follows from this that if a cosmological theory claims that Space
measured by certain rods has an overall positive curvature, we cannot
reinterpret it so that it claims that Space is Euclidean but that 'causal
anomalies' occur. Hence what I have called in Chapter 4 (pp. 77ft) the
two interpretations of the General Theory of Relativity are in fact two
different theories making two different sets of predictions. The original
interpretation was that the geometry of Space was Euclidean, but that
all rods were deformed near massive bodies. The later and now
traditional interpretation is that very short rods of negligible thickness
preserve everywhere the same length, but that the geometry of Space is
affected by the presence of massive bodies, being, Einstein argued in
1917, on the cosmic scale of positive curvature. On a smaller scale both
theories make the same predictions, but on the cosmic scale - given
Einstein's 1917 argument - they do not. For the later interpretation
claims that the topology of Space is non-Euclidean, that Space is closed,
while the former interpretation claims that it is Euclidean. The original
interpretation claims that all rods are deformed by the presence of
massive bodies, but, as we have seen, however they are deformed, they
will measure the true topology of Space. Hence this interpretation
predicts that measuring rods will measure a Euclidean topology, while
the later interpretation claims that they will measure a non-Euclidean
topology. The grounds for choosing between the two interpretations
are the normal grounds for choosing between scientific theories. I shall
henceforward understand by the General Theory of Relativity the
interpretation which involves the claim that very short rods of negligible
thickness preserve everywhere the same length, since this is the one
normally used by physicists today.
In the light of the argument of this chapter, it is now clear how the
question raised at the end of Chapter 3 whether Space is M-absolute or
. M-relative is to be settled. To say that it is M-absolute is to say that its
geometrical properties are independent of the physical objects which it
contains; to say that it is M-relative is to say that they are dependent on
them. If the geometry of Space varies from place to place, dependent on
the concentration of physical objects, then Space is M-relative; if it does
not, then Space is M-absolute. The geometry of Space is the geometry
112 Space and Time
which would be measured by a rod corrected for all deforming
influences. To account for coincidences between approximately rigid
rods not being preserved under transport, we have to postulate
differential influences, Ifit adds to the simplicity of the physical system,
we must also postulate universal influences, so long as our ordinary
understanding of when a rod has approximately preserved its length is
not thereby violated,
In this chapter we have considered what it means to say that Space has
a certain geometry. The Space in which we live is clearly far too large for
investigations into its geometry to be carried out in terms of rigid rods.
The question therefore arises how we can determine by. less direct
methods the geometry of our Space. This question will be considered in
Chapter 14.
BIBLIOGRAPHY
'.For precise modern formulation of Menger's definition see [I], p'.24, For the
history of the definition see [I] Chapter I,
116 Space and Time
of 'distance' or 'direction' will also serve, since a measurement of a
'direction' is a measurement of the ratios of 'distances'.) This restriction
is evidently tacitly implied in claims about the dimensionality of physical
space which relied on the original definition. To say that physical space
is three-dimensional is to say that three measurements of distance are
necessary and sufficient to identify any point. If we replace the triad of
nwnbers which identify a point by a single nwnber, that single number
cannot measure the number of units of distance (or direction) at which
the point lies from parts of some frame.
This can be seen by a feature of the Cantorian method for putting the
points of a plane P and a line L into one-one correspondence. The points
of P and L can be put into one-one correspondence, but not into one-one
continuous correspondence. This means that in general the straight lines
of P will not correspond to continuous segments of L. Hence all points
of P which are very close to each other can have associated with them
pairs of coordinates, each mem ber of which differs but slightly from the
corresponding member of the other pair. But if the points are given
single coordinates by being associated with L, this will not in general
hold. Very close points will have wildly different coordinates. The points
with the single coordinates 2 and I might very well be at far greater
distance from each other than the points I and 1000. If single
coordinates are used for identification of points of a Euclidean plane,
they cannot be measurements of distance.
The older definition with the restriction runs as follows: a 'space' is n-
dimensional if and only if n measurements of 'distance' (or 'direction')
are necessary and sufficient for unique identification of any point. With
the restriction the older definition and the topologist's definition will
yield the same dimensions as each other for Euclidean spaces and many
other spaces including the elliptic and hyperbolic spaces discussed in the
last chapter, viz. if such a space is n-dimensional on one definition, it will
be n-dimensional on the other.
Which definition are we to choose? Clearly, that definition which
brings out best what we ordinarily mean when we say that a Euclidean
plane has two dimensions or that physical space has three dimensions.
Since the topologist's definition was only developed in the twentieth
century! and the other definition is much older and is the only one which
most men learn when they learn geometry, it would seem that the older
! The same is true of any other definitions which may be proposed, e.g. that
provided in [I] p. 24, and hence the argument which I give against the
topologists' definition, if valid, holds against .them too.
The Dimensions of Space 117
definition is a better elucidation of our ordinary usage. This suggests
that if the older definition and the topologist's definition were for some
space to yield different results, we would say that the dimensions of the
space were those yielded by the older definition. True, the original
definition as amended is only of use for a metric 'space', viz. one in
which an analogous concept to that of distance can be introduced, and
the topologist's wider definition is of more use for his own purposes. But
in dealing with the dimensions of physical space, we are of course
dealing with a metric space.
I would suggest then that to say that a space has n-dimensions is to say
that it is possible uniquely to identify every point by n measurements of
distance or direction from some frame of reference and not possible by
less than n measurements. For many metrics including a Euclidean one,
the following system of measurement will give unique identification of
any point P. In a one-dimensional space you need only specify distance
from a given point along the line. In a two-dimensional space you need
one straight line given in advance. A straight line is then drawn from P to
the given line perpendicular to it. Distances are then measured from an
origin to the point of intersection of the lines, and from the point of
interesection to P. In a space of three dimensions a line is drawn from P
perpendicular to a given plane. Two measurements of distance are then
needed to locate the point on the plane where it is cut by the line from P;
and another measurement to measure distance along the line from the
plane to P. So three measurements of distance uniquely identify P. In a
space offour dimensions, a line is drawn from P perpendicular to a given
hyperplane, a hyperplane being a three-dimensional entity produced by
moving a plane kept parallel to itself in a direction perpendicular to
itself. Three measurements are necessary to locate the point of
perpendicular intersection with the hyperplane of a line from the point,
and a further measurement to measure the distance along the intersect-
ing line. Four measurements of distance in all are thus necessary
uniquely to identify a point; and so on for spaces of higher dimensions:
We can always substitute a measurement of direction for a measurement
of distance. Thus spherical polar coordinates use one measurement of
distance (the distance along the straight line joining the origin 0 to the
point under investigation P) and two measurements of direction (the
angle which this line makes with a given plane, and the angle which a
certain projection of the line on to the plane makes with a given line).
In a different metric from the Euclidean it might be necessary to make
measurements from a frame of reference in a different way. Thus in a
plane of some more complicated metric we might need to make one
118 Space and Time
measurement along a given circle and the other measurement along a
line of a certain curvature joining the circle and the point whose location
we are measuring. We might need to do this because the space might be
such that no line from certain pointsinteresected at right-angles a given
straight line such as we would use for measurements in a Euclidean
space. But to claim that the space is n-dimensional is to claim that a way
of measuring can be laid down in advance so that every point can be
uniquely identified by giving n coordinates which represent measure-
ments of distance and cannot be uniquely identified by giving less than n
coordinates.
Clearly the dimension of a space, as we have defined it, is a property
thereof invariant under continuous deformation and hence a topologi-
cal property. For suppose an n-dimensional space S to be changed into a
space S' by a series of continuous deformations D, 0',0", etc. Now ifby
measuring along certain lines n measurements were previously necess-
ary, then n measurements will now be necessary if we measure along the
lines obtained from the previous ones by the deformations D, 0', 0", etc.
Hence if n was the minimum number of measurements necessary for
unique identification of points in S, points in S can be uniquely
identified by n measurements. Hence continuous deformation cannot
increase the dimensions of a space. Nor therefore can it (lecrease the
dimensions of a space; for, if it could, the reverse deformation would
increase the dimensions.
Having now clarified what it means to say of a 'space' in the pure
geometer's sense that it has, for some n, n-dimensions, we must now
revert to applied geometry and inquire whether it is a logically necessary
truth that physical space is three-dimensional.
Kant claimed 'that complete space (which is not itself the boundary of
another space) has three dimensions, and that space in general cannot
have more, is built on the proposition that not more than three lines can
intersect at right-angles in a point. This proposition cannot be shown
from concepts, but rests immediately on intuition, and indeed because it
is apodictically certain, on pure intuition a priori' ( [4] § 12). All Kant's
philosophical predecessors who discussed the question also argued for
the necessity in a stronger sense than 'physical' of the tridimensionality
of space. Aristotle simply stated that bodies were divjsible in three ways
and that nothing was divisible in more than three ways.! Ptolemy,
according to Simplicius, argued that space must be three-dimensional,
because distances are measured along perpendicular lines and that only
I Simplicius, Commentary on Aristotle's 'On the Heavens', 7a, 33. Quoted in [2],
p,l.
2 Galileo Galilei, Dialogue Concerning the two Chief World Systems (trans. S.
Drake), (Berkeley and Los Angeles, 1962), p. 14.
3 In his discussion Kant [4] also refers to the fact that similar but incongruent
two-dimensional counterparts, viz. a shape and its mirror image, can be made
congruent by rotation in a three-dimensional space; but similar but incongruent
three-dimensional counterparts, such as a left and a right hand, cannot be
brought into congruence by any known physical procedure. It was subsequently
proved in the nineteenth century that any two (n - I )-dimensional objects
metrically symmetrical about an (n - 2)-dimensional object can only be brought
into congruence in an n-dimensional space. The fact that the left and right hand
cannot be brought into congruence by any known physical procedure is only
further evidence for and consequence of the tridimensionality of physical space,
not a proof of the logical necessity or contingency of this fact.
120 Space and Time
Both these latter conditions hold in Euclidean geometry if the first
condition holds, but this is not so in all pure geometries. Given the latter
conditions and given that only three lines can be mutually per-
pendicular, it follows deductively that space must be three-dimensional.
But to prove that the conclusion holds of logical necessity, we need first
to show that geometries in which the two latter conditions stated above
do not hold when the first does hold are not logically possible
geometries. And secondly we need to prove that granted that in fact in
our Universe no more than three straight lines can be mutually
perpendicular at a point, this must be so of any universe. The logical
necessity of these conditions seems no more obvious than that space be
three-dimensional..Ptolemy has shown no firmer foundation for our
proposition.
In a different tradition some seem to have reasoned somewhat vaguely
that since for any n, there is a perfectly consistent n-dimensional pure
geometry, the fact that points of physical space need three and only three
coordinates for unique identification is a merely empirical matter. This
of course does not follow, for it would have to be shown that there is
nothing in the nature ofa point, in the sense elucidated in Chapters 2 and
6, which limits its necessary identifying coordinates to three. No.
proposition about the possibilities of pure geometry could prove that.
What is needed is an examination of the properties of points.
With the arrival of multi-dimensional geometries in the nineteenth
century and even earlier, a number of attempts were made to show that
the tridimensionality of space was contingent by attempting to explain it
as the conseq uence of the operation of some scientific law. For if it can
be shown that the tridimensionality of space is a consequence of some
scientific and so logically contingent law, then if a different law held
space would have different dimensions. Since any given scientific law
might, it is logically possible, not have held, the dimensions of space
would be a logically contingent matter. The prototype of such expla-
nation was, interestingly, given by Kant in a phase of his thought prior
to the writing of the Critique and the Prolegomena. He noted that
'substances in the existing· world so act upon one another that the
strength of the action holds inversely as the square of the distances' ( [3]
§ 10). Gravitational effect is propagated in inverse proportion to the
square of distances of bodies affected; and, as was discovered a few years
after Kant had written this, so too is magnetostatic and electrostatic
effect. Now let a source of attraction or repulsion be surrounded by
particles all lying at some specified distance from it and completely
enclosing the source. Let us call the sum of the changes of momentwn
The Dimensions of Space 121
produced by the source in all these particles at the distance the total
effect of the source at that distance. If we assume that the total effect of
some source of force is the same at any distance, and if we assume a
Euclidean geometry, as Kant of course did, then in a three-dimensional
space an inverse square law of attraction or repulsion must apply. For
the particles completely enclosing the source all lying at some specified
distance from it will form a two-dimensional surface, the area of which is
proportional to the square of the distance from the source. Conversely if
the attraction or repulsion on any particle is inversely proportional to
the square of the distance, and the total effect at any distance is the same
and geometry is Euclidean, space must be three-dimensional. Though
Kant does not making his argument completely clear, it seems to be as I
have stated it, and others ' used later the argument which I have stated.
This argument is perfectly valid, but the conclusion which the early Kant
wished to draw from its conclusion - that the tridimensionality of space
was logically contingent - does not follow. Kant claimed that owing to
the inverse square law 'the whole which thence arises has the property of
threefold dimension, that this law is arbitrary, and that God could have
chosen another, for instance the inverse threefold relation; and lastly
that from a different law an extension with other properties and
dimensions would have arisen' ([3] §. 10). This further argument is that
since the existence of inverse square laws as opposed to inverse cube laws
is clearly contingent, then if we suppose a different law to hold, e.g. an
inverse cube law, given also that the total effect of a force at any distance
from a source was the same and space was Euclidean, space would have
other dimensions. Hence the tridimensionality of space is a contingent
consequence of the existence of inverse square laws. But although the
existence of inverse cube laws is clearly logically possible, the question
arises whether the existence of inverse cube laws is logically compatible
with the operation of the principle of total effect in a Euclidean space.
Only if it could be shown that the three suppositions together describe a
logically possible universe, could the tridimensionality of our space be
attributed to the operation of inverse square laws. Only then would it
follow that God could have brought about 'an extension with other
properties and dimensions'. The later Kant no doubt saw this point.
Similar arguments purporting to prove that the tridimensionality of
space was the consequence of some scientific principles and so con-
tingent have been much used ever since Kant. 2 They mostly have the
o~------c o~------c
Figure 7
Such a world is, I now argue, inconceivable. U such a world were
conceivable, then an inhabitant or it, ir it were to exist, would judge that
our world was· rperely the surrace or a four-dimensional world, that the
material objects or our world seemed to us to have only three dimensions
but in ract had a rourth. Because it is not physically possible that our
objects move in a direction other than the three ramiliar to us we-
ralsely, according to such an outside observer - would claim that no
other type or movement was logically possible. But physical impossi-
bility would have, he would say, closed our eyes to logical possibility. If
there were a four-dimensional universe, this claim or its inhabitant
would be correct. For his argument would have exactly the same
structure as our argument to prove the impossibility or a two-
dimensional world. He would be aware or logical possibilities to which
our eyes are closed.
The Dimensions of Space 127
Now if our 'space' is in fact the boundary of a four-dimensional space,
there must be directions in our Universe other than those which are
combinations of the directions given by three mutually perpendicular
rods. Where are these directions? An object should be able to move in
directions other than back and forward, to this side and that, up and
down. But how could it? The suggestion that it might does not describe
anything which we can conceive, of which we can make sense. No
motion is conceivable which takes us 'outside' our three-dimensional
world and enables us to look at it 'thence'.
A second objection, similar to the first, is this. An observer outside our
hyperplane would be able to observe at any instant all the 'inside' of any
three-dimensional object known to us, such as a cube, just as we can
observe at an instant the inside of any two-dimensional object. This
initially seems conceivable. We imagine ourselves seeing the inside of a
cube if we look into it through a number of transparent layers. But this is
not seeing the 'inside' of the cube in the way that we see the inside of a
surface. We can see the inside of a surface without looking through
transparent layers of the surface from the edge. But what would it mean
to see the 'inside' of a cube without looking through its Dounding
surfaces? No sense seems to pertain to such a notion.
Thirdly, if our world was a four-dimensional world, a man ought to be
able to pass across a cube without passing across its two-dimensional
bounding planes; as we can pass across a two-dimensional surface
without passing over its one-dimensional edges. Yet no meaning would
seem to attach to such a purported description. For these reasons I
conclude that no sense can be given to the thesis that we live in a four-
dimensional space and hence Jo the thesis that space could be four-
dimensional. My objections apply afortiori to the suggestion that space
might have dimensions greater than four. ] therefore conclude that space
must of logical necessity have three dimensions.
Some have attenpted to make sense of a four-dimensional space by
picturing as in Figure 7 four rods mutually at right-angles. We imagine
ourselves measuring the six angles LAOB, LBOe, LeOD, LAOe,
LAOD, LBOD and finding that they each measured a quarter ofa circle.
But even if we discover that n and no more than n rods fit mutually at
right-angles, that 90es not by itself mean that we have found an n-
dimensional universe. It will only do so, as we have noted on pp. 119, if
further conditions are satisfied. We noted two conditions true in our
universe, the joint satisfaction of which guarantees that if four and no
more than four rods can meet mutually at right-angles, space is four-
dimensional. It is not, however, logically possible that one of these
128 Space and Time
conditions be satisfied if four rods meet mutually at right-angles. This is
the condition which states that each point of space P can be. uniquely
identified by its perpendicular distance from some point P' on some
object constructed by the procedure described on p. 119 from (n - I )
perpendicular lines but not from the object constructed by that pro-
cedure from (n - 2) perpendicular lines. This means that we would
have to be able to construct a three-dimensional object at some point 0
by moving "the plane OAB parallel to itself along the line ~C, and
measure distance along a line perpendicular to this object, viz. 0 D. Hence
OD would have to pass through 0 without touching the object at any
neighbouring point - for if it did touch the object at a point next to but
distinct from 0, it would not be perpendicular to the object. That OD
touch the object at no neighbouring point is clearly incQnceivable. Hence
it is inconceivable that any point can be identified by its perpendicular
distance from any object constructed from three mutually perpendicular
lines by the procedure described on p. 119. So the further condition
would not be satisfied. Even if four rods can be placed mutually at right-
angles,l that does not guarantee l\ four-dimensional geometry.
The position then with purported two- (and one-)2 dimensional
spaces is that a certain world which many might wish to describe as a
me Honor Brotman [6] has argued that it is conceivable that four rods meet
mutually at right-angles. However she has not shown this because in order to
establish that the angles are right-angles (in her terminology 'square angles') she
uses as a criterion for the: straightness ofa rod ( [6] p. 254) 'The look of the thing'.
But clearly a rod may look straight without being straight. The tests for the
straightness of a rod are that it lies along a straight line, the tests for which I have
described on pp. 63f. However, she may well be right that it is possible for four
rods to meet mutually at right-angles. But I claim to have shown that if four rods
could be arranged mutually aC right-angles this would only mean that the
Universe was four-dimensional if further sets of conditions were satisfied, and
that the most likely of these of logical necessity cannot be satisfied if four rods
meet mutually at right-angles.
2 Though there are difficulties with a one-dimensional space which do not arise
with a two-dimensional space. Given that - by definition - material objects
cannot be penetrated by other material objects, no measurements could take
place in a 'one-dimensional space' since 'material objects' could not there be
circumvented by measuring rods. Measuring rods which are not material objects
would not measure distance as we have analysed distance. So how could sense be
given to the notion of something being at a certain distance from something else?
Abbott's description of , Line land' in [8} sections 13-14 does not really meet this
difficulty. He replaclls the concept of distance by something bearing little
resemblance to it.
The Dimensions of Space 129
two-dimensional world was conceivable but that the description was
mistaken. For purported spaces of more than three dimensions the
position is that the situation which we would wish to describe as a
four-dimensional world is not a conceivable one. The concepts
such as straightness and angle by which we determine the distance
and direction of objects are such that it is not possible that a point
could not be identified uniquely by three measurements of distance
and direction. Of course one could so define concepts of 'straight' and
'angle' and 'place"that this was possible and so also that space be four-
dimensional. But to say this is no more informative about physical space
that it would be informative to be told about mathematics that with
suitable definitions of '2', '+' and ' = ' one could ensure that
'2 + 2 = 5'. With such concepts of straightness and angle as we have,
space must be three-dimensional.
But, it may be objected, how do we know that we are not in the same
situation as the inhabitants of the purported two-dimensional world
described earlier? It seemed to them that only two lines could be
mutually perpendicular. They made this mistake because it was not
physically possible for the 'objects' with which they were familiar to
move outside their surface. Might not we be making a similar mistake in
supposing that our space is three-dimensional because it is not
physically possible for the objects with which we are familiar to move
outside the three-dimensional hyperplane? It must be admitted that we
might be making just this mistake. But the concession that what we have
apparently proved to be clearly inconceivable might, just might, be
conceivable, is a concession that must be made, as similar concessions
must be made for all purported proofs in philosophy and mathematics
and in other fields. There might, just might, be a surface which was both
red and green all over at the same time, despite the apparent
inconceivability of such a surface. All the human race might be subject
to some odd mental blockage in failing to realise that such a surface was
a logical possibility. So too our proofs that a four-dimensional (or two-
dimensional) space is inconceivable might be wrong. But I urge that the
difficulties of giving sense to these notions are apparently overwhelming,
and hence the onus is on him who wishes to deny our thesis to overcome
these.
BIBLIOGRAPHY
Suppose we come across a tribe who have the following custom. Every
second year, the young men of the tribe are sent, as part of their
initiation ritual, on a lion hunt: they have to prove their manhood.
They travel for two days, hunt lions for two days and spend two days
on the return journey; observers gowith them, and report to the chief
on their return whether the young men acquitted themselves with
bravery or not ... while the young men are away from the village the
chief performs ceremonies - dances, let us say - intended to cause
the young men to act bravely. We notice that he continues to perform
these dances for the whole six days that the party is away, that is to
say, for two days during which the events that the dancing is supposed
to influence have already taken place ( [9], pp. 348f).
Is it logically possible that the chief's dancing of the last two days could
affect whether the young men had been brave? Suppose there is a strong
correlation between bravery of the young men and the chief dancing in
the last two days. We should naturally say in that case that the bravery
was the cause of the dancing and not vice versa, But suppose that ~ to
alI appearances - it is in the chief's power to dance or not. Whenever
the chief does dance, the evidence shows that the young men have been
brave. If the chief dances after the observers' report that they had not
been brave, the observers subsequently admit to 'lying'. ([9], p, 354),
Dummett concludes that in this situation we should say that the chief's
1 Both Ayer [2] and Russell [I] seem to have claimed that it is logically possible
that men should affect the past. Both have urged that it is only the fact that we
know so much about the past that makes us think that we Cannot affect it. But,
they claim, our relatively greater knowledge of the past than of the future is a
contingent matter.
136 Space and Time
dancing subsequent to the hunt produced the bravery of the young men,
made the young,men have been brave. He says that normally we assume
that a man can know whether a past event occurred independently of his
present intentions. This sort of case, he argues, would, however, force us
to reconsider this assumption.
There are here two issues which must be kept sharply distinct. One is
the main issue of whether it is logically possible that an action such as the
chief's dancing could have an effect at an earlier time. The other is
whether anyone could ever have any evidence that an action such as the
chief's dancing had an effect at an earlier time. A verificationist (see
p. 6) will hold that a negative answer to the latter question will entail a
negative answer to the former question; for him claims are only logically
possible factual claims if there can be evidence for or against them. I do
not wish to take the verificationist line, and so I will produce one
argument to show that there could never be any evidence that an action
had a past effect (without begging the question whether it is logically
possible that it could), and a second argument to show that it is not
logically possible that an action have a past effect. Someone who was not
convinced by the latter argument might still accept the former one,
Our grounds for believing that a past action has a future effect is that
before the action our evidence suggests that one ~hing is going to happen
in future, and after the action our evidence suggests that a different thing
is going to happen. Analogously any defender of the claim that there
could be evidence that an action had had a past effect must claim, as
Dummett does, that the action must produce changes in the evidence
about what happened in the past. Otherwise there would be no ground
for supposing that the action had any effect in the past at all. Now I shall
urge that no change in the evidence could possibly substantiate the claim
that some one had affected the past.
The evidence about what happened need not take the form solely or at
all of memory-claims by observers of the past event. It may alternatively
take the form of traces of the past event. Some present state of the world
is a trace of a former state (or event) if well-substantiated scientific laws
show that such a state (or event) normally precedes such a state. The
change in the evidence may therefore consist either, as in Dummett's
example, of a change in the memory claims, or of a change in the traces.
Suppose then that the advocate of the possibility of affecting the past
claims that an action at 12 , an instant subsequent to I l ' had the result that
the evidence about what happened at t 1 (that is, the memory claims and
observed traces about what happened) was different subsequently to 12 ,
let us say at t 3' from what it was at 11 and up to 12 . The evidence at 11 (and
Pasl and Fulure 137
up to 12) and the evidence at 13 (and at all instants since 12) suggest
ditTerent occurrences at I I' Then there are two alternatives. Either we
realise at 13 that a change has taken place at 12 in the evidence or we do
not. In the latter case we could never have any grounds for believing that
we had atTected the past, for we would have no knowledge of any change
in the evidence as to what had happened. So there would be no place in
our beliefs about the world for a belief that we could atTect the past. In
the former case we realise at 13 thatwe did something at 12 as a result of
which the observed evidence at 13 about what happened at I I is ditTerent
from what it was at I p one or other evidence having been atTected by the
action. This suggestion we must now examine more fully.
Which is the better evidence a bout what happened at I I - the
evidence observed at I I or that observed at 13? The first possibility (one
which Dummett does not consider) is that the evidence at I I might be
better and the evidence at 13 worse. But this claim cannot be made by an
observer at 13' For in claiming that the evidence at I I indicated more
truly than that at 13 what happened at I I' he is claiming that the evidence
available to him shows that the evidence at I I shows better what
happened at I I than does the evidence at 13 , But the evidence available to
the observer is the evidence at 13 , And what that shows is that what
happened at I I is what was indicated by the evidence at 13' Hence the
suggestion that the evidence at I I might be the better evidence proves
self-defeating.
The other possibility (the one which Dummett does consider) is that
the evidence at 13 might be the better evidence. This could arise for the
reason that the evidence observed at I I was not typical (viz. had more
evidence been observed at I I' different conclusions could have been
drawn). But in that case the past has not been atTected by the action at 12 ,
The only etTect of the latter is to improve the condition and appreciation
of the evidence. But if it be said that the evidence observed at 13 is better
evidence despite the fact that the evidence observed at (I was typical of
the total possible evidence, then the evidence at I I and at 13 cannot be
really about what,happened at II' For, we concluded earlier, evidence
about what happened at (I could always - it is logically possible - be
obtained at (I better than at any other instant. If the evidence at (3 is
more to be relied on than any amount of evidence at I I' then what it is
evidence of cannot really be what happened at I I: the description
allegedly of what happened at I I must have a covert future reference.
Let us apply these points to Dummett's example. The young men
return and the observers report that they have not been brave. The chief
then dances. The observers then say that they previously gave untrue
138 Space and Time
reports. The young men really had been brave after all. Now the
observers may believe that they lied, that they knew all along that the
young men had been brave. In which case they would have no grounds
for saying that the chief's dance had affected the past; the only effect of
the chief's dance was, they must conclude, to make them now tell the
truth. If others who heard the two reports believed that the observers
had lied, they too must conclude that the only effect of the chief's dance
was to elicit truth from lying observers. On the other hand the observers
might now believe that the young men had been brave without believing
that they had believed this when they gave their first report. They would
not in that case, if they were now being honest, say that they had lied
previously, but that they had been mistaken. Despite Dummett's choice
of the word 'lying' to describe what happened, a defender of the
possibility of affecting the past might describe the situation in these
terms. Now the observers or anyone else who accepted this description
of what happened might be making one of two possible claims. The first
is that at the instant of the occurrence, the observers were deceived. The
evidence at the instant of the occurrence was that the young men had
been brave, but that evidence was not fully observed or was insufficiently
appreciated at that instant. The observed evidence was untypical of the
total possible evidence and hence the first report of the observers was
mistaken. So the effect of the dance is to give a correct estimation of the
previous experience. It enabled the observers to see that they had been
subject to a deception. The dance affects the subsequent state of mind of
the observers. There is again no question of it affecting the past.
But the observers' assertion might be interpreted in a different way.
When the observers claimed that they were mistaken in their previous
report, they might be claiming that while the first report was' based
properly on the available evidence and that the available evidence was a
representative sample of the total (logically) possible evidence; neverthe-
Jess they now judged on subsequent evidence that the young men had
been brave, and that this latter evidence was to be preferred. But in this
case the statement 'the young men were brave' is not a statement solely
about what happened on the lion hunt, but, at any rate partly, about
something subsequent, e.g. about what the observers were naturally
inclined to claim about what happened on the lion hunt. The earlier
distinction in this case delimits 'the young men were brave' as a
statement with a covert future reference. But in that case the fact thaUhe
chief's action affects its truth value is no more puzzling than the fact that
future occurrences will affect the truth-value of'Father Jones baptised in
1964 the next century's greatest pianist'. For in both cases the influence
PaSI and Fulure 139
of the affecting action is really in the future. I conclude that on none of
the alternative interpretations do the observers or anyone else have
reason to suppose that the chief's dance affects what is past and gone.
And generally no one can have any reason to suppose that an action at 12
can affect what happens at an earlier time I •.
Note that the above argument has no tendency to yield the absurd
consequence that we cannot have reason to believe that an action at 12
cannot have an effect at 13. For after 13 we may havejust such reason. We
may know that there was a change at 12 in what the evidence suggests is
going to happen at 13. We take the evidence at, and around, 13 as
showing what in fact happened at 13 and therefore as better evidence
than the evidence at I •. The discrepancy shows that what happened at 12
made something to happen at 13 which would not otherwise happen.
But even if we would never have reason to believe that an action Cat 12
brought about an event E at an earlier time I., is it not logically possible
that it should bring it about all the same? I do not think so. For if C
brings about E, then the performance of C must make the difference to
whether or not E occurs. But it is logically possible that any event such as
C should have no cause, that an agent should freely choose at 12 to bring
about C without any cause making him to choose. In that case it could
not be predetermined before 12 that E occurs yet some one could observe
E happen at I. and know as surely as could be known that E was
happening then; and still it would be not yet predetermined whether or
not Ewould happen; it could be that a man's future choice might make E
happen or alternatively make it not happen. Here there does seem to be a
contradiction. It cannot be the case both that E is now happening and
that it is yet to be determined whether E will happen. And so it is not
logically possible that a man should know the former if the latter is true.
It is therefore not logically possible that a man could affect a past state
by his present action. Actions can only have present and future effects-
and they could only have present effects if influences could be
propagated with infinite velocity.
Now granted that an agent cannot by his action affect what is past and
gone but only what is to come, it follows that no state (or event) can have
its cause subsequent to it.' For if we say that in certain circumstances an
event A was the cause of a state B, we thereby commit ourselves to the
proposition that a man could have made B happen by making A happen.
1 Several recent philosophical articles have made this point, producing different
arguments in support of it. See [4], [5], [7] and [II]. See [3],[9] and (12] for the
opposite point of view.
140 Space and Time
But since, as we have seen, it is not logically possible that a man can
make B happen by making A happen if B is prior to A, then A cannot be
the cause of B if B is prior to A.
One might attempt to avoid this conclusion by denying that the
implication of the last paragraph holds in cases where A is outside
human powers of control. One might thus claim that to say that in
certain circumstances A is the cause of B does not commit you in cases
where A is a kind of state (or event) outside human control to saying that
if a man were to make A happen in those circumstances he would
thereby bring about the occurrence of B. One would thus be claiming
that what is involved in saying that A is the cause of B is different in cases
where A is something lying outside human control. Thus 'cause' would
have a different meaning in The explosion of the bomb was the cause of
the devastation' from 'The explosion of a star was the cause of the gas
cloud', because we can make bombs but not stars explode. But this is
not plausible, for the boundaries of what humans can do are continu-
ally extending without the meaning of 'cause' thereby changing.
'Chromosomal mutations cause the appearance of new phenotypic
characteristics in organisms' did not change its meaning when men·
became able to produce chromosomal mutations. There is no reason to
suppose that the meaning of ' The explosion ofa star was the cause of the
cloud' will change when humans learn to make stars explode. Rather
'cause' retains its meaning, while the boundaries of what humans can
cause change. I conclude that oflogical necessity no cause can follow its
effect and hence that of logical necessity the past is determinate.
States of objects at any instant t 2 of logical necessity cannot, we have
shown, be affected by states subsequent to t 2' Hence either they are
uncaused (they just happen, nothing produces them) or they are
produced by states at t 2 or prior to t 2' Now consider the state or the
Universe in a region R[ at a time t p which we will calI state Sp and the
state of the Universe in that or some other region R2 at a later time t 2'
which we will call S2' Now, as we have. seen, it is not logically possible
that S 2 be the cause of S [. But it is always logically possible that S [ be the
cause of S2' We can conceive of evidence from other states at other
periods of time which would confirm the theory that S[ was the cause of
S2' The evidence would be observations that states similjrt to S[ in
regions similar to R[ are always followed by states similar to S2 in
regions similar to R 2 , the regions having to each other the relation that
R [ had to R 2 and the states occurring a t the time interval t 2 - t [; or the
evidence could be more indirect evidence showing that such connections
were to be expected in nature. Whether or not S [ is the cause of S2 is a
Past and Future- 141
logically contingent matter and depends on what are the laws of nature
governing the succession of states. This being so, it is always logically
possible that any specified future state of a region of the Universe be
caused by any specified present state; whereas, as we have seen, it is not
logically possible that any past state of a region of the Universe could be
caused by a present state.
From these basic principles a number of important necessary truths
about time will be proved in this and subsequent chapters. The first of
these which I shall demonstrate here is that the present instant cannot
ever return, that is that time cannot ever be closed. If the present instant
t 1 will return, then the next instant subsequent to this one, t 2' will be
both before and after t l' Yet in virtue of being before t 2 the state of any
region of the Universe at t 1 is of logical necessity un affectable by any
state at t 2' Whereas in virtue of its being after t 2' it is logically possible
that any state at t 1 could be affected by the state at t 2' Hence the
supposition of a cyclical or reversing time leads to self-contradiction. 1
This is not to deny that it makes sense to talk of a cyclical or reversing
Universe. To say that the Universe is cyclical is to say that after so many
years its state is exactly similar to what it was and an exactly similar
series of events take place again. S 1 is followed by S2' S2 by S3' S3 by Sp
S 1 by S2 and so on ad infinitum. The Stoics believed in just such a cyclical
Universe. But the point is that S 1 comes again at a later temporal
instant, not at the same instant. So with a reversing Universe. To say
that the Universe is reversing is to say that after so many years an exactly
similar series of states of the Universe to the series which has occurred
1 This point suffices to rule out on purely logical grounds one cosmological
model of the Universe developed from the General Theory of Relativity, the
Godel model. The GOOel model has closed time-lines allowed the physical
possibility of an agent influencing the past. Gooel claimed that it would be, at any
rate in 1949, practicany impossible for an agent to achieve a sufficiently high
velocity relative to his galactic cluster (at least ~2 x the speed of light) to have
effects in the past, and seemes to think that this avoids the logical difficulties of
his model. But ifhis model allows as physically possible something which cannot
be physically possible because it is logically impossible, the model must be wrong.
For the. mathematical presentation of his model see K. Godel, •An Example of a
New Type of Cosmological Solutions of Einstein's Field Equations of
Gravitation' in Reviews of Modern Physics, 1949,21,447-50. For his attempt to
deal with the logical difficulties of the model see K. Gooel, •A Remark about the
Relationship Between Relativity Theory and Idealistic Philosophy' in Albert
Einstein: Philosopher-Scientist (ed. P. A. Schilpp) New York, 1949 and -1951, pp.
555--62. See especially pp. 56Of.
142 Space and Time
occurs in the reverse order to that of its original occurrence. S, is
followed by S2' S2 by S3' S3 by S2' S2 by S,. But the point, as before, is
that S, comes again at a later instant, not at the same instant'.
BIBLIOGRAPHY
Our most usual grounds for taking some particular state or event as a
trace of some other state or event is that observers have reported that
states or events similar to the first are normally accompanied by statj!s or
events similar to the second. I take a present state A I as a trace of a past
state B I because observers, including possibly myself, report that
. whenever an A occurs, a B has always or almost always preceded it (I use
the capital letters without subscript for a type of event or state, and the
letter with subscript for an individual event ot state of the type referred
tp by the capital letter). The footprint is a trace that someone has walked
on the sand r'ecently because in the experience of men footprints (in' the
·sense of marks in the shape of human feet) are normally preceded by
someone walking on the sand. We take a present stakel as a trace of a
future state DI because observers report that at other times whenever a
C occurs, a D always or almost always follows it. The fall in the
barometer is a trace of impending rain because in the experience of men
fal!s of the barometer are normally followed by rain. The announcement
in The Times is a sign that the meeting is now taking place because men
have experienced that announcements in The Times of events for the day
of publication are normally correlated with the occurrence of those
events. In so far as the reports are many and corroborated, and the
reports are of invariable correlations (e.g. Cs bej.ng· followed by Ds
without exception), by so much the trace is better evidence of what it
indicates. We may use more complicated combinations of evidence of
the kind described. We may take an event FI as a sign of a precede~t
event HI by using a chain of evidence. We may never hav.e observed Fs
being preceded by Hs, but we may have observed that Fs are normally
preceded by Gs, and Gs by Hs. It is in this way that we conclude that
some present geological state is evidence of a long-past state.
Our grounds fOf taking some event as a trace of another may be more
remote. Various miscellaneous observed connections between events
may be most simply explained by setting up some scientific theory, a
deductive consequence of which is the law 'Al1 A s produce Bs' .This law
then allows us to take an A I as a trace ofa future B I . No one may ever
have observed any A being followed by a B, but the evidence for the
truth of the law may be more indirect. Thus by studying stellar spectra
and masses, we may obtain evidence for a theory of stellar evolution.
One law derivable from this theory might be the law that stars of a
certain type explode after reaching a certain stage of evolution, viz. the
law that a certain state of a star A produces a subsequent event of an
148 .Space and Time
explosion B. We establish the law without ever having observed stars of
the type in question exploding. Hence when we observed a particular
star in a state A 1 we would take this as a trace of a future explosion HI'
Yet by whichever way we learn to take A 1 as a trace of HI our'
justifi.cation for doing so lies in the reliability of the reports of observers.
For this reason reports of observers provide the ultimate evidence of
what happens at other places and temporal instants. The procedure of'
postulating that observed cortnections or inferred laws operate beyond
the range over which they have been observed to operate is the
procedure which I have called in earlier chapters extrapolation.
There is an important asymmetry between the use of scientific laws to
establish traces of future events and their use to establish traces of past
events, between prediction and retrodiction. Scientific laws state how
events and states of one type, As, cause events or states of another type,
Bs. They may not use the word 'cause', but this is their forin. They state
the consequences of tpe occurrence of an A, however the A may be
brought about. Consequently. they are of no use as they stand for
retrodiction; without a further premiss. We may have good evidence
that As cause Bs but that by itself does not allow us to take BIas a trace
'of a precedent A l ' For many other states than As may cause Bs. Only if
we have grounds for believing that Bl had a cause and that any other
cause of Bs than As would have produced other effects which As do not .
produce and theSe are not observed, can we take BIas a trace of
precedent A I ' Our grounds for supposing that B 1 had a cause is that
science has shown us events usually do have causes, and explanations
can be provided for the previously mysterious. But we would have.
reason to believe that BI did not have a cause if we could show that only
a certain number of states could bring about Bs and that none of them
were operative on this occasion. How we could show this will be
illustrated at some length in Chapter 15 when we shall be discussing an
important example of a state which, it is suggested, had no cause. If then
we suppose that B I has a cause, in order to show that it was A 1 we have'
to show that any other cause of Bs than As would have produced other
effects as well, but that these. have not occurred. But we can only
investigate the posiible occurrence of effects of a finite number of
possible causes. So only if we have reason to believe that only one of a
small finite number ofpossible causes was responsible for BI can we test
to show that A I is responsible. But what reason could we have for
supposing that only one of a certain small number of possible states C l '
D 1 , £1' F 1 , etc. could have brought about HI? One reason we could have
is that we (or other observers) have observed the state of the Universe in
Logical Limits to Spatio-Temporal Knowledge 149
I. We believe that human heads tum into skulls in this condition after
a million years. This is a.well, albeit indirectly, established scientific law.
Hence we have a possible cause.
2. .A few other known possible causes of the skull being in that
condition have left no other effects; viz. there is no evidence of forgery of
a k~owri.type.
3. We believe that other possible causes of the skull in that condition
are seldom if ever operative.
But our only grounds for the latter beliefis that we have never-observed
them in operation. These are strong grounds, but they do not show at all
conclusively that another cause could not have been operative in this
instance. On this occasion perhaps an unknown cause, viz. a cause at the
description .of which we c;mnot guess, may have brought about H._ We
cannot conduct any experimental test to refute this suggestion - for
reasons of logic.
If someone suggests that an unknown effect is caused by some event i
H. in certain circumstances, we can - it is logically, if not always
practically possible - do an experiment to ~onfirm or refute the
suggestion. We could produce an H in similar circumstances and see
what happened. But for reasons oflogic, as we saw in the last chapter, we
cannot do a comparable experiment to' test whether an unknown cause
brought about a certain effect. For we cannot make things to have
happened. Hence there is of logical necessity a weakness involved in
150 Space and Time
retrodiction which is not involved in prediction. Scientists do not
postulate backward-moving laws because they could never be experimen-
tally tested, that is, tested in the way just described.
Now it is true that some forward-moving scientific laws cannot be
experimentally tested, and this is true in particular of cosmological laws',
laws predicting a future state of the universe millions of years ahead
caused by a present state. This cannot be tested experimentally because
humans cannot bring about states of the Universe. But there are two
differences from the case of any purported backward-moving laws. First
th~ limit is physical or practical, not logical. It is juSt a contingent matter
of fact that we cannot bring about a certain state of the Universe and see
what happens. Secondly, cosmological theories can be subjected to
. indirect experimental test, because a law forming part of a more general
theory, viz. normally the General Theory of Relativity, from which'they
are deductively derived by the addition of astronomical data, can be
subjected to experimental test on the small scale. But no purported
backward-moving law' could be subjected even to indirect experimental
test, for the only theoretical law of which it would be a deductive
consequence would be a backward-moving law and this again - for
reasons of logic-could not be subjected to experimental test.
The use of scientific laws to establish that a state is a trace of some
other state at the same temporal instant, suffers from ttre same weakness
as the use of scientific laws for retrodiction and for the same reason-
given, as scientific evidence shows, that no effects are propagated with
infinite velocity. For given this, no law stating that one state is correlated
with a contemporaneous one could ~e experimentally tested. Hence
science does not purport to put forward any real laws of co-presence,
any more than. it puts forward backward-moving laws. Certainly there
are a number of well-established scientific principles, stating how
various contemporaneous characteristics are correlated. Biology is full
of such principles as that tortoise-shell cats must be female. But the form
of such principles is misleading. The zoologist in stating that tortoise-
shell cats must be female does not really mean that of physical necessity
if a genetic engineer makes a cat to be tortoise~shell, he thereby causes it
to be female. What he means is that tortoise-shell cats bred from the
current stock of the cat spedes by normal processes must be female. But
in that case the principle is not a law of co-presence. The principle states
that whatever arrangement of a genotype drawn from the genetic pool
currently availa~le to the feline species produces cats of tortoise-shell
colour also produces the normal female characteri~tics in them. The
principle is in fact a summary of two forward-moving laws and a
•
Logical Limits to Spatio- Temporal Knowledge 151
historical statement about the genetic pool currently available to the
feline species. .
Any proposed real law of co-presence would be unable to be tested
experimentally, just as any proposed backward-moving law would be
unable to be tested experimentally. For suppose a scientist claims as a
law of nature that a state A is always co-present with a state B. Now
given that effects are never propagated with infinite velocity, the state A
cannot be the cause of the state B or conversely. Hence if the one state is
correlated invariably with the other, it must be because the two states
have a common cause. But then'one could only prove the correlation to
be invariable if one could prove that it was physically necessary that
there be a common :cause. But for the reasons given above one cannot
prove at alfconclusively that some effect must have a certain cause, and
so one cannot prove at all conclusively that two characteristics must be
correlated. Genetic engineers of the next century may well produce male
tortoise-shell cats.
So then we may use traces to infer the occurrence of past, present, or
future states. We may take a'state as a trace of another on the basis of
directly observed connections or on the basis of an inferred -scientific
law. If the scientific' law established is true and concerns invariable
correlations (and includes all the necessary qualifications for surround-
ing circumstances) then the trace of the future state which is predicted by
the law will be absolutely reliable. Traces of both past and present states
could not be proved to be such by experimental test - for reasons of
. logic; and hence there are logical limits to the kind of confirmation
which they could receive of their status. Proof that a trace is a trace of a
past or present state depends at a crucial point on mere observational
correlations, about which it cannot be tested experimentally whether
they hold universally.
Evidence from one trace may, conflict with evidence from another
,trace. We may have good reason to take an observed present state A 1 as
evidence of another present state Bland also good reason to take an
observed present state C 1 as evidence of another present state DI which
could not co-exist with B I' Here of course we take the best substantiated
connection as providing the most reliable indication. If the connection
of As and Bs is better established than the connection of Cs and Ds, then
the two observed states A 1 and C 1 taken together are traces of a present
state B I' They show that the connection between Cs and Ds does not
hold universally. We saw this process at work in Chapter 5 (pp. 94ff)
where estimates of the distance of distant objects which assumed one
regular connection to hold universally were opposed to other estimates
152 Space and Time
which assumed a different connection to hold universally. The issue
turned on which connection was best established. One which was a
consequence of a well-established theory of mechanics was better
established than one which depended on mere observation of a few
instances.
The supposition that observed connections hold beyond the range for
which they have been tested becomes less and less reliable, the greater
the difference between the range studied and the range in which they are
supposed to bold. We have'teason to assume that what is a trace within
the observed region is a trace in a region outside that observed - so long
-as the unobserved region does not have characteristics, including spati()-
temporal location, .too radically different from the observed one.
Suppose that I was born at a place P in the Sahara desert and that my
knowledge of the Earth is confined to a region of radius fifty miles
surrounding P. This I have explored, but I have no knowledge by
personal exploration or from reports of othertFavellers of regions more
distant. I thus conclude that a square mile of sandy desert at some
distance from P is an unfailing trace that there is another square mile of
sandy desert at a distance from P greater by one mile. Clearly J am
justified in using these traces to reach conclusions' about the geography
of regions up to, say, sixty or seventy miles from P, that there is nothing
but sand there. I reach such conclusions by taking an observed square
mile of sand as a trace of a square mile of sand beyond it and the latter, to
which I have inferred, as a trace of a square mile of sand beyond it, and
so on. But clearly the further away I claim to have knowledge of the'
geography of the Earth, the less justified are my claims. Claims to·
knowledge of the geography of regions 1000 miles from P on the basisof
an examination of regions within fifty miles of P would clearly be very
weakly justified. The further from P I stake my claim to knowledge,
relatively to the size of the region which I have examined, the -less
justified my claim will be. This is because the region to the properties of
which I infer will differ more and more, at any rate in respect of spatial
location, from the region which I have studied. The possibility looms
larger and larger that the region which I have examined is untypical of
the regions of the Earth about which I am making inferences. Traces
claimed to be reliable on the basis of examining a small region become
less and less reliable, the further they are extrapolated. Similar
arguments apply to the reliability of inferences to e~entsat different
temporal instants.
The observations which wemake in order to reach_conclusions about
a temporal succession (though not about co-presence) must be obser-
Logical Limits to Spatio-Temporal Knowledge 153
vations of temporal successions. Only on the basis of directly perceived
temporal ,successions can we infer to others. Some writers on the
philosophy of science have, however, wanted to urge that knowing that
certain events happened we could arrange those events in temporal
order without basing that arrangement on any direct perception of
temporal succession, but simply by means of the principle enunciated in
the last chapter that causes precede their effects. Now certainly of any
two particular events A 1 and B l' if we can say that B 1 is cause and Al
effect, Jhen we can say that B\ happened before A l' ,But how can we
know that B\ is cause and A 1 effect unless we have observeq that in
similar circumstances As normally follow Bs, but Bs do not normally
follow As, or had more complicated'evidence of temporal succession,
such as that Gs follow Hs, but Hs do not follow Gs, from which we
constructed a theory, a deductive consequence of which is that As cause
Bs? If all we knew was that As and Bs or some other phenomena
occurred we could suppose that they had occurred either in one order or
in the reverse order. On one supposition we would conclude that Al
must precede Bp on the other that Bp must precede A l' We have to
observe straight off many temporal successions in order to infer 6the~s.
All the writers who have attempted to derive conclusions about
temporal order from premisses not referring to it and thus set up what
they have called a 'causal theory .of time' are guilty of some pelifio
principii. The most celebrated example is Reichenbach. Reichenbach
claimed that we could find out which of two events was cause and which
effect by the following principle of temporal ordering: 'If E 1 is the cause
of E 2, then a small variation (a mark) in E1 is associated with a small
variation in E 2, whereas small variations in E2 are not associated with
a
variations in E l' ( [I] p. 136). Thus suppose we have beam of light
passing between one slit (its presence there being the event E I) and
another (its presence there being the event E 2 ). We wish to ascertain-
which way the light is streaming, viz. whether E\ is the cause of E2 orvice
versa. We therefore put a piece of green glass in the way of the light at
one instant at the first slit (viz. modify E1 by making the light at the first
slit green) and then at another instant at the second slit (viz. modify E 2 •
by making the light at the second slit green). In this way, Reichenbach
claimed, we could discover which of Eland E2 were cause and effect
without relying on any direct, viz. non-inferred, perception of temporal
order. For, indicating marked events (that is light made green) with an
asterisk: 'We observe only the combinations E1 E 2, E1* E/, E\ E2 * and
.never the combination E 1* E 2' ([I ], p. 137). Hence we conclude that E 1
is the cause; because if it is affected, E2 is also - butthe converse does
154 Space and Time
not occur. The fallacy should be apparent. I How are the pairs grouped?
The events Ep E 1· , E2 and E 2 • have occurred a number of times. So
much we know without presupposing direct perception Of temporal
order. But in judging that E I • went in a 'combination' with E 2· rather
than with E2 which also occurred. ~e are relying on the fact that we
perceived E 2· to occur immediately before or after E I · whereas we
observed E2 to occur on another occasion. .
So then some judgements of temporal precedence must be made in
virtue of direct perception of temporal order by observ<:rs. However.
once we have made some judgements of temporal priority from direct
observation, we can make other judgements of temporal priority on the.
basis of the first judgements. Thus if I have noticed that in other cases
craters of a certain type are always preceded by'bomb explosions, but
that the occurrence of such craters is not generally followed by bomb
explosions, then, knowing that within a small temporal period there was
a crater formed and also an explosion at a certain place, I can judge that
the explosion preceded the formation of the crater.
It is, we have argued, by traces and reports that we learn about events
at other temporal instants and places. If we learn .about a past event or
state of some object by observing a trace of it, where the trace is an effect
produced by it, and the trace is one from which we become accllstomed
to make inference to the past event, than we may be said, in a derivative
sense of the term, to observe the past event or state.
An object or property or event is observed by an observer in the usual
and obvious sense if he has knowledge of it as a result of stimuli from it
impinging on his eye or other sense organ. An obj¢Ct or property or
event is observed in a derivative sense if the observer observes in the
usual sense effects produced by it which can be readily interpreted by the
observer so that he knows about the object, property, or event. Thus we
are said to observe the structure of the cell wall of the amoeba when we
observe in the usual sense a photograph of it produced by an electron-
microscope. Distant galaxies are said to be observed if we observe in the
usual sense photographs of them produced with the aid of a radio-
telescope. To say that something is observable is to say that either it is
observable in the usual sense or it is observable in the derivative sense, by
which is meant that either its effects can now be readily interpreted or
that a machine by which the effects could be turned into traces which
could tx! easily interpreted as effects of that thing could be 'constructed.
Temporal instants t, t2 t3 t4 t5 t6
Experiences E, E2 E3 E4 E5 E6
Corroboratable reports S, S2 S3 S4 S5 S6
(supported by traces)
BIBLIOGRAPHY
I 'Different times are but parts of one and the same time.' [1], B.47.
166 Space and Time
knows about the event because he has observed a trace of it, the event
must be prior to, simultaneous with, or posterior to the trace. For, as we
saw in the last chapter, our usual grounds for taking an event A 1 as a
trace of another event B I, are that As always follow, are simultaneous
with, or precede Bs. In these cases respectively A 1 will follow, be
simultaneous with or precede B I' If our grounds for taking A 1 as a trace
of BI are less direct, they will be that it is the consequence of a welJ-
established scientific theory that As are correlated with Bs and the same
conclusion will follow. For a scientific theory, as we saw in the last
chapter, states the consequences of some occurrence, what it brings
about. Hence all the theory can tell us is that some event will follow
another one, or two events will follow a common predecessor or
something of that sort. So any event which is known, as a consequence of
the adoption of a scientific theory, to be a trace of another event will be
prior to, simultaneous with, or subsequent to it. Hence all events about
which at a given instant an observer has knowledge occur at instants
connected with the present instant by a temporal chain, viz. are before or
after an instant which is before or after an instant ... which is before or
after the present instant. So if the relation of being temporally related is a
transitive one, every event of which at instant t 1 a man has knowledge
must be temporally related to, viz. prior to, simultaneous with, or
posterior to possession of the knowledge at t I' Further, if the relation of
being temporally related is a transitive one, then since every event of
which an observer has knowledge at t 1 is temporally related to his
possession of the knowledge at t I, every such event will be temporally
related to each other. Hence the claim that there were events not
temporally related to each other could have no evidence produced in its
favour. For evidence would be evidence about events at other instants
and the only ways in which we could learn about those events would be
ways which presuppose that the events are temporally related to the
event of learning about them.
If on the other hand, it is not a logically necessary truth that being
temporally related is a transitive relation, then there could be three
events or states E 1 , E 2, and E3 connected by temporal chains such that
EI was after or before both E2 and E3 but E2 and E3 were not temporally
related.
The question therefore arises whether it is a logically necessary truth
that being temporally related is a transitive relation. We argued (p. 28),
that being spatially related is of logical necessity a transitive relation. If a
place A is spatially related to a place Band B to a place C, then A and C
must be spatialJy related. For if A is spatially related to B, a traveller
Times and the Topology oj'Time 167
could, it is logically possible, travel by local motion from A to B, and if B
is spatially related to C, a traveller could, it is logically possible, travel by
local motion from B to C. Hence it is logically possible that a traveller
could travel by local motion from A to C, if need be, by B. Hence A and C
must be spatially related. A similar argument will not work, however, in
the case of time. One does not travel by local motion between instants of
. time; and there is no obvious analogue in the case of time for this process.
Since, as we have seen, all instants about which at a given instant we
have knowledge must be connected with it by temporal chains, evidence
that there are two times could only have the form of evidence that a
present instant will be followed or has been preceded by two instants
which are neither prior to nor subsequent to one another. Could one ever
have such evidence? To make plausible the suggestion that one might
have such evidence, I shall recount a myth on the basis of which it might
appear that men could claim with justification that a present instant had
been preceded by instants not temporally related to each other and they
by a common instant. If we accepted that the myth showed the latter, we
would accept that it showed that there were two times; and hence we
would have shown that it was logically possible that there could be two
times. I shall, however, suggest that we ought not to accept this account
of the myth, and that there is a conclusive objection to our doing so. I
shall then urge that this objection must apply to any other two-time
myth that could be constructed, and hence that we could never have
knowledge of events belonging to two different times.
The myth is as follows. Two warring tribes, the Okku and Bokku, live
in the land of U g: The land of U g is the only known inhabited land. The
two tribes are assembled before their seer. He says to them: 'You have
been fighting together too long and you clearly cannot live at peace with
one another in this land of ours. I am reluctant to banish either of you
from the land you love so dearly. However I have decided that while you
may live in the same place as before, you must live completely separate
existences. This will go on for the next twenty years as far as the Okku are
concerned and for the next thirty years as far as the Bokku are concerned,
the present members of the Bokku tribe being the more quarrelsome.'
The seer then waved his wand. At this point, according to the Okku, the
Bokku gradually disappeared from sight and the Okku inhabited the
whole land and began to quarrel among themselves about who should
own the territories suddenly vacated by the Bokku. The Bokku story is
that the Okku gradually disappeared from sight and the Bokku
inhabited the whole land and began to quarrel among themst:lves about
Who should own the territories vacated by the Okku. Later {twenty years
168 Space and Time
later by Okku clocks and thirty years later by Bokku clocks), there
happened a mysterious event differently described by the two tribes.
According to the Bokku. the Okku suddenly reappeared in their midst;
while the Okku claim that the Bokku suddenly reappeared in their midst.
The members of the two tribes in the intervals of fighting over disputed
territory then exchange their stories.
At the instant of reunion, according to both tribes, certain changes
occurred in their buildings and possessions and to some extent natural
environment. The changes, however. are differently described by the two
tribes. The result of the changes is a compromise between the world
known to the Bokku immediately before the reunion and that known to
the Okku. During the thirty years since the separation of the tribes
during which the Bokku lived, according to their account, in the land of
U g, they effected, on their account, certain changes to it. So, according to
them, did the Okku, during the twenty years between the moments of
separation and reunion when they inhabited the land ofUg. Both tribes
claim to have built new houses and cut down old trees. Some of the
houses claimed to have been built by each tribe remain after the reunion
and some of the differences in the environment claimed to have been
effected by each remain. Others do not.
Now notice the important features of this myth. Each tribe has a
perfectly coherent, well-corroborated, and unbroken account of its
history. There is no question of either account being the account of a
mere dream. The Okku, for instance, have, they all say, always lived in
the land of Ug. The landmarks are the ones that their fathers and
grandfathers have told them about. Many years ago, the Okku say, the
Bokku lived there too but they were so troublesome that a seer waved
them away. Unfortunately he brought them back again only recently.
The Bokku also claim a continuous history in the land of Ug, while
claiming that it was the Okku who disappeared for a period. The detailed
accounts given by the two tribes differ only in respect of the events of the
intennediate period. Yet each account is well-authenticated, for all the
members of each tribe agree with each other as to what happened.
Now what would be the evidence available to us as observers in the
land of Ug after the reunion about where and when the events of the
'intermediate period' occurre'd? How are they to be fitted on to the
spatio-temporal framework? The evidence of the Bokku and the Okku is
that both series of events occurred in the land of U g. If we accept this and
also suppose that there was only one period of time, one series of
temporal instants, between the instant of separation and the instant of
reunion then we have to postulate the operation of some extraordinary
Times and the Topology of Time 169
physical laws. What happened at the instant of separation, a holder of
this interpretation would urge, was that each tribe became invisible and
intangible to the other and that implements wielded by one tribe had no
affect on the other. The interpretation, however, would have to become
very complex to deal with some phenomena. For instance, if an Okku
chops down a tree at some instant (Ion Okku clocks, no Okku can lean
on it after (1- However, a Bokku can lean on it until the reunion. By all
Okku clocks the period of separation lasted twenty years; by all Bokku
clocks thirty years. We could describe in detail how certain physical laws
ensured that all the observers of one tribe suffered various visual and
tactual illusions. But th~ story would clearly be tremendously
complicated.
To avoid this awkward interpretation we might postulate that the two
series of events occurred at different places from each other. This
interpretation is not, however, a natural one. For the Okku and the
Bokku are at all instants surrounded by qualitatively similar objects to
those of the land of Ug before the instant of separation e:J(cept for the
members of the other tribe, and hence we have little grounds for saying
that one or both tribes have changed their place. Normally, by the
criterion of similarity (p. 21), if we find ourselves 'surrounded by
qualitatively similar objects to those by which we were previously
surrounded, except for a few which are no longer present, we judge (at
least provisionally) that the few objects had moved, but that we ourselves
had remained in the same place. In both stories indeed there are two
mysterious events. At one instant, the other tribe disappears; at a second
instant it reappears and there are a few odd physical changes to the
environment at the same time. But the occurrence of two mysterious
events in its history does not mean that a tribe has moved to a new place.
A further awkwardness for the two-place interpretation would arise if
the Okku and Bokku both had evidence (in the way outlined in
Chapter 2) that if the other tribe was situated at another place during the
period of separation, that place would not be spatially related to their
own place. We would then have to postulate that each lived in a separate
space during the period of separation. Further, if we did really wish to
claim that at the instant of separation one tribe moved to another space
and returned to the land of Ug at the instant of reunion, would we not
have to make the same claim about the other tribe? For the fact that both
tribes experience equal upsets means that we have no justification for
claiming that one tribe, say the Okku, had moved to another space, while
the other t.ribe, the Bokku, remained put. In that case we would have to
postulate the existence of three qualitatively virtually identical spaces. To
170 Space and Time
claim that there must be two or three different spaces solely because a
place ceased to be inhabited by a certain group of people seems to stretch
the criteria of evidence for identity of place beyond those described in
Chapter 1.
If we accept that to postulate that new and very odd laws of physics
operate in the intermediate period or that the events experienced by the
two tribes occur in two different places are both too complicated
hypotheses, an alternative description in terms of two times is unavoid-
able. Time, on this interpretation, became split at the instant of
separation into two streams Okku-time and Bokku-time, which were
subsequently reunited. Each instant in Okku-time would be temporally
related to each other such instant and to instants in Ug-time (viz. instants
at which events occurred in the land of Ug before the separation and
after the reunion) but not to any instant in Bokku-time; and conversely
for Bokku-time.
If we refuse to allow the different physics or the different place
interpretations, the two-time interpretation of the myth is inevitable. For
take some event 0 in the story told by the Okku and another event B in
the story told by the Bokku. Both events occur, we assume, in exactly the
same place in the land of U g. Now 0 cannot occur at the same instant as
B -- since whatever occurs at the same place and at the same instant as
another event would (unless a radically different physics is postulated for
the intermediate period), if it is the sort of thing such an observer
observes (viz. the observer must have and use the senses and the
categories appropriate to observing the event), be observed by any
observer of the second event. Yet, ex hypothesi, 0 cannot be observed by
any observer of B. Nor can 0 have occurred later than B. For in that case,
if an observer of B stayed long enough at the place with his eyes skinned
he would observe O. But, ex hypothesi, he would not. The same
argument, mutatis mutandis, holds against the suggestion that 0
occurred earlier than B. So of two well-authenticated events 0 and B, 0
occurred neither before, nor at the same temporal instant as, nor after B.
So, to generalise, 0 and all events in the Okku history of the intermediate
period are not temporally related to B and all other events in the Bokku
history of the intermediate period.
Now we have seen that in some respects the different physics and
different place interpretations of the myth are very awkward, and the
principle of simplicity might seem to lead us to choose the two-time
interpretation. But I believe that to accept the two-time interpretation
would be to deny a logically necessary truth and hence one of the other
interpretations would have to be taken.
Times and the Topology 01 Time 171
I Kant claimed that this was a synthetic a priori truth - [[], 8.47.
Times and the Topology of Time 173
unbounded - after some instant t I there may be no Universe (this issue
will be discussed in Chapter 15). But this is only true if there is some
period after t I during which there is no Universe.
We saw in Chapter 8 that of logical necessity the same temporal
instant never returns. Since time is of logical necessity unbounded, it
must therefore of logical necessity be infinite. Since before every period
of time having a beginning and after every period of time having an end
there must be another period, and since the same instant and so period
never returns, there is no limit to time:1t has gone on and will go on
forever. Space, as we saw in Chapter 6, is different. It is of logical
necessity unbounded, but, as it may be the case that the series of places of
some finite size is finite for in every series of such places we may always
come to the same place again, space mayor may not be infinite.
Space, we concluded in Chapter 3, would not exist if there were no
physical objects in space. We have now reached the conclusion that time
would exist without physical objects. I As Shoemaker [4] has argued,
awareness of the passage of time involves awareness of change in or of
objects. Minimally, this awareness may be simply awareness of a change
in ourselves (e.g. me passing from a state of knowledge that it was a short
time since so and so happened, to knowledge that it was a long time). But
normally of course it will be awareness of things changing in the public
world. Hence there could not be awareness of the passing of time without
there being change in the world. But it does seem logically possible that
there should be periods of time in which nothing changed and,
Shoemaker argues, it would be logically possible to have evidence before
or after its occurrence that there was a period of the world's history
during which nothing changed. Shoemaker imagines a world divided
into three regions, A, B, and C. In A something becomes motionless
('freezes') for a year every three years; this can be observed on most
occasions from the other regions. In B everything freezes for a year every
four years, and in C everything freezes for a year every five years; and
normally these freezes can be observed from other regions. The
observation of such regular patterns of change over many years would
give observers good inductive evidence that these freezes would coincide
in all three regions every s.ixty years; and so that there would be a year in
which there was no change at all over the whole world.
In a similar way it seems logically possible that there should be a
lOur conclusion runs contrary to Augustine's claim that time began with the
beginning of the Universe. He argued that Time could not exist, if there were no
creatures', Confessions, II. 30.
174 Space and Time
period of time in which there was nothing existent, preceded and
followed by periods in which physical objects existed. One could have
inductive evidence for the existence of such periods in a way analogous to
the way in which Shoemaker suggested that one could have evidence for
the occurrence of periods of time without change. (This has been
suggested by Newton-Smith [6].) There could be a world, divided into
three regions, A, B, C. On A physical objects vanish for a year every three
years, after which objects similar to those which disappeared reappear.
The objects in B vanish for a year every four years, and those in C for a
year every five years, similar objects reappearing in the two regions after
the year. These cycles of disappearance in one region can usually be
observed from the other regions. The cycles of disappearance will
coincide every sixty years. There would then be a period of a year in
which there was nothing existent. Observers would have inductive
evidence of the existence of such a period. The law which determined the
appearance of new objects would state that objects similar to those
which disappeared would appear in a place with the same spatial
relations to qualitatively similar objects as those possessed by the old
objects. (It could not state that they appear in the same place, relative to
other objects, as the old objects which, had disappeared; because every.
sixty years there would be no other objects, and so no such place.) Talk of
periods of time during which or instants at which there are no physical
objects makes sense because those periods and instants can be picked out
by their temporal distance from periods at which there were changing
physical objects. But in such periods in which there are no physical
objects points of space cannot be picked out, because there are no frames
of reference by reference to which points can be identified. It was for this
reason that I argued in Chapter 3 that there could be no space without
physical objects then existent; but I now argue that there could be time
without such.
Time, then, being of logical necessity unique, one-dimensional, and
infinite, has of logical necessity a unique topology. Instants have to each
other the neighbourhood relations of points on a line of infinite length.
Time having this topology cannot properly be said to have a metrical
geometry as well. For a metrical geometry states, as we saw in Chapter 6,
what relations of distance and direction and what propositions about
area and volume are entailed by other relations of distance and direction.
The temporal analogue to a point is an instant, to distance is temporal
interval, to direction is being before or after. Time being one-
dimensional, there is no temporal analogue to area and volume. Further,
when we have identified temporal instants by their distance and direction
Times and the Topology of Time 175
from some.specified instant, it becomes a matter oflogical necessity what
relations of distance and direction these have to each other. Thus for
three instants a, b, and c, if b was an interval TI after a, and c was an
interval T2 after b, then c was after a an interval TI + T2. That this
conclusion foIlows of logical necessity from the premisses is a con-
sequence of the fact demonstrated in this chapter that·if we know of two
instants a and c both temporally related to an instant b, they must be
temporally related to each other in a one-dimensional infinite time.
Hence the only temporal path between two such points a and c must pass
through b, and so the distance along it must be that from a to b plus that
from b to c. But in contrast there are many paths between any two points
of Space and hence it becomes an empirical matter where the one
lies satisfying the criteria described in Chapter 4 for being a straight
line.
We discussed in Chapter 9 methods for arranging events in temporal
order, that is for ascertaining which occurs before another, which occurs
at an instant prior to another. We do not, however, merely arrange events
in temporal order, but measure the temporal interval between them. To
do this we need a procedure for measuring temporal intervals, and such a
procedure I shall call a time scale. Indeed it is often physically impossible
to ascertain which of two events Eland E2 was the earlier without the use
of a time scale. Suppose for instance that Eland E2 are spatially separate
events, so far apart that it would be physically impossible for anyone who
observed EI also to observe E~ or to receive before observing EI signals
from a man who had observed E 2 , and conversely. In that case observers
must judge which occurred first by judging which would have occurred
first as indicated by readings on clocks at the place of each event,
previously synchronised and ticking at the same rate. How time intervals
are to be measured is the main theme of the next chapter.
BIBLIOGRAPHY
1 It has normally been held that the supposition that there could be particles or
other signals which had a velocity greater than that oflight would contradict the
Special Theory of Relativity. However recently physicists have speculated
whether after all there could be such particles to be called tachyons. But it looks
rather as if the existence of tachyons would only be compatible with the
universal applicability of Special Relativity if causes could follow their effects,
and we have seen reason to suppose that that is not logically possible, or iflogic
is violated in some other way. See the discussion in John Earman, 'Implications
of Causal Propagation outside the Null Cone', Australian Journal of Philosophy,
1972, SO, 222-37, and the articles referred to therein.
Time Measurement and Absolute Time 183
material objects that - at any rate in the region of our galactic c1uster-
its mean velocity has the same value in all inertial frames. By this is meant
that when a light signal is sent between any two points P and Q and back
again (stationary or in motion relative to each other), and that passage is
marked as a passage on inertial frames moving relative to P or Q with any
velocity, then the mean velocity of the signal will be the same in all such
frames.
That the mean velocity of light has the same value in all inertial frames
is the most natural extrapolation from a very large number of
experimental results, the most celebrated of which is the Michelson-
Morley experiment. It was assumed by Einstein in formulating the
Special Theory of Relativity and confirmed experimentally in 1913 that
the velocity of light received at E from a distant source S is independent
of the velocity of S relative to E. Light from each of a pair of double stars,
one of which is receding from the Earth and one of which is approaching
it (judged by the criteria of Chapter 5) has the same one-way velocity
over the surface of the Earth. This can be shown by reflecting flashes of
light from the two stars which arrived simultaneously at P on Earth to
another point Q on Earth. The two light flashes will arrive at Q at the
same instant. So, the Earth being to all appearances a typical near-
inertial frame, either the mean velocity of light is the same in all inertial
frames or it varies with the velocity of the receiving frame E relative to
something else other than the source. During the nineteenth century it
was supposed by many that light had a constant one-way velocity relative
to an underlying medium, ether, and that its velocity relative to'E varied
with .the velocity of E relative to the ether. The Michelson-Morley
experiment was undertaken in 1887 to ascertain the velocity of the Earth
relative to the ether. A light signal was sent from a source (0) stationary
on Earth a distance L (relative to the Earth)
in perpendicular directions to mirrors M 1
and M 2 stationary on Earth and reflected
back again to the source (0). Suppose that
along 0 M 1 the Earth has a velocity V
relative to the ether. Then if light has a L L
constant one-way velocity c relative to the
ether, and it takes an interval T2 to travel in
L
a direction perpendicular to the direction of ~-----........ M,
the Earth's velocity relative to the ether and o v~
I By the Ives-Stilwell effect and cosmic ray phenomena. For details see [3] pp.
213-1 S.
2 See A. Einstein, 'On the Electrodynamics of Moving Bodies' (originally
published 19O5) in A. Einstein et al., The Principle of Relativity (W. Perrett and
Time Measurement and Absolute Time 187
that frame synchronised by the light signal method show the same
reading when the two events occur. So E I and E 2 may be simultaneous in
F, but not in F'. But no contradition arises, since it does not follow from
E I and E 2 being simultaneous in F that they will be simultaneous in F' or
any other frame ..
The alternative is to deny that, except possibly in one preferred frame,
the light signal method measures simultaneity. We could hold for
instance that simultaneity is properly judged by the light signal method
in a certain preferred frame F only, and that two events simultaneous in
F are truly simultaneous. Then to find out what the time interval is
between any two events we have to find out what judgement· about time
interval would be made by an observer on F using the light signal
method. This means assuming that only in F is the velocity of a reflected
light signal constant in both directions. In all other inertial frames it will
have some velocity j (different for each frame) in a certain direction and
G. B. Jeffery) (London, 1923) pp. 42f. 'We see that we cannot attach any absolute
signification to the concept of simultaneity, but that two events which, viewed
from a system of coordinates are simUltaneous, can no longer be looked upon as
simultaneous events when envisaged from a system which is in motion relatively
to that system.'
1 For the first stages of such a reinterpretation and demonstration of its
consistency see Grunbaum [4] ch. 12, section B.
188 Space and Time
basic frame of reference, we ought to take this as a preferred frame (since
the laws of physics ought to be referred to most basic frames) and so we
would have a coherent standard of simultaneity, a method of synchroni-
sation which does not lead to confusion.
So much for the signal method, the circumstances in which it does and
the circumstances in which it does not lead to confusion. We must now
consider the clock-transport method which seems perhaps more ob-
viously a primary test for indirect synchronisation than the signal
method. This method, it will be recalled, for synchronising clocks at P
and Q is to have two true clocks synchronised at P, and then to move one
to Q. The clock at Q is then synchronised with the clock moved from P,
and the clocks at P and Q are then said to show the same time. If we
moved clocks from P to Q at regular intervals we could then see whether
clocks remaining at P and Q yielded the same time scale.
Now the clock-transport method normally leads to confusion, and
this for two reasons. The first is that clocks showing at P the same
reading taken from P to Q by different routes or- at different velocities
will record on arrival at Q different readings. The second difficulty is that
a clock moved from P to Q and then taken back again will not in general
record the same passage of time as a clock which remained at P, but a
shorter time. (Here the celebrated twin 'paradox' that if one twin remains
on the Earth while the other travels in a space rocket on a long journey
and then returns to Earth, the latter will have aged less than the former.
For his ageing mechanism would be expected to operate at the rate of a
clock transported with him.) Hence either when moved from P to Q or
when moved from Q to Pit must have shown on arrival a smaller reading
than that shown at the same instant on the clock at the point which it left.
So we can only use the clock-transport method if there is a route and
velocity of moving a clock which is such that moving a clock from P to
Q and back again has the consequence that on arrival back at P it
records the same time as the clock which stays at P, and which is such
that we have reason to suppose that it ticks at the same rate on its journey
to Q as on its journey back. Fortunately, given the Special Theory of
Relativity, within anyone inertial frame there is such a method of
moving clocks so that this holds when the smallest of corrections is made
to the clock. The method is to move the clock (by any route) very slowly,
relative to that inertial frame. Noting that when we move the clock from
P to Q and back very slowly, its reading on arrival back at P diverges very
little from the reading on the clock which stayed at P, and the more
slowly it is moved, the less it diverges, we can work out the limit of this
Time Measurement and Absolute Time 189
process - i.e. to talk slightly metaphorically, what it would read if it were
moved infinitely slowly. That reading will exaCtly coincide with the
reading on the clock which stayed at P. The simplest hypothesis is to
suppose that the clock ticks at the same rate when moved from P to Q as
when moved in the other direction. Hence we have reason to use the
clock-transport method of infinitely slow transport in a straight line in
order to synchronise a clock at Q with one at P. Ellis and Bowman [9]
point out that, given the Special Theory of Relativity, this method gives
exactly the same results as the signal method so far described; and in
opposition to the 'conventionalist' approach of [10], Ellis [11] rightly
urges that all this is good reason for adopting the judgements of
simultaneity which these methods yield. However, as we have seen, all
this holds only within a given inertial frame; and if we are to use a
standard of simultaneity, we need grounds for preferring one inertial
frame to others.
Leaving for the moment these difficulties about synchronisation, let us
next present and give the evidence for the overall picture of the Universe,
given by modern cosmology. In so doing we shall at last present the
evidence for the cosmologist's thesis about basic frames which I set
forward in Chapter 3.
An observer on Earth notes that, on the evidence of their Doppler
shift, all other clusters are in recessiOn from his own. All other clusters at
any given distance (estimated by the methods described in Chapter 5) in
whatever direction have, when allowance is made for small random
velocities, .the same velocity of recession from our cluster, a velocity
which increases with distance (possibly uniformly), viz. the observed
region of the Universe is, relative to our cluster, approximately isotropic.
All observed clusters belong to one of a few types and the different types
are spread throughout the observable region. Now is it only relative to
our cluster at this instant on its clocks that this isotropic recession of
observed clusters occurs, or does it occur relative to the other clusters at
all instants on the clocks of each? The supposition that only relative to
our cluster at this instant is there isotropic recession seems unreasonable,
and the principle of simplicity dictates us to postulate that for all clusters
at all instants on the clocks of each the observable part of the Universe is
approximately isotropic (viz. observed clusters at any given distance
from any observed cluster recede from it with approximately the same
velocity). We can avoid the awkwardness that the isotropy is only
approximate by postulating an imaginary frame of reference in the
vicinity of each cluster, relative to which the recession of other such
190 Space and Time
frames is uniform. Such a frame is termed in the literature of cosmology a
fundamental particle, and an observer imagined as situated on one a
fundamental observer l .
The fundamental particles are then in uniform recession relative to
each other. A cluster may rotate relative to its fundamental particle, but
any velocity of recession of the cluster from the particle will be small and
temporary. Motion of a cluster relative to its fundamental particle is
random and thus equally likely to be in any direction. The supposition
that we can postulate associated with each cluster a fundamental particle
its motion relative to which satisfies these conditions, and that relative to
any fundamental particle at any instant the observable Universe is
isotropic, may be called the principle of isotropy. That it holds is an
empirical postulate which might turn out to be false. Wireless messages
might one day be received from astronomers in a distant cluster
reporting that relative to them the observable Universe is very far from
isotropic. But until such messages are received it seems reasonable to
adopt the principle.
If then the observable Universe is always isotropic relative to each
fundamental observer, it seems a further simplification to postulate that
the laws of physics as formulated by an observer on each fundamental
particle, using his measures of distance and clocks, are the same; and so
that they are the same on each c1usfer, if we make allowances for any
random motion or rotation of the cluster. This principle we shall call the
principle of equivalent laws. By it the law that a certain machine will tick
always at the same rate will hold on every cluster and so true clocks on
every cluster will have the same construction. The principle is an
empirical postulate which might turn out to be false. There is, however,
some empirical evidence that the principle is true, at any rate for local
laws, that is laws governing the behaviour of matter on each cluster. One
piece of evidence is that light spectra of distant clusters can most
reasonably be interpreted, as we saw on p. 91, as spectra of elements
I Some recent research suggests that the isotropy may indeed be only appro-
I If one of these two properties holds of the Universe, Weyl's postulate will be
1 The claim that the density, energy, and distribution patterns of physical objects
(viz, the clustering of galaxies and their velocities of mutual recession) are
approximately the same in all spatial regions, if regions of sufficient size be
taken, at any given cosmic instant is often known as the cosmological principle.
The claim that they are the same at all cosmic instants is often known as the
perfect cosmological principle, This latter principle, as we shall see in Chapter 14
(see p. 241) formed the basis of Steady State Theory. These principles
196 Space and Time
random velo.cities of recession of clusters from their fundamental
particles virtually cancel out). For by the principle of equivalent laws the
law governing the recession of the clusters will be the same relative to
every fundamental particle, and so approximately the same relative to
every cluster. Hence at any given cosmic instant the clusters isotropically
surrounding any given cluster will be at approximately the same
distances from it as those surrounding any other cluster are from it, ~nd
their velocities.of recession from it be approximately the same as those of
similarly positioned clusters from any other cluster. This_ claim is
summarised by cosmologists as the claim that the Universe is always
homogeneous.
The supposition that there is a cosmic time scale would have to be
abandoned if new evidence suggested that the principle of similar clocks
was false. Any evidence tending to show that the principle of isotropy or
the principle of equivalent laws were false would tend'ceteris paribus to
show that the principle of similar clocks was false, since the arguments
which I have stated currently adduced for its truth are arguments from
the former principles. Other more complicated ways in which obser-
vational evidence might be relevant to the truth of tliese principles and
the homogeneity of the Universe wiII be mentioned in Chapter 14.
So then between heavenly bodies within a galactic cluster, both
synchronisation tests appear to give the same results ~ given those
methods of applying them which are based on the laws of nature which
result from the simplest extrapolation from observations (e.g. the law
which states that light has the same one-way velocity in all directions).
Between clusters the simplest extrapolation from observations indicates
(via the principle of similar clocks) a reliable method of applying the
clock-transport test. Given that simplicity is evidence of truth, there are
1 See [3] Chapter 5, for an account of the work of Robertson, Walker, and
Rindler in proving the equations to be stated without proof in the next two
pages.
Time Measurement and Absolute Time 199
are points on the path of a ray of light ds = 0.) Here e is-to a high
degree of approximation - the" one-way velocity of light relative to each
fundamental particle through which it passes, and the mean velocity of
light relative to other frames in its neighbourhood moving with uniform
rectilinear velocity relative to that particle, as shown by Special Relativity
which applies to a high degree of approximation in the neighbourhood
of each fundamental particle. k is a constant having values k = 0 for
Euclidean Space, k = + 1 for elliptic space (and also for spherical space),
k = ~ 1 for hyperbolic space. l From the Robertson-Walker metric
cosmojogists develop various more specific 'models" that is cosmological
theories, by giving a specific value to k and 'making some specific
supposition about the equation governing the value of R(t).
The adoption of the Robertson-Walker metric may commit us to a
prediction that the mean velocity of light on a cosmic scale is not the
same relative to all fundamental observers at -all instants. The proper
distance from a cluster C a at the origin, of a photon, the particle of light,
emitted from a cluster C b having u-coordinate ub at cosmic instant t I in
the direction of C a is, at cosmic instant t:
I = R(t) { I'
a(u b ) - "
cdt }
R(t) ,
BIBLIOGRAPHY
1 See, e.g. H, Reichenbach. The Direction of Time (Berkeley and Los Angeles,
/966). He asks (p. 269) the question 'What is the relation between the time of
physics and the time of our experience'?'
Time Measurement and Absolute Time 205
[3] G. J. Whitrow, The Natural Philosophy of Time, London, 1961 ,chs
4 and 5. (This work includes a much more detailed account of the
physical and mathematical basis of cosmological developments
leading to the formulation' of the Robertson~ Walker metric than I
have given.)
[4] A. Grunbaum, PhilosophicaJ Problems o/Space and Time, London,
1964, ch. 12.
[5] J. D. North, The Measure of the Universe, Oxford, 1965, especially
chs 14 and 16.
For details of time scales used for measuring time at anyone place see:
[6] William Markowitz, article on 'Time Measurement' in Enc.y-
c10paedia Britannica, 1964 ed, vol. 22.
For details of the physical and mathematical basis of the Special Theory
of Relativity, see:
[7] W. Rind1er, Special Relatirity, Edinburgh, 1960.
[8]0. Bohm, The Special Theory of Relatit'ity, New York, 1965.
On simultaneity within an inertial frame see:
[9] Brian Ellis and Peter Bowman, 'Conventionality in Distant
Simultaneity', Philosophy of Science, 1967, 34, 116~36.
[10] Adolf Griinbaum, Wesley C. Salmon, Bas C. van Fraassen and
Allen I. Janis, 'A Panel Discussion of Simultaneity by Slow Clock-
Transport in the Special and General Theories of Relativity',
Philosophy of Science, 1969, 36, 1~81.
[II] Brian Ellis, 'On Conventionality and Simultaneity', Australasian
Journal of Philosophy, 1971, 49, 177 ~ 203.
'-,
12 Physical Limits to·
Knowledge of the
Universe- (i) Horizons
So far in this work we have mainly been investigating the meaning of
spatial and temporal terms and of propositions about Space and Time,
although in the last chapter in order to examine the application of our
criteria of simultaneity, we had to set forward the evidence for certain
empirical truths about the Universe. The task of the concluding chapters
is to examine more fully what kind of conclusions science can hope to
reach about the general spatio-temporal character of the Universe.
A necessary preliminary to this task is to inquire whether there are any
necessary limits to our knowledge of the Universe at other places and
temporal instants. In Chapter 9 we considered whether there are any
logical limits to such knowledge. In this and the next chapter we must
consider whether the laws of physics'impose any limitations. Initially it
might seem that, within the logical limits discussed in Chapter 9, it is a
merely practical matter that we do not know about events at distant
places and remote temporal instants. For any event at a distant place or
remote temporal instant it is easy to imagine the constituents of the
Universe being different so that we might learn about it. If only ancient
papyri had not been accidentally destroyed, would we not be able to read
all the plays of Sophocles? Could we not learn about events on a certain
galaxy at a very remote instant if we built a radio telescope big enough to
detect waves emanating from it? However, it has recently been suggested
that in two important ways the laws of nature have certain characteristics
which prevent our acquiring knowledge of events at distant places or
remote instants. These suggestions we must consider in this and the next
chapter. In this 'chapter we shall consider suggested limits to knowledge
of states of distant galactic clusters, brought about by their fast velocity
of recession from ourselves; in the next chapter we shall consider
suggested limits to knowledge brought about by states being more
reliable signs of past states than of future states.
Physical Limits to Knowledge of the Universe - (i) Horizons 207
The more distant a galactic cluster is from the Earth or any other
heavenly body, the faster it recedes from it. Such is the well substantiated
evidence of astronomy. It has been suggested for two distinct reasons
that the law of recession leads to a limit to what we can detect by
observation.
The first reason is that clusters may recede so fast from each other that
light emitted from a cluster will never reach some clusters, since they are
. receding from it as fast as or faster than the ligh t is approaching them. It
is clear that if the Special TheoFY of Relativity were applicable on a
cosmic scale this could never happen. For on this theory light has the
same one-way velocity relative to all inertial frames of reference, and a
cluster is to some degree of approximation an inertial frame. Hence light
emitted from a distant cluster towards the Earth will have,.relative to the
Earth, the same one-way velocity as it has relative to the distant cluster,
viz. c; and so will eventually reach the Earth, however distant the distant
cluster be from the Earth. However, if we suppose, as we have done so far
for reasons given in the last chapter, that it is not Special Relativity but
the Robertson Walker line element that applies to the Universe on a
cosmological scale, clusters may recede from each other with the
consequences described. Since we have reason to believe that no signal
travels faster than light, then iflight from a distant cluster never reaches a
certain cluster, the former cluster will be unobservable by an observer on
the latter one.
Cosmologists discuss two kinds of 'horizon' which arise from high
velocity of mutual recession. A horizon is a surface in Space marking
some limit to the knowledge by observation available to an observer of
events occurring beyond the surface. (Horizons are sometimes depicted
as surfaces in space-time, but they can be represented, and I shall
represent them as surfaces in Space.) In a classic paper [1], analysing the
properties of cosmological horizons, Rindler defined 'an event-horizon
for a given fundamental observer A', as 'a (hyper-) surface in space-time
which divides all events into two non-empty classes: those that have been,
are, or will be observable by A, and those that are for ever outside A's
possible powers of observation.' If we consider as we shall do in future,
the horizon as a surface in space, not space-time, an event horizon will be
'for a given fundamental observer A and cosmic instant to a surface in
instantaneous 3-space t = to which divides all fundamental particles into
two non-empty classes: those, events occurring on which at to will be
observable by A at some future cosmic instant, an:d those, events
occurring on which at to will never be observable by A.' (By '3::Space'
Rindler means physical Space.) Having made and used earlier the
208 Space and Time
distinction between fundamental particles and galactic clusters, I shall
henceforward - to make discussion simple - often ignore the differ-
ence between them and make statements about clusters which strictly
speaking apply to fundamental particles, and are only approximately
true of galactic clusters. Hence I shall say that an observer who remains
on the Earth is a fundamental observer because he remains on his
galactic cluster. The Earth's eve!1t horizon then marks a limit to
knowledge by observation available to an observer who remains on the
Earth.
Rindler distinguished the event-horiion from the particle horizon. He
defined 'a particle horizon for any given fundamental observer A and
cosmic instant to' as 'a surface in instantaneous 3-space t = to, which
divides all fundamental particles into two non-empty classes: those that
have already been observable by A at a time to and those that have not.'
An event-horizon arises if any two clusters recede from each other at
an ever-increasing rate, so that light emitted from one towards the other
after some instant never reaches the other. Clusters lying beyond the
Earth's horizon are those, events currently occurring on which will never
be observable on Earth. Events which occurred on them previously may
have been observed on Earth. Clusters lying beyond the Earth's particle
horizon are ones which have never yet been observable on Earth. Such a ,
horizon will arise if clusters recede from each other at an initial period of
the existence of the Universe with a velocity greater than c (and also
under certain other highly unlikely physical circumstances). Of the
various 'models' developed from the Robertson-Walker line element
considered by cosmologists as possible theories of the Universe, some
have an event-horizon, some have a particle horizon and some have both.
In his article Rindler showed that the necessary and sufficient condition
for an event-horizon to exist at cosmic instant to, in a given model is,
given the correctness of the Robertson-Walker line element, that the
integral foc d(t converge to a finite limit, where R(t) is the 'radius of the
'0 R t)
Universe' at cosmic instant t. Thus the De Sitter model,where R(t) = e,!T,
and Page's model where R(t) = at 2 (a and T being constants) have such
horizons. The necessary and sufficient conditions for a particle horizon
to exist at cosmic instant to is, given the Robertson-Walker line element,
f'o
the convergence of the integral
1 '0 dt
- - (or
o R(t) -x
-dt- , where the
R(t) .
definition of R(t) extends to negatively unbounded values of t). Thus the
Einstein-De Sitter model where R(t) = at 2 / 3 (a being a constant) has.a
particle horizon.! Models not consistent with the Robertson-Walker
Physical Limits to Knowledge of the Universe - (i) Horizons 209
line element may also have particle or event-horizons. The con-
siderations involved in saying that a model gives a correct theory of the
Universe will be examined in detail in Chapter 14.2
The question which I wish to examine in connection with such
horizons is, if they existed, to what extent would they limit our
knowledge of the Universe at other places. They would be limits to
knowledge by observation. The particle horizon would be a barrier only
to present knowledge, not to future knowledge by observation. More
fundamental, therefore, is the event-horizon. If a cluster lies beyond the
Earth's event-horizon, its present and future states would be therefore,
unobservable by an observer on Earth. He could,· however, have
observed its past states and use them as traces of its present and future
states.
The grounds which the cosmologist has for taking some state of a
physical object within the range of his observation as a trace of a physical
object outside that range are, as we have seen, that this sort of connection
has been observed locally or that the simplest scientific theory com-
patible with observations postulates that it will be. It is for these reasons,
as we saw in the last chapter, that cosmologists infer that clusters of
galaxies are distributed uniformly throughout Space and recede from
each other with a velocity which is a function of their distance apart.
Astronomers also find by observation, as we shall see in detail in Chapter
14, other features of the Universe which are the same in' all observed
Oor R(O
-00
~. If an EP horizon exists at one cosmic instant, it will exist at
all others. One model, which Rindler develops in detail, which has an EP horizon
is the model with k = 0, ). positive and R (I) = a(cosh bt·- 1)\, a and b being
constants. If any cluster lies beyond both the event and particle horizons it will lie
beyond the EP horizon (hence my name for it), and conversely. To obtain
knowledge of clusters lying beyond the Earth's EP horizon we must observe
states of other clusters, viz. ones lying within it, and uses these as traces of states
of the former.
2\0 Space and Time'
regions. For such features F, if a spatial region has that feature, then it is
observed that the next region beyond it also has it. Hence F in one region
is a trace of F in the region beyond it, and the latter of F in a region
beyond it and so on. Thereby we can obtain knowledge of the Universe
beyond the limited range, if the range is limited, which it is physically
possible that we might study by observation.
Now if we had reason to believe that there was an event-horizon, a
surface which could be marked in Space which was a limit to our
knowledge by observation, it might appear that although knowledge
could be obtained of the Universe beyond that surface, that knowledge
would be shaky and unreliable and hence Jhat that surface was the
boundary to reliable knowledge of the Universe. But for two reasons the
position cannot be like this. For first the very fact that we can infer- for
whatever reasons - the existence of an event-horizon means that we
have two ,important pieces of information about the physical objects
lying beyond the horizon. One is that there is at least one such physical
object. The other is that all physical objects beyond the event-horizon are
receding from us at speeds so fast that the fastest influence which they
can now transmit towards us will never reach us. These assertions follow
from the definition which I gave of the event-horizon. If we have reason
to know that there is an event-horizon, we have reason to know this
much about objects beyond it.
But in the second place we must have used and have needed to use
traces in reaching the conclusion that there was an event-horizon. The
cosmologists who adopt models with horizons are only justified in doing
so if these models are a justifiable extrapolation to all spatial regions of
properties observed in near~by regions. We observe the distribution of
galaxies and their rates of recession and other features of the Universe
and from these infer the laws of cosmology, as has been shown to some
extent in Chapter 11 and will be shown in more detail in Chapter 14. This
enables us to take spatial regions occupied by observed clusters as traces
of more distant spatial regions occupied by other clusters. Hence we
learn that in all spatial regions beyond a certain distance from us there
are clusters spread out homogeneously through space, all receding from
each other and receding from us with such velocities that signals now
sent from them could never reach us, and no physical objects of which
the opposite is true. If we had reason to believe that traces were shaky
and unreliable, we would not be justified in concluding that there was an
event-horizon. We only have reason for believing that there is an event-
horizon and so a limit to knowledge of the Universe by observation
because we have knowledge by the use of traces of regions beyond
Physical Limits to Knowledge of the Universe -(i) Horizons 211
observation. If the use of traces to ascertain the state of the Universe
beyond the observed realm is unreliable, then our conclusion that there is
an event-horizon, a limit to knowledge of the Universe by observation, is
also unreliable.
However, though the existence of an event-horizon does not mean we
can have no reliable knowledge of objects beyond the horizon, it does
mean a limit to the type of knowledge possible about those objects. The
cosmologist uses as traces of the states of unobservable regions those
properties of observable regions which have proved, in so far as this can
be tested, to be reliable indicators of the properties of other observed
regions. These are the properties of observable regions which remain
constant with the spatio-temporal distance of a region from ourselves or
which vary in a systematic way with that distance. Hence the cosmologist
can have knowledge of the density of matter and rate of recession of the
galaxies in regions for any reason unobservable. But he cannot use as
traces those properties of the regions which are neither constant nor vary
in a systematic way. What those properties are, astronomers are not yet
in a position to tell us. But it might turn out that, within certain limits,
average cluster size or the average number of exploding supernovae per
galaxy per year Were neither constant nor varied in any systematic way
with the spatio-temporal distance from ourselves of the regions to which
they belonged. In that case we could not use traces to learn, within those
limits, about average cluster size or average number of exploding
supernovae per galaxy per year in regions unobserved. The fact or the
existen<;.e of an event-horizon would mean that events on the Earth
would be entirely unaffected by events now or subsequently occurring on
objects lying beyond the event-horizon. Hence we would know nothing
about average cluster size or the average number of exploding super-
novae per galaxy per year in regions beyond the event-horizon. Whereas
if there was no event-horizon, all physical objects could produce effects
on the Earth enabling us to detect such properties. (Of course the more
distant an object of given size, the more refined the astronomical
instruments needed to detect any properties thereof.) So my conclusion is
that the existence of an event-horizon for an observer on the Earth would
not mean that such an observer could know nothing of regions beyond
that horizon but that he could only have knowledge of those properties
of those regions which, within the range of our observation, were
constant or varied in a systematic way with spatio-temporal distance
from the Earth. Ifit is a fact that we can only have knowledge by traces of
some distant regions, that means that we can only know certain
properties of those regions. Knowledge by traces which are not effects
212 Space and Time
does not include knowledge or peculiarities in the way that knowledge by
observation does.
The supposed existence or an event-horizon has somc;!times led
scientists to suggest a definition or the Universe not, as we have done, as
all the physical objects that there are spatially related to ourselves, but as
all the physical objects within that horizon. Bondi relates that many
consider the Universe as the set or all events which could have affected us
in the past and 'which may affec~ us at sometime in the ruture and all
events which have been or wiII be affected by us'. (Bondi adds that 'some
authors ... consider a different set, viz. the largest set to which our
physical laws (extrapolated in some manner or other) can be ap-
plied .... The physical significance or this "universe" is not very clear'
([2] p. to).
Now while it may be reasonable to define a space-time Universe as
consisting or events, the Universe or ordinary language is most naturally
said to consist or physical objects. U we are to define a Universe in terms
or physical objects, the definition or it best satisrying the positivistic urge
behind the suggested definition would seem to be or the Universe as all
the physical objects with which at any instant we could have causal links,
that is, all physical objects within the event-horizon. U there were no
event-horizon, all physical objects could affect and be affected by events
or the Earth, and hence the Universe would be all the physical objects
that there were. However, ir there was an event-horizon, and we had
good reason to believe that there was, we would have good reason to
know about objects and their states beyond that horizon, as we have
noted. Hence, on the suggested definition, irthere were an event-horizon,
there would be physical objects spatially related to ourselves outside the
Universe about which we could claim knowledge on grounds or scientific
inrerence - which is absurd. I conclude that the suggested definition is
unacceptable, and a more reasonable definition elucidating ordinary
usage is that the Universe is all the physical objects (spatially related to
ourselves) that there are.
I must make one final modification. I have been discussing the limits or
knowledge or distant regions or the Universe ror an observer in the
Earth. We saw that the event-horizon marked a limit to his knowledge by
observation. However, in order to make claims about the possible
knowledge or the Universe available by observation to an observer now
on the Earth, we rieed to consider the possibility or that observer
travelling in space. Hence it would be more appropriate to define the
rollowing horizon which I shall call the EThorizon. I define it as rollows.
The EThorizon, ror any observer A on a rundamental particle at cosmic
Physical Limits to Knowledge of the Universe - (i) Horizons 213
instant to, is a surface in instantaneous 3-space t = to which divides all
fundamental particles into two non-empty classes: those, events occur-
ring on which up to to have been obselVable by A on his fundamental par-
ticle up to to or will be observable thereafter ifhe travels in space with some
velocity possible for matter leaving his fundamental particle, and those
not so observable. No events now occurring on objects lying beyond his
E T horizon could ever be observed by an observer now on Earth: If we
wish to consider the limitations to knowledge produced by the existence
of horizons for an observer who is allowed to travel in space, itis the ET
horizon that marks the limit to his knowledgepf the Universe by
observation. It must be noted that while an observer now on Earth may
obtain knowledge by observation of events now occurring in any region
within the EThorizon by travelling in space, he qlnnot necessarily obtain
knowledge of events now occurring in all such regions - if he travels in
one -direction, events now occurring on some clusters situated in the
other direction within the ET horizon may not ever be observable by
him. The reasons for believing in the existence of an ET horizon will be
similar to those for believing in the existence of an event-horizon (though
some cosmological models may have an event-horizon without having
an ET horizon) and the extent of limitation of knowledge is similar.
So much for t he extent and kind of limit to knowledge which would
be produced by the existence of horizons I. Such horizons will, how-
ever, only arise if photons do not have on the cosmic scale the same
one-way veloCity relative to all inertial frames. There is, however, another
I It is sometimes suggested (see e.g. [4] p. 1(0), that 'black holes' have event-
horizons. A black hole is a large star or other massine body which has collapsed
under gravitational self-attraction into a small v.olume so dense that all matter or
radiation within a critical radius of the body cannot escape but is pulled
continually closer to the centre of the body. Astrophysicis1S have calculated from
General Relativity that the final stage in the evolution of stars having more than
three times the mass of the Sun is to become black holes, and there now seems to
be some observational evidence'of the existence of black holes. This evidence is
that of stars moving like components of a binary star system under the
gravitational influence of an invisibLe body, losing matter which is streaming in
the direction of the postulated invisible body and emitting X-rays in the process.
A black hole will not be observable by detecting photons emitted from it, but it
can be observed so long as it causes effects of some kind which can be deteCted by
an instrument. (See my pp. 154f). The mass, angular momentum, and electric
charge of a black hole cause effects outside the critical radius; and instruments
can be constructed which provide information about the state of the black hole
by measuring these effects. It is because of these effects that astronomers can
locate black holes.
214 . Sp~ce and Time
limit to knowledge by observation for an observer who remains on his
cluster, which arises even if photons did have the same one-way velocity
(c) relative to all inertial frames. This will arise because of the Heisenberg
indeterminacy principle. 1 Because of this principle very little infor-
mation can be received on a cluster from another cluster whose velocity
of recession from it approaches c. (Given the universal application of
. Special Relativity, velocities of mutual recess'ion could not equal or
exceed c.) For as the velocity of recession approaches c, so the wave-
.' . ( A 1 + Vrlc h'
length 0 f ~he recelvedhght approaches 00 Ao = (1 _ V;
Ic 2)1/2 ' were It
is the wave-length of the received light, Ao of the emitted light, and so the
h 'f z
Dopp Ier SIt = A- Ao an
-A-' 'd Vr t h
e veioCltyo
' f recesSIOn.
. ) But by
BIBLIOGRAPHY
By contrast, when galactic clusters recede from each other too fast, no effects at
all are propagated from one to the other. I conclude that black holes do not have
event-horizons. (For more details on black holes see [4] pp. 91-1.02.)
. 1 I use here the argument of [3]. .
)
13 Physical Limits to'
Knowledge of the
Universe - (ii) Past/
Futur~_ Sign Asymmetry
We considered in the last chapter the extent to which the fast velocity of
mutual recession of galactic clusters limited the knowledge that we on
Ear!h could have of their states. In this chapter we must consider how far
physical laws are responsible for our much greater knowledge of past
than of future.
We noted in Chapter 9 that the amount of our knowledge of the past
was very much greater than our knowledge of the future, and that this
was ih small part a logical and in much larger part an empirical matter. If
there are to be beings with knowledge they must remember and so be
able to report the past stages of their arguments and must be ignorant of
the kind of investigations which they will conduct into the truth of
propositions; and of the results of their investigations. But given these
limits, there seem no other logical factors responsible for our much
greater knowledge of the past' than of the futlire. Consequently the
question arises as t.o th.e empirical source of this asymmetry. Do we just
happen to be ignorant ofthe future in this way or is some fundamental
physical law responsible?
We saw in Chapter 9 that our greater knowledge of the past is due to
the fact that men can report what they have observed but not what they
will observe, and that traces allowing any inference are much more
frequently of the past than of the future. Now the physical source of
men's ability to report straight otT on what happened rather than on
what will happen must lie in the characteristics of human brains since,
psychologists and physiologists assure us, human memory and speech
abilities depend on the constitution of the brain rather than of any other
part of the body. Hence the brains of organisms must be such that their
states differ in respect of ditTer~nt past macrosc'opically distinguishable
216 Space and Time
events and states. Thus my brain is in some state B 1 now. It would be in a
differerit state, say B 2 , had I just bumped into you in the corridor. "
Whereas it will normally be in the same state BI whether or not I am
about to bump into you in the corridor. If brains did not have this
property we would not be able to report what had happened more
readily than what -.yill happen. (I neglect as scientincally implausible the
possibility that brain states could be correlated equally easily with future
iQteractions as with past interactions but that this had no effect on
human ability to report about the future.) What forms these states take
are currently being investigated by physiologists - the states may
be chemical deposits or electrochemical rhythms.
The physical source of the asymmetry of ease of prediction and
, retrodiction from traces lies in the existence of traces allowing easy
inference to the past rather than to the future. This means that
macroscopically distinguishable events and states of objects are in
human experience more reliably correia table with past than with future
macroscopically distinguishable events and states of objects. If we
consider only the case where an object is a trace of its own past or future
state, this means that men find that things generally begin only in one or
few recognisably distinct ways but may end in many recognisably
"distinct ways. Men are born only in a womb as the result of fertilisation,
but they die everywhere and from manifold causes. And so it is with other
things. Houses are built by men, but they may be destroyed by
earthquakes or floods or hurricanes or bombs. Washing machines begin
their lives in a washing machine factory, but they may end them on a
municipal rubbish heap or a metal scrap-yard, in a back garden or by a
main road. .
Now to say that there are easily recognisable traces allowing
inferences to the past much more reliable than inferences to the future is
to say (see p. 146) that states whiCh men can easily recognise are, in
human experience, signs of past states but not of future states. I made a
distinction in Chapter 9 between the two terms which I introduced,
'traces' and 'signs'. A stateA; is a trace of another state Iii if men have'
g<,>od reason to believe it to be a sign of B l' it I is a sign of B 1 if it is a
member of a class of events As, the occurrence of members of which isin
fact correlated to a very high degree with the occurrence of Bs.
Now to what extent is this asymmetry of ease of prediction and
retrodiction a function of the signs which men can recognise easily and
from which they can make ready predictions, and to what extent is iran
objeCtive feature of the world? Does it depend on the psychology and
Limits to Know/edge of Universe -- (ii) Sign Asymmetry 217
physiology of humans or would any organism however constructed find
the same asymmetry?
Our ability to predict and retrodict clearly depends·on our ability to
recognise certain events and states. If an organism can recognise a certain
state, I shall say that it has the concept of that state. Now the fact that we
can infer from a footprint to a foot which made it depends in part on the
fact that we have and can easily apply the relevant concepts. That is, we
. recognise as instances bf the same concept (footprint) marks of different
. sizes and shapes in the sand. Many of the lower orgal}isms cannot do this.
Also we recogniseasinstances.ofthe same concept ('man walking on the
sand') you walking on the sand, me walking on the sand, and so on .
. Unless humans were like this, they could not make the relevant
inferences. That they are like this depends on their brains and sense
organs. So might there not be <,:reatures with other brains and sense
organs who had concepts enabling them. to predict better than to
retrodict?Such creatures might be able to recognise states which would
subsequently normally be loilowed by.a man walking on the sand instead
of states normaily preceded by a man. walking on the sand. Against this I
would urge that whatever the brains and sense organs of creatures the
asymmetry of ease of prediction and retrodiction would very probably
hold, and this because of a very general objective feature of the world,
which I shall .call past/future .sign asymmetry. This feature is the
following.
. Any state of a system (that is, collection of physical ohjects) A I is
normally a sign of some past state BI but not of any future state CI of
thai system. By this I mean that (As) are very highly correlated with some
past states (Bs) but.not with any future st~tes (Cs). A st.ate ofa system is,
however, normally a sign ofa future state of some specifiable sub-system
within it. (States of systems are very much less often signs of states.of
other systems than of the same system.) It is supposed throughout this
description of past/future sign asymmetry that states are described in
equal detail in ail cases .
.Thus the readings on weather instruments or the state of the ground in
Hull today are signs of the state of the .weather in Huil yesterday.
Whereas no state of Hull today is a sign of tomorrow~s weather in Hull.
On the other hand the state of a much wider system t<.>day -- various
readings on recording instruments throughout the world - is a sig'.1
of tomorrow's weather in Hull.
a
A loose way of putting this poiht'is to say that state of an object is
normally a ~ufficient condition of a prior state of the whole object, but
218 Space and Time
only a sufficient condition of a future state of small patt of the original
object, not of a future state of the whole object. This is a loose way of
putting the point because only if the correlation of the sign with the thing
signified is invariable' is the sign a sufficient condition of the thing
signified. Hence our more precise formulation.
. The evidence that in the sense described past/future sign asymmetry is
an objective feature of the world is that more detailed scientific
investigation supports our superficial impression that states and changes
of state are very highly correlated with past but not with future states and
changes of state of the same system. The most detailed investigation into
the Mture of states of systems (viz. the most detailed analysis of the
nature and states of particles of systems) shows no evidence of high
correlations with the future.
It should be noted thal past/future sign asymmetry exists in almost
every possible region of inquiry (one 'possible exception will be
considered on p. 227). A counter-example which is often urged is that of
planetary astronomy. It is urged that the present positions and velocities
of planets are as reliable signs of their future pQsitio'n as of their 'past,
positions. Now it is certainly true that we are able to predict the future
positions of planets as easily as we are able to retrodict their past
positions. But, we arti only able to predict future planetary positions
because we assllme that no large heavenly body will come near to the'
planets and disturb their courses in the immediate future. We make thi~
assumption because we have observed a much larger region of the
hei\.vens than that occupied by fhe planets and seen that no large
heavenly body is, likely to intrude. We make our predictions, in other
. words, by observing the s\ate ofa large region of Space in the middle of
which lie the planets and inferring the future state of the sub-system of
the planets. One state of the large region is ind~ed a sign of the future
state of the small region. But the present state of the region occupied by
the' planets is a sign of the past state of the same region. If the planetary
system had been interfered with in the past, there would be evidence of
that fact within the system. '
Now given past/future sign asymmetry in the objective sense defined,
whatever the construction of its brain and sense organs, any organism is
almost bound to be able to retrodict from traces better than to predict,
For there is a lot more information to be had in'the world about the past
than about the future, and the organism would have to have a very
peculiar psychology ~nd physiology not to benefit from iI, ,while being
able to predict. Thus to take the weather example, he wQuld have to
recogJlise cloud patterns as traces of future weather, but not the state of
Limits to Knowledge of Universe~(ii) Sign Asymm.etry 219
ground or vegetation as traces of past weather; to read weather-
predicting but not weather-recording instruments.
A further consequence of past/future sign asymmetry is that the states
of a human brain are signs of past but not of future states of that brain.
Now, as we saw in Chapter 9, if a man observes something, stimuli from
that thing must affect him. The evidence of physiologist's and psycho-
logists is that the stimuli affect the brain. Hence the states of a man's
brain are signs of the states which he has observed but not of those which
he is going to observe. Hence men are able to report what they have
observed but not what they' will observe. '
'Hence past/future sign asymmetry explains the existence of cor-
roborated reports of the past but not of the future, as well as explaining
the fact that traces allowing easy inference are much more frequently of
the past than of the future. Our greater knowledge of the past than of the
future is~within the logical limits set forward ill Chapter 9~du.e to
this physical feature, past/future sign asymmetry. It is clearly a most
important feature of the physical world, and the question arises whether
we can trace it to the operation of anyone scientific law.
There has been a well-known attempt to account for past/future sign
asymmetry in terms of the second law bfthermodynamics. Reichenbach
[1] traced the asymmetry in detail to the operation of this law and the
existence of what he termed 'branch systems'. I shall argue that
Reichenbach's explanation of the asymmetry is seriously inadequate;
but, before doing so, I had better set forward the second law of
thermodynamics. .
The first law of thermodynamics states that a body gains energy,
including heat which is a form of energy, only if another body loses
energy, possibly in the form of heat. If you put a cube of ice on a warm
surface, the ice will melt only if the surface loses heat. The second law of
thermodynamics states that energy flows from systems possessing more
to systems possessing less; and so heat flows from hot bodies to cold
bodies. The first law would be satisfied if the ice grew colder and the
surface grew warmer; heat would have been conserved. But the second
law states that this never happens ~ temperatures even out. The ice gets
warmer and the surface grows colder. A kettle' of water on the fire gets
warmer and the fire gets colder. The second la w gave an exact description
of this observable fact by defining a quantity known as entropy (S) and
stating that the entropy of a closed 'system always remains constant or
increases, never decreases. The entropy of a closed system is the sum of
the entropy of its parts in thermodynamic equilibrium (viz. at the same
temperature through<?ut), A system is closed if it is not subject to
220 -Space and Time
influences from outside, and only a closed system has entropy. The
entropy of a system in thermodynamic equilibrium is a function of its
internal energy (kinetic and potential) and volume. The internal kinetic
energy of a dosed sy~tem is given by its temperature (T). Thus the I
the temperatures will have evened out, both volumes of gas having a
3T .
temperature of 2-'
In the later nineteenth century Maxwell and Boltzmann gave an
. explanation of the second law itself. Any system can be in anyone of a
large number of microstates. A description of a microstate is a
description of certain parameters of all the particles of the system, these
parameters being position and momentum or perhaps kinetic energy.
Two microstates of the system differ if the parameters of any of the
particles differ within a range- e.g. if a particle is in this small box rather
than in that small box, or has a velocity in this band rather than in that
band. (The size of boxes or bands which we take does not in general affect
the conclusion thatthe second law holds - on this see [3].) Given a fixed
total energy and volume of the system, there are very many possible
microstates of the system. These are assumed to be equiprobable-that
is, the system is at any instant, given no furthei' information, as likely to
be in anyone microstate as in any other. A large number of different
microstates all corr!!spond to one single macrostate. A macrostate Of the
system is defined by the temperature of the different parts of the system.
A system is at one instant in a different macrostate from what it is at
another irtstant if the various parts of the system at one instant have
noticeably different temperatures from the same parts at the other
Limits to Knowledge of Universe---,-(ii) Sign Asymmetry 221
instant. The temperature of a part which has the same temperature
throughout is defined in terms of the mean kinetic energy of the, particles
in the part. It can then be shown that for normal sys~ems (e.g. volumes of
gas) in our universe almost all the microstates correspond to the single
macrostate of an approximately even distribution of temperature
throughout the system, -and so are microstates which are ilOt macro-
scopically distinguishable, and that fewer and. fewer microstates cor-
respond to macrostates of lower and lower entropy. It can be shown
from this that the 'entropy of a system which passed through different
, macrostates would very probably get larger and larger until it reached
maximum, at which with very slight fluctuations the sys~em would;,
remain.
The conclusion depends on the assumption that all microstates are
equiprobable. Microstates are so defintXl that this assumption holds.
Different types of statistics are defined by the different microstates which
they distinguish and term equiprobable. The Maxwell-Boltzmann
statistics assumes that individual particles can be distinguished. Hence a
microstate of particle a l being in position PI with velocity VI' while al is
at Pl with velocity V 1 , is a different microstate from al at PI with VI and al
at Pl with V 1 . On the Bose-Einstein statistics particles are not dis-
tinguishable, and hence the number of possible microstates is much
smaller. On the Fermi-Dirac statistics no two particles can be in the same
energy state, and so the number of possible microstates will be much
smaller still. Again these statistiCs can be subdivided according to
whether the equiprobable microstates are states of energy or momen-
tum; and whether we assume an infinite range of possible energy or
momentum states, or, following quantum theory, only a finite number.
Which statistics is to be applied to a given system is a matter for empirical
investigation. Thus the Fermi-Dirac statistics, applying to all particles
subject to the Pauli exclusion principle, applies to electrons and protons;
whereas the Bose-Einstein' statistics applies to molecules having even
numbers of protons, electrons, and neutrons. We must adopt that
system of statistics, the macroscopic consequences of vyhich show that
its microstates are equiprobable.
In all these forms the basic law is that entropy ~emains constant or
increases- but is now in a merely statistical form. The law now states
that it is very very improbable that entropy decreases by any significant
amount, but it nevertheless remains just possible that it will. Very,
'very occasionally a kettle of water put on a fire will freeze. The entropy
graph of a closed system with initial low entropy is thus somewhat as
. follows:
222 Space and Time
\
S
(entropy)
I (time)
Figure 9
from present to future states; this limit is riot detectable on the macroscopic
scale. Anyway it is doubtful whether Quantum Theory imposes greater limits on
the accuracy of predictions than on the accuracy of retrodictions (for this, see [I] .
ch. v).
, 228 Space and Time
process of a closed system passing from a state S I on a certain path to a
state S2 is A-reversible if,. given the occurrence of SI and the rate of
change of state at the instant of the occurrence of S I, the passage from S I
to S2 is as probable as the passage along the reverse path from S2 to SI'
gi~en the occurrence of S 2 and that the rate of change of state at the
instant of the occurrence of S2 is the opposite of that on the original
passage, If the laws are deterministic and, given S I and a certain rate of
change at the instant of the occurrence of S I' the system inevitably passes
to a state S2' then if the process is A-reversible, the reverse passage from
S2 to SI will be equally inevitable, given S2 and the opposite rate of
change at the instant of the occurrence of S2 to that on the original
passage. Another way of putting the claim that a law describes an A-
reversible process is to say that the law accounts equally well for the
succession of states if we substitute - t for + t in the equations
describing the states. The same law would account for the process
apparently depicted by a film of the original process played in reverse.
The law will be A-irreversible if one passage is more probable to occur
than the reverse passage, given S I and the rate of change at the instant of
its occurrence for the first passage, and for the reverse passage S2 and the
opposite rate of change of state of S2 at the instant of its occurrence to
that at the instant of its occurrence on the first passage.
All the laws of classical physics discovered before the second law of
thermodynamics are A-reversible laws. Thus if we have a state S I of four .
particles at distances each of one foot from a central point, and S2 those
. four particles at distances each of two feet from the central point along
the same radii as before, then the law of inertia allows equally passage
from SI to Sl> and from S2 to SI' Given certain velocities of the particle
at the instant of the occurrence of S I, the system inevitably passes from
SI to S2' If we reverse the velocities of the particles at S2, the reverse
passage of the system is equally inevitable.
Now many changes which we meet in chemistry and biology are
apparently A-irreversible - marigold seeds turn into marigolds but a
film of the process in reverse would not appear to depict a physically
realisa,ble pro~ess. The first A-irreversible law t9 be studied by science
was the second law of thermodynamics, and it is to the A-irreversibility
of this law that the A-irreversibility of many of the changes of chemistry
and biology referred to have been attributed. In its original pre-statistical
form the second law was A-irreversible because it stated that increase of
entropy could occur but decrease could .not. In its statistical form the
second law is A-ir~eversible because it-states that it is \l1ore probable th~t
a closed system in a state of increasing low entropy should pass along
•
1 For the original papers see R. S. Casella Physical Review Letters, 1968, 21,
1128-1131, and 1969,22,534-6.
230 Space and Time
law is an isolated case, and seems unlikely. to have many effects on a
macroscopic scale. In general the bask laws of nature seem to be A-
reversible.
I distinguish laws describing A-irreversible processes from laws
describing B-irreversible processes. A process of passage between a state
S I and a state S2 is, on my definition, B-reversible if the occurrence of
passage from S I to S2 is as probable as the occurrence of passage from S2
to S I, and B-:irreversible if this condition does not hold. The difference
from A-reversible processes is that I do not write here 'given the
occurrence of S I and the initial rate of change of state' or 'given the
occurrence of S 2 and its rate of change being the opposite of that on the
original passage'. Now on this definition, passage from a state of high
entropy to one of low entropy is not a B-irreversible process. This is
because, as I noted on p. 222, in a syste'm closed for an infinite period,
passage from a state of high entropy to one of low entropy is just as
probable an occurrence as the reverse passage. But the second law of
thermodynamics describes an A-irreversible process because if one
system is (as is highly improbable) in a state of low entropy, it is very
likely to pass quickly to a state of high entropy, whereas if (as is highly
probable) the system is in a state of high entropy, it is very unlikely to
pass quickly to a state oflow entropy. Although it is highly probable that
once you have them, marigold seeds will turn into marigolds and highly
improbable that once you have them, marigolds will turn into marigold
seeds, a state of an eternal Universe in which one of the processes is
occurring is just as likely. as one in which the other process is occurring.
However there- are B-irreversible processes in nature. Examples
recently discussed are the examples of light waves and water waves. I
Suppose a source oflight d to send photons radially in ~II directions until
they illuminate the walls of a container surrounding the source. S I is the
state of the system when the source emits photons; S 2 its state as the walls
are illuminated, Now the passage from S I to S2 is an A-reversible
sequence. You can push photons back from widely separated positions
to a central one just as you can p'ush billiard balls back. 'But the
occurrence of the passage S2 to S I is highly improbable. This is because,
unless the container is spherical with internal reflecting walls, the only
way to initiate the reverse passage is fora clever lighting engineer to rig
up the container so that its walls are light sources emitting photons at
such different instan'ts as to illuminate 0 simultaneously. The existence
BIBLIOGRAPHY'
J Thus Reichenbach (ch [1], p. 139) supposed that there might be 'a galaxy in
which time goes in a direction opposite to that of our galaxy'. But he only
supposed this because he had defined ([ 1], p. 127) 'the direction of time' as 'th<:
direction in which most thermodynamical processes occur'.
14 The Size and Geometry
of the Universe
In Chapter 6we considered what it meant to say of Space that it had a
certain geometry, was finite or infinite and, if finite, had a certain volume.
In Chapter 9 and in the last two chapters we have examined the amount
of knowledge we can have about states of objects at. other places and
temporal instants and the degree of its reliability. We are now in a
position to examine how we can reach a conclusion about the size and
geometry of our Universe.
We should understand by the Universe, as was explained in Chapter I,
all the physical objects that there are spatially related to the Earth. The
concept of the size of the Universe is ambiguous. We may understand by
it the size of the space in which those objects are situated. I shall call this
the s-size of the Universe. But we could understand by the size of the
Universe the size of the smallest volume of space which enclosed all the
physical objects that there were. (Since the constituents of the Universe
are separated by empty space, to make this concept precise, some
convention would have to be laid down as to how the enclosing surface
was to be drawn round the edge of the outermost objects.) I shall call the
latter concept the m-size of the Universe. The difference between the two
concepts can be easily seen. From the seventeenth century onward men
believed without question that the s-size of the Universe was infinite.
However far you went along a straight line, you could always go on,
coming to new regions of space getting all the time further from your
starting-point. But they argued much about whether the visible Universe
of stars and planets formed an island in an otherwise empty space or
whether, however far you went along any straight line, you would come
to new stars and planets. Their argument was about the m-size of the
Universe. To say that the Universe is s-finite or s-infinite is to say that the
space in which its constituents are situated is finite or infinite. The
meaning of the latter statements was discussed in Chapter 6. To say that
the Universe is m-finite is to say that the smallest volume containing all
the constituents of the Universe is finite. To say that the Universe is m-
234 Space and Time
infinite is to say that no finite volume can contain all its constituents. If
the Universe is s-finite, then-of logical necessity-it is m-finite, and if
it is m-infinite, then ---' of logical necessity - it is's-infinite, but the
converse relations do not hold. Finally, I understand by the geometry of
the Universe the geometry of the space in which the objects of the
Universe are situated. This seems the clear meaning of an unam biguous
expression.
We discussed in Chapter 6 arguments to show that Space must of
logical necessity have a certain geometry or be infinite, and showed their
worthlessness. The s-size and geometry of the Universe is a matter for
empirical investigation. Arguments to prove that of logical necessity the
Universe is m-finite or m-infinite are equally worthless. Such arguments
on these issues I ,shall term a priori arguments.
Kant [1] put forward what "he considered to be two very good a priori·
arguments, one to show that the Universe was m-finite and the other to
show that it was m-infinite. The argument that the Universe is finite (viz.
m-finite) is set forward as Kant's thesis. In order to conceive of the
Universe being infinite, he argues, we should have to conceive of an
infinite number of parts of the Universe, viz. physical objects composing
it, having been enumerated. But to conceive of this we should have to
imagine ourselves having counted the parts of the Universe for an
infinite time. 'This, however, is impossible. An infinite aggregate of actual
things cannot therefore be viewed as a given whole, nor, consequently, as
simultaneously given' ([1], B.456) and hence the Universe must be finite.
But in answer to Kant we may say that certainly, we cannot picture in our
mind's eye an infinite Universe, but we can understand what is meant by
saying that there is no limit to the number of physical objects of finite
volume which can be counted; and in the way to be outlined in this
chapter, we can coherently describe evidence which would support this
claim. Hence the alternative to the finite Universe is not inconceivable.
The argument that the Universe is infinite is set forward as Kant's
antithesis. To suppose that the Universe is finite, he urges, is to suppose a
finite world surrounded by empty space. But empty space is not a thing.
So we suppose a relation between something and nothing - which is
meaningless. So, he argues, a finite Universe is inconceivable, and hence'
the Universe must be infinite. But we have only to phrase the claim that
the Universe is finite a little more clearly to avoid this objection. To say
that the Universe is m-finite is to say that there are no physical objects
spatially related to the Earth, except those within a certain finite distance
from the Earth. In this form of the proposition we are not relating
The Size and Geometry of the Universe 235
something to nothing. I conclude that neither the argument of Kant's
thesis nor that of his antithesis is valid. 1
Kant claimed that, given that talk of the Universe as a whole was
proper, both his conclusions were correct. Since, however, the con-
clusions contradicted each other, this only showed that all talk about the
Universe as a whole was improper. He urged a similar conclusion from
the alleged fact (to be discussed in our next chapter) that he could prove'
both the finitude and infinitude of the age of the Universe. We shall argue
that Kant's arguments on that topic also are bad arguments. Kant has
not substantiated his claim that all talk of the Universe as a whole is
improper. He put forward his claim that an antinomy (viz. contradictory
conclusions) arose if we talked of the Universe as a whole, to substantiate
his general claim that any talk of objects which could not be 'given
adequately in experience' was improper. The Universe, he claimed, was
such an object, because it consisted, not of a certain number of (to use his
term) 'phenomena', but of all the phenomena that there a~e. If taken
seriously Kant would seem tt be claiming that we cannot make proper
assertions about all members of a class, all the so-and-sos that there arc.
But quite obviously we can talk of all swans, or all pieces of iron - even
though we cannot picture in our mind's eye the totality of these. If a man
claims that, although all talk about all swans and all pieces of iron is
legitimate, nevertheless talk about all physical objects spatially related to
the Earth is not, the onus is.onhim who makes this claim to prove it, since
the logical status of the two kinds of talk is very similar. Scientists discuss
coherently whether all physical objects (meaning physical objects
spatially related to the Earth) have mass or have a minimum volume.
WhY should they not discuss whether there is a limit to the number of
such objects?
I conclude in the absence of counter-arguments that talk about the
. Universe as a whole is proper but that arguments to prove that the
Universe of logical necessity is m- or s-finite or infinite or has some
'specified geometry fail. The size and geometry of the Universe is an
empirical matter. Our Universe is clearly very large and we on Earth can
only observe a very small portion of it. Hence we must use observed
states as traces of unobserved states and thereby, if we can do so, learn
about all the unobserved objects that there are and so reach a conclusion
that the Universe was s-finite and so m-finite which are of mere, historical
interest: On the Heavens, 276a-279b.
236 Space and Time .
about the size and geometry of the Universe. The scientific cosmologist
argues from such evidence about parts of the Universe distant in space
and time as he can collect (within the limits, logical, and physical,
discussed in Chapters 9,12, and 13) to thestructure of the whole. Such
arguments from particular features of the Universe to its general
character I will term a posteriori arguments. The procedure is to use
traces to infer the density of matter and the geometry of Space and othe,r
features in regions immediately beyond observation and to use these
traces to infer the density of matter and the geometry of Space in more
distant regions, and so ad infinitum. Thus we may ascertain the metrical
geometry of Space as a whole and the density of matter throughout the
Universe. The geometry of Space may prove to be such that if we take a
certain finite number of spatial regions of finite size beyond those which
we observe, we eventually return to our starting-poinLThen these will be
all the regions that there are. As we saw in Chapter 6, if we examine all of.
many observed regions and find that the geometry is elliptic of uniform
curvature, and so we use each region of a certain finite size with that
geometry as a trace that beyond it there is a region with the same
geometry,.~e should have to conclude there were only so many regions
of that finite size. The evidence from traces is then that Space is finite, that
is, the Universe is s-finite. The evidence from traces may, however, be that
however many spatial regions of finite size we consider, there is always
another one. The evidence is then that Space is infinite, that is the
Universe is s-infinite. If the evidence from traces is that there is nolimit to
the number of regions of finite size occupied by matter, then it shows that
the Universe is m-infinite. If the evidence is that there is a limit, then the
evidence is that the Universe is m-finite.lfwe were to observe that in each
examined region of Space of volume Vat some distance d from the Earth
the quantity of matter wask -d + Ik -d I <lk -dl means the positive
value of the difference between k and d), k being a constant, then the
evidence is that after a distance k = d from the Earth there is no matter
and hence that the Universe is m-finite. The evidence, however, which we
have sketched in Chapter 11 is that the Universe is homogeneous, that in
each regiori of sufficient size there is the same volume of matter. Hence
we have reason to believe that if the Universe is s-infinite, it is m,infinite,
and that if it is m-finite. it is s-finite.
The question therefore is whether the Universe is s-infinite, whether
Space is infinite. To answer this question we must ascertain the geometry
of distant regions and so see whether there is a limit to the number of
regions of finite size. To do this we need a cosmological theory showing
how the geometry of Space varies, if it does, from region to region,
The Size and Geometry of the Universe 237
whether dependent on or independent of the density and distribution of
physical objects. With its aid we can use the states of observed regions as
traces of the states of regions beyond them.
The cosmologist endeavours to put forward a scientific theory which
will describe and explain the overall behaviour of the Universe, the
interactions and distribution of its constitutents on the large scale. His
data fall into two groups. One group is the results of observations and
experiments near the surface of the Earth explained at a lower level by
such physical theories as the electromagnetic theory and the Special
Theory of Relativity. The other group is the results of telescopic
observation into parts of the Universe distant in space and time from
ourselves. The cosmologist endeavours to build a coherent theory of the
Universe from which the observed data can be derived and new
predictions made which can be tested.
Most modern cosmological theories derive from the General Theory·
of Relativity. This was formulated by Einstein in 1916 [2] solely on the
basis of the first class of data. Einstein's Special Theory of Relativity had
shown, as we saw in Chapter 3, that the laws of physics referred to any
inertial system were the same. Einstein felt that it was somehow arbitrary
to refer the laws of nature to inertial systems; and he sought, as we noted,
to set them forward in a way so that they could be referred to any E-
frame of reference whatsover. They should hold just as truly on. a frame
rotating relative to an inertial frame as on an inertial frame. The
requirement that the laws of nature have the same form when referred to
any coordinate system (that is, any E-frame) whatever is known as the
principle of covariance. Further, it seemed a weakness of previous
mechanics that the accelerated motion of a body under the force of
gravity was described IlS motion under a force. Since all bodies are
equally subject to, gravity, why not describe it as a natural motion (viz.
one not due to the operation of a force)? Einstein's principle. of
equivalence states that to describe the effect of a gravitational field is
equivalent to referring the motion of the body to an accelerated frame of
reference. Since by the principle of covariance the laws of nature are to be
the same in all coordinate frames, the body's motion is thus a natural one.
Aiming to generalise mechanics and to some extent optics in these ways,
Einstein set forward what he considered to be the simplest most general
equations or motion. These were that free particles move along what he
called the geodesics of space-time. (Space-time is our three-dimensional
space with a fourth dimension added for time. The use of this fo~r
dimensional geometry is, as noted in Chapter 7, a mere convenient
calculating device. Geodesics are paths in space-time analogous to the
238 Space and Time
straight lines of ordinary physical space. A special kind of geodesic is a
null geodesic, along which light must pass.) Einstein then set forward a
formula relating the geometry of space-time to the distribution of matter
which would make possible calculation of geodesic paths. This formula
was chosen as the most simple possible form of such a principle
compatible with the data of mechanics and optics available to date. The
formula summarising in tensor notation ten equations, known as the
field equations, is as follows:
k and R(t) have the role described on pp.198f. R'(t) is the first derivative of
R(t) with respect to time (t) (viz. it measures the rate of change of R(t)
over time) and so is a measure of the rate of mutual recession of the
clusters. R" (t) is the secQnd derivative of R (t) with respect to t (viz.
measures the rate of change of R'(t) over time). p is the mean internal
pressure of matter (and radiation), and p its mean density. G is the
constant of gravitation, and c the velocity of light, as before. A is the
cosmological constant, whose value is only calculable through the field
equations. Thus G and c are known constants; k and A u.nknown
constants; p, p, R(t), R' (t) andR" (t) are 'time-dependent variables. Since
the Universe is homogeneous, and geometry is determined by the
presence of physical objects, the geometry of every region of sufficient
size (viz. a region in which there are enough clusters for their random
motions and distribution to cancel out) will be the same as that of any
other such region. This is measured by k. If k = 0 or - 1 we can
extrapolate without limit; beyond every region of finite size there will be
another one and so the Universe will be infinite. If k = + 1, there will be
onl'y ~finite number of regions and the geometry will be elliptical (or
spherical). The volume of an elliptic universe is n 2 R (t)3. (Of a spherical
universe it is 2n 2 R(t)3.)
Mathematical cosmologists have produced from the simplified field
equations various more specific 'models', by making assumptions about
the values of the constants and for the present cosmic instant of the
variables referred to above, thereby presenting specific cosmological
theories r-eady for confirmation or rejection by the results of astronomi-
cal observation. Thus if we suppose, as Einstein reasonably did in 1917
before the evidence from the red-shift of the expansion of the Universe
a
nad been produced, static Universe, R' (t) = R"(t) = O. The equations
above then reduced to:
240 Space and Time
8nG ,k
. 7P = A - R (t)2
8nG _ A 3k
- 7P - - R(t)2
BIBLIOGRAPHY
Kant claims that, but for the inadmissibility of talking about the
Universe as a whole, this argument would be valid. This, however, seems
plainly false, and so there is no justification for taking Kant's escape,
route. The argument fails for the following reason. In considering series
we normally suppose the first member identified and then go on to see
what can be said about the other members or the sum. Hence it seems
reasonable enough to claim that a completed infinity is impossible. But if
it makes sense (which seems dubious) to talk about a series with a last,
but not a first, member identified, the impossibility of a completed
infinite series of this type is in no way obvious. It seems more natural,
however, to say that the series to be considered in discussing our
question is a series of causes, not effects, and that the first member is the
present state of the Universe. In this case there is no difficulty in
supposing that th~ series may be infinite, foi-,it may have no last member
and so it would not be completed. _
The best~known a priori argument for the eternity of the Universe
derives from Aristotle. 1 The medieval theologians discussed it and
claimed to have refuted it, and Kant did the same. The argument is that
if the Universe had a beginn'ing at an instant of time, before that instant
there must have been nothing at all. But then nothing cannot give birth
to something. The creation of the Universe needs to be explained by
some preceding state which brought it about. But if all there was before
creation was void, then there was no state'to explain the creation.
Therefore creation cannot have occurred. This is roughly the argument
I Aquinas argued that although it was a necessary truth that every state had a
cause, that cause need not be another state of physical objects. He allowed that
instead an agent might be the cause ([3] 2.35-7). Science, however, operates
with state or event causality and attempts to explain all states in terms of other
states which bring them about. The suggestion that agent causality is a different
type of causality from state or event causality, but that explanation in terms of it
. is equally good explanation, is a difficult and much disputed one. To discuss,
however, would take us too far afield. We have accepted Aquinas's claim that is
it not a logically necessary truth that every state has a precedent state as a cause;
which alone is the relevant point here. For the argument of Kant's antithesis
depends on the latter being a logically necessary truth.
252 Space and Time
Alternatively the cosmologist might succeed in giving an explanation
of every state of the Universe in terms of a preceding state, and that
would show that that sort of explanation could be given.
All this means that the claim that every state Of the Universe is caused
by a preceding state needs substantiating by detailed scientific cosmo-
logy. So we must turn from the a priori arguments to the a poste,iori
arguments about the beginning of the Universe, these being ones which
include among their premisses statements of particular observable
features of the Universe. We must look, in other words, at models of the
Universe developed from the General Theory of Relativity. or other
general theories of mechanics. We investigated in the last chapter the
evidence available now and the type of evidence which we could have
subsequently in support of some one of such models. Before examining
cosmological models, however, we had better clarify the structure of the
a posteriori arguments used in both cases, to prove that the Univer~e had
a beginning and to prove its eternity.
The a posteriori arguments to prove thatthe Universe had a beginning
are developed by popularisers of science or theology from what I may
term creation theories of science. All creation theories have the
following common structure. The Universe is said to be now in a state
Spo It is further claimed that a fundamental law of nature controls the
evolution of its states, such that if it is now in state Sp, then so many
years ago it must have been in state Sm and in so many years' time it will
be in state Sn. Sp is thus a trace of those states. On the basis of this law,
inference is made that at to, some finite number of years ago on all
admissible time scales, viz. normally on a cosmic time scale, the Universe
must have been in a state So. But So is either a state known, on the basis
of this or other laws of nature, to be physically impossible or is a state in
which there is nothing physical existent. But if the state is physically
impossible or there is nothing physical existent, the popular argument
takes over from the scientific, then the Universe carinot have existed at
that instant, but must have come into existence subsequently.
Scientific laws, we saw in Chapter 9, are forward moving; they state
the future consequences of some present state. Hence if a law is to be
used for retrQdiction we have to have reason to believe that any state
which could have caused the present state other than the one stated by
. the law did not occur at the relevant time. However, in the case of a
purported cosmological law , that is, a purported basic law governing the
succession of states of the Universe, given that there can be no cause
lying outside the Universe of a state of the Universe, then unless the
purported law is not completely true?r there is an uncaused state of the
The Beginning and End of the Universe 253
Universe, it can be used for retrodiction. For the Universe is all the
physical objects that there are spatially related to the Earth. If there is a
cause of a state of the Universe which lies outside it, it could only be a
state of an object not spatially related to the Earth. But we would hardly
be justified in postulating the existence of such objects solely to avoid the
conclusion that an uncaused state occurred. If we suppose as a working
hypothesis - to avoid arguments about theology - that we have no
evidence of the existence of such states, we must look for causes
of states of the Universe within the Universe. But in that case if some
, state of the Universe Sm has a cause, it must be a precedent state of the
Universe, say Sk. But then either the purported cosmological law in
question permits Sk to have brought about Smorit does not. Ifit does not
it cannot be the tru~ law governing the evolution of states of the
Universe. If we have evidence that a state of the Universe is brought
about other than in the way which we could retrodict by using a
suggested cosmological law, then it shows that the cosmological law
must be wrong. The case with cosmological laws is thus radically
different from the case with other laws. Men walking on sand produce
footprints. We see certain footprints and infer that men have walked.
But suppose we have evidence that these footprints have other causes .
(say, animals walking with man's shoes on them). That has no tendency
to show that the purported law 'men walking on sand produce
footprints' is false. It is otherwise with cosmological laws (given the non-
existence of other spaces); for states of the Universe can only h~ve other
states of the Universe as causes. If the purported cosmological law
compels us to retrodict Sj from Sm, but we have evidence that Sb not Sj
brought about Sm, that shows that the purported law is wrong. The only
way in which we can avoid the retrodictive consequences of accepting a
cosmological law is by postulating that some state of the Universe had
no preceding state as ca1,lse.
The prototype of the a posteriori argument to a beginning of the
Universe is the' thermodynamic argument used by various popularisers
of science and theology during the last hundred years. The argument
runs as follows. The Universe is ,at present in a state of considerable
thermodynamic disequilibrium, and entropy lies between its minimum
and maximum. By the second law of thermodynamics, as originally
formulated, entropy increases or remains constant, never decreases i'n a
closed system. The Universe is a closed system since it consists of all the
physical objects that there are spatially related to the Earth. Therefore
. some finite number of years ago, n years, entropy must have been at its
minimum. But it could not'have been like this for ever previously,. since a
254 Space and Time
long continuing state of minimum entropy (So) is a physically im-
possible state. Hence there could not have been a Universe more than
n-or perhaps n + i-years ago. Now this argument is not much in
vogue today because it begs many important scientific questions - for
example, it may only work for a spatially finite Universe, for the
existence of which it produces no evidence. Further, on the stat~tical
version of the second law (see Chapter i3) decreasing entropy is not
completely ruled out and it is ~s probable that an eternally closed system'
should be at any instant in a state of decreasing entropy as that it should
be in a state of increasing entropy. But I cite the argument because it
illustrates clearly and simply the a posteriori arguments for a finite age of
the Universe which are in vogue today.
The a posteriori arguments in vogue today are based on the
cosmological 'models', viz. theories of the Universe substantiated by
various empirical observations in the way described in Chapter 14. They
are, as we' noted, often constructed by giving various values to the
constants and variables of equations developed from the General
Theory of Relativity or rival theories of mechanics. The resulting
solutions yield various values of R(t), the scale factor such that if the
distance between two galactic clusters at cosmic instant t 1 is R(t Jx, then
the distance between them at t 2 will be R(t 2)X. R'(t) is the rate of
expansion of the Universe. We get oscillating Universes, and universes
'with constant or variable rates of expansion. Some of the 'models' yield
an instant to, a finite time ago at which R(t 0) = O. They show, that is,
expansion from an infinitely dense Universe. But an infinitely dense
Universe is a physically impossible Universe. Hence the Universe must
have come into existence after to' The argument here is more subtle than.
the thermodynamic argument, but its structure is the same. The
Universe is now in a certain state - Sp, that is, in this case, has a certain
scale. The law stating its expansion is a fundamental law of physics.
Hence a first state of the Universe must have been subsequent to to' This
is the conclusion to which we are led if a model yielding an instant to at
which R(t 0) = 0 is very well substantiated. For if the state of the
Universe at the instant immediately after to' Slat t l' was brought about
by a causal process, it cannot have been in accordance with the
fundamental theory stated. If an explanation of So is to be provided. it.
can only be done by having a difTerent cosmological theory. The new
. theory would need justification, more justification than that the old
theory could not explain So. For if the latter sort of fact were always
sufficient justification for accepting a new theory, then 'every state has a
cause' must be a necessary truth - and I have argued that it is not. If the
The Beginning and End of the Universe 255
mere failure of a theory to provide a cause for s·omething within its
domain shows that the theory is mistaken thenthe false consequence
follows.
Hence we can only maintain the existence of the Universe before 10 by
a supposition of entirely different fundamental laws of the Universe for
which ther~ is ~ex hypothesi - no other evidence from its states. The
criterion of simplicity demands simple laws with one unexplained state
rather than complex laws explaining all states. Scientific evidence in
consequence points to S I being a state uncaused by a precedent state. If
S I was not caused, it could not have been preceded. For given the truth
of the current cosmological theory, any preceding state would have
brought about some other state than S I ' But S I occurred. Therefore it
cannot have been preceded. So the Universe must have had a beginning
a finite time ago.
It is not mere failure to achieve an explanation of S I ' that provides
evidence that SI could not be explained. Science has often failed to
explain something for centuries and then succeeded. But the evidence
that S I could not be explained would be that such explanation was ruled
out if you accepted the very successful theory which so brilliantly
accounted for all other cosmological phenomena (viz. all the other states
of the Universe). This point is brought out by the similar case of
Quantum Theory. It is not the mere fact that scientists have not .yet
succeeded in giving a complete explanation of certain phenomena within
the quantum range which makes them want to say that those phenom-
ena are not fully explicable. It is the fact that they have succeeded in
explaining very well to a certain degree of accuracy a whole range of.
atomic phenomena, and that the theory by which they explain those
phenomena rules out any complete explanation of phenomena within
the quantum range.
Arguments to prove that the Universe has lasted eternally have the
following common pattern. The Universe is said to be now in a state Sp.
It is claimed that a fundamental law of nature controls the evolution of
that state, such that if the Universe is now in state Sp, then so many years
ago it must have been in state S'" and in so many years' time it will be:in
state Sn. On the basis of this law retrodiction is possible to physically
possible states as long ago on all admissible time scales, viz. normally the
cosmic time scale, as you choose. So Sp is best explained by postulating a
previous state Sn" and SOl is best explained by postulating a prior state Sj
and so on ad infinitum. If you accept that the evidence shows this, then
you accept that the best explanation of the present state of the Universe
is as a member of an infinite series of such states.
256 Space and Time
Various cosmological theories do permit infinite regress of expla-
nation. The best known is Steady State Theory. As explained in Chapter
14, according to this theory the Universe has always consisted of clusters
of galaxies at roughly the distance apart that they are now from each
other. The increasing distance apart of any two clusters is compensated
by the continual 'creation' of matter in the space between clusters. This
new matter then condenses into new clusters, and hence average distan'ce
between clusters remains roughly constant. Other theories permitting
infinite regress of explanation include oscillating Universe theories (that
is, theories of the Universe oscillating between physically possible states,
vii. ones both having a finite positive value of R(t». According to
oscillating universe theories the Universe is now expanding, was.
previously contracting and so on into the infinite past or future. On all
such theories a present state of the Universe is best explained by a
preceding state and so ad infinitum.
Now alI a posteriori arguments to the eternity or to a beginning of the
Universe, like a posteriori arguments to its spatial finitude or infinitude,
depend on the truth of a cosmological theory which, as a scientific
theory, may be chalIenged on the usual scientific grounds. It may b~
urged, first, that the Universe isnot now in the state Sp- that the claim
that it is represents an illegitimate interpretation of observations. We
analysed in the last chapter the difficulties involved in interpreting
astronomical observations. Secondly - as is most usually done - it
may be claimed that the proposed law of change of S, the proposed
cosmological theory, is wrongly stated; or, if correctly stated, is not a
fundamental law of nature, but applies only over a certain temporal
period. The latter is the claim that its application in certain conditions is
a derivative consequence of more fundamental laws of nature; and since
these conditions only hold for a certain period, the law in question is
only operative for that period. All proposed cosmological theories are
chalIenged on this ground. Thus G. C. McVittie claimed of models
having R = 0 at an initial instant t that: 'Going backwards in time we
can legitimately expect that, long before R = 0 is reached, conditions
similar to those that prevail at· present in the 'observed universe would
have broken down. Thus the uniform model universes would cease to .
apply.'! The conditions under which the proposed cosmological theory
applies are-matter for normal scientific argument. What must be shown
is that the theory is the simplest theory of the Universe consistent with
observations so far made, and hence that there are grounds for
BIBLIOGRAPHY
[I] Aristotle, On the Heavens, Book i, ch. 10; Book ii, ch. 't.
[2] St Bonaventure, Commentary on the Sentences, Book ii, dist. i,
pars i, articulus i, quaestio ii.
[3] St Thomas Aquinas, Summa Contra Gentiles, 2.32-8.
[4] I. Kant, The Critiq~e of Pure Reason, Transcendental Dialectic,
ch. 2, 'The Antinomy of Pure Reason' especially B.454-61 .•
For discussion of Kant's argument, see:
[5] P. F. Strawson, The Bounds of Sense, London, 1966, part iii, ch. 3.
262 Space and Time
For description of the world-models of modern cosmology, see
references in the Bibliography to Chapter 14.
For modern philosophical criticism of the coherence of talk about the
Universe as a whole and of the possibility of reaching a conclusion about
whether or not it had a beginning, see, among many writers who have a
similar approacb:
[6] M. Scriven, 'The Age of the Universe', British Journal for the
Philosophy of Science, 1954,5,181-90.
[7] Milton K. Munitz, Space, Time and Creation, Glencoe, Illinois,
1957.
[8] R. Harre, 'Philosophical Aspects of Cosmology', British Journal
for the Philosophy of Science, 1962, 13, 104-19.
I have not produced detailedcounter-arg~ments to deal with each of
these criticisms individually, but consider that I have met all the
criticisms in this chapter and in Chapters 1 and 14 of this book. For a
general attempt to show that many such criticisms are invalid see:
[9] G. H. Bird, 'The Beginning of the Universe', Proceedings of the
Aristotelian Society, Supplementary Volume, 1966,40, 139-50 .
..
.Index
Abbott, E. A. (and his novel Cantor, G., 115f
Flatland), 124, 128, 130 Carnap, R., 113, 180,245
Absolute Space, motion, acceleration, Casella, R. S., 229
42-60, 66, 202 cause, prior to eITect, proof of logical
Absolute Time, 202 necessity of, 132-40
analytic statements, see logically 'every event has a cause', 250f, 254f
necessary statements Cepheid stars, 89ff
a posteriori statement, definition of, 3f Chisholm, R. M., 143
a priori statement, definition of, 3f Clarke, S., 49f, 58, 59
Aquinas, St Thomas, 248-51, 261 €Iemence, G. M., 48
Aristotle, 10,24, 46f, 57,59, 118f, 131, clock, true, concept of, 177
23"5, 248ff, 261 clocks, atomic, 179f
Armstrong, D. M., 23, 25, 66 clocks, principle of ~imilar, 192-6
Augustine, St., 173 clusters, see galaxies, clusters of
Ayer, A. J., 135, 142 Coburn, R. c., 25, 41
contingently dependent occupants of
backward moving laws, impossibility Space, 17
of, 149f' conventionalism, 80IT, 109IT, 180f,
Bennett, J. F., 9, 15 20If
Berenda, C. W., 214 Copernicus, 47f
Berkeley, G., 52, 59 . Coprescnce, laws of, impossibility of,
Berry, M., 97, 214, 246 150f
Bessel, F. W., 89 cosmic radiation, 190,258
big bang, 258 . cosmic time scale, 195-202, 247f
Bird, G. H., 262· cosmological models, see cosmologi-
Black, M., 142f cal theories
black holes, 213f cosmological principle, 195f
Bohm, D., 205 - cosmological theories, 50f, 94f, 150,
Bokku, myth of Okku and, 167-71 208-13, 237-45, 254-61
Bolyai, J., 102 covariance, Einstein's principle of,
Boltzmann, L., 220 237 ,
Bonaventure, St., 248, 261 'creation' of matter, 241, 256
. Bondi, H., 212, 214, 241 curvature of Space (K), 104, 199
Bonola, R., 112
Bose-Einstein statistics, 221 deformation of rods, 69-83, 106, III,
Bosworth, R. C. L., 232 118
Bowman, P., 189, 205 De Sitter model, 208, 240
Brotman, H., 128, 130 diameter, method of apparent, 92
Brouwer, L. E. J., 115 differential forces, see universal forces
differential influences, see universal
caesium radiation, frequency of, 179f influences
264 Index
direction, definition of, 64f more basic, most basic frame, de·
distance, II, 61 -97, 197 finition of, 42-6
distance, by apparent size, definition free mobility, axiom of, 105
of, 96 Freedman, D. Z., 240f
distance, coordinate, definition of, 104 fundamental particle, definition of,
distance in a frame, 66-9 54, 189f
distance, luminosity, definition of, 96 see also galaxies, clusters of
distance, parallax, definition of, 96
distance, proper, definition of, 67. galaxies, clusters of (and fundamental
Doppler shift, see red shift particles) 51f, 68f, 91f, 189~200,
Dray, W., 143 206-12,238-44 '
Duinmett, M. A. E., 135-9, 142f Gale, R. M., 143
Dwyer, L., 164 - Galileo, 47f, 59, 119, 184
geometrodynamics, 59,60
Earman, J., 60, 182 geometry of Space, of the Universe,
Eddington, A. S., 81 56f, 72ff, 98-113, 197f, 233-46
Einstein, A., 42f, 50, 56f, 59, 78f, 83, congruence geometries, 105, 197f
106, III, 112, 179, 186f, 237ff, elliptic, Euclidean, hyperbolic, and
245 spherical geometries, main pro-
Einstein·De Sitter model, 208, 240, perties of, 98 -I 06
259 pure and physical, distinguished,-
Ellis, B., 77, 189,205 98-101
entailment, 4ff Gold, T., 241
entropy, definition of, 219f Godel, K., 141
wide concept 0(, 222f Gorowitz', S., 143
see also thermodynamics, second Graves, J. C, 60
law of Grice, H.P., 9
equivalence, Einstein's principle of, Grimaldi, F. M" 85
237 Griinbaum, A., 72, 81ff, 142f, 154,
equivalent laws, principle of, 190ff, 164,187,205; 231f
195f, 198 Guthrie, W. K. C, 46
ether, 183f
Euclid, 99f, 112, 131 Harre, R., 262
see also geometry, Euclidean Harte, E., viii
experimental test, defihition of, 149f Hausdorff Space, 98
extrapolation, 87 -96,108f; 147f, I 5 If, Heaven, 40
209-12,227, 244f, 257-61 Heisenberg indeterminacy principle,
214
factual statement, see logically con- Hesse, M. B., viii, 245
tingent statement Hill, E. L., 231 f
Fermi~Dirac statistics, 221 Hirsch, E., 25
Fine, A. I., 73, 83 homogeneity of the Universe, 196,
Flew, A. G. N., 143 236, 239, 243f
foreknowledge, "Iaims to, 144-8, Hooker, CA., 60
157-64, 215f, 219 horizons, cosmological, 206-14
van Fraassen, Bas c., 176, 205 EP horizon, definition of, 209
frame of reference, definition of, II f ET horizon, definition of, 213f
E·frames, 23f, 237' ' event horizon, definition of, 207f
equibasic frame, definition of, 51 particle horizon, definition of, 208
Index 265
Hoyle, F., 241 Lucas, J. R., 143, 175
Hubble, E. P., 93 luminosity, absolute and apparent,
Hume, D., 4 defined, 90
Huntington, E. V., 41 luminosity distance, see distance,
Hurewicz, W., 129f luminosity
Huygens, C, 48f
hyperplane, definition of, 117 M-absolute Space, see Space, M-
absolute
inertial frame, definition of, 48· M-relative Space, See Space,M-
instant of time, concept of, 131 , relative
irreversible laws and processes, see Mach, E., 52-6, 59
reversible laws and processes Mackie., J. L., viii, 143
isotropy, principle of, 189ff, 195f magnitude, absolute and apparent,
Ives-Stilwell effect, 186 90-3
Manning, H. P., 112, 130
Jammer, M., 130 Markowitz, W., 205
Janis, A. I.; 205 material objects, 13-15, 123-7
criteria for identity of, 19-24,31-7
k, definition of, \99 Maxwell, J. C, 220
Kant, I., 2, 4, 8,17,28,40,101,104, Maxwell-Boltzmann statistics, 221
1I0f, 112, 118-23, 130f, 165,172, McGechie, J. E., 164
175, 234f, 245, 248-51. 261 McVittie, G. C, 96f, 246, 2·56
Kepler, J., 43, 47f, 88 memory claims, 144-8, 157-64, 215f
Klein, F., 102f Menger, K., 115
knowledge of past greater than of Mercury, advance of the perihelion of,
future, 157-64,215-27 238,240
Krimsky, S.,.60 . mere physical objects, 15f
krypton light, wave length of, 75f meson decay, 229f
metrical geometry, definition of, 99
Leibniz, G. W., 49f, 57f, 59 Michelson-Morley experiment, 183r,
Lewis, D., 164 200
light, deflection of, near the surface of Milne, E. A., 180, 247f
the Sun, 238 Minkowski,H., 114
diffraction of, 84f MO-invariant groups, 23f
rectilinear propagation of, 84f, 87, Montague, R. D. L., viii
198, 238 Muller, R. A., 190
velocity of, 145, 182-8, 191, 199f, Munitz~ M. K., 262
203f, 207, 242
line, concept of, 98 necessity, see logically necessary state-
Lobachevsky, N. I., 102 ments, physical necessity, and
local motion, 39 ' practical necessity
Locke, J., 14f negation of a statement, definition of,
logically contingent statements, 2-7 2
logically dependent occupants of Nerlich, G. C, 83
Space, 16f Newton, I., 42, 48ff, 52ff, 57f, 59, 179,
logically impossible statements, 2-7 202,204
logically necessary statements, 2-7 Newton~Smith"W. R, 41, 173f, 176
logically possible statements, 2-7 van Nieuwenhuizen, P., 240f
Lorenz transformations, 185f North, J. D., 59, 97, 205
266 Index
observation, 154-7 quasars, 93
Okku, myth of Bokku and, 167-71 Quine, W. V. 0., 4, 8
oscillating universe theories, 256 Quinton, A. M., 13,25,28-31,37-41,
175
Page's model, 208
parallax distance, see distance, radius of the Universe, R (I), de-
parallax finition of, 198
parallax, method of, 87fT, 95 red shift, (Doppler shift) 89, 91fT,
Parfit, D., 35f, 41 189fT, 214, 239f, 242, 245
past, changing the, 134f gravitational red shift, 91
past, determinate, 132-40 Reichenbach, H., 44, 54JT, 59, 77, 80,
see also cause, prior to efTect 82f, 106, 109f, 112f, 122fT, 130,
. past/future sign asymmetry, 215-27, 153f, 164, 180,204,219,223-7,
259f 232
past, statements really about the, de- Relativity, General Theory of, If, 50,
finition of, 132fT 54, 77fT, 91,95, III, 141, 150, 198,
Pauli exclusion principle, 221 201,213,237-41
peT invariance, 229 Mechanical, Principle of, 48
perfect cosmological principle, 195f, Principle of, 50
241 Special Theory of, If, 50f, 54, 66fT,
Perry, J., 41 '184-8, 191f, 198-201,207,214,
personal identity, 31-':7 237f
physical object, concept of, 16 retrodiction, 145-50, 157, 215-27,
physical impossibility, necessity, and 252-8
possibility, 7f reversible and irreversible laws and
place, concept of, 10-12 processes, 227 -32
criteria of identity of, 20-4, 169f Riemann, B., 102f
plane, concept of, 63f Riemannian geometry, 103
Playfair's axiom, 102 rigidity, 62, 69-82
Poincare, H., 106, 112, 115 , Rindler, W., 198,205, 207f, 214
point, concept of, 26, 98 Robertson, H. P., 198 .
Popper, K. R., 54, 230JT Robertson- Walker line element, 198-
possibility, see logically possible 202, 207JT, 239, 240
statements, physical possibility, Russell, B., 135, 1.42
and practical possibility
practical impossibility, necessity, and Salmon, W. C, 205
possibility, 8 Sanford, D., 25
primary place, definition of, 10. Schlegel, R., 232
primary qualities, 14f . Scriven,. M., 262
primary tests, definition of, 6\ Searle, J. R., 164
propagation of efTects, finite vefocity secondary qualities, 14f
of, 145, 182f secondary tests, definition of, 61
proper distance, see distance, proper sentences distinguished from state-
Ptolemy, 47, 118JT ments,3f
Putnam, H., 83 Shoemaker, S., 173, 176
sign, definition of, 146
Quantum Theory, 1,7, 226[,.J50f, 255 similarity, criterion of, 2If, 32, 36, 169
see also Heisenberg indeterminacy simplicity, principle of, expoullded,
principle· 21,43fT
Index 267