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THE NATURE
OF THE
PHYSICAL WORLD
THE NATURE
OF THE
PHYSICAL WORLD
by
A. S. EDDINGTON
M.A., LL.D., D.SC, F.R.S.
Plumian Professor of Astronomy
in the
University of Cambridge
THE
GIFFORD LECTURES
1927
NEW YORK :
THE MACMILLAN COMPANY
CAMBRIDGE, ENGLAND:
AT THE UNIVERSITY PRESS
I929
All rights reserved
Copyright, 1928,
By THE MACMILLAN COMPANY.
Set up and electrotyped.
Published November, 1928.
Reprinted February, 1929.
Twice. March, 1929.
Reprinted April, 1929
SET UP BY BROWN BROTHERS LINOTYPERS
PRINTED IN THE UNITED STATES OF AMERICA
BY THE FERRIS PRINTING COMPANY
PREFACE
This book is substantially the course of Gifford Lectures
which I delivered in the University of Edinburgh in
January to March 1927. It treats of the philosophical
outcome of the great changes of scientific thought which
have recently come about. The theory of relativity and
the quantum theory have led to strange new conceptions
of the physical world; the progress of the principles of
thermodynamics has wrought more gradual but no less
profound change. The first eleven chapters are for the
most part occupied with the new physical theories, with
the reasons which have led to their adoption, and es-
pecially with the conceptions which seem to underlie
them. The aim is to make clear the scientific view of
the world as it stands at the present day, and, where it
is incomplete, to judge the direction in which modern
ideas appear to be tending. In the last four chapters I
consider the position which this scientific view should
occupy in relation to the wider aspects of human ex-
perience, including religion. The general spirit of the
inquiry followed in the lectures is stated in the concluding
paragraph of the Introduction (p. xviii).
I hope that the scientific chapters may be read with
interest apart from the later applications in the book;
'but they are not written quite on the lines that would
have been adopted had they been wholly independent.
It would not serve my purpose to give an easy intro-
duction to the rudiments of the relativity and quantum
theories; it was essential to reach the later and more
recondite developments in which the conceptions of great-
est philosophical significance are to be found. Whilst
much of the book should prove fairly easy reading, argu-
vi PREFACE
ments of considerable difficulty have to be taken in their
turn.
My principal aim has been to show that these scien-
tific developments provide new material for the philoso-
pher. I have, however, gone beyond this and indicated
how I myself think the material might be used. I realise
that the philosophical views here put forward can only
claim attention in so far as they are the direct outcome
of a study and apprehension of modern scientific work.
General ideas of the nature of things which I may have
formed apart from this particular stimulus from science
are of little moment to anyone but myself. But although
the two sources of ideas were fairly distinct in my mind
when I began to prepare these lectures they have become
inextricably combined in the effort to reach a coherent
outlook and to defend it from probable criticism. For
that reason I would like to recall that the idealistic tinge
in my conception of the physical world arose out of math-
ematical researches on the relativity theory. In so far as
I had any earlier philosophical views, they were of an
entirely different complexion.
From the beginning I have been doubtful whether it
was desirable for a scientist to venture so far into extra-
scientific territory. The primary justification for such
an expedition is that it may afford a better view of his
own scientific domain. In the oral lectures it did not
seem a grave indiscretion to speak freely of the various
suggestions I had to offer. But whether they should be
recorded permanently and given a more finished appear-
ance has been difficult to decide. I have much to fear
from the expert philosophical critic, but I am filled with
even more apprehension at the thought of readers who
may look to see whether the book is u on the side of the
angels" and judge its trustworthiness accordingly. Dur-
PREFACE vii
ing the year which has elapsed since the delivery of the
lectures I have made many efforts to shape this and other
parts of the book into something with which I might feel
better content. I release it now with more diffidence than
I have felt with regard to former books.
The conversational style of the lecture-room is gen-
erally considered rather unsuitable for a long book, but
I decided not to modify it. A scientific writer, in for-
going the mathematical formulae which are his natural
and clearest medium of ｅxpression, may perhaps claim
some concession from the reader in return. Many parts
of the subject are intrinsically so difficult that my only
hope of being understood is to explain the points as I
would were I face to face with an inquirer.
It may be necessary to remind the American reader
that our nomenclature for large numbers differs from
his, so that a billion here means a million million.
A. S. E.
August 192S
INTRODUCTION
I have settled down to the task of writing these lectures
and have drawn up my chairs to my two tables. Two
tables! Yes; there are duplicates of every object about
me — two tables, two chairs, two pens.
This is not a very profound beginning to a course
which ought to reach transcendent levels of scientific
philosophy. But we cannot touch bedrock immediately;
we must scratch a bit at the surface of things first. And
whenever I begin to scratch the first thing I strike is —
my two tables.
one of them has been familiar to me from earliest
years. It is a commonplace object of that environment
which I call the world. How shall I describe it? It has
extension; it is comparatively permanent; it is coloured;
above all it is substantial. By substantial I do not merely
mean that it does not collapse when I lean upon it ; I mean
that it is constituted of "substance" and by that word
I am trying to convey to you some conception of its
intrinsic nature. It is a thing; not like space, which is
a mere negation; nor like time, which is — Heaven
knows what ! But that will not help you to my meaning
because it is the distinctive characteristic of a "thing"
to have this substantiality, and I do not think substan-
tiality can be described better than by saying that it is
the kind of nature exemplified by an ordinary table. And
so we go round in circles.^ After all if you are a plain
commonsense man, not too much worried with scien-
tific scruples, you will be confident that you understand
the nature of an ordinary table. I have even heard
of plain men who had the idea that they could better
understand the mystery of their own nature if scientists
ix
x INTRODUCTION
would discover a way of explaining it in terms of the
easily comprehensible nature of a table.
Table No. 2 is my scientific table. It is a more recent
acquaintance and I do not feel so familiar with it. It
does not belong to the world previously mentioned —
that world which spontaneously appears around me when
I open my eyes, though how much of it is objective and
how much subjective I do not here consider. It is part
of a world which in more devious ways has forced
itself on my attention. My scientific table is mostly
emptiness. Sparsely scattered in that emptiness are !
numerous electric charges rushing about with great
speed; but their combined bulk amounts to less than a
billionth of the bulk of the table itself. Notwithstanding
its strange construction it turns out to be an entirely
efficient table. It supports my writing paper as satisfac-
torily as table No. 1 ; for when I lay the paper on it the
little electric particles with their headlong speed keep
on hitting the underside, so that the paper is maintained
in shuttlecock fashion at a nearly steady level. If I lean
upon this table I shall not go through; or, to be strictly
accurate, the chance of my scientific elbow going through
my scientific table is so excessively small that it can be
neglected in practical life. Reviewing their properties
one by one, there seems to be nothing to choose between
the two tables for ordinary purposes; but when ab-
normal circumstances befall, then my scientific table
shows to advantage. If the house catches fire my sci-
entific table will dissolve quite naturally into scientific
smoke, whereas my familiar table undergoes a metamor-
phosis of its substantial nature which I can only regard
as miraculous.
There is nothing substantial about my second table.
It is nearly all empty space — space pervaded, it is true,
INTRODUCTION xi
by fields of force, but these are assigned to the category
of "influences", not of "things". Even in the minute
part which is not empty we must not transfer the old
notion of substance. In dissecting matter into electric
charges we have travelled far from that picture of it
which first gave rise to the conception of substance, and
the meaning of that conception — if it ever had any —
has been lost by the way. The whole trend of modern
scientific views is to break down the separate categories
of "things", "influences", "forms", etc., and to substi-
tute a common background of all experience. Whether
we are studying a material object, a magnetic field, a
geometrical figure, or a duration of time, our scientific
information is summed up in measures ; neither the appa-
ratus of measurement nor the mode of using it suggests
that there is anything essentially different in these prob-
lems. The measures themselves afford no ground for
a classification by categories. We feel it necessary to
concede some background to the measures — an external
world; but the attributes of this world, except in so far
as they 'are reflected in the measures, are outside scien-
tific scrutiny. Science has at last revolted against
attaching the exact knowledge contained in these meas-
urements to a traditional picture-gallery of conceptions
which convey no authentic information of the back-
ground and obtrude irrelevancies into the scheme of
knowledge.
I will not here stress .further the non-substantiality
of electrons, since it is scarcely necessary to the present
line of thought. Conceive them as substantially as you
will, there is a vast difference between my scientific table
with its substance (if any) thinly scattered in specks
in a region mosdy empty and the table of everyday
conception which we regard as the type of solid reality
xii INTRODUCTION
— an incarnate protest against Berkleian subjectivism.
It makes all the difference in the world whether the
paper before me is poised as it were on a swarm of flies
and sustained in shuttlecock fashion by a series of tiny
blows from the swarm underneath, or whether it is sup-
ported because there is substance below it, it being the
intrinsic nature of substance to occupy space to the exclu-
sion of other substance; all the difference in conception
at least, but no difference to my practical task of writing
on the paper.
I need not tell you that modern physics has by deli-
cate test and remorseless logic assured me that my sec-
ond scientific table is the only one which is really there —
wherever "there" may be. on the other hand I need
not tell you that modern physics will never succeed in
exorcising that first table — strange compound of external
nature, mental imagery and inherited prejudice — which
lies visible to my eyes and tangible to my grasp. We
must bid good-bye to it for the present for we are about
to turn from the familiar world to the scientific world
revealed by physics. This is, or is intended to be, a
wholly external world.
"You speak paradoxically of two worlds. Are they
not really two aspects or two interpretations of one and
the same world?"
Yes, no doubt they are ultimately to be identified
after some fashion. But the process by which the ex-
ternal world of physics is transformed into a world of
familiar acquaintance in human consciousness is outside
the scope of physics. And so the world studied accord-
ing to the methods of physics remains detached from
the world familiar to consciousness, until after the
physicist has finished his labours upon it. Provisionally,
therefore, we regard the table which is the subject of
INTRODUCTION xiii
physical research as altogether separate from the familiar
table, without prejudging the question of their ultimate
identification. It is true that the whole scientific
inquiry starts from the familiar world and in the end it
must return to the familiar world; but the part of the
journey over which the physicist has charge is in foreign
territory.
Until recently there was a much closer linkage; the
physicist used to borrow the raw material of his world
from the familiar world, but he does so no longer. His
raw materials are aether, electrons, quanta, potentials,
Hamiltonian functions, etc., and he is nowadays scrupu-
lously careful to guard these from contamination by con-
ceptions borrowed from the other world. There is a
familiar table parallel to the scientific table, but there is
no familiar electron, quantum or potential parallel to the
scientific electron, quantum or potential. We do not even
desire to manufacture a familiar counterpart to these
things or, as we should commonly say, to "explain" the
electron. After the physicist has quite finished his world-
building a linkage or identification is allowed; but prema-
ture attempts at linkage have been found to be entirely
mischievous.
Science aims at constructing a world which shall be
symbolic of the world of commonplace experience. It
is not at all necessary that every individual symbol that
is used should represent something in common experi-
ence or even something explicable in terms of com-
mon experience. The man in the street is always mak-
ing this demand for concrete explanation of the things
referred to in science; but of necessity he must be
disappointed. It is like our experience in learning to
read. That which is written in a book is symbolic of a
story in real life. The whole intention of the book is
xiv INTRODUCTION
that ultimately a reader will identify some symbol, say
BREAD, with one of the conceptions of familiar life. But
it is mischievous to attempt such identifications prema-
turely, before the letters are strung into words and
the words into sentences. The symbol A is not the
counterpart of anything in familiar life. To the child
the letter A would seem horribly abstract; so we give
him a familiar conception along with it. "A was an
Archer who shot at a frog." This tides over his imme-
diate difficulty; but he cannot make serious progress with
word-building so long as Archers, Butchers, Captains,
dance round the letters. The letters are abstract, and
sooner or later he has to realise it. In physics we have
outgrown archer and apple-pie definitions of the funda-
mental symbols. To a request to explain what an electron
really is supposed to be we can only answer, "It is part
of the A B c of physics".
The external world of physics has thus become a world
of shadows. In removing our illusions we have removed
the substance, for indeed we have seen that substance is
one of the greatest of our illusions. Later perhaps
we may inquire whether in our zeal to cut out all that is
unreal we may not have used the knife too ruthlessly.
Perhaps, indeed, reality is a child which cannot survive
without its nurse illusion. But if so, that is of little con-
cern to the scientist, who has good and sufficient reasons
for pursuing his investigations in the world of shadows
and is content to leave to the philosopher the determina-
tion of its exact status in regard to reality. In the world
of physics we watch a shadowgraph performance of
the drama of familiar life. The shadow of my
elbow rests on the shadow table as the shadow ink
flows over the shadow paper. It is all symbolic, and
as a symbol the physicist leaves it. Then comes the
INTRODUCTION xv
alchemist Mind who transmutes the symbols. The
sparsely spread nuclei of electric force become a tangible
solid; their restless agitation becomes the warmth of
summer; the octave of aethereal vibrations becomes a
gorgeous rainbow. Nor does the alchemy stop here. In
the transmuted world new significances arise which are
scarcely to be traced in the world of symbols; so that
it becomes a world of beauty and purpose — and, alas, suf-
fering and evil.
The frank realisation that physical science is con-
cerned with a world of shadows is one of the most sig-
nificant of recent advances. I do not mean that physicists
are to any extent preoccupied with the philosophical impli-
cations of this. From their point of view it is not so much
a withdrawal of untenable claims as an assertion of free-
dom for autonomous development. At the moment I am
not insisting on the shadowy and symbolic character of
the world of physics because of its bearing on philosophy,
but because the aloofness from familiar conceptions will
be apparent in the scientific theories I have to describe.
If you are not prepared for this aloofness you are
likely to be out of sympathy with modern scientific
theories, and may even think them ridiculous — as, I
daresay, many people do.
It is difficult to school ourselves to treat the physical
world as purely symbolic. We are always relapsing and
mixing with the symbols incongruous conceptions taken
from the world of consciousness. Untaught by long
experience we stretch a hand to grasp the shadow,
instead of accepting its shadowy nature. Indeed, unless
we confine ourselves altogether to mathematical sym-
bolism it is hard to avoid dressing our symbols in deceit-
ful clothing. When I think of an electron there
rises to my mind a hard x red, tiny ball; the proton simi-
xvi INTRODUCTION
larly is neutral grey. Of course the colour is absurd —
perhaps not more absurd than the rest of the conception —
but I am incorrigible. I can well understand that the
younger minds are finding these pictures too concrete
and are striving to construct the world out of Hamil-
tonian functions and symbols so far removed from
human preconception that they do not even obey
the laws of orthodox arithmetic. For myself I find some
difficulty in rising to that plane of thought; but I am
convinced that it has got to come.
In these lectures I propose to discuss some of the
results of modern study of the physical world which
give most food for philosophic thought. This will include
new conceptions in science and also new knowledge. In
both respects we are led to think of the material uni-
verse in a way very different from that prevailing at the
end of the last century. I shall not leave out of
sight the ulterior object which must be in the mind of
a Gifford Lecturer, the problem of relating these
purely physical discoveries to the wider aspects and
interests of our human nature. These relations can-
not but have undergone change, since our whole concep-
tion of the physical world has radically changed. I am
convinced that a just appreciation of the physical
world as it is understood to-day carries with it a feeling
of open-mindedness towards a wider significance tran-
scending scientific measurement, which might have
seemed illogical a generation ago; and in the later
lectures I shall try to focus that feeling and make
inexpert efforts to find where it leads. But I should
be untrue to science if I did not insist that its study is
an end in itself. The path of science must be pursued
for its own sake, irrespective of the views it may afford
of a wider landscape; in this spirit we must follow the
INTRODUCTION xvii
path whether it leads to the hill of vision or the tunnel
of obscurity. Therefore till the last stage of the course
is reached you must be content to follow with me the
beaten track of science, nor scold me too severely for
loitering among its wayside flowers. That is to be the
understanding between us. Shall we set forth?
CONTENTS
Preface vii
Introduction xi
Chapter I. The Downfall of Classical Physics I
II. Relativity 20
III. Time 36
IV. The Running-Down of the Universe 63
V. "Becoming" 87
VI. Gravitation — the Law 111
VII. Gravitation — the Explanation 138
VIII. Man's Place in the Universe 163
IX. The Quantum Theory 179
X. The New Quantum Theory 200
XI. World Building 230
XII. Pointer Readings 247
XIII. Reality . 273
XIV. Causation 293
XV. Science and Mysticism 316
Conclusion 343
Index 355
THE NATURE
OF THE
PHYSICAL WORLD
Chapter I
THE DOWNFALL OF CLASSICAL PHYSICS
The Structure of the Atom. Between 1905 and 1908 Ein-
stein and Minkowski introduced fundamental changes in
our ideas of time and space. In 191 1 Rutherford intro-
duced the greatest change in our idea of matter since the
time of Democritus. The reception of these two changes
was curiously different. The new ideas of space and time
were regarded on all sides as revolutionary; they were
received with the greatest enthusiasm by some and
the keenest opposition by others. The new idea of mat-
ter underwent the ordinary experience of scientific dis-
covery; it gradually proved its worth, and when the
evidence became overwhelmingly convincing it quietly
supplanted previous theories. No great shock was felt.
And yet when I hear to-day protests against the Bolshev-
ism of modern science and regrets for the old-established
order, I am inclined to think that Rutherford, not Ein-
stein, is the real villain of the piece. When we compare
the universe as it is now supposed to be with the universe
as we had ordinarily preconceived it, the most arresting
change is not the rearrangement of space and time by
Einstein but the dissolution of all that we regard as most
solid into tiny specks floating in void. That gives an
abrupt jar to those who think that things are more or
less what they seem. The revelation by modern physics
of the void within the atom is more disturbing than
the revelation by astronomy of the immense void of
interstellar space.
The atom is as porous as the solar system. If we
eliminated all the unfilled space in a man's body and
2 DOWNFALL OF CLASSICAL PHYSICS
collected his protons and electrons into one mass, the
man would be reduced to a speck just visible with a
magnifying glass.
This porosity of matter was not foreshadowed in the
atomic theory. Certainly it was known that in a gas
like air the atoms are far separated, leaving a great deal
of empty space; but it was only to be expected that mate-
rial with the characteristics of air should have rela-
tively little substance in it, and "airy nothing" is a com-
mon phrase for the insubstantial. In solids the atoms
are packed tightly in contact, so that the old atomic
theory agreed with our preconceptions in regard-
ing solid bodies as mainly substantial without much
interstice.
The electrical theory of matter which arose towards
the end of the nineteenth century did not at first alter
this view. It was known that the negative electricity
was concentrated into unit charges of very small bulk;
but the other constituent of matter, the positive elec-
tricity, was pictured as a sphere of jelly of the same
dimensions as the atom and having the tiny negative
charges embedded in it. Thus the space inside a solid
was still for the most part well filled.
But in 191 1 Rutherford showed that the positive
electricity was also concentrated into tiny specks. His
scattering experiments proved that the atom was able to
exert large electrical forces which would be impossible
unless the positive charge acted as a highly concentrated
source of attraction; it must be contained in a nucleus
minute in comparison with the dimensions of the atom.
Thus for the first time the main volume of the atom was
entirely evacuated, and a "solar system" type of atom
was substituted for a substantial "billiard-ball". Two
years later Niels Bohr developed his famous theory on
THE STRUCTURE OF THE ATOM 3
the basis of the Rutherford atom, and since then rapid
progress has been made. Whatever further changes of
view are in prospect, a reversion to the old substantial
atoms is unthinkable.
The accepted conclusion at the present day is that all
varieties of matter are ultimately composed of two ele-
mentary constituents — protons and electrons. Electrically
these are the exact opposites of one another, the proton
being a charge of positive electricity and the electron
a charge of negative electricity. But in other respects
their properties are very different. The proton has 1840
times the mass of the electron, so that nearly all the
mass of matter is due to its constituent protons.
The proton is not found unadulterated except in hydro-
gen, which seems to be the most primitive form of mat-
ter, its atom consisting of one proton and one electron.
In other atoms a number of protons and a lesser
number of electrons are cemented together to form
a nucleus; the electrons required to make up the bal-
ance are scattered like remote satellites of the nucleus,
and can even escape from the atom and wander freely
through the material. The diameter of an electron is
about 1/50,000 of the diameter of an atom; that of the
nucleus is not very much larger; an isolated proton is
supposed to be much smaller still.
Thirty years ago there was much debate over the ques-
tion of aether-drag — whether the earth moving round
the sun drags the aether with it. At that time the solidity
of the atom was unquestioned, and it was difficult
to believe that matter could push its way through the
aether without disturbing it. It was surprising and per-
plexing to find as the result of experiments that no
convection of the aether occurred. But we now realise
that the aether can slip through the atoms as easily as
4 DOWNFALL OF CLASSICAL PHYSICS
through the solar system, and our expectation is all the
other way.
We shall return to the "solar system" atom in later
chapters. For the present the two things which concern
us are (i) its extreme emptiness, and (2) the fact that it
is made up of electrical charges.
Rutherford's nuclear theory of the atom is not usually
counted as one of the scientific revolutions of the present
century. It was a far-reaching discovery, but a discovery
falling within the classical scheme of physics. The nature
and significance of the discovery could be stated in plain
terms, i.e. in terms of conceptions already current in
science. The epithet "revolutionary" is usually reserved
for two great modern developments — the Relativity
Theory and the Quantum Theory. These are not
merely new discoveries as to the content of the world;
they involve changes in our mode of thought about the
world. They cannot be stated immediately in plain
terms because we have first to grasp new conceptions
undreamt of in the classical scheme of physics.
I am not sure that the phrase "classical physics" has
ever been closely defined. But the general idea is that
the scheme of natural law developed by Newton in the
Principia provided a pattern which all subsequent devel-
opments might be expected to follow. Within the four
corners of the scheme great changes of outlook were
possible; the wave-theory of light supplanted the cor-
puscular theory; heat was changed from substance (calo-
ric) to energy of motion; electricity from continuous
fluid to nuclei of strain in the aether. But this was all
allowed for in the elasticity of the original scheme.
Waves, kinetic energy, and strain already had their
place in the scheme; and the application of the same
conceptions to account for a wider range of phenomena
THE FITZGERALD CONTRACTION 5
was a tribute to the comprehensiveness of Newton's
original outlook.
We have now to see how the classical scheme broke
down.
The FitzGerald Contraction. We can best start from
the following fact. Suppose that you have a rod moving
at very high speed. Let it first be pointing transverse
to its line of motion. Now turn it through a right angle
so that it is along the line of motion. The rod contracts.
It is shorter when it is along the line of motion than
when it is across the line of motion.
This contraction, known as the FitzGerald contrac-
tion, is exceedingly small in all ordinary circumstances.
It does not depend at all on the material of the rod but
only on the speed. For example, if the speed is 19 miles
a second — the speed of the earth round the sun — the
contraction of length is 1 part in 200,000,000, or 2^4
inches in the diameter of the earth.
This is demonstrated by a number of experiments of
different kinds of which the earliest and best known is
the Michelson-Morley experiment first performed in
1887, repeated more accurately by Morley and Miller
in 1905, and again by several observers within the last
year or two. I am not going to describe these experi-
ments except to mention that the convenient way of
giving your rod a large velocity is to carry it on the
earth which moves at high^ speed round the sun. Nor
shall I discuss here how complete is the proof afforded
by these experiments. It is much more important that
you should realise that the contraction is just what would
be expected from our current knowledge of a material
rod.
You are surprised that the dimensions of a moving,
6 DOWNFALL OF CLASSICAL PHYSICS
rod can be altered merely by pointing it different ways.
You expect them to remain unchanged. But which rod
are you thinking of? (You remember my two tables.)
If you are thinking of continuous substance, extending in
space because it is the nature of substance to occupy
space, then there seems to be no valid cause for a change
of dimensions. But the scientific rod is a swarm of
electrical particles rushing about and widely separated
from one another. The marvel is that such a swarm
should tend to preserve any definite extension. The
particles, however, keep a certain average spacing so
that the whole volume remains practically steady; they
exert electrical forces on one another, and the volume
which they fill corresponds to a balance between the
forces drawing them together and the diverse motions
tending to spread them apart. When the rod is set in
motion these electrical forces change. Electricity in
motion constitutes an electric current. But electric
currents give rise to forces of a different type from those
due to electricity at rest, viz. magnetic forces. More-
over these forces arising from the motion of electric
charges will naturally be of different intensity in the
directions along and across the line of motion.
By setting in motion the rod with all the little electric
charges contained in it we introduce new magnetic forces
between the particles. Clearly the original balance is
upset, and the average spacing between the particles
must alter until a new balance is found. And so the
extension of the swarm of particles — the length of the
rod — alters.
There is really nothing mysterious about the Fitz-
Gerald contraction. It would be an unnatural property
of a rod pictured in the old way as continuous substance
occupying space in virtue of its substantiality; but it is
THE FITZGERALD CONTRACTION 7
an entirely natural property of a swarm of particles held
in delicate balance by electromagnetic forces, and occu-
pying space by buffeting away anything that tries to
enter. Or you may look at it this way: your expecta-
tion that the rod will keep its original length presup-
poses, of course, that it receives fair treatment and
is not subjected to any new stresses. But a rod in motion
is subjected to a new magnetic stress, arising not from
unfair outside tampering but as a necessary consequence
of its own electrical constitution; and under this stress
the contraction occurs. Perhaps you will think that if
the rod were rigid enough it might be able to resist the
compressing force. That is not so; the FitzGerald con-
traction is the same for a rod of steel and for a rod of
india-rubber; the rigidity and the compressing stress are
bound up with the constitution in such a way that if
one is large so also is the other. It is necessary to rid
our minds of the idea that this failure to keep a constant
length is an imperfection of the rod; it is only imperfect
as compared with an imaginary "something" which has
not this electrical constitution — and therefore is not
material at all. The FitzGerald contraction is not an
imperfection but a fixed and characteristic property of
matter, like inertia.
We have here drawn a qualitative inference from the
electrical structure of matter; we must leave it to the
mathematician to calculate the quantitative effect. The
problem was worked out by Lorentz and Larmor about
1900. They calculated the change in the average spacing
of the particles required to restore the balance after it
had been upset by the new forces due to the change of
motion of the charges. This calculation was found to
give precisely the FitzGerald contraction, i.e. the amount
already inferred from the experiments above mentioned.
8 DOWNFALL OF CLASSICAL PHYSICS
Thus we have two legs to stand on. Some will prefer to
trust the results because they seem to be well established
by experiment; others will be more easily persuaded by
the knowledge that the FitzGerald contraction is a
necessary consequence of the scheme of electromag-
netic laws universally accepted since the time of Max-
well. Both experiments and theories sometimes go
wrong; so it is just as well to have both alternatives.
Consequences of the Contraction. This result alone,
although it may not quite lead you to the theory of rela-
tivity, ought to make you uneasy about classical physics.
The physicist when he wishes to measure a length —
and he cannot get far in any experiment without meas-
uring a length — takes a scale and turns it in the direc-
tion needed. It never occurred to him that in spite
of all precautions the scale would change length when
he did this; but unless the earth happens to be at rest
a change must occur. The constancy of a measur-
ing scale is the rock on which the whole structure of
physics has been reared; and that rock has crumbled
away. You may think that this assumption cannot have
betrayed the physicist very badly; the changes of length
cannot be serious or they would have been noticed.
Wait and see.
Let us look at some of the consequences of the Fitz-
Gerald contraction. First take what may seem to be a
rather fantastic case. Imagine you are on a planet mov-
ing very fast indeed, say 161,000 miles a second. For
this speed the contraction is one-half. Any solid con-
tracts to half its original length when turned from across
to along the line of motion. A railway journey between
two towns which was 100 miles at noon is shortened to
50 miles at 6 p.m. when the planet has turned through
CONSEQUENCES OF THE CONTRACTION 9
a right angle. The inhabitants copy Alice in Wonder-
land; they pull out and shut up like a telescope.
I do not know of a planet moving at 161,000 miles
a second, but I could point to a spiral nebula far away
in space which is moving at 1000 miles a second. This
may well contain a planet and (speaking unprofession-
ally) perhaps I shall not be taking too much licence if
I place intelligent beings on it. At 1000 miles a second
the contraction is not large enough to be appreciable in
ordinary affairs; but it is quite large enough to be appre-
ciable in measurements of scientific or even of engi-
neering accuracy. one of the most fundamental pro-
cedures in physics is to measure lengths with a scale
moved about in any way. Imagine the consternation of
the physicists on this planet when they learn that they
have made a mistake in supposing that their scale is a
constant measure of length. What a business to go back
over all the experiments ever performed, apply the
corrections for orientation of the scale at the time, and
then consider de novo the inferences and system of
physical laws to be deduced from the amended data !
How thankful our own physicists ought to be that they
are not in this runaway nebula but on a decently slow-
moving planet like the earth !
But stay a moment. Is it so certain that we are on
a slow-moving planet? I can imagine the astronomers
in that nebula observing far away in space an insignifi-
cant star attended by an insignificant planet called
Earth. They observe too that it is moving with the
huge velocity of 1000 miles a second; because naturally
if we see them receding from us at 1000 miles a second
they will see us receding from them at 1000 miles a
second. "A thousand miles a second!" exclaim the
nebular physicists, "How unfortunate for the poor
io DOWNFALL OF CLASSICAL PHYSICS
physicists on the Earth! The FitzGerald contraction
will be quite appreciable, and all their measures with
scales will be seriously wrong. What a weird system of
laws of Nature they will have deduced, if they have over-
looked this correction !"
There is no means of deciding which is right — to
which of us the observed relative velocity of iooo
miles a second really belongs. Astronomically the gal-
axy of which the earth is a member does not seem to
be more important, more central, than the nebula.
The presumption that it is we who are the more nearly
at rest has no serious foundation; it is mere self-
flattery.
"But", you will say, "surely if these appreciable
changes of length occurred on the earth, we should
detect them by our measurements." That brings me to
the interesting point. We could not detect them by any
measurement; they may occur and yet pass quite un-
noticed. Let me try to show how this happens.
This room, we will say, is travelling at 161,000 miles
a second vertically upwards. That is my statement, and
it is up to you to prove it wrong. I turn my arm from
horizontal to vertical and it contracts to half its original
length. You don't believe me? Then bring a yard-
measure and measure it. First, horizontally, the result
is 30 inches; now vertically, the result is 30 half-inches.
You must allow for the fact that an inch-division of the
scale contracts to half an inch when the yard-measure
is turned vertically.
"But we can see that your arm does not become
shorter; can we not trust our own eyes?"
Certainly not, unless you remember that when you
got up this morning your retina contracted to half its
original width in the vertical direction; consequently it
CONSEQUENCES OF THE CONTRACTION n
is now exaggerating vertical distances to twice the scale
of horizontal distances.
"Very well", you reply, "I will not get up. I will lie
in bed and watch you go through your performance in
an inclined mirror. Then my retina will be all right,
but I know I shall still see no contraction."
But a moving mirror does not give an undistorted
image of what is happening. The angle of reflection of
light is altered by motion of a mirror, just as the angle
of reflection of a billiard-ball would be altered if the
cushion were moving. If you will work out by the
ordinary laws of optics the effect of moving a mirror at
161,000 miles a second, you will find that it introduces
a distortion which just conceals the contraction of my
arm.
And so on for every proposed test. You cannot
disprove my assertion, and, of course, I cannot prove
it; I might equally well have chosen and defended any
other velocity. At first this seems to contradict what
I told you earlier — that the contraction. had been proved
and measured by the Michelson-Morley and other experi-
ments — but there is really no contradiction. They were
all null experiments, just as your experiment of watch-
ing my arm in an inclined mirror was a null experiment.
Certain optical or electrical consequences of the earth's
motion were looked for of the same type as the
distortion of images by a moving mirror; these would
have been observed unless a contraction occurred of
just the right amount to compensate them. They
were not observed; therefore the compensating contrac-
tion had occurred. There was just one alternative; the
earth's true velocity through space might happen to
have been nil. This was ruled out by repeating the
experiment six months later, since the earth's motion
12 DOWNFALL OF CLASSICAL PHYSICS
could not be nil on both occasions. Thus the contraction
was demonstrated and its law of dependence on velocity
verified. But the actual amount of contraction on either
occasion was unknown, since the earth's true velocity
(as distinct from its orbital velocity with respect to the
sun) was unknown. It remains unknown because the
optical and electrical effects by which we might hope
to measure it are always compensated by the contraction.
I have said that the constancy of a measuring scale is
the rock on which the structure of physics has been
reared. The structure has also been supported by sup-
plementary props because optical and electrical devices
can often be used instead of material scales to ascertain
lengths and distances. But we find that all these are
united in a conspiracy not to give one another away.
The rock has crumbled and simultaneously all the other
supports have collapsed.
Frames of Space. We can now return to the quarrel
between the nebular physicists and ourselves. one of us
has a large velocity and his scientific measurements are
seriously affected by the contraction of his scales. Each
has hitherto taken it for granted that it is the other
fellow who is making the mistake. We cannot settle
the dispute by appeal to experiment because in every
experiment the mistake introduces two errors which just
compensate one another.
It is a curious sort of mistake which always carries
with it its own compensation. But remember that the
compensation only applies to phenomena actually ob-
served or capable of observation. The compensation
does not apply to the intermediate part of our deduc-
tion — that system of inference from observation which
forms the classical physical theory of the universe.
FRAMES OF SPACE 13
Suppose that we and the nebular physicists survey
the world, that is to say we allocate the surrounding
objects to their respective positions in space. one
party, say the nebular physicists, has a large velocity;
their yard-measures will contract and become less than
a yard when they measure distances in a certain direc-
tion; consequently they will reckon distances in that
direction too great. It does not matter whether they
use a yard-measure, or a theodolite, or merely judge
distances with the eye; all methods of measurement
must agree. If motion caused a disagreement of any
kind, we should be able to determine the motion by
observing the amount of disagreement; but, as we have
already seen, both theory and observation indicate that
there is complete compensation. If the nebular physi-
cists try to construct a square they will construct an
oblong. No test can ever reveal to them that it is not a
square; the greatest advance they can make is to recog-
nise that there are people in another world who have got
it into their heads that it is an oblong, and they may be
broadminded enough to admit that this point of view, ab-
surd as it seems, is really as defensible as their own. It
is clear that their whole conception of space is distorted
as compared with ours, and ours is distorted as com-
pared with theirs. We are regarding the same universe,
but we have arranged it in different spaces. The original
quarrel as to whether they or we are moving with the
speed of 1000 miles a second has made so deep a cleavage
between us that we cannot even use the same space.
Space and time are words conveying more than one
meaning. Space is an empty void; or it is such and such
a number of inches, acres, pints. Time is an ever-rolling
stream; or it is something signalled to us by wireless.
The physicist has no use for vague conceptions; he often
i 4 DOWNFALL OF CLASSICAL PHYSICS
has them, alas ! but he cannot make real use of them.
So when he speaks of space it is always the inches or
pints that he should have in mind. It is from this point
of view that our space and the space of the nebular
physicists are different spaces; the reckoning of inches
and pints is different. To avoid possible misunder-
standing it is perhaps better to say that we have different
frames of space — different frames to which we refer the
location of objects. Do not, however, think of a frame
of space as something consciously artificial; the frame
of space comes into our minds with our first perception of
space. Consider, for example, the more extreme case
when the FitzGerald contraction is one-half. If a man
takes a rectangle 2"Xi" to be a square it is clear that
space must have dawned on his intelligence in a way very
different from that in which we have apprehended it.
The frame of space used by an observer depends only
on his motion. Observers on different planets with the
same velocity (i.e. having zero relative velocity) will
agree as to the location of the objects of the universe;
but observers on planets with different velocities have
different frames of location. You may ask, How can
I be so confident as to the way in which these imaginary
beings will interpret their observations? If that objec-
tion is pressed I shall not defend myself; but those who
dislike my imaginary beings must face the alternative
of following the argument with mathematical symbols.
Our purpose has been to express in a conveniently
apprehensible form certain results which follow from
terrestrial experiments and calculations as to the effect
of motion on electrical, optical and metrical phenomena.
So much careful work has been done on this subject
that science is in a position to state what will be the
consequence of making measurements with instruments
FRAMES OF SPACE 15
travelling at high speed — whether instruments of a
technical kind or, for example, a human retina. In only
one respect do I treat my nebular observer as more than
a piece of registering apparatus; I assume that he is
subject to a common failing of human nature, viz. he
takes it for granted that it was his planet that God
chiefly had in mind when the universe was created.
Hence he is (like my reader perhaps?) disinclined to
take seriously the views of location of those people who
are so misguided as to move at 1000 miles a second
relatively to his parish pump.
An exceptionally modest observer might take some
other planet than his own as the standard of rest. Then
he would have to correct all his measurements for the
FitzGerald contraction due to his own motion with
respect to the standard, and the corrected measures
would give the space-frame belonging to the standard
planet as the original measures gave the space-frame of
his own planet. For him the dilemma is even more
pressing, for there is nothing to guide him as to the
planet to be selected for the standard of rest. once
he gives up the naive assumption that his own frame is
the one and only right frame the question arises, Which
then of the innumerable other frames is right? There
is no answer, and so far as we can see no possibility of
an answer. Meanwhile all his experimental measure-
ments are waiting unreduced, because the corrections
to be applied to them depend on the answer. I am
afraid our modest observer will get rather left behind
by his less humble colleagues.
The trouble that arises is not that we have found
anything necessarily wrong with the frame of location
that has been employed in our system of physics; it has
not led to experimental contradictions. The only thing
16 DOWNFALL OF CLASSICAL PHYSICS
known to be "wrong" with it is that it is not unique.
If we had found that our frame was unsatisfactory and
another frame was preferable, that would not have
caused a great revolution of thought; but to discover
that ours is one of many frames, all of which are equally
satisfactory, leads to a change of interpretation of the
significance of a frame of location.
"Commonsense" Objections. Before going further I must
answer the critic who objects in the name of common-
sense. Space — his space — is so vivid to him. "This
object is obviously here; that object is just there. I know
it; and I am not going to be shaken by any amount of sci-
entific obscurantism about contraction of measuring rods."
We have certain preconceived ideas about location
in space which have come down to us from ape-like
ancestors. They are deeply rooted in our mode of
thought, so that it is very difficult to criticise them
impartially and to realise the very insecure foundation
on which they rest. We commonly suppose that each
of the objects surrounding us has a definite location in
space and that we are aware of the right location. The
objects in my study are actually in the positions where
I am "aware" that they are; and if an observer (on
another star) surveying the room with measuring rods,
etc., makes out a different arrangement of location, he
is merely spinning a scientific paradox which does not
shake the real facts of location obvious to any man
of commonsense. This attitude rejects with contempt
the question, How am I aware of the location? If the
location is determined by scientific measurements with
elaborate precautions, we are ready enough to sug-
gest all sorts of ways in which the apparatus might
have misbehaved; but if the knowledge of location is
a
COMMONSENSE" OBJECTIONS 17
obtained with no precautions, if it just comes into our
heads unsought, then it is obviously true and to doubt
it would be flying in the face of commonsense ! We
have a sort of impression (although we do not like
to acknowledge it) that the mind puts out a feeler into
space to ascertain directly where each familiar object is.
That is nonsense; our commonsense knowledge of location
is not obtained that way. Strictly it is sense knowledge,
not commonsense knowledge. It is partly obtained
by touch and locomotion; such and such an object
is at arm's length or a few steps away. Is there
any essential difference (other than its crudity) between
this method and scientific measurements with a scale?
It is partly obtained by vision — a crude version of
scientific measurement with a theodolite. Our common
knowledge of where things are is not a miraculous
revelation of unquestionable authority; it is inference
from observations of the same kind as, but cruder than,
those made in a scientific survey. Within its own limits
of accuracy the scheme of location of objects that I am
instinctively "aware" of is the same as my scientific
scheme of location, or frame of space.
When we use a carefully made telescope lens and a
sensitised plate instead of the crystalline lens and retina
of the eye we increase the accuracy but do not alter the
character of our survey of space. It is by this increase
of refinement that we have become "aware" of certain
characteristics of space which were not known to our
ape-like ancestor when he instituted the common ideas
that have come down to us. His scheme of location
works consistently so long as there is no important
change in his motion (a few miles a second makes no
appreciable difference) ; but a large change involves a
transition to a different system of location which is like-
18 DOWNFALL OF CLASSICAL PHYSICS
wise self-consistent, although it is inconsistent with the
original one. Having any number of these systems of
location, or frames of space, we can no longer pretend
that each of them indicates "just where things are".
Location is not something supernaturally revealed to the
mind; it is a kind of conventional summary of those
properties or relations of objects which condition certain
visual and tactual sensations.
Does not this show that "right" location in space
cannot be nearly so important and fundamental as it is
made out to be in the Newtonian scheme of things?
The different observers are able to play fast and loose
with it without ill effects.
Suppose that location is, I will not say entirely a
myth, but not quite the definite thing it is made out to
be in classical physics; that the Newtonian idea of
location contains some truth and some padding, and it
is not the truth but the padding that our observers are
quarrelling over. That would explain a great deal. It
would explain, for instance, why all the forces of Nature
seem to have entered into a conspiracy to prevent our
discovering the definite location of any object (its posi-
tion in the "right" frame of space) ; naturally they
cannot reveal it, if it does not exist.
This thought will be followed up in the next chapter.
Meanwhile let us glance back over the arguments that
have led to the present situation. It arises from the
failure of our much-trusted measuring scale, a failure
which we can infer from strong experimental evidence
or more simply as an inevitable consequence of accepting
the electrical theory of matter. This unforeseen be-
haviour is a constant property of all kinds of matter and
is even shared by optical and electrical measuring devices.
SUMMARY 19
Thus it is not betrayed by any kind of discrepancy in
applying the usual methods of measurement. The dis-
crepancy is revealed when we change the standard
motion of the measuring appliances, e.g. when we com-
pare lengths and distances as measured by terrestrial
observers with those which would be measured by
observers on a planet with different velocity. Provision-
ally we shall call the measured lengths which contain
this discrepancy "fictitious lengths".
According to the Newtonian scheme length is definite
and unique; and each observer should apply corrections
(dependent on his motion) to reduce his fictitious lengths
to the unique Newtonian length. But to this there are
two objections. The corrections to reduce to Newtonian
length are indeterminate; we know the corrections
necessary to reduce our own fictitious lengths to those
measured by an observer with any other prescribed
motion, but there is no criterion for deciding which
system is the one intended in the Newtonian scheme.
Secondly, the whole of present-day physics has been
based on lengths measured by terrestrial observers
without this correction, so that whilst its assertions
ostensibly refer to Newtonian lengths they have actually
been proved for fictitious lengths.
The FitzGerald contraction may seem a little thing
to bring the whole structure of classical physics tumbling
down. But few indeed are the experiments contributing
to our scientific knowledge which would not be invali-
dated if our methods of measuring lengths were funda-
mentally unsound. We now find that there is no
guarantee that they are not subject to a systematic kind
of error. Worse still we do not know if the error
occurs or not, and there is every reason to presume
that it is impossible to know.
Chapter II
RELATIVITY
Einstein's Principle. The modest observer mentioned in
the first chapter was faced with the task of choosing
between a number of frames of space with nothing to
guide his choice. They are different in the sense that they
frame the material objects of the world, including the
observer himself, differently; but they are indistinguish-
able in the sense that the world as framed in one space
conducts itself according to precisely the same laws
as the world framed in another space. Owing to the
accident of having been born on a particular planet
our observer has hitherto unthinkingly adopted one of
the frames; but he realises that this is no ground for
obstinately asserting that it must be the right frame.
Which is the right frame?
At this juncture Einstein comes forward with a sug-
gestion —
"You are seeking a frame of space which you call
the right frame. In what does its rightness consist?"
You are standing with a label in your hand before a
row of packages all precisely similar. You are worried
because there is nothing to help you decide which of
the packages it should be attached to. Look at the label
and see what is written on it. Nothing.
"Right" as applied to frames of space is a blank label.
It implies that there is something distinguishing a right
frame from a wrong frame; but when we ask what is
this distinguishing property, the only answer we receive
is "Rightness", which does not make the meaning clearer
or convince us that there is a meaning.
20
EINSTEIN'S PRINCIPLE 21
I am prepared to admit that frames of space in spite
of their present resemblance may in the future turn out
to be not entirely indistinguishable. (I deem it unlikely,
but I do not exclude it.) The future physicist might
find that the frame belonging to Arcturus, say, is unique
as regards some property not yet known to science. Then
no doubt our friend with the label will hasten to affix
it. "I told you so. I knew I meant something
when I talked about a right frame." But it does not
seem a profitable procedure to make odd noises on
the off-chance that posterity will find a significance to
attribute to them. To those who now harp on a right
frame of space we may reply in the words of Bottom the
weaver —
"Who would set his wit to so foolish a bird? Who
would give a bird the lie, though he cry 'cuckoo' never
so?"
And so the position of Einstein's theory is that the
question of a unique right frame of space does not arise.
There is a frame of space relative to a terrestrial ob-
server, another frame relative to the nebular observers,
others relative to other stars. Frames of space are rela-
tive. Distances, lengths, volumes — all quantities of
space-reckoning which belong to the frames — are likewise
relative. A distance as reckoned by an observer on one
star is as good as the distance reckoned by an observer
on another star. We must not expect them to agree;
the one is a distance relative, to one frame, the other is
a distance relative to another frame. Absolute distance,
not relative to some special frame, is meaningless.
The next point to notice is that the other quantities
of physics go along with the frame of space, so that they
also are relative. You may have seen one of those tables
of "dimensions" of physical quantities showing how
22 RELATIVITY
they are all related to the reckoning of length, time and
mass. If you alter the reckoning of length you alter the
reckoning of other physical quantities.
Consider an electrically charged body at rest on the
earth. Since it is at rest it gives an electric field but no
magnetic field. But for the nebular physicist it is a
charged body moving at iooo miles a second. A moving
charge constitutes an electric current which in accordance
with the laws of electromagnetism gives rise to a mag-
netic field. How can the same body both give and
not give a magnetic field? on the classical theory we
should have had to explain one of these results as an
illusion. (There is no difficulty in doing that; only there
is nothing to indicate which of the two results is the one
to be explained away.) on the relativity theory both
results are accepted. Magnetic fields are relative.
There is no magnetic field relative to the terrestrial
frame of space; there is a magnetic field relative to
the nebular frame of space. The nebular physicist will
duly detect the magnetic field with his instruments
although our instruments show no magnetic field. That
is because he uses instruments at rest on his planet and
we use instruments at rest on ours; or at least we correct
our observations to accord with the indications of instru-
ments at rest in our respective frames of space.
Is there really a magnetic field or not? This is like
the previous problem of the square and the oblong.
There is one specification of the field relative to one
planet, another relative to another. There is no abso-
lute specification.
It is not quite true to say that all the physical quan-
tities are relative to frames of space. We can construct
new physical quantities by multiplying, dividing, etc.;
thus we multiply mass and velocity to give momentum,
RELATIVE AND ABSOLUTE QUANTITIES 23
divide energy by time to give horse-power. We can set
ourselves the mathematical problem of constructing in
this way quantities which shall be invariant, that is to
say, shall have the same measure whatever frame of
space may be used. one or two of these invariants
turn out to be quantities already recognised in pre-
relativity physics; "action" and "entropy" are the best
known. Relativity physics is especially interested in
invariants, and it has discovered and named a few more.
It is a common mistake to suppose that Einstein's theory
of relativity asserts that everything is relative. Actually
it says, "There are absolute things in the world but
you must look deeply for them. The things that first
present themselves to your notice are for the most part
relative."
Relative and Absolute Quantities. I will try to make
clear the distinction between absolute and relative quan-
tities. Number (of discrete individuals) is absolute. It
is the result of counting, and counting is an absolute
operation. If two men count the number of people in
this room and reach different results, one of them must
be wrong.
The measurement of distance is not an absolute
operation. It is possible for two men to measure the
same distance and reach different results, and yet neither
of them be wrong.
I mark two dots on the^ blackboard and ask two
students to measure very accurately the distance between
them. In order that there may be no possible doubt as
to what I mean by distance I give them elaborate
instructions as to the standard to be used and the pre-
cautions necessary to obtain an accurate measurement
of distance. They bring me results which differ. I ask
24 RELATIVITY
them to compare notes to find out which of them is
wrong, and why? Presently they return and say: "It
was your fault because in one respect your instructions
were not explicit. You did not mention what motion
the scale should have when it was being used." one
of them without thinking much about the matter had
kept the scale at rest on the earth. The other had
reflected that the earth was a very insignificant planet of
which the Professor had a low opinion. He thought it
would be only reasonable to choose some more impor-
tant body to regulate the motion of the scale, and so he
had given it a motion agreeing with that of the enor-
mous star Betelgeuse. Naturally the FitzGerald contrac-
tion of the scale accounted for the difference of results.
I am disinclined to accept this excuse. I say severely,
"It is all nonsense dragging in the earth or Betel-
geuse or any other body. You do not require any
standard external to the problem. I told you to measure
the distance of two points on the blackboard; you should
have made the motion of the scale agree with that of
the blackboard. Surely it is commonsense to make your
measuring scale move with what you are measuring.
Remember that next time."
A few days later I ask them to measure the wave-
length of sodium light — the distance from crest to crest
of the light waves. They do so and return in triumphal
agreement: ''The wave-length is infinite". I point out
to them that this does not agree with the result given
in the book (.000059 cm.). "Yes", they reply, u we
noticed that; but the man in the book did not do it
right. You told us always to make the measuring scale
move with the thing to be measured. So at great trouble
and expense we sent our scales hurtling through the
laboratory at the same speed as the light." At this speed
RELATIVE AND ABSOLUTE QUANTITIES 25
the FitzGerald contraction is infinite, the metre rods
contract to nothing, and so it takes an infinite number
of them to fill up the interval from crest to crest of the
waves.
My supplementary rule was in a way quite a good
rule; it would always give something absolute — some-
thing on which they would necessarily agree. only
unfortunately it would not give the length or distance.
When we ask whether distance is absolute or relative,
we must not first make up our minds that it ought to
be absolute and then change the current significance of
the term to make it so.
Nor can we altogether blame our predecessors for
having stupidly made the word "distance" mean some-
thing relative when they might have applied it to a
result of spatial measurement which was absolute and
unambiguous. The suggested supplementary rule has
one drawback. We often have to consider a system
containing a number of bodies with different motions;
it would be inconvenient to have to measure each body
with apparatus in a different state of motion, and we
should get into a terrible muddle in trying to fit the
different measures together. Our predecessors were
wise in referring all distances to a single frame of space,
even though their expectation that such distances would
be absolute has not been fulfilled.
As for the absolute quantity given by the proposed
supplementary rule, we may set it alongside distances
relative to the earth and distances relative to Betelgeuse,
etc., as a quantity of some interest to study. It is called
"proper-distance". Perhaps you feel a relief at getting
hold of something absolute and would wish to follow
it up. Excellent. But remember this will lead you away
from the classical scheme of physics which has chosen
26 RELATIVITY
the relative distances to build on. The quest of the
absolute leads into the four-dimensional world.
A more familiar example of a relative quantity is
"direction" of an object. There is a direction of Cam-
bridge relative to Edinburgh and another direction rela-
tive to London, and so on. It never occurs to us to
think of this as a discrepancy, or to suppose that there
must be some direction of Cambridge (at present undis-
coverable) which is absolute. The idea that there ought
to be an absolute distance between two points contains
the same kind of fallacy. There is, of course, a differ-
ence of detail; the relative direction above mentioned is
relative to a particular position of the observer, whereas
the relative distance is relative to a particular velocity
of the observer. We can change position freely and
so introduce large changes of relative direction; but
we cannot change velocity appreciably — the 300 miles
an hour attainable by our fastest devices being too
insignificant to count. Consequently the relativity of
distance is not a matter of common experience as the
relativity of direction is. That is why we have unfor-
tunately a rooted impression in our minds that distance
ought to be absolute.
A very homely illustration of a relative quantity is
afforded by the pound sterling. Whatever may have
been the correct theoretical view, the man in the street
until very recently regarded a pound as an absolute
amount of wealth. But dire experience has now con-
vinced us all of its relativity. At first we used to cling
to the idea that there ought to be an absolute pound
and struggle to express the situation in paradoxical state-
ments — the pound had really become seven-and-six-
pence. But we have grown accustomed to the situation
and continue to reckon wealth in pounds as before,
NATURE'S PLAN OF STRUCTURE 27
merely recognising that the pound is relative and there-
fore must not be expected to have those properties that
we had attributed to it in the belief that it was absolute.
You can form some idea of the essential difference in
the outlook of physics before and after Einstein's
principle of relativity by comparing it with the difference
in economic theory which comes from recognising the
•relativity of value of money. I suppose that in stable
times the practical consequences of this relativity are
manifested chiefly in the minute fluctuations of foreign
exchanges, which may be compared with the minute
changes of length affecting delicate experiments like the
Michelson-Morley experiment. Occasionally the con-
sequences may be more sensational — a mark-exchange
soaring to billions, a high-speed 8 particle contracting
to a third of its radius. But it is not these casual mani-
festations which are the main outcome. Clearly an
economist who believes in the absoluteness of the pound
has not grasped the rudiments of his subject. Similarly
if we have conceived the physical world as intrinsically
constituted out of those distances, forces and masses
which are now seen to have reference only to our own
special reference frame, we are far from a proper under-
standing of the nature of things.
Nature's Plan of Structure. Let us now return to the
observer who was so anxious to pick out a "right"
frame of space. I suppose that what he had in mind
was to find Nature's own frame — the frame on which
Nature based her calculations when she poised the
planets under the law of gravity, or the reckoning of
symmetry which she used when she turned the electrons
on her lathe. But Nature has been too subtle for him;
she has not left anything to betray the frame which she
28 RELATIVITY
used. Or perhaps the concealment is not any particular
subtlety; she may have done her work without employing
a frame of space. Let me tell you a parable.
There was once an archaeologist who used to com-
pute the dates of ancient temples from their orientation.
He found that they were aligned with respect to the
rising of particular stars. Owing to precession the
star no longer rises in the original line, but the date
when it was rising in the line of the temple can be
calculated, and hence the epoch of construction of the
temple is discovered. But there was one tribe for
which this method would not work; they had built only
circular temples. To the archaeologist this seemed a
manifestation of extraordinary subtlety on their part;
they had hit on a device which would conceal entirely
the date when their temples were constructed. one
critic, however, made the ribald suggestion that per-
haps this particular tribe was not enthusiastic about
astronomy.
Like the critic I do not think Nature has been par-
ticularly subtle in concealing which frame she prefers.
It is just that she is not enthusiastic about frames of
space. They are a method of partition which w T e have
found useful for reckoning, but they play no part in
the architecture of the universe. Surely it is absurd to
suppose that the universe is planned in such a way as to
conceal its plan. It is like the schemes of the White
Knight —
But I was thinking of a plan
To dye one's whiskers green,
And always use so large a fan
That they could not be seen.
If this is so we shall have to sweep away the frames
of space before we can see Nature's plan in its real
NATURE'S PLAN OF STRUCTURE 29
significance. She herself has paid no attention to them,
and they can only obscure the simplicity of her scheme.
I do not mean to suggest that we should entirely rewrite
physics, eliminating all reference to frames of space or
any quantities referred to them; science has many tasks
to perform, besides that of apprehending the ultimate
plan of structure of the world. But if we do wish to
have insight on this latter point, then the first step is to
make an escape from the irrelevant space-frames.
This will involve a great change from classical con-
ceptions, and important developments will follow from
our change of attitude. For example, it is known that
both gravitation and electric force follow approximately
the law of inverse-square of the distance. This law
appeals strongly to us by its simplicity; not only is it
mathematically simple but it corresponds very naturally
with the weakening of an effect by spreading out in
three dimensions. We suspect therefore that it is
likely to be the exact law of gravitational and electric
fields. But although it is simple for us it is far from
simple for Nature. Distance refers to a space-frame;
it is different according to the frame chosen. We cannot
make sense of the law of inverse-square of the distance
unless we have first fixed on a frame of space; but
Nature has not fixed on any one frame. Even if by
some self-compensation the law worked out so as to give
the same observable consequences whatever space-frame
we might happen to choose (which it does not) we should
still be misapprehending its real mode of operation. In
chapter VI we shall try to gain a new insight into the
law (which for most practical applications is so nearly
expressed by the inverse-square) and obtain a picture
of its working which does not drag in an irrelevant frame
of space. The recognition of relativity leads us to
30 RELATIVITY
seek a new way of unravelling the complexity of natural
phenomena.
Velocity through the Aether. The theory of relativity is
evidently bound up with the impossibility of detecting
absolute velocity; if in our quarrel with the nebular
physicists one of us had been able to claim to be
absolutely at rest, that would be sufficient reason for
preferring the corresponding frame. This has some-
thing in common with the well-known philosophic belief
that motion must necessarily be relative. Motion is
change of position relative to something-, if we try to
think of change of position relative to nothing the whole
conception fades away. But this does not completely
settle the physical problem. In physics we should not
be quite so scrupulous as to the use of the word absolute.
Motion with respect to aether or to any universally sig-
nificant frame would be called absolute.
No aethereal frame has been found. We can only
discover motion relative to the material landmarks
scattered casually about the world; motion with respect
to the universal ocean of aether eludes us. We say,
"Let V be the velocity of a body through the aether",
and form the various electromagnetic equations in which
V is scattered liberally. Then we insert the observed
values, and try to eliminate everything that is unknown
except V. The solution goes on famously; but just as
we have got rid of the other unknowns, behold! V dis-
appears as well, and we are left with the indisputable
but irritating conclusion —
This is a favourite device that mathematical equations
resort to, when we propound stupid questions. If we
tried to find the latitude and longitude of a point north-
VELOCITY THROUGH THE AETHER 31
east from the north pole we should probably receive
the same mathematical answer. "Velocity through
aether" is as meaningless as "north-east from the north
pole".
This does not mean that the aether is abolished. We
need an aether. The physical world is not to be analysed
into isolated particles of matter or electricity with
featureless interspace. We have to attribute as much
character to the interspace as to the particles, and in
present-day physics quite an army of symbols is required
to describe what is going on in the interspace. We
postulate aether to bear the characters of the interspace
as we postulate matter or electricity to bear the charac-
ters of the particles. Perhaps a philosopher might ques-
tion whether it is not possible to admit the characters
alone without picturing anything to support them — thus
doing away with aether and matter at one stroke. But
that is rather beside the point.
In the last century it was widely believed that aether
was a kind of matter, having properties such as mass,
rigidity, motion, like ordinary matter. It would be
difficult to say when this view died out. It probably
lingered longer in England than on the continent, but
I think that even here it had ceased to be the orthodox
view some years before the advent of the relativity
theory. Logically it was abandoned by the numerous
nineteenth-century investigators who regarded matter
as vortices, knots, squirts, etc., in the aether; for clearly
they could not have supposed that aether consisted of
vortices in the aether. But it may not be safe to assume
that the authorities in question were logical.
Nowadays it is agreed that aether is not a kind of
matter. Being non-material, its properties are sui generis.
We must determine them by experiment; and since we
32 RELATIVITY
have no ground for any preconception, the experimental
conclusions can be accepted without surprise or mis-
giving. Characters such as mass and rigidity which we
meet with in matter will naturally be absent in aether;
but the aether will have new and definite characters of
its own. In a material ocean we can say that a particu-
lar particle of water which was here a few moments ago
is now over there; there is no corresponding assertion
that can be made about the aether. If you have been
thinking of the aether in a way which takes for granted
this property of permanent identification of its particles,
you must revise your conception in accordance with the
modern evidence. We cannot find our velocity through
the aether; we cannot say whether the aether now in this
room is flowing out through the north wall or the south
wall. The question would have a meaning for a mate-
rial ocean, but there is no reason to expect it to have a
meaning for the non-material ocean of aether.
The aether itself is as much to the fore as ever it was,
in our present scheme of the world. But velocity through
aether has been found to resemble that elusive lady
Mrs. Harris; and Einstein has inspired us with the
daring scepticism — "I don't believe there's no sich a
person".
Is the FitzGerald Contraction Real? I am often asked
whether the FitzGerald contraction really occurs. It
was introduced in the first chapter before the idea of
relativity was mentioned, and perhaps it is not quite
clear what has become of it now that the theory of
relativity has given us a new conception of what is going
on in the world. Naturally my first chapter, which
describes the phenomena according to the ideas of
classical physics in order to show the need for a new
IS FITZGERALD CONTRACTION REAL? 33
theory, contains many statements which we should
express differently in relativity physics.
Is it really true that a moving rod becomes shortened
in the direction of its motion? It is not altogether easy
to give a plain answer. I think we often draw a dis-
tinction between what is true and what is really true. A
statement which does not profess to deal with anything
except appearances may be true; a statement which is
not only true but deals with the realities beneath the
appearances is really true.
You receive a balance-sheet from a public company
and observe that the assets amount to such and such a
figure. Is this true? Certainly; it is certified by a
chartered accountant. But is it really true? Many
questions arise; the real values of items are often very
different from those which figure in the balance-sheet.
I am not especially referring to fraudulent companies.
There is a blessed phrase "hidden reserves"; and gen-
erally speaking the more respectable the company the
more widely does its balance-sheet deviate from reality.
This is called sound finance. But apart from deliberate
use of the balance-sheet to conceal the actual situation,
it is not well adapted for exhibiting realities, because
the main function of a balance-sheet is to balance
and everything else has to be subordinated to that
end.
The physicist who uses a frame of space has to
account for every millimetre of space — in fact to draw
up a balance-sheet, and make it balance. Usually there
is not much difficulty. But suppose that he happens to
be concerned with a man travelling at 161^000 miles
a second. The man is an ordinary 6-foot man. So far
as reality is concerned the proper entry in the balance-
sheet would appear to be 6 feet. But then the balance-
34 RELATIVITY
sheet would not balance. In accounting for the rest of
space there is left only 3 feet between the crown of his
head and the soles of his boots. His balance-sheet
length is therefore "written down" to 3 feet.
The writing-down of lengths for balance-sheet pur-
poses is the FitzGerald contraction. The shortening of
the moving rod is true, but it is not really true. It is not
a statement about reality (the absolute) but it is a true
statement about appearances in our frame of reference.*
An object has different lengths in the different space-
frames, and any 6-foot man will have a length 3 feet in
some frame or other. The statement that the length of
the rapid traveller is 3 feet is true, but it does not indicate
any special peculiarity about the man; it only indicates
that our adopted frame is the one in which his length is
3 feet. If it hadn't been ours, it would have been some-
one else's.
Perhaps you will think we ought to alter our method
of keeping the accounts of space so as to make them
directly represent the realities. That would be going to
a lot of trouble to provide for what are after all rather
rare transactions. But as a matter of fact we have
managed to meet your desire. Thanks to Minkowski
a way of keeping accounts has been found which
exhibits realities (absolute things) and balances. There
has been no great rush to adopt it for ordinary purposes
because it is a four-dimensional balance-sheet.
Let us take a last glance back before we plunge into
*The proper-length (p. 25) is unaltered; but the relative length is
shortened. We have already seen that the word "length" as currently-
used refers to relative length, and in confirming the statement that the
moving rod changes its length we are, of course, assuming that the word
is used with its current meaning.
SUMMARY 35
four dimensions. We have been confronted with some-
thing not contemplated in classical physics — a multi-
plicity of frames of space, each one as good as any
other. And in place of a distance, magnetic force,
acceleration, etc., which according to classical ideas
must necessarily be definite and unique, we are con-
fronted with different distances, etc., corresponding to
the different frames, with no ground for making a choice
between them. Our simple solution has been to give
up the idea that one of these is right and that the others
are spurious imitations, and to accept them en bloc; so
that distance, magnetic force, acceleration, etc., are
relative quantities, comparable with other relative quan-
tities already known to us such as direction or velocity.
In the main this leaves the structure of our physical
knowledge unaltered; only we must give up certain
expectations as to the behaviour of these quantities, and
certain tacit assumptions which were based on the belief
that they are absolute. In particular a law of Nature
which seemed simple and appropriate for absolute quan-
tities may be quite inapplicable to relative quantities and
therefore require some tinkering. Whilst the structure of
our physical knowledge is not much affected, the change
in the underlying conceptions is radical. We have trav-
elled far from the old standpoint which demanded
mechanical models of everything in Nature, seeing that
we do not now admit even a definite unique distance
between two points. The relativity of the current scheme
of physics invites us to search deeper and find the abso-
lute scheme underlying it, so that we may see the world
in a truer perspective.
Chapter III
TIME
Astronomer Royal's Time. I have sometimes thought it
would be very entertaining to hear a discussion between
the Astronomer Royal and, let us say, Prof. Bergson on
the nature of time. Prof. Bergson's authority on the
subject is well known; and I may remind you that the
Astronomer Royal is entrusted with the duty of finding
out time for our everyday use, so presumably he has
some idea of what he has to find. I must date the
discussion some twenty years back, before the spread of
Einstein's ideas brought about a rapprochement. There
would then probably have been a keen disagreement,
and I rather think that the philosopher would have had
the best of the verbal argument. After showing that
the Astronomer Royal's idea of time was quite non-
sensical, Prof. Bergson would probably end the dis-
cussion by looking at his watch and rushing off to catch
a train which was starting by the Astronomer Royal's
time.
Whatever may be time de ]ure } the Astronomer
Royal's time is time de facto. His time permeates every
corner of physics. It stands in no need of logical de-
fence; it is in the much stronger position of a vested
interest. It has been woven into the structure of the
classical physical scheme. "Time" in physics means
Astronomer Royal's time. You may be aware that it is
revealed to us in Einstein's theory that time and space
are mixed up in a rather strange way. This is a great
stumbling-block to the beginner. He is inclined to say,
"That is impossible. I feel it in my bones that time and
36
ASTRONOMER ROYAL'S TIME 37
space must be of entirely different nature. They cannot
possibly be mixed up." The Astronomer Royal com-
placently retorts, "It is not impossible. / have mixed
them up." Well, that settles it. If the Astronomer
Royal has mixed them, then his mixture will be the
groundwork of present-day physics.
We have to distinguish two questions which are not
necessarily identical. First, what is the true nature of
time? Second, what is the nature of that quantity which
has under the name of time become a fundamental part
of the structure of classical physics? By long history
of experiment and theory the results of physical inves-
tigation have been woven into a scheme which has on
the whole proved wonderfully successful. Time — the
Astronomer Royal's time — has its importance from the
fact that it is a constituent of that scheme, the binding
material or mortar of it. That importance is not les-
sened if it should prove to be only imperfectly repre-
sentative of the time familiar to our consciousness. We
therefore give priority to the second question.
But I may add that Einstein's theory, having cleared
up the second question, having found that physical
time is incongruously mixed with space, is able to pass
on to the first question. There is a quantity, unrecog-
nised in pre-relativity physics, which more directly
represents the time known to consciousness. This is
called proper-time or interval. It is definitely separate
from and unlike proper-space. Your protest in the
name of commonsense against a mixing of time and
space is a feeling which I desire to encourage. Time and
space ought to be separated. The current representa-
tion of the enduring world as a three-dimensional space
leaping from instant to instant through time is an
unsuccessful attempt to separate them. Come back with
38 TIME
me into the virginal four-dimensional world and we will
carve it anew on a plan which keeps them entirely
distinct. We can then resurrect the almost forgotten
time of consciousness and find that it has a gratifying
importance in the absolute scheme of Nature.
But first let us try to understand why physical time
has come to deviate from time as immediately perceived.
We have jumped to certain conclusions about time and
have come to regard them almost as axiomatic, although
they are not really justified by anything in our immediate
perception of time. Here is one of them.
If two people meet twice they must have lived the
same time between the two meetings, even if one of
them has travelled to a distant part of the universe and
back in the interim.
An absurdly impossible experiment, you will say.
Quite so; it is outside all experience. Therefore, may
I suggest that you are not appealing to your experience
of time when you object to a theory which denies the
above statement? And yet if the question is pressed
most people would answer impatiently that of course
the statement is true. They have formed a notion of
time rolling on outside us in a way which makes this
seem inevitable. They do not ask themselves whether
this conclusion is warranted by anything in their actual
experience of time.
Although we cannot try the experiment of sending a
man to another part of the universe, we have enough
scientific knowledge to compute the rates of atomic and
other physical processes in a body at rest and a body
travelling rapidly. We can say definitely that the bodily
processes in the traveller occur more slowly than the
corresponding processes in the man at rest (i.e. more
slowly according to the Astronomer Royal's time). This
ASTRONOMER ROYAL'S TIME 39
is not particularly mysterious; it is well known both
from theory and experiment that the mass or inertia of
matter increases when the velocity increases. The re-
tardation is a natural consequence of the greater inertia.
Thus so far as bodily processes are concerned the fast-
moving traveller lives more slowly. His cycle of diges-
tion and fatigue; the rate of muscular response to stim-
ulus; the development of his body from youth to age;
the material processes in his brain which must more or
less keep step with the passage of thoughts and emo-
tions; the watch which ticks in his waistcoat pocket; all
these must be slowed down in the same ratio. If the
speed of travel is very great we may find that, whilst
the stay-at-home individual has aged 70 years, the trav-
eller has aged 1 year. He has only found appetite for
365 breakfasts, lunches, etc.; his intellect, clogged by a
slow-moving brain, has only traversed the amount of
thought appropriate to one year of terrestrial life. His
watch, which gives a more accurate and scientific reck-
oning, confirms this. Judging by the time which con-
sciousness attempts to measure after its own rough
fashion — and, I repeat, this is the only reckoning of time
which we have a right to expect to be distinct from
space — the two men have not lived the same time
between the two meetings.
Reference to time as estimated by consciousness is
complicated by the fact that the reckoning is very erratic.
"I'D tell you who Time ambles withal, who Time trots
withal, who Time gallops withal, and who he stands
still withal." I have not been referring to these sub-
jective variations. I do not very willingly drag in
so unsatisfactory a time-keeper; only I have to deal
with the critic who tells me what "he feels in his bones"
about time, and I would point out to him that the basis
40 TIME
of that feeling is time lived, which we have just seen
may be 70 years for one individual and 1 year for
another between their two meetings. We can reckon
"time lived' 5 quite scientifically, e.g. by a watch travel-
ling with the individual concerned and sharing his
changes of inertia with velocity. But there are obvious
drawbacks to the general adoption of "time lived". It
might be useful for each individual to have a private
time exactly proportioned to his time lived; but it would
be extremely inconvenient for making appointments.
Therefore the Astronomer Royal has adopted a uni-
versal time-reckoning which does not follow at all strictly
the time lived. According to it the time-lapse does not
depend on how the object under consideration has
moved in the meanwhile. I admit that this reckoning
is a little hard on our returned traveller, who will be
counted by it as an octogenarian although he is to all
appearances still a boy in his teens. But sacrifices must
be made for the general benefit. In practice we have
not to deal with human beings travelling at any great
speed; but we have to deal with atoms and electrons
travelling at terrific speed, so that the question of pri-
vate time-reckoning versus general time-reckoning is a
very practical one.
Thus in physical time (or Astronomer Royal's time)
two people are deemed to have lived the same time
between two meetings, whether or not that accords with
their actual experience. The consequent deviation from
the time of experience is responsible for the mixing
up of time and space, which, of course, would be
impossible if the time of direct experience had been
rigidly adhered to. Physical time is, like space, a kind
of frame in which we locate the events of the external
world. We are now going to consider how in practice
LOCATION OF EVENTS
4i
external events are located in a frame of space and time.
We have seen that there is an infinite choice of alter-
native frames; so, to be quite explicit, I will tell you
how / locate events in my frame.
Location of Events, In Fig. 1 you see a collection of
events, indicated by circles. They are not at present in
FUTURE
O
u
Ld
cc
q:
LxJ
O
©HERE-NOW
O
O
Ld
I
X
Ld
Ld
if>
O
<o
-J
^m^
^+^
-J
Ld
O
°o
Ld
.
O
PAST
Fig. 1
their right places; that is the job before me — to put
them into proper location in my frame of space and
time. Among them I can immediately recognise and
label the event Here-Now, viz. that which is happening
in this room at this moment. The other events are at
varying degrees of remoteness from Here-Now, and it
42 TIME
is obvious to me that the remoteness is not only of
different degrees but of different kinds. Some events
spread away towards what in a general way I call the
Past; I can contemplate others which are distant in
the Future; others are remote in another kind of way
towards China or Peru, or in general terms Elsewhere.
In this picture I have only room for one dimension of
Elsewhere; another dimension sticks out at right angles
to the paper; and you must imagine the third dimension
as best you can.
Now we must pass from this vague scheme of location
to a precise scheme. The first and most important thing
is to put Myself into the picture. It sounds egotistical;
but, you see, it is my frame of space that will be used,
so it all hangs round me. Here I am — a kind of four-
dimensional worm (Fig. 2). It is a correct portrait;
I have considerable extension towards the Past and
presumably towards the Future, and only a moderate
extension towards Elsewhere. The "instantaneous me",
i.e. myself at this instant, coincides with the event Here-
Now. Surveying the world from Here-Now, I can see
many other events happening now. That puts it into my
head that the instant of which I am conscious here must
be extended to include them; and I jump to the con-
clusion that Now is not confined to Here-Now. I there-
fore draw the instant Now, running as a clean section
across the world of events, in order to accommodate all
the distant events which are happening now. I select
the events which I see happening now and place them
on this section, which I call a moment of time or an
"instantaneous state of the world". I locate them on
Now because they seem to be Now.
This method of location lasted until the year 1667,
when it was found impossible to make it work consist-
LOCATION OF EVENTS
43
ently. It was then discovered by the astronomer Roemer
that what is seen now cannot be placed on the instant
Now. (In ordinary parlance — light takes time to travel.)
That was really a blow to the whole system of world-
wide instants, which were specially invented to accommo-
date these events. We had been mixing up two distinct
FUT
URE
Id
Ld
^ NOW
c
N HERE-NOW /-., NOW *^
Ld " ' ""
^A
A ^ '" K..' " Ld
1 ^*'*'
id-**-
"* Id
</>
<n
_i
-j
Ld
u.
_J
111
(0
>
5
id
PAST
Fig. 2^
events; there was the original event somewhere out in
the external world and there was a second event, viz.
the seeing by us of the first event. The second event
was in our bodies Here-Now; the first event was neither
Here nor Now. The experience accordingly gives no
indication of a Now which is not Here; and we might
44 TIME
well have abandoned the idea that we have intuitive
recognition of a Now other than Here-Now, which was
the original reason for postulating world-wide instants
Now.
However, having become accustomed to world-wide
instants, physicists were not ready to abandon them.
And, indeed, they have considerable usefulness pro-
vided that we do not take them too seriously. They were
left in as a feature of the picture, and two Seen-Now
lines were drawn, sloping backwards from the Now line,
on which events seen now could be consistently placed.
The cotangent of the angle between the Seen-Now lines
and the Now line was interpreted as the velocity of light.
Accordingly when I see an event in a distant part of
the universe, e.g. the outbreak of a new star, I locate it
(quite properly) on the Seen-Now line. Then I make a
certain calculation from the measured parallax of the
star and draw my Now line to pass, say, 300 years in
front of the event, and my Now line of 300 years ago
to pass through the event. By this method I trace the
course of my Now lines or world-wide instants among
the events, and obtain a frame of time-location for
external events. The auxiliary Seen-Now lines, having
served their purpose, are rubbed out of the picture.
That is how / locate events; how about youf We
must first put You into the picture (Fig. 3). We shall
suppose that you are on another star moving with
different velocity but passing close to the earth at the
present moment. You and I were far apart in the past
and will be again in the future, but we are both Here-
Now. That is duly shown in the picture. We survey
the world from Here-Now, and of course we both see
the same events simultaneously. We may receive rather
different impressions of them; our different motions
LOCATION OF EVENTS
45
will cause different Doppler effects, FitzGerald con-
tractions, etc. There may be slight misunderstandings
until we realise that what you describe as a red square
is what I would describe as a green oblong, and so on.
But, allowing for this kind of difference of description,
FUTURE
id
cc MY NQW
Id Y oUR N°^
Id
HERE-NOW
v0 ornow.
MY NOW
**- s £V
2f?
W
Id
cr
Ld
I
U
<0
_J
Id
PAST
Fig. 3
it will soon become clear that we are looking at the same
events, and we shall agree entirely as to how the Seen-
Now lines lie with respect to the events. Starting from
our common Seen-Now lines, you have next to make the
calculations for drawing your Now line among the
events, and you trace it as shown in Fig. 3.
46 TIME
How is it that, starting from the same Seen-Now
lines, you do not reproduce my Now line? It is because
a certain measured quantity, viz. the velocity of light,
has to be employed in the calculations; and naturally
you trust to your measures of it as I trust to mine.
Since our instruments are affected by different Fitz-
Gerald contractions, etc., there is plenty of room for
divergence. Most surprisingly we both find the same
velocity of light, 299,796 kilometres per second. But
this apparent agreement is really a disagreement; be-
cause you take this to be the velocity relative to your
planet and I take it to be the velocity relative to mine.*
Therefore our calculations are not in accord, and your
Now line differs from mine.
If we believe our world-wide instants or Now lines
to be something inherent in the world outside us, we
shall quarrel frightfully. To my mind it is ridiculous
that you should take events on the right of the picture
which have not -happened yet and events on the left
which are already past and call the combination an
instantaneous condition of the universe. You are
equally scornful of my grouping. We can never agree.
Certainly it looks from the picture as though my
instants were more natural than yours; but that is
because / drew the picture. You, of course, would
redraw it with your Now lines at right angles to your-
self.
* The measured velocity of light is the average to-and-fro velocity.
The velocity in one direction singly cannot be measured until after the
Now lines have been laid down and therefore cannot be used in laying
down the Now lines. Thus there is a deadlock in drawing the Now lines
which can only be removed by an arbitrary assumption or convention.
The convention actually adopted is that (relative to the observer) the
velocities of light in the two opposite directions are equal. The resulting
Now lines must therefore be regarded as equally conventional.
ABSOLUTE PAST AND FUTURE 47
But we need not quarrel if the Now lines are merely
reference lines drawn across the world for convenience
in locating events — like the lines of latitude and longi-
tude on the earth. There is then no question of a right
way and a wrong way of drawing the lines; we draw
them as best suits our convenience. World-wide instants
are not natural cleavage planes of time; there is nothing
equivalent to them in the absolute structure of the world;
they are imaginary partitions which we find it con-
venient to adopt.
We have been accustomed to regard the world — the
enduring world — as stratified into a succession of in-
stantaneous states. But an observer on another star
would make the strata run in a different direction from
ours. We shall see more clearly the real mechanism of
the physical world if we can rid our minds of this
illusion of stratification. The world that then stands
revealed, though strangely unfamiliar, is actually much
simpler. There is a difference between simplicity and
familiarity. A pig may be most familiar to us in the
form of rashers, but the unstratified pig is a simpler
object to the biologist who wishes to understand how
the animal functions.
Absolute Past and Future. Let us now try to attain this
absolute view. We rub out all the Now lines. We rub
out Yourself and Myself, since we are no longer
essential to the world. But the Seen-Now lines are left.
They are absolute, since all observers from Here-Now
agree about them. The flat picture is a section; you
must imagine it rotated (twice rotated in fact, since
there are two more dimensions outside the picture). The
Seen-Now locus is thus really a cone ; or by taking account
of the prolongation of the lines into the future a double
48 TIME
cone or hour-glass figure (Fig. 4). These hour-glasses
(drawn through each point of the world considered in
turn as a Here-Now) embody what we know of the abso-
lute structure of the world so far as space and time are
concerned. They show how the "grain" of the world
runs.
Father Time has been pictured as an old man with
a scythe and an hour-glass. We no longer permit him
to mow instants through the world with his scythe; but
we leave him his hour-glass.
ABSOLUTE FUTURE
ABSOLUTE ^**^rv^^ ABSOLUTE
ELSEWHERE ^^^itMtrnvw ELSEWHERE
ABSOLUTE PAST
«*g£^~- — -^W
Fig. 4
Since the hour-glass is absolute its two cones provide
respectively an Absolute Future and an Absolute Past
for the event Here-Now. They are separated by a
wedge-shaped neutral zone which (absolutely) is neither
past nor future. The common impression that relativity
turns past and future altogether topsy-turvy is quite
false. But, unlike the relative past and future, the
absolute past and future are not separated by an in-
finitely narrow present. It suggests itself that the
ABSOLUTE PAST AND FUTURE 49
neutral wedge might be called the Absolute Present ; but I
do not think that is a good nomenclature. It is much
better described as Absolute Elsewhere. We have
abolished the Now lines, and in the absolute world the
present (Now) is restricted to Here-Now.
Perhaps I may illustrate the peculiar conditions
arising from the wedge-shaped neutral zone by a rather
hypothetical example. Suppose that you are in love with
a lady on Neptune and that she returns the sentiment.
It will be some consolation for the melancholy separation
if you can say to yourself at some — possibly pre-
arranged — moment, "She is thinking of me now".
Unfortunately a difficulty has arisen because we have
had to abolish Now. There is no absolute Now, but
only the various relative Nows differing according to
the reckoning of different observers and covering the
whole neutral wedge which at the distance of Neptune
is about eight hours thick. She will have to think of
you continuously for eight hours on end in order to
circumvent the ambiguity of "Now".
At the greatest possible separation on the earth the
thickness of the neutral wedge is no more than a tenth
of a second; so that terrestrial synchronism is not
seriously interfered with. This suggests a qualification
of our previous conclusion that the absolute present is
confined to Here-Now. It is true as regards instan-
taneous events (point-events). But in practice the
events we notice are of more than infinitesimal duration.
If the duration is sufficient to cover the width of the
neutral zone, then the event taken as a whole may fairly
be considered to be Now absolutely. From this point
of view the "nowness" of an event is like a shadow cast
by it into space, and the longer the event the farther
will the umbra of the shadow extend.
50 TIME
As the speed of matter approaches the speed of light
its mass increases to infinity, and therefore it is impos-
sible to make matter travel faster than light. This
conclusion is deduced from the classical laws of physics,
and the increase of mass has been verified by experiment
up to very high velocities. In the absolute world this
means that a particle of matter can only proceed from
Here-Now into the absolute future — which, you will
agree, is a reasonable and proper restriction. It cannot
travel into the neutral zone; the limiting cone is the
track of light or of anything moving with the speed of
light. We ourselves are attached to material bodies, and
therefore we can only go on into the absolute future.
Events in the absolute future are not absolutely
Elsewhere. It would be possible for an observer to
travel from Here-Now to the event in question in time
to experience it, since the required velocity is less than
that of light; relative to the frame of such an observer
the event would be Here. No observer can reach an
event in the neutral zone, since the required speed is too
great. The event is not Here for any observer (from
Here-Now) ; therefore it is absolutely Elsewhere.
The Absolute Distinction of Space and Time. By divid-
ing the world into Absolute Past and Future on the one
hand and Absolute Elsewhere on the other hand, our
hour-glasses have restored a fundamental differentiation
between time and space. It is not a distinction between
time and space as they appear in a space-time frame, but
a distinction between temporal and spatial relations.
Events can stand to us in a temporal relation (absolutely
past or future) or a spatial relation (absolutely else-
where), but not in both. The temporal relations radiate
into the past and future cones and the spatial relations
DISTINCTION OF SPACE AND TIME 51
into the neutral wedge; they are kept absolutely sepa-
rated by the Seen-Now lines which we have identified with
the grain of absolute structure in the world. We have
recovered the distinction which the Astronomer Royal
confused when he associated time with the merely arti-
ficial Now lines.
I would direct your attention to an important differ-
ence in our apprehension of time-extension and space-
extension. As already explained our course through the
world is into the absolute future, i.e. along a sequence
of time-relations. We can never have a similar experi-
ence of a sequence of space-relations because that
would involve travelling with velocity greater than light.
Thus we have immediate experience of the time-relation
but not of the space-relation. Our knowledge of space-
relations is indirect, like nearly all our knowledge of the
external world — a matter of inference and interpretation
of the impressions which reach us through our sense-
organs. We have similar indirect knowledge of the
time-relations existing between the events in the world
outside us; but in addition we have direct experience
of the time-relations that we ourselves are traversing —
a knowledge of time not coming through external sense-
organs, but taking a short cut into our consciousness.
When I close my eyes and retreat into my inner mind,
I feel myself enduring, I do not feel myself extensive. It
is this feeling of time as affecting ourselves and not
merely as existing in the relations of external events
which is so peculiarly characteristic of it; space on the
other hand is always appreciated as something external.
That is why time seems to us so much more mysteri-
ous than space. We know nothing about the intrinsic
nature of space, and so it is quite easy to conceive it
satisfactorily. We have intimate acquaintance with the
52 TIME
nature of time and so it baffles our comprehension. It
is the same paradox which makes us believe we under-
stand the nature of an ordinary table whereas the nature
of human personality is altogether mysterious. We
never have that intimate contact with space and tables
which would make us realise how mysterious they are;
we have direct knowledge of time and of the human
spirit which makes us reject as inadequate that merely
symbolic conception of the world which is so often mis-
taken for an insight into its nature.
The Four-Dimensional World. I do not know whether
you have been keenly alive to the fact that for some time
now we have been immersed in a four-dimensional
world. The fourth dimension required no introduction;
as soon as we began to consider events it was there.
Events obviously have a fourfold order which we can
dissect into right or left, behind or in front, above or
below, sooner or later — or into many alternative sets of
fourfold specification. The fourth dimension is not a
difficult conception. It is not difficult to. conceive of
events as ordered in four dimensions; it is impossible to
conceive them otherwise. The trouble begins when we
continue farther along this line of thought, because by
long custom we have divided the world of events into
three-dimensional sections or instants, and regarded the
piling of the instants as something distinct from a
dimension. That gives us the usual conception of a
three-dimensional world floating in the stream of time.
This pampering of a particular dimension is not entirely
without foundation; it is our crude appreciation of the
absolute separation of space-relations and time-relations
by the hour-glass figures. But the crude discrimination
has to be replaced by a more accurate discrimination.
THE FOUR-DIMENSIONAL WORLD 53
The supposed planes of structure represented bj
Now lines separated one dimension from the other
three; but the cones of structure given by the hour-
glass figures keep the four dimensions firmly pinned
together.*
We are accustomed to think of a man apart from his
duration. When I portrayed "Myself" in Fig. 2, you
were for the moment surprised that I should include
my boyhood and old age. But to think of a man without
his duration is just as abstract as to think of a man
without his inside. Abstractions are useful, and a man
without his inside (that is to say, a surface) is a well-
known geometrical conception. But we ought to realise
what is an abstraction and what is not. The "four-
dimensional worms" introduced in this chapter seem to
many people terribly abstract. Not at all; they are un-
familiar conceptions but not abstract conceptions. It is
the section of the worm (the man Now) which is an
abstraction. And as sections may be taken in somewhat
different directions, the abstraction is made differently
by different observers who accordingly attribute different
FitzGerald contractions to it. The non-abstract man
enduring through time is the common source from which
the different abstractions are made.
The appearance of a four-dimensional world in this
subject is due to Minkowski. Einstein showed the rela-
tivity of the familiar quantities of physics; Minkowski
showed how to recover the absolute by going back to
their four-dimensional origin and searching more deeply.
* In Fig. 4 the scale is such that a second of time corresponds to
70,000 miles of space. If we take a more ordinary scale of experience,
say a second to a yard, the Seen-Now lines become almost horizontal;
and it will easily be understood why the cones which pin the four
dimensions together have generally been mistaken for sections separating
them.
54 TIME
The Velocity of Light. A feature of the relativity
theory which seems to have aroused special interest
among philosophers is the absoluteness of the velocity of
light. In general velocity is relative. If I speak of a
velocity of 40 kilometres a second I must add "relative
to the earth", "relative to Arcturus", or whatever refer-
ence body I have in mind. No one will understand any-
thing from my statement unless this is added or implied.
But it is a curious fact that if I speak of a velocity
of 299,796 kilometres a second it is unnecessary to
add the explanatory phrase. Relative to what? Rela-
tive to any and every star or particle of matter in the
universe.
It is no use trying to overtake a flash of light;
however fast you go it is always travelling away from
you at 186,000 miles a second. Now from one point
of view this is a rather unworthy deception that Nature
has practised upon us. Let us take our favourite observer
who travels at 161,000 miles a second and send him in
pursuit of the flash of light. It is going 25,000 miles
a second faster than he is; but that is not what he will
report. Owing to the contraction of his standard scale
his miles are only half-miles; owing to the slowing down
of his clocks his seconds are double-seconds. His
measurements would therefore make the speed 100,000
miles a second (really half-miles per double-second).
He makes a further mistake in synchronising the clocks
with which he records the velocity. (You will remember
that he uses a different Now line from ours.). This
brings the speed up to 186,000 miles a second. From
his own point of view the traveller is lagging hopelessly
behind the light; he does not realise what a close race
he is making of it, because his measuring appliances
have been upset. You will note that the evasiveness of
THE VELOCITY OF LIGHT 55
the light-flash is not in the least analogous to the
evasiveness of the rainbow.
But although this explanation may help to reconcile
us to what at first seems a blank impossibility, it is not
really the most penetrating. You will remember that
a Seen-Now line, or track of a flash of light, represents
the grain of the world-structure. Thus the peculiarity
of a velocity of 299,796 kilometres a second is that it
coincides with the grain of the world. The four-
dimensional worms representing material bodies must
necessarily run across the grain into the future cone, and
we have to introduce some kind of reference frame to
describe their course. But the flash of light is exactly
along the grain, and there is no need of any artificial
system of partitions to describe this fact.
The number 299,796 (kilometres per second) is,
so to speak, a code-number for the grain of the wood.
Other code-numbers correspond to the various worm-
holes which may casually cross the grain. We have
different codes corresponding to different frames of
space and time; the code-number of the grain of the
wood is the only one which is the same in all codes.
This is no accident; but I do not know that any deep
inference is to be drawn from it, other than that our
measure-codes have been planned rationally so as to turn
on the essential and not on the casual features of world-
structure.
The speed of 299,796 kilometres per second which
occupies a unique position in every measure-system is
commonly referred to as the speed of light. But it is
much more than that; it is the speed at which the mass
of matter becomes infinite, lengths contract to zero,
clocks stand still. Therefore it crops up in all kinds of
problems whether light is concerned or not.
56 TIME
The scientist's interest in the absoluteness of this
velocity is very great; the philosopher's interest has
been, I think, largely a mistaken interest. In asserting
its absoluteness scientists mean that they have assigned
the same number to it in every measure-system; but
that is a private arrangement of their own — an un-
witting compliment to its universal importance.* Turn-
ing from the measure-numbers to the thing described
by them, the "grain" is certainly an absolute feature
of the wood, but so also are the "worm-holes"
(material particles). The difference is that the grain is
essential and universal, the worm-holes casual. Science
and philosophy have often been at cross-purposes in
discussing the Absolute — a misunderstanding which is
I am afraid chiefly the fault of the scientists. In science
we are chiefly concerned with the absoluteness or relativity
of the descriptive terms we employ; but when the term
absolute is used with reference to that which is being
described it has generally the loose meaning of "uni-
versal" as opposed to "casual".
Another point on which there has sometimes been a
misunderstanding is the existence of a superior limit to
velocity. It is not permissible to say that no velocity can
exceed 299,796 kilometres per second. For example,
imagine a search-light capable of sending an accurately
parallel beam as far as Neptune. If the search-light is
made to revolve once a minute, Neptune's end of the
beam will move round a circle with velocity far greater
than the above limit. This is an example of our habit
of creating velocities by a mental association of states
* In the general relativity theory (chapter vi) measure-systems are
employed in which the velocity of light is no longer assigned the same
constant value, but it continues to correspond to the grain of absolute
world-structure.
THE VELOCITY OF LIGHT 57
which are not themselves in direct causal connection.
The assertion made by the relativity theory is more
restricted, viz. —
Neither matter, nor energy, nor anything capable of
being used as a signal can travel faster than 299,796
kilometres per second, provided that the velocity is
referred to one of the frames of space and time con-
sidered in this chapter.*
The velocity of light in matter can under certain
circumstances (in the phenomenon of anomalous dis-
persion) exceed this value. But the higher velocity is
only attained after the light has been passing through
the matter for some moments so as to set the molecules
in sympathetic vibration. An unheralded light-flash
travels more slowly. The speed, exceeding 299,796
kilometres a second, is, so to speak, achieved
by prearrangement, and has no application in sig-
nalling.
We are bound to insist on this limitation of the speed
of signalling. It has the effect that it is only possible to
signal into the Absolute Future. The consequences of
being able to transmit messages concerning events
Here-Now into the neutral wedge are too bizarre to
contemplate. Either the part of the neutral wedge that
can be reached by the signals must be restricted in a
way which violates the principle of relativity; or it will
be possible to arrange for a confederate to receive the
messages which we shall send him to-morrow, and to
retransmit them to us so that we receive them to-dav^'
The limit to the velocity of signals is our bulwark
* Some proviso of this kind is clearly necessary. We often employ
for special purposes a frame of reference rotating with the earth; in this
frame the stars describe circles once a day, and are therefore ascribed
enormous velocities.
58 TIME
against that topsy-turvydom of past and future, of which
Einstein's theory is sometimes wrongfully accused.
Expressed in the conventional way this limitation of
the speed of signalling to 299,796 kilometres a
second seems a rather arbitrary decree of Nature. We
almost feel it as a challenge to find something that goes
faster. But if we state it in the absolute form that
signalling is only possible along a track of temporal
relation and not along a track of spatial relation the
restriction seems rational. To violate it we have not
merely to find something which goes just 1 kilometre
per second better, but something which overleaps that
distinction of time and space — which, we are all con-
vinced, ought to be maintained in any sensible theory.
Practical Applications. In these lectures I am concerned
more with the ideas of the new theories than with their
practical importance for the advancement of science.
But the drawback of dwelling solely on the underlying
conceptions is that it is likely to give the impression that
the new physics is very much u up in the air". That is
by no means true, and the relativity theory is used in
a businesslike way in the practical problems to which
it applies. I can only consider here quite elementary
problems which scarcely do justice to the power of the
new theory in advanced scientific research. Two
examples must suffice.
1. It has often been suggested that the stars will be
retarded by the back-pressure of their own radiation.
The idea is that since the star is moving forward the
emitted radiation is rather heaped up in front of it and
thinned out behind. Since radiation exerts pressure the
pressure will be stronger on the front surface than on
the rear, Therefore there is a force retarding the star
PRACTICAL APPLICATIONS 59
tending to bring it gradually to rest. The effect might
be of great importance in the study of stellar motions;
it would mean that on the average old stars must have
lower speeds than young stars — a conclusion which, as
it happens, is contrary to observation.
But according to the theory of relativity "coming to
rest" has no meaning. A decrease of velocity relative
to one frame is an increase relative to another frame.
There is no absolute velocity and no absolute rest for
the star to come to. The suggestion may therefore be
at once dismissed as fallacious.
2. The B particles shot out by radioactive substances
are electrons travelling at speeds not much below
the speed of light. Experiment shows that the mass
of one of these high-speed electrons is considerably
greater than the mass of an electron at rest. The theory
of relativity predicts this increase and provides the
formula for the dependence of mass on velocity. The
increase arises solely from the fact that mass is a relative
quantity depending by definition on the relative quan-
tities length and time.
Let us look at a 3 particle from its own point of view.
It is an ordinary electron in no wise different from any
other. But it is travelling with unusually high speed?
"No", says the electron, "That is your point of view.
I contemplate with amazement your extraordinary
speed of 100,000 miles a second with which you are
shooting past me. I wonder what it feels like to move
so quickly. However, it is no business of mine." So
the p particle, smugly thinking itself at rest, pays no
attention to our goings on, and arranges itself with the
usual mass, radius and charge. It has just the standard
mass of an electron, 9.10" 28 grams. But mass and
radius are relative quantities, and in this case the frame
60 TIME
to which they are referred is evidently the frame appro-
priate to an electron engaged in self-contemplation, viz.
the frame in which it is at rest But when we talk about
mass we refer it to the frame in which we are at rest.
By the geometry of the four-dimensional world we can
calculate the formulae for the change of reckoning of
mass in two different frames, which is consequential on
the change of reckoning of length and time; we find
in fact that the mass is increased in the same ratio as the
length is diminished (FitzGerald factor). The increase
of mass that we observe arises from the change of
reckoning between the electron's own frame and our
frame.
All electrons are alike from their own point of view.
The apparent differences arise in fitting them into our
own frame of reference which is irrelevant to their
structure. Our reckoning of their mass is higher than
their own reckoning, and increases with the difference
between our respective frames, i.e. with the relative
velocity between us.
We do not bring forward these results to demonstrate
or confirm the truth of the theory, but to show the use
of the theory. They can both be deduced from the
classical electromagnetic theory of Maxwell coupled (in
the second problem) with certain plausible assumptions
as to the conditions holding at the surface of an electron.
But to realise the advantage of the new theory we must
consider not what could have been but what was deduced
from the classical theory. The historical fact is that the
conclusions of the classical theory as to the first prob-
lem were wrong; an important compensating factor
escaped notice. Its conclusions as to the second problem
were (after some false starts) entirely correct numer-
ically. But since the result was deduced from the electro-
SUMMARY 61
magnetic equations of the electron it was thought that
it depended on the fact that an electron is an electrical
structure; and the agreement with observation was
believed to confirm the hypothesis that an electron is
pure electricity and nothing else. Our treatment above
makes no reference to any electrical properties of the
electron, the phenomenon having been found to arise
solely from the relativity of mass. Hence, although
there may be other good reasons for believing that an
electron consists solely of negative electricity, the in-
crease of mass with velocity is no evidence one way or
the other.
In this chapter the idea of a multiplicity of frames of
space has been extended to a multiplicity of frames of
space and time. The system of location in space, called
a frame of space, is only a part of a fuller system of
location of events in space and time. Nature provides
no indication that one of these frames is to be preferred
to the others. The particular frame in which we are
relatively at rest has a symmetry with respect to us
which other frames do not possess, and for this reason
we have drifted into the common assumption that it is
the only reasonable and proper frame; but this egocen-
tric outlook should now be abandoned, and all frames
treated as on the same footing. By considering time
and space together we have been able to understand
how the multiplicity of frames arises. They correspond
to different directions of section of the four-dimensional
world of events, the sections being the "world-wide
instants". Simultaneity (Now) is seen to be relative.
The denial of absolute simultaneity is intimately con-
nected with the denial of absolute velocity; knowledge
of absolute velocity would enable us to assert that
62 TIME
certain events in the past or future occur Here but not
Now; knowledge of absolute simultaneity would tell us
that certain events occur Now but not Here. Removing
these artificial sections, we have had a glimpse of the
absolute world-structure with its grain diverging and
interlacing after the plan of the hour-glass figures. By
reference to this structure we discern an absolute dis-
tinction between space-like and time-like separation of
events — a distinction which justifies and explains our
instinctive feeling that space and time are fundamentally
different. Many of the important applications of the
new conceptions to the practical problems of physics
are too technical to be considered in this book; one of
the simpler applications is to determine the changes of
the physical properties of objects due to rapid motion.
Since the motion can equally well be described as a
motion of ourselves relative to the object or of the
object relative to ourselves, it cannot influence the abso-
lute behaviour of the object. The apparent changes in
the length, mass, electric and magnetic fields, period of
vibration, etc., are merely a change of reckoning intro-
duced in passing from the frame in which the object is
at rest to the frame in which the observer is at rest.
Formulae for calculating the change of reckoning of
any of these quantities are easily deduced now that the
geometrical relation of the frames has been ascertained.
Chapter IV
THE RUNNING-DOWN OF THE UNIVERSE
Shuffling. The modern outlook on the physical world is
not composed exclusively of conceptions which have
arisen in the last twenty-five years; and we have now
to deal with a group of ideas dating far back in the last
century which have not essentially altered since the
time of Boltzmann. These ideas display great activity
and development at the present time. The subject is
relevant at this stage because it has a bearing on the
deeper aspects of the problem of Time; but it is so
fundamental in physical theory that we should be bound
to deal with it sooner or later in any comprehensive
survey.
If you take a pack of cards as it comes from the
maker and shuffle it for a few minutes, all trace of
the original systematic order disappears. The order wiil
never come back however long you shuffle. Something
has been done which cannot be undone, namely, the intro-
duction of a random element in place of arrangement.
Illustrations may be useful even when imperfect, and
therefore I have slurred over two points, which affect
the illustration rather than the application which we are
about to make. It was scarcely true to say that the
shuffling cannot be undone. You can sort out the cards
into their original order if you like. But in considering
the shuffling which occurs in the physical world we are
not troubled by a deus ex machina like you. I am not
prepared to say how far the human mind is bound by the
conclusions we shall reach. So I exclude you — at least
I exclude that activity of your mind which you employ
63
64 THE RUNNING-DOWN OF THE UNIVERSE
in sorting the cards. I allow you to shuffle them because
you can do that absent-mindedly.
Secondly, it is not quite true that the original order
never comes back. There is a ghost of a chance that
some day a thoroughly shuffled pack will be found to
have come back to the original order. That is because
of the comparatively small number of cards in the pack.
In our applications the units are so numerous that this
kind of contingency can be disregarded.
We shall put forward the contention that —
Whenever anything happens which cannot be undone,
it is always reducible to the introduction of a random
element analogous to that introduced by shuffling.
Shuffling is the only thing which Nature cannot undo.
When Humpty Dumpty had a great fall —
All the king's horses and all the king's men
Cannot put Humpty Dumpty together again.
Something had happened which could not be undone.
The fall could have been undone. It is not necessary to
invoke the king's horses and the king's men; if there
had been a perfectly elastic mat underneath, that would
have sufficed. At the end of his fall Humpty Dumpty
had kinetic energy which, properly directed, was just
sufficient to bounce him back on to the wall again. But,
the elastic mat being absent, an irrevocable event hap-
pened at the end of the fall — namely, the introduction
of a random element into Humpty Dumpty.
But why should we suppose that shuffling is the only
process that cannot be undone?
The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety and Wit
Can lure it back to cancel half a Line.
SHUFFLING 65
When there is no shuffling, is the Moving Finger stayed?
The answer of physics is unhesitatingly Yes. To judge
of this we must examine those operations of Nature in
which no increase of the random element can possibly
occur. These fall into two groups. Firstly, we can
study those laws of Nature which control the behaviour
of a single unit. Clearly no shuffling can occur in these
problems; you cannot take the King of Spades away
from the pack and shuffle him. Secondly, we can study
the processes of Nature in a crowd which is already so
completely shuffled that there is no room for any further
increase of the random element. If our contention is
right, everything that occurs in these conditions is
capable of being undone. We shall consider the first
condition immediately; the second must be deferred
until p. 78.
Any change occurring to a body which can be treated
as a single unit can be undone. The laws of Nature
admit of the undoing as easily as of the doing. The
earth describing its orbit is controlled by laws of motion
and of gravitation; these admit of the earth's actual mo-
tion, but they also admit of the precisely opposite
motion. In the same field of force the earth could retrace
its steps; it merely depends on how it was started off. It
may be objected that we have no right to dismiss the
starting-off as an inessential part of the problem; it may
be as much a part of the coherent scheme of Nature as
the laws controlling the subsequent motion. Indeed, as-
tronomers have theories explaining why the eight planets
all started to move the same way round the sun. But
that is a problem of eight planets, not of a single
individual — a problem of the pack, not of the isolated
card. So long as the earth's motion is treated as an
isolated problem, no one would dream of putting into
66 THE RUNNING-DOWN OF THE UNIVERSE
the laws of Nature a clause requiring that it must go
this way round and not the opposite.
There is a similar reversibility of motion in fields of
electric and magnetic force. Another illustration can be
given from atomic physics. The quantum laws admit
of the emission of certain kinds and quantities of light
from an atom; these laws also admit of absorption of
the same kinds and quantities, i.e. the undoing of the
emission. I apologise for an apparent poverty of illus-
tration; it must be remembered that many properties of
a body, e.g. temperature, refer to its constitution as a
large number of separate atoms, and therefore the laws
controlling temperature cannot be regarded as control-
ling the behaviour of a single individual.
The common property possessed by laws governing
the individual can be stated more clearly by a reference
to time. A certain sequence of states running from past
to future is the doing of an event; the same sequence
running from future to past is the undoing of it — be-
cause in the latter case we turn round the sequence so as
to view it in the accustomed manner from past to future.
So if the laws of Nature are indifferent as to the doing
and undoing of an event, they must be indifferent as
to a direction of time from past to future. That is their
common feature, and it is seen at once when (as
usual) the laws are formulated mathematically. There
is no more distinction between past and future than
between right and left. In algebraic symbolism, left is
— x, right is +*; past is — f, future is +J. This holds
for all laws of Nature governing the behaviour of non-
composite individuals — the "primary laws", as we shall
call them. There is only one law of Nature — the second
law of thermodynamics — which recognises a distinction
between past and future more profound than the
SHUFFLING 67
difference of plus and minus. It stands aloof from all
the rest. But this law has no application to the behaviour
of a single individual, and as we shall see later its sub-
ject-matter is the random element in a crowd.
Whatever the primary laws of physics may say, it is
obvious to ordinary experience that there is a distinction
between past and future of a different kind from the
distinction of left and right. In The Plattner Story
H. G. Wells relates how a man strayed into the fourth
dimension and returned with left and right interchanged.
But we notice that this interchange is not the theme of
the story; it is merely a corroborative detail to give an
air of verisimilitude to the adventure. In itself the
change is so trivial that even Mr. Wells cannot weave
a romance out of it. But if the man had come back with
past and future interchanged, then indeed the situation
would have been lively. Mr. Wells in The Time-Machine
and Lewis Carroll in Sylvie and Bruno give us a glimpse
of the absurdities which occur when time runs back-
wards. If space is "looking-glassed" the world con-
tinues to make sense; but looking-glassed time has an
inherent absurdity which turns the world-drama into
the most nonsensical farce.
Now the primary laws of physics taken one by one
all declare that they are entirely indifferent as to which
way you consider time to be progressing, just as they
are indifferent as to whether you view the world from
the right or the left. This -is true of the classical laws,
the relativity laws, and even of the quantum laws. It
is not an accidental property; the reversibility is inherent
in the whole conceptual scheme in which these laws
find a place. Thus the question whether the world does
or does not "make sense" is outside the range of these
laws. We have to appeal to the one outstanding law —
68 THE RUNNING-DOWN OF THE UNIVERSE
the second law of thermodynamics — to put some sense
into the world. It opens up a new province of know-
ledge, namely, the study of organisation; and it is in con-
nection with organisation that a direction of time-flow
and a distinction between doing and undoing appears for
the first time.
Time's Arrow, The great thing about time is that it goes
on. But this is an aspect of it which the physicist some-
times seems inclined to neglect. In the four-dimensional
world considered in the last chapter the events past and
future lie spread out before us as in a map. The events
are there in their proper spatial and temporal relation;
but there is no indication that they undergo what has
been described as "the formality of taking place", and
the question of their doing or undoing does not arise.
We see in the map the path from past to future or from
future to past; but there is no signboard to indicate that
it is a one-way street. Something must be added to
the geometrical conceptions comprised in Minkowski's
world before it becomes a complete picture of the world
as we know it. We may appeal to consciousness to suffuse
the whole — to turn existence into happening, being into
becoming. But first let us note that the picture as it
stands is entirely adequate to represent those primary
laws of Nature which, as we have seen, are indifferent
to a direction of time. Objection has sometimes been
felt to the relativity theory because its four-dimensional
picture of the world seems to overlook the directed
character of time. The objection is scarcely logical, for
the theory is in this respect no better and no worse than
its predecessors. The classical physicist has been using
without misgiving a system of laws which do not
TIME'S ARROW 69
recognise a directed time; he is shocked that the new
picture should expose this so glaringly.
Without any mystic appeal to consciousness it is
possible to find a direction of time on the four-dimen-
sional map by a study of organisation. Let us draw an
arrow arbitrarily. If as we follow the arrow we find
more and more of the random element in the state of
the world, then the arrow is pointing towards the future;
if the random element decreases the arrow points
towards the past. That is the only distinction known to
physics. This follows at once if our fundamental con-
tention is admitted that the introduction of randomness
is the only thing which cannot be undone.
I shall use the phrase "time's arrow" to express this
one-way property of time which has no analogue in
space. It is a singularly interesting property from a
philosophical standpoint. We must note that —
( 1 ) It is vividly recognised by consciousness.
(2) It is equally insisted on by our reasoning faculty,
which tells us that a reversal of the arrow would render
the external world nonsensical.
(3) It makes no appearance in physical science except
in the study of organisation of a number of individuals.
Here the arrow indicates the direction of progressive
increase of the random element.
Let us now consider in detail how a random element
brings the irrevocable into the world. When a stone
falls it acquires kinetic energy, and the amount of the
energy is just that which would be required to lift the
stone back to its original height. By suitable arrange-
ments the kinetic energy can be made to perform this
task; for example, if the stone is tied to a string it can
alternately fall and reascend like a pendulum. But if
the stone hits an obstacle its kinetic energy is converted
70 THE RUNNING-DOWN OF THE UNIVERSE
into heat-energy. There is still the same quantity of
energy, but even if we could scrape it together and put
it through an engine we could not lift the stone back
with it. What has happened to make the energy no
longer serviceable?
Looking microscopically at the falling stone we see
an enormous multitude of molecules moving downwards
with equal and parallel velocities — an organised motion
like the march of a regiment. We have to notice two
things, the energy and the organisation of the energy.
To return to its original height the stone must preserve
both of them.
When the stone falls on a sufficiently elastic surface
the motion may be reversed without destroying the
organisation. Each molecule is turned backwards and the
whole array retires in good order to the starting-point —
The famous Duke of York
With twenty thousand men,
He marched them up to the top of the hill
And marched them down again.
History is not made that way. But what usually happens
at the impact is that the molecules suffer more or less
random collisions and rebound in all directions. They
no longer conspire to make progress in any one direc-
tion; they have lost their organisation. Afterwards they
continue to collide with one another and keep changing
their directions of motion, but they never again find a
common purpose. Organisation cannot be brought
about by continued shuffling. And so, although the
energy remains quantitatively sufficient (apart from un-
avoidable leakage which we suppose made good), it
cannot lift the stone back. To restore the stone we must
supply extraneous energy which has the required
amount of organisation.
COINCIDENCES 71
Here a point arises which unfortunately has no
analogy in the shuffling of a pack of cards. No one
(except a conjurer) can throw two half-shuffled packs
into a hat and draw out one pack in its original order
and one pack fully shuffled. But we can and do put
partly disorganised energy into a steam-engine, and
draw it out again partly as fully organised energy of
motion of massive bodies and partly as heat-energy in
a state of still worse disorganisation. Organisation of
energy is negotiable, and so is the disorganisation or
random element; disorganisation does not for ever
remain attached to the particular store of energy which
first suffered it, but may be passed on elsewhere. We
cannot here enter into the question why there should be
a difference between the shuffling of energy and the
shuffling of material objects; but it is necessary to use
some caution in applying the analogy on account of this
difference. As regards heat-energy the temperature is
the measure of its degree of organisation; the lower the
temperature, the greater the disorganisation.
Coincidences, There are such things as chance coinci-
dences; that is to say, chance can deceive us by bringing
about conditions which look very unlike chance. In
particular chance might imitate organisation, whereas
we have taken organisation to be the antithesis o{
chance or, as we have called it, the "random element".
This threat to our conclusions is, however, not very
serious. There is safety in numbers.
Suppose that you have a vessel divided by a partition
into two halves, one compartment containing air and
the other empty. You withdraw the partition. For the
moment all the molecules of air are in one half of the
vessel; a fraction of a second later they are spread over
72 THE RUNNING-DOWN OF THE UNIVERSE
the whole vessel and remain so ever afterwards. The
molecules will not return to one half of the vessel; the
spreading cannot be undone — unless other material is
introduced into the problem to serve as a scapegoat for
the disorganisation and carry off the random element
elsewhere. This occurrence can serve as a criterion to
distinguish past and future time. If you observe first
the molecules spread through the vessel and (as it
seems to you) an instant later the molecules all in one
half of it — then your consciousness is going backwards,
and you had better consult a doctor.
Now each molecule is wandering round the vessel
with no preference for one part rather than the other.
on the average it spends half its time in one compart-
ment and half in the other. There is a faint possibility
that at one moment all the molecules might in this way
happen to be visiting the one half of the vessel. You
will easily calculate that if n is the number of molecules
(roughly a quadrillion) the chance of this happening is
( y 2 ) n . The reason why we ignore this chance may be seen
by a rather classical illustration. If I let my fingers
wander idly over the keys of a typewriter it might
happen that my screed made an intelligible sentence.
If an army of monkeys were strumming on typewriters
they might write all the books in the British Museum.
The chance of their doing so is decidedly more favour-
able than the chance of the molecules returning to one
half of the vessel.
When numbers are large, chance is the best warrant
for certainty. Happily in the study of molecules and
energy and radiation in bulk we have to deal with a
vast population, and we reach a certainty which does
not always reward the expectations of those who court
the fickle goddess.
COINCIDENCES 73
In one sense the chance of the molecules returning
to one half of the vessel is too absurdly small to think
about. Yet in science we think about it a great deal,
because it gives a measure of the irrevocable mischief
we did when we casually removed the partition. Even
if we had good reasons for wanting the gas to fill the
vessel there was no need to waste the organisation; as
we have mentioned, it is negotiable and might have been
passed on somewhere where it was useful.* When the
gas was released and began to spread across the vessel,
say from left to right, there was no immediate increase
of the random element. In order to spread from left to
right, left-to-right velocities of the molecules must have
preponderated, that is to say the motion was partly
organised. Organisation of position was replaced by
organisation of motion. A moment later the molecules
struck the farther wall of the vessel and the random
element began to increase. But, before it was destroyed,
the left-to-right organisation of molecular velocities was
the exact numerical equivalent of the lost organisation
in space. By that we mean that the chance against the
left-to-right preponderance of velocity occurring by
accident is the same as the chance against segregation
in one half of the vessel occurring by accident.
The adverse chance here mentioned is a preposterous
number which (written in the usual decimal notation)
would fill all the books in the world many times over.
We are not interested in it as a practical contingency;
but we are interested in the fact that it is definite. It
raises "organisation" from a vague descriptive epithet
to one of the measurable quantities of exact science.
We are confronted with many kinds of organisation.
* If the gas in expanding had been made to move a piston, the
organisation would have passed into the motion of the piston.
74 THE RUNNING-DOWN OF THE UNIVERSE
The uniform march of a regiment is not the only form
of organised motion; the organised evolutions of a
stage chorus have their natural analogue in sound waves.
A common measure can now be applied to all forms
of organisation. Any loss of organisation is equitably
measured by the chance against its recovery by an acci-
dental coincidence. The chance is absurd regarded as
a contingency, but it is precise as a measure.
The practical measure of the random element which
can increase in the universe but can never decrease is
called entropy. Measuring by entropy is the same as
measuring by the chance explained in the last paragraph,
only the unmanageably large numbers are transformed
(by a simple formula) into a more convenient scale of
reckoning. Entropy continually increases. We can,
by isolating parts of the world and postulating rather
idealised conditions in our problems, arrest the increase,
but we cannot turn it into a decrease. That would
involve something much worse than a violation of an
ordinary law of Nature, namely, an improbable coinci-
dence. The law that entropy always increases — the
second law of thermodynamics — holds, I think, the
supreme position among the laws of Nature. If someone
points out to you that your pet theory of the universe
is in disagreement with Maxwell's equations — then so
much the worse for Maxwell's equations. If it is found
to be contradicted by observation — well, these experi-
mentalists do bungle things sometimes. But if your
theory is found to be against the second law of thermo-
dynamics I can give you no hope; there is nothing for
it but to collapse in deepest humiliation. This exaltation
of the second law is not unreasonable. There are other
laws which we have strong reason to believe in, and we
feel that a hypothesis which violates them is highly
PRIMARY AND SECONDARY LAW] 75
improbable; but the improbability is vague and does
not confront us as a paralysing array of figures, whereas
the chance against a breach of the second law (i.e.
against a decrease of the random element) can be stated
in figures which are overwhelming.
I wish I could convey to you the amazing power of
this conception of entropy in scientific research. From
the property that entropy must always increase, practical
methods of measuring it have been found. The chain
of deductions from this simple law have been almost
illimitable; and it has been equally successful in con-
nection with the most recondite problems of theoretical
physics and the practical tasks of the engineer. Its
special feature is that the conclusions are independent
of the nature of the microscopical processes that are
going on. It is not concerned with the nature of the
individual; it is interested in him only as a component
of a crowd. Therefore the method is applicable in
fields of research where our ignorance has scarcely begun
to lift, and we have no hesitation in applying it to prob-
lems of the quantum theory, although the mechanism
of the individual quantum process is unknown and at
present unimaginable.
Primary and Secondary Law. I have called the laws
controlling the behaviour of single individuals "primary
laws", implying that the second law of thermodynamics,
although a recognised law of Nature, is in some sense a
secondary law. This distinction can now be placed on
a regular footing. Some things never happen in the
physical world because they are impossible; others
because they are too improbable. The laws which forbid
the first are the primary laws; the laws which forbid the
second are the secondary laws. It has been the convic-
76 THE RUNNING-DOWN OF THE UNIVERSE
tion of nearly all physicists* that at the root of every-
thing there is a complete scheme of primary law govern-
ing the career of every particle or constituent of the
world with an iron determinism. This primary scheme
is all-sufficing, for, since it fixes the history of every
constituent of the world, it fixes the whole world-
history.
But for all its completeness primary law does not
answer every question about Nature which we might
reasonably wish to put. Can a universe evolve back-
wards, i.e. develop in the opposite way to our own
system? Primary law, being indifferent to a time-
direction, replies, "Yes, it is not impossible". Secondary
law replies, "No, it is too improbable". The answers are
not really in conflict; but the first, though true, rather
misses the point. This is typical of some much more
commonplace queries. If I put this saucepan of water
on this fire, will the water boil? Primary law can answer
definitely if it is given the chance; but it must be
understood that "this" translated into mathematics
means a specification of the positions, motions, etc., of
some quadrillions of particles and elements of energy.
So in practice the question answered is not quite the
one that; is asked: If I put a saucepan resembling this
one in a few major respects on a fire, will the water
boil? Primary law replies, "It may boil; it may freeze; it
may do pretty well anything. The details given are insuf-
ficient to exclude any result as impossible." Secondary
law replies plainly, "It will boil because it is too im-
probable that it should do anything else." Secondary
law is not in conflict with primary law, nor can we re-
gard it as essential to complete a scheme of law already
♦There are, however, others beside myself who have recently begun to
question it.
THERMODYNAMICAL EQUILIBRIUM 77
complete in itself. It results from a different '(and
rather more practical) conception of the aim of our
traffic with the secrets of Nature.
The question whether the second law of thermo-
dynamics and other statistical laws are mathematical
deductions from the primary laws, presenting their
results in a conveniently usable form, is difficult to
answer; but I think it is generally considered that there
is an unbridgeable hiatus. At the bottom of all the
questions settled by secondary law there is an elusive
conception of u a priori probability of states of the
world" which involves an essentially different attitude to
knowledge from that presupposed in the construction of
the scheme of primary law.
Thermo dynamical Equilibrium. Progress of time intro-
duces more and more of the random element into the
constitution of the world. There is less of chance about
the physical universe to-day than there will be to-mor-
row. It is curious that in this very matter-of-fact
branch of physics, developed primarily because of its
importance for engineers, we can scarcely avoid express-
ing ourselves in teleological language. We admit that
the world contains both chance and design, or at any
rate chance and the antithesis of chance. This antithe-
sis is emphasised by our method of measurement of
entropy; we assign to the organisation or non-chance
element a measure which is,* so to speak, proportional
to the strength of our disbelief in a chance origin for it.
"A fortuitous concourse of atoms" — that bugbear of
the theologian — has a very harmless place in orthodox
physics. The physicist is acquainted with it as a much-
prized rarity. Its properties are very distinctive, and
unlike those of the physical world in general. The
78 THE RUNNING-DOWN OF THE UNIVERSE
scientific name for a fortuitous concourse of atoms is
"thermodynamical equilibrium".
Thermodynamical equilibrium is the other case
which we promised to consider in which no increase in
the random element can occur, namely, that in which the
shuffling is already as thorough as possible. We must
isolate a region of the universe, arranging that no
energy can enter or leave it, or at least that any bound-
ary effects are precisely compensated. The conditions
are ideal, but they can be reproduced with sufficient
approximation to make the ideal problem relevant to
practical experiment. A region in the deep interior of
a star is an almost perfect example of thermodynamical
equilibrium. Under these isolated conditions the energy
will be shuffled as it is bandied from matter to aether
and back again, and very soon the shuffling will be
complete.
The possibility of the shuffling becoming complete
is significant. If after shuffling the pack you tear each
card in two, a further shuffling of the half-cards becomes
possible. Tear the cards again and again; each time
there is further scope for the random element to
increase. With infinite divisibility there can be no end
to the shuffling. The experimental fact that a definite
state of equilibrium is rapidly reached indicates that
energy is not infinitely divisible, or at least that it is not
infinitely divided in the natural processes of shuffling.
Historically this is the result from which the quantum
theory first arose. We shall return to it in a later
chapter.
In such a region we lose time's arrow. You remember
that the arrow points in the direction of increase of
the random element. When the random element has
reached its limit and become steady the arrow does not
THERMODYNAMICAL EQUILIBRIUM 79
know which way to point. It would not be true to say
that such a region is timeless; the atoms vibrate as usual
like little clocks; by them we can measure speeds and
durations. Time is still there and retains its ordinary
properties, but it has lost its arrow; like space it extends,
but it does not "go on".
This raises the important question, Is the random
element (measured by the criterion of probability
already discussed) the only feature of the physical world
which can furnish time with an arrow? Up to the
present we have concluded that no arrow can be found
from the behaviour of isolated individuals, but there is
scope for further search among the properties of crowds
beyond the property represented by entropy. To give
an illustration which is perhaps not quite so fantastic
as it sounds, Might not the assemblage become more
and more beautiful (according to some agreed aesthetic
standard) as time proceeds?* The question is answered
by another important law of Nature which runs —
Nothing in the statistics of an assemblage can distin-
guish a direction of time when entropy fails to distinguish
one.
I think that although this law was only discovered in
the last few years there is no serious doubt as to its
truth. It is accepted as fundamental in all modern
studies of atoms and radiation and has proved to be one
of the most powerful weapons of progress in such
researches. It is, of course, one of the secondary laws.
It does not seem to be rigorously deducible from the
second law of thermodynamics, and presumably must
be regarded as an additional secondary law.t
* In a kaleidoscope the shuffling is soon complete and all the patterns
are equal as regards random element, but they differ greatly in elegance.
f The law is so much disguised in the above enunciation that I must
explain to the advanced reader that I am referring to "the Principle of
80 THE RUNNING-DOWN OF THE UNIVERSE
The conclusion is that whereas other statistical
characters besides entropy might perhaps be used to
discriminate time's arrow, they can only succeed when
it succeeds and they fail when it fails. Therefore they
cannot be regarded as independent tests. So far as
physics is concerned time's arrow is a property of
entropy alone.
Are Space and Time Infinite? I suppose that everyone
has at some time plagued his imagination with the
question, Is there an end to space? If space comes to
an end, what is beyond the end? on the other hand the
idea that there is no end, but space beyond space for
ever, is inconceivable. And so the imagination is tossed
to and fro in a dilemma. Prior to the relativity theory
the orthodox view was that space is infinite. No one
can conceive infinite space; w r e had to be content to
admit in the physical world an inconceivable concep-
tion — disquieting but not necessarily illogical. Einstein's
theory now offers a way out of the dilemma. Is space
infinite, or does it come to an end? Neither. Space
is finite but it has no end; "finite but unbounded"
is the usual phrase.
Infinite space cannot be conceived by anybody;
finite but unbounded space is difficult to conceive but
not impossible. I shall not expect you to conceive it;
but you can try. Think first of a circle; or, rather, not
Detailed Balancing." This principle asserts that to every type of process
(however minutely particularised) there is a converse process, and in
thermodynamical equilibrium direct and converse processes occur with
equal frequency. Thus every statistical enumeration of the processes is
unaltered by reversing the time-direction, i.e. interchanging direct and
converse processes. Hence there can be no statistical criterion for a
direction of time when there is thermodynamical equilibrium, i.e. when
entropy is steady and ceases to indicate time's arrow.
ARE SPACE AND TIME INFINITE? 81
the circle, but the line forming its circumference. This
is a finite but endless line. Next think of a sphere — the
surface of a sphere — that also is a region which is
finite but unbounded. The surface of this earth never
comes to a boundary; there is always some country
beyond the point you have reached; all the same there
is not an infinite amount of room on the earth. Now go
one dimension more; circle, sphere — the next thing.
Got that? Now for the real difficulty. Keep a tight hold
of the skin of this hypersphere and imagine that the
inside is not there at all — that the skin exists without
the inside. That is finite but unbounded space.
No; I don't think you have quite kept hold of the
conception. You overbalanced just at the end. It was
not the adding of one more dimension that was the real
difficulty; it was the final taking away of a dimension
that did it. I will tell you what is stopping you. You
are using a conception of space which must have
originated many million years ago and has become
rather firmly embedded in human thought. But the
space of physics ought not to be dominated by this
creation of the dawning mind of an enterprising ape.
Space is not necessarily like this conception; it is like —
whatever we find from experiment it is like. Now the
features of space which we discover by experiment are
extensions, i.e. lengths and distances. So space is like
a network of distances. Distances are linkages whose
intrinsic nature is inscrutable; we do not deny the
inscrutability when we apply measure numbers to them
— 2 yards, 5 miles, etc. — as a kind of code distinction.
We cannot predict out of our inner consciousness the
laws by which code-numbers are distributed among the
different linkages of the network, any more than we
can predict how the code-numbers for electromagnetic
82 THE RUNNING-DOWN OF THE UNIVERSE
force are distributed. Both are a matter for experi-
ment.
If we go a very long way to a point A in one direction
through the universe and a very long way to a point B
in the opposite direction, it is believed that between
A and B there exists a linkage of the kind indicated by
a very small code-number; in other words these points
reached by travelling vast distances in opposite directions
would be found experimentally to be close together.
Why not? This happens when we travel east and west
on the earth. It is true that our traditional inflexible
conception of space refuses to admit it; but there was
once a traditional conception of the earth which refused
to admit circumnavigation. In our approach to the
conception of spherical space the difficult part was to
destroy the inside of the hypersphere leaving only its
three-dimensional surface existing. I do not think that
is so difficult when we conceive space as a network of
distances. The network over the surface constitutes a
self-supporting system of linkage which can be con-
templated without reference to extraneous linkages. We
can knock away the constructional scaffolding which
helped us to approach the conception of this kind of
network of distances without endangering the con-
ception.
We must realise that a scheme of distribution of
inscrutable relations linking points to one another is not
bound to follow any particular preconceived plan, so
that there can be no obstacle to the acceptance of any
scheme indicated by experiment.
We do not yet know what is the radius of spherical
space; it must, of course, be exceedingly great com-
pared with ordinary standards. on rather insecure
evidence it has been estimated to be not many times
ARE SPACE AND TIME INFINITE? 83
greater than the distance of the furthest known nebulae.
But the boundlessness has nothing to do with the
bigness. Space is boundless by re-entrant form not by
great extension. That which is is a shell floating in the
infinitude of that which is not. We say with Hamlet,
"I could be bounded in a nutshell and count myself a
king of infinite space".
But the nightmare of infinity still arises in regard to
time. The world is closed in its space dimensions like
a sphere, but it is open at both ends in the time dimen-
sion. There is a bending round by which East ulti-
mately becomes West, but no bending by which Before
ultimately becomes After.
I am not sure that I am logical but I cannot feel the
difficulty of an infinite future time very seriously. The
difficulty about A.D. co will not happen until we reach
A.D. 00, and presumably in order to reach A.D. 00 the
difficulty must first have been surmounted. It should
also be noted that according to the second law of thermo-
dynamics the whole universe will reach thermodynamical
equilibrium at a not infinitely remote date in the future.
Time's arrow will then be lost altogether and the whole
conception of progress towards a future fades away.
But the difficulty of an infinite past is appalling. It
is inconceivable that we are the heirs of an infinite time
of preparation; it is not less inconceivable that there was
once a moment with no moment preceding it.
This dilemma of the beginning of time would worry
us more were it not shut out by another overwhelming
difficulty lying between us and the infinite past. We
have been studying the running-down of the universe;
if our views are right, somewhere between the beginning
of time and the present day we must place the winding
up of the universe.
$4 THE RUNNING-DOWN OF THE UNIVERSE
Travelling backwards into the past we find a world
with more and more organisation. If there is no barrier
to stop us earlier we must reach a moment when the
energy of the world was wholly organised with none of
the random element in it. It is impossible to go back
any further under the present system of natural law.
I do not think the phrase "wholly organised" begs the
question. The organisation, we are concerned with is
exactly definable, and there is a limit at which it becomes
perfect. There is not an infinite series of states of
higher and still higher organisation; nor, I think, is the
limit one which is ultimately approached more and more
slowly. Complete organisation does not tend to be
more immune from loss than incomplete organisation.
There is no doubt that the scheme of physics as it
has stood for the last three-quarters of a century postu-
lates a date at which either the entities of the universe
were created in a state of high organisation, or pre-
existing entities were endowed with that organisation
which they have been squandering ever since. More-
over, this organisation is admittedly the antithesis of
chance. It is something which could not occur for-
tuitously.
This has long been used as an argument against a
too aggressive materialism. It has been quoted as
scientific proof of the intervention of the Creator at a
time not infinitely remote from to-day. But I am not
advocating that we drew any hasty conclusions from it.
Scientists and theologians alike must regard as some-
what crude the naive theological doctrine which
(suitably disguised) is at present to be found in every
textbook of thermodynamics, namely that some billions
of years ago God wound up the material universe and
has left it to chance ever since. This should be regarded
ARE SPACE AND TIME INFINITE? 85
as the working-hypothesis of thermodynamics rather
than its declaration of faith. It is one of those conclu-
sions from which we can see no logical escape — only
it suffers from the drawback that it is incredible. As a
scientist I simply do not believe that the present order
of things started off with a bang; unscientifically I feel
equally unwilling to accept the implied discontinuity in
the divine nature. But I can make no suggestion to
evade the deadlock.
Turning again to the other end of time, there is one
school of thought which finds very repugnant the idea
of a wearing out of the world. This school is attracted
by various theories of rejuvenescence. Its mascot is the
Phoenix. Stars grow cold and die out. May not two
dead stars collide, and be turned by the energy of the
shock into fiery vapour from which a new sun — with
planets and with life — is born? This theory very
prevalent in the last century is no longer contemplated
seriously by astronomers. There is evidence that the
present stars at any rate are products of one evolutionary
process which swept across primordial matter and
caused it to aggregate ; they were not formed individually
by haphazard collisions having no particular time con-
nection with one another. But the Phoenix complex is
still active. Matter, we believe, is gradually destroyed
and its energy set free in radiation. Is there no coun-
ter-process by which radiation collects in space, evolves
into electrons and protons, and begins star-building all
over again? This is pure speculation and there is not
much to be said on one; side or the other as to its truth.
But I would mildly criticise the mental outlook which
wishes it to be true. However much we eliminate the
minor extravagances of Nature, we do not by these
theories stop the inexorable running-down of the world
86 THE RUNNING-DOWN OF THE UNIVERSE
by loss of organisation and increase of the random
element. Whoever wishes for a universe which can
continue indefinitely in 'activity must lead a crusade
against the second law of thermodynamics; the possi-
bility of re-formation of matter from radiation is not
crucial and we can await conclusions with some indif-
ference.
At present we can see no way in which an attack on
the second law of thermodynamics could possibly
succeed, and I confess that personally I have no great
desire that it should succeed in averting the final
running-down of the universe. I am no Phoenix
worshipper. This is a topic on which science is silent,
and all that one can say is prejudice. But since prejudice
in favour of a never-ending cycle of rebirth of matter
and worlds is often vocal, I may perhaps give voice to
the opposite prejudice. I would feel more content that
the universe should accomplish some great scheme of
evolution and, having achieved whatever may be
achieved, lapse back into chaotic changelessness, than
that its purpose should be banalised by continual
repetition. I am an Evolutionist, not a Multiplicationist.
It seems rather stupid to keep doing the same thing
over and over again.
Chapter V
"BECOMING"
Linkage of Entropy with Becoming. When you say to
yourself, "Every day I grow better and better", science
churlishly replies —
"I see no signs of it. I see you extended as a four-
dimensional worm in space-time; and, although goodness
is not strictly within my province, I will grant that one
end of you is better than the other. But whether you
grow better or worse depends on which way up I hold
you. There is in your consciousness an idea of growth
or 'becoming' which, if it is not illusory, implies that
you have a label 'This side up'. I have searched for
such a label all through the physical world and can find
no trace of it, so I strongly suspect that the label is
non-existent in the world of reality."
That is the reply of science comprised in primary
law. Taking account of secondary law, the reply is
modified a little, though it is still none too gracious —
"I have looked again and, in the course of studying
a property called entropy, I find that the physical world
is marked with an arrow which may possibly be in-
tended to indicate which way up it should be regarded.
With that orientation I find that you really do grow
better. Or, to speak precisely, your good end is in the
part of the world with most entropy and your bad end
in the part with least. Why this arrangement should
be considered more creditable than that of your neigh-
bour who has his good and bad ends the other way
round, I cannot imagine."
A problem here rises before us concerning the
87
88 "BECOMING"
linkage of the symbolic world of physics to the world
of familiar experience. As explained in the Introduction
this question of linkage remains over at the end of the
strictly physical investigations. Our present problem
is to understand the linkage between entropy which
provides time's arrow in the symbolic world and the
experience of growing or becoming which is the inter-
pretation of time's arrow in the familiar world. We
have, I think, shown exhaustively in the last chapter that
the former is the only scientific counterpart to the latter.
But in treating change of entropy as a symbolic
equivalent for the moving on of time familiar to our
minds a double difficulty arises. Firstly, the symbol
seems to be of inappropriate nature; it is an elaborate
mathematical construct, whereas we should expect so
fundamental a conception as "becoming" to be among
the elementary indefinables — the A B C of physics.
Secondly, a symbol does not seem to be quite what is
wanted; we want a significance which can scarcely be
conveyed by a symbol of the customary metrical type —
the recognition of a dynamic quality in external Nature.
We do not "put sense into the world" merely by
recognising that one end of it is more random than the
other; we have to put a genuine significance of "be-
coming" into it and not an artificial symbolic substitute.
The linkage of entropy-change to "becoming"
presents features unlike every other problem of paral-
lelism of the scientific and familiar worlds. The usual
relation is illustrated by the familiar perception of
colour and its scientific equivalent electromagnetic wave-
length. Here there is no question of resemblance
between the underlying physical cause and the mental
sensation which arises. All that we can require of
the symbolic counterpart of colour is that it shall be
ENTROPY AND BECOMING 89
competent to pull the trigger of a (symbolic) nerve. The
physiologist can trace the nerve mechanism up to the
brain; but ultimately there is a hiatus which no one
professes to fill up. Symbolically we may follow the
influences of the physical world up to the door of the
mind; they ring the door-bell and depart.
But the association of "becoming" with entropy-
change is not to be understood in the same way. It
is clearly not sufficient that the change in the random
element of the world should deliver an impulse at the
end of a nerve, leaving the mind to create in response
to this stimulus the fancy that it is turning the reel of
a cinematograph. Unless we have been altogether
misreading the significance of the world outside us —
by interpreting it in terms of evolution and progress,
instead of a static extension — we must regard the
feeling of "becoming" as (in some respects at least) a
true mental insight into the physical condition which
determines it. It is true enough that whether we are
dealing with the experience of "becoming" or with the
more typical sense-experiences of light, sound, smell,
etc., there must always be some point at which we lose
sight of the physical entities ere they arise in new dress
above our mental horizon. But if there is any experience
in which this mystery of mental recognition can be
interpreted as insight rather than image-building, it
should be the experience of "becoming"; because in this
case the elaborate nerve mechanism does not intervene.
That which consciousness is reading off when it feels the
passing moments lies just outside its door. Whereas,
even if we had reason to regard our vivid impression
of colour as insight, it could not be insight into the
electric waves, for these terminate at the retina far from
the seat of consciousness.
9 o "BECOMING"
I am afraid that the average reader will feel impa-
tient with the long-winded discussion I am about to give
concerning the dynamic character of the external world.
"What is all the bother about? Why not make at once
the hypothesis that 'becoming' is a kind of one-way
texture involved fundamentally in the structure of
Nature? The mind is cognisant of this texture (as it is
cognisant of other features of the physical world) and
apprehends it as the passing on of time — a fairly correct
appreciation of its actual nature. As a result of this
one-way texture the random element increases steadily
in the direction of the grain, and thus conveniendy
provides the physicist with an experimental criterion for
determining the way of the grain; but it is the grain
and not this particular consequence of it which is the
direct physical counterpart of 'becoming'. It may be
difficult to find a rigorous proof of this hypothesis; but
after all we have generally to be content with hypotheses
that rest only on plausibility."
This is in fact the kind of idea which I wish to
advocate; but the "average reader" has probably not
appreciated that before the physicist can admit it, a
delicate situation concerning the limits of scientific
method and the underlying basis of physical law has to
be faced. It is one thing to introduce a plausible
hypothesis in order to explain observational phenomena;
it is another thing to introduce it in order to give the
world outside us a significant or purposive meaning,
however strongly that meaning may be insisted on by
something in our conscious nature. From the side of
scientific investigation we recognise only the progressive
change in the random element from the end of the world
with least randomness to the end with most; that in itself
gives no ground for suspecting any kind of dynamical
ENTROPY AND BECOMING 91
meaning. The view here advocated is tantamount to an
admission that consciousness, looking out through a pri-
vate door, can learn by direct insight an underlying char-
acter of the world which physical measurements do not
betray.
In any attempt to bridge the domains of experience
belonging to the spiritual and physical sides of our na-
ture, Time occupies the key position. I have already re-
ferred to its dual entry into our consciousness — through
the sense organs which relate it to the other entities of
the physical world, and directly through a kind of pri-
vate door into the mind. The physicist, whose method
of inquiry depends on sharpening up our sense organs by
auxiliary apparatus of precision, naturally does not look
kindly v on private doors, through which all formsl of
superstitious fancy might enter unchecked. But is he
ready to forgo that knowledge of the going on of time
which has reached us through the door, and content
himself with the time inferred from sense-impressions
which is emaciated of all dynamic quality?
No doubt some will reply that they are content; to
these I would say — Then show your good faith by
reversing the dynamic quality of time (which you may
freely do if it has no importance in Nature), and, just
for a change, give us a picture of the universe passing
from the more random to the less random state, each
step showing a gradual victory of antichance over
chance. If you are a biologist, teach us how from Man
and a myriad other primitive forms of life, Nature in
the course of ages achieved the sublimely simple struc-
ture of the amoeba. If you are an astronomer, tell how
waves of light hurry in from the depths of space and
condense on to the stars; how the complex solar system
unwinds itself into the evenness of a nebula. Is this the
92 "BECOMING"
enlightened outlook which you wish to substitute for
the first chapter of Genesis? If you genuinely believe
that a contra-evolutionary theory is just as true and as
significant as an evolutionary theory, surely it is time that
a protest should be made against the entirely one-sided
version currently taught.
Dynamic Quality of the External World. But for our
ulterior conviction of the dynamic quality of time, it
would be possible to take the view that ''becoming" is
purely subjective — that there is no "becoming" in the
external world which lies passively spread out in the
time-dimension as Minkowski pictured it. My con-
sciousness then invents its own serial order for the sense
impressions belonging to the different view-points along
the track in the external world, occupied by the four-
dimensional worm who is in some mysterious way
Myself; and in focussing the sensations of a particular
view-point I get the illusion that the corresponding
external events are "taking place". I suppose that this
would be adequate to account for the observed phe-
nomena. The objections to it hinge on the fact that it
leaves the external world without any dynamic quality
intrinsic to it.
It is useful to recognise how some of our most ele-
mentary reasoning tacitly assumes the existence of this
dynamic quality or trend; to eradicate it would almost
paralyse our faculties of inference. In the operation of
shuffling cards it seems axiomatic that the cards must
be in greater disarrangement at a later instant. Can
you conceive Nature to be such that this is not obviously
true? But what do we here mean by "later"? So far
as the axiomatic character of the conclusion is concerned
DYNAMIC QUALITY OF THE WORLD 93
(not its experimental verification) we cannot mean
"later" as judged by consciousness; its obviousness is
not bound up with any speculations as to the behaviour
of consciousness. Do we then mean "later" as judged
by the physical criterion of time's arrow, i.e. corre-
sponding to a greater proportion of the random element?
But that would be tautological — the cards are more
disarranged when there is more of the random element.
We did not mean a tautology; we unwittingly accepted
as a basis for our thought about the question an unam-
biguous trend from past to future in the space-time where
the operation of shuffling is performed.
The crux of the matter is that, although a change
described as sorting is the exact opposite to a change
described as shuffling we cannot imagine a cause of
sorting to be the exact opposite of a cause of shuffling.
Thus a reversal of the time-direction which turns
shuffling into sorting does not make the appropriate
transformation of their causes. Shuffling can have in-
organic causes, but sorting is the prerogative of mind or
instinct. We cannot believe that it is merely an orienta-
tion with respect to the time-direction which differentiates
us from inorganic nature. Shuffling is related to sorting
(so far as the change of configuration is concerned) as
plus is to minus; but to say that the cause of shuffling
is related to the cause of sorting in the same way would
seem equivalent to saying that the activities of matter
and mind are related like plus and minus — which
surely is nonsense. Hence if we view the world from
future to past so that shuffling and sorting are inter-
changed, their causes do not follow suit, and the rational
connection is broken. To restore coherency we must
postulate that by this change of direction something
else has been reversed, viz. the trend in world-texture
94 "BECOMING"
spoken of above; "becoming" has been turned into
"unbecoming". If we like we can now go on to account,
not for things becoming unshuffled, but for their un-
becoming shuffled — and, if we wish to pursue this aspect
further, we must discuss not the causes but the un-
causes. But, without tying ourselves into verbal knots,
the meaning evidently is that "becoming" gives a
texture to the world which it is illegitimate to reverse.
Objectivity of Becoming. In general we should describe
the familiar world as subjective and the scientific world
as objective. Take for instance our former example of
parallelism, viz. colour in the familiar world and its
counterpart electromagnetic wave-length in the scientific
world. Here we have little hesitation in describing the
waves as objective and the colour as subjective. The
wave is the reality — or the nearest we can get to a
description of reality; the colour is mere mind-spinning.
The beautiful hues which flood our consciousness under
stimulation of the waves have no relevance to the ob-
jective reality. For a colour-blind person the hues are
different; and although persons of normal sight make
the same distinctions of colour, we cannot ascertain
whether their consciousness of red, blue, etc. is just like
our own. Moreover, we recognise that the longer and
shorter electromagnetic waves which have no visual
effect associated with them are just as real as the col-
oured waves. In this and other parallelisms we find the
objective in the scientific world and the subjective in the
familiar world.
But in the parallelism between entropy-gradient and
"becoming" the subjective and objective seem to have
got on to the wrong sides. Surely "becoming" is a
reality — or the nearest we can get to a description of
OBJECTIVITY OF BECOMING 95
reality. We are convinced that a dynamic character
must be attributed to the external world; making all
allowance for mental imagery, I do not see how the
essence of "becoming" can be much different from
what it appears to us to be. on the other side we have
entropy which is frankly of a much more subjective
nature than most of the ordinary physical qualities.
Entropy is an appreciation of arrangement and or-
ganisation; it is subjective in the same sense that the
constellation Orion is subjective. That which is arranged
is objective, so too are the stars composing the con-
stellation; but the association is the contribution of the
mind which surveys. If colour is mind-spinning, so also
is entropy a mind-spinning — of the statistician. It has
about as much objectivity as a batting average.
Whilst the physicist would generally say that the
matter of this familiar table is really a curvature of
space, and its colour is really electromagnetic wave-
length, I do not think he would say that the familiar
moving on of time is really an entropy-gradient. I am
quoting a rather loose way of speaking; but it reveals
that there is a distinct difference in our attitude towards
the last parallelism. Having convinced ourselves that
the two things are connected, we must conclude that
there is something as yet ungrasped behind the notion
of entropy — some mystic interpretation, if you like —
which is not apparent in the definition by which we
introduced it into physics. " In short we strive to see
that entropy-gradient may really be the moving on of
time (instead of vice versa).
Before passing on I would note that this exceptional
appearance of subjective and objective apparently in
their wrong worlds gives food for thought. It may
prepare us for a view of the scientific world adopted in
96 "BECOMING"
the later chapters which is much more subjective than
that usually held by science.
The more closely we examine the association of
entropy with "becoming" the greater do the obstacles
appear. If entropy were one of the elementary in-
definables of physics there would be no difficulty. Or
if the moving on of time were something of which we
were made aware through our sense organs there would
be no difficulty. But the actual combination which we
have to face seems to be unique in its difficulty.
Suppose that we had had to identify "becoming"
with electrical potential-gradient instead of with en-
tropy-change. We discover potential through the
readings of a voltmeter. The numerical reading stands
for something in the condition of the world, but we form
no picture of what that something is. In scientific
researches we only make use of the numerical value —
a code-number attached to a background outside all
conception. It would be very interesting if we could
relate this mysterious potential to any of our familiar
conceptions. Clearly, if we could identify the change
of potential with the familiar moving on of time, we
should have made a great step towards grasping its
intrinsic nature. But turning from supposition to fact,
we have to identify potential-gradient with force. Now
it is true that we have a familiar conception of force —
a sensation of muscular effort. But this does not give
us any idea of the intrinsic nature of potential-gradient;
the sensation is mere mind-spinning provoked by
nervous impulses which have travelled a long way from
the seat of the force. That is the way with all physical
entities which affect the mind through the sense organs.
The interposed nerve-mechanism would prevent any close
association of the mental image with the physical cause,
OBJECTIVITY OF BECOMING 97
even if we were disposed to trust our mental insight when
it has a chance of operating directly.
Or suppose that we had had to identify force with
entropy-gradient. That would only mean that entropy-
gradient is a condition which stimulates a nerve, which
thereupon transmits an impulse to the brain, out of
which the mind weaves its own peculiar impression of
force. No one would feel intuitive objection to the
hypothesis that the muscular sensation of force is
associated with change of organisation of the molecules
of the muscle.
Our trouble is that we have to associate two things,
both of which we more or less understand, and, so far
as we understand them, they are utterly different. It
is absurd to pretend that we are in ignorance of the
nature of organisation in the external world in the same
way that we are ignorant of the intrinsic nature of
potential. It is absurd to pretend that we have no
justifiable conception of "becoming" in the external
world. That dynamic quality — that significance which
makes a development from past to future reasonable
and a development from future to past farcical — has to
do much more than pull the trigger of a nerve. It is so
welded into our consciousness that a moving on of
time is a condition of consciousness. We have direct
insight into "becoming" which sweeps aside all sym-
bolic knowledge as on an inferior plane. If I grasp the
notion of existence because I myself exist, I grasp the no-
tion of becoming because I myself become. It is the in-
nermost Ego of all which is and becomes.
The incongruity of symbolising this fundamental
intuition by a property of arrangement of the micro-
scopic constituents of the world, is evident. What this
difficulty portends is still very obscure. But it is not
9 8 "BECOMING"
irrelevant to certain signs of change which we may
discern in responsible scientific opinion with regard to
the question of primary and secondary law. The cast-
iron determinism of primary law is, I think, still widely
accepted but no longer unquestioningly. It now seems
clear that we have not yet got hold of any primary law
— that all those laws at one time supposed to be primary
are in reality statistical. No doubt it will be said that
that was only to be expected; we must be prepared for
a very long search before we get down to ultimate
foundations, and not be disappointed if new discoveries
reveal unsuspected depths beneath. But I think it might
be said that Nature has been caught using rather unfair
dodges to prevent our discovering primary law — that
kind of artfulness which frustrated our efforts to discover
velocity relative to the aether.* I believe that Nature is
honest at heart, and that she only resorts to these ap-
parent shifts of concealment when we are looking for
something which is not there. It is difficult to see now
any justification for the strongly rooted conviction in the
ultimate re-establishment of a deterministic scheme of
law except a supposed necessity of thought. Thought
has grown accustomed to doing without a great many
"necessities" in recent years.
one would not be surprised if in the reconstruction
of the scheme of physics which the quantum theory is
now pressing on us, secondary law becomes the basis
and primary law is discarded. In the reconstructed
world nothing is impossible though many things are
improbable. The effect is much the same, but the kind
of machinery that we must conceive is altogether
different. We shall have further glimpses of this problem
and I will not here pursue it. Entropy, being a quantity
* See p. 23i.
OUR DUAL RECOGNITION OF TIME 99
introduced in connection with secondary law will now
exist, so to speak, in its own right instead of by its
current representation as arrangement of the quantities
in the abandoned primary scheme; and in that right it
may be more easily accepted as the symbol for the
dynamic quality of the world. I cannot make my
meaning more precise, because I am speaking of a
still hypothetical change of ideas which no one has been
able to bring about.
Our Dual Recognition of Time, Another curiosity which
strikes us is the divorce in physics between time and
time's arrow. A being from another world who wishes
to discover the temporal relation of two events in this
world has to read two different indicators. He must
read a clock in order to find out how much later one
event is than the other, and he must read some arrange-
ment for measuring the disorganisation of energy (e.g. a
thermometer) in order to discover which event is the
later.* The division of labour is especially striking
when we remember that our best clocks are those in
which all processes such as friction, which introduce
disorganisation of energy, are eliminated as far as
possible. The more perfect the instrument as a meas-
urer of time, the more completely does it conceal time's
arrow.
* To make the test strictly from another world he must not assume
that the figures marked on the clock-dial necessarily go the right way
round; nor must he assume that the progress of his consciousness has
any relation to the flow of time in our world. He has, therefore, merely
two dial-readings for the two events without knowing whether the
difference should be reckoned plus or minus. The thermometer would
be used in conjunction with a hot and cold body in contact. The differ-
ence of the thermometer readings for the two bodies would be taken at
the moment of each event. The event for which the difference is smaller
is the later.
ioo "BECOMING"
This paradox seems to be explained by the fact
pointed out in chapter in that time comes into our
consciousness by two routes. We picture the mind like
an editor in his sanctum receiving through the nerves
scrappy messages from all over the outside world, and
making a story of them with, I fear, a good deal of
editorial invention. Like other physical quantities time
enters in that way as a particular measurable relation
between events in the outside world; but it comes in
without its arrow. In addition our editor himself ex-
periences a time in his consciousness — the temporal
relation along his own track through the world. This
experience is immediate, not a message from outside,
but the editor realises that what he is experiencing is
equivalent to the time described in the messages. Now
consciousness declares that this private time possesses
an arrow, and so gives a hint to search further for the
missing arrow among the messages. The curious thing
is that, although the arrow is ultimately found among
the messages from outside, it is not found in the mes-
sages from clocks, but in messages from thermometers
and the like instruments which do not ordinarily pretend
to measure time.
Consciousness, besides detecting time's arrow, also
roughly measures the passage of time. It has the right
idea of time-measurement, but is a bit of a bungler in
carrying it out. Our consciousness somehow manages
to keep in close touch with the material world, and we
must suppose that its record of the flight of time is the
reading of some kind of a clock in the material of the
brain — possibly a clock which is a rather bad time-
keeper. I have generally had in mind in this connection
an analogy with the clocks of physics designed for good
time-keeping; but I am now inclined to think that a
OUR DUAL RECOGNITION OF TIME 101
better analogy would be an entropy-clock, i.e. an in-
strument designed primarily for measuring the rate of
disorganisation of energy, and only very roughly keep-
ing pace with time.
A typical entropy-clock might be designed as follows.
An electric circuit is composed of two different metals
with their two junctions embedded respectively in a
hot and cold body in contact. The circuit contains a
galvanometer which constitutes the dial of the entropy-
clock. The thermoelectric current in the circuit is
proportional to the difference of temperature of the two
bodies; so that as the shuffling of energy between them
proceeds, the temperature difference decreases and the
galvanometer reading continually decreases. This clock
will infallibly tell an observer from another world which
of two events is the later. We have seen that no ordi-
nary clock can do this. As to its time-keeping qualities
we can only say that the motion of the galvanometer
needle has some connection with the rate of passage of
time — which is perhaps as much as can be said for the
time-keeping qualities of consciousness.
It seems to me, therefore, that consciousness with its
insistence on time's arrow and its rather erratic ideas of
time measurement may be guided by entropy-clocks in
some portion of the brain. That avoids the unnatural
assumption that we consult two different cells of the
material brain in forming our ideas of duration and of
becoming, respectively. Entropy-gradient is then the
direct equivalent of the time of consciousness in both
its aspects. Duration measured by physical clocks (time-
like interval) is only remotely connected.
Let us try to clear up our ideas of time by a summary
of the position now reached. Firstly, physical time is a
102 "BECOMING"
system of partitions in the four-dimensional world
(world-wide instants). These are artificial and relative
and by no means correspond to anything indicated to
us by the time of consciousness. Secondly, we recognise
in the relativity theory something called a temporal
relation which is absolutely distinct from a spatial
relation. one consequence of this distinction is that the
mind attached to a material body can only traverse a
temporal relation; so that, even if there is no closer
connection, there is at least a one-to-one correspondence
between the sequence of phases of the mind and a
sequence of points in temporal relation. Since the mind
interprets its own sequence as a time of consciousness, we
can at least say that the temporal relation in physics
has a connection with the time of consciousness which
the spatial relation does not possess. I doubt if the
connection is any closer. I do not think the mental
sequence is a "reading off" of the physical temporal
relation, because in physics the temporal relation is
arrowless. I think it is a reading off of the physical
entropy-gradient, since this has the necessary arrow.
Temporal relation and entropy-gradient, both rigorously
defined in physics, are entirely distinct and in general
are not numerically related. But, of course, other things
besides time can "keep time"; and there is no reason
why the generation of the random element in a special
locality of the brain should not proceed fairly uniformly.
In that case there will not be too great a divergence
between the passage of time in consciousness and the
length of the corresponding temporal relation in the
physical world.
THE REACTION FROM ANALYSIS 103
The Scientific Reaction from Microscopic Analysis. From
the point of view of philosophy of science the con-
ception associated with entropy must I think be ranked
as the great contribution of the nineteenth century to
scientific thought. It marked a reaction from the view
that everything to which science need pay attention is
discovered by a microscopic dissection of objects. It
provided an alternative standpoint in which the centre
of interest is shifted from the entities reached by the
customary analysis (atoms, electric potentials, etc.) to
qualities possessed by the system as a whole, which
cannot be split up and located — a little bit here, and a
little bit there. The artist desires to convey significances
which cannot be told by microscopic detail and accord-
ingly he resorts to impressionist painting. Strangely
enough the physicist has found the same necessity; but
his impressionist scheme is just as much exact science
and even more practical in its application than his micro-
scopic scheme.
Thus in the study of the falling stone the microscopic
analysis reveals myriads of separate molecules. The
energy of the stone is distributed among the molecules,
the sum of the energies of the molecules making up the
energy of the stone. But we cannot distribute in that
way the organisation or the random element in the
motions. It would be meaningless to say that a particu-
lar fraction of the organisation is located in a par-
ticular molecule.
There is one ideal of survey which would look into
each minute compartment of space in turn to see what
it may contain and so make what it would regard as
a complete inventory of the world. But this misses
any world-features which are not located in minute
compartments. We often think that when we have
104 "BECOMING"
completed our study of one we know all about two> be-
cause "two" is one and one". We forget that we have
still to make a study of "and". Secondary physics is
the study of "and" — that is to say, of organisation.
Thanks to clear-sighted pioneers in the last century
science became aware that it was missing something of
practical importance by following the inventory method
of the primary scheme of physics. Entropy became
recognised although it was not found in any of the com-
partments. It was discovered and exalted because it was
essential to practical applications of physics, not to
satisfy any philosophic hungering. But by it science
has been saved from a fatal narrowness. If we had kept
entirely to the inventory method, there would have been
nothing to represent "becoming" in the physical world.
And science, having searched high and low, would
doubtless have reported that "becoming" is an un-
founded mental illusion — like beauty, life, the soul, and
other things which it is unable to inventory.
I think that doubts might well have been entertained
as to whether the newcomer was strictly scientific.
Entropy was not in the same category as the other
physical quantities recognised in science, and the ex-
tension — as we shall presently see — was in a very
dangerous direction. once you admit attributes of
arrangement as subject-matter of physics, it is difficult
to draw the line. But entropy had secured a firm place
in physics before it was discovered that it was a measure
of the random element in arrangement. It was in great
favour with the engineers. Their sponsorship was the
highest testimonial to its good character; because at that
time it was the general assumption that the Creation was
the work of an engineer (not of a mathematician, as is
the fashion nowadays).
THE REACTION FROM ANALYSIS 105
Suppose that we were asked to arrange the following
in two categories —
distance, mass, electric force, entropy, beauty, melody.
I think there are the strongest grounds for placing
entropy alongside beauty and melody and not with the
first three. Entropy is only found when the parts are
viewed in association, and it is by viewing or hearing
the parts in association that beauty and melody are
discerned. All three are features of arrangement. It is
a pregnant thought that one of these three associates
should be able to figure as a commonplace quantity of
science. The reason why this stranger can pass itself
off among the aborigines of the physical world is, that
it is able to speak their language, viz. the language of
arithmetic. It has a measure-number associated with it
and so is made quite at home in physics. Beauty and
melody have not the arithmetical pass-word and so are
barred out. This teaches us that what exact science looks
out for is not entities of some particular category, but
entities with a metrical aspect. We shall see in a later
chapter that when science admits them it really admits
only their metrical aspect and occupies itself solely with
that. It would be no use for beauty, say, to fake up a
few numerical attributes (expressing for instance the
ideal proportions of symmetry) in the hope of thereby
gaining admission into the portals of science and carrying
on an aesthetic crusade within. It would find that the
numerical aspects were duly admitted, but the aesthetic
significance of them left outside. So also entropy is
admitted in its numerical aspect; if it has as we faintly
suspect some deeper significance touching that which
appears in our consciousness as purpose (opposed to
chance) , that significance is left outside. These fare no
106 "BECOMING"
worse than mass, distance, and the like which surely
must have some significance beyond mere numbers; if
so, that significance is lost on their incorporation into
the scientific scheme — the world of shadows.
You may be inclined to regard my insistence that
entropy is something excluded from the inventory of
microscopic contents of the world as word-splitting. If
you have all the individuals before you, their associations,
arrangement and organisation are automatically before
you. If you have the stars, you have the constellations.
Yes; but if you have the stars, you do not take the
constellations seriously. It had become the regular
outlook of science, closely associated with its materialistic
tendencies, that constellations are not to be taken
seriously, until the constellation of entropy made a
solitary exception. When we analyse the picture into
a large number of particles of paint, we lose the aes-
thetic significance of the picture. The particles of paint
go into the scientific inventory, and it is claimed that
everything that there really was in the picture is kept.
But this way of keeping a thing may be much the same
as losing it. The essence of a picture (as distinct from
the paint) is arrangement. Is arrangement kept or lost?
The current answer seems inconsistent. In so far as
arrangement signifies a picture, it is lost; science has
to do with paint, not pictures. In so far as arrangement
signifies organisation it is kept; science has much to do
with organisation. Why should we (speaking now as
philosophers, not scientists) make a discrimination
between these two aspects of arrangement? The dis-
crimination is made because the picture is no use to the
scientist — he cannot get further with it. As impartial
judges it is our duty to point out that likewise entropy
is no use to the artist — he cannot develop his outlook
with it.
INSUFFICIENCY OF PRIMARY LAW 107
I am not trying to argue that there is in the external
world an objective entity which is the picture as distinct
from the myriads of particles into which science has
analysed it. I doubt if the statement has any meaning;
nor, if it were true, would it particularly enhance my
esteem of the picture. What I would say is this:
There is a side of our personality which impels us to
dwell on beauty and other aesthetic significances in
Nature, and in the work of man, so that our environ-
ment means to us much that is not warranted by any-
thing found in the scientific inventory of its struc-
ture. An overwhelming feeling tells us that this is
right and indispensable to the purpose of our existence.
But is it rational? How can reason regard it otherwise
than as a perverse misrepresentation of what is after all
only a collection of atoms, aether-waves and the like,
going about their business? If the physicist as advocate
for reason takes this line, just whisper to him the word
Entropy.
Insufficiency of Primary Law. I daresay many of my
physical colleagues will join issue with me over the
status I have allowed to entropy as something foreign
to the microscopic scheme, but essential to the physical
world. They would regard it rather as a labour-saving
device, useful but not indispensable. Given any practical
problem ordinarily solved by introducing the conception
of entropy, precisely the same result could be reached
(more laboriously) by following out the motion of each
individual particle of matter or quantum of energy under
the primary microscopic laws without any reference to
entropy explicit or implicit. Very well ; let us try. There's
a problem for you —
[A piece of chalk was thrown on the lecture table
where it rolled and broke into two pieces.]
108 "BECOMING"
You are given the instantaneous position and velocity*
of every molecule, or if you like every proton and
electron, in those pieces of chalk and in as much of the
table and surrounding air as concerns you. Details of
the instantaneous state of every element of energy are
also given. By the microscopic (primary) laws of mo-
tion you can trace the state from instant to instant.
You can trace how the atoms moving aimlessly within
the lumps of chalk gradually form a conspiracy so that
the lumps begin to move as a whole. The lumps bounce
a little and roll on the table; they come together and
join up; then the whole piece of chalk rises gracefully
in the air, describes a parabola, and comes to rest be-
tween my fingers. I grant that you can do all that with-
out requiring entropy or anything outside the limits of
microscopic physics. You have solved the problem.
But, have you quite got hold of the significance of your
solution? Is it quite a negligible point that what you
have described from your calculations is an unhappen-
ingf There is no need to alter a word of your descrip-
tion so far as it goes; but it does seem to need an
addendum which would discriminate between a trick
worthy of Mr. Maskelyne and an ordinary everyday
unoccurrence.
The physicist may say that the addendum asked for
relates to significance, and he has nothing to do with
significances; he is only concerned that his calculations
shall agree with observation. He cannot tell me whether
the phenomenon has the significance of a happening or
an unhappening; but if a clock is included in the
* Velocities are relative to a frame of space and time. Indicate which
frame you prefer, and you will be given velocity relative to that frame.
(This throws on you the responsibility for any labelling of the frame —
left, right, past future* etc.)
INSUFFICIENCY OF PRIMARY LAW 109
problem he can give the readings of the clock at each
stage. There is much to be said for excluding the whole
field of significance from physics; it is a healthy reaction
against mixing up with our calculations mystic con-
ceptions that (officially) we know nothing about.
I rather envy the pure physicist his impregnable position.
But if he rules significances entirely outside his scope,
somebody has the job of discovering whether the physi-
cal world of atoms, aether and electrons has any signifi-
cance whatever. Unfortunately for me I am expected in
these lectures to say how the plain man ought to regard
the scientific world when it comes into competition with
other views of our environment. Some of my audience
may not be interested in a world invented as a mere
calculating device. Am I to tell them that the scientific
world has no claim on their consideration when the eter-
nal question surges in the mind, What is it all about? I
am sure my physical colleagues will expect me to put up
some defence of the scientific world in this connection.
I am ready to do so; only I must insist as a preliminary
that we should settle which is the right way up of it.
I cannot read any significance into a physical world
when it is held before me upside down, as happened
just now. For that reason I am interested in entropy
not only because it shortens calculations which can be
made by other methods, but because it determines an
orientation which cannot be found by other methods.
The scientific world is, as I have often repeated, a
shadow-world, shadowing a world familiar to our con-
sciousness. Just how much do we expect it to shadow?
We do not expect it to shadow all that is in our mind,
emotions, memory, etc. In the main we expect it to
shadow impressions which can be traced to external
sense-organs. But time makes a dual entry and thus
no "BECOMING"
t
forms an intermediate link between the internal and the
external. This is shadowed partially by the scientific
world of primary physics (which excludes time's ar-
row), but fully when we enlarge the scheme to include
entropy. Therefore by the momentous departure in the
nineteenth century the scientific world is not confined to
a static extension around which the mind may spin a
romance of activity and evolution; it shadows that
dynamic quality of the familiar world which cannot be
parted from it without disaster to its significance.
In sorting out the confused data of our experience it
has generally been assumed that the object of the quest
is to find out all that really exists. There is another
quest not less appropriate to the nature of our experience
— to find out all that really becomes.
Chapter VI
GRAVITATION— THE LAW
You sometimes speak of gravity as essential and inherent to matter. Pray
do not ascribe that notion to me; for the cause of gravity is what I do not
pretend to know, and therefore would take more time to consider of
it. . . .
Gravity must be caused by some agent acting constantly according
to certain laws; but whether this agent be material or immaterial I have
left to the consideration of my readers.
Newton, Letters to Bentley.
The Man in the Lift. About 19 15 Einstein made a
further development of his theory of relativity extending
it to non-uniform motion. The easiest way to approach
this subject is by considering the Man in the Lift.
Suppose that this room is a lift. The support breaks
and down we go with ever-increasing velocity, falling
freely.
Let us pass the time by performing physical experi-
ments. The lift is our laboratory and we shall start at
the beginning and try to discover all the laws of Nature
— that is to say, Nature as interpreted by the Man in
the Lift. To a considerable extent this will be a repeti-
tion of the history of scientific discovery already made
in the laboratories on terra firma. But there is one
notable difference.
I perform the experiment of dropping an apple held
in the hand. The apple cannot fall any more than it
was doing already. You remember that our lift and all
things contained in it are falling freely. Consequently
the apple remains poised 'by my hand. There is one
incident in the history of science which will not repeat
itself to the men in the lift, viz. Newton and the apple
tree. The magnificent conception that the agent which
in
ii2 GRAVITATION— THE LAW
guides the stars in their courses is the same as that
which in our common experience causes apples to drop,
breaks down because it is our common experience in the
lift that apples do not drop.
I think we have now sufficient evidence to prove that
in all other respects the scientific laws determined in
the lift will agree with those determined under more
orthodox conditions. But for this one omission the men
in the lift will derive all the laws of Nature with which
w r e are acquainted, and derive them in the same form
that we have derived them. only the force which
causes apples to fall is not present in their scheme.
I am crediting our observers in the lift with the usuai
egocentric attitude, viz. the aspect of the world to me
is its natural one. It does not strike them as odd to
spend their lives falling in a lift; they think it much
more odd to be perched on the earth's surface. There-
fore although they perhaps have calculated that to beings
supported in this strange way apples would seem to
have a perplexing habit of falling, they do not take our
experience of the ways of apples any more seriously
than we have hitherto taken theirs.
Are we to take their experience seriously? Or to put
it another way — What is the comparative importance to
be attached to a scheme of natural laws worked out by
observers in the falling lift and one worked out by
observers on terra ftrmal Is one truer than the other?
Is one superior to the other? Clearly the difference if
any arises from the fact that the schemes are referred
to different frames of space and time. Our frame is a
frame in which the solid ground is at rest; similarly their
frame is a frame in which their lift is at rest. We have
had examples before of observers using different frames,
but those frames differed by a uniform velocity. The
THE MAN IN THE LIFT 113
velocity of the lift is ever-increasing — accelerated. Can
we extend to accelerated frames our principle that
Nature is indifferent to frames of space and time, so
that no one frame is superior to any other? I think we
can. The only doubt that arises is whether we should
not regard the frame of the man in the lift as superior
to, instead of being merely coequal with, our usual
frame.
When we stand on the ground the molecules of the
ground support us by hammering on the soles of our
boots with force equivalent to some ten stone weight.
But for this we should sink through the interstices of
the floor. We are being continuously and vigorously
buffeted. Now this can scarcely be regarded as the ideal
condition for a judicial contemplation of our natural
surroundings, and it would not be surprising if our
senses suffering from this treatment gave a jaundiced
view of the world. Our bodies are to be regarded as
scientific instruments used to survey the world. We
should not willingly allow anyone to hammer on a
galvanometer when it was being used for observation;
and similarly it is preferable to avoid a hammering on
one's body when it is being used as a channel of scien-
tific knowledge. We get rid of this hammering when
we cease to be supported.
Let us then take a leap over a precipice so that we
may contemplate Nature undisturbed. Or if that seems
to you an odd way of convincing yourself that bodies do
not fall,* let us enter the runaway lift again. Here
nothing need be supported; our bodies, our galvano-
* So far as I can tell (without experimental trial) the man who jumped
over a precipice would soon lose all conception of falling; he would only
notice that the surrounding objects were impelled past him with ever-
increasing speed.
ii 4 GRAVITATION— THE LAW
meters, and all measuring apparatus are relieved of
hammering and their indications can be received without
misgiving. The space- and time-frame of the falling lift
is the frame natural to observers who are unsupported;
and the laws of Nature determined in these favourable
circumstances should at least have not inferior status to
those established by reference to other frames.
I perform another experiment. This time I take two
apples and drop them at opposite ends of the lift. What
will happen? Nothing much at first; the apples remain
poised where they were let go. But let us step outside
the lift for a moment to watch the experiment. The two
apples are pulled by gravity towards the centre of the
earth. As they approach the centre their paths con-
verge and they will meet at the centre. Now step back
into the lift again. To a first approximation the apples
remain poised above the floor of the lift; but presently
we notice that they are drifting towards one another,
and they will meet at the moment when (according to
an outside observer) the lift is passing through the
centre of the earth. Even though apples (in the lift)
do not tend to fall to the floor there is still a mystery
about their behaviour; and the Newton of the lift may
yet find that the agent which guides the stars in their
courses is to be identified with the agent which plays
these tricks with apples nearer home.
It comes to this. There are both relative and absolute
features about gravitation. The feature that impresses
us most is relative — relative to a frame that has no
special importance apart from the fact that it is the one
commonly used by us. This feature disappears alto-
gether in the frame of the man in the lift and we ought
to disregard it in any attempt to form an absolute pic-
ture of gravitation. But there always remains something
A NEW PICTURE OF GRAVITATION 115
absolute, of which we must try to devise an appropriate
picture. For reasons which I shall presently explain we
find that it can be pictured as a curvature of space and
time.
A New Picture of Gravitation. The Newtonian picture
of gravitation is a tug applied to the body whose path
is disturbed. I want to explain why this picture must
be superseded. I must refer again to the famous incident
in which Newton and the apple-tree were concerned.
The classical conception of gravitation is based on New-
ton's account of what happened; but it is time to hear
what the apple had to say. The apple with the usual
egotism of an observer deemed itself to be at rest;
looking down it saw the various terrestrial objects includ-
ing Newton rushing upwards with accelerated velocity
to meet it. Does it invent a mysterious agency or tug
to account for their conduct? No; it points out that
the cause of their acceleration is quite evident. Newton
is being hammered by the molecules of the ground
underneath him. This hammering is absolute — no ques-
tion of frames of reference. With a powerful enough
magnifying appliance anyone can see the molecules at
work and count their blows. According to Newton's
own law of motion this must give him an acceleration,
which is precisely what the apple has observed. New-
ton had to postulate a mysterious invisible force pulling
the apple down; the apple can point to an evident cause
propelling Newton up.
The case for the apple's view is so overwhelming that
I must modify the situation a little in order to give
Newton a fair chance; because I believe the apple is
making too much of a merely accidental advantage. I
will place Newton at the centre of the earth where
u6 GRAVITATION— THE LAW
gravity vanishes, so that he can remain at rest without
support — without hammering. He looks up and sees
apples falling at the surface of the earth, and as before
ascribes this to a mysterious tug which he calls gravita-
tion. The apple looks down and sees Newton approach-
ing it; but this time it cannot attribute Newton's accelera-
tion to any evident hammering. It also has to invent
a mysterious tug acting on Newton.
We have two frames of reference. In one of them
Newton is at rest and the apple is accelerated; in the
other the apple is at rest and Newton accelerated. In
neither case is there a visible cause for the acceleration;
in neither is the object disturbed by extraneous ham-
mering. The reciprocity is perfect and there is no ground
for preferring one frame rather than the other. We
must devise a picture of the disturbing agent which will
not favour one frame rather than the other. In this
impartial humour a tug will not suit us, because if we
attach it to the apple we are favouring Newton's frame
and if we attach it to Newton we are favouring the
apple's frame.* The essence or absolute part of gravi-
tation cannot be a force on a body, because we are en-
tirely vague as to the body to which it is applied. We
must picture it differently.
* It will probably be objected that since the phenomena here dis-
cussed are evidently associated with the existence of a massive body (the
earth), and since Newton makes his tugs occur symmetrically about that
body whereas the apple makes its tugs occur unsymmetrically (vanishing
where the apple is, but strong at the antipodes), therefore Newton's
frame is clearly to be preferred. It would be necessary to go deeply into
the theory to explain fully why we do not regard this symmetry as of
first importance ; we can only say here that the criterion of symmetry
proves to be insufficient to pick out a unique frame and does not draw
a sharp dividing line between the frames that it would admit and those
it would have us reject. After all we can appreciate that certain frames
are more symmetrical than others without insisting on calling the sym-
metrical ones '"right"' and unsymmetrical ones 'wrong'.
A NEW PICTURE OF GRAVITATION 117
The ancients believed that the earth was flat. The
small part which they had explored could be represented
without serious distortion on a flat map. When new
countries were discovered it would be natural to think
that they could be added on to the flat map. A familiar
example of such a flat map is Mercator's projection, and
you will remember that in it the size of Greenland
appears absurdly exaggerated. (In other projections
directions are badly distorted.) Now those who adhered
to the flat-earth theory must suppose that the map gives
the true size of Greenland — that the distances shown in
the map are the true distances. How then w r ould they
explain that travellers in that country reported that the
distances seemed to be much shorter than they "really"
were ? They would, I suppose, invent a theory that there
was a demon living in Greenland who helped travellers
on their way. Of course no scientist would use so crude
a word; he would invent a Graeco-Latin polysyllable to
denote the mysterious agent which made the journeys
seem so short; but that is only camouflage. But now
suppose the inhabitants of Greenland have developed
their own geography. They find that the most important
part of the earth's surface (Greenland) can be repre-
sented without serious distortion on a flat map. But
when they put in distant countries such as Greece the
size must be exaggerated; or, as they would put it, there
is a demon active in Greece who makes the journeys
there seem different from what the flat map clearly
shows them to be. The demon is never where you are;
it is always the other fellow who is haunted by him.
We now understand that the true explanation is that the
earth is curved, and the apparent activities of the demon
arise from forcing the curved surface into a flat map
and so distorting the simplicity of things.
u8 GRAVITATION— THE LAW
What has happened to the theory of the earth has
happened also to the theory of the world of space-time.
An observer at rest at the earth's centre represents what
is happening in a frame of space and time constructed
on the usual conventional principles which give what
is called a flat space-time. He can locate the events in
his neighbourhood without distorting their natural sim-
plicity. Objects at rest remain at rest; objects in uni-
form motion remain in uniform motion unless there is
some evident cause of disturbance such as hammering;
light travels in straight lines. He extends this flat frame
to the surface of the earth where he encounters the
phenomenon of falling apples. This new phenomenon has
to be accounted for by an intangible agency or demon
called gravitation which persuades the apples to deviate
from their proper uniform motion. But we can also start
with the frame of the falling apple or of the man in the
lift. In the lift-frame bodies at rest remain at rest; bodies
in uniform motion remain in uniform motion. But, as we
have seen, even at the corners of the lift this simplicity
begins to fail; and looking further afield, say to the
centre of the earth, it is necessary to postulate the acti-
vity of a demon urging unsupported bodies upwards
(relatively to the lift-frame). As we change from one
observer to another — from one flat space-time frame to
another — the scene of activity of the demon shifts. It
is never where our observer is, but always away yonder.
Is not the solution now apparent? The demon is sim-
ply the complication which arises when we try to fit a
curved world into a flat frame. In referring the world
to a flat frame of space-time we distort it so that the
phenomena do not appear in their original simplicity.
Admit a curvature of the world and the mysterious
agency disappears. Einstein has exorcised the demon.
A NEW LAW OF GRAVITATION 119
Do not imagine that this preliminary change of con-
ception carries us very far towards an explanation of
gravitation. We are not seeking an explanation; we
are seeking a picture. And this picture of world-
curvature (hard though it may seem) is more graspable
than an elusive tug which flits from one object to
another according to the point of view chosen.
A New Law of Gravitation. Having found a new pic-
ture of gravitation, we require a new law of gravitation;
for the Newtonian law told us the arcounr. of the tug
and there is now no tug to be considered. Since the
phenomenon is now pictured as curvature the new law
must say something about curvature. Evidently it must
be a law governing and limiting the possible curvature
of space-time.
There are not many things which can be said about
curvature — not many of a general character. So that
when Einstein felt this urgency to say something about
curvature, he almost automatically said the right thing.
I mean that there was only one limitation or law that
suggested itself as reasonable, and that law has proved
to be right when tested by observation.
Some of you may feel that you could never bring your
minds to conceive a curvature of space, let alone of
space-time; others may feel that, being familiar with
the bending of a two-dimensional surface, there is no
insuperable difficulty in imagining something similar for
three or even four dimensions. I rather think that
the former have the best of it, for at least they escape
being misled by their preconceptions. I have spoken of
a "picture", but it is a picture that has to be described
analytically rather than conceived vividly. Our ordinary
conception of curvature is derived from surfaces, i.e.
120 GRAVITATION— THE LAW
two-dimensional manifolds embedded in a three-dimen-
sional space. The absolute curvature at any point is
measured by a single quantity called the radius of spheri-
cal curvature. But space-time is a four-dimensional
manifold embedded in — well, as many dimensions as it
can find new ways to twist about in. Actually a four-
dimensional manifold is amazingly ingenious in discover-
ing new kinds of contortion, and its invention is not
exhausted until it has been provided with six extra
dimensions, making ten dimensions in all. Moreover,
twenty distinct measures are required at each point to
specify the particular sort and amount of twistiness
there. These measures are called coefficients of curva-
ture. Ten of the coefficients stand out more prominently
than the other ten.
Einstein's law of gravitation asserts that the ten prin-
cipal coefficients of curvature are zero in empty space.
If there were no curvature, i.e. if all the coefficients
were zero, there would be no gravitation. Bodies would
move uniformly in straight lines. If curvature were
unrestricted, i.e. if all the coefficients had unpredictable
values, gravitation would operate arbitrarily and with-
out law. Bodies would move just anyhow. Einstein
takes a condition midway between; ten of the coefficients
are zero and the other ten are arbitrary. That gives
a world containing gravitation limited by a law. The
coefficients are naturally separated into two groups of
ten, so that there is no difficulty in choosing those which
are to vanish.
To the uninitiated it may seem surprising that an
exact law of Nature should leave some of the coefficients
arbitrary. But we need to leave something over to be
settled when we have specified the particulars of the
problem to which it is proposed to apply the law. A
A NEW LAW OF GRAVITATION 121
general law covers an infinite number of special cases.
The vanishing of the ten principal coefficients occurs
everywhere in empty space whether there is one gravi-
tating body or many. The other ten coefficients vary
according to the special case under discussion. This may
remind us that after reaching Einstein's law of gravi-
tation and formulating it mathematically, it is still a very
long step to reach its application to even the simplest
practical problem. However, by this time many hun-
dreds of readers must have gone carefully through the
mathematics; so we may rest assured that there has
been no mistake. After this work has been done it
becomes possible to verify that the law agrees with
observation. It is found that it agrees with Newton's
law to a very close approximation so that the main
evidence for Einstein's law is the same as the evidence
for Newton's law; but there are three crucial astro-
nomical phenomena in which the difference is large
enough to be observed. In these phenomena the obser-
vations support Einstein's law against Newton's.*
It is essential to our faith in a theory that its predic-
tions should accord with observation, unless a reasonable
explanation of the discrepancy is forthcoming; so that
it is highly important that Einstein's law should have
survived these delicate astronomical tests in which New-
ton's law just failed. But our main reason for reject-
ing Newton's law is not its imperfect accuracy as shown
by these tests; it is because it does not contain the
kind of information about Nature that we want to
know now that we have an ideal before us which was
not in Newton's mind at all. We can put it this way.
* one of the tests — a shift of the spectral lines to the red in the sun
and stars as compared with terrestrial sources — is a test of Einstein's
theory rather than of his law.
122 GRAVITATION— THE LAW
Astronomical observations show that within certain
limits of accuracy both Einstein's and Newton's laws
are true. In confirming (approximately) Newton's law,
we are confirming a statement as to what the appear-
ances would be when referred to one particular space-
time frame. No reason is given for attaching any
fundamental importance to this frame. In confirming
(approximately) Einstein's law, we are confirming a
statement about the absolute properties of the world,
true for all space-time frames. For those who are try-
ing to get beneath the appearances Einstein's statement
necessarily supersedes Newton's; it extracts from the
observations a result with physical meaning as opposed
to a mathematical curiosity. That Einstein's law has
proved itself the better approximation encourages us in
our opinion that the quest of the absolute is the best way
to understand the relative appearances; but had the suc-
cess been less immediate, we could scarcely have turned
our back on the quest.
I cannot but think that Newton himself would rejoice
that after 200 years the "ocean of undiscovered truth"
has rolled back another stage. I do not think of him as
censorious because we will not blindly apply his formula
regardless of the knowledge that has since accumulated
and in circumstances that he never had the opportunity
of considering.
I am not going to describe the three tests here, since
they are now well known and will be found in any of
the numerous guides to relativity; but I would refer to
the action of gravitation on light concerned in one of
them. Light-waves in passing a massive body such as
the sun are deflected through a small angle. This is
additional evidence that the Newtonian picture of
gravitation as a tug is inadequate. You cannot deflect
THE LAW OF MOTION 123
waves by tugging at them, and clearly another repre-
sentation of the agency which deflects them must be
found.
The Law of Motion. I must now ask you to let your
mind revert to the time of your first introduction to
mechanics before your natural glimmerings of the truth
were sedulously uprooted by your teacher. You were
taught the First Law of Motion —
"Every body continues in its state of rest or uniform
motion in a straight line, except in so far as it may be
compelled to change that state by impressed forces."
Probably you had previously supposed that motion
was something which would exhaust itself; a bicycle
stops of its own accord if you do not impress force to
keep it going. The teacher rightly pointed out the
resisting forces which tend to stop the bicycle; and he
probably quoted the example of a stone skimming over
ice to show that when these interfering forces are re-
duced the motion lasts much longer. But even ice offers
some frictional resistance. Why did not the teacher do
the thing thoroughly and abolish resisting forces alto-
gether, as he might easily have done by projecting the
stone into empty space? Unfortunately in that case
its motion is not uniform and rectilinear; the stone
describes a parabola. If you raised that objection you
would be told that the projectile was compelled to
change its state of uniform motion by an invisible force
called gravitation. How do we know that this invisible
force exists? Why! because if the force did not exist
the projectile would move uniformly in a straight line.
The teacher is not playing fair. He is determined to
have his uniform motion in a straight line, and if we
point out to him bodies which do not follow his rule
124 GRAVITATION— THE LAW
he blandly invents a new force to account for the devia-
tion. We can improve on his enunciation of the First
Law of Motion. What he really meant was —
"Every body continues in its state of rest or uniform
motion in a straight line, except in so far as it doesn't."
Material frictions and reactions are visible and abso-
lute interferences which can change the motion of a
body. I have nothing to say against them. The mole-
cular battering can be recognised by anyone who looks
deeply into the phenomenon no matter what his frame
of reference. But when there is no such indication of
disturbance the whole procedure becomes arbitrary. on
no particular grounds the motion is divided into two
parts, one of which is attributed to a passive tendency
of the body called inertia and the other to an interfer-
ing field of force. The suggestion that the body really
wanted to go straight but some mysterious agent made
it go crooked is picturesque but unscientific. It makes
two properties out of one; and then we wonder why they
are always proportional to one another — why the gravi-
tational force on different bodies is proportional to
their inertia or mass. The dissection becomes untenable
when we admit that all frames of reference are on the
same footing. The projectile which describes a parabola
relative to an observer on the earth's surface describes
a straight line relative to the man in the lift. Our
teacher will not easily persuade the man in the lift who
sees the apple remaining where he released it, that the
apple really would of its own initiative rush upwards
were it not that an invisible tug exactly counteracts this
tendency.*
Einstein's Law of Motion does not recognise this
dissection. There are certain curves which can be
* The reader will verify tkat this is the doctrine the teacher would have
to inculcate if he went as a missionary to the men in the lift.
THE LAW OF MOTION 125
defined on a curved surface without reference to any
frame or system of partitions, viz. the geodesies or
shortest routes from one point to another. The geo-
desies of our curved space-time supply the natural tracks
which particles pursue if they are undisturbed.
We observe a planet wandering round the sun in an
elliptic orbit. A little consideration will show that if we
add a fourth dimension (time), the continual moving on
in the time-dimension draws out the ellipse into a helix.
Why does the planet take this spiral track instead of
going straight? It is because it is following the shortest
track; and in the distorted geometry of the curved
region round the sun the spiral track is shorter than any
other between the same points. You see the great
change in our view. The Newtonian scheme says that
the planet tends to move in a straight line, but the sun's
gravity pulls it away. Einstein says that the planet tends
to take the shortest route and does take it.
That is the general idea, but for the sake of accuracy
I must make one rather trivial correction. The planet
takes the longest route.
You may remember that points along the track of
any material body (necessarily moving with a speed less
than the velocity of light) are in the absolute past or
future of one another; they are not absolutely ''else-
where". Hence the length of the track in four dimensions
is made up of time-like relations and must be measured
in time-units. It is in fact the number of seconds
recorded by a clock carried on a body which describes
the track.* This may be different from the time re-
* It may be objected that you cannot make a clock follow an arbitrary
curved path without disturbing it by impressed forces (e.g. molecular
hammering). But this difficulty is precisely analogous to the difficulty
of measuring the length of a curve with a rectilinear scale, and is sur-
mounted in the same way. The usual theory of "rectification of curves"
applies to these time-tracks as well as to space-curves.
126 GRAVITATION— THE LAW
corded by a clock which has taken some other route
between the same terminal points. on p. 39 we con-
sidered two individuals whose tracks had the same
terminal points; one of them remained at home on the
earth and the other travelled at high speed to a distant
part of the universe and back. The first recorded a
lapse of 70 years, the second of one year. Notice that
it is the man who follows the undisturbed track of the
earth who records or lives the longest time. The man
whose track was violently dislocated when he reached
the limit of his journey and started to come back again
lived only one year. There is no limit to this reduction;
as the speed of the traveller approaches the speed of
light the time recorded diminishes to zero. There is no
unique shortest track; but the longest track is unique.
If instead of pursuing its actual orbit the earth made a
wide sweep which required it to travel with the velocity
of light, the earth could get from 1 January 1927 to 1
January 1928 in no time, i.e. no time as recorded by an
observer or clock travelling with it, though it would be
reckoned as a year according to "Astronomer Royal's
time". The earth does not do this, because it is a rule
of the Trade Union of matter that the longest possible
time must be taken over every job.
Thus in calculating astronomical orbits and in similar
problems two laws are involved. We must first cal-
culate the curved form of space-time by using Einstein's
law of gravitation, viz. that the ten principal curva-
tures are zero. We next calculate how the planet moves
through the curved region by using Einstein's law of
motion, viz. the law of the longest track. Thus far the
procedure is analogous to calculations made with New-
ton's law of gravitation and Newton's law of motion.
But there is a remarkable addendum which applies only
THE LAW OF MOTION 127
to Einstein's laws. Einstein's law of motion can be
deduced from his law of gravitation. The prediction of
the track of a planet although divided into two stages
for convenience rests on a single law.
I should like to show you in a general way how it is
possible for a law controlling the curvature of empty
space to determine the tracks of particles without being
supplemented by any other conditions. Two "particles" in
the four-dimensional world are shown in Fig. 5, namely
yourself and myself. We are not empty space so there is
— ^-"^
Fig. 5
no limit to the kind of curvature entering into our com-
position; in fact our unusual sort of curvature is what
distinguishes us from empty space. We are, so to
speak, ridges in the four-dimensional world where it is
gathered into a pucker. The pure mathematician in his
unflattering language would describe us as "singulari-
ties". These two non-empty ridges are joined by empty
space, which must be free from those kinds of curva-
ture described by the ten principal coefficients. Now
it is common experience that if we introduce local
puckers into the material of a garment, the remainder
has a certain obstinacy and will not lie as smoothly as
128 GRAVITATION— THE LAW
we might wish. You will realise the possibility that,
given two ridges as in Fig. 5, it may be impossible to
join them by an intervening valley without the illegal
kind of curvature. That turns out to be the case. Two
perfectly straight ridges alone in the world cannot be
properly joined by empty space and therefore they can-
not occur alone. But if they bend a little towards one
another the connecting region can lie smoothly and sat-
isfy the law of curvature. If they bend too much the
illegal puckering reappears. The law of gravitation is
a fastidious tailor who will not tolerate wrinkles (except
of a limited approved type) in the main area of the
garment; so that the seams are required to take courses
which will not cause wrinkles. You and I have to sub-
mit to this and so our tracks curve towards each other.
An onlooker will make the comment that here is an
illustration of the law that two massive bodies attract
each other.
We thus arrive at another but equivalent conception
of how the earth's spiral track through the four-dimen-
sional world is arrived at. It is due to the necessity of
arranging two ridges (the solar track and the earth's
track) so as not to involve a wrong kind of curvature in
the empty part of the world. The sun as the more
pronounced ridge takes a nearly straight track; but the
earth as a minor ridge on the declivities of the solar
ridge has to twist about considerably.
Suppose the earth were to defy the tailor and take a
straight track. That would make a horrid wrinkle in the
garment; and since the wrinkle is inconsistent with the
laws of empty space, something must be there — where
the wrinkle runs. This "something" need not be matter
in the restricted sense. The things which can occupy
space so that it is not empty in the sense intended in
RELATIVITY OF ACCELERATION 129
Einstein's law, are mass (or its equivalent energy)
momentum and stress (pressure or tension). In this case
the wrinkle might correspond to stress. That is reason-
able enough. If left alone the earth must pursue its
proper curved orbit; but if some kind of stress or pres-
sure were inserted between the sun and earth, it might
well take another course. In fact if we were to observe
one of the planets rushing off in a straight track, New-
tonians and Einsteinians alike would infer that there
existed a stress causing this behaviour. It is true that
causation has apparently been turned topsy-turvy; ac-
cording to our theory the stress seems to be caused by
the planet taking the wrong track, whereas we usually
suppose that the planet takes the wrong track because it
is acted on by the stress. But that is a harmless accident
common enough in primary physics. The discrimination
between cause and effect depends on time's arrow and
can only be settled by reference to entropy. We need
not pay much attention to suggestions of causation aris-
ing in discussions of primary laws which, as likely as
not, are contemplating the world upside down.
Although we are here only at the beginning of Ein-
stein's general theory I must not proceed further into
this very technical subject. The rest of this chapter will
be devoted to elucidation of more elementary points.
Relativity of Acceleration. The argument in this chapter
rests on the relativity of acceleration. The apple had an
acceleration of 32 feet per second per second relative to
the ordinary observer, but zero acceleration relative to
the man in the lift. We ascribe to it one acceleration or
the other according to the frame we happen to be using,
but neither is to be singled out and labelled "true"
or absolute acceleration. That led us to reject the
i 3 o GRAVITATION— THE LAW
Newtonian conception which singled out 32 feet per
second per second as the true acceleration and invented
a disturbing agent of this particular degree of strength.
It will be instructive to consider an objection brought,
I think, originally by Lenard. A train is passing through
a station at 60 miles an hour. Since velocity is relative,
it does not matter whether we say that the train is
moving at 60 miles an hour past the station or the
station is moving at 60 miles an hour past the train.
Now suppose, as sometimes happens in railway acci-
dents, that this motion is brought to a standstill in a
few seconds. There has been a change of velocity or
acceleration — a term which includes deceleration. If
acceleration is relative this may be described indiffer-
ently as an acceleration of the train (relative to the sta-
tion) or an acceleration of the station (relative to the
train). Why then does it injure the persons in the train
and not those in the station?
Much the same point was put to me by one of my
audience. "You must find the journey between Cam-
bridge and Edinburgh very tiring. I can understand
the fatigue, if you travel to Edinburgh; but why should
you get tired if Edinburgh comes to you?" The answer
is that the fatigue arises from being shut up in a box
and jolted about for nine hours; and it makes no differ-
ence whether in the meantime I move to Edinburgh or
Edinburgh moves to me. Motion does not tire anybody.
With the earth as our vehicle we are travelling at 20
miles a second round the sun; the sun carries us at 12
miles a second through the galactic system; the galactic
system bears us at 250 miles a second amid the spiral
nebulae; the spiral nebulae. ... If motion could tire,
we ought to be dead tired.
Similarly change of motion or acceleration does not
RELATIVITY OF ACCELERATION 131
injure anyone, even when it is (according to the New-
tonian view) an absolute acceleration. We do not even
feel the change of motion as our earth takes the curve
round the sun. We feel something when a railway train
takes a curve, but what we feel is not the change of
motion nor anything which invariably accompanies
change of motion; it is something incidental to the
curved track of the train but not to the curved track of
the earth. The cause of injury in the railway accident
is easily traced. Something hit the train; that is to say,
the train was bombarded by a swarm of molecules and
the bombardment spread all the way along it. The
cause is evident — gross, material, absolute — recognised
by everyone, no matter what his frame of reference,
as occurring in the train not the station. Besides injur-
ing the passengers this cause also produced the relative
acceleration of the train and station — an effect which
might equally well have been produced by molecular
bombardment of the station, though in this case it was
not
The critical reader will probably pursue his objection.
"Are you not being paradoxical when you say that a
molecular bombardment of the train can cause an accel-
eration of the station — and in fact of the earth and the
rest of the universe? To put it mildly, relative accelera-
tion is a relation with two ends to it, and we may at
first seem to have an option which end we shall grasp
it by; but in this case the causation (molecular bom-
bardment) clearly indicates the right end to take hold
of, and you are merely spinning paradoxes when you
insist on your liberty to take hold of the other."
If there is an absurdity in taking hold of the wrong
end of the relation it has passed into our current
speech and thought. Your suggestion is in fact more
132 GRAVITATION— THE LAW
revolutionary than anything Einstein has ventured to
advocate. Let us take the problem of a falling stone.
There is a relative acceleration of 32 feet per second
per second — of the stone relative to ourselves or of our-
selves relative to the stone. Which end of the relation
must we choose? The one indicated by molecular bom-
bardment? Well, the stone is not bombarded; it is
falling freely in vacuo. But we are bombarded by the
molecules of the ground on which we stand. Therefore
it is we who have the acceleration; the stone has zero
acceleration, as the man in the lift supposed. Your sug-
gestion makes out the frame of the man in the lift to
be the only legitimate one; I only went so far as to
admit it to an equality with our own customary frame.
Your suggestion would accept the testimony of the
drunken man who explained that u the paving-stone got
up and hit him" and dismiss the policeman's account of
the incident as "merely spinning paradoxes". What
really happened was that the paving-stone had been
pursuing the man through space with ever-increasing
velocity, shoving the man in front of it so that they kept
the same relative position. Then, through an unfor-
tunate wobble of the axis of the man's body, he failed
to increase his speed sufficiently, with the result that
the paving-stone overtook him and came in contact with
his head. That, please understand, is your suggestion;
or rather the suggestion which I have taken the liberty
of fathering on you because it is the outcome of a very
common feeling of objection to the relativity theory.
Einstein's position is that whilst this is a perfectly
legitimate way of looking at the incident the more usual
account given by the policeman is also legitimate; and
he endeavours like a good magistrate to reconcile them
both.
TIME GEOMETRY 133
Time Geometry. Einstein's law of gravitation controls
a geometrical quantity curvature in contrast to Newton's
law which controls a mechanical quantity force. To
understand the origin of this geometrisation of the world
in the relativity theory we must go back a little.
The science which deals with the properties of space
is called geometry. Hitherto geometry has not included
time in its scope. But now space and time are so inter-
locked that there must be one science — a somewhat
extended geometry — embracing them both. Three-
dimensional space is only a section cut through four-
dimensional space-time, and moreover sections cut in
different directions form the spaces of different
observers. We can scarcely maintain that the study of
a section cut in one special direction is the proper sub-
ject-matter of geometry and that the study of slightly
different sections belongs to an altogether different
science. Hence the geometry of the world is now con-
sidered to include time as well as space. Let us follow
up the geometry of time.
You will remember that although space and time are
mixed up there is an absolute distinction between a
spatial and a temporal relation of two events. Three
events will form a space-triangle if the three sides
correspond to spatial relations — if the three events are
absolutely elsewhere with respect to one another.*
Three events will form a time-triangle if the three sides
correspond to temporal relations — if the three events
are absolutely before or after one another. (It is pos-
sible also to have mixed triangles with two sides time-like
and one space-like, or vice versa.) A well-known law
of the space-triangle is that any two sides are together
* This would be an instantaneous space-triangle. An enduring triangle
is a kind of four-dimensional prism.
134 GRAVITATION— THE LAW
greater than the third side. There is an analogous, but
significantly different, law for the time-triangle, viz. two
of the sides (not any two sides) are together less than
the third side. It is difficult to picture such a triangle
but that is the actual fact.
Let us be quite sure that we grasp the precise mean-
ing of these geometrical propositions. Take first the
space-triangle. The proposition refers to the lengths of
the sides, and it is well to recall my imaginary discus-
sion with two students as to how lengths are to be
measured (p. 23). Happily there is no ambiguity
now, because the triangle of three events determines a
plane section of the world, and it is only for that mode
of section that the triangle is purely spatial. The propo-
sition then expresses that
"If you measure with a scale from A to B and from
B to C the sum of your readings will be greater than the
reading obtained by measuring with a scale from A to C."
For a time-triangle the measurements must be made
with an instrument which can measure time, and the
proposition then expresses that
"If you measure with a clock from A to B and from
B to C the sum of your readings will be less than the
reading obtained by measuring with a clock from A to C."
In order to measure from an event A to an event B
with a clock you must make an adjustment of the clock
analogous to orienting a scale along the line AB. What
is this analogous adjustment? The purpose in either
case is to bring both A and B into the immediate
neighbourhood of the scale or clock. For the clock that
means that after experiencing the event A it must travel
with the appropriate velocity needed to reach the locality
of B just at the moment that B happens. Thus the
velocity of the clock is prescribed. one further point
TIME GEOMETRY 135
should be noticed. After measuring with a scale from
A to B you can turn your scale round and measure from
B to A, obtaining the same result. But you cannot turn
a clock round, i.e. make it go backwards in time. That
is important because it decides which two sides are less
than the third side. If you choose the wrong pair the
enunciation of the time proposition refers to an im-
possible kind of measurement and becomes meaningless.
You remember the traveller (p. 39) who went off
to a distant star and returned absurdly young. He was
a clock measuring two sides of a time-triangle. He
recorded less time than the stay-at-home observer who
was a clock measuring the third side. Need I defend
my calling him a clock? We are all of us clocks whose
faces tell the passing years. This comparison was simply
an example of the geometrical proposition about time-
triangles (which in turn is a particular case of Einstein's
law of longest track). The result is quite explicable in
the ordinary mechanical way. All the particles in the
traveller's body increase in mass on account of his high
velocity according to the law already discussed and
verified by experiment. This renders them more slug-
gish, and the traveller lives more slowly according to
terrestrial time-reckoning. However, the fact that the
result is reasonable and explicable does not render it the
less true as a proposition of time geometry.
Our extension of geometry to include time as well as
space will not be a simple addition of an extra dimension
to Euclidean geometry, because the time propositions,
though analogous, are not identical with those which
Euclid has given us for space alone. Actually the dif-
ference between time geometry and space geometry is
not very profound, and the mathematician easily glides
over it by a discrete use of the symbol V-i. We still
i 3 6 GRAVITATION— THE LAW
call (rather loosely) the extended geometry Euclidean;
or, if it is necessary to emphasise the distinction, we
call it hyperbolic geometry. The term non-Euclidean
geometry refers to a more profound change, viz. that
involved in the curvature of space and time by which
we now represent the phenomenon of gravitation. We
start with Euclidean geometry of space, and modify it
in a comparatively simple manner when the time-dimen-
sion is added; but that still leaves gravitation to be
reckoned with, and wherever gravitational effects are
observable it is an indication that the extended Euclidean
geometry is not quite exact, and the true geometry is a
non-Euclidean one — appropriate to a curved region as
Euclidean geometry is to a flat region.
Geometry and Mechanics. The point that deserves special
attention is that the proposition about time-triangles is
a statement as to the behaviour of clocks moving with
different velocities. We have usually regarded the
behaviour of clocks as coming under the science of
mechanics. We found that it was impossible to confine
geometry to space alone, and we had to let it expand a
little. It has expanded with a vengeance and taken a
big slice out of mechanics. There is no stopping it, and
bit by bit geometry has now swallowed up the whole of
mechanics. It has also made some tentative nibbles at
electromagnetism. An ideal shines in front of us, far
ahead perhaps but irresistible, that the whole of our
knowledge of the physical world may be unified into a
single science which will perhaps be expressed in terms
of geometrical or quasi-geometrical conceptions. Why
not? All the knowledge is derived from measurements
made with various instruments. The instruments used
in the different fields of inquiry are not fundamentally
GEOMETRY AND MECHANICS 137
unlike. There is no reason to regard the partitions of
the sciences made in the early stages of human thought
as irremovable.
But mechanics in becoming geometry remains none
the less mechanics. The partition between mechanics
and geometry has broken down and the nature of each
of them has diffused through the whole. The apparent
supremacy of geometry is really due to the fact that it
possesses the richer and more adaptable vocabulary;
and since after the amalgamation we do not need the
double vocabulary the terms employed are generally
taken from geometry. But besides the geometrisation of
mechanics there has been a mechanisation of geometry.
The proposition about the space-triangle quoted above
was seen to have grossly material implications about the
behaviour of scales which would not be realised by any-
one who thinks of it as if it were a proposition of pure
mathematics.
We must rid our minds of the idea that the word
space in science has anything to do with void. As pre-
viously explained it has the other meaning of distance,
volume, etc., quantities expressing physical measure-
ment just as much as force is a quantity expressing
physical measurement. Thus the (rather crude) state-
ment that Einstein's theory reduces gravitational force
to a property of space ought not to arouse misgiving.
In any case the physicist does not conceive of space
as void. Where it is empty of all else there is still the
aether. Those who for some reason dislike the word
aether, scatter mathematical symbols freely through the
vacuum, and I presume that they must conceive some
kind of characteristic background for these symbols. I
do not think any one proposes to build even so relative
and elusive a thing as force out of entire nothingness.
Chapter VII
GRAVITATION— THE EXPLANATION
The Law of Curvature. Gravitation can be explained.
Einstein's theory is not primarily an explanation of
gravitation. When he tells us that the gravitational field
corresponds to a curvature of space and time he is giv-
ing us a picture. Through a picture we gain the insight
necessary to deduce the various observable consequences.
There remains, however, a further question whether
any reason can be given why the state of things pictured
should exist. It is this further inquiry which is meant
when we speak of "explaining" gravitation in any far-
reaching sense.
At first sight the new picture does not leave very
much to explain. It shows us an undulating hum-
mocky world, whereas a gravitationless world would be
plane and uniform. But surely a level lawn stands more
in need of explanation than an undulating field, and a
gravitationless world would be more difficult to account
for than a world with gravitation. We are hardly called
upon to account for a phenomenon which could only
be absent if (in the building of the world) express pre-
cautions were taken to exclude it. If the curvature were
entirely arbitrary this would be the end of the explana-
tion; but there is a law of curvature — Einstein's law of
gravitation — and on this law our further inquiry must
be focussed. Explanation is needed for regularity, not
for diversity; and our curiosity is roused, not by the
diverse values of the ten subsidiary coefficients of curva-
ture which differentiate the world from a flat world,
but by the vanishing everywhere of the ten principal
coefficients.
138
THE LAW OF CURVATURE 139
All explanations of gravitation on Newtonian lines
have endeavoured to show why something (which I have
disrespectfully called a demon) is present in the world.
An explanation on the lines of Einstein's theory must
show why something (which we call principal curvature)
is excluded from the world.
In the last chapter the law of gravitation was stated
in the form — the ten principal coefficients of curvature
vanish in empty space. I shall now restate it in a slightly
altered form —
The radius of spherical* curvature of every three-di-
mensional section of the world, cut in any direction at any
point of empty space, is always the same constant length.
Besides the alteration of form there is actually a little
difference of substance between the two enunciations;
the second corresponds to a later and, it is believed, more
accurate formula given by Einstein a year or two after
his first theory. The modification is. made necessary by
our realisation that space is finite but unbounded (p.
80). The second enunciation would be exactly equiva-
lent to the first if for "same constant length" we read
"infinite length". Apart from very speculative esti-
mates we do not know the constant length referred to,
but it must certainly be greater than the distance of the
furthest nebula, say io 20 miles. A distinction between
so great a length and infinite length is unnecessary in
most of our arguments and investigations, but it is
necessary in the present chapter.
♦Cylindrical curvature of the world has nothing to do with gravita-
tion, nor so far as we know with any other phenomenon. Anything
drawn on the surface of a cylinder can be unrolled into a flat map without
distortion, but the curvature introduced in the last chapter was intended
to account for the distortion which appears in our customary flat map; it
is therefore curvature of the type exemplified by a sphere, not a cylinder.
140 GRAVITATION— THE EXPLANATION
We must try to reach the vivid significance which
lies behind the obscure phraseology of the law. Suppose
that you are ordering a concave mirror for a telescope.
In order to obtain what you want you will have to
specify two lengths (i) the aperture, and (2) the radius
of curvature. These lengths both belong to the mirror —
both are necessary to describe the kind of mirror you
want to purchase — but they belong to it in different
ways. You may order a mirror of 100 foot radius of
curvature and yet receive it by parcel post. In a certain
sense the 100 foot length travels with the mirror, but
it does so in a way outside the cognizance of the postal
authorities. The 100 foot length belongs especially to
the surface of the mirror, a two-dimensional continuum;
space-time is a four-dimensional continuum, and you will
see from this analogy that there can be lengths belonging
in this way to a chunk of space-time — lengths having
nothing to do with the largeness or smallness of the
chunk, but none the less part of the specification of the
particular sample. Owing to the two extra dimensions
there are many more such lengths associated with space-
time than with the mirror surface. In particular, there
is not only one general radius of spherical curvature, but
a radius corresponding to any direction you like to take.
For brevity I will call this the "directed radius" of the
world. Suppose now that you order a chunk of space-
time with a directed radius of 500 trillion miles in one
direction and 800 trillion miles in another. Nature
replies "No. We do not stock that. We keep a wide
range of choice as regards other details of specification;
but as regards directed radius we have nothing different
in different directions, and in fact all our goods have the
one standard radius, x trillion miles." I cannot tell you
what number to put for x because that is still a secret
of the firm.
RELATIVITY OF LENGTH 141
The fact that this directed radius which, one would
think, might so easily differ from point to point and
from direction to direction, has only one standard value
in the world is Einstein's law of gravitation. From it
we can by rigorous mathematical deduction work out the
motions of planets and predict, for example, the eclipses
of the next thousand years; for, as already explained,
the law of gravitation includes also the law of motion.
Newton's law of gravitation is an approximate adapta-
tion of it for practical calculation. Building up from
the law all is clear; but what lies beneath it? Why is
there this unexpected standardisation? That is what we
must now inquire into.
Relativity of Length. There is no such thing as absolute
length; we can only express the length of one thing in
terms of the length of something else.* And so when
we speak of the length of the directed radius we mean
its length compared with the standard metre scale.
Moreover, to make this comparison, the two lengths
must lie alongside. Comparison at a distance is as un-
thinkable as action at a distance; more so, because com-
parison is a less vague conception than action. We must
either convey the standard metre to the site of the
length we are measuring, or we must use some device
which, we are satisfied, will give the same result as if we
actually moved the metre rod.
Now if we transfer the metre rod to another point of
space and time, does it necessarily remain a metre long?
Yes, of course it does; so long as it is the standard of
length it cannot be anything else but a metre. But does
it really remain the metre that it was? I do not know
* This relativity with respect to a standard unit is, of course, addi-
tional to and independent of the relativity with respect to the observer's
motion treated in chapter n.
1 42 GRAVITATION— THE EXPLANATION
what you mean by the question; there is nothing by
reference to which we could expose delinquencies of the
standard rod, nothing by reference to which we could
conceive the nature of the supposed delinquencies. Still
the standard rod was chosen with considerable care; its
material was selected to fulfil certain conditions — to be
affected as little as possible by casual influences such
as temperature, strain or corrosion, in order that its
extension might depend only on the most essential char-
acteristics of its surroundings, present and past.* We
cannot say that it was chosen to keep the same absolute
length since there is no such thing known; but it was
* In so far as these casual influences are not entirely eliminated by
the selection of material and the precautions in using the rod, appropriate
corrections must be applied. But the rod must not be corrected for
essential characteristics of the space it is measuring. We correct the
reading of a voltmeter for temperature, but it would be nonsensical to
correct it for effects of the applied voltage. The distinction between
casual and essential influences — those to be eliminated and those to be
left in — depends on the intention of the measurements. The measuring
rod is intended for surveying space, and the essential characteristic of
space is "metric". It would be absurd to correct the readings of our
scale to the values they would have had if the space had some other
metric. The region of the world to which the metric refers may also
contain an electric field; this will be regarded as a casual characteristic
since the measuring rod is not intended for surveying electric fields.
I do not mean that from a broader standpoint the electric field is any less
essential to the region than its peculiar metric. It would be hard to say
in what sense it would remain the same region if any of its qualities were
other than they actually are. This point does not trouble us here, because
there are vast regions of the world practically empty of all characteristics
except metric, and it is to these that the law of gravitation is applied both
in theory and in practice. It has seemed, however, desirable to dwell on
this distinction between essential and casual characteristics because there
are some who, knowing that we cannot avoid in all circumstances cor-
rections for casual influences, regard that as license to adopt any arbi-
trary system of corrections — a procedure which would merely have the
effect of concealing what the measures can teach us about essential
characteristics.
RELATIVITY OF LENGTH 143
chosen so that it might not be prevented by casual in-
fluences from keeping the same relative length — relative
to what? Relative to some length inalienably associated
with the region in which it is placed. I can conceive
of no other answer. An example of such a length
inalienably associated with a region is the directed radius.
The long and short of it is that when the standard
metre takes up a new position or direction it measures
itself against the directed radius of the world in that
region and direction, and takes up an extension which
is a definite fraction of the directed radius. I do not
see what else it could do. We picture the rod a little
bewildered in its new surroundings wondering how
large it ought to be — how much of the unfamiliar terri-
tory its boundaries ought to take in. It wants to do
just what it did before. Recollections of the chunk of
space that it formerly filled do not help, because there
is nothing of the nature of a landmark. The one thing
it can recognise is a directed length belonging to the
region where it finds itself; so it makes itself the same
fraction of this directed length as it did before.
If the standard metre is always the same fraction of
the directed radius, the directed radius is always the
same number of metres. Accordingly the directed
radius is made out to have the same length for all
positions and directions. Hence we have the law of
gravitation.
When we felt surprise at finding as a law of Nature
that the directed radius of curvature was the same for
all positions and directions, we did not realise that our
unit of length had already made itself a constant fraction
of the directed radius. The whole thing is a vicious
circle. The law of gravitation is — a put-up job.
144 GRAVITATION— THE EXPLANATION
This explanation introduces no new hypothesis. In
saying that a material system of standard specification
always occupies a constant fraction of the directed radius
of the region where it is, we are simply reiterating
Einstein's law of gravitation — stating it in the inverse
form. Leaving aside for the moment the question
whether this behaviour of the rod is to be expected or
not, the law of gravitation assures us that that is the
behaviour. To see the force of the explanation we
must, however, realise the relativity of extension. Exten-
sion which is not relative to something in the surround-
ings has no meaning. Imagine yourself alone in the
midst of nothingness, and then try to tell me how large
you are. The definiteness of extension of the standard
rod can only be a definiteness of its ratio to some other
extension. But we are speaking now of the extension
of a rod placed in empty space, so that every standard
of reference has been removed except extensions be-
longing to and implied by the metric of the region. It
follows that one such extension must appear from our
measurements to be constant everywhere (homogeneous
and isotropic) on account of its constant relation to what
we have accepted as the unit of length.
We approached the problem from the point of view
that the actual world with its ten vanishing coefficients
of curvature (or its isotropic directed curvature) has a
specialisation which requires explanation; we were then
comparing it in our minds with a world suggested by
the pure mathematician which has entirely arbitrary
curvature. But the fact is that a world of arbitrary
curvature is a sheer impossibility. If not the directed
radius, then some other directed length derivable from
the metric, is bound to be homogeneous and isotropic.
In applying the ideas of the pure mathematician we
RELATIVITY OF LENGTH 145
overlooked the fact that he was imagining a world
surveyed from outside with standards foreign to it
whereas we have to do with a world surveyed from
within with standards conformable to it.
The explanation of the law of gravitation thus lies in
the fact that we are dealing with a world surveyed from
within. From this broader standpoint the foregoing
argument can be generalised so that it applies not only
to a survey with metre rods but to a survey by optical
methods, which in practice are generally substituted as
equivalent. When we recollect that surveying apparatus
can have no extension in itself but only in relation to the
world, so that a survey of space is virtually a self-com-
parison of space, it is perhaps surprising that such a
self-comparison should be able to show up any hetero-
geneity at all. It can in fact be proved that the metric
of a two-dimensional or a three-dimensional world sur-
veyed from within is necessarily uniform. With four or
more dimensions heterogeneity becomes possible, but it
is a heterogeneity limited by a law which imposes some
measure of homogeneity.
I believe that this has a close bearing on the rather
heterodox views of Dr. Whitehead on relativity. He
breaks away from Einstein because he will not admit
the non-uniformity of space-time involved in Einstein's
theory. "I deduce that our experience requires and
exhibits a basis of uniformity, and that in the case of
nature this basis exhibits itself as the uniformity of
spatio-temporal relations. This conclusion entirely cuts
away the casual heterogeneity of these relations which
is the essential of Einstein's later theory."* But we now
see that Einstein's theory asserts a casual heterogeneity
*A. N. Whitehead, The Principle of Relativity, Preface.
146 GRAVITATION—THE EXPLANATION
of only one set of ten coefficients and complete uniform-
ity of the other ten. It therefore does not leave us with-
out the basis of uniformity of which Whitehead in his
own way perceived the necessity. Moreover, this uni-
formity is not the result of a law casually imposed on the
world; it is inseparable from the conception of survey of
the world from within — which is, I think, just the con-
dition that Whitehead would demand. If the world of
space-time had been of two or of three dimensions
Whitehead would have been entirely right; but then
there could have been no Einstein theory of gravitation
for him to criticise. Space-time being four-dimensional,
we must conclude that Whitehead discovered an im-
portant truth about uniformity but misapplied it.
The conclusion that the extension of an object in any
direction in the four-dimensional world is determined by
comparison with the radius of curvature in that direction
has one curious consequence. So long as the direction
in the four-dimensional world is space-like, no difficulty
arises. But when we pass over to time-like directions
(within the cone of absolute past or future) the directed
radius is an imaginary length. Unless the object
ignores the warning symbol V — i it has no standard
of reference for settling its time extension. It has no
standard duration. An electron decides how large it
ought to be by measuring itself against the radius of
the world in its space-directions. It cannot decide how
long it ought to exist because there is no real radius of
the world in its time-direction. Therefore it just goes on
existing indefinitely. This is not intended to be a rigor-
ous proof of the immortality of the electron — subject
always to the condition imposed throughout these
arguments that no agency other than metric interferes
with the extension. But it shows that the electron
PREDICTIONS FROM THE LAW 147
behaves in the simple way which we might at least hope
to find.*
Predictions from the Law. I suppose that it is at first
rather staggering to find a law supposed to control the
movements of stars and planets turned into a law
finicking with the behaviour of measuring rods. But
there is no prediction made by the law of gravitation in
which the behaviour of measuring appliances does not
play an essential part. A typical prediction from the law
is that pn a certain date 384,400,000 metre rods laid
end to end would stretch from the earth to the moon.
We may use more circumlocutory language, but that is
what is meant. The fact that in testing the prediction
we shall trust to indirect evidence, not carrying out the
whole operation literally, is not relevant; the prophecy
is made in good faith and not with the intention of tak-
ing advantage of our remissness in checking it.
We have condemned the law of gravitation as a put-
up job. You will want to know how after such a dis-
creditable exposure it can still claim to predict eclipses
and other events which come off.
A famous philosopher has said —
"The stars are not pulled this way and that by
mechanical forces; theirs is a free motion. They go on
their way, as the ancients said, like the blessed gods." f
This sounds particularly foolish even for a philo-
sopher; but I believe that there is a sense in which it is
true.
* on the other hand a quantum (see chapter ix) has a definite
periodicity associated with it, so that it must be able to measure itself
against a time-extension. Anyone who contemplates the mathematical
equations of the new quantum theory will see abundant evidence of the
battle with the intervening symbol V — *•
t Hegel, Werke (1842 Ed.), Bd. 7, Abt. 1, p. 97.
i 4 8 GRAVITATION— THE EXPLANATION
We have already had three versions of what the earth
is trying to do when it describes its elliptic orbit around
the sun.
(i) It is trying to go in a straight line but it is
roughly pulled away by a tug emanating from the sun.
(2) It is taking the longest possible route through
the curved space-time around the sun.
(3) It is accommodating its track so as to avoid
causing any illegal kind of curvature in the empty space
around it.
We now add a fourth version.
(4) The earth goes anyhow it likes.
It is not a long step from the third version to the
fourth now that we have seen that the mathematical
picture of empty space containing "illegal" curvature
is a sheer impossibility in a world surveyed from within.
For if illegal curvature is a sheer impossibility the earth
will not have to take any special precautions to avoid
causing it, and can do anything it likes. And yet the
non-occurrence of this impossible curvature is the law
(of gravitation) by which we calculate the track of the
earth!
The key to the paradox is that we ourselves, our
conventions, the kind of thing that attracts our interest,
are much more concerned than we realise in any account
we give of how the objects of the physical world are
behaving. And so an object which, viewed through our
frame of conventions, anay seem to be behaving in a
very special and remarkable way may, viewed according
to another set of conventions, be doing nothing to excite
particular comment. This will be clearer if we consider
a practical illustration, and at the same time defend
version (4).
PREDICTIONS FROM THE LAW 149
You will say that the earth must certainly get into
the right position for the eclipse next June (1927); so
it cannot be free to go anywhere it pleases. I can put
that right. I hold to it that the earth goes anywhere it
pleases. The next thing is that we must find out where
it has been pleased to go. The important question for us
is not where the earth has got to in the inscrutable
absolute behind the phenomena, but where we shall
locate it in our conventional background of space and
time. We must take measurements of its position, for
Fig. 6
example, measurements of its distance from the sun.
In Fig. 6, SSx shows the ridge in the world which we
recognise as the sun; I have drawn the earth's ridge in
duplicate (EE 1} EE 2 ) because I imagine it as still un-
decided which track it will take. If it takes EE ± we lay
our measuring rods end to end down the ridges and
across the valley from S ± to E ± , count up the number,
and report the result as the earth's distance from the
sun. The measuring rods, you will remember, adjust
their lengths proportionately to the radius of curvature
of the world. The curvature along this contour is rather
150 GRAVITATION— THE EXPLANATION
large and the radius of curvature small. The rods
therefore are small, and there will be more of them in
$i£i than the picture would lead you to expect. If the
earth chooses to go to E 2 the curvature is less sharp;
the greater radius of curvature implies greater length
of the rods. The number needed to stretch from S ± to
E 2 will not be so great as the diagram at first suggests;
it will not be increased in anything like the proportion
of S ± E 2 to S ± E X in the figure. We should not be sur-
prised if the number turned out to be the same in both
cases. If so, the surveyor will report the same distance
of the earth from the sun whether the track is EE ± or
EE 2 . And the Superintendent of the Nautical Almanac
who published this same distance some years in advance
will claim that he correctly predicted where the earth
would go.
And so you see that the earth can play truant to any
extent but our measurements will still report it in the
place assigned to it by the Nautical Almanac. The
predictions of that authority pay no attention to the
vagaries of the god-like earth; they are based on what
will happen when we come to measure up the path that
it has chosen. We shall measure it with rods that adjust
themselves to the curvature of the world. The mathe-
matical ｅxpression of this fact is the law of gravitation
used in the predictions.
Perhaps you will object that astronomers do not in
practice lay measuring rods end to end through inter-
planetary space in order to find out where the planets
are. Actually the position is deduced from the light
rays. But the light as it proceeds has to find out what
course to take in order to go "straight", in much the
same way as the metre rod has to find out how far to
extend. The metric or curvature is a sign-post for the
PREDICTIONS FROM THE LAW 151
light as it is a gauge for the rod. The light track is in
fact controlled by the curvature in such a way that it is
incapable of exposing the sham law of curvature. And
so wherever the sun, moon and earth may have got to,
the light will not give them away. If the law of curva-
ture predicts an eclipse the light will take such a track
that there is an eclipse. The law of gravitation is not a
stern ruler controlling the heavenly bodies; it is a kind-
hearted accomplice who covers up their delinquencies.
I do not recommend you to try to verify from Fig. 6
that the number of rods in SiE t (full line) and SJL 2
(dotted line) is the same. There are two dimensions of
space-time omitted in the picture besides the extra dimen-
sions in which space-time must be supposed to be bent;
moreover it is the spherical, not the cylindrical, curvature
which is ,the gauge for the length. It might be an
instructive, though very laborious, task to make this
direct verification, but we know beforehand that the
measured distance of the earth from the sun must be
the same for either track. The law of gravitation, ex-
pressed mathematically by G^ u — Xg^ t means nothing
more nor less than that the unit of length everywhere
is a constant fraction of the directed radius of the world
at that point. And as the astronomer who predicts the
future position of the earth does not assume anything
more about what the earth will choose to do than is
expressed in the law G tLV ^=Xg lxl/i so we shall find the
same position of the earth, if we assume nothing more
than that the practical unit of length involved in measure-
ments of the position is a constant fraction of the directed
radius. We do not need to decide whether the track is
to be represented by EE ± or EE 2 , and it would convey
no information as to any observable phenomena if we
knew the representation.
152 GRAVITATION— THE EXPLANATION
I shall have to emphasise elsewhere that the whole of
our physical knowledge is based on measures and that
the physical world consists, so to speak, of measure-
groups resting on a shadowy background that lies
outside the scope of physics. Therefore in conceiving
a world which had existence apart from the measure-
ments that we make of it, I was trespassing outside the
limits of what we call physical reality. I would not
dissent from the view that a vagary which by its very
nature could not be measurable has no claim to a physical
existence. No one knows what is meant by such a
vagary. I said that the earth might go anywhere it
chose, but did not provide a "where" for it to choose;
since our conception of "where" is based on space
measurements which were at that stage excluded. But
I do not think I have been illogical. I am urging that,
do what it will, the earth cannot get out of the track
laid down for it by the law of gravitation. In order to
show this I must suppose that the earth has made the
attempt and stolen nearer to the sun; then I show that
our measures conspire quietly to locate it back in its
proper orbit. I have to admit in the end that the earth
never was out of its proper orbit;* I do not mind that,
because meanwhile I have proved my point. The fact
that a predictable path through space and time is laid
down for the earth is not a genuine restriction on its
conduct, but is imposed by the formal scheme in which
we draw up our account of its conduct.
* Because I can attach no meaning to an orbit other than an orbit in
space and time, i.e. as located by measures. But I could not assume that
the alternative orbit would be meaningless (inconsistent with possible
measures) until I tried it.
NON-EMPTY SPACE 153
Non-Empty Space. The law that the directed radius is
constant does not apply to space which is not completely
empty. There is no longer any reason to expect it to
hold. The statement that the region is not empty means
that it has other characteristics besides metric, and the
metre rod can then find other lengths besides curvatures
to measure itself against. Referring to the earlier (suf-
ficiently approximate) ｅxpression of the law, the ten
principal coefficients of curvature are zero in empty
space but have non-zero values in non-empty space. It
is therefore natural to use these coefficients as a measure
of the fullness of space.
one of the coefficients corresponds to mass (or
energy) and in most practical cases it outweighs the
others in importance. The old definition of mass as
"quantity of matter" associates it with a fullness of
space. Three other coefficients make up the momentum
— a directed quantity with three independent com-
ponents. The remaining six coefficients of principal
curvature make up the stress or pressure-system. Mass,
momentum and stress accordingly represent the non-
emptiness of a region in so far as it is able to disturb
the usual surveying apparatus with which we explore
space — clocks, scales, light-rays, etc. It should be
added, however, that this is a summary description and
not a full account of the non-emptiness, because we
have other exploring apparatus — magnets, electroscopes,
etc. — which provide further details. It is usually con-
sidered that when we use these we are exploring not
space, but a field in space. The distinction thus created
is a rather artificial one which is unlikely to be accepted
permanently. It would seem that the results of ex-
ploring the world with a measuring scale and a magnetic
compass respectively ought to be welded together into
154 GRAVITATION— THE EXPLANATION
a unified description, just as we have welded together
results of exploration with a scale and a clock. Some
progress has been made towards this unification. There
is, however, a real reason for admitting a partially
separate treatment; the one mode of exploration deter-
mines the symmetrical properties and the other the
antisymmetrical properties of the underlying world-
structure.*
Objection has often been taken, especially by philo-
sophical writers, to the crudeness of Einstein's initial
requisitions, viz. a clock and a measuring scale. But the
body of experimental knowledge of the world which
Einstein's theory seeks to set in order has not come into
our minds as a heaven-sent inspiration; it is the result
of a survey in which the clock and the scale have actually
played the leading part. They may seem very gross
instruments to those accustomed to the conceptions of
atoms and electrons, but it is correspondingly gross
knowledge that we have been discussing in the chapters
concerned with Einstein's theory. As the relativity
theory develops, it is generally found desirable to replace
the clock and scale by the moving particle and light-
ray as the primary surveying appliances; these are test
bodies of simpler structure. But they are still gross
compared with atomic phenomena. The light-ray, for
instance, is not applicable to measurements so refined
that the diffraction of light must be taken into account.
Our knowledge of the external world cannot be divorced
from the nature of the appliances with which we have
obtained the knowledge. The truth of the law of gravi-
tation cannot be regarded as subsisting apart from the
experimental procedure by which we have ascertained
its truth.
* See p. 236.
NON-EMPTY SPACE 155
The conception of frames of space and time, and of the
non-emptiness of the world described as energy, momen-
tum, etc., is bound up with the survey by gross ap-
pliances. When they can no longer be supported by
such a survey, the conceptions melt away into meaning-
lessness. In particular the interior of the atom could
not conceivably be explored by a gross survey. We
cannot put a clock or a scale into the interior of an atom.
It cannot be too strongly insisted that the terms dis-
tance, period of time, mass, energy, momentum, etc.,
cannot be used in a description of an atom with the
same meanings that they have in our gross experience.
The atomic physicist who uses these terms must find
his own meanings for them — must state the appliances
which he requisitions when he imagines them to be
measured. It is sometimes supposed that (in addition
to electrical forces) there is a minute gravitational
attraction between an atomic nucleus and the satellite
electrons, obeying the same law as the gravitation
between the sun and its planets. The supposition seems
to me fantastic; but it is impossible to discuss it without
any indication as to how the region within the atom is
supposed to have been measured up. Apart from such
measuring up the electron goes as it pleases "like the
blessed gods".
We have reached a point of great scientific and philo-
sophic interest. The ten principal coefficients of cur-
vature of the world are not strangers to us; they are
already familiar in scientific discussion under other
names (energy, momentum, stress). This is comparable
with a famous turning-point in the development of elec-
tromagnetic theory. The progress of the subject led to
the consideration of waves of electric and magnetic force
travelling through the aether; then it flashed upon
156 GRAVITATION— THE EXPLANATION
Maxwell that these waves were not strangers but were
already familiar in our experience under the name of
light. The method of identification is the same. It is
calculated that electromagnetic waves will have just
those properties which light is observed to have; so too
it is calculated that the ten coefficients of curvature have
just those properties which energy, momentum and stress
are observed to have. We refer here to physical pro-
perties only. No physical theory is expected to explain
why there is a particular kind of image in our minds
associated with light, nor why a conception of substance
has arisen in our minds in connection with those parts
of the world containing mass.
This leads to a considerable simplification, because
identity replaces causation. on the Newtonian theory
no explanation of gravitation would be considered com-
plete unless it described the mechanism by which a piece
of matter gets a grip on the surrounding medium and
makes it the carrier of the gravitational influence radi-
ating from the matter. Nothing corresponding to this
is required in the present theory. We do not ask how
mass gets a grip on space-time and causes the curvature
which our theory postulates. That would be as super-
fluous as to ask how light gets a grip on the electro-
magnetic medium so as to cause it to oscillate. The
light is the oscillation; the mass is the curvature. There
is no causal effect to be attributed to mass; still less is
there any to be attributed to matter. The conception
of matter, which we associate with these regions of un-
usual contortion, is a monument erected by the mind to
mark the scene of conflict. When you visit the site of a
battle, do you ever ask how the monument that com-
memorates it can have caused so much carnage?
The philosophic outcome of this identification will
NON-EUCLIDEAN GEOMETRY 157
occupy us considerably in later chapters. Before leaving
the subject of gravitation I wish to say a little about
the meaning of space-curvature and non-Euclidean
geometry.
Non-Euclidean Geometry. I have been encouraging you
to think of space-time as curved; but I have been careful
to speak of this as a picture, not as a hypothesis. It is
a graphical representation of the things we are talking
about which supplies us with insight and guidance.
What we glean from the picture can be expressed in a
more non-committal way by saying that space-time has
non-Euclidean geometry. The terms "curved space"
and "non-Euclidean space" are used practically synony-
mously; but they suggest rather different points of view.
When we were trying to conceive finite and unbounded
space (p. 81) the difficult step was the getting rid of
the inside and the outside of the hypersphere. There is
a similar step in the transition from curved space to
non-Euclidean space — the dropping of all relations to
an external (and imaginary) scaffolding and the holding
on to those relations which exist within the space itself.
If you ask what is the distance from Glasgow to New
York there are two possible replies. one man will tell
you the distance measured over the surface of the
ocean; another will recollect that there is a still shorter
distance by tunnel through the earth. The second man
makes use of a dimension which the first had put out
of mind. But if two men do not agree as to distances,
they will not agree as to geometry; for geometry treats
of the laws of distances. To forget or to be ignorant of
a dimension lands us into a different geometry. Dis-
tances for the second man obey a Euclidean geometry
of three dimensions; distances for the first man obey
158 GRAVITATION— THE EXPLANATION
a non-Euclidean geometry of two dimensions. And so
if you concentrate your attention on the earth's surface
so hard that you forget that there is an inside or an
outside to it, you will say that it is a two-dimensional
manifold with non-Euclidean geometry; but if you
recollect that there is three-dimensional space all round
which affords shorter ways of getting from point to
point, you can fly back to Euclid after all. You will then
"explain away" the non-Euclidean geometry by saying
that what you at first took for distances were not the
proper distances. This seems to be the easiest way of
seeing how a non-Euclidean geometry can arise —
through mislaying a dimension — but we must not infer
that non-Euclidean geometry is impossible unless it arises
from this cause.
In our four-dimensional world pervaded by gravitation
the distances obey a non-Euclidean geometry. Is this
because we are concentrating attention wholly on its
four dimensions and have missed the short cuts through
regions beyond? By the aid of six extra dimensions we
can return to Euclidean geometry; in that case our usual
distances from point to point in the world are not the
"true" distances, the latter taking shorter routes through
an eighth or ninth dimension. To bend the world in a
super-world of ten dimensions so as to provide these
short cuts does, I think, help us to form an idea
of the properties of its non-Euclidean geometry; at any
rate the picture suggests a useful vocabulary for de-
scribing those properties. But we are not likely to accept
these extra dimensions as a literal fact unless we regard
non-Euclidean geometry as a thing which at all costs
must be explained away.
Of the two alternatives — a curved manifold in a
Euclidean space of ten dimensions or a manifold with
NON-EUCLIDEAN GEOMETRY 159
non-Euclidean geometry and no extra dimensions —
which is right? I would rather not attempt a direct
answer, because I fear I should get lost in a fog of
metaphysics. But I may say at once that I do not take
the ten dimensions seriously; whereas I take the non-
Euclidean geometry of the world very seriously, and
I do not regard it as a thing which needs explaining
away. The view, which some of us were taught at
school, that the truth of Euclid's axioms can be seen in-
tuitively, is universally rejected nowadays. We can no
more settle the laws of space by intuition than we can
settle the laws of heredity. If intuition is ruled out, the
appeal must be to experiment — genuine open-minded ex-
periment unfettered by any preconception as to what the
verdict ought to be. We must not afterwards go back
on the experiments because they make out space to be
very slightly non-Euclidean. It is quite true that a way
out could be found. By inventing extra dimensions we
can make the non-Euclidean geometry of the world
depend on a Euclidean geometry of ten dimensions; had
the world proved to be Euclidean we could, I believe,
have made its geometry depend on a non-Euclidean
geometry of ten dimensions. No one would treat the
latter suggestion seriously, and no reason can be given
for treating the former more seriously.
I do not think that the six extra dimensions have any
stalwart defenders; but we. often meet with attempts to
reimpose Euclidean geometry on the world in another
way. The proposal, which is made quite unblushingly,
is that since our measured lengths do not obey Euclidean
geometry we must apply corrections to them — cook them
— till they do. A closely related view often advocated
is that space is neither Euclidean nor non-Euclidean;
it is all a matter of convention and we are free to
i6o GRAVITATION— THE EXPLANATION
adopt any geometry we choose.* Naturally if we hold
ourselves free to apply any correction we like to our
experimental measures we can make them obey any
law; but was it worth while saying this? The asser-
tion that any kind of geometry is permissible could only
be made on the assumption that lengths have no fixed
value — that the physicist does not (or ought not to)
mean anything in particular when he talks of length.
I am afraid I shall have a difficulty in making my
meaning clear to those who start from the assumption
that my words mean nothing in particular; but for those
who will accord them some meaning I will try to remove
any possible doubt. The physicist is accustomed to state
lengths to a great number of significant figures; to
ascertain the significance of these lengths we must notice
how they are derived; and we find that they are derived
from a comparison with the extension of a standard of
specified material constitution. (We may pause to notice
that the extension of a standard material configuration
may rightly be regarded as one of the earliest subjects
of inquiry in a physical survey of our environment.)
These lengths are a gateway through which knowledge
of the world around us is sought. Whether or not they
will remain prominent in the final picture of world-
structure will transpire as the research proceeds; we do
not prejudge that. Actually we soon find that space-
lengths or time-lengths taken singly are relative, and only
* As a recent illustration of this attitude I may refer to Bertrand
Russell's Analysis of Matter, p. 78 — a book with which I do not often
seriously disagree. "Whereas Eddington seems to regard it as necessary
to adopt Einstein's variable space, Whitehead regards it as necessary
to reject it. For my part, I do not see why we should agree with either
view; the matter seems to be one of convenience in the interpretation of
formulae." Russell's view is commended in a review by C. D. Broad.
See also footnote, p. 142.
NON-EUCLIDEAN GEOMETRY 161
a combination of them could be expected to appear
even in the humblest capacity in the ultimate world-
structure. Meanwhile the first step through the gate-
way takes us to the geometry obeyed by these lengths
— very nearly Euclidean, but actually non-Euclidean and,
as we have seen, a distinctive type of non-Euclidean
geometry in which the ten principal coefficients of cur-
vature vanish. We have shown in this chapter that
the limitation is not arbitrary; it is a necessary property
of lengths expressed in terms of the extension of a ma-
terial standard, though it might have been surprising if
it had occurred in lengths defined otherwise. Must we
stop to notice the interjection that if we had meant
something different by length we should have found a
different geometry? Certainly we should; and if we
had meant something different by electric force we should
have found equations different from Maxwell's equations.
Not only empirically but also by theoretical reasoning,
we reach the geometry which we do because our lengths
mean what they do.
I have too long delayed dealing with the criticism of
the pure mathematician who is under the impression
that geometry is a subject that belongs entirely to him.
Each branch of experimental knowledge tends to have
associated with it a specialised body of mathematical
investigations. The pure mathematician, at first called in
as servant, presently likes to assert himself as master;
the connexus of mathematical propositions becomes for
him the main subject, and he does not ask permission
from Nature when he wishes to vary or generalise the
original premises. Thus he can arrive at a geometry
unhampered by any restriction from actual space meas-
ures; a potential theory unhampered by any question
as to how gravitational and electrical potentials really
1 62 GRAVITATION— THE EXPLANATION
behave; a hydrodynamics of perfect fluids doing things
which it would be contrary to the nature of any material
fluid to do. But it seems to be only in geometry that
he has forgotten that there ever was a physical subject
of the same name, and even resents the application of
the name to anything but his network of abstract math-
ematics. I do not think it can be disputed that, both
etymologically and traditionally, geometry is the science
of measurement of the space around us; and however
much the mathematical superstructure may now over-
weigh the observational basis, it is properly speaking an
experimental science. This is fully recognised in the
"reformed" teaching of geometry in schools; boys are
taught to verify by measurement that certain of the
geometrical propositions are true or nearly true. No
one questions the advantage of an unfettered develop-
ment of geometry as a pure mathematical subject; but
only in so far as this subject is linked to the quantities
arising out of observation and measurement, will it find
mention in a discussion of the Nature of the Physical
World.
Chapter VIII
MAN'S PLACE IN THE UNIVERSE
The Sidereal Universe. The largest telescopes reveal
about a thousand million stars. Each increase in tele-
scopic power adds to the number and we can scarcely
set a limit to the multitude that must exist. Nevertheless
there are signs of exhaustion, and it is clear that the
distribution which surrounds us does not extend uni-
formly through infinite space. At first an increase in
light-grasp by one magnitude brings into view three
times as many stars; but the factor diminishes so that
at the limit of faintness reached by the giant telescopes
a gain of one magnitude multiplies the number of stars
seen by only 1.8, and the ratio at that stage is rapidly
decreasing. It is as though we are approaching a limit
at which increase of power will not bring into view very
many additional stars.
Attempts have been made to find the whole number
of stars by a risky extrapolation of these counts, and
totals ranging from 3000 to 30,000 millions are some-
times quoted. But the difficulty is that the part of the
stellar universe which we mainly survey is a local con-
densation or star-cloud forming part of a much greater
system. In certain directions in the sky our telescopes
penetrate to the limits of the system, but in other direc-
tions the extent is too great for us to fathom. The
Milky Way, which on a dark night forms a gleaming
belt round the sky, shows the direction in which there
lie stars behind stars until vision fails. This great
flattened distribution is called the Galactic System. It
forms a disc of thickness small compared to its areal
163
164 MAN'S PLACE IN THE UNIVERSE
extent. It is partly broken up into subordinate con-
densations, which are probably coiled in spiral form like
the spiral nebulae which are observed in great numbers
in the heavens. The centre of the galactic system lies
somewhere in the direction of the constellation Sagit-
tarius; it is hidden from us not only by great distance but
also to some extent by tracts of obscuring matter (dark
nebulosity) which cuts off the light of the stars behind.
We must distinguish then between our local star-
cloud and the great galactic system of which it is a part.
Mainly (but not exclusively) the star-counts relate to
the local star-cloud, and it is this which the largest
telescopes are beginning to exhaust. It too has a flat-
tened form — flattened nearly in the same plane as the
galactic system. If the galactic system is compared to
a disc, the local star-cloud may be compared to a bun,
its thickness being about one-third of its lateral ex-
tension. Its size is such that light takes at least 2000
years to cross from one side to the other; this measure-
ment is necessarily rough because it relates to a vague
condensation which is probably not sharply separated
from other contiguous condensations. The extent of
the whole spiral is of the order 100,000 light years. It
can scarcely be doubted that the flattened form of the
system is due to rapid rotation, and indeed there is
direct evidence of strong rotational velocity; but it is
one of the unexplained mysteries of evolution that
nearly all celestial bodies have come to be endowed with
fast rotation.
Amid this great population the sun is a humble unit.
It is a very ordinary star about midway in the scale of
brilliancy. We know of stars which give at least 10,000
times the light of the sun; we know also of stars which
give 1/10,000 of its light. But those of inferior light
THE SIDEREAL UNIVERSE 165
greatly outnumber those of superior light. In mass, in
surface temperature, in bulk, the sun belongs to a very
common class of stars; its speed of motion is near the
average; it shows none of the more conspicuous phe-
nomena such as variability which excite the attention
of astronomers. In the community of stars the sun
corresponds to a respectable middle-class citizen. It
happens to be quite near the centre of the local star-
cloud; but this apparently favoured position is dis-
counted by the fact that the star-cloud itself is placed
very eccentrically in relation to the galactic system, being
in fact near the confines of it. We cannot claim to be
at the hub of the universe.
The contemplation of the galaxy impresses us with
the insignificance of our own little world; but we have
to go still lower in the valley of humiliation. The
galactic system is one among a million or more spiral
nebulae. There seems now to be no doubt that, as has
long been suspected, the spiral nebulae are "island uni-
verses" detached from our own. They too are great
systems of stars — or systems in the process of developing
into stars — built on the same disc-like plan. We see
some of them edgeways and can appreciate the flatness
of the disc; others are broadside on and show the ar-
rangement of the condensations in the form of a double
spiral. Many show the effects of dark nebulosity
breaking into the regularity -and blotting out the star-
light. In a few of the nearest spirals it is possible to
detect the brightest of the stars individually; variable
stars and novae (or "new stars") are observed as in our
own system. From the apparent magnitudes of the stars
of recognisable character (especially the Cepheid vari-
ables) it is possible to judge the distance. The nearest
spiral nebula is 850,000 light years away.
166 MAN'S PLACE IN THE UNIVERSE
From the small amount of data yet collected it would
seem that our own nebula or galactic system is ex-
ceptionally large; it is even suggested that if the spiral
nebulae are "islands" the galactic system is a "con-
tinent". But we can scarcely venture to claim premier
rank without much stronger evidence. At all events
these other universes are aggregations of the order of
ioo million stars.
Again the question raises itself, How far does this
distribution extend? Not the stars this time but uni-
verses stretch one behind the other beyond sight. Does
this distribution too come to an end? It may be that
imagination must take another leap, envisaging super-
systems which surpass the spiral nebulae as the spiral
nebulae surpass the stars. But there is one feeble gleam
of evidence that perhaps this time the summit of the
hierarchy has been reached, and that the system of
the spirals is actually the whole world. As has already
been explained the modern view is that space is finite —
finite though unbounded. In such a space light which
has travelled an appreciable part of the way "round the
world" is slowed down in its vibrations, with the result
that all spectral lines are displaced towards the red.
Ordinarily we interpret such a red displacement as sig-
nifying receding velocity in the line of sight. Now it is
a striking fact that a great majority of the spirals which
have been measured show large receding velocities often
exceeding iooo kilometres per second. There are only
two serious exceptions, and these are the largest spirals
which must be nearer to us than most of the others.
on ordinary grounds it would be difficult to explain why
these other universes should hurry away from us so fast
and so unanimously. Why should they shun us like
a plague? But the phenomenon is intelligible if what
THE SCALE OF TIME 167
has really been observed is the slowing down of vibra-
tions consequent on the light from these objects having
travelled an appreciable part of the way round the world.
on that theory the radius of space is of the order twenty
times the average distance of the nebulae observed, or
say 100 million light years. That leaves room for a
few million spirals; but there is nothing beyond. There
is no beyond — in spherical space "beyond" brings us
back towards the earth from the opposite direction.*
The Scale of Time. The corridor of time stretches back
through the past. We can have no conception how it
all began. But at some stage we imagine the void to
have been filled with matter rarified beyond the most
tenuous nebula. The atoms sparsely strewn move hither
and thither in formless disorder.
Behold the throne
Of Chaos and his dark pavilion spread
Wide on the wasteful deep.
Then slowly the power of gravitation is felt. Centres
of condensation begin to establish themselves and draw
in other matter. The first partitions are the star-systems
such as our galactic system; sub-condensations separate
the star-clouds or clusters; these divide again to give
the stars.
Evolution has not reached the same development in
*A very much larger radius of space (io 11 light years) has recently
been proposed by Hubble; but the basis of his calculation, though con-
cerned with spiral nebulae, is different and to my mind unacceptable.
It rests on an earlier theory of closed space proposed by Einstein which
has generally been regarded as superseded. The theory given above (due
to W. de Sitter) is, of course, very speculative, but it is the only clue we
possess as to the dimensions of space.
168 MAN'S PLACE IN THE UNIVERSE
all parts. We observe nebulae and clusters in different
stages of advance. Some stars are still highly diffuse;
others are concentrated like the sun with density greater
than water; others, still more advanced, have shrunk to
unimaginable density. But no doubt can be entertained
that the genesis of the stars is a single process of evolu-
tion which has passed and is passing over a primordial
distribution. Formerly it was freely speculated that the
birth of a star was an individual event like the birth of
an animal. From time to time two long extinct stars
would collide and be turned into vapour by the energy of
the collision; condensation would follow and life as a
luminous body would begin all over again. We can
scarcely affirm that this will never occur and that the
sun is not destined to have a second or third innings;
but it is clear from the various relations traced among
the stars that the present stage of existence of the
sidereal universe is the first innings. Groups of stars are
found which move across the sky with common proper
motion; these must have had a single origin and cannot
have been formed by casual collisions. Another aban-
doned speculation is that lucid stars may be the excep-
tion, and that there may exist thousands of dead stars
for every one that is seen shining. There are ways of
estimating the total mass in interstellar space by its
gravitational effect on the average speed of the stars;
it is found that the lucid stars account for something
approaching the total mass admissible and the amount
left over for dark stars is very limited.
Biologists and geologists carry back the history of the
earth some thousand million years. Physical evidence
based on the rate of transmutation of radioactive sub-
stances seems to leave no escape from the conclusion
that the older (Archaean) rocks in the earth's crust were
PLURALITY OF WORLDS 169
laid down 1200 million years ago. The sun must have
been burning still longer, living (we now think) on its
own matter which dissolves bit by bit into radiation.
According to the theoretical time-scale, which seems
best supported by astronomical evidence, the beginning
of the sun as a luminous star must be dated five billion
(5-I0 12 ) years ago. The theory which assigns this date
cannot be trusted confidently, but it seems a reasonably
safe conclusion that the sun's age does not exceed this
limit. The future is not so restricted and the sun may
continue as a star of increasing feebleness for 50 or
500 billion years. The theory of sub-atomic energy
has prolonged the life of a star from millions to bil-
lions of years, and we may speculate on processes
of rejuvenescence which might prolong the exist-
ence of the sidereal universe from billions to trillions
of years. But unless we can circumvent the second
law of thermodynamics — which is as much as to
say unless we can find cause for time to run back-
wards — the ultimate decay draws surely nearer and the
world will at the last come to a state of uniform
changelessness.
Does this prodigality of matter, of space, of time,
find its culmination in Man?
Plurality of Worlds. I will here put together the present
astronomical evidence as tQ the habitability of other
worlds. The popular idea that an answer to this ques-
tion is one of the main aims of the study of celestial
objects is rather disconcerting to the astronomer. Any-
thing that he has to contribute is of the nature of frag-
mentary hints picked up in the course of investigations
with more practicable and commonplace purposes.
Nevertheless, the mind is irresistibly drawn to play with
170 MAN'S PLACE IN THE UNIVERSE
the thought that somewhere in the universe there may
be other beings "a little lower than the angels" whom
Man may regard as his equals — or perhaps his
superiors.
It is idle to guess the forms that life might take in con-
ditions differing from those of our planet. If I have
rightly understood the view of palaeontologists, mam-
malian life is the third terrestrial dynasty — Nature's
third attempt to evolve an order of life sufficiently flex-
ible to changing conditions and fitted to dominate the
earth. Minor details in the balance of circumstances
must greatly affect the possibility of life and the type of
organism destined to prevail. Some critical branch-
point in the course of evolution must be negotiated be-
fore life can rise to the level of consciousness. All this
is remote from the astronomer's line of study. To avoid
endless conjecture I shall assume that the required con-
ditions of habitability are not unlike those on the earth,
and that if such conditions obtain life will automatically
make its appearance.
We survey first the planets of the solar system; of
these only Venus and Mars seem at all eligible. Venus,
so far as we know, would be well adapted for life
similar to ours. It is about the same size as the earth,
nearer the sun but probably not warmer, and it possesses
an atmosphere of satisfactory density. Spectroscopic ob-
servation has unexpectedly failed to give any indication of
oxygen in the upper atmosphere and thus suggests a
doubt as to whether free oxygen exists on the planet;
but at present we hesitate to draw so definite an infer-
ence. If transplanted to Venus we might perhaps con-
tinue to live without much derangement of habit —
except that I personally would have to find a new pro-
fession, since Venus is not a good place for astronomers.
PLURALITY OF WORLDS 171
It is completely covered with cloud or mist. For this
reason no definite surface markings can be made out,
and it is still uncertain how fast it rotates on its axis
and in which direction the axis lies. one curious theory
may be mentioned though it should perhaps not be taken
too seriously. It is thought by some that the great
cavity occupied by the Pacific Ocean is a scar left by the
moon when it was first disrupted from the earth. Evi-
dently this cavity fulfils an important function in drain-
ing away superfluous water, and if it were filled up
practically all the continental area would be submerged.
Thus indirectly the existence of dry land is bound up
with the existence of the moon. But Venus has no moon,
and since it seems to be similar to the earth in other
respects, it may perhaps be inferred that it is a world
which is all ocean — where fishes are supreme. The
suggestion at any rate serves to remind us that the
destinies of organic life may be determined by what
are at first sight irrelevant accidents.
The sun is an ordinary star and the earth is an
ordinary planet, but the moon is not an ordinary satel-
lite. No other known satellite is anything like so large
in proportion to the planet which it attends. The moon
contains about 1/80 part of the mass of the earth which
seems a small ratio; but it is abnormally great compared
with other satellites. The next highest ratio is found
in the system of Saturn whose largest satellite Titan has
1/4000 of the planet's mass. Very special circum-
stances must have occurred in the history of the earth
to have led to the breaking away of so unusual a frac-
tion of the mass. The explanation proposed by Sir
George Darwin, which is still regarded as most prob-
able, is that a resonance in period occurred between
the solar tides and the natural free period of vibration
172 MAN'S PLACE IN THE UNIVERSE
of the globe of the earth. The tidal deformation of the
earth thus grew to large amplitude, ending in a cata-
clysm which separated the great lump of material that
formed the moon. Other planets escaped this dangerous
coincidence of period, and their satellites separated by-
more normal development. If ever I meet a being who
has lived in another world, I shall feel very humble in
most respects, but I expect to be able to boast a little
about the moon.
Mars is the only planet whose solid surface can be
seen and studied; and it tempts us to consider the possi-
bility of life in more detail. Its smaller size leads to
considerably different conditions; but the two essentials,
air and water, are both present though scanty. The
Martian atmosphere is thinner than our own but it is
perhaps adequate. It has been proved to contain oxy-
gen. There is no ocean; the surface markings repre-
sent, not sea and land, but red desert and darker ground
which is perhaps moist and fertile. A conspicuous fea-
ture is the white cap covering the pole which is clearly
a deposit of snow; it must be quite shallow since it melts
away completely in the summer. Photographs show
from time to time indubitable clouds which blot out
temporarily large areas of surface detail; clear weather,
however, is more usual. The air, if cloudless, is slightly
hazy. W. H. Wright has shown this very convincingly
by comparing photographs taken with light of dif-
ferent wave-lengths. Light of short wave-length is
much scattered by haze and accordingly the ordinary
photographs are disappointingly blurry. Much sharper
surface-detail is shown when visual yellow light is
employed (a yellow screen being commonly used to
adapt visual telescopes for photography) ; being of
longer wave-length the visual rays penetrate the haze
PLURALITY OF WORLDS 173
more easily.* Still clearer detail is obtained by photo-
graphing with the long infra-red waves.
Great attention has lately been paid to the deter-
mination of the temperature of the surface of Mars; it
is possible to find this by direct measurement of the heat
rediated to us from different parts of the surface. The
results, though in many respects informative, are
scarcely accurate and accordant enough to give a defi-
nite idea of the climatology. Naturally the tempera-
ture varies a great deal between day and night and in
different latitudes; but on the average the conditions
are decidedly chilly. Even at the equator the tempera-
ture falls below freezing point at sunset. If we accepted
the present determinations as definitive we should have
some doubt as to whether life could endure the con-
ditions.
In one of Huxley's Essays there occurs the passage
"Until human life is longer and the duties of the
present press less heavily I do not think that wise men
will occupy themselves with Jovian or Martian natural
history." To-day it would seem that Martian natural
history is not altogether beyond the limits of serious
science. At least the surface of Mars shows a seasonal
change such as we might well imagine the forest-clad
earth would show to an outside onlooker. This seasonal
change of appearance is very conspicuous to the atten-
tive observer. As the spring in one hemisphere advances
(I mean, of course, the Martian spring), the darker
areas, which are at first few and faint, extend and
deepen in contrast. The same regions darken year after
* It seems to have been a fortunate circumstance that the pioneers
of Martian photography had no suitable photographic telescopes and
had to adapt visual telescopes — thus employing visual (yellow) light
which, as it turned out, was essential for good results.
i 7 4 MAN'S PLACE IN THE UNIVERSE
year at nearly the same date in the Martian calendar.
It may be that there is an inorganic explanation; the
spring rains moisten the surface and change its colour.
But it is perhaps unlikely that there is enough rain
to bring about this change as a direct effect. It is
easier to believe that we are witnessing the annual
awakening of vegetation so familiar on our own
planet.
The existence of oxygen in the Martian atmosphere
supplies another argument in support of the existence
of vegetable life. Oxygen combines freely with many
elements, and the rocks in the earth's crust are thirsty
for oxygen. They would in course of time bring about
its complete disappearance from the air, were it not that
the vegetation extracts it from the soil and sets it free
again. If oxygen in the terrestrial atmosphere is main-
tained in this way, it would seem reasonable to assume
that vegetable life is required to play the same part on
Mars. Taking this in conjunction with the evidence of
the seasonal changes of appearance, a rather strong case
for the existence of vegetation seems to have been made
out.
If vegetable life must be admitted, can we exclude
animal life? I have come to the end of the astronomical
data and can take no responsibility for anything further
that you may infer. It is true that the late Prof. Lowell
argued that certain more or less straight markings on
the planet represent an artificial irrigation system and
are the signs of an advanced civilisation; but this theory
has not, I think, won much support. In justice to the
author of this speculation it should be said that his own
work and that of his observatory have made a magni-
ficent contribution to our knowledge of Mars; but few
would follow him all the way on the more picturesque
FORMATION OF PLANETARY SYSTEMS 175
side of his conclusions.* Finally we may stress one
point. Mars has every appearance of being a planet
long past its prime; and it is in any case improbable that
two planets differing so much as Mars and the Earth
would be in the zenith of biological development con-
temporaneously.
Formation of Planetary Systems. If the planets of the
solar system should fail us, there remain some thousands
of millions of stars which we have been accustomed to
regard as suns ruling attendant systems of planets. It
has seemed a presumption, bordering almost on impiety,
to deny to them life of the same order of creation as
ourselves. It would indeed be rash to assume that
nowhere else in the universe has Nature repeated the
strange experiment which she has performed on the
earth. But there are considerations which must hold us
back from populating the universe too liberally.
on examining the stars with a telescope we are sur-
prised to find how many of those which appear single
points to the eye are actually two stars close together.
When the telescope fails to separate them the spectro-
scope often reveals two stars in orbital revolution round
each other. At least one star in three is double — a pair
of self-luminous globes both comparable in dimensions
with the sun. The single supreme sun is accordingly
not the only product of evolution; not much less fre-
quently the development has taken another turn and
resulted in two suns closely associated. We may prob-
ably rule out the possibility of planets in double stars.
♦Mars is not seen under favourable conditions except from low lati-
tudes and high altitudes. Astronomers who have not these advantages
are reluctant to form a decided opinion on the many controversial points
that have arisen.
176 MAN'S PLACE IN THE UNIVERSE
Not only is there a difficulty in ascribing to them per-
manent orbits under the more complicated field of gravi-
tation, but a cause for the formation of planets seems
to be lacking. The star has satisfied its impulse to
fission in another manner; it has divided into two nearly
equal portions instead of throwing off a succession of
tiny fragments.
The most obvious cause of division is excessive rota-
tion. As the gaseous globe contracts it spins fast and
faster until a time may come when it can no longer hold
together, and some kind of relief must be found. Ac-
cording to the nebular hypothesis of Laplace the sun
gained relief by throwing off successively rings of matter
which have formed the planets. But were it not for
this one instance of a planetary system which is known
to us, we should have concluded from the thousands of
double stars in the sky that the common consequence of
excessive rotation is to divide the star into two bodies
of equal rank.
It might still be held that the ejection of a planetary
system and the fission into a double star are alternative
solutions of the problem arising from excessive rotation,
the star taking one course or the other according to
circumstances. We know of myriads of double stars
and of only one planetary system; but in any case it is
beyond our power to detect other planetary systems if
they exist. We can only appeal to the results of theo-
retical study of rotating masses of gas; the work pre-
sents many complications and the results may not be
final; but the researches of Sir J. H. Jeans lead to the
conclusion that rotational break-up produces a double
star and never a system of planets. The solar system is
not the typical product of development of a star; it is
not even a common variety of development; it is a freak.
FORMATION OF PLANETARY SYSTEMS 177
By elimination of alternatives it appears that a con-
figuration resembling the solar system would only be
formed if at a certain stage of condensation an unusual
accident had occurred. According to Jeans the accident
was the close approach of another star casually pursuing
its way through space. This star must have passed
within a distance not far outside the orbit of Neptune;
it must not have passed too rapidly, but have slowly
overtaken or been overtaken by the sun. By tidal dis-
tortion it raised big protuberances on the sun, and caused
it to spurt out filaments of matter which have condensed
to form the planets. That was more than a thousand
million years ago. The intruding star has since gone on
its way and mingled with the others; its legacy of a
system of planets remains, including a globe habitable
by man.
Even in the long life of a star encounters of this kind
must be extremely rare. The density of distribution of
stars in space has been compared to that of twenty
tennis-balls roaming the whole interior of the earth.
The accident that gave birth to the solar system may be
compared to the casual approach of two of these balls
within a few yards of one another. The data are too
vague to give any definite estimate of the odds against
this occurence, but I should judge that perhaps not one
in a hundred millions of stars can have undergone this
experience in the right stage and conditions to result in
the formation of a system of planets.
However doubtful this conclusion as to the rarity of
solar systems may be, it is a useful corrective to the view
too facilely adopted which looks upon every star as a
likely minister 'to life. We know the prodigality of
Nature. How many acorns are scattered for one that
grows to an oak? And need she be more careful of her
i;8 MAN'S PLACE IN THE UNIVERSE
stars than of her acorns? If indeed she has no grander
aim than to provide a home for her greatest experiment,
Man, it would be just like her methods to scatter a mil-
lion stars whereof one might haply achieve her purpose.
The number of possible abodes of life severely
restricted in this way at the outset may no doubt be
winnowed down further. on our house-hunting expedi-
tion we shall find it necessary to reject many apparently
eligible mansions on points of detail. Trivial circum-
stances may decide whether organic forms originate at
all; further conditions may decide whether life ascends
to a complexity like ours or remains in a lower form.
I presume, however, that at the end of the weeding
out there will be left a few rival earths dotted here and
there about the universe.
A further point arises if we have especially in mind
contemporaneous life. The time during which man has
been on the earth is extremely small compared with the
age of the earth or of the sun. There is no obvious
physical reason why, having once arrived, man should
not continue to populate the earth for another ten billion
years or so; but — well, can you contemplate it? Assum-
ing that the stage of highly developed life is a very
small fraction of the inorganic history of the star, the
rival earths are in general places where conscious life
has already vanished or is yet to come. I do not think
that the whole purpose of the Creation has been staked
on the one planet where we live; and in the long run we
cannot deem ourselves the only race that has been or
will be gifted with the mystery of consciousness. But
I feel inclined to claim that at the present time our race
is supreme; and not one of the profusion of stars in
their myriad clusters looks down on scenes comparable
to those which are passing beneath the rays of the sun.
Chapter IX
THE QUANTUM THEORY
The Origin of the Trouble. Nowadays whenever en-
thusiasts meet together to discuss theoretical physics the
talk sooner or later turns in a certain direction. You
leave them conversing on their special problems or the
latest discoveries; but return after an hour and it is any
odds that they will have reached an all-engrossing topic
— the desperate state of their ignorance. This is not a
pose. It is not even scientific modesty, because the atti-
tude is often one of naive surprise that Nature should
have hidden her fundamental secret successfully from
such powerful intellects as ours. It is simply that we
have turned a corner in the path of progress and our
ignorance stands revealed before us, appalling and insist-
ent. There is something radically wrong with the pres-
ent fundamental conceptions of physics and we do not
see how to set it right.
The cause of all this trouble is a little thing called h
which crops up continually in a wide range of experi-
ments. In one sense we know just what h is, because
there are a variety of ways of measuring it; h is
.0000000000000000000000000065 5 erg-seconds.
That will (rightly) suggest to you that h is something
very small; but the most important information is con-
tained in the concluding phrase erg-seconds. The erg
is the unit of energy and the second is the unit of time;
so that we learn that h is of the nature of energy multi-
plied by time.
Now in practical life it does not often occur to us to
179
180 THE QUANTUM THEORY
multiply energy by time. We often divide energy by
time. For example, the motorist divides the output of
energy of his engine by time and so obtains the horse-
power. Conversely an electric supply company multi-
plies the horse-power or kilowatts by the number of
hours of consumption and sends in its bill accordingly.
But to multiply by hours again would seem a very odd
sort of thing to do.
But it does not seem quite so strange when we look
at it in the absolute four-dimensional world. Quantities
such as energy, which we think of as existing at an
instant, belong to three-dimensional space, and they need
to be multiplied by a duration to give them a thickness
before they can be put into the four-dimensional world.
Consider a portion of space, say Great Britain; we
should describe the amount of humanity in it as 40
million men. But consider a portion of space-time, say
Great Britain between 19 15 and 1925; we must describe
the amount of humanity in it as 400 million man-years.
To describe the human content of the world from a
space-time point of view we have to take a unit which is
limited not only in space but in time. Similarly if some
other kind of content of space is described as so many
ergs, the corresponding content of a region of space-time
will be described as so many erg-seconds.
We call this quantity in the four-dimensional world
which is the analogue or adaptation of energy in the
three-dimensional world by the technical name action. The
name does not seem to have any special appropriateness,
but we have to accept it. Erg-seconds or action belongs
to Minkowski's world which is common to all observers,
and so it is absolute. It is one of the very few absolute
quantities noticed in pre-relativity physics. Except for
action and entropy (which belongs to an entirely different
THE ORIGIN OF THE TROUBLE 181
class of physical conceptions) all the quantities promi-
nent in pre-relativity physics refer to the three-dimen-
sional sections which are different for different observers.
Long before the theory of relativity showed us that
action was likely to have a special importance in the
scheme of Nature on account of its absoluteness, long
before the particular piece of action h began to turn up
in experiments, the investigators of theoretical dynamics
were making great use of action. It was especially the
work of Sir William Hamilton which brought it to the
fore; and since then very extensive theoretical develop-
ments of dynamics have been made on this basis. I
need only refer to the standard treatise on Analytical
Dynamics by your own (Edinburgh) Professor*, which
fairly reeks of it. It was not difficult to appreciate the
fundamental importance and significance of the main
principle; but it must be confessed that to the non-
specialist the interest of the more elaborate develop-
ments did not seem very obvious — except as an ingenious
way of making easy things difficult. In the end the
instinct which led to these researches has justified itself
emphatically. To follow any of the progress in the
quantum theory of the atom since about 19 17, it is
necessary to have plunged rather deeply into the Hamil-
tonian theory of dynamics. It is remarkable that just
as Einstein found ready prepared by the mathematicians
the Tensor Calculus which he needed for developing
his great theory of gravitation, so the quantum physicists
found ready for them an extensive action-theory of
dynamics without which they could not have made head-
way.
But neither the absolute importance of action in the
four-dimensional world, nor its earlier prominence in
* Prof. E. T. Whittaker.
1 82 THE QUANTUM THEORY
Hamiltonian dynamics, prepares us for the discovery
that a particular lump of it can have a special import-
ance. And yet a lump of standard size 6-55. io~ 27 erg-
seconds is continually turning up experimentally. It is
all very well to say that we must think of action as
atomic and regard this lump as the atom of action. We
cannot do it. We have been trying hard for the last
ten years. Our present picture of the world shows
action in a form quite incompatible with this kind of
atomic structure, and the picture will have to be redrawn.
There must in fact be a radical change in the funda-
mental conceptions on which our scheme of physics is
founded; the problem is to discover the particular
change required. Since 1925 new ideas have been
brought into the subject which seem to make the dead-
lock less complete, and give us an inkling of the nature
of the revolution that must come; but there has been
no general solution of the difficulty. The new ideas will
be the subject of the next chapter. Here it seems best
to limit ourselves to the standpoint of 1925, except at
the very end of the chapter, where we prepare for the
transition.
The Atom of Action. Remembering that action has two
ingredients, namely, energy and time, we must look about
in Nature for a definite quantity of energy with which
there is associated some definite period of time. That is
the way in which without artificial section a particular
lump of action can be separated from the rest of the action
which fills the universe. For example, the energy of consti-
tution of an electron is a definite and known quantity; it
is an aggregation of energy which occurs naturally in all
parts of the universe. But there is no particular duration
of time associated with it that we are aware of, and so it
THE ATOM OF ACTION 183
does not suggest to us any particular lump of action.
We must turn to a form of energy which has a definite
and discoverable period of time associated with it, such
as a train of light-waves; these carry with them a unit
of time, namely, the period of their vibration. The
yellow light from sodium consists of aethereal vibrations
of period 510 billions to the second. At first sight we
seem to be faced with the converse difficulty; we have
now our definite period of time; but how are we to cut
up into natural units the energy coming from a sodium
flame? We should, of course, single out the light pro-
ceeding from a single atom, but this will not break up
into units unless the atom emits light discontinuously.
It turns out that the atom does emit light discontin-
uously. It sends out a long train of waves and then
stops. It has to be restarted by some kind of stimula-
tion before it emits again. We do not perceive this
intermittence in an ordinary beam of light, because there
are myriads of atoms engaged in the production.
The amount of energy coming away from the sodium
atom during any one of these discontinuous emissions
is found to be 3-4. io -12 ergs. This energy is, as we
have seen, marked by a distinctive period 1-9. io~ 15 sees.
We have thus the two ingredients necessary for a
natural lump of action. Multiply them together, and
we obtain 6-55. io~ 27 erg-seconds. That is the quan-
tity h.
The remarkable law of Nature is that we are con-
tinually getting the same numerical results. We may
take another source of light — hydrogen, calcium, or any
other atom. The energy will be a different number of
ergs; the period will be a different number of seconds;
but the product will be the same number of erg-seconds.
The same applies to X-rays, to gamma rays and to other
1 84 THE QUANTUM THEORY
forms of radiation. It applies to light absorbed by an
atom as well as to light emitted, the absorption being
discontinuous also. Evidently h is a kind of atom —
something which coheres as one unit in the processes of
radiation; it is not an atom of matter but an atom or,
as we usually call it, a quantum of the more elusive
entity action. Whereas there are 92 different kinds of
material atoms there is only one quantum of action —
the same whatever the material it is associated with.
I say the same without reservation. You might perhaps
think that there must be some qualitative difference
between the quantum of red light and the quantum of
blue light, although both contain the same number of
erg-seconds; but the apparent difference is only relative
to a frame of space and time and does not concern the
absolute lump of action. By approaching the light-
source at high speed we change the red light to blue
light in accordance with Doppler's principle; the energy
of the waves is also changed by being referred to a
new frame of reference. A sodium flame and a hydro-
gen flame are throwing out at us the same lumps of
action, only these lumps are rather differently orientated
with respect to the Now lines which we have drawn
across the four-dimensional world. If we change our
motion so as to alter the direction of the Now lines,
we can see the lumps of sodium origin under the same
orientation in which we formerly saw the lumps of
hydrogen origin and recognise that they are actually the
same.
We noticed in chapter iv that the shuffling of energy
can become complete, so that a definite state is reached
known as thermodynamical equilibrium; and we re-
marked that this is only possible if indivisible units are
being shuffled. If the cards can be torn into smaller and
CONFLICT WITH WAVE-THEORY 185
smaller pieces without limit there is no end to the
process of shuffling. The indivisible units in the shuf-
fling of energy are the quanta. By radiation absorp-
tion and scattering energy is shuffled among the different
receptacles in matter and aether, but only a whole
quantum passes at each step. It was in fact this definite-
ness of thermodynamical equilibrium which first put
Prof. Max Planck on the track of the quantum; and the
magnitude of h was first calculated by analysis of the
observed composition of the radiation in the final state
of randomness. Progress of the theory in its adolescent
stage was largely due to Einstein so far as concerns the
general principles and to Bohr as regards its connection
with atomic structure.
The paradoxical nature of the quantum is that
although it is indivisible it does not hang together. We
examined first a case in which a quantity of energy was
obviously cohering together, viz. an electron, but we did
not find h; then we turned our attention to a case in which
the energy was obviously dissolving away through space,
viz. light-waves, and immediately h appeared. The
atom of action seems to have no coherence in space;
it has a unity which overleaps space. How can such a
unity be made to appear in our picture of a world
extended through space and time?
Conflict with the Wave-Theory of Light. The pursuit of
the quantum leads to many surprises; but probably none
is more outrageous to our preconceptions than the
regathering of light and other radiant energy into
A-units, when all the classical pictures show it to be
dispersing more and more. Consider the light-waves
which are the result of a single emission by a single atom
on the star Sirius. These bear away a certain amount of
1 86 THE QUANTUM THEORY
energy endowed with a certain period, and the product
of the two is //. The period is carried by the waves
without change, but the energy spreads out in an ever-
widening circle. Eight years and nine months after the
emission the wave-front is due to reach the earth. A
few minutes before the arrival some person takes it into
his head to go out and admire the glories of the heavens
and — in short — to stick his eye in the way. The light-
waves when they started could have had no notion what
they were going to hit; for all they knew they were
bound on a journey through endless space, as most of
their colleagues were. Their energy would seem to be
dissipated beyond recovery over a sphere of 50 billion
miles' radius. And yet if that energy is ever to enter
matter again, if it is to work those chemical changes in
the retina which give rise to the sensation of light, it
must enter as a single quantum of action h. Just
6-55. 1 o -27 erg-seconds must enter or none at all. Just
as the emitting atom regardless of all laws of classical
physics is determined that whatever goes out of it shall
be just /*, so the receiving atom is determined that what-
ever comes into it shall be just h. Not all the light-
waves pass by without entering the eye; for somehow
we are able to see Sirius. How is it managed? Do the
ripples striking the eye send a message round to the
back part of the wave, saying, "We have found an eye.
Let's all crowd into it!"
Attempts to account for this phenomenon follow two
main devices which we may describe as the "collection-
box" theory and the "sweepstake" theory, respectively.
Making no effort to translate them into scientific
language, they amount to this: In the first the atom
holds a collection-box into which each arriving group
of waves pays a very small contribution; when the
CONFLICT WITH WAVE-THEORY 187
amount in the box reaches a whole quantum, it enters
the atom. In the second the atom uses the small frac-
tion of a quantum offered to it to buy a ticket in a
sweepstake in which the prizes are whole quanta; some
of the atoms will win whole quanta which they can
absorb, and it is these winning atoms in our retina
which tell us of the existence of Sirius.
The collection-box explanation is not tenable. As
Jeans once said, not only does the quantum theory
forbid us to kill two birds with one stone; it will not
even let us kill one bird with two stones. I cannot go
fully into the reasons against this theory, but may
illustrate one or two of the difficulties. one serious
difficulty would arise from the half-filled collection-
boxes. We shall see this more easily if, instead of
atoms, we consider molecules which also absorb only
full quanta. A molecule might begin to collect the
various kinds of light which it can absorb, but before it
has collected a quantum of any one kind it takes part
in a chemical reaction. New compounds are formed
which no longer absorb the old kinds of light; they have
entirely different absorption spectra. They would have
to start afresh to collect the corresponding kinds of
light. What is to be done with the old accumulations
now useless, since they can never be completed? one
thing is certain; they are not tipped out into the aether
when the chemical change occurs.
A phenomenon which seems directly opposed to any
kind of collection-box explanation is the photoelectric
effect. When light shines on metallic films of sodium,
potassium, rubidium, etc., free electrons are discharged
from the film. They fly away at high speed, and it is
possible to measure experimentally their speed or
energy. Undoubtedly it is the incident light which
188 THE QUANTUM THEORY
provides the energy of these explosions, but the phe-
nomenon is governed by a remarkable rule. Firstly, the
speed of the electrons is not increased by using more
powerful light. Concentration of the light produces
more explosions but not more powerful explosions.
Secondly, the speed is increased by using bluer light, i.e.
light of shorter period. For example, the feeble light
reaching us from Sirius will cause more powerful ejec-
tions of electrons than full sunlight, because Sirius is
bluer than the sun; the remoteness of Sirius does not
weaken the ejections though it reduces their number.
This is a straightforward quantum phenomenon.
Every electron flying out of the metal has picked up just
one quantum from the incident light. Since the /z-rule
associates the greater energy with the shorter vibration
period, bluer light gives the more intense energy.
Experiments show that (after deducting a constant
"threshold" energy used up in extricating the electron
from the film) each electron comes out with a kinetic
energy equal to the energy of the quantum of incident
light.
The film can be prepared in the dark; but on ex-
posure to feeble light electrons immediately begin to
fly out before any of the collection-boxes could have
been filled by fair means. Nor can we appeal to any
trigger action of the light releasing an electron already
loaded up with energy for its journey; it is the nature
of the light which settles the amount of the load. The
light calls the tune, therefore the light must pay the
piper. only classical theory does not provide light with
a pocket to pay from.
It is always difficult to make a fence of objections so
thorough as to rule out all progress along a certain line
of explanation. But even if it is still possible to wriggle
CONFLICT WITH WAVE-THEORY 189
on, there comes a time when one begins to perceive that
the evasions are far-fetched. If we have any instinct
that can recognise a fundamental law of Nature when
it sees one, that instinct tells us that the interaction of
radiation and matter in single quanta is something lying
at the root of world-structure and not a casual detail in
the mechanism of the atom. Accordingly we turn to the
"sweepstake" theory, which sees in this phenomenon a
starting-point for a radical revision of the classical con-
ceptions.
Suppose that the light-waves are of such intensity that,
according to the usual reckoning of their energy, one-
millionth of a quantum is brought within range of each
atom. The unexpected phenomenon is that instead of
each atom absorbing one-millionth of a quantum, one
atom out of every million absorbs a whole quantum.
That whole quanta are absorbed is shown by the photo-
electric experiments already described, since each of the
issuing electrons has managed to secure the energy of a
whole quantum.
It would seem that what the light-waves were really
bearing within reach of each atom was not a millionth
of a quantum but a millionth chance of securing a whole
quantum. The wave-theory of light pictures and
describes something evenly distributed over the whole
wave-front which has usually been identified with energy.
Owing to well-established phenomena such as interfer-
ence and diffraction it seems impossible to deny this uni-
formity, but we must give it another interpretation; it
is a uniform chance of energy. Following the rather
old-fashioned definition of energy as "capacity for doing
work" the waves carry over their whole front a uniform
chance of doing work. It is the propagation of a chance
which the wave-theory studies.
190 THE QUANTUM THEORY
Different views may be held as to how the prize-
drawing is conducted on the sweepstake theory. Some
hold that the lucky part of the wave-front is already
marked before the atom is reached. In addition to the
propagation of uniform waves the propagation of a
photon or "ray of luck" is involved. This seems to me
out of keeping with the general trend of the modern
quantum theory; and although most authorities now take
this view, which is said to be indicated definitely by
certain experiments, I do not place much reliance on the
stability of this opinion.
Theory of the Atom. We return now to further experi-
mental knowledge of quanta. The mysterious quantity
h crops up inside the atom as well as outside it. Let us
take the simplest of all atoms, namely, the hydrogen
atom. This consists of a proton and an electron, that is
to say a unit charge of positive electricity and a unit
charge of negative electricity. The proton carries nearly
all the mass of the atom and remains rock-like at the
centre, whilst the nimble electron moves round in a
circular or elliptic orbit under the inverse square-law
of attraction between them. The system is thus very
like a sun and a planet. But whereas in the solar system
the planet's orbit may be of any size and any eccentricity,
the electron's orbit is restricted to a definite series of
sizes and shapes. There is nothing in the classical
theory of electromagnetism to impose such a restriction;
but the restriction exists, and the law imposing it has
been discovered. It arises because the atom is arranging
to make something in its interior equal to h. The inter-
mediate orbits are excluded because they would involve
fractions of /*, and h cannot be divided.
But there is one relaxation. When wave-energy is
THEORY OF THE ATOM 191
sent out from or taken into the atom, the amount and
period must correspond exactly to h. But as regards
its internal arrangements the atom has no objection to
2/*, 3/*, 4/i, etc.; it only insists that fractions shall be
excluded. That is why there are many alternative orbits
for the electron corresponding to different integral mul-
tipliers of h. We call these multipliers quantum num-
bers, and speak of 1 -quantum orbits, 2-quantum orbits,
etc. I will not enter here into the exact definition of
what it is that has to be an exact multiple of h; but it
is something which, viewed in the four-dimensional
world, is at once seen to be action though this may not
be so apparent when we view it in the ordinary way in
three-dimensional sections. Also several features of the
atom are regulated independently by this rule, and
accordingly there are several quantum numbers — one for
each feature; but to avoid technical complication I shall
refer only to the quantum numbers belonging to one
leading feature.
According to this picture of the atom, which is due
to Niels Bohr, the only possible change of state is the
transfer of an electron from one quantum orbit to
another. Such a jump must occur whenever light is
absorbed or emitted. Suppose then that an electron which
has been travelling in one of the higher orbits jumps
down into an orbit of less energy. The atom will then
have a certain amount of surplus energy that must be got
rid of. The lump of energy is fixed, and it remains to
settle the period of vibration that it shall have when it
changes into aether-waves. It seems incredible that the
atom should get hold of the aether and shake it in any
other period than one of those in which it is itself
vibrating. Yet it is the experimental fact that, when the
atom by radiating sets the aether in vibration, the
192 THE QUANTUM THEORY
periods of its electronic circulation are ignored and the
period of the aether-waves is settled not by any pictur-
able mechanism but by the seemingly artificial h-rulc. It
would seem that the atom carelessly throws overboard
a lump of energy which, as it glides into the aether,
moulds itself into a quantum of action by taking on the
period required to make the product of energy and
period equal to h. If this unmechanical process of emis-
sion seems contrary to our preconceptions, the exactly
converse process of absorption is even more so. Here
the atom has to look out for a lump of energy of the
exact amount required to raise an electron to the higher
orbit. It can only extract such a lump from aether-
waves of particular period — not a period which has
resonance with the structure of the atom, but the period
which makes the energy into an exact quantum.
As the adjustment between the energy of the orbit jump
and the period of the light carrying away that energy so
as to give the constant quantity h is perhaps the most
striking evidence of the dominance of the quantum, it
will be worth while to explain how the energy of an
orbit jump in an atom can be measured. It is possible to
impart to a single electron a known amount of energy by
making it travel along an electric field with a measured
drop of potential. If this projectile hits an atom it may
cause one of the electrons circulating in the atom to
jump to an upper orbit, but, of course, only if its energy
is sufficient to supply that required for the jump; if the
electron has too little energy it can do nothing and must
pass on with its energy intact. Let us fire a stream of
electrons all endowed with the same known energy
into the midst of a group of atoms. If the energy is
below that corresponding to an orbit jump, the stream
will pass through without interference other than
CLASSICAL AND QUANTUM LAWS 193
ordinary scattering. Now gradually increase the energy
of the electrons; quite suddenly we find that the electrons
are leaving a great deal of their energy behind. That
means that the critical energy has been reached and
orbit jumps are being excited. Thus we have a means
of measuring the critical energy which is just that of the
jump — the difference of energy of the two states of the
atom. This method of measurement has the advantage
that it does not involve any knowledge of the constant h,
so that there is no fear of a vicious circle when we use
the measured energies to test the h rule.* Incidentally
this experiment provides another argument against the
collection-box theory. Small contributions of energy are
not thankfully received, and electrons which offer any-
thing less than the full contribution for a jump are not
allowed to make any payment at all.
Relation of Classical Laws to Quantum Laws. To fol-
low up the verification and successful application of the
quantum laws would lead to a detailed survey of the
greater part of modern physics — specific heats, mag-
netism, X-rays, radioactivity, and so on. We must leave
this and return to a general consideration of the rela-
tion between classical laws and quantum laws. For at
least fifteen years we have used classical laws and quan-
tum laws alongside one another notwithstanding the
irreconcilability of their conceptions. In the model atom
the electrons are supposed to traverse their orbits under
the classical laws of electrodynamics; but they jump
from one orbit to another in a way entirely incon-
sistent with those laws. The energies of the orbits
* Since the h rule is now well established the energies of different
states of the atoms are usually calculated by its aid; to use these to test
the rule would be a vicious circle.
194 THE QUANTUM THEORY
in hydrogen are calculated by classical laws; but one
of the purposes of the calculation is to verify the
association of energy and period in the unit /*, which is
contrary to classical laws of radiation. The whole
procedure is glaringly contradictory but conspicuously
successful.
In my observatory there is a telescope which con-
denses the light of a star on a film of sodium in a photo-
electric cell. I rely on the classical theory to conduct
the light through the lenses and focus it in the cell; then
I switch on to the quantum theory to make the light
fetch out electrons from the sodium film to be collected
in an electrometer. If I happen to transpose the two
theories, the quantum theory convinces me that the light
will never get concentrated in the cell and the classical
theory shows that it is powerless to extract the elec-
trons if it does get in. I have no logical reason for
not using the theories this way round; only experience
teaches me that I must not. Sir William Bragg was not
overstating the case when he said that we use the classi-
cal theory on Mondays, Wednesday and Fridays, and
the quantum theory on Tuesdays, Thursdays and Satur-
days. Perhaps that ought to make us feel a little sym-
pathetic towards the man whose philosophy of the uni-
verse takes one form on weekdays and another form on
Sundays.
In the last century — and I think also in this — there
must have been many scientific men who kept their
science and religion in watertight compartments. one
set of beliefs held good in the laboratory and another set
of beliefs in church, and no serious effort was made to
harmonise them. The attitude is defensible. To discuss
the compatibility of the beliefs would lead the scientist
into regions of thought in which he was inexpert; and
any answer he might reach would be undeserving of
CLASSICAL AND QUANTUM LAWS 195
strong confidence. Better admit that there was some
truth both in science and religion; and if they must fight,
let it be elsewhere than in the brain of a hard-working
scientist. If we have ever scorned this attitude, Nemesis
has overtaken us. For ten years we have had to divide
modern science into two compartments; we have one set
of beliefs in the classical compartment and another set
of beliefs in the quantum compartment. Unfortunately
our compartments are not watertight.
We must, of course, look forward to an ultimate
reconstruction of our conceptions of the physical world
which will embrace both the classical laws and the
quantum laws in harmonious association. There are still
some who think that the reconciliation will be effected
by a development of classical conceptions. But the
physicists of what I may call "the Copenhagen school"
believe that the reconstruction has to start at the other
end, and that in the quantum phenomena we are getting
down to a more intimate contact with Nature's way of
working than in the coarse-grained experience which
has furnished the classical laws. The classical school
having become convinced of the existence of these uni-
form lumps of action, speculates on the manufacture of
the chopper necessary to carve off uniform lumps; the
Copenhagen school on the other hand sees in these
phenomena the insubstantial pageant of space, time and
matter crumbling into grains of action. I do not think
that the Copenhagen school has been mainly influenced
by the immense difficulty of constructing a satisfactory
chopper out of classical material; its view arises espe-
cially from a study of the meeting point of quantum and
classical laws.
The classical laws are the limit to which the quantum
laws tend when states of very high quantum number are
concerned.
i 9 6 THE QUANTUM THEORY
This is the famous Correspondence Principle enun-
ciated by Bohr. It was at first a conjecture based on
rather slight hints; but as our knowledge of quantum
laws has grown, it has been found that when we apply
them to states of very high quantum number they con-
verge to the classical laws, and predict just what the
classical laws would predict.
For an example, take a hydrogen atom with its elec-
tron in a circular orbit of very high quantum number,
that is to say far away from the proton. on Monday,
Wednesday and Friday it is governed by classical laws.
These say that it must emit a feeble radiation continu-
ously, of strength determined by the acceleration it is
undergoing and of period agreeing with its own period
of revolution. Owing to the gradual loss of energy it
will spiral down towards the proton. on Tuesday,
Thursday and Saturday it is governed by quantum laws
and jumps from one orbit to another. There is a
quantum law that I have not mentioned which prescribes
that (for circular orbits only) the jump must always be
to the circular orbit next lower, so that the electron
comes steadily down the series of steps without skipping
any. Another law prescribes the average time between
each jump and therefore the average time between the
successive emissions of light. The small lumps of
energy cast away at each step form light-waves of period
determined by the h rule.
"Preposterous! You cannot seriously mean that the
electron does different things on different days of the
week!"
But did I say that it does different things? I used
different words to describe its doings. I run down the
stairs on Tuesday and slide down the banisters on
Wednesday; but if the staircase consists of innumerable
CLASSICAL AND QUANTUM LAWS 197
infinitesimal steps, there is no essential difference in
my mode of progress on the two days. And so it makes
no difference whether the electron steps from one orbit
to the next lower or comes down in a spiral when the
number of steps is innumerably great. The succession
of lumps of energy cast overboard merges into a con-
tinuous outflow. If you had the formulae before you,
you would find that the period of the light and the
strength of radiation are the same whether calculated by
the Monday or the Tuesday method — but only when
the quantum number is infinitely great. The disagree-
ment is not very serious when the number is moderately
large; but for small quantum numbers the atom cannot
sit on the fence. It has to decide between Monday
(classical) and Tuesday (quantum) rules. It chooses
Tuesday rules.
If, as we believe, this example is typical, it indicates
one direction which the reconstruction of ideas must
take. We must not try to build up from classical con-
ceptions, because the classical laws only become true and
the conceptions concerned in them only become defined
in the limiting case when the quantum numbers of the
system are very large. We must start from new con-
ceptions appropriate to low as well as to high numbered
states; out of these the classical conceptions should
emerge, first indistinctly, then definitely, as the number
of the state increases, and the classical laws become
more and more nearly true. " I cannot foretell the result
of this remodelling, but presumably room must be
found for a conception of "states", the unity of a
state replacing the kind of tie expressed by classical
forces. For low numbered states the current vocabulary
of physics is inappropriate; at the moment we can
scarcely avoid using it, but the present contradictoriness
1 98 THE QUANTUM THEORY
of our theories arises from this misuse. For such states
space and time do not exist — at least I can see no reason
to believe that they do. But it must be supposed that
when high numbered states are considered there will
be found in the new scheme approximate counterparts
of the space and time of current conception — some-
thing ready to merge into space and time when the
state numbers are infinite. And simultaneously the inter-
actions described by transitions of states will merge
into classical forces exerted across space and time. So
that in the limit the classical description becomes an
available alternative. Now in practical experience we
have generally had to deal with systems whose ties are
comparatively loose and correspond to very high quan-
tum numbers; consequently our first survey of the
world has stumbled across the classical laws and our
present conceptions of the world consist of those enti-
ties which only take definite shape for high quantum
numbers. But in the interior of the atom and molecule,
in the phenomena of radiation, and probably also in the
constitution of very dense stars such as the Companion
of Sirius, the state numbers are not high enough to
admit this treatment. These phenomena are now forcing
us back to the more fundamental conceptions out of
which the classical conceptions (sufficient for the other
types of phenomena) ought to emerge as one extreme
limit.
For an example I will borrow a quantum conception
from the next chapter. It may not be destined to sur-
vive in the present rapid evolution of ideas, but at any
rate it will illustrate my point. In Bohr's semi-classical
model of the hydrogen atom there is an electron de-
scribing a circular or elliptic orbit. This is only a model;
the real atom contains nothing of the sort. The real
CLASSICAL AND QUANTUM LAWS 199
atom contains something which it has not entered into
the mind of man to conceive, which has, however, been
described symbolically by Schrodinger. This "some-
thing" is spread about in a manner by no means com-
parable to an electron describing an orbit. Now excite
the atom into successively higher and higher quantum
states. In the Bohr model the electron leaps into higher
and higher orbits. In the real atom Schrodinger's
"something" begins to draw itself more and more
together until it begins sketchily to outline the Bohr
orbit and even imitates a condensation running round.
Go on to still higher quantum numbers, and Schro-
dinger's symbol now represents a compact body moving
round in the same orbit and the same period as the
electron in Bohr's model, and moreover radiating
according to the classical laws of an electron. And so
when the quantum number reaches infinity, and the
atom bursts, a genuine classical electron flies out. The
electron, as it leaves the atom, crystallises out of Schro-
dinger's mist like a genie emerging from his bottle.
Chapter X
THE NEW QUANTUM THEORY
The conflict between quantum theory and classical
theory becomes especially acute in the problem of the
propagation of light. Here in effect it becomes a con-
flict between the corpuscular theory of light and the
wave theory.
In the early days it was often asked, How large is a
quantum of light? one answer is obtained by examining
a star image formed with the great ioo-inch reflector
at Mt. Wilson. The diffraction pattern shows that each
emission from each atom must be filling the whole mir-
ror. For if one atom illuminates one part only and
another atom another part only, we ought to get the
same effect by illuminating different parts of the mirror
by different stars (since there is no particular virtue in
using atoms from the same star) ; actually the diffraction
pattern then obtained is not the same. The quantum
must be large enough to cover a ioo-inch mirror.
But if this same star-light without any artificial con-
centration falls on a film of potassium, electrons will
fly out each with the whole energy of a quantum. This
is not a trigger action releasing energy already stored in
the atom, because the amount of energy is fixed by the
nature of thi light, not by the nature of the atom. A
whole quantum of light energy must have gone into the
atom and blasted away the electron. The quantum
must be small enough to enter an atom.
I do not think there is much doubt as to the ultimate
origin of this contradiction. We must not think about
space and time in connection with an individual quan-
200
WAVE-THEORY OF MATTER 201
turn; and the extension of a quantum in space has no
real meaning. To apply these conceptions to a single
quantum is like reading the Riot Act to one man. A
single quantum has not travelled 50 billion miles from
Sirius; it has not been 8 years on the way. But when
enough quanta are gathered to form a quorum there
will be found among them statistical properties which
are the genesis of the 50 billion miles' distance of Sirius
and the 8 years' journey of the light.
Wave-Theory of Matter. It is comparatively easy to
realise what we have got to do. It is much more diffi-
cult to start to do it. Before we review the attempts
in the last year or two to grapple with this problem we
shall briefly consider a less drastic method of progress
initiated by De Broglie. For the moment we shall be
content to accept the mystery as a mystery. Light, we
will say, is an entity with the wave property of spread-
ing out to fill the largest object glass and with all the
well-known properties of diffraction and interference;
simultaneously it is an entity with the corpuscular or
bullet property of expending its whole energy on one
very small target. We can scarcely describe such an
entity as a wave or as a particle; perhaps as a com-
promise we had better call it a "wavicle".
There is nothing new under the sun, and this latest.
volte-face almost brings us back to Newton's theory of
light — a curious mixture of corpuscular and wave-theory.
There is perhaps a pleasing sentiment in this "return
to Newton". But to suppose that Newton's scientific
reputation is especially vindicated by De Broglie's
theory of light, is as absurd as to suppose that it is
shattered by Einstein's theory of gravitation. There
was no phenomenon known to Newton which could not
202 THE NEW QUANTUM THEORY
be amply covered by the wave-theory; and the clearing
away of false evidence for a partly corpuscular theory,
which influenced Newton, is as much a part of scientific
progress as the bringing forward of the (possibly) true
evidence, which influences us to-day. To imagine that
Newton's great scientific reputation is tossing up and
down in these latter-day revolutions is to confuse science
with omniscience.
To return to the wavicle. — If that which we have
commonly regarded as a wave partakes also of the
nature of a particle, may not that which we have com-
monly regarded as a particle partake also of the nature
of a wave? It was not until the present century that
experiments were tried of a kind suitable to bring out
the corpuscular aspect of the nature of light; perhaps
experiments may still be possible which will bring out
a wave aspect of the nature of an electron.
So, as a first step, instead of trying to clear up the
mystery we try to extend it. Instead of explaining how
anything can possess simultaneously the incongruous
properties of wave and particle we seek to show experi-
mentally that these properties are universally associated.
There are no pure waves and no pure particles.
The characteristic of a wave-theory is the spreading
of a ray of light after passing through a narrow aper-
ture — a well-known phenomenon called diffraction. The
scale of the phenomenon is proportional to the wave-
length of the light. De Broglie has shown us how to
calculate the lengths of the waves (if any) associated
with an electron, i.e. considering it to be no longer a pure
particle but a wavicle. It appears that in some circum-
stances the scale of the corresponding diffraction effects
will not be too small for experimental detection. There
are now a number of experimental results quoted as
WAVE-THEORY OF MATTER 203
verifying this prediction. I scarcely know whether they
are yet to be considered conclusive, but there does seem
to be serious evidence that in the scattering of electrons
by atoms phenomena occur which would not be pro-
duced according to the usual theory that electrons are
purely corpuscular. These effects analogous to the
diffraction and interference of light carry us into the
stronghold of the wave-theory. Long ago such phe-
nomena ruled out all purely corpuscular theories of
light; perhaps to-day we are finding similar phenomena
which will rule out all purely corpuscular theories of
matter.*
A similar idea was entertained in a "new statistical
mechanics" developed by Einstein and Bose — at least
that seems to be the physical interpretation of the highly
abstract mathematics of their theory. As so often hap-
pens the change from the classical mechanics, though
far-reaching in principle, gave only insignificant cor-
rections when applied to ordinary practical problems.
Significant differences could only be expected in matter
much denser than anything yet discovered or imagined.
Strange to say, just about the time when it was realised
that very dense matter might have strange properties
different from those expected according to classical
conceptions, very dense matter was found in the uni-
verse. Astronomical evidence seems to leave practically
no doubt that in the so-called white dwarf stars the
density of matter far transcends anything of which we
have terrestrial experience; in the Companion of Sirius,
for example, the density is about a ton to the cubic inch.
This condition is explained by the fact that the high
temperature and correspondingly intense agitation of
*The evidence is much stronger now than when the lectures were
delivered.
204 THE NEW QUANTUM THEORY
the material breaks up (ionises) the outer electron sys-
tems of the atoms, so that the fragments can be packed
much more closely together. At ordinary temperatures
the minute nucleus of the atom is guarded by outposts
of sentinel electrons which ward off other atoms from
close approach even under the highest pressures; but at
stellar temperatures the agitation is so great that the
electrons leave their posts and run all over the place.
Exceedingly tight packing then becomes possible under
high enough pressure. R. H. Fowler has found that in
the white dwarf stars the density is so great that classi-
cal methods are inadequate and the new statistical
mechanics must be used. In particular he has in this
way relieved an anxiety which had been felt as to their
ultimate fate; under classical laws they seemed to be
heading towards an intolerable situation — the star could
not stop losing heat, but it would have insufficient energy
to be able to cool down!*
Transition to a New Theory. By 1925 the machinery
of current theory had developed another flaw and was
urgently calling for reconstruction; Bohr's model of the
atom had quite definitely broken down. This is the
model, now very familiar, which pictures the atom as
a kind of solar system with a central positively charged
nucleus and a number of elecrons describing orbits about
it like planets, the important feature being that the
possible orbits are limited by the rules referred to on
p. 190. Since each line in the spectrum of the atom is
emitted by the jump of an electron between two par-
* The energy is required because on cooling down the matter must
regain a more normal density and this involves a great expansion of
volume of the star. In the expansion work has to be done against the
force of gravity.
TRANSITION TO A NEW THEORY 205
ticular orbits, the classification of the spectral lines must
run parallel with the classification of the orbits by their
quantum numbers in the model. When the spectro-
scopists started to unravel the various series of lines in
the spectra they found it possible to assign an orbit
jump for every line — they could say what each line
meant in terms of the model. But now questions of
finer detail have arisen for which this correspondence
ceases to hold. one must not expect too much from a
model, and it would have been no surprise if the model
had failed to exhibit minor phenomena or if its accuracy
had proved imperfect. But the kind of trouble now
arising was that only two orbit jumps were provided
in the model to represent three obviously associated
spectral lines; and so on. The model which had been
so helpful in the interpretation of spectra up to a point,
suddenly became altogether misleading; and spectro-
scopists were forced to turn away from the model and
complete their classification of lines in a way which
ignored it. They continued to speak of orbits and
orbit jumps but there was no longer a complete one-
to-one correspondence with the orbits shown in the
model.*
The time was evidently ripe for the birth of a new
theory. The situation then prevailing may be summar-
ised as follows :
(1) The general working rule was to employ the
classical laws with the supplementary proviso that
whenever anything of the nature of action appears it
*Each orbit or state of the atom requires three (or, for later refine-
ments, four) quantum numbers to define it. The first two quantum
numbers are correctly represented in the Bohr model ; but the third
number which discriminates the different lines forming a doublet or
multiplet spectrum is represented wrongly — a much more serious failure
than if it were not represented at all.
206 THE NEW QUANTUM THEORY
must be made equal to h ) or sometimes to an integral mul-
tiple of h.
(2) The proviso often led to a self-contradictory use
of the classical theory. Thus in the Bohr atom the
acceleration of the electron in its orbit would be gov-
erned by classical electrodynamics whilst its radiation
would be governed by the h rule. But in classical elec-
trodynamics the acceleration and the radiation are indis-
solubly connected.
(3) The proper sphere of classical laws was known.
They are a form taken by the more general laws in a
limiting case, viz. when the number of quanta concerned
is very large. Progress in the investigation of the com-
plete system of more general laws must not be ham-
pered by classical conceptions which contemplate only the
limiting case.
(4) The present compromise involved the recognition
that light has both corpuscular and wave properties.
The same idea seems to have been successfully extended
to matter and confirmed by experiment. But this success
only renders the more urgent some less contradictory
way of conceiving these properties.
(5) Although the above working rule had generally
been successful in its predictions, it was found to give
a distribution of electron orbits in the atom differing in
some essential respects from that deduced spectroscopi-
cally. Thus a reconstruction was required not only to
remove logical objections but to meet the urgent de-
mands of practical physics.
Development of the New Quantum Theory. The "New
Quantum Theory" originated in a remarkable paper
by Heisenberg in the autumn of 1925. I am writing
the first draft of this lecture just twelve months after
DEVELOPMENT OF NEW THEORY 207
the appearance of the paper. That does not give long
for development; nevertheless the theory has already
gone through three distinct phases associated with the
names of Born and Jordan, Dirac, Schrodinger. My
chief anxiety at the moment is lest another phase of
reinterpretation should be reached before the lecture
can be delivered. In an ordinary way we should describe
the three phases as three distinct theories. The pioneer
work of Heisenberg governs the whole, but the three
theories show wide differences of thought. The first
entered on 'the new road in a rather matter-of-fact
way; the second was highly transcendental, almost
mystical; the third seemed at first to contain a reac-
tion towards classical ideas, but that was probably a
false impression. You will realise the anarchy of
this branch of physics when three successive pre-
tenders seize the throne in twelve months; but you
will not realise the steady progress made in that time
unless you turn to the mathematics of the subject.
As regards philosophical ideas the three theories are
poles apart; as regards mathematical content they are
one and the same. Unfortunately the mathematical
content is just what I am forbidden to treat of in these
lectures.
I am, however, going to transgress to the extent of
writing down one mathematical formula for you to con-
template; I shall not be so unreasonable as to expect
you to understand it. All authorities seem to be agreed
that at, or nearly at, the root of everything in the phy-
sical world lies the mystic formula
qp—pq = ih/2Tz
We do not yet understand that; probably if we could
understand it we should not think it so fundamental.
208 THE NEW QUANTUM THEORY
Where the trained mathematician has the advantage is
that he can use it, and in the past year or two it has
been used in physics with very great advantage indeed.
It leads not only to those phenomena described by the
older quantum laws such as the h rule, but to many
related phenomena which the older formulation could
not treat.
on the right-hand side, besides h (the atom of action)
and the merely numerical factor 2tt, there appears i (the
square root of — i) which may seem rather mystical.
But this is only a well-known subterfuge; and far back
in the last century physicists and engineers were well
aware that V — i in their formulae was a kind of sig-
nal to look out for waves or oscillations. The right-
hand side contains nothing unusual, but the left-hand side
baffles imagination. We call q and p co-ordinates and mo-
menta, borrowing our vocabulary from the world of
space and time and other coarse-grained experience;
but that gives no real light on their nature, nor does
it explain why qp is so ill-behaved as to be unequal
to pq.
It is here that the three theories differ most essen-
tially. Obviously q and p cannot represent simple
numerical measures, for then qp — pq would be zero.
For Schrodinger p is an operator. His "momentum"
is not a quantity but a signal to us to perform a certain
mathematical operation on any quantities which may
follow. For Born and Jordan p is a matrix — not one
quantity, nor several quantities, but an infinite number
of quantities arranged in systematic array. For Dirac
p is a symbol without any kind of numerical interpreta-
tion; he calls it a ^-number, which is a way of saying
that it is not a number at all.
I venture to think that there is an idea implied in
DEVELOPMENT OF NEW THEORY 209
Dirac^s treatment which may have great philosophical
significance, independently of any question of success in
this particular application. The idea is that in digging
deeper and deeper into that which lies at the base of
physical phenomena we must be prepared to come to
entities which, like many things in our conscious experi-
ence, are not measurable by numbers in any way; and
further it suggests how exact science, that is to say the
science of phenomena correlated to measure-numbers,
can be founded on such a basis.
one of the greatest changes in physics between the
nineteenth century and the present day has been the
change in our ideal of scientific explanation. It was the
boast of the Victorian physicist that he would not claim
to understand a thing until he could make a model of
it; and by a model he meant something constructed of
levers, geared wheels, squirts, or other appliances
familiar to an engineer. Nature in building the universe
was supposed to be dependent on just the same kind of
resources as any human mechanic; and when the physi-
cist sought an explanation of phenomena his ear was
straining to catch the hum of machinery. The man who
could make gravitation out of cog-wheels would have
been a hero in the Victorian age.
Nowadays we do not encourage the engineer to build
the world for us out of his material, but we turn to the
mathematician to build it out of his material. Doubtless
the mathematician is a loftier being than the engineer,
but perhaps even he ought not to be entrusted with the
Creation unreservedly. We are dealing in physics with
a symbolic world, and we can scarcely avoid employing
the mathematician who is the professional wielder of
symbols; but he must rise to the full opportunities of the
responsible task entrusted to him and not indulge too
210 THE NEW QUANTUM THEORY
freely his own bias for symbols with an arithmetical
interpretation. If we are to discern controlling laws of
Nature not dictated by the mind it would seem neces-
sary to escape as far as possible from the cut-and-dried
framework into which the mind is so ready to force
everything that it experiences.
I think that in principle Dirac's method asserts this
kind of emancipation. He starts with basal entities
inexpressible by numbers or number-systems and his
basal laws are symbolic ｅxpressions unconnected with
arithmetical operations. The fascinating point is that
as the development proceeds actual numbers are exuded
from the symbols. Thus although p and q individually
have no arithmetical interpretation, the combination
qp — pq has the arithmetical interpretation expressed by
the formula above quoted. By furnishing numbers,
though itself non-numerical, such a theory can well be
the basis for the measure-numbers studied in exact
science. The measure-numbers, which are all that we
glean from a physical survey of the world, cannot be
the whole world; they may not even be so much of it
as to constitute a self-governing unit. This seems the
natural interpretation of Dirac's procedure in seeking
the governing laws of exact science in a non-arithmetical
calculus.
I am afraid it is a long shot to predict anything like
this emerging from Dirac's beginning; and for the
moment Schrodinger has rent much of the mystery from
the />'s and qs by showing that a less transcendental
interpretation is adequate for present applications. But
I like to think that we may have not yet heard the last
of the idea.
Schrodinger's theory is now enjoying the full tide
of popularity, partly because of intrinsic merit, but also,
OUTLINE OF SCHRODINGER'S THEORY 211
I suspect, partly because it is the only one of the three
that is simple enough to be misunderstood. Rather
against my better judgment I will try to give a rough
impression of the theory. It would probably be wiser
to nail up over the door of the new quantum theory a
notice, "Structural alterations in progress — No admit-
tance except on business", and particularly to warn the
doorkeeper to keep out prying philosophers. I will,
however, content myself with the protest that, whilst
Schrodinger's theory is guiding us to sound and rapid
progress in many of the mathematical problems con-
fronting us and is indispensable in its practical utility,
I do not see the least likelihood that his ideas will sur-
vive long in their present form.
Outline of Schrodinger's Theory. Imagine a sub-aether
whose surface is covered with ripples. The oscillations
of the ripples are a million times faster than those of
visible light — too fast to come within the scope of our
gross experience. Individual ripples are beyond our
ken; what we can appreciate is a combined effect — when
by convergence and coalescence the waves conspire to
create a disturbed area of extent large compared with
individual ripples but small from our own Brobding-
nagian point of view. Such a disturbed area is recog-
nised as a material particle; in particular it can be an
electron.
The sub-aether is a dispersive medium, that is to say
the ripples do not all travel with the same velocity; like
water-ripples their speed depends on their wave-length
or period. Those of shorter period travel faster. More-
over the speed may be modified by local conditions.
This modification is the counterpart in Schrodinger's
theory of a field of force in classical physics. It will
212 THE NEW QUANTUM THEORY
readily be understood that if we are to reduce all phe-
nomena to a propagation of waves, then the influence
of a body on phenomena in its neighbourhood (com-
monly described as the field of force caused by its
presence) must consist in a modification of the propa-
gation of waves in the region surrounding it.
We have to connect these phenomena in the sub-
aether with phenomena in the plane of our gross ex-
perience. As already stated, a local stormy region is
detected by us as a particle; to this we now add that the
frequency (number of oscillations per second) of the
waves constituting the disturbance is recognised by us
as the energy of the particle. We shall presently try to
explain how the period manages to manifest itself to us
in this curiously camouflaged way; but however it comes
about, the recognition of a frequency in the sub-aether
as an energy in gross experience gives at once the con-
stant relation between period and energy which we have
called the h rule.
Generally the oscillations in the sub-aether are too
rapid for us to detect directly; their frequency reaches
the plane of ordinary experience by affecting the speed
of propagation, because the speed depends (as already
stated) on the wave-length or frequency. Calling the
frequency v, the equation expressing the law of propa-
gation of the ripples will contain a term in v. There will
be another term expressing the modification caused by
the "field of force" emanating from the bodies present
in the neighbourhood. This can be treated as a kind of
spurious v, since it emerges into our gross experience
by the same method that v does. If v produces those
phenomena which make us recognise it as energy, the
spurious v will produce similar phenomena correspond-
ing to a spurious kind of energy. Clearly the latter will
OUTLINE OF SCHRODINGER'S THEORY 213
be what we call potential energy, since it originates from
influences attributable to the presence of surrounding
objects.
Assuming that we know both the real v and the
spurious or potential v for our ripples, the equation of
wave-propagation is settled, and we can proceed to solve
any problem concerning wave-propagation. In particular
we can solve the problem as to how the stormy areas
move about. This gives a remarkable result which
provides the first check on our theory. The stormy
areas (if small enough) move under precisely the same
laws that govern the motions of particles in classical
mechanics. The equations for the motion of a wave-
group with given frequency and potential frequency are
the same as the classical equations of motion of a par-
ticle with the corresponding energy and potential energy.
It has to be noticed that the velocity of a stormy area
or group of waves is not the same as the velocity of an
individual wave. This is well known in the study of
water-waves as the distinction between group-velocity
and wave-velocity. It is the group-velocity that is ob-
served by us as the motion of the material particle.
We should have gained very little if our theory did
no more than re-establish the results of classical me-
chanics on this rather fantastic basis. Its distinctive
merits begin to be apparent when we deal with pheno-
mena not covered by classical mechanics. We have
considered a stormy area of so small extent that its
position is as definite as that of a classical particle, but
we may also consider an area of wider extent. No
precise delimitation can be drawn between a large area
and a small area, so that we shall continue to associate
the idea of a particle with it; but whereas a small
concentrated storm fixes the position of the particle
214 THE NEW QUANTUM THEORY
closely, a more extended storm leaves it very vague. If
we try to interpret an extended wave-group in classical
language we say that it is a particle which is not at any
definite point of space, but is loosely associated with a
wide region.
Perhaps you may think that an extended stormy area
ought to represent diffused matter in contrast to a con-
centrated particle. That is not Schrodinger's theory.
The spreading is not a spreading of density; it is an
indeterminacy of position, or a wider distribution of the
probability that the particle lies within particular limits
of position. Thus if we come across Schrodinger waves
uniformly filling a vessel, the interpretation is not that
the vessel is filled with matter of uniform density, but
that it contains one particle which is equally likely to be
anywhere.
The first great success of this theory was in repre-
senting the emission of light from a hydrogen atom —
a problem far outside the scope of classical theory. The
hydrogen atom consists of a proton and electron which
must be translated into their counterparts in the sub-
aether. We are not interested in what the proton is
doing, so we do not trouble about its representation by
waves; what we want from it is its field of force, that is
to say, the spurious v which it provides in the equation
of wave-propagation for the electron. The waves
travelling in accordance with this equation constitute
Schrodinger's equivalent for the electron; and any solu-
tion of the equation will correspond to some possible
state of the hydrogen atom. Now it turns out that
(paying attention to the obvious physical limitation that
the waves must not anywhere be of infinite amplitude)
solutions of this wave-equation only exist for waves with
particular frequencies. Thus in a hydrogen atom the
OUTLINE OF SCHRODINGER'S THEORY 215
sub-aethereal waves are limited to a particular discrete
series of frequencies. Remembering that a frequency
in the sub-aether means an energy in gross experience,
the atom will accordingly have a discrete series of pos-
sible energies. It is found that this series of energies
is precisely the same as that assigned by Bohr from his
rules of quantisation (p. 191). It is a considerable
advance to have determined Jiese energies by a wave-
theory instead of by an inexplicable mathematical rule.
Further, when applied to more complex atoms Schro-
dinger's theory succeeds on those points where the Bohr
model breaks down; it always gives the right number of
energies or "orbits" to provide one orbit jump for each
observed spectral line.
It is, however, an advantage not to pass from wave-
frequency to classical energy at this stage, but to follow
the course of events in the sub-aether a little farther.
It would be difficult to think of the electron as having
two energies (i.e. being in two Bohr orbits) simultane-
ously; but there is nothing to prevent waves of two dif-
ferent frequencies being simultaneously present in the
sub-aether. Thus the wave-theory allows us easily to
picture a condition which the classical theory could only
describe in paradoxical terms. Suppose that two sets
of waves are present. If the difference of frequency is
not very great the two systems of waves will produce
"beats". If two broadcasting stations are transmitting
on wave-lengths near together we hear a musical note
or shriek resulting from the beats of the two carrier
waves; the individual oscillations are too rapid to affect
the ear, but they combine to give beats which are slow
enough to affect the ear. In the same way the individual
wave-systems in the sub-aether are composed of oscilla-
tions too rapid to affect our gross senses ; but their beats
216 THE NEW QUANTUM THEORY
are sometimes slow enough to come within the octave
covered by the eye. These beats are the source of the
light coming from the hydrogen atom, and mathematical
calculation shows that their frequencies are precisely
those of the observed light from hydrogen. Hetero-
dyning of the radio carrier waves produces sound;
heterodyning of the sub-aethereal waves produces light.
Not only does this theory give the periods of the dif-
ferent lines in the spectra, but it also predicts their in-
tensities — a problem which the older quantum theory had
no means of tackling. It should, however, be under-
stood that the beats are not themselves to be identified
with light-waves; they are in the sub-aether, whereas
light-waves are in the aether. They provide the oscil-
lating source which in some way not yet traced sends out
light-waves of its own period.
What precisely is the entity which we suppose to be
oscillating when we speak of the waves in the sub-
aether? It is denoted by op, and properly speaking we
should regard it as an elementary indefinable of the
wave-theory. But can we give it a classical interpreta-
tion of any kind? It seems possible to interpret it as a
probability. The probability of the particle or electron
being within a given region is proportional to the amount
of ip in that region. So that if ip is mainly concentrated
in one small stormy area, it is practically certain that
the electron is there; we are then able to localise it
definitely and conceive of it as a classical particle. But
the ip-waves of the hydrogen atom are spread about
all over the atom; and there is no definite localisation of
the electron, though some places are more probable than
others.*
* The probability is often stated to be proportional to ty 2 , instead of
\p, as assumed above. The whole interpretation is very obscure, but it
OUTLINE OF SCHRODINGER'S THEORY 217
Attention must be called to one highly important
consequence of this theory. A small enough stormy
area corresponds very nearly to a particle moving about
under the classical laws of motion; it would seem there-
fore that a particle definitely localised as a moving point
is stricdy the limit when the stormy area is reduced to
a point. But curiously enough by continually reducing
the area of the storm we never quite reach the ideal
classical particle; we approach it and then recede from
it again. We have seen that the wave-group moves like
a particle (localised somewhere within the area of the
storm) having an energy corresponding to the frequency
of the waves; therefore to imitate a particle exactly, not
only must the area be reduced to a point but the group
must consist of waves of only one frequency. The two
conditions are irreconcilable. With one frequency we
can only have an infinite succession of waves not ter-
minated by any boundary. A boundary to the group is
provided by interference of waves of slightly different
length, so that while reinforcing one another at the
centre they cancel one another at the boundary. Roughly
speaking, if the group has a diameter of 1000 wave-
lengths there must be a range of wave-length of o-i per
cent., so that 1000 of the longest waves and 1001 of
the shortest occupy the same distance. If we take a
more concentrated stormy area of diameter 10 wave-
seems to depend on whether you are considering the probability after
you know what has happened or the probability for the purposes of
prediction. The ijj 2 is obtained by introducing two symmetrical systems
of ij>-waves travelling in opposite directions in time; one of these must
presumably correspond to probable inference from what is known (or
is stated) to have been the condition at a later time. Probability neces-
sarily means "probability in the light of certain given information", so
that the probability cannot possibly be represented by the same function
in different classes of problems with different initial data.
2i 8 THE NEW QUANTUM THEORY
lengths the range is increased to 10 per cent.; 10 of
the longest and 1 1 of the shortest waves must extend the
same distance. In seeking to make the position of the
particle more definite by reducing the area we make its
energy more vague by dispersing the frequencies of the
waves. So our particle can never have simultaneously
a perfectly definite position and a perfectly definite
energy; it always has a vagueness of one kind or the
other unbefitting a classical particle. Hence in delicate
experiments we must not under any circumstances expect
to find particles behaving exactly as a classical particle
was supposed to do — a conclusion which seems to be in
accordance with the modern experiments on diffraction
of electrons already mentioned.
We remarked that Schrodinger's picture of the hy-
drogen atom enabled it to possess something that would
be impossible on Bohr's theory, viz. two energies at
once. For a particle or electron this is not merely per-
missive, but compulsory — otherwise we can put no limits
to the region where it may be. You are not asked to
imagine the state of a particle with several energies;
what is meant is that our current picture of an electron
as a particle with single energy has broken down, and
we must dive below into the sub-aether if we wish to
follow the course of events. The picture of a particle
may, however, be retained when we are not seeking high
accuracy; if we do not need to know the energy more
closely than I per cent., a series of energies ranging
over i per cent, can be treated as one definite energy.
Hitherto I have only considered the waves correspond-
ing to one electron; now suppose that we have a prob-
lem involving two electrons. How shall they be repre-
sented? "Surely, that is simple enough! We have only
to take two stormy areas instead of one." I am afraid
OUTLINE OF SCHRODINGER'S THEORY 219
not. Two stormy areas would correspond to a single
electron uncertain as to which area it was located in.
So long as there is the faintest probability of the first
electron being in any region, we cannot make the Schro-
dinger waves there represent a probability belonging to
a second electron. Each electron wants the whole of
three-dimensional space for its waves; so Schrodinger
generously allows three dimensions for each of them.
For two electrons he requires a six-dimensional sub-
aether. He then successfully applies his method on the
same lines as before. I think you will see now that
Schrodinger has given us what seemed to be a com-
prehensible physical picture only to snatch it away again.
His sub-aether does not exist in physical space; it is in
a "configuration space" imagined by the mathematician
for the purpose of solving his problems, and imagined
afresh with different numbers of dimensions according
to the problem proposed. It was only an accident
that in the earliest problems considered the configu-
ration space had a close correspondence with physical
space, suggesting some degree of objective reality
of the waves. Schrodinger's wave-mechanics is not
a physical theory but a dodge — and a very good dodge
too.
The fact is that the almost universal applicability of
this wave-mechanics spoils all chance of our taking it
seriously as a physical theory. A delightful illustration
of this occurs incidentally in the work of Dirac. In one
of the problems, which he solves by Schrodinger waves,
the frequency of the waves represents the number of
systems of a given kind. The wave-equation is formu-
lated and solved, and (just as in the problem of the
hydrogen atom) it is found that solutions only exist for
a series of special values of the frequency. Consequently
220 THE NEW QUANTUM THEORY
the number of systems of the kind considered must
have one of a discrete series of values. In Dirac's
problem the series turns out to be the series of integers.
Accordingly we infer that the number of systems must
be either i, 2, 3, 4, . . ., but can never be 2% f° r
example. It is satisfactory that the theory should give
a result so well in accordance with our experience !
But we are not likely to be persuaded that the true
explanation of why we count in integers is afforded by a
system of waves.
Principle of Indeterminacy. My apprehension lest a
fourth version of the new quantum theory should
appear before the lectures were delivered was not ful-
filled; but a few months later the theory definitely
entered on a new phase. It was Heisenberg again who
set in motion the new development in the summer of
1927, and the consequences were further elucidated by
Bohr. The outcome of it is a fundamental general
principle which seems to rank in importance with the
principle of relativity. I shall here call it the "principle
of indeterminacy".
The gist of it can be stated as follows : a particle may
have position or it may have velocity but it cannot in any
exact sense have both.
If we are content with a certain margin of inaccuracy
and if we are content with statements that claim no
certainty but only high probability, then it is possible
to ascribe both position and velocity to a particle. But
if we strive after a more accurate specification of position
a very remarkable thing happens; the greater accuracy
can be attained, but it is compensated by a greater
inaccuracy in the specification of the velocity. Similarly
if the specification of the velocity is made more accurate
the position becomes less determinate.
PRINCIPLE OF INDETERMINACY 221
Suppose for example that we wish to know the posi-
tion and velocity of an electron at a given moment.
Theoretically it would be possible to fix the position with
a probable error of about 1/1000 of a millimetre and
the velocity with a probable error of 1 kilometre per
second. But an error of 1/1000 of a millimetre is large
compared with that of some of our space measurements;
is there no conceivable way of fixing the position to
1/10,000 of a millimetre? Certainly; but in that case it
will only be possible to fix the velocity with an error of
10 kilometres per second.
The conditions of our exploration of the secrets of
Nature are such that the more we bring to light the
secret of position the more the secret of velocity is
hidden. They are like the old man and woman in the
weather-glass; as one comes out of one door, the other
retires behind the other door. When we encounter un-
expected obstacles in finding out something which we
wish to know, there are two possible courses to take. It
may be that the right course is to treat the obstacle
as a spur to further efforts; but there is a second
possibility — that we have been trying to find some-
thing which does not exist. You will remember that
that was how the relativity theory accounted for the
apparent concealment of our velocity through the
aether.
When the concealment is found to be perfectly sys-
tematic, then we must banish the corresponding entity
from the physical world. There is really no option.
The link with our consciousness is completely broken.
When we cannot point to any causal effect on anything
that comes into our experience, the entity merely becomes
part of the unknown — undifferentiated from the rest of
the vast unknown. From time to time physical discover-
ies are made; and new entities, coming out of the un-
222 THE NEW QUANTUM THEORY
known, become connected to our experience and are duly
named. But to leave a lot of unattached labels floating
in the as yet undifferentiated unknown in the hope that
they may come in useful later on, is no particular sign
of prescience and is not helpful to science. From this
point of view we assert that the description of the posi-
tion and velocity of an electron beyond a limited num-
ber of places of decimals is an attempt to describe some-
thing that does not exist; although curiously enough the
description of position or of velocity if it had stood alone
might have been allowable.
Ever since Einstein's theory showed the importance
of securing that the physical quantities which we talk
about are actually connected to our experience, we have
been on our guard to some extent against meaningless
terms. Thus distance is defined by certain operations of
measurement and not with reference to nonsensical con-
ceptions such as the "amount of emptiness" between
two points. The minute distances referred to in atomic
physics naturally aroused some suspicion, since it is not
always easy to say how the postulated measurements
could be imagined to be carried out. I would not like
to assert that this point has been cleared up; but at any
rate it did not seem possible to make a clean sweep of
all minute distances, because cases could be cited in which
there seemed no natural limit to the accuracy of deter-
mination of position. Similarly there are ways of
determining momentum apparently unlimited in accuracy.
What escaped notice was that the two measurements
interfere with one another in a systematic way, so that
the combination of position with momentum, legitimate
on the large scale, becomes indefinable on the small
scale. The principle of indeterminacy is scientifically
stated as follows: if q is a co-ordinate and p the corre-
PRINCIPLE OF INDETERMINACY 223
sponding momentum, the necessary uncertainty of our
knowledge of q multiplied by the uncertainty of p is of
the order of magnitude of the quantum constant h.
A general kind of reason for this can be seen without
much difficulty. Suppose it is a question of knowing
the position and momentum of an electron. So long as
the electron is not interacting with the rest of the uni-
verse we cannot be aware of it. We must take our
chance of obtaining knowledge of it at moments when it
is interacting with something and thereby producing
effects that can be observed. But in any such interaction
a complete quantum is involved; and the passage of this
quantum, altering to an important extent the conditions
at the moment of our observation, makes the information
out of date even as we obtain it.
Suppose that (ideally) an electron is observed under
a powerful microscope in order to determine its position
with great accuracy. For it to be seen at all it must be
illuminated and scatter light to reach the eye. The least
it can scatter is one quantum. In scattering this it re-
ceives from the light a kick of unpredictable amount;
we can only state the respective probabilities of kicks
of different amounts. Thus the condition of our ascer-
taining the position is that we disturb the electron in an
incalculable way which will prevent our subsequently as-
certaining how much momentum it had. However, we
shall be able to ascertain the momentum with an uncer-
tainty represented by the kick, and if the probable kick
is small the probable error will be small. To keep the
kick small we must use a quantum of smali energy, that
is to say, light of long wave-length. But to use long
wave-length reduces the accuracy of our microscope.
The longer the waves, the larger the diffraction images.
And it must be remembered that it takes a great many
224 THE NEW QUANTUM THEORY
quanta to outline the diffraction image; our one scattered
quantum can only stimulate one atom in the retina of
the eye, at some haphazard point within the theoretical
diffraction image. Thus there will be an uncertainty in
our determination of position of the electron propor-
tional to the size of the diffraction image. We are in a
dilemma. We can improve the determination of the
position with the microscope by using light of shorter
wave-length, but that gives the electron a greater kick
and spoils the subsequent determination of momentum.
A picturesque illustration of the same dilemma is
afforded if we imagine ourselves trying to see one of the
electrons in an atom. For such finicking work it is no
use employing ordinary light to see with; it is far too
gross, its wave-length being greater than the whole
atom. We must use fine-grained illumination and train
our eyes to see with radiation of short wave-length —
with X-rays in fact. It is well to remember that X-rays
have a rather disastrous effect on atoms, so we had better
use them sparingly. The least amount we can use is one
quantum. Now, if we are ready, will you watch, whilst
I flash one quantum of X-rays on to the atom? I may
not hit the electron the first time; in that case, of course,
you will not see it. Try again; this time my quantum
has hit the electron. Look sharp, and notice where it is.
Isn't it there? Bother! I must have blown the electron
out of the atom.
This is not a casual difficulty; it is a cunningly
arranged plot — a plot to prevent you from seeing
something that does not exist, viz. the locality of the
electron within the atom. If I use longer waves which
do no harm, they will not define the electron sharply
enough for you to see where it is. In shortening the wave-
length, just as the light becomes fine enough its quan-
A NEW EPISTEMOLOGY 225
turn becomes too rough and knocks the electron out of
the atom.
Other examples of the reciprocal uncertainty have
been given, and there seems to be no doubt that it is
entirely general. The suggestion is that an association
of exact position with exact momentum can never be
discovered by us because there is no such thing in Nature.
This is not inconceivable. Schrodinger's model of the
particle as a wave-group gives a good illustration of how
it can happen. We have seen (p. 217) that as the posi-
tion of a wave-group becomes more defined the energy
(frequency) becomes more indeterminate, and vice versa.
I think that that is the essential value of Schrodinger's
theory; it refrains from attributing to a particle a kind
of determinacy which does not correspond to anything
in Nature. But I would not regard the principle of
indeterminacy as a result to be deduced from Schro-
dinger's theory; it is the other way about. The principle
of indeterminacy, like the principle of relativity, repre-
sents the abandonment of a mistaken assumption which
we never had sufficient reason for making. Just as we
were misled into untenable ideas of the aether through
trusting to an analogy with the material ocean, so we
have been misled into untenable ideas of the attributes
of the microscopic elements of world-structure through
trusting to analogy with gross particles.
A New Epistemology. The principle of indeterminacy
is epistemological. It reminds us once again that the
world of physics is a world contemplated from within
surveyed by appliances which are part of it and subject
to its laws. What the world might be deemed like if
probed in some supernatural manner by appliances not
furnished by itself we do not profess to know.
226 THE NEW QUANTUM THEORY
There is a doctrine well known to philosophers that
the moon ceases to exist when no one is looking at it.
I will not discuss the doctrine since I have not the least
idea what is the meaning of the word existence when
used in this connection. At any rate the science of as-
tronomy has not been based on this spasmodic kind of
moon. In the scientific world (which has to fulfil func-
tions less vague than merely existing) there is a moon
which appeared on the scene before the astronomer; it
reflects sunlight when no one sees it; it has mass when
no one is measuring the mass; it is distant 240,000 miles
from the earth when no one is surveying the distance;
and it will eclipse the sun in 1999 even if the human race
has succeeding in killing itself off before that date. The
moon — the scientific moon — has to play the part of a
continuous causal element in a world conceived to be all
causally interlocked.
What should we regard as a complete description of
this scientific world? We must not introduce anything
like velocity through aether, which is meaningless since
it is not assigned any causal connection with our ex-
perience. on the other hand we cannot limit the de-
scription to the immediate data of our own spasmodic
observations. The description should include nothing
that is unobservable but a great deal that is actually
unobserved. Virtually we postulate an infinite army of
watchers and measurers. From moment to moment they
survey everything that can be surveyed and measure
everything that can be measured by methods which we
ourselves might conceivably employ. Everything they
measure goes down as part of the complete description
of the scientific world. We can, of course, introduce
derivative descriptions, words expressing mathematical
combinations of the immediate measures which may give
A NEW EPISTEMOLOGY 227
greater point to the description — so that we may not
miss seeing the wood for the trees.
By employing the known physical laws expressing
the uniformities of Nature we can to a large extent
dispense with this army of watchers. We can afford to
let the moon out of sight for an hour or two and deduce
where it has been in the meantime. But when I assert
that the moon (which I last saw in the west an hour ago)
is now setting, I assert this not as my deduction but as
a true fact of the scientific world. I am still postulating
the imaginary watcher; I do not consult him, but I
retain him to corroborate my statement if it is chal-
lenged. Similarly, when we say that the distance of
Sirius is 50 billion miles we are not giving a merely con-
ventional interpretation to its measured parallax; we in-
tend to give it the same status in knowledge as if some-
one had actually gone through the operation of laying
measuring rods end to end and counted how many were
needed to reach to Sirius; and we should listen patiently
to anyone who produced reasons for thinking that our
deductions did not correspond to the "real facts", i.e.
the facts as known to our army of measurers. If we
happen to make a deduction which could not conceivably
be corroborated or disproved by these diligent measur-
ers, there is no criterion of its truth or falsehood and it
is thereby a meaningless deduction.
This theory of knowledge is primarily intended to
apply to our macroscopic or large-scale survey of the
physical world, but it has usually been taken for granted
that it is equally applicable to a microscopic study. We
have at last realised the disconcerting fact that though
it applies to the moon it does not apply to the
electron.
It does not hurt the moon to look at it. There is no
228 THE NEW QUANTUM THEORY
inconsistency in supposing it to have been under the
surveillance of relays of watchers whilst we were asleep.
But it is otherwise with an electron. At certain times,
viz. when it is interacting with a quantum, it might be
detected by one of our watchers; but between whiles it
virtually disappears from the physical world, having no
interaction with it. We might arm our observers with
flash-lamps to keep a more continuous watch on its
doings; but the trouble is that under the flashlight it
will not go on doing what it was doing in the dark.
There is a fundamental inconsistency in conceiving the
microscopic structure of the physical world to be under
continuous survey because the surveillance would itself
wreck the whole machine.
I expect that at first this will sound to you like a
merely dialectical difficulty. But there is much more in
it than that. The deliberate frustration of our efforts to
bring knowledge of the microscopic world into orderly
plan, is a strong hint to alter the plan.
It means that we have been aiming at a false ideal of
a complete description of the world. There has not yet
been time to make serious search for a new epistemology
adapted to these conditions. It has become doubtful
whether it will ever be possible to construct a physical
world solely out of the knowable — the guiding principle
in our macroscopic theories. If it is possible, it involves
a great upheaval of the present foundations. It seems
more likely that we must be content to admit a mixture
of the knowable and unknowable. This means a denial
of determinism, because the data required for a pre-
diction of the future will include the unknowable ele-
ments of the past. I think it was Heisenberg who said,
u The question whether from a complete knowledge of
the past we can predict the future, does not arise because
A NEW EPISTEMOLOGY 229
a complete knowledge of the past involves a self-con-
tradiction."
It is only through a quantum action that the outside
world can interact with ourselves and knowledge of it
can reach our minds. A quantum action may be the
means of revealing to us some fact about Nature, but
simultaneously a fresh unknown is implanted in the womb
of Time. An addition to knowledge is won at the ex-
pense of an addition to ignorance. It is hard to empty
the well of Truth with a leaky bucket.
Chapter XI
WORLD BUILDING
We have an intricate task before us. We are going to
build a World — a physical world which will give a
shadow performance of the drama enacted in the world
of experience. We are not very expert builders as yet;
and you must not expect the performance to go off
without a hitch or to have the richness of detail which a
critical audience might require. But the method about
to be described seems to give the bold outlines; doubt-
less we have yet to learn other secrets of the craft of
world building before we can complete the design.
The first problem is the building material. I remem-
ber that as an impecunious schoolboy I used to read
attractive articles on how to construct wonderful con-
trivances out of mere odds and ends. Unfortunately
these generally included the works of an old clock, a
few superfluous telephones, the quicksilver from a
broken barometer, and other oddments which happened
not to be forthcoming in my lumber room. I will try
not to let you down like that. I cannot make the world
out of nothing, but I will demand as little specialised
material as possible. Success in the game of World
Building consists in the greatness of the contrast
between the specialised properties of the completed
structure and the unspecialised nature of the basal
material.
Relation Structure. We take as building material rela-
tions and relata. The relations unite the relata; the
relata are the meeting points of the relations. The one
230
RELATION STRUCTURE 231
is unthinkable apart from the other. I do not think that
a more general starting-point of structure could be
conceived.
To distinguish the relata from one another we assign
to them monomarks. The monomark consists of four
numbers ultimately to be called "co-ordinates". But
co-ordinates suggest space and geometry and as yet there
is no such thing in our scheme; hence for the present
we shall regard the four identification numbers as no
more than an arbitrary monomark. Why four numbers?
We use four because it turns out that ultimately the
structure can be brought into better order that way;
but we do not know why this should be so. We have
got so far as to understand that if the relations insisted
on a threefold or a fivefold ordering it would be much
more difficult to build anything interesting out of them;
but that is perhaps an insufficient excuse for the
special assumption of fourfold order in the primitive
material.
The relation between two human individuals in its
broadest sense comprises every kind of connection or
comparison between them — consanguinity, business trans-
actions, comparative stature, skill at golf — any kind of
description in which both are involved. For generality
we shall suppose that the relations in our world-material
are likewise composite and in no way expressible in nu-
merical measure. Nevertheless there must be some kind
of comparability or likeness of relations, as there is in
the relations of human individuals; otherwise there
would be nothing more to be said about the world than
that everything in it was utterly unlike everything else.
To put it another way, we must postulate not only rela-
tions between the relata but some kind of relation of
likeness between some of the relations. The slightest
232 WORLD BUILDING
concession in this direction will enable us to link the
whole into a structure.
We assume then that, considering a relation between
two relata, it will in general be possible to pick out two
other relata close at hand which stand to one another
in a "like" relation. By "like" I do not mean "like in
every respect", but like in respect to one of the aspects
of the composite relation. How is the particular aspect
selected? If our relata were human individuals different
judgments of likeness would be made by the geneal-
ogist, the economist, the psychologist, the sportsman,
etc.; and the building of structure would here diverge
along a number of different lines. Each could build his
own world-structure from the common basal material
of humanity. There is no reason to deny that a similar
diversity of worlds could be built out of our postulated
material. But all except one of these worlds will be
stillborn. Our labour will be thrown away unless the
world we have built is the one which the mind chooses
to vivify into a world of experience. The only definition
we can give of the aspect of the relations chosen for the
criterion of likeness, is that it is the aspect which will
ultimately be concerned in the getting into touch of mind
w T ith the physical world. But that is beyond the province
of physics.
This one-to-one correspondence of "likeness" is only
supposed to be definite in the limit when the relations
are very close together in the structure. Thus we avoid
any kind of comparison at a distance which is as
objectionable as action at a distance. Let me confess at
once that I do not know what I mean here by "very
close together". As yet space and time have not been
built. Perhaps we might say that only a few of the
relata possess relations whose comparability to the first
RELATION STRUCTURE
233
is definite, and take the definiteness of the comparability
as the criterion of contiguity. I hardly know. The
building at this point shows some cracks, but I think it
should not be beyond the resources of the mathematical
logician to cement them up. We should also arrange at
this stage that the monomarks are so assigned as to give
an indication of contiguity.
Fig. 7
Let us start with a relatum A and a relation AP
radiating from it. Now step to a contiguous relatum
B and pick out the "like" relation BQ. Go on to
another contiguous relatum C and pick out the relation
CR which is like BQ. (Note that since C is farther
from A than from B } the relation at C which is like
AP is not so definite as the relation which is like BQ.)
Step by step we may make the comparison round a
route AEFA which returns to the starting-point. There
is nothing to ensure that the final relation AP' which
234 WORLD BUILDING
has, so to speak, been carried round the circuit will be
the relation AP with which we originally started.
We have now two relations AP, AP' radiating from
the first relatum, their difference being connected with
a certain circuit in the world AEFA. The loose ends of
the relations P and P have their monomarks, and we
can take the difference of the monomarks (i.e. the
difference of the identification numbers comprised in
them) as the code ｅxpression for the change introduced
by carrying AP round the circuit. As we vary the circuit
and the original relation, so the change PP' varies; and
the next step is to find a mathematical formula express-
ing this dependence. There are virtually four things to
connect, the circuit counting double since, for example,
a rectangular circuit would be described by specifying
two sides. Each of them has to be specified by four
identification numbers (either monomarks or derived
from monomarks) ; consequently, to allow for all com-
binations, the required mathematical formula contains
4 4 or 256 numerical coefficients. These coefficients give
a numerical measure of the structure surrounding the
initial relatum.
This completes the first part of our task to introduce
numerical measure of structure into the basal material.
The method is not so artificial as it appears at first sight.
Unless we shirk the problem by putting the desired
physical properties of the world directly into the original
relations and relata, we must derive them from the
structural interlocking of the relations; and such
interlocking is naturally traced by following circuits
among the relations. The axiom of comparability of
contiguous relations only discriminates between like
and unlike, and does not initially afford any means
of classifying various decrees and kinds of unlikeness;
RELATION STRUCTURE 235
but we have found a means of specifying the kind
of unlikeness of AP and AP' by reference to a circuit
which "transforms" one into the other. Thus we have
built a quantitative study of diversity on a definition of
similarity.
The numerical measures of structure will be dependent
on, and vary according to, the arbitrary code of mono-
marks used for the identification of relata. This, how-
ever, renders them especially suitable for building the
ordinary quantities of physics. When the monomarks
become co-ordinates of space and time the arbitrary
choice of the code will be equivalent to the arbitrary
choice of a frame of space and time; and it is in accord-
ance with the theory of relativity that the measures of
structure and the physical quantities to be built from
them should vary with the frame of space and time.
Physical quantities in general have no absolute value,
but values relative to chosen frames of reference or
codes of monomarks.
We have now fashioned our bricks from the primitive
clay and the next job is to build with them. The 256
measures of structure varying from point to point of
the world are somewhat reduced in number when dupli-
cates are omitted; but even so they include a great deal
of useless lumber which we do not require for the
building. That seems to have worried a number of the
most eminent physicists; but I do not quite see why.
Ultimately it is the mind that decides what is lumber —
which part of our building will shadow the things of
common experience, and which has no such counterpart.
It is no part of our function as purveyors of building
material to anticipate what will be chosen for the
palace of the mind. The lumber will now be dropped as
irrelevant in the further operations, but I do not agree
236 WORLD BUILDING
with those who think it a blemish on the theory that
the lumber should ever have appeared in it.
By adding together certain of the measures of struc-
ture in a symmetrical manner and by ignoring others
we reduce the really important measures to 16.* These
can be divided into 10 forming a symmetrical scheme
and 6 forming an antisymmetrical scheme. This is the
great point of bifurcation of the world.
Symmetrical coefficients (10). Out of these we find it
possible to construct Geometry and Mechanics. They
are the ten potentials of Einstein (g,J). We derive
from them space, time, and the world-curvatures re-
presenting the mechanical properties of matter, viz.
momentum, energy, stress, etc.
Antisymmetrical coefficients (6). Out of these we con-
struct Electromagnetism. They are the three com-
ponents of electric intensity and three components of
magnetic force. We derive electric and magnetic
potential, electric charge and current, light and other
electric waves.
We do not derive the laws and phenomena of
atomicity. Our building operation has somehow been
too coarse to furnish the microscopic structure of the
world, so that atoms, electrons and quanta are at present
beyond our skill.
But in regard to what is called field-physics the
construction is reasonably complete. The metrical,
gravitational and electromagnetic fields are all included.
We build the quantities enumerated above; and they
obey the great laws of field-physics in virtue of the way
in which they have been built. That is the special fea-
ture; the field laws — conservation of energy, mass, mo-
* Mathematically we contract the original tensor of the fourth rank
to one of the second rank.
IDENTICAL LAWS 237
mentum and of electric charge, the law of gravitation,
Maxwell's equations — are not controlling laws.* They
are truisms. Not truisms when approached in the way
the mind looks out on the world, but truisms when we
encounter them in a building up of the world from a
basal structure. I must try to make clear our new
attitude to these laws.
Identical Laws. Energy momentum and stress, which
we have identified with the ten principal curvatures of
the world, are the subject of the famous laws of con-
servation of energy and momentum. Granting that the
identification is correct, these laws are mathematical
identities. Violation of them is unthinkable. Perhaps
I can best indicate their nature by an analogy.
An aged college Bursar once dwelt secluded in his
rooms devoting himself entirely to accounts. He realised
the intellectual and other activities of the college only
as they presented themselves in the bills. He vaguely
conjectured an objective reality at the back of it all —
some sort of parallel to the real college — though he
could only picture it in terms of the pounds, shillings
and pence which made up what he would call "the
commonsense college of everyday experience". The
method of account-keeping had become inveterate habit
handed down from generations of hermit-like bursars;
he accepted the form of accounts as being part of the
nature of things. But he was of a scientific turn and he
wanted to learn more about the college. one day in
looking over his books he discovered a remarkable law.
* one law commonly grouped with these, viz. the law of pondero-
motive force of the electric field, is not included. It seems to be impos-
sible to get at the origin of this law without tackling electron structure
which is beyond the scope of our present exercise in world-building.
238 WORLD BUILDING
For every item on the credit side an equal item appeared
somewhere else on the debit side. "Ha I" said the
Bursar, "I have discovered one of the great laws con-
trolling the college. It is a perfect and exact law of the
real world. Credit must be called plus and debit minus;
and so we have the law of conservation of £ s. d. This
is the true way to find out things, and there is no limit
to what may ultimately be discovered by this scientific
method. I will pay no more heed to the superstitions
held by some of the Fellows as to a beneficent spirit
called the King or evil spirits called the University
Commissioners. I have only to go on in this way and
I shall succeed in understanding why prices are always
going up."
I have no quarrel with the Bursar for believing that
scientific investigation of the accounts is a road to exact
(though necessarily partial) knowledge of the reality
behind them. Things may be discovered by this method
which go deeper than the mere truism revealed by his
first effort. In any case his life is especially concerned
with accounts and it is proper that he should discover
the laws of accounts whatever their nature. But I would
point out to him that a discovery of the overlapping of
the different aspects in which the realities of the college
present themselves in the world of accounts, is not a
discovery of the laws controlling the college; that he
has not even begun to find the controlling laws. The
college may totter but the Bursar's accounts still balance.
The law of conservation of momentum and energy
results from the overlapping of the different aspects in
which the "non-emptiness of space" presents itself to
our practical experience. once again we find that a
fundamental law of physics is no controlling law but a
"put-up job" as soon as we have ascertained the nature
SELECTIVE INFLUENCE OF THE MIND 239
of that which is obeying it. We can measure certain
forms of energy with a thermometer, momentum with
a ballistic pendulum, stress with a manometer. Com-
monly we picture these as separate physical entities
whose behaviour towards each other is controlled by
a law. But now the theory is that the three instruments
measure different but slightly overlapping aspects of a
single physical condition, and a law connecting their
measurements is of the same tautological type as a "law"
connecting measurements with a metre-rule and a foot-
rule.
I have said that violation of these laws of conserva-
tion is unthinkable. Have we then found physical laws
which will endure for all time unshaken by any future
revolution? But the proviso must be remembered,
"granting that the identification [of their subject
matter] is correct". The law itself will endure as long
as two and two make four; but its practical importance
depends on our knowing that which obeys it. We
think we have this knowledge, but do not claim in-
fallibility in this respect. From a practical point of view
the law would be upset, if it turned out that the thing
conserved was not that which we are accustomed to
measure with the above-mentioned instruments but
something slightly different.
Selective Influence of the Mind. This brings us very
near to the problem of bridging the gulf between the
scientific world and the world of everyday experience.
The simpler elements of the scientific world have no
immediate counterparts in everyday experience; we
use them to build things which have counterparts.
Energy, momentum and stress in the scientific world
shadow well-known features of the familiar world.
240 WORLD BUILDING
I feel stress in my muscles; one form of energy gives me
the sensation of warmth; the ratio of momentum to mass
is velocity, which generally enters into my experience
as change of position of objects. When I say that I feel
these things I must not forget that the feeling, in so far
as it is located in the physical world at all, is not in the
things themselves but in a certain corner of my brain.
In fact, the mind has also invented a craft of world-
building; its familiar world is built not from the dis-
tribution of relata and relations but by its own peculiar
interpretation of the code messages transmitted along
the nerves into its sanctum.
Accordingly we must not lose sight of the fact that
the world which physics attempts to describe arises
from the convergence of two schemes of world-building.
If we look at it only from the physical side there is
inevitably an arbitrariness about the building. Given
the bricks — the 16 measures of world-structure — there
are all sorts of things we might build. Or we might
take up again some of the rejected lumber and build a
still wider variety of things. But we do not build
arbitrarily; we build to order. The things we build have
certain remarkable properties; they have these pro-
perties in virtue of the way they are built, but they also
have them because such properties were ordered. There
is a general description which covers at any rate most
of the building operations needed in the construction
of the physical world; in mathematical language the
operation consists in Hamiltonian differentiation of an
invariant function of the 16 measures of structure. I do
not think that there is anything in the basal relation-
structure that cries out for this special kind of com-
bination; the significance of this process is not in
inorganic nature. Its significance is that it corresponds
SELECTIVE INFLUENCE OF THE MIND 241
to an outlook adopted by the mind for its own reasons;
and any other building process would not converge to
the mental scheme of world-building. The Hamiltonian
derivative has just that kind of quality which makes it
stand out in our minds as an active agent against a
passive extension of space and time; and Hamiltonian
differentiation is virtually the symbol for creation of an
active world out of the formless background. Not once
in the dim past, but continuously by conscious mind is
the miracle of the Creation wrought.
By following this particular plan of building we
construct things which satisfy the law of conservation,
that is to say things which are permanent. The law of
conservation is a truism for the things which satisfy it;
but its prominence in the scheme of law of the physical
world is due to the mind having demanded permanence.
We might have built things which do not satisfy this
law. In fact we do build one very important thing
"action" which is not permanent; in respect to "action"
physics has taken the bit in her teeth, and has insisted
on recognising this as the most fundamental thing of all,
although the mind has not thought it worthy of a place
in the familiar world and has not vivified it by any
mental image or conception. You will understand that
the building to which I refer is not a shifting about of
material; it is like building constellations out of stars.
The things which we might have built but did not, are
there just as much as those we did build. What we have
called building is rather a selection from the patterns
that weave themselves.
The element of permanence in the physical world,
which is familiarly represented by the conception of
substance, is essentially a contribution of the mind to
the plan of building or selection. We can see this
242 WORLD BUILDING
selective tendency at work in a comparatively simple
problem, viz. the hydrodynamical theory of the ocean.
At first sight the problem of what happens when the
water is given some initial disturbance depends solely on
inorganic laws; nothing could be more remote from the
intervention of conscious mind. In a sense this is true;
the laws of matter enable us to work out the motion
and progress of the different portions of the water; and
there, so far as the inorganic world is concerned, the
problem might be deemed to end. But actually in
hydrodynamical textbooks the investigation is diverted
in a different direction, viz. to the study of the motions
of waves and wave-groups. The progress of a wave is
not progress of any material mass of water, but of a
form which travels over the surface as the water heaves
up and down; again the progress of a wave-group is not
the progress of a wave. These forms have a certain
degree of permanence amid the shifting particles of
water. Anything permanent tends to become dignified
with an attribute of substantiality. An ocean traveller
has even more vividly the impression that the ocean is
made of waves than that it is made of water.* Ulti-
mately it is this innate hunger for permanence in our
minds which directs the course of development of
hydrodynamics, and likewise directs the world-building
out of the sixteen measures of structure.
Perhaps it will be objected that other things besides
mind can appreciate a permanent entity such as mass;
a weighing machine can appreciate it and move a
pointer to indicate how much mass there is. I do not
think that is a valid objection. In building the physical
world we must of course build the measuring appliances
* This was not intended to allude to certain consequential effects of
the waves; it is true, I think, of the happier impressions of the voyage.
SELECTIVE INFLUENCE OF THE MIND 243
which are part of it; and the measuring appliances
result from the plan of building in the same way as the
entities which they measure. If, for example, we had
used some of the "lumber" to build an entity x, we
could presumably construct from the same lumber an
appliance for measuring x. The difference is this — if the
pointer of the weighing machine is reading 5 lbs. a
human consciousness is in a mysterious way (not yet
completely traced) aware of the fact, whereas if the
measuring appliance for x reads 5 units no human mind
is aware of it. Neither x nor the appliance for measur-
ing x have any interaction with consciousness. Thus the
responsibility for the fact that the scheme of the scientific
world includes mass but excludes x rests ultimately with
the phenomena of consciousness.
Perhaps a better way of expressing this selective
influence of mind on the laws of Nature is to say that
values are created by the mind. All the "light and shade"
in our conception of the world of physics comes in this
way from the mind, and cannot be explained without
reference to the characteristics of consciousness.
The world which we have built from the relation-
structure is no doubt doomed to be pulled about a good
deal as our knowledge progresses. The quantum theory
shows that some radical change is impending. But I
think that our building exercise has at any rate widened
our minds to the possibilities and has given us a different
orientation towards the idea of physical law. The points
which I stress are:
Firstly, a strictly quantitative science can arise from
a basis which is purely qualitative. The comparability
that has to be assumed axiomatically is a merely quali-
tative discrimination of likeness and unlikeness.
Secondly, the laws which we have hitherto regarded
244 WORLD BUILDING
as the most typical natural laws are of the nature of
truisms, and the ultimate controlling laws of the basal
structure (if there are any) are likely to be of a differ-
ent type from any yet conceived.
Thirdly, the mind has by its selective power fitted
the processes of Nature into a frame of law of a pattern
largely of its own choosing; and in the discovery of this
system of law the mind may be regarded as regaining
from Nature that which the mind has put into
Nature.
Three Types of Law. So far as we are able to judge, the
laws of Nature divide themselves into three classes:
(i) identical laws, (2) statistical laws, (3) transcenden-
tal laws. We have just been considering the identical
laws, i.e. the laws obeyed as mathematical identities in
virtue of the way in which the quantities obeying them
are built. They cannot be regarded as genuine laws of
control of the basal material of the world. Statistical
laws relate to the behaviour of crowds, and depend on
the fact that although the behaviour of each individual
may be extremely uncertain average results can be
predicted with confidence. Much of the apparent uni-
formity of Nature is a uniformity of averages. Our
gross senses only take cognisance of the average effect of
vast numbers of individual particles and processes; and
the regularity of the average might well be compatible
with a great degree of lawlessness of the individual. I do
not think it is possible to dismiss statistical laws (such
as the second law of thermodynamics) as merely mathe-
matical adaptations of the other classes of law to certain
practical problems. They involve a peculiar element
of their own connected with the notion of a priori proba-
bility; but we do not yet seem able to find a place for
THREE TYPES OF LAW 245
this in any of the current conceptions of the world sub-
stratum.
If there are any genuine laws of control of the physical
world they must be sought in the third group — the
transcendental laws. The transcendental laws comprise
all those which have not become obvious identities im-
plied in the scheme of world-building. They are con-
cerned with the particular behaviour of atoms, electrons
and quanta — that is to say, the laws of atomicity of
matter, electricity and action. We seem to be mak-
ing some progress towards formulating them, but it is
clear that the mind is having a much harder struggle to
gain a rational conception of them than it had with the
classical field-laws. We have seen that the field-laws,
especially the laws of conservation, are indirecdy imposed
by the mind which has, so to speak, commanded a plan of
world-building to satisfy them. It is a natural suggestion
that the greater difficulty in elucidating the transcenden-
tal laws is due to the fact that we are no longer engaged
in recovering from Nature what we have ourselves put
into Nature, but are at last confronted with its own in-
trinsic system of government. But I scarcely know what
to think. We must not assume that the possible develop-
ments of the new attitude towards natural law have been
exhausted in a few short years. It may be that the laws
of atomicity, like the laws of conservation, arise only in
the presentation of the world to us and can be recognised
as identities by some extension of the argument we have
followed. But it is perhaps as likely that after we have
cleared away all the superadded laws which arise solely
in our mode of apprehension of the world about us, there
will be left an external world developing under genuine
laws of control.
At present we can notice the contrast that the laws
246 WORLD BUILDING
which we now recognise as man-made are characterised
by continuity, whereas the laws to which the mind as
yet lays no claim are characterised by atomicity. The
quantum theory with its avoidance of fractions and
insistence on integral units seems foreign to any scheme
which we should be likely subconsciously to have im-
posed as a frame for natural phenomena. Perhaps our
final conclusion as to the world of physics will resemble
Kronecker's view of pure mathematics.
u God made the integers, all else is the work of man."*
* Die ganzen Zahlen hat Gott gemacht; alles anderes ist Menschen-
werk.
Chapter XII
POINTER READINGS
Familiar Conceptions and Scientific Symbols. We have
said in the Introduction that the raw material of the
scientific world is not borrowed from the familiar world.
It is only recently that the physicist has deliberately cut
himself adrift from familiar conceptions. He did not
set out to discover a new world but to tinker with the
old. Like everyone else he started with the idea that
things are more or less what they seem, and that our
vivid impression of our environment may be taken as
a basis to work from. Gradually it has been found that
some of its most obvious features must be rejected. We
learn that instead of standing on a firm immovable earth
proudly rearing our heads towards the vault of heaven,
we are hanging by our feet from a globe careering
through space at a great many miles a second. But this
new knowledge can still be grasped by a rearrangement
of familiar conceptions. I can picture to myself quite
vividly the state of affairs just described; if there is any
strain, it is on my credulity, not on my powers of con-
ception. Other advances of knowledge can be accommo-
dated by that very useful aid to comprehension — "like
this only more so". For example, if you think of some-
thing like a speck of dust only more so you have the
atom as it was conceived up to a fairly recent date.
In addition to the familiar entities the physicist had
to reckon with mysterious agencies such as gravitation
or electric force; but this did not disturb his general
outlook. We cannot say what electricity is "like"j but
247
248 POINTER READINGS
at first its aloofness was not accepted as final. It was
taken to be one of the main aims of research to discover
how to reduce these agencies to something describable
in terms of familiar conceptions — in short to "explain"
them. For example, the true nature of electric force
might be some kind of displacement of the aether.
(Aether was at that time a familiar conception — like
some extreme kind of matter only more so.) Thus
there grew up a waiting-list of entities which should
one day take on their rightful relation to conceptions
of the familiar world. Meanwhile physics had to
treat them as best it could without knowledge of their
nature.
It managed surprisingly well. Ignorance of the nature
of these entities was no bar to successful prediction of
behaviour. We gradually awoke to the fact that the
scheme of treatment of quantities on the waiting-list
was becoming more precise and more satisfying than
our knowledge of familiar things. Familiar conceptions
did not absorb the waiting-list, but the waiting-list
began to absorb familiar conceptions. Aether, after
being in turn an elastic solid, a jelly, a froth, a con-
glomeration of gyrostats, was denied a material and
substantial nature and put back on the waiting-list. It
was found that science could accomplish so much with
entities whose nature was left in suspense that it began
to be questioned whether there was any advantage in
removing the suspense. The crisis came when we began
to construct familiar entities such as matter and light
out of things on the waiting-list. Then at last it was seen
that the linkage to familiar concepts should be through
the advanced constructs of physics and not at the be-
ginning of the alphabet. We have suffered, and we still
suffer, from expectations that electrons and quanta must
SCIENTIFIC SYMBOLS 249
be in some fundamental respects like materials or forces
familiar in the workshop — that all we have got to do is
to imagine the usual kind of thing on an infinitely smaller
scale. It must be our aim to avoid such prejudgments,
which are surely illogical; and since we must cease to
employ familiar concepts, symbols have become the only
possible alternative.
The synthetic method by which we build up from
its own symbolic elements a world which will imitate
the actual behaviour of the world of familiar experience
is adopted almost universally in scientific theories. Any
ordinary theoretical paper in the scientific journals
tacitly assumes that this approach is adopted. It has
proved to be the most successful procedure; and it is the
actual procedure underlying the advances set forth in
the scientific part of this book. But I would not claim
that no other way of working is admissible. We agree
that at the end of the synthesis there must be a linkage
to the familiar world of consciousness, and we are not
necessarily opposed to attempts to reach the physical
world from that end. From the point of view of philo-
sophy it is desirable that this entrance should be
explored, and it is conceivable that it may be fruitful
scientifically. If I have rightly understood Dr. White-
head's philosophy, that is the course which he takes. It
involves a certain amount of working backwards (as
we should ordinarily describe it) ; but his method of
"extensive abstraction" is intended to overcome some
of the difficulties of such a procedure. I am not qualified
to form a critical judgment of this work, but in principle
it appears highly interesting. Although this book may
in most respects seem diametrically opposed to Dr.
Whitehead's widely read philosophy of Nature, I think
it would be truer to regard him as an ally who from the
250 POINTER READINGS
opposite side of the mountain is tunnelling to meet his
less philosophically minded colleagues. The important
thing is not to confuse the two entrances.
Nature of Exact Science. one of the characteristics of
physics is that it is an exact science, and I have generally
identified the domain of physics with the domain of
exact science. Strictly speaking the two are not synony-
mous. We can imagine a science arising which has no
contact with the usual phenomena and laws of physics,
which yet admits of the same kind of exact treatment.
It is conceivable that the Mendelian theory of heredity
may grow into an independent science of this kind, for
it would seem to occupy in biology the same position
that the atomic theory occupied in chemistry a hundred
years ago. The trend of the theory is to analyse com-
plex individuals into "unit characters". These are like
indivisible atoms with affinities and repulsions; their
matings are governed by the same laws of chance which
play so large a part in chemical thermodynamics; and
numerical statistics of the characters of a population are
predictable in the same way as the results of a chemical
reaction.
Now the effect of such a theory on our philosophical
views of the significance of life does not depend on
whether the Mendelian atom admits of a strictly physical
explanation or not. The unit character may be contained
in some configuration of the physical molecules of the
carrier, and perhaps even literally correspond to a chem-
ical compound; or it may be something superadded which
is peculiar to living matter and is not yet comprised in
the schedule of physical entities. That is a side-issue.
We are drawing near to the great question whether there
is any domain of activity — of life, of consciousness, of
NATURE OF EXACT SCIENCE 251
deity — which will not be engulfed by the advance of
exact science; and our apprehension is not directed
against the particular entities of physics but against all
entities of the category to which exact science can apply.
For exact science invokes, or has seemed to invoke, a
type of law inevitable and soulless against which the
human spirit rebels. If science finally declares that man
is no more than a fortuitous concourse of atoms, the
blow will not be softened by the explanation that the
atoms in question are the Mendelian unit characters
and not the material atoms of the chemist.
Let us then examine the kind of knowledge which is
handled by exact science. If we search the examination
papers in physics and natural philosophy for the more
intelligible questions we may come across one beginning
something like this: "An elephant slides down a
grassy hillside. . . ." The experienced candidate knows
that he need not pay much attention to this; it is only
put in to give an impression of realism. He reads on:
"The mass of the elephant is two tons." Now we are
getting down to business; the elephant fades out of the
problem and a mass of two tons takes its place. What
exactly is this two tons, the real subject-matter of the
problem? It refers to some property or condition which
we vaguely describe as "ponderosity" occurring in a
particular region of the external world. But we shall not
get much further that way; the nature of the external
world is inscrutable, and we shall only plunge into a
quagmire of indescribables. Never mind what two tons
refers to; what is it? How has it actually entered in so
definite a way into our experience? Two tons is the
reading of the pointer when the elephant was placed
on a weighing-machine. Let us pass on. "The slope
of the hill is 6o°." Now the hillside fades out of the
252 POINTER READINGS
problem and an angle of 6o° takes its place. What is
6o°? There is no need to struggle with mystical con-
ceptions of direction; 6o° is the reading of a plumb-line
against the divisions of a protractor. Similarly for the
other data of the problem. The softly yielding turf on
which the elephant slid is replaced by a coefficient of
friction, which though perhaps not direcdy a pointer
reading is of kindred nature. No doubt there are more
roundabout ways used in practice for determining the
weights of elephants and the slopes of hills, but these
are justified because it is known that they give the same
results as direct pointer readings.
And so we see that the poetry fades out of the prob-
lem, and by the time the serious application of exact
science begins we are left with only pointer readings.
If then only pointer readings or their equivalents are
put into the machine of scientific calculation, how can
we grind out anything but pointer readings? But that
is just what we do grind out. The question presumably
was to find the time of descent of the elephant, and the
answer is a pointer reading on the seconds' dial of our
watch.
The triumph of exact science in the foregoing problem
consisted in establishing a numerical connection between
the pointer reading of the weighing-machine in one
experiment on the elephant and the pointer reading of
the watch in another experiment. And when we examine
critically other problems of physics we find that this is
typical. The whole subject-matter of exact science
consists of pointer readings and similar indications.
We cannot enter here into the definition of what are
to be classed as similar indications. The observation of
approximate coincidence of the pointer with a scale-
division can generally be extended to include the
NATURE OF EXACT SCIENCE 253
observation of any kind of coincidence — or, as it is
usually expressed in the language of the general rela-
tivity theory, an intersection of world-lines. The
essential point is that, although we seem to have very
definite conceptions of objects in the external world,
those conceptions do not enter into exact science and
are not in any way confirmed by it. Before exact science
can begin to handle the problem they must be replaced
by quantities representing the results of physical meas-
urement.
Perhaps you will object that although only the pointer
readings enter into the actual calculation it would make
nonsense of the problem to leave out all reference to
anything else. The problem necessarily involves some
kind of connecting background. It was not the pointer
reading of the weighing-machine that slid down the
hill! And yet from the point of view of exact science the
thing that really did descend the hill can only be de-
scribed as a bundle of pointer readings. (It should be
remembered that the hill also has been replaced by
pointer readings, and the sliding down is no longer an
active adventure but a functional relation of space and
time measures.) The word elephant calls up a certain
association of mental impressions, but it is clear that
mental impressions as such cannot be the subject
handled in the physical problem. We have, for example,
an impression of bulkiness. To this there is presumably
some direct counterpart in the external world, but that
counterpart must be of a nature beyond our appre-
hension, and science can make nothing of it. Bulkiness
enters into exact science by yet another substitution;
we replace it by a series of readings of a pair of cali-
pers. Similarly the greyish black appearance in our
mental impression is replaced in exact science by the read-
254 POINTER READINGS
ings of a photometer for various wave-lengths of light.
And so on until all the characteristics of the elephant
are exhausted and it has become reduced to a schedule
of measures. There is always the triple correspond-
ence —
(a) a mental image, which is in our minds and not in
the external world;
(b) some kind of counterpart in the external world,
which is of inscrutable nature;
(<:) a set of pointer readings, which exact science can
study and connect with other pointer readings.
And so we have our schedule of pointer readings
ready to make the descent. And if you still think that
this substitution has taken away all reality from the
problem, I am not sorry that you should have a foretaste
of the difficulty in store for those who hold that exact
science is all-sufficient for the description of the universe
and that there is nothing in our experience which cannot
be brought within its scope.
I should like to make it clear that the limitation of
the scope of physics to pointer readings and the like is
not a philosophical craze of my own but is essentially
the current scientific doctrine. It is the outcome of a
tendency discernible far back in the last century but
only formulated comprehensively with the advent of
the relativity theory. The vocabulary of the physicist
comprises a number of words such as length, angle,
velocity, force, potential, current, etc., which we call
"physical quantities". It is now recognised as essential
that these should be defined according to the way in
which we actually recognise them when confronted with
them, and not according to the metaphysical significance
which we may have anticipated for them. In the old
textbooks mass was defined as "quantity of matter";
NATURE OF EXACT SCIENCE 255
but when it came to an actual determination of mass, an
experimental method was prescribed which had no
bearing on this definition. The belief that the quantity
determined by the accepted method of measurement
represented the quantity of matter in the object was
merely a pious opinion. At the present day there is no
sense in which the quantity of matter in a pound of lead
can be said to be equal to the quantity in a pound of
sugar. Einstein's theory makes a clean sweep of these
pious opinions, and insists that each physical quantity
should be defined as the result of certain operations of
measurement and calculation. You may if you like
think of mass as something of inscrutable nature to
which the pointer reading has a kind of relevance. But
in physics at least there is nothing much to be gained
by this mystification, because it is the pointer reading
itself which is handled in exact science; and if you
embed it in something of a more transcendental nature,
you have only the extra trouble of digging it out
again.
It is quite true that when we say the mass is two tons
we have not specially in mind the reading of the particu-
lar machine on which the weighing was carried out.
That is because we do not start to tackle the problem of
the elephant's escapade ab initio as though it were the
first inquiry we had ever made into the phenomena of
the external world. The examiner would have had to be
much more explicit if he had not presumed a general
acquaintance with the elementary laws of physics, i.e.
laws which permit us to deduce the readings of other
indicators from the reading of one. // is this connec-
tivity of pointer readings, expressed by physical laws,
which supplies the continuous background that any realis-
tic problem demands.
256 POINTER READINGS
It is obviously one of the conditions of the problem
that the same elephant should be concerned in the
weighing experiment and in the tobogganing experi-
ment. How can this identity be expressed in a descrip-
tion of the world by pointer readings only? Two
readings may be equal, but it is meaningless to inquire
if they are identical; if then the elephant is a bundle of
pointer readings, how can we ask whether it is continu-
ally the identical bundle ? The examiner does not confide
to us how the identity of the elephant was ensured; we
have only his personal guarantee that there was no
substitution. Perhaps the creature answered to its name
on both occasions; if so the test of identity is clearly
outside the present domain of physics. The only test
lying purely in the domain of physics is that of con-
tinuity; the elephant must be watched all the way from
the scales to the hillside. The elephant, we must remem-
ber, is a tube in the four-dimensional world demarcated
from the rest of space-time by a more or less abrupt
boundary. Using the retina of his eye as an indicator
and making frequent readings of the oudine of the
image, the observer satisfied himself that he was fol-
lowing one continuous and isolated world-tube from
beginning to end. If his vigilance was intermittent he
took a risk of substitution, and consequently a risk of
the observed time of descent failing to agree with the
time calculated.* Note that we do not infer that there
is any identity of the contents of the isolated world-tube
throughout its length; such identity would be meaning-
* A good illustration of such substitution is afforded by astronomical
observations of a certain double star with two components of equal
brightness. After an intermission of observation the two components
were inadvertently interchanged, and the substitution was not detected
until the increasing discrepancy between the actual and predicted orbits
was inquired into.
LIMITS OF PHYSICAL KNOWLEDGE 257
less in physics. We use instead the law of conserva-
tion of mass (either as an empirical law or deduced
from the law of gravitation) which assures us that,
provided the tube is isolated, the pointer reading on
the schedule derived from the weighing-machine type
of experiment has a constant value along the tube.
For the purpose of exact science "the same object"
becomes replaced by "isolated world-tube". The con-
stancy of certain properties of the elephant is not
assumed as self-evident from its sameness, but is an
inference from experimental and theoretical laws re-
lating to world-tubes which are accepted as well
established.
Limitations of Physical Knowledge, Whenever we state
the properties of a body in terms of physical quantities
we are imparting knowledge as to the response of
various metrical indicators to its presence, and nothing
more. After all, knowledge of this kind is fairly com-
prehensive. A knowledge of the response of all kinds
of objects — weighing-machines and other indicators —
would determine completely its relation to its environ-
ment, leaving only its inner un-get-atable nature un-
determined. In the relativity theory we accept this as
full knowledge, the nature of an object in so far as it is
ascertainable by scientific inquiry being the abstraction
of its relations to all surrounding objects. The progress
of the relativity theory has been largely due to the
development of a powerful mathematical calculus for
dealing compendiously with an infinite scheme of
pointer readings, and the technical term tensor used so
largely in treatises on Einstein's theory may be translated
schedule of pointer readings. It is part of the aesthetic
appeal of the mathematical theory of relativity that the
258 POINTER READINGS
mathematics is so closely adapted to the physical con-
ceptions. It is not so in all subjects. For example, we
may admire the triumph of patience of the mathemati-
cian in predicting so closely the positions of the moon,
but aesthetically the lunar theory is atrocious; it is
obvious that the moon and the mathematician use dif-
ferent methods of finding the lunar orbit. But by the
use of tensors the mathematical physicist precisely de-
scribes the nature of his subject-matter as a schedule of
indicator readings; and those accretions of images and
conceptions which have no place in physical science are
automatically dismissed.
The recognition that our knowledge of the objects
treated in physics consists solely of readings of pointers
and other indicators transforms our view of the status
of physical knowledge in a fundamental way. Until
recently it was taken for granted that we had knowledge
of a much more intimate kind of the entities of the ex-
ternal world. Let me give an illustration which takes
us to the root of the great problem of the relations
of matter and spirit. Take the living human brain
endowed with mind and thought. Thought is one of the
indisputable facts of the world. I know that I think,
with a certainty which I cannot attribute to any of my
physical knowledge of the world. More hypothetically,
but pn fairly plausible evidence, I am convinced that
you have minds which think. Here then is a world fact
to be investigated. The physicist brings his tools and
commences systematic exploration. All that he dis-
covers is a collection of atoms and electrons and fields of
force arranged in space and time, apparently similar to
those found in inorganic objects. He may trace other
physical characteristics, energy, temperature, entropy.
None of these is identical with thought. He might set
LIMITS OF PHYSICAL KNOWLEDGE 259
down thought as an illusion — some perverse interpreta-
tion of the interplay of the physical entities that he has
found. Or if he sees the folly of calling the most un-
doubted element of our experience an illusion, he will
have to face the tremendous question, How can this col-
lection of ordinary atoms be a thinking machine? But
what knowledge have we of the nature of atoms which
renders it at all incongruous that they should constitute
a thinking object? The Victorian physicist felt that he
knew just what he was talking about when he used such
terms as matter and atoms. Atoms were tiny billiard
balls, a crisp statement that was supposed to tell you
all about their nature in a way which could never be
achieved for transcendental things like consciousness,
beauty or humour. But now we realise that science has
nothing to say as to the intrinsic nature of the atom. The
physical atom is, like everything else in physics, a
schedule of pointer readings. The schedule is, we agree,
attached to some unknown background. Why not then
attach it to something of spiritual nature of which a
prominent characteristic is thought. It seems rather silly
to prefer to attach it to something of a so-called "con-
crete" nature inconsistent with thought, and then to
wonder where the thought comes from. We have dis-
missed all preconception as to the background of our
pointer readings, and for the most part we can discover
nothing as to its nature. But in one case — namely, for
the pointer readings of my own brain — I have an in-
sight which is not limited to the evidence of the pointer
readings. That insight shows that they are attached to
a background of consciousness. Although I may expect
that the background of other pointer readings in physics
is of a nature continuous with that revealed to me in this
particular case, I do not suppose that it always has the
260 POINTER READINGS
more specialised attributes of consciousness.* But in
regard to my one piece of insight into the background
no problem of irreconcilability arises; I have no other
knowledge of the background with which to reconcile it.
In science we study the linkage of pointer readings
with pointer readings. The terms link together in endless
cycle with the same inscrutable nature running through
the whole. There is nothing to prevent the assemblage
of atoms constituting a brain from being of itself a
thinking object in virtue of that nature which physics
leaves undetermined and undeterminable. If we must
embed our schedule of indicator readings in some kind
of background, at least let us accept the only hint we
have received as to the significance of the background —
namely that it has a nature capable of manifesting itself
as mental activity.
Cyclic Method of Physics. I must explain this reference
to an endless cycle of physical terms. I will refer again
to Einstein's law of gravitation. I have already ex-
pounded it to you more than once and I hope you gained
some idea of it from the explanation. This time I am
going to expound it in a way so complete that there is
not much likelihood that anyone will understand it.
Never mind. We are not now seeking further light on
the cause of gravitation; we are interested in seeing
* For example, we should most of us assume (hypothetically) that
the dynamical quality of the world referred to in chapter v is characteris-
tic of the whole background. Apparently it is not to be found in the
pointer readings, and our only insight into it is in the feeling of "becom-
ing" in our consciousness. "Becoming" like "reasoning" is known to us
only through its occurrence in our own minds; but whereas it would be
absurd to suppose that the latter extends to inorganic aggregations of
atoms, the former may be (and commonly is) extended to the inorganic
world, so that it is not a matter of indifference whether the progress of
the inorganic world is viewed from past to future or from future to past.
CYCLIC METHOD OF PHYSICS 261
what would really be involved in a complete explanation
of anything physical.
Einstein's law in its analytical form is a statement that
in empty space certain quantities called potentials obey
certain lengthy differential equations. We make a
memorandum of the word ''potential" to remind us
that we must later on explain what it means. We might
conceive a world in which the potentials at every moment
and every place had quite arbitrary values. The actual
world is not so unlimited, the potentials being restricted
to those values which conform to Einstein's equations.
The next question is, What are potentials? They can
be defined as quantities derived by quite simple mathe-
matical calculations from certain fundamental quantities
called intervals. (Mem. Explain "interval".) If we
know the values of the various intervals throughout the
world definite rules can be given for deriving the values
of the potentials. What are intervals? They are rela-
tions between pairs of events which can be measured with
a scale or a clock or with both. (Mem. Explain "scale"
and "clock".) Instructions can be given for the correct
use of the scale and clock so that the interval is given by
a prescribed combination of their readings. What are
scales and clocks? A scale is a graduated strip of mat-
ter which. . . . (Mem. Explain "matter".) on second
thoughts I will leave the rest of the description as "an
exercise to the reader" since \t would take rather a long
time to enumerate all the properties and niceties of
behaviour of the material standard which a physicist
would accept as a perfect scale or a perfect clock. We
pass on to the next question, What is matter? We have
dismissed the metaphysical conception of substance. We
might perhaps here describe the atomic and electrical
structure of matter, but that leads to the microscopic
262 POINTER READINGS
aspects of the world, whereas we are here taking the
macroscopic outlook. Confining ourselves to mechanics,
which is the subject in which the law of gravitation
arises, matter may be defined as the embodiment of three
related physical quantities, mass (or energy), momentum
and stress. What are "mass", "momentum" and
"stress"? It is one of the most far-reaching achieve-
ments of Einstein's theory that it has given an exact
answer to this question. They are rather formidable
looking ｅxpressions containing the potentials and their
first and second derivatives with respect to the co-
ordinates. What are the potentials? Why, that is just
what I have been explaining to you!
The definitions of physics proceed according to the
method immortalised in "The House that Jack built" :
This is the potential, that was derived from the interval,
that was measured by the scale, that was made from the
matter, that embodied the stress, that. . . . But instead
of finishing with Jack, whom of course every youngster
must know without need for an introduction, we make
a circuit back to the beginning of the rhyme: . . . that
worried the cat, that killed the rat, that ate the malt,
that lay in the house, that was built by the priest all
shaven and shorn, that married the man. . . . Now we
can go round and round for ever.
But perhaps you have already cut short my explana-
tion of gravitation. When we reached matter you had
had enough of it. "Please do not explain any more,
I happen to know what matter is." Very well; matter
is something that Mr. X knows. Let us see how it goes :
This is the potential that was derived from the interval
that was measured by the scale that was made from the
matter that Mr. X knows. Next question, What is Mr. X?
Well, it happens that physics is not at all anxious to
CYCLIC METHOD OF PHYSICS 263
pursue the question, What is Mr. X? It is not disposed
to admit that its elaborate structure of a physical uni-
verse is ''The House that Mr. X built". It looks upon
Mr. X — and more particularly the part of Mr. X that
knows — as a rather troublesome tenant who at a late
stage of the world's history has come to inhabit a
Potential
Stress m •Interval
Matter • < • Scale
Mr. X •
Fig. 8
structure which inorganic Nature has by slow evolutionary
progress contrived to build. And so it turns aside from
the avenue leading to Mr. X — and beyond — and closes
up its cycle leaving him out in the cold.
From its own point of view physics is entirely jus-
tified. That matter in some indirect way comes within
the purview of Mr. X's mind is not a fact of any utility
264 POINTER READINGS
for a theoretical scheme of physics. We cannot embody
it in a differential equation. It is ignored; and the
physical properties of matter and other entities are
expressed by their linkages in the cycle. And you can
see how by the ingenious device of the cycle physics
secures for itself a self-contained domain for study with
no loose ends projecting into the unknown. All other
physical definitions have the same kind of interlocking.
Electric force is defined as something which causes
motion of an electric charge ; an electric charge is some-
thing which exerts electric force. So that an electric
charge is something that exerts something that produces
motion of something that exerts something that produces
. . . ad infinitum.
But I am not now writing of pure physics, and from
a broader standpoint I do not see how we can leave out
Mr. X. The fact that matter is "knowable to Mr. X"
must be set down as one of the fundamental attributes
of matter. I do not say that it is very distinctive, since
other entities of physics are also knowable to him; but
the potentiality of the whole physical world for awaking
impressions in consciousness is an attribute not to be
ignored when we compare the actual world with worlds
which, we fancy, might have been created. There seems
to be a prevalent disposition to minimise the importance
of this. The attitude is that "knowableness to Mr. X"
is a negligible attribute, because Mr. X is so clever that
he could know pretty much anything that there was to
know. I have already urged the contrary view — that
there is a definitely selective action of the mind; and
since physics treats of what is knowable to mind * its
* This is obviously true of all experimental physics, and must be
true of theoretical physics if it is (as it professes to be) based on experi-
ment.
ACTUALITY 265
subject-matter has undergone, and indeed retains evi-
dences of, this process of selection.
Actuality. "Knowableness to mind" is moreover a
property which differentiates the actual world of our
experience from imaginary worlds in which the same
general laws of Nature are supposed to hold true.
Consider a world — Utopia, let us say — governed by all
the laws of Nature known and unknown which govern
our own world, but containing better stars, planets,
cities, animals, etc. — a world which might exist, but
it just happens that it doesn't. How can the physicist
test that Utopia is not the actual world? We refer to
a piece of matter in it; it is not real matter but it attracts
any other piece of (unreal) matter in Utopia according
to the law of gravitation. Scales and clocks constructed
of this unreal matter will measure wrong intervals, but
the physicist cannot detect that they are wrong unless
he has first shown the unreality of the matter. As soon
as any element in it has been shown to be unreal Utopia
collapses; but so long as we keep to the cycles of physics
we can never find the vulnerable point, for each element
is correctly linked to the rest of the cycle, all our laws
of Nature expressed by the cycle being obeyed in Utopia
by hypothesis. The unreal stars emit unreal light which
falls on unreal retinas and ultimately reaches unreal
brains. The next step takes it outside the cycle and gives
the opportunity of exposing the whole deception. Is
the brain disturbance translated into consciousness?
That will test whether the brain is real or unreal. There
is no question about consciousness being real or not;
consciousness is self-knowing and the epithet real adds
nothing to that. Of the infinite number of worlds
which are examples of what might be possible under the
266 POINTER READINGS
laws of Nature, there is one which does something more
than fulfil those laws of Nature. This property, which
is evidently not definable with respect to any of the laws
of Nature, we describe as "actuality" — generally using
the word as a kind of halo of indefinite import. We
have seen that the trend of modern physics is to reject
these indefinite attributions and to define its terms
according to the way in which we recognise the pro-
perties w T hen confronted by them. We recognise the
actuality of a particular world because it is that world
alone with which consciousness interacts. However
much the theoretical physicist may dislike a reference to
consciousness, the experimental physicist uses freely
this touchstone of actuality. He would perhaps prefer
to believe that his instruments and observations are certi-
fied as actual by his material sense organs; but the final
guarantor is the mind that comes to know the indications
of the material organs. Each of us is armed with this
touchstone of actuality; by applying it we decide that
this sorry world of ours is actual and Utopia is a dream.
As our individual consciousnesses are different, so our
touchstones are different; but fortunately they all agree
in their indication of actuality — or at any rate those
which agree are in sufficient majority to shut the others
up in lunatic asylums.
It is natural that theoretical physics in its formulation
of a general scheme of law should leave out of account
actuality and the guarantor of actuality. For it is just
this omission which makes the difference between a law
of Nature and a particular sequence of events. That
which is possible (or not "too improbable") is the
domain of natural science; that which is actual is the
domain of natural history. We need scarcely add that
the contemplation in natural science of a wider domain
ACTUALITY 267
than the actual leads to a far better understanding of
the actual.
From a broader point of view than that of elaborating
the physical scheme of law we cannot treat the connection
with mind as merely an incident in a self-existent inor-
ganic world. In saying that the differentiation of the
actual from the non-actual is only expressible by reference
to mind I do not mean to imply that a universe without
conscious mind would have no more status than Utopia.
But its property of actuality would be indefinable since
the one approach to a definition is cut off. The actuality
of Nature is like the beauty of Nature. We can scarcely
describe the beauty of a landscape as non-existent when
there is no conscious being to witness it; but it is through
consciousness that we can attribute a meaning to it.
And so it is with the actuality of the world. If actuality
means "known to mind" then it is a purely subjective
character of the world; to make it objective we must
substitute "knowable to mind". The less stress we lay
on the accident of parts of the world being known at
the present era to particular minds, the more stress we
must lay on the potentiality of being known to mind as
a fundamental objective property of matter, giving it
the status of actuality whether individual consciousness
is taking note of it or not.
In the diagram Mr. X has been linked to the cycle at
a particular point in deference to his supposed claim
that he knows matter; but a little reflection will show
that the point of contact of mind with the physical
universe is not very definite. Mr. X knows a table; but
the point of contact with his mind is not in the material
of the table. Light waves are propagated from the
table to the eye; chemical changes occur in the retina;
propagation of some kind occurs in the optic nerves;
268 POINTER READINGS
atomic changes follow in the brain. Just where the
final leap into consciousness occurs is not clear. We do
not know the last stage of the message in the physical
world before it became a sensation in consciousness.
This makes no difference. The physical entities have
a cyclic connection, and whatever intrinsic nature we
attribute to one of them runs as a background through
the whole cycle. It is not a question whether matter or
electricity or potential is the direct stimulus to the mind;
in their physical aspects these are equally represented
as pointer readings or schedules of pointer readings.
According to our discussion of world building they are
the measures of structure arising from the comparability
of certain aspects of the basal relations — measures which
by no means exhaust the significance of those relations.
I do not believe that the activity of matter at a certain
point of the brain stimulates an activity of mind; my
view is that in the activity of matter there is a metrical
description of certain aspects of the activity of mind.
The activity of the matter is our way of recognising a
combination of the measures of structure; the activity
of the mind is our insight into the complex of relations
whose comparability gives the foundation of those
measures.
"What is Mr. X?" In the light of these considerations
let us now see what we can make of the question, What
is Mr. X? I must undertake the inquiry single-handed;
I cannot avail myself of your collaboration without first
answering or assuming an answer to the equally difficult
question, What are you? Accordingly the whole in-
quiry must take place in the domain of my own con-
sciousness. I find there certain data purporting to
relate to this unknown X; and I can (by using powers
"WHAT IS MR. X?" 269
which respond to my volition) extend the data, i.e. I can
perform experiments on X. For example I can make
a chemical analysis. The immediate result of these
experiments is the occurrence of certain visual or
olfactory sensations in my consciousness. Clearly it is
a long stride from these sensations to any rational in-
ference about Mr. X. For example, I learn that Mr. X
has carbon in his brain, but the immediate knowledge
was of something (not carbon) in my own mind. The
reason why I, on becoming aware of something in my
mind, can proceed to assert knowledge of something
elsewhere, is because there is a systematic scheme of
inference which can be traced from the one item of
knowledge to the other. Leaving aside instinctive or
commonsense inference — the crude precursor of scien-
tific inference — the inference follows a linkage, which
can only be described symbolically, extending from the
point in the symbolic world where I locate myself to the
point where I locate Mr. X.
one feature of this inference is that I never discover
what carbon really is. It remains a symbol. There is
carbon in my own brain-mind; but the self-knowledge
of my mind does not reveal this to me. I can only know
that the symbol for carbon must be placed there by
following a route of inference through the external
world similar to that used in discovering it in Mr. X; and
however closely associated this carbon may be with my
thinking powers, it is as a symbol divorced from any
thinking capacity that I learn of its existence. Carbon
is a symbol definable only in terms of the other symbols
belonging to the cyclic scheme of physics. What I have
discovered is that, in order that the symbols describing
the physical world may conform to the mathematical
formulae which they are designed to obey, it is necessary
270 POINTER READINGS
to place the symbol for carbon (amongst others) in the
locality of Mr. X. By similar means I can make an
exhaustive physical examination of Mr. X and discover
the whole array of symbols to be assigned to his
locality.
Will this array of symbols give me the whole of
Mr. X? There is not the least reason to think so. The
voice that comes to us over the telephone wire is not the
whole of what is at the end of the wire. The scientific
linkage is like the telephone wire; it can transmit just
what it is constructed to transmit and no more.
It will be seen that the line of communication has
two aspects. It is a chain of inference stretching from
the symbols immediately associated with the sensations
in my mind to the symbols descriptive of Mr. X; and
it is a chain of stimuli in the external world starting
from Mr. X and reaching my brain. Ideally the steps
of the inference exactly reverse the steps of the physical
transmission which brought the information. (Naturally
we make many short cuts in inference by applying
accumulated experience and knowledge.) Commonly
we think of it only in its second aspect as a physical
transmission; but because it is also a line of inference
it is subject to limitations which we should not necessarily
expect a physical transmission to conform to.
The system of inference employed in physical in-
vestigation reduces to mathematical equations governing
the symbols, and so long as we adhere to this procedure
we are limited to symbols of arithmetical character
appropriate to such mathematical equations.* Thus
there is no opportunity for acquiring by any physical
* The solitary exception is, I believe, Dirac's generalisation which
introduces g-numbers (p. 210). There is as yet no approach to a general
system of inference on a non-numerical basis.
"WHAT IS MR. X?" 271
investigation a knowledge of Mr. X other than that
which can be expressed in numerical form so as to
be passed through a succession of mathematical
equations.
Mathematics is the model of exact inference; and
in physics we have endeavoured to replace all cruder
inference by this rigorous type. Where we cannot
complete the mathematical chain we confess that we are
wandering in the dark and are unable to assert real
knowledge. Small wonder then that physical science
should have evolved a conception of the world consisting
of entities rigorously bound to one another by mathe-
matical equations forming a deterministic scheme. This
knowledge has all been inferred and it was bound there-
fore to conform to the system of inference that was used.
The determinism of the physical laws simply reflects
the determinism of the method of inference. This soulless
nature of the scientific world need not worry those who
are persuaded that the main significances of our en-
vironment are of a more spiritual character. Anyone
who studied the method of inference employed by the
physicist could predict the general characteristics of
the world that he must necessarily find. What he could
not have predicted is the great success of the method —
the submission of so large a proportion of natural
phenomena to be brought into the prejudged scheme.
But making all allowance for future progress in develop-
ing the scheme, it seems to be flying in the face of
obvious facts to pretend that it is all comprehensive,
Mr. X is one of the recalcitrants. When sound-waves
impinge on his ear he moves, not in accordance with a
mathematical equation involving the physical measure
numbers of the waves, but in accordance with the
meaning that those sound-waves are used to convey. To
272 POINTER READINGS
know what there is about Mr. X which makes him
behave in this strange way, we must look not to a
physical system of inference, but to that insight beneath
the symbols which in our own minds we possess. It is
by this insight that we can finally reach an answer to
our question, What is Mr. X?
Chapter XIII
REALITY
The Real and the Concrete. one of our ancestors, taking
arboreal exercise in the forest, failed to reach the bough
intended and his hand closed on nothingness. The
accident might well occasion philosophical reflections
on the distinctions of substance and void — to say nothing
of the phenomenon of gravity. However that may be,
his descendants down to this day have come to be
endowed with an immense respect for substance arising
we know not how or why. So far as familiar experience
is concerned, substance occupies the centre of the stage,
rigged out with the attributes of form, colour, hardness,
etc., which appeal to our several senses. Behind it is a
subordinate background of space and time permeated
by forces and unconcrete agencies to minister to the
star performer.
Our conception of substance is only vivid so long as
we do not face it. It begins to fade when we analyse it.
We may dismiss many of its supposed attributes which
are evidently projections of our sense-impressions out-
wards into the external world. Thus the colour which is
so vivid to us is in our minds and cannot be embodied
in a legitimate conception of the substantial object itself.
But in any case colour is no part of the essential nature
of substance. Its supposed nature is that which we try
to call to mind by the word "concrete", which is
perhaps an outward projection of our sense of touch.
273
274 REALITY
When I try to abstract from the bough everything but
its substance or concreteness and concentrate on an
effort to apprehend this, all ideas elude me; but the
effort brings with it an instinctive tightening of the
fingers — from which perhaps I might infer that my
conception of substance is not very different from my
arboreal ancestor's.
So strongly has substance held the place of leading
actor on the stage of experience that in common usage
concrete and real are almost synonymous. Ask any man
who is not a philosopher or a mystic to name something
typically real; he is almost sure to choose a concrete
thing. Put the question to him whether Time is real;
he will probably decide with some hesitation that it
must be classed as real, but he has an inner feeling that
the question is in some way inappropriate and that he is
being cross-examined unfairly.
In the scientific world the conception of substance
is wholly lacking, and that which most nearly replaces
it, viz. electric charge, is not exalted as star-performer
above the other entities of physics. For this reason the
scientific world often shocks us by its appearance of
unreality. It offers nothing to satisfy our demand for
the concrete. How should it, when we cannot formu-
late that demand? I tried to formulate it; but nothing
resulted save a tightening of the fingers. Science does
not overlook the provision for tactual and muscular
sensation. In leading us away from the concrete, science
is reminding us that our contact with the real is more
varied than was apparent to the ape-mind, to whom the
bough which supported him typified the beginning and
end of reality.
It is not solely the scientific world that will now
occupy our attention. In accordance with the last
THE REAL AND THE CONCRETE 275
chapter we are takfng a larger view in which the cyclical
schemes of physics are embraced with much besides.
But before venturing on this more risky ground I have
to emphasise one conclusion which is definitely scien-
tific. The modern scientific theories have broken away
from the common standpoint which identifies the real
with the concrete. I think we might go so far as to say
that time is more typical of physical reality than matter,
because it is freer from those metaphysical associations
which physics disallows. It would not be fair, being
given an inch, to take an ell, and say that having gone
so far physics may as well admit at once that reality is
spiritual. We must go more warily. But in approaching
such questions we are no longer tempted to take up the
attitude that everything which lacks concreteness is
thereby self-condemned.
The cleavage between the scientific and the extra-
scientific domain of experience is, I believe, not a
cleavage between the concrete and the transcendental
but between the metrical and the non-metrical. I am
at one with the materialist in feeling a repugnance
towards any kind of pseudo-science of the extra-
scientific territory. Science is not to be condemned as
narrow because it refuses to deal with elements of
experience which are unadapted to its own highly
organised method ; nor can it be blamed for looking super-
ciliously on the comparative disorganisation of our knowl-
edge and methods of reasoning about the non-metrical
part of experience. But I think we have not been guilty
of pseudo-science in our attempt to show in the last two
chapters how it comes about that within the whole
domain of experience a selected portion is capable of
that exact metrical representation which is requisite for
development by the scientific method.
276 REALITY
Mind-Stuff. I will try to be as definite as I can as to the
glimpse of reality which we seem to have reached. only
I am well aware that in committing myself to details
I shall probably blunder. Even if the right view has
here been taken of the philosophical trend of modern
science, it is premature to suggest a cut-and-dried
scheme of the nature of things. If the criticism is made
that certain aspects are touched on which come more
within the province of the expert psychologist, I must
admit its pertinence. The recent tendencies of science
do, I believe, take us to an eminence from which we
can look down into the deep waters of philosophy; and
if I rashly plunge into them,, it is not because I have
confidence in my powers of swimming, but to try to
show that the water is really deep.
To put the conclusion crudely — the stuff of the world
is mind-stuff. As is often the way with crude statements,
I shall have to explain that by "mind" I do not here
exactly mean mind and by "stuff" I do not at all mean
stuff. Still this is about as near as we can get to the idea
in a simple phrase. The mind-stuff of the world is, of
course, something more general than our individual
conscious minds; but we may think of its nature as not
altogether foreign to the feelings in our consciousness.
The realistic matter and fields of force of former
physical theory are altogether irrelevant — except in so
far as the mind-stuff has itself spun these imaginings.
The symbolic matter and fields of force of present-day
theory are more relevant, but they bear to it the same
relation that the bursar's accounts bear to the activity
of the college. Having granted this, the mental activity
of the part of the world constituting ourselves occasions
no surprise; it is known to us by direct self-knowledge,
and we do not explain it away as something other than
MIND-STUFF 277
we know it to be — or, rather, it knows itself to be. It
is the physical aspects of the world that we have to
explain, presumably by some such method as that set
forth in our discussion on world-building. Our bodies
are more mysterious than our minds — at least they
would be, only that we can set the mystery on one side
by the device of the cyclic scheme of physics, which
enables us to study their phenomenal behaviour without
ever coming to grips with the underlying mystery.
The mind-stuff is not spread in space and time; these
are part of the cyclic scheme ultimately derived out of
it. But we must presume that in some other way or
aspect it can be differentiated into parts. only here and
there does it rise to the level of consciousness, but from
such islands proceeds all knowledge. Besides the direct
knowledge contained in each self-knowing unit, there
is inferential knowledge. The latter includes our know-
ledge of the physical world. It is necessary to keep
reminding ourselves that all knowledge of our environ-
ment from which the world of physics is constructed,
has entered in the form of messages transmitted along
the nerves to the seat of consciousness. Obviously the
messages travel in code. When messages relating to a
table are travelling in the nerves, the nerve-disturbance
does not in the least resemble either the external table
that originates the mental impression or the conception
of the table that arises in consciousness.* In the central
clearing station the incoming messages are sorted and
decoded, partly by instinctive image-building inherited
*I mean, resemble in intrinsic nature. It is true (as Bertrand Russell
has emphasised) that the symbolic description of structure will be iden-
tical for the t table in the external world and for the conception of the
table in consciousness if the conception is scientifically correct. If the
physicist does not attempt to penetrate beneath the structure he is in-
different as to which of the two we imagine ourselves to be discussing.
278 REALITY
from the experience of our ancestors, partly by scientific
comparison and reasoning. By this very indirect and
hypothetical inference all our supposed acquaintance
with and our theories of a world outside us have been
built up. We are acquainted with an external world
because its fibres run into our consciousness; it is only
our own ends of the fibres that we actually know; from
those ends we more or less successfully reconstruct the
rest, as a palaeontologist reconstructs an extinct monster
from its footprint.
The mind-stuff is the aggregation of relations and
relata which form the building material for the physical
world. Our account of the building process shows,
however, that much that is implied in the relations is
dropped as unserviceable for the required building.
Our view is practically that urged in 1875 by W. K.
Clifford—
"The succession of feelings which constitutes a man's
consciousness is the reality which produces in our minds
the perception of the motions of his brain."
That is to say, that which the man himself knows as
a succession of feelings is the reality which when probed
by the appliances of an outside investigator affects their
readings in such a way that it is identified as a configura-
tion of brain-matter. Again Bertrand Russell writes — *
What the physiologist sees when he examines a brain is in the
physiologist, not in the brain he is examining. What is in the
brain by the time the physiologist examines it if it is dead, I do
not profess to know; but while its owner was alive, part, at least,
of the contents of his brain consisted of his percepts, thoughts,
and feelings. Since his brain also consisted of electrons, we are
compelled to conclude that an electron is a grouping of events,
* Analysis of Matter, p. 320.
MIND-STUFF 279
and that if the electron is in a human brain, some of the events
composing it are likely to be some of the "mental states" of the
man to whom the brain belongs. Or, at any rate, they are likely
to be parts of such "mental states" — for it must not be assumed
that part of a mental state must be a mental state. I do not wish
to discuss what is meant by a "mental state"; the main point for
us is that the term must include percepts. Thus a percept is an
event or a group of events, each of which belongs to one or more
of the groups constituting the electrons in the brain. This,
I think, is the most concrete statement that can be made about
electrons; everything else that can be said is more or less abstract
and mathematical.
I quote this partly for the sake of the remark that it
must not be assumed that part of a mental state must
necessarily be a mental state. We can no doubt analyse
the content of consciousness during a short interval of
time into more or less elementary constituent feelings;
but it is not suggested that this psychological analysis
will reveal the elements out of whose measure-numbers
the atoms or electrons are built. The brain-matter is a
partial aspect of the whole mental state; but the analysis
of the brain-matter by physical investigation does not
run at all parallel with the analysis of the mental state
by psychological investigation. I assume that Russell
meant to warn us that, in speaking of part of a mental
state, he was not limiting himself to parts that would
be recognised as such psychologically, and he was ad-
mitting a more abstract kind of dissection.
This might give rise to some difficulty if we were
postulating complete identity of mind-stuff with con-
sciousness. But we know that in the mind there are
memories not in consciousness at the moment but
capable of being summoned into consciousness. We
are vaguely aware that things we cannot recall are lying
somewhere about and may come into the mind at any
280 REALITY
moment. Consciousness is not sharply defined, but
fades into subconsciousness; and beyond that we must
postulate something indefinite but yet continuous with
our mental nature. This I take to be the world-stuff.
We liken it to our conscious feelings because, now that
we are convinced of the formal and symbolic character of
the entities of physics, there is nothing else to liken it to.
It is sometimes urged that the basal stuff of the world
should be called "neutral stuff" rather than "mind-
stuff", since it is to be such that both mind and matter
originate from it. If this is intended to emphasise that
only limited islands of it constitute actual minds, and
that even in these islands that which is known mentally
is not equivalent to a complete inventory of all that may
be there, I agree. In fact I should suppose that the
self-knowledge of consciousness is mainly or wholly a
knowledge which eludes the inventory method of de-
scription. The term "mind-stuff" might well be amended;
but neutral stuff seems to be the wrong kind of amend-
ment. It implies that we have two avenues of approach
to an understanding of its nature. We have only one
approach, namely, through our direct knowledge of
mind. The supposed approach through the physical
world leads only into the cycle of physics, where we run
round and round like a kitten chasing its tail and never
reach the world-stuff at all.
I assume that we have left the illusion of substance
so far behind that the word "stuff" will not cause any
misapprehension. I certainly do not intend to materialise
or substantialise mind. Mind is — but you know what
mind is like, so why should I say more about its nature?
The word "stuff" has reference to the function it has
to perform as a basis of world-building and does not
imply any modified view of its nature.
MIND-STUFF 281
It is difficult for the matter-of-fact physicist to accept
the view that the substratum of everything is of mental
character. But no one can deny that mind is the first
and most direct thing in our experience, and all else is
remote inference — inference either intuitive or deli-
berate. Probably it would never have occurred to us
(as a serious hypothesis) that the world could be based
on anything else, had we not been under the impression
that there was a rival stuff with a more comfortable kind
of "concrete" reality — something too inert and stupid
to be capable of forging an illusion. The rival turns
out to be a schedule of pointer readings; and though a
world of symbolic character can well be constructed from
it, this is a mere shelving of the inquiry into the nature
of the world of experience.
This view of the relation of the material to the
spiritual world perhaps relieves to some extent a tension
between science and religion. Physical science has
seemed to occupy a domain of reality which is self-
sufficient, pursuing its course independently of and
indifferent to that which a voice within us asserts to be
a higher reality. We are jealous of such independence.
We are uneasy that there should be an apparently self-
contained world in which God becomes an unnecessary
hypothesis. We acknowledge that the ways of God are
inscrutable; but is there not still in the religious mind
something of that feeling of the prophets of old, who
called on God to assert his kingship and by sign or
miracle proclaim that the forces of Nature are subject
to his command? And yet if the scientist were to repent
and admit that it was necessary to include among
the agents controlling the stars and the electrons an omni-
present spirit to whom we trace the sacred things of con-
sciousness, would there not be even graver apprehension ?
282 REALITY
We should suspect an intention to reduce God to a system
of differential equations, like the other agents which at
various times have been introduced to restore order in the
physical scheme. That fiasco at any rate is avoided. For
the sphere of the differential equations of physics is the
metrical cyclic scheme extracted out of the broader
reality. However much the ramifications of the cycles
may be extended by further scientific discovery, they
cannot from their very nature trench on the background
in which they have their being — their actuality. It is
in this background that our own mental consciousness lies;
and here, if anywhere, we may find a Power greater than
but akin to consciousness. It is not possible for the con-
trolling laws of the spiritual substratum, which in so far
as it is known to us in consciousness is essentially non-
metrical, to be analogous to the differential and other
mathematical equations of physics which are meaningless
unless they are fed with metrical quantities. So that the
crudest anthropomorphic image of a spiritual deity can
scarcely be so wide of the truth as one conceived in terms
of metrical equations.
The Definition of Reality. It is time we came to grips
with the loose terms Reality and Existence, which we
have been using without any inquiry into what they are
meant to convey. I am afraid of this word Reality, not
connoting an ordinarily definable characteristic of the
things it is applied to but used as though it were some
kind of celestial halo. I very much doubt if any one of
us has the faintest idea of what is meant by the reality
or existence of anything but our own Egos. That is a
bold statement, which I must guard against misinter-
pretation. It is, of course, possible to obtain consistent
use of the word "reality" by adopting a conventional
THE DEFINITION OF REALITY 283
definition. My own practice would probably be covered
by the definition that a thing may be said to be real if
it is the goal of a type of inquiry to which I personally
attach importance. But if I insist on no more than this
I am whittling down the significance that is generally
assumed. In physics we can give a cold scientific
definition of reality which is free from all sentimental
mystification. But this is not quite fair play, because the
word "reality" is generally used with the intention of
evoking sentiment. It is a grand word for a peroration.
"The right honourable speaker went on to declare that
the concord and amity for which he had unceasingly
striven had now become a reality (loud cheers). " The
conception which it is so troublesome to apprehend is
not "reality" but "reality (loud cheers)".
Let us first examine the definition according to the
purely scientific usage of the word, although it will not
take us far enough. The only subject presented to me
for study is the content of my consciousness. You are
able to communicate to me part of the content of your
consciousness which thereby becomes accessible in my
own. For reasons which are generally admitted, though
I should not like to have to prove that they are conclusive,
I grant your consciousness equal status with my own;
and I use this second-hand part of my consciousness to
"put myself in your place". Accordingly my subject of
study becomes differentiated into the contents of many
consciousnesses, each content constituting a view-point.
There then arises the problem of combining the view-
points, and it is through this that the external world of
physics arises. Much that is in any one consciousness
is individual, much is apparently alterable by volition;
but there is a stable element which is common to other'
consciousnesses. That common element we desire to
284 REALITY
study, to describe as fully and accurately as possible,
and to discover the laws by which it combines now with
one view-point, now with another. This common ele-
ment cannot be placed in one man's consciousness
rather than in another's; it must be in neutral ground —
an external world.
It is true that I have a strong impression of an external
world apart from any communication with other con-
scious beings. But apart from such communication
I should have no reason to trust the impression. Most
of our common impressions of substance, world-wide
instants, and so on, have turned out to be illusory, and
the externality of the world might be equally untrust-
worthy. The impression of externality is equally strong
in the world that comes to me in dreams; the dream-
world is less rational, but that might be used as an argu-
ment in favour of its externality as showing its dissocia-
tion from the internal faculty of reason. So long as we
have to deal with one consciousness alone, the hypothesis
that there is an external world responsible for part of
what appears in it is an idle one. All that can be asserted
of this external world is a mere duplication of the know-
ledge that can be much more confidently asserted of the
world appearing in the consciousness. The hypothesis
only becomes useful when it is the means of bringing
together the worlds of many consciousnesses occupying
different view-points.
The external world of physics is thus a symposium
of the worlds presented to different view-points. There
is general agreement as to the principles on which the
symposium should be formed. Statements made about
this external world, if they are unambiguous, must be
either true or false. This has often been denied by
philosophers. It is quite commonly said that scientific
THE DEFINITION OF REALITY 285
theories about the world are neither true nor false but
merely convenient or inconvenient. A favourite phrase
is that the gauge of value of a scientific theory is that it
economises thought. Certainly a simple statement is
preferable to a circumlocutory one; and as regards any
current scientific theory, it is much easier to show that
it is convenient or that it economises thought than that
it is true. But whatever lower standards we may apply
in practice we need not give up our ideals; and so long
as there is a distinction between true and false theories
our aim must be to eliminate the false. For my part
I hold that the continual advance of science is not a
mere utilitarian progress; it is progress towards ever
purer truth. only let it be understood that the truth
we seek in science is the truth about an external world
propounded as the theme of study, and is not bound up
with any opinion as to the status of that world — whether
or not it wears the halo of reality, whether or not it is
deserving of "loud cheers".
Assuming that the symposium has been correctly
carried out, the external world and all that appears in it
are called real without further ado. When we (scientists)
assert of anything in the external world that it is real
and that it exists, we are expressing our belief that the
rules of the symposium have been correctly applied —
that it is not a false concept introduced by an error in
the process of synthesis, or a iiallucination belonging to
only one individual consciousness, or an incomplete
representation which embraces certain view-points but
conflicts with others. We refuse to contemplate the
awful contingency that the external world, after all our
care in arriving at it, might be disqualified by failing
to exist; because we have no idea what the supposed
qualification would consist in, nor in what way the
286 REALITY
prestige of the world would be enhanced if it passed
the implied test. The external world is the world that
confronts that experience which we have in common,
and for us no other world could fill the same role, no
matter how high honours it might take in the qualifying
examination.
This domestic definition of existence for scientific
purposes follows the principle now adopted for all other
definitions in science, namely, that a thing must be
defined according to the way in which it is in practice
recognised and not according to some ulterior signi-
ficance that we imagine it to possess. Just as matter
must shed its conception of substantiality, so existence
must shed its halo, before we can admit it into physical
science. But clearly if we are to assert or to question
the existence of anything not comprised in the external
world of physics, we must look beyond the physical
definition. The mere questioning of the reality of the
physical world implies some higher censorship than the
scientific method itself can supply.
The external world of physics has been formulated
as an answer to a particular problem encountered in
human experience. Officially the scientist regards it as
a problem which he just happened across, as he might
take up a cross-word problem encountered in a news-
paper. His sole business is to see that the problem is
correctly solved. But questions may be raised about a
problem which play no part and need not be considered
in connection with the solving of the problem. The
extraneous question naturally raised about the problem
of the external world is whether there is some higher
justification for embarking on this world-solving com-
petition rather than on other problems which our
experience might suggest to us. Just what kind of
THE DEFINITION OF REALITY 287
justification the scientist would claim for his quest is not
very clear, because it is not within the province of science
to formulate such a claim. But certainly he makes
claims which do not rest on the aesthetic perfection of
the solution or on material benefits derived from scien-
tific research. He would not allow his subject to be
shoved aside in a symposium on truth. We can scarcely
say anything more definite than that science claims a
"halo" for its world.
If we are to find for the atoms and electrons of the
external world not merely a conventional reality but
"reality (loud cheers)" we must look not to the end but
to the beginning of the quest. It is at the beginning that
we must find that sanction which raises these entities
above the mere products of an arbitrary mental exercise.
This involves some kind of assessment of the impulse
which sets us forth on the voyage of discovery. How
can we make such assessment? Not by any reasoning
that I know of. Reasoning would only tell us that the
impulse might be judged by the success of the adventure
— whether it leads in the end to things which really
exist and wear the halo in their own right; it takes us
to and fro like a shuttle along the chain of inference in
vain search for the elusive halo. But, legitimately or not,
the mind is confident that it can distinguish certain
quests as sanctioned by indisputable authority. We
may put it in different ways ;- the impulse to this quest
is part of our very nature; it is the ｅxpression of a
purpose which has possession of us. Is this precisely
what we meant when we sought to affirm the reality of
the external world? It goes some way towards giving
it a meaning but is scarcely the full equivalent. I doubt
if we really satisfy the conceptions behind that demand
unless we make the bolder hypothesis that the quest
288 REALITY
and all that is reached by it are of worth in the eyes of
an Absolute Valuer.
Whatever justification at the source we accept to
vindicate the reality of the external world, it can scarcely
fail to admit on the same footing much that is outside
physical science. Although no long chains of regularised
inference depend from them we recognise that other
fibres of our being extend in directions away from
sense-impressions. I am not greatly concerned to borrow
words like "existence" and "reality" to crown these
other departments of the soul's interest. I would rather
put it that any raising of the question of reality in its
transcendental sense (whether the question emanates
from the world of physics or not) leads us to a perspective
from which we see man not as a bundle of sensory
impressions, but conscious of purpose and responsi-
bilities to which the external world is subordinate.
From this perspective we recognise a spiritual world
alongside the physical world. Experience — that is to
say, the self cum environment — comprises more than
can be embraced in the physical world, restricted as it
is to a complex of metrical symbols. The physical world
is, we have seen, the answer to one definite and urgent
problem arising in a survey of experience; and no other
problem has been followed up with anything like the
same precision and elaboration. Progress towards an
understanding of the non-sensory constituents of our
nature is not likely to follow similar lines, and indeed
is not animated by the same aims. If it is felt that this
difference is so wide that the phrase spiritual world is a
misleading analogy, I will not insist on the term. All
I would claim is that those who in the search for truth
start from consciousness as a seat of self-knowledge with
interests and responsibilities not confined to the material
PHYSICAL ILLUSTRATIONS 289
plane, are just as much facing the hard facts of experi-
ence as those who start from consciousness as a device
for reading the indications of spectroscopes and micro-
meters.
Physical Illustrations. If the reader is unconvinced that
there can be anything indefinite in the question whether
a thing exists or not, let him glance at the following
problem. Consider a distribution of matter in Einstein's
spherical "finite but unbounded" space. Suppose that
the matter is so arranged that every particle has an
exactly similar particle at its antipodes. (There is some
reason to believe that the matter would necessarily have
this arrangement in consequence of the law of gravita-
tion; but this is not certain.) Each group of particles
will therefore be exactly like the antipodal group not
only in its structure and configuration but in its entire
surroundings; the two groups will in fact be indis-
tinguishable by any possible experimental test. Starting
on a journey round the spherical world we come across
a group A, and then after going half round we come to
an exactly similar group A' indistinguishable by any
test; another half circle again brings us to an exactly
similar group, which, however, we decide is the original
group A. Now let us ponder a little. We realise that
in any case by going on far enough we come back to the
same group. Why do we not accept the obvious con-
clusion that this happened when we reached A'; every-
thing was exactly as though we had reached the starting-
point again? We have encountered a succession of
precisely similar phenomena but for some arbitrary
reason have decided that only the alternate ones are
really the same. There is no difficulty in identifying all
of them; in that case the space is "elliptical" instead of
29Q REALITY
"spherical". But which is the real truth? Disregard
the fact that I introduced A and A' to you as though
they were not the same particles, because that begs the
question; imagine that you have actually had this
adventure in a world you had not been told about. You
cannot find out the answer. Can you conceive what the
question means ? I cannot. All that turns on the answer
is whether we shall provide two separate haloes for A
and A' or whether one will suffice.
Descriptions of the phenomena of atomic physics
have an extraordinary vividness. We see the atoms with
their girdles of circulating electrons darting hither and
thither, colliding and rebounding. Free electrons torn
from the girdles hurry away a hundred times faster,
curving sharply round the atoms with side slips and
hairbreadth escapes. The truants are caught and
attached to the girdles and the escaping energy shakes
the aether into vibration. X-rays impinge on the atoms
and toss the electrons into higher orbits. We see these
electrons falling back again, sometimes by steps, some-
times with a rush, caught in a cul-de-sac of metasta-
bility, hesitating before "forbidden passages". Behind
it all the quantum h regulates each change with mathe-
matical precision. This is the sort of picture that appeals
to our understanding — no insubstantial pageant to fade
like a dream.
The spectacle is so fascinating that we have perhaps
forgotten that there was a time when we wanted to be
told what an electron is. The question was never
answered. No familiar conceptions can be woven round
the electron; it belongs to the waiting list. Similarly
the description of the processes must be taken with a
grain of salt. The tossing up of the electron is a con-
ventional way of depicting a particular change of state
PHYSICAL ILLUSTRATIONS 291
of the atom which cannot really be associated with
movements in space as macroscopically conceived.
Something unknown is doing we don't know what — that is
what our theory amounts to. It does not sound a par-
ticularly illuminating theory. I have read something
like it elsewhere —
The slithy toves
Did gyre and gimble in the wabe.
There is the same suggestion of activity. There is the
same indefiniteness as to the nature of the activity and
of what it is that is acting. And yet from so unpromising
a beginning we really do get somewhere. We bring
into order a host of apparently unrelated phenomena;
we make predictions, and our predictions come off.
The reason — the sole reason — for this progress is that
our description is not limited to unknown agents
executing unknown activities, but numbers are scattered
freely in the description. To contemplate electrons
circulating in the atom carries us no further; but by
contemplating eight circulating electrons in one atom
and seven circulating electrons in another we begin to
realise the difference between oxygen and nitrogen.
Eight slithy toves gyre and gimble in the oxygen wabe;
seven in nitrogen. By admitting a few numbers even
"Jabberwocky" may become scientific. We can now
venture on a prediction; if one of its toves escapes,
oxygen will be masquerading in a garb properly be-
longing to nitrogen. In the stars and nebulae we do
find such wolves in sheep's clothing which might
otherwise have startled us. It would not be a bad
reminder of the essential unknownness of the funda-
mental entities of physics to translate it into "Jabber-
wocky"; provided all numbers — all metrical attributes
292 REALITY
— are unchanged, it does not suffer in the least. Out
of the numbers proceeds that harmony of natural law
which it is the aim of science to disclose. We can grasp
the tune but not the player. Trinculo might have been
referring to modern physics in the words, "This is the
tune of our catch, played by the picture of Nobody".
Chapter XIV
CAUSATION
In the old conflict between freewill and predestination
it has seemed hitherto that physics comes down heavily
on the side of predestination. Without making ex-
travagant claims for the scope of natural law, its moral
sympathy has been with the view that whatever the
future may bring forth is already foretold in the con-
figurations of the past —
Yea, the first Morning of Creation wrote
What the Last Dawn of Reckoning shall read.
I am not so rash as to invade Scotland with a solution
of a problem which has rent her from the synod to the
cottage. Like most other people, I suppose, I think it
incredible that the wider scheme of Nature which
includes life and consciousness can be completely
predetermined; yet I have not been able to form a
satisfactory conception of any kind of law or causal
sequence which shall be other than deterministic. It
seems contrary to our feeling of the dignity of the mind
to suppose that it merely registers a dictated sequence
of thoughts and emotions; but it seems equally con-
trary to its dignity to put it at the mercy of impulses
with no causal antecedents. I shall not deal with this
dilemma. Here I have to set forth the position of
physical science on this matter so far as it comes into
her territory. It does come into her territory, because
that which we call human will cannot be entirely
dissociated from the consequent motions of the muscles
and disturbance of the material world. on the scientific
293
294 CAUSATION
side a new situation has arisen. It is a consequence of
the advent of the quantum theory that physics is no
longer pledged to a scheme of deterministic law. Deter-
minism has dropped out altogether in the latest for-
mulations of theoretical physics and it is at least open
to doubt whether it will ever be brought back.
The foregoing paragraph is from the manuscript of
the original lecture delivered in Edinburgh. The attitude
of physics at that time was one of indifference to deter-
minism. If there existed a scheme of strictly causal law
at the base of phenomena the search for it was not at
present practical politics, and meanwhile another ideal
was being pursued. The fact that a causal basis had
been lost sight of in the new theories was fairly well
known; many regretted it, and held that its restoration
was imperative.*
In rewriting this chapter a year later I have had to
mingle with this attitude of indifference an attitude
more definitely hostile to determinism which has arisen
from the acceptance of the Principle of Indeterminacy
(p. 220). There has been no time for more than a hur-
ried examination of the far-reaching consequences of this
principle; and I should have been reluctant to include
"stop-press" ideas were it not that they appear to clinch
the conception towards which the earlier developments
were leading. The future is a combination of the causal
influences of the past together with unpredictable ele-
ments — unpredictable not merely because it is im-
* A few days after the course of lectures was completed, Einstein
wrote in his message on the Newton Centenary, "It is only in the quan-
tum theory that Newton's differential method becomes inadequate, and
indeed strict causality fails us. But the last word has not yet been said.
May the spirit of Newton's method give us the power to restore unison
between physical reality and the profoundest characteristic of Newton's
teaching — strict causality." (Nature, 1927, March 26, p. 467.)
CAUSATION AND TIME'S ARROW 295
practicable to obtain the data of prediction, but because
no data connected causally with our experience exist.
It will be necessary to defend so remarkable a change of
opinion at some length. Meanwhile we may note that
science thereby withdraws its moral opposition to free-
will. Those who maintain a deterministic theory of
mental activity must do so as the outcome of their study
of the mind itself and not with the idea that they are
thereby making it more conformable with our experi-
mental knowledge of the laws of inorganic nature.
Causation and Time's Arrow. Cause and effect are closely
bound up with time's arrow; the cause must precede
the effect. The relativity of time has not obliterated this
order. An event Here-Now can only cause events in the
cone of absolute future; it can be caused by events in
the cone of absolute past; it can neither cause nor be
caused by events in the neutral wedge, since the neces-
sary influence would in that case have to be transmitted
with a speed faster than light. But curiously enough this
elementary notion of cause and effect is quite incon-
sistent with a strictly causal scheme. How can I cause
an event in the absolute future, if the future was pre-
determined before I was born? The notion evidently
implies that something may be born into the world at
the instant Here-Now, which has an influence extending
throughout the future cone but no corresponding
linkage to the cone of absolute past. The primary laws
of physics do not provide for any such one-way linkage;
any alteration in a prescribed state of the world implies
alterations in its past state symmetrical with the altera-
tions in its future state. Thus in primary physics, which
knows nothing of time's arrow, there is no discrimina-
tion of cause and effect; but events are connected by a
296 CAUSATION
symmetrical causal relation which is the same viewed
from either end.
Primary physics postulates a strictly causal scheme,
but the causality is a symmetrical relation and not the
one-way relation of cause and effect. Secondary physics
can distinguish cause and effect but its foundation does
not rest on a causal scheme and it is indifferent as to
whether or not strict causality prevails.
The lever in a signal box is moved and the signal
drops. We can point out the relation of constraint
which associates the positions of lever and signal; we
can also find that the movements are not synchronous,
and calculate the time-difference. But the laws of
mechanics do not ascribe an absolute sign to this time-
difference; so far as they are concerned we may quite
well suppose that the drop of the signal causes the motion
of the lever. To settle which is the cause, we have two
options. We can appeal to the signalman who is con-
fident that he made the mental decision to pull the lever;
but this criterion will only be valid if we agree that there
was a genuine decision between two possible courses
and not a mere mental registration of what was already
predetermined. Or we can appeal to secondary law
which takes note of the fact that there was more of the
random element in the world when the signal dropped
than when the lever moved. But the feature of secon-
dary law is that it ignores strict causation; it concerns
itself not with what must happen but with what is
likely to happen. Thus distinction of cause and effect
has no meaning in the closed system of primary laws
of physics; to get at it we have to break into the scheme,
introducing considerations of volition or of probability
which are foreign to it. This is rather analogous to the
ten vanishing coefficients of curvature which could only
CAUSATION AND TIME'S ARROW 4&)
be recognised if the closed system of the world were
broken into by standards foreign to it.
For convenience I shall call the relation of effect to
cause causation, and the symmetrical relation which does
not distinguish between cause and effect causality. In
primary physics causality has completely replaced
causation. Ideally the whole world past and future is
connected into a deterministic scheme by relations of
causality. Up till very recently it was universally held
that such a determinate scheme must exist (possibly
subject to suspension by supernatural agencies outside
the scope of physics) ; we may therefore call this the
"orthodox" view. It was, of course, recognised that we
were only acquainted with part of the structure of this
causal scheme, but it was the settled aim of theoretical
physics to discover the whole.
This replacement in orthodox science of causation by
causality is important in one respect. We must not let
causality borrow an intuitive sanction which really
belongs only to causation. We may think we have an
intuition that the same cause cannot have two alternative
effects; but we do not claim any intuition that the same
effect may not spring from two alternative causes. For
this reason the assumption of a rigid determinateness
enforced by relations of causality cannot be said to be
insisted on by intuition.
What is the ground for so much ardent faith in the
orthodox hypothesis that physical phenomena rest ulti-
mately on a scheme of completely deterministic laws?
I think there are two reasons —
(i) The principal laws of Nature which have been
discovered are apparently of this deterministic type,
and these have furnished the great triumphs of physical
prediction. It is natural to trust to a line of progress
298 CAUSATION
which has served us well in the past. Indeed it is a
healthy attitude to assume that nothing is beyond the
scope of scientific prediction until the limits of prediction
actually declare themselves.
(2) The current epistemology of science presupposes
a deterministic scheme of this type. To modify it in-
volves a much deeper change in our attitude to natural
knowledge than the mere abandonment of an untenable
hypothesis.
In explanation of the second point we must recall
that knowledge of the physical world has to be inferred
from the nerve-messages which reach our brains, and
the current epistemology assumes that there exists a
determinate scheme of inference (lying before us
as an ideal and gradually being unravelled). But, as has
already been pointed out, the chains of inference are
simply the converse of the chains of physical causality
by which distant events are connected to the nerve-
messages. If the scheme of transmission of these mes-
sages through the external world is not deterministic
then the scheme of inference as to their source cannot
be deterministic, and our epistemology has been based
on an impossible ideal. In that case our attitude to the
whole scheme of natural knowledge must be profoundly
modified.
These reasons will be considered at length, but it is
convenient to state here our answers to them in equally
summary form.
(1) In recent times some of the greatest triumphs of
physical prediction have been furnished by admittedly
statistical laws which do not rest on a basis of causality.
Moreover the great laws hitherto accepted as causal
appear on minuter examination to be of statistical
character.
PREDICTABILITY OF EVENTS 299
(2) Whether or not there is a causal scheme at the
base of atomic phenomena, modern atomic theory is not
now attempting to find it; and it is making rapid prog-
ress because it no longer sets this up as a practical aim.
We are in the position of holding an epistemological
theory of natural knowledge which does not correspond
to actual aim of current scientific investigation.
Predictability of Events. Let us examine a typical case
of successful scientific prediction. A total eclipse of the
sun visible in Cornwall is prophesied for 1 1 August
1999. It is generally supposed that this eclipse is
already predetermined by the present configuration of
the sun, earth and moon. I do not wish to arouse
unnecessary misgiving as to whether the eclipse will
come off. I expect it will; but let us examine the grounds
of expectation. It is predicted as a consequence of the
law of gravitation — a law which we found in chapter vil
to be a mere truism. That does not diminish the value
of the prediction; but it does suggest that we may not be
able to pose as such marvellous prophets when we come
up against laws which are not mere truisms. I might
venture to predict that 2 + 2 will be equal to 4 even in
1999; but if this should prove correct it will not help
to convince anyone that the universe (or, if you like, the
human mind) is governed by laws of deterministic type.
I suppose that in the most erratically governed world
something can be predicted if truisms are not ex-
cluded.
But we have to look deeper than this. The law of
gravitation is only a truism when regarded from a
macroscopic point of view. It presupposes space, and
measurement with gross material or optical arrange-
ments. It cannot be refined to an accuracy beyond the
300 CAUSATION
limits of these gross appliances; so that it is a truism
with a probable error — small, but not infinitely small.
The classical laws hold good in the limit when exceed-
ingly large quantum numbers are involved. The system
comprising the sun, earth and moon has exceedingly
high state-number (p. 198); and the predictability of
its configurations is not characteristic of natural pheno-
mena in general but of those involving great numbers
of atoms of action — such that we are concerned not
with individual but with average behaviour.
Human life is proverbially uncertain; few things are
more certain than the solvency of a life-insurance com-
pany. The average law is so trustworthy that it may be
considered predestined that half the children now born
will survive the age of x years. But that does not tell us
whether the span of life of young A. McB. is already
written in the book of fate, or whether there is still time
to alter it by teaching him not to run in front of motor-
buses. The eclipse in 1999 is as safe as the balance of
a life-insurance company; the next quantum jump of an
atom is as uncertain as your life and mine.
We are thus in a position to answer the main argu-
ment for a predetermination of the future, viz. that
observation shows the laws of Nature to be of a type
which leads to definite predictions of the future, and it
is reasonable to expect that any laws which remain
undiscovered will conform to the same type. For when
we ask what is the characteristic of the phenomena that
have been successfully predicted, the answer is that they
are effects depending on the average configurations of vast
numbers of individual entities. But averages are pre-
dictable because they are averages, irrespective of the
type of government of the phenomena underlying
them.
PREDICTABILITY OF EVENTS 301
Considering an atom alone in the world in State 3,
the classical theory would have asked, and hoped to
answer, the question, What will it do next? The quan-
tum theory substitutes the question, Which will it do
next? Because it admits only two lower states for the
atom to go to. Further, it makes no attempt to find a
definite answer, but contents itself with calculating the
respective odds on the jumps to State 1 and State 2.
The quantum physicist does not fill the atom with
gadgets for directing its future behaviour, as the classical
physicist would have done; he fills it with gadgets de-
termining the odds on its future behaviour. He studies
the art of the bookmaker not of the trainer.
Thus in the structure of the world as formulated in
the new quantum theory it is predetermined that of
500 atoms now in State 3, approximately 400 will go
on to State 1 and 100 to State 2 — in so far as anything
subject to chance fluctuations can be said to be pre-
determined. The odds of 4 to 1 find their appropriate
representation in the picture of the atom; that is to say,
something symbolic of a 4 : 1 ratio is present in each of
the 500 atoms. But there are no marks distinguishing
the atoms belonging to the group of 100 from the 400.
Probably most physicists would take the view that
although the marks are not yet shown in the picture,
they are nevertheless present in Nature; they belong to
an elaboration of the theory which will come in good
time. The marks, of course, need not be in the atom
itself; they may be in the environment which will
interact with it. For example, we may load dice in such
a way that the odds are 4 to 1 on throwing a 6. Both
those dice which turn up 6 and those which do not
have these odds written in their constitution — by a
displaced position of the centre of gravity. The result
302 CAUSATION
of a particular throw is not marked in the dice; never-
theless it is strictly causal (apart perhaps from the
human element involved in throwing the dice) being de-
termined by the external influences which are concerned.
Our own position at this stage is that future develop-
ments of physics may reveal such causal marks (either
in the atom or in the influences outside it) or it may not.
Hitherto whenever we have thought we have detected
causal marks in natural phenomena they have always
proved spurious, the apparent determinism having come
about in another way. Therefore we are inclined to
regard favourably the possibility that there may be no
causal marks anywhere.
But, it will be said, it is inconceivable that an atom
can be so evenly balanced between two alternative
courses that nowhere in the world as yet is there any
trace of the ultimately deciding factor. This is an ap-
peal to intuition and it may fairly be countered with
another appeal to intuition. I have an intuition much
more immediate than any relating to the objects of the
physical world; this tells me that nowhere in the world
as yet is there any trace of a deciding factor as to
whether I am going to lift my right hand or my left.
It depends on an unfettered act of volition not yet made
or foreshadowed.* My intuition is that the future is
able to bring forth deciding factors which are not
secretly hidden in the past.
The position is that the laws governing the micro-
scopic elements of the physical world — individual
atoms, electrons, quanta — do not make definite pre-
dictions as to what the individual will do next. I am
* It is fair to assume the trustworthiness of this intuition in answering
an argument which appeals to intuition; the assumption would beg the
question if we were urging the argument independently.
THE NEW EPISTEMOLOGICAL OUTLOOK 303
here speaking of the laws that have been actually dis-
covered and formulated on the old quantum theory and
the new. These laws indicate several possibilities in the
future and state the odds on each. In general the odds
are moderately balanced and are not tempting to an
aspiring prophet. But short odds on the behaviour of
individuals combine into very long odds on suitably
selected statistics of a number of individuals; and the
wary prophet can find predictions of this kind on which
to stake his credit — without serious risk. All the success-
ful predictions hitherto attributed to causality are trace-
able to this. It is quite true that the quantum laws for
individuals are not incompatible with causality; they
merely ignore it. But if we take advantage of this
indifference to reintroduce determinism at the basis of
world structure it is because our philosophy predisposes
us that way, not because we know of any experimental
evidence in its favour.
We might for illustration make a comparison with
the doctrine of predestination. That theological doc-
trine, whatever may be said against it, has hitherto
seemed to blend harmoniously with the predetermination
of the material universe. But if we were to appeal to
the new conception of physical law to settle this question
by analogy the answer would be : — The individual is not
predestined to arrive at either of the two states, which
perhaps may here be sufficiently discriminated as
State 1 and State 2; the most that can be considered
already settled is the respective odds on his reaching
these states.
The New Epistemological Outlook. Scientific investiga-
tion does not lead to knowledge of the intrinsic nature
of things. "Whenever we state the properties of a body
304 CAUSATION
in terms of physical quantities we are imparting know-
ledge of the response of various metrical indicators to
its presence and nothing more" (p. 257). But if a body-
is not acting according to strict causality, if there is an
element of uncertainty as to the response of the indica-
tors, we seem to have cut away the ground for this kind of
knowledge. It is not predetermined what will be the
reading of the weighing-machine if the body is placed
on it, therefore the body has no definite mass; nor where
it will be found an instant hence, therefore it has no
definite velocity; nor where the rays now being reflected
from it will converge in the microscope, therefore it has
no definite position; and so on. It is no use answering
that the body really has a definite mass, velocity,
position, etc., which we are unaware of; that statement,
if it means anything, refers to an intrinsic nature of
things outside the scope of scientific knowledge. We
cannot infer these properties with precision from any-
thing that we can be aware of, because the breach of
causality has broken the chain of inference. Thus our
knowledge of the response of indicators to the presence
of the body is non-existent; therefore we cannot assert
knowledge of it at all. So what is the use of talking
about it? The body which was to be the abstraction of
all these (as yet unsettled) pointer readings has become
superfluous in the physical world. That is the dilemma
into which the old epistemology leads us as soon as we
begin to doubt strict causality.
In phenomena on a gross scale this difficulty can be
got round. A body may have no definite position but
yet have within close limits an extremely probable
position. When the probabilities are large the substitu-
tion of probability for certainty makes little difference;
it adds only a negligible haziness to the world. But
THE NEW EPISTEMOLOGICAL OUTLOOK 305
though the practical change is unimportant there are
fundamental theoretical consequences. All probabilities
rest on a basis of a priori probability, and we cannot say
whether probabilities are large or small without having
assumed such a basis. In agreeing to accept those of our
calculated probabilities which are very high as virtually
equivalent to certainties on the old scheme, we are as it
were making our adopted basis of a priori probability
a constituent of the world-structure — adding to the
world a kind of symbolic texture that cannot be ex-
pressed on the old scheme.
on the atomic scale of phenomena the probabilities
are in general well-balanced, and there are no "naps"
for the scientific punter to put his shirt on. If a body is
still defined as a bundle of pointer readings (or highly
probable pointer readings) there are no "bodies" on
the atomic scale. All that we can extract is a bundle of
probabilities. That is in fact just how Schrodinger tries
to picture the atom — as a wave centre of his probability
entity i|>.
We commonly have had to deal with probabilities
which arise through ignorance. With fuller knowledge
we should sweep away the references to probability and
substitute the exact facts. But it appears to be a funda-
mental point in Schrodinger's theory that his probabili-
ties are not to be replaced in that way. When his ip is
sufficiently concentrated it indicates the point where the
electron is; when it is diffused it gives only a vague
indication of the position. But this vague indication is
not something which ideally ought to be replaced by
exact knowledge; it is ip itself which acts as the source
of the light emitted from the atom, the period of the
light being that of the beats of i|>. I think this means
that the spread of ty is not a symbol for uncertainty aris-
306 CAUSATION
ing through lack of information; it is a symbol for
causal failure — an indeterminacy of behaviour which is
part of the character of the atom.
We have two chief ways of learning about the interior
of the atom. We can observe electrons entering or
leaving, and we can observe light entering or leaving.
Bohr has assumed a structure connected by strictly
causal law with the first phenomenon, Heisenberg and
his followers with the second. If the two structures were
identifiable then the atom w r ould involve a complete
causal connection of the two types of phenomena. But
apparently no such causal linkage exists. Therefore we
have to be content with a correlation in which the
entities of the one model represent probabilities in the
second model. There are perhaps details in the two
theories which do not quite square with this; but it
seems to express the ideal to be aimed at in describing
the laws of an incompletely causal world, viz. that the
causal source of one phenomenon shall represent the
probability of causal source of another phenomenon.
Schrodinger's theory has given at least a strong hint
that the actual world is controlled on this plan.
The Principle of Indeterminacy. Thus far we have
shown that modern physics is drifting away from the
postulate that the future is predetermined, ignoring it
rather than deliberately rejecting it. With the discovery
of the Principle of Indeterminacy (p. 220) its attitude
has become more definitely hostile.
Let us take the simplest case in which we think we
can predict the future. Suppose that we have a particle
with known position and velocity at the present instant.
Assuming that nothing interferes with it we can predict
the position at a subsequent instant. (Strictly the non-
THE PRINCIPLE OF INDETERMINACY 307
interference would be a subject for another prediction,
but to simplify matters we shall concede it.) It is just
this simple prediction which the principle of indeter-
minacy expressly forbids. It states that we cannot know
accurately both the velocity and position of a particle
at the present instant.
At first sight there seems to be an inconsistency.
There is no limit to the accuracy with which we may
know the position, provided that we do not want to
know the velocity also. Very well; let us make a highly
accurate determination of position now, and after
waiting a moment make another highly accurate deter-
mination of position. Comparing the two accurate
positions we compute the accurate velocity — and snap
our fingers at the principle of indeterminacy. This
velocity, however, is of no use for prediction, because in
making the second accurate determination of position
we have rough-handled the particle so much that it no
longer has the velocity we calculated. // is a purely
retrospective velocity. The velocity does not exist in the
present tense but in the future perfect; it never exists,
it never will exist, but a time may come when it will have
existed. There is no room for it in Fig. 4 which contains
an Absolute Future and an Absolute Past but not an
Absolute Future Perfect.
The velocity which we attribute to a particle now
can be regarded as an anticipation of its future positions.
To say that it is unknowable (except with a certain
degree of inaccuracy) is to say that the future cannot be
anticipated. Immediately the future is accomplished,
so that it is no longer an anticipation, the velocity be-
comes knowable.
The classical view that a particle necessarily has a
definite (but not necessarily knowable) velocity now,
308 CAUSATION
amounts to disguising a piece of the unknown future as
an unknowable element of the present. Classical physics
foists a deterministic scheme on us by a trick; it smuggles
the unknown future into the present, trusting that we
shall not press an inquiry as to whether it has become
any more knowable that way.
The same principle extends to every kind of pheno-
menon that we attempt to predict, so long as the need
for accuracy is not buried under a mass of averages. To
every co-ordinate there corresponds a momentum, and
by the principle of indeterminacy the more accurately
the co-ordinate is known the less accurately the momen-
tum is known. Nature thus provides that knowledge
of one-half of the world will ensure ignorance of the
other half — ignorance which, we have seen, may be
remedied later when the same part of the world is con-
templated retrospectively. We can scarcely rest content
with a picture of the world which includes so much that
cannot be known. We have been trying to get rid of
unknowable things, i.e. all conceptions which have no
causal connection with our experience. We have elimi-
nated velocity through aether, "right" frames of space,
etc., for this reason. This vast new unknowable element
must likewise be swept out of the Present. Its proper
place is in the Future because then it will no
longer be unknowable. It has been put in prematurely
as an anticipation of that which cannot be antici-
pated.
In assessing whether the symbols which the physicist
has scattered through the external world are adequate to
predetermine the future, we must be on our guard
against retrospective symbols. It is easy to prophesy
after the event.
NATURAL AND SUPERNATURAL 309
Natural and Supernatural. A rather serious consequence
of dropping causality in the external world is that it
leaves us with no clear distinction between the Natural
and the Supernatural. In an earlier chapter I compared
the invisible agent invented to account for the tug of
gravitation to a "demon". Is a view of the world which
admits such an agent any more scientific than that of a
savage who attributes all that he finds mysterious in
Nature to the work of invisible demons? The New-
tonian physicist had a valid defence. He could point
out that his demon Gravitation was supposed to act
according to fixed causal laws and was therefore not to
be compared with the irresponsible demons of the
savage. once a deviation from strict causality is ad-
mitted the distinction melts away. I suppose that the
savage would admit that his demon was to some extent
a creature of habit and that it would be possible to make
a fair guess as to what he would do in the future; but
that sometimes he would show a will of his own. It is
that imperfect consistency which formerly disqualified
him from admission as an entity of physics along with
his brother Gravitation.
That is largely why there has been so much bother
about "me"; because I have, or am persuaded that I
have, "a will of my own". Either the physicist must
leave his causal scheme at the mercy of supernatural
interference from me, or he must explain away my
supernatural qualities. In self-defence the materialist
favoured the latter course; he decided that I was not
supernatural — only complicated. We on the other hand
have concluded that there is no strict causal behaviour
anywhere. We can scarcely deny the charge that in
abolishing the criterion of causality we are opening the
door to the savage's demons. It is a serious step, but
310 CAUSATION
I do not think it means the end of all true science. After
all if they try to enter we can pitch them out again, as
Einstein pitched out the respectable causal demon who
called himself Gravitation. It is a privation to be no
longer able to stigmatise certain views as unscientific
superstition; but we are still allowed, if the circumstances
justify it, to reject them as bad science.
Volition. From the philosophic point of view it is of deep
interest to consider how this affects the freedom of the
human mind and spirit. A complete determinism of
the material universe cannot be divorced from deter-
minism of the mind. Take, for example, the prediction
of the weather this time next year. The prediction is
not likely ever to become practicable, but "orthodox"
physicists are not yet convinced that it is theoretically
impossible; they hold that next year's weather is already
predetermined. We should require extremely detailed
knowledge of present conditions, since a small local
deviation can exert an ever-expanding influence.
We must examine the state of the sun so as to predict
the fluctuations in the heat and corpuscular radiation
which it sends us. We must dive into the bowels of the
earth to be forewarned of volcanic eruptions which may
spread a dust screen over the atmosphere as Mt. Katmai
did some years ago. But further we must penetrate into
the recesses of the human mind. A coal strike, a great
war, may directly change the conditions of the atmo-
sphere; a lighted match idly thrown away may cause
deforestation which will change the rainfall and climate.
There can be no fully deterministic control of inorganic
phenomena unless the determinism governs mind itself.
Conversely if we wish to emancipate mind we must to
some extent emancipate the material world also. There
VOLITION 311
appears to be no longer any obstacle to this emanci-
pation.
Let us look more closely into the problem of how the
mind gets a grip on material atoms so that movements
of the body and limbs can be controlled by its volition.
I think we may now feel quite satisfied that the volition
is genuine. The materialist view was that the motions
which appear to be caused by our volition are really
reflex actions controlled by the material processes in the
brain, the act of will being an inessential side pheno-
menon occurring simultaneously with the physical
phenomena. But this assumes that the result of apply-
ing physical laws to the brain is fully determinate. It is
meaningless to say that the behaviour of a conscious
brain is precisely the same as that of a mechanical brain
if the behaviour of a mechanical brain is left undeter-
mined. If the laws of physics are not strictly causal the
most that can be said is that the behaviour of the
conscious brain is one of the possible behaviours of a
mechanical brain. Precisely so; and the decision between
the possible behaviours is what we call volition.
Perhaps you will say, When the decision of an atom
is made between its possible quantum jumps, is that
also "volition"? Scarcely; the analogy is altogether too
remote. The position is that both for the brain and the
atom there, is nothing in the physical world, i.e. the
world of pointer readings, to predetermine the decision;
the decision is a fact of the physical world with con-
sequences in the future but not causally connected to
the past. In the case of the brain we have an insight
into a mental world behind the world of pointer readings
and in that world we get a new picture of the fact of
decision which must be taken as revealing its real
nature — if the words real nature have any meaning.
312 CAUSATION
For the atom we have no such insight into what is
behind the pointer readings. We believe that behind
all pointer readings there is a background continuous
with the background of the brain; but there is no more
ground for calling the background of the spontaneous
behaviour of the atom "volition" than for calling the
background of its causal behaviour "reason". It should
be understood that we are not attempting to reintroduce
in the background the strict causality banished from
the pointer readings. In the one case in which we have
any insight — the background of the brain — we have
no intention of giving up the freedom of the mind and
will. Similarly we do not suggest that the marks of
predestination of the atom, not found in the pointer
readings, exist undetectable in the unknown back-
ground. To the question whether I would admit that
the cause of the decision of the atom has something in
common with the cause of the decision of the brain,
I would simply answer that there is no cause. In the
case of the brain I have a deeper insight into the
decision; this insight exhibits it as volition, i.e. some-
thing outside causality.
A mental decision to turn right or turn left starts one
of two alternative sets of impulses along the nerves to
the feet. At some brain centre the course of behaviour
of certain atoms or elements of the physical world is
directly determined for them by the mental decision —
or, one may say, the scientific description of that be-
haviour is the metrical aspect of the decision. It would
be a possible though difficult hypothesis to assume that
very few atoms (or possibly only one atom) have this
direct contact with the conscious decision, and that
these few atoms serve as a switch to deflect the material
world from one course to the other. But it is physically
MIND AND STATISTICAL LAWS 313
improbable that each atom has its duty in the brain so
precisely allotted that the control of its behaviour would
prevail over all possible irregularities of the other atoms.
If I have at all rightly understood the processes of my
own mind, there is no finicking with individual atoms.
I do not think that our decisions are precisely
balanced on the conduct of certain key-atoms. Could
we pick out one atom in Einstein's brain and say that
if it had made the wrong quantum jump there would
have been a corresponding flaw in the theory of rela-
tivity? Having regard to the physical influences of
temperature and promiscuous collision it is impossible
to maintain this. It seems that we must attribute to the
mind power not only to decide the behaviour of atoms
individually but to affect systematically large groups —
in fact to tamper with the odds on atomic behaviour.
This has always been one of the most dubious points
in the theory of the interaction of mind and matter.
Interference with Statistical Laws. Has the mind power
to set aside statistical laws which hold in inorganic
matter? Unless this is granted its opportunity of inter-
ference seems to be too circumscribed to bring about
the results which are observed to follow from mental
decisions. But the admission involves a genuine
physical difference between inorganic and organic (or,
at any rate, conscious) matter. I would prefer to avoid
this hypothesis, but it is necessary to face the issue
squarely. The indeterminacy recognised in modern
quantum theory is only a partial step towards freeing
our actions from deterministic control. To use an
analogy — we have admitted an uncertainty which may
take or spare human lives; but we have yet to find an
uncertainty which may upset the expectations of a life-
3H CAUSATION
insurance company. Theoretically the one uncertainty
might lead to the other, as when the fate of millions
turned on the murders at Sarajevo. But the hypothesis
that the mind operates through two or three key-atoms
in the brain is too desperate a way of escape for us, and
I reject it for the reasons already stated.
It is one thing to allow the mind to direct an atom
between two courses neither of which would be im-
probable for an inorganic atom; it is another thing to
allow it to direct a crowd of atoms into a configuration
which the secondary laws of physics would set aside as
"too improbable". Here the improbability is that a
large number of entities each acting independently
should conspire to produce the result; it is like the
improbability of the atoms finding themselves by chance
all in one half of a vessel. We must suppose that in the
physical part of the brain immediately affected by a
mental decision there is some kind of interdependence
of behaviour of the atoms which is not present in
inorganic matter.
I do not wish to minimise the seriousness of admitting
this difference between living and dead matter. But
I think that the difficulty has been eased a little, if it
has not been removed. To leave the atom constituted as
it was but to interfere with the probability of its un-
determined behaviour, does not seem quite so drastic
an interference with natural law as other modes of
mental interference that have been suggested. (Perhaps
that is only because we do not understand enough about
these probabilities to realise the heinousness of our
suggestion.) Unless it belies its name, probability can
be modified in ways which ordinary physical entities
would not admit of. There can be no unique probability
attached to any event or behaviour; we can only speak
MIND AND STATISTICAL LAWS 315
of "probability in the light of certain given informa-
tion", and the probability alters according to the extent
of the information. It is, I think, one of the most un-
satisfactory features of the new quantum theory in its
present stage that it scarcely seems to recognise this
fact, and leaves us to guess at the basis of information
to which its probability theorems are supposed to refer.
Looking at it from another aspect — if the unity of
a man's consciousness is not an illusion, there must be
some corresponding unity in the relations of the mind-
stuff which is behind the pointer readings. Applying
our measures of relation structure, as in chapter XI,
we shall build matter and fields of force obeying
identically the principal field-laws; the atoms will
individually be in no way different from those which
are without this unity in the background. But it seems
plausible that when we consider their collective be-
haviour we shall have to take account of the broader
unifying trends in the mind-stuff, and not expect the
statistical results to agree with those appropriate to
structures of haphazard origin.
I think that even a materialist must reach a conclusion
not unlike ours if he fairly faces the problem. He will
need in the physical world something to stand for a
symbolic unity of the atoms associated with an individual
consciousness, which does not exist for atoms not so
associated — a unity which naturally upsets physical
predictions abased on the hypothesis of random dis-
connection. For he has not only to translate into
material configurations the multifarious thoughts and
images of the mind, but must surely not neglect to find
some kind of physical substitute for the Ego.
Chapter XV
SCIENCE AND MYSTICISM
one day I happened to be occupied with the subject
of "Generation of Waves by Wind". I took down the
standard treatise on hydrodynamics, and under that
heading I read —
The equations (12) and (13) of the preceding Art. enable us
to examine a related question of some interest, viz. the generation
and maintenance of waves against viscosity, by suitable forces
applied to the surface.
If the external forces p' yy , p'^ be given multiples of «***+**,
where k and a are prescribed, the equations in question determine
A and C, and thence, by (9) the value of tj. Thus we find
P'vv _ (^ + 2yffflS + Q 2 ) A - i ((T 2 + 2vkma) C
gprj "" gk(J- iC) l
£*v___a 2hk 2 J + (a + 2yg| C
gprj-gk' (J-iQ "
where o 2 has been written for gk -\- T r k z as before. . . .
And so on for two pages. At the end it is made clear
that a wind of less than half a mile an hour will leave
the surface unruffled. At a mile an hour the surface is
covered with minute corrugations due to capillary waves
which decay immediately the disturbing cause ceases.
At two miles an hour the gravity waves appear. As
the author modestly concludes, "Our theoretical investi-
gations give considerable insight into the incipient stages
of wave-formation".
on another occasion the same subject of "Generation
316
SCIENCE AND MYSTICISM 317
of Waves by Wind" was in my mind; but this time
another book was more appropriate, and I read —
There are waters blown by changing winds to laughter
And lit by the rich skies, all day. And after,
Frost, with a gesture, stays the waves that dance
And wandering loveliness. He leaves a white
Unbroken glory, a gathered radiance,
A width, a shining peace, under the night.
The magic words bring back the scene. Again we
feel Nature drawing close to us, uniting with us, till
we are filled with the gladness of the waves dancing in
the sunshine, with the awe of the moonlight on the
frozen lake. These were not moments when we fell
below ourselves. We do not look back on them and say,
"It was disgraceful for a man with six sober senses and
a scientific understanding to let himself be deluded in
that way. I will take Lamb's Hydrodynamics with me
next time". It is good that there should be such
moments for us. Life would be stunted and narrow if
we could feel no significance in the world around us
beyond that which can be weighed and measured with
the tools of the physicist or described by the metrical
symbols of the mathematician.
Of course it was an illusion. We can easily expose
the rather clumsy trick that was played on us. Aethereal
vibrations of various wave-lengths, reflected at different
angles from the disturbed interface between air and
water, reached our eyes, and by photoelectric action
caused appropriate stimuli to travel along the optic
nerves to a brain-centre. Here the mind set to work to
weave an impression out of the stimuli. The incoming
material was somewhat meagre; but the mind is a great
storehouse of associations that could be used to clothe
318 SCIENCE AND MYSTICISM
the skeleton. Having woven an impression the mind
surveyed all that it had made and decided that it was
very good. The critical faculty was lulled. We ceased
to analyse and were conscious only of the impression
as a whole. The warmth of the air, the scent of the
grass, the gentle stir of the breeze, combined with the
visual scene in one transcendent impression, around us
and within us. Associations emerging from their store-
house grew bolder. Perhaps we recalled the phrase
"rippling laughter". Waves — ripples — laughter — glad-
ness — the ideas jostled one another. Quite illogically we
were glad; though what there can possibly be to be glad
about in a set of aethereal vibrations no sensible person
can explain. A mood of quiet joy suffused the whole
impression. The gladness in ourselves was in Nature,
in the waves, everywhere. That's how it was.
It was an illusion. Then why toy with it longer?
These airy fancies which the mind, when we do not
keep it severely in order, projects into the external world
should be of no concern to the earnest seeker after truth.
Get back to the solid substance of things, to the material
of the water moving under the pressure of the wind and
the force of gravitation in obedience to the laws of
hydrodynamics. But the solid substance of things is
another illusion. It too is a fancy projected by the mind
into the external world. We have chased the solid
substance from the continuous liquid to the atom, from
the atom to the electron, and there we have lost it. But
at least, it will be said, we have reached something real
at the end of the chase — the protons and electrons. Or
if the new quantum theory condemns these images as
too concrete and leaves us with no coherent images at
all, at least we have symbolic co-ordinates and momenta
and Hamiltonian functions devoting themselves with
SCIENCE AND MYSTICISM 319
single-minded purpose to ensuring that qp — pq shall be
equal to ih/m.
In a previous chapter I have tried to show that by
following this course we reach a cyclic scheme which
from its very nature can only be a partial ｅxpression of
our environment. It is not reality but the skeleton of
reality. "Actuality" has been lost in the exigencies of
the chase. Having first rejected the mind as a worker
of illusion we have in the end to return to the mind and
say, "Here are worlds well and truly built on a basis
more secure than your fanciful illusions. But there is
nothing to make any one of them an actual world.
Please choose one and weave your fanciful images into
it. That alone can make it actual". We have torn away
the mental fancies to get at the reality beneath, only to
find that the reality of that which is beneath is bound
up with its potentiality of awakening these fancies. It
is because the mind, the weaver of illusion, is also the
only guarantor of reality that reality is always to be
sought at the base of illusion. Illusion is to reality as
the smoke to the fire. I will not urge that hoary un-
truth "There is no smoke without fire". But it is
reasonable to inquire whether in the mystical illusions of
man there is not a reflection of an underlying reality.
To put a plain question — Why should it be good for
us to experience a state of self-deception such as I have
described? I think everyone admits that it is good to
have a spirit sensitive to the influences of Nature, good
to exercise an appreciative imagination and not always
to be remorselessly dissecting our environment after
the manner of the mathematical physicists. And it is
good not merely in a utilitarian sense, but in some
purposive sense necessary to the fulfilment of the life
that is given us. It is not a dope which it is expedient
320 SCIENCE AND MYSTICISM
to take from time to time so that we may return with
greater vigour to the more legitimate employment of
the mind in scientific investigation. Just possibly it
might be defended on the ground that it affords to the
non-mathematical mind in some feeble measure that
delight in the external world which would be more
fully provided by an intimacy with its differential
equations. (Lest it should be thought that I have
intended to pillory hydrodynamics, I hasten to say in
this connection that I would not rank the intellectual
(scientific) appreciation on a lower plane than the
mystical appreciation; and I know of passages written
in mathematical symbols which in their sublimity might
vie with Rupert Brooke's sonnet.) But I think you will
agree with me that it is impossible to allow that the one
kind of appreciation can adequately fill the place of the
other. Then how can it be deemed good if there is
nothing in it but self-deception? That would be an
upheaval of all our ideas of ethics. It seems to me that
the only alternatives are either to count all such sur-
render to the mystical contact of Nature as mischievous
and ethically wrong, or to admit that in these moods
we catch something of the true relation of the world to
ourselves — a relation not hinted at in a purely scientific
analysis of its content. I think the most ardent material-
ist does not advocate, or at any rate does not practice,
the first alternative; therefore I assume the second alter-
native, that there is some kind of truth at the base of the
illusion.
But we must pause to consider the extent of the
illusion. Is it a question of a small nugget of reality
buried under a mountain of illusion? If that were so it
would be our duty to rid our minds of some of the
illusion at least, and try to know the truth in purer form.
SYMBOLIC AND INTIMATE KNOWLEDGE 321
But I cannot think there is much amiss with our appre-
ciation of the natural scene that so impresses us. I do
not think a being more highly endowed than ourselves
would prune away much of what we feel. It is not so
much that the feeling itself is at fault as that our
introspective examination of it wraps it in fanciful
imagery. If I were to try to put into words the essen-
tial truth revealed in the mystic experience, it would be
that our minds are not apart from the world; and the
feelings that we have of gladness and melancholy and
our yet deeper feelings are not of ourselves alone, but
are glimpses of a reality transcending the narrow limits
of our particular consciousness — that the harmony and
beauty of the face of Nature is at root one with the
gladness that transfigures the face of man. We try to
express much the same truth when we say that the
physical entities are only an extract of pointer readings
and beneath them is a nature continuous with our own.
But I do not willingly put it into words or subject it to
introspection. We have seen how in the physical world
the meaning is greatly changed when we contemplate
it as surveyed from without instead of, as it essentially
must be, from within. By introspection we drag out the
truth for external survey; but in the mystical feeling
the truth is apprehended from within and is, as it should
be, a part of ourselves.
Symbolic Knowledge and Intimate Knowledge. May I
elaborate this objection to introspection? We have two
kinds of knowledge which I call symbolic knowledge
and intimate knowledge. I do not know whether it
would be correct to say that reasoning is only applicable
to symbolic knowledge, but the more customary forms
of reasoning have been developed for symbolic know-
322 SCIENCE AND MYSTICISM
ledge only. The intimate knowledge will not submit to
codification and analysis; or, rather, when we attempt
to analyse it the intimacy is lost and it is replaced by
symbolism.
For an illustration let us consider Humour. I suppose
that humour can be analysed to some extent and the
essential ingredients of the different kinds of wit
classified. Suppose that we are offered an alleged joke.
We subject it to scientific analysis as we would a chemical
salt of doubtful nature, and perhaps after careful con-
sideration of all its aspects we are able to confirm that
it really and truly is a joke. Logically, I suppose, our
next procedure would be to laugh. But it may certainly
be predicted that as the result of this scrutiny we shall
have lost all inclination we may ever have had to laugh
at it. It simply does not do to expose the inner workings
of a joke. The classification concerns a symbolic know-
ledge of humour which preserves all the characteristics
of a joke except its laughableness. The real appreciation
must come spontaneously, not introspectively. I think
this is a not unfair analogy for our mystical feeling for
Nature, and I would venture even to apply it to our
mystical experience of God. There are some to whom
the sense of a divine presence irradiating the soul is one
of the most obvious things of experience. In their view
a man without this sense is to be regarded as we regard
a man without a sense of humour. The absence is a kind
of mental deficiency. We may try to analyse the ex-
perience as we analyse humour, and construct a theology,
or it may be an atheistic philosophy, which shall put
into scientific form what is to be inferred about it. But
let us not forget that the theology is symbolic knowledge
whereas the experience is intimate knowledge. And as
laughter cannot be compelled by the scientific exposition
DEFENCE OF MYSTICISM 323
of the structure of a joke, so a philosophic discussion
of the attributes of God (or an impersonal substitute)
is likely to miss the intimate response of the spirit which
is the central point of the religious experience.
Defence of Mysticism. A defence of the mystic might
run something like this. We have acknowledged that the
entities of physics can from their very nature form only
a partial aspect of the reality. How are we to deal with
the other part? It cannot be said that that other part
concerns us less than the physical entities. Feelings,
purpose, values, make up our consciousness as much as
sense-impressions. We follow up the sense-impressions
and find that they lead into an external world discussed
by science; we follow up the other elements of our
being and find that they lead — not into a world of space
and time, but surely somewhere. If you take the view
that the whole of consciousness is reflected in the dance
of electrons in the brain, so that each emotion is a
separate figure of the dance, then all features of con-
sciousness alike lead into the external world of physics.
But I assume that you have followed me in rejecting
this view, and that you agree that consciousness as a
whole is greater than those quasi-metrical aspects of it
which are abstracted to compose the physical brain.
We have then to deal with those parts of our being
unamenable to metrical specification, that do not make
contact — jut out, as it were — into space and time. By
dealing with them I do not mean make scientific in-
quiry into them. The first step is to give acknowledged
status to the crude conceptions in which the mind invests
them, similar to the status of those crude conceptions
which constitute the familiar material world.
Our conception of the familiar table was an illusion.
324 SCIENCE AND MYSTICISM
But if some prophetic voice had warned us that it was
an illusion and therefore we had not troubled to investi-
gate further we should never have found the scientific
table. To reach the reality of the table we need to be
endowed with sense-organs to weave images and illusions
about it. And so it seems to me that the first step in a
broader revelation to man must be the awakening of
image-building in connection with the higher faculties
of his nature, so that these are no longer blind alleys
but open out into a spiritual world — a world partly of
illusion, no doubt, but in which he lives no less than in
the world, also of illusion, revealed by the senses.
The mystic, if haled before a tribunal of scientists,
might perhaps end his defence on this note. He would
say, The familiar material world of everyday conception,
though lacking somewhat in scientific truth, is good
enough to live in; in fact the scientific world of pointer
readings would be an impossible sort of place to inhabit.
It is a symbolic world and the only thing that could live
comfortably in it would be a symbol. But I am not
a symbol; I am compounded of that mental activity
which is from your point of view a nest of illusion, so
that to accord with my own nature I have to transform
even the world explored by my senses. But I am not
merely made up of senses; the rest of my nature has to
live and grow. I have to render account of that environ-
ment into which it has its outlet. My conception of
my spiritual environment is not to be compared with
your scientific world of pointer readings; it is an every-
day world to be compared with the material world of
familiar experience. I claim it as no more real and no
less real than that. Primarily it is not a world to be
analysed, but a world to be lived in."
Granted that this takes us outside the sphere of
DEFENCE OF MYSTICISM 325
exact knowledge, and that it is difficult to imagine that
anything corresponding to exact science will ever be
applicable to this part of our environment, the mystic
is unrepentant. Because we are unable to render exact
account of our environment it does not follow that it
would be better to pretend that we live in a vacuum.
If the defence may be considered to have held good
against the first onslaught, perhaps the next stage of the
attack will be an easy tolerance. "Very well. Have it
your own way. It is a harmless sort of belief — not like
a more dogmatic theology. You want a sort of spiritual
playground for those queer tendencies in man's nature,
which sometimes take possession of him. Run away
and play then; but do not bother the serious people who
are making the world go round." The challenge now
comes not from the scientific materialism which pro-
fesses to seek a natural explanation of spiritual power,
but from the deadlier moral materialism which despises
it. Few deliberately hold the philosophy that the forces
of progress are related only to the material side of our
environment, but few can claim that they are not more
or less under its sway. We must not interrupt the
"practical men", these busy moulders of history carry-
ing us at ever-increasing pace towards our destiny as
an ant-heap of humanity infesting the earth. But is it
true in history that material forces have been the
most potent factors? Call it of God, of the Devil,
fanaticism, unreason; but do not underrate the power of
the mystic. Mysticism may be fought as error or believed
as inspired, but it is no matter for easy tolerance —
We are the music-makers
And we are the dreamers of dreams
Wandering by lone sea-breakers
And sitting by desolate streams ;
326 SCIENCE AND MYSTICISM
World-losers and world-forsakers,
on whom the pale moon gleams:
Yet we are the movers and shakers
Of the world for ever, it seems.
Reality and Mysticism, But a defence before the scien-
tists may not be a defence to our own self-questionings.
We are haunted by the word reality. I have already tried
to deal with the questions which arise as to the meaning
of reality; but it presses on us so persistently that, at the
risk of repetition, I must consider it once more from
the standpoint of religion. A compromise of illusion
and reality may be all very well in our attitude towards
physical surroundings; but to admit such a compromise
into religion would seem to be a trifling with sacred
things. Reality seems to concern religious beliefs much
more than any others. No one bothers as to whether
there is a reality behind humour. The artist who tries
to bring out the soul in his picture does not really care
whether and in what sense the soul can be said to exist.
Even the physicist is unconcerned as to whether atoms
or electrons really exist; he usually asserts that they do,
but, as we have seen, existence is there used in a
domestic sense and no inquiry is made as to whether
it is more than a conventional term. In most subjects
(perhaps not excluding philosophy) it seems sufficient
to agree on the things that we shall call real, and after-
wards try to discover what we mean by the word. And
so it comes about that religion seems to be the one field
of inquiry in which the question of reality and existence
is treated as of serious and vital importance.
But it is difficult to see how such an inquiry can be
profitable. When Dr. Johnson felt himself getting tied
up in argument over "Bishop Berkeley's ingenious
sophistry to prove the non-existence of matter, and that
REALITY AND MYSTICISM 327
everything in the universe is merely ideal", he answered,
"striking his foot with mighty force against a large
stone, till he rebounded from it, — 'I refute it thus* ".
Just what that action assured him of is not very obvious;
but apparently he found it comforting. And to-day the
matter-of-fact scientist feels the same impulse to recoil
from these flights of thought back to something kick-
able, although he ought to be aware by this time that
what Rutherford has left us of the large stone is scarcely
worth kicking.
There is still the tendency to use "reality" as a word
of magic comfort like the blessed word "Mesopotamia".
If I were to assert the reality of the soul or of God,
I should certainly not intend a comparison with
Johnson's large stone — a patent illusion — or even with
the p's and qs of the quantum theory — an abstract
symbolism. Therefore I have no right to use the word
in religion for the purpose of borrowing on its behalf
that comfortable feeling which (probably wrongly) has
become associated with stones and quantum co-ordi-
nates.
Scientific instincts warn me that any attempt to
answer the question "What is real?" in a broader sense
than that adopted for domestic purposes in science, is
likely to lead to a floundering among vain words and
high-sounding epithets. We all know that there are
regions of the human spirit untrammelled by the world
of physics. In the mystic sense of the creation around
us, in the ｅxpression of art, in a yearning towards God,
the soul grows upward and finds the fulfilment of
something implanted in its nature. The sanction for
this development is within us, a striving born with our
consciousness or an Inner Light proceeding from a
greater power than ours. Science can scarcely question
328 SCIENCE AND MYSTICISM
this sanction, for the pursuit of science springs from a
striving which the mind is impelled to follow, a ques-
tioning that will not be suppressed. Whether in the
intellectual pursuits of science or in the mystical pur-
suits of the spirit, the light beckons ahead and the
purpose surging in our nature responds. Can we not
leave it at that? Is it really necessary to drag in the
comfortable word "reality" to be administered like a
pat on the back?
The problem of the scientific world is part of a
broader problem — the problem of all experience. Ex-
perience may be regarded as a combination of self
and environment, it being part of the problem to
disentangle these two interacting components. Life,
religion, knowledge, truth are all involved in this
problem, some relating to the finding of ourselves, some
to the finding of our environment from the experience
confronting us. All of us in our lives have to make
something of this problem; and it is an important
condition that we who have to solve the problem are
ourselves part of the problem. Looking at the very
beginning, the initial fact is the feeling of purpose in
ourselves which urges us to embark on the problem.
We are meant to fulfil something by our lives. There
are faculties with which we are endowed, or which we
ought to attain, which must find a status and an outlet
in the solution. It may seem arrogant that we should in
this way insist on moulding truth to our own nature;
but it is rather that the problem of truth can only spring
from a desire for truth which is in our nature.
A rainbow described in the symbolism of physics is
a band of aethereal vibrations arranged in systematic
order of wave-length from about -000040 cm. to
•000072 cm. From one point of view we are paltering
SIGNIFICANCE AND VALUES 329
with the truth whenever we admire the gorgeous bow
of colour, and should strive to reduce our minds to such
a state that we receive the same impression from the
rainbow as from a table of wave-lengths. But although
that is how the rainbow impresses itself on an impersonal
spectroscope, we are not giving the whole truth and
significance of experience — the starting-point of the
problem — if we suppress the factors wherein we our-
selves differ from a spectroscope. We cannot say that
the rainbow, as part of the world, was meant to convey
the vivid effects of colour; but we can perhaps say that
the human mind as part of the world was meant to
perceive it that way.
Significance and Values. When we think of the sparkling
waves as moved with laughter we are evidently attri-
buting a significance to the scene which was not there.
The physical elements of the water — the scurrying
electric charges — were guiltless of any intention to
convey the impression that they were happy. But so
also were they guiltless of any intention to convey the
impression of substance, of colour, or of geometrical
form of the waves. If they can be held to have had any
intention at all it was to satisfy certain differential
equations — and that was because they are the creatures
of the mathematician who has a partiality for differential
equations. The physical no less than the mystical
significance of the scene is not there; it is here — in the
mind.
What we make of the world must be largely de-
pendent on the sense-organs that we happen to possess.
How the world must have changed since man came to
rely on his eyes rather than his nose ! You are alone on
the mountains wrapt in a great silence ; but equip yourself
330 SCIENCE AND MYSTICISM
with an extra artificial sense-organ and, lo! the aether is
hideous with the blare of the Savoy bands. Or —
The isle is full of noises,
Sounds, and sweet airs, that give delight, and hurt not.
Sometimes a thousand twangling instruments
Will hum about mine ears ; and sometimes voices.
So far as broader characteristics are concerned wc
see in Nature what we look for or are equipped to look
for. Of course, I do not mean that we can arrange the
details of the scene; but by the light and shade of our
values we can bring out things that shall have the broad
characteristics we esteem. In this sense the value placed
on permanence creates the world of apparent substance;
in this sense, perhaps, the God within creates the God
in Nature. But no complete view can be obtained so
long as we separate our consciousness from the world
of which it is a part. We can only speak speculatively
of that which I have called the "background of the
pointer readings"; but it would at least seem plausible
that if the values which give the light and shade of the
world are absolute they must belong to the background,
unrecognised in physics because they are not in the
pointer readings but recognised by consciousness which
has its roots in the background. I have no wish to put
that forward as a theory; it is only to emphasise that,
limited as we are to a knowledge of the physical world
and its points of contact with the background in isolated
consciousness, we do not quite attain that thought of
the unity of the whole which is essential to a complete
theory. Presumably human nature has been specialised
to a considerable extent by the operation of natural
selection; and it might well be debated whether its
valuation of permanence and other traits now apparently
SIGNIFICANCE AND VALUES 331
fundamental are essential properties of consciousness or
have been evolved through interplay with the external
world. In that case the values given by mind to the
external world have originally come to it from the
external world-stuff. Such a tossing to and fro of values
is, I think, not foreign to our view that the world-stuff
behind the pointer readings is of nature continuous with
the mind.
In viewing the world in a practical way values for
normal human consciousness may be taken as standard.
But the evident possibility of arbitrariness in this
valuation sets us hankering after a standard that could
be considered final and absolute. We have two alter-
natives. Either there are no absolute values, so that the
sanctions of the inward monitor in our consciousness are
the final court of appeal beyonu which it is idle to in-
quire. Or there are absolute values; then we can only
trust optimistically that our values are some pale
reflection of those of the Absolute Valuer, or that we
have insight into the mind of the Absolute from whence
come those strivings and sanctions whose authority we
usually forbear to question.
I have naturally tried to make the outlook reached in
these lectures as coherent as possible, but I should not
be greatly concerned if under the shafts of criticism it
becomes very ragged. Coherency goes with finality;
and the anxious question is whether our arguments have
begun right rather than whether^ they have had the good
fortune to end right. The leading points which have
seemed to me to deserve philosophic consideration may
be summarised as follows:
(1) The symbolic nature of the entities of physics
is generally recognised; and the scheme of physics is
now formulated in such a way as to make it almost
332 SCIENCE AND MYSTICISM
self-evident that it is a partial aspect of something
wider.
(2) Strict causality is abandoned in the material
world. Our ideas of the controlling laws are in process
of reconstruction and it is not possible to predict what
kind of form they will ultimately take; but all the in-
dications are that strict causality has dropped out
permanently. This relieves the former necessity of
supposing that mind is subject to deterministic law or
alternatively that it can suspend deterministic law in the
material world.
(3) Recognising that the physical world is entirely
abstract and without "actuality" apart from its linkage
to consciousness, we restore consciousness to the funda-
mental position instead of representing it as an in-
essential complication occasionally found in the midst
of inorganic nature at a late stage of evolutionary
history.
(4) The sanction for correlating a "real" physical
world to certain feelings of which we are conscious does
not seem to differ in any essential respect from the
sanction for correlating a spiritual domain to another
side of our personality.
It is not suggested that there is anything new in this
philosophy. In particular the essence of the first point
has been urged by many writers, and has no doubt won
individual assent from many scientists before the recent
revolutions of physical theory. But it places a somewhat
different complexion on the matter when this is not
merely a philosophic doctrine to which intellectual
assent might be given, but has become part of the
scientific attitude of the day, illustrated in detail in the
current scheme of physics.
CONVICTION 333
Conviction. Through fourteen chapters you have fol-
lowed with me the scientific approach to knowledge.
I have given the philosophical reflections as they have
naturally arisen from the current scientific conclusions,
I hope without distorting them for theological ends. In
the present chapter the standpoint has no longer been
predominantly scientific; I started from that part of our
experience which is not within the scope of a scientific
survey, or at least is such that the methods of physical
science would miss the significance that we consider it
essential to attribute to it. The starting-point of belief
in mystical religion is a conviction of significance or,
as I have called it earlier, the sanction of a striving in
the consciousness. This must be emphasised because
appeal to intuitive conviction of this kind has been the
foundation of religion through all ages and I do not
wish to give the impression that we have now found
something new and more scientific to substitute. I re-
pudiate the idea of proving the distinctive beliefs of
religion either from the data of physical science or by
the methods of physical science. Presupposing a
mystical religion based not on science but (rightly or
wrongly) on a self-known experience accepted as fun-
damental, we can proceed to discuss the various criti-
cisms which science might bring against it or the
possible conflict with scientific views of the nature of
experience equally originating from self-known data.
It is necessary to examine further the nature of the
conviction from which religion arises; otherwise we may
seem to be countenancing a blind rejection of reason as
a guide to truth. There is a hiatus in reasoning, we must
admit; but it is scarcely to be described as a rejection
of reasoning. There is just the same hiatus in reasoning
about the physical world if we go back far enough. We
334 SCIENCE AND MYSTICISM
can only reason from data and the ultimate data must
be given to us by a non-reasoning process — a self-
knowledge of that which is in our consciousness. To
make a start we must be aware of something. But that
is not sufficient; we must be convinced of the signifi-
cance of that awareness. We are bound to claim for
human nature that, either of itself or as inspired by a
power beyond, it is capable of making legitimate
judgments of significance. Otherwise we cannot even
reach a physical world.*
Accordingly the conviction which we postulate is
that certain states of awareness in consciousness have
at least equal significance with those which are called
sensations. It is perhaps not irrelevant to note that time
by its dual entry into our minds (p. 51) to some extent
bridges the gap between sense-impressions and these
other states of awareness. Amid the latter must be
found the basis of experience from which a spiritual
religion arises. The conviction is scarcely a matter to
be argued about, it is dependent on the forcefulness of
the feeling of awareness.
But, it may be said, although we may have such a
department of consciousness, may we not have mis-
understood altogether the nature of that which we
believe we are experiencing? That seems to me to be
rather beside the point. In regard to our experience of
the physical world we have very much misunderstood
the meaning of our sensations. It has been the task of
science to discover that things are very different from
* We can of course solve the problem arising from certain data
without being convinced of the significance of the data — the "official"
scientific attitude as I have previously called it But a physical world
which has only the status of the solution of a problem, arbitrarily chosen
to pass an idle hour, is not what is intended here.
CONVICTION 335
what they seem. But we do not pluck out our eyes
because they persist in deluding us with fanciful
colourings instead of giving us the plain truth about
wave-length. It is in the midst of such misrepresenta-
tions of environment (if you must call them so) that we
have to live. It is, however, a very one-sided view of
truth which can find in the glorious colouring of our
surroundings nothing but misrepresentation — which
takes the environment to be all important and the
conscious spirit to be inessential. In our scientific
chapters we have seen how the mind must be regarded
as dictating the course of world-building; without it
there is but formless chaos. It is the aim of physical
science, so far as its scope extends, to lay bare the
fundamental structure underlying the world; but science
has also to explain if it can, or else humbly to accept,
the fact that from this world have arisen minds capable
of transmuting the bare structure into the richness of
our experience. It is not misrepresentation but rather
achievement — the result perhaps of long ages of bio-
logical evolution — that we should have fashioned a
familiar world out of the crude basis. It is a fulfilment
of the purpose of man's nature. If likewise the spiritual
world has been transmuted by a religious colour beyond
anything implied in its bare external qualities, it may
be allowable to assert with equal conviction that this
is not misrepresentation but the achievement of a divine
element in man's nature.
May I revert again to the analogy of theology with
the supposed science of humour which (after consulta-
tion with a classical authority) I venture to christen
"geloeology". Analogy is not convincing argument, but
it must serve here. Consider the proverbial Scotchman
with strong leanings towards philosophy and incapable
336 SCIENCE AND MYSTICISM
of seeing a joke. There is no reason why he should not
take high honours in geloeology, and for example write
an acute analysis of the differences between British and
American humour. His comparison of our respective
jokes would be particularly unbiased and judicial, seeing
that he is quite incapable of seeing the point of either.
But it would be useless to consider his views as to which
was following the right development; for that he would
need a sympathetic understanding — he would (in the
phrase appropriate to the other side of my analogy) need
to be converted. The kind of help and criticism given
by the geloeologist and the philosophical theologian is
to secure that there is method in our madness. The
former may show that our hilarious reception of a
speech is the result of a satisfactory dinner and a good
cigar rather than a subtle perception of wit; the latter
may show that the ecstatic mysticism of the anchorite
is the vagary of a fevered body and not a transcendent
revelation. But I do not think we should appeal to
either of them to discuss the reality of the sense with
which we claim to be endowed, nor the direction of its
right development. That is a matter for our inner sense
of values which we all believe in to some extent, though
it may be a matter of dispute just how far it goes. If we
have no such sense then it would seem that not only
religion, but the physical world and all faith in reasoning
totter in insecurity.
I have sometimes been asked whether science cannot
now furnish an argument which ought to convince any
reasonable atheist. I could no more ram religious con-
viction into an atheist than I could ram a joke into the
Scotchman. The only hope of "converting" the latter
is that through contact with merry-minded companions
he may begin to realise that he is missing something
CONVICTION 337
in life which is worth attaining. Probably in the recesses
of his solemn mind there exists inhibited the seed of
humour, awaiting an awakening by such an impulse.
The same advice would seem to apply to the propagation
of religion; it has, I believe, the merit of being entirely
orthodox advice.
We cannot pretend to offer proofs. Proof is an idol
before whom the pure mathematician tortures himself.
In physics we are generally content to sacrifice before
the lesser shrine of Plausibility. And even the pure
mathematician — that stern logician — reluctantly allows
himself some prejudgments; he is never quite convinced
that the scheme of mathematics is flawless, and mathe-
matical logic has undergone revolutions as profound as
the revolutions of physical theory. We are all alike
stumblingly pursuing an ideal beyond our reach. In
science we sometimes have convictions as to the right
solution of a problem which we cherish but cannot
justify; we are influenced by some innate sense of the
fitness of things. So too there may come to us convic-
tions in the spiritual sphere which our nature bids us
hold to. I have given an example of one such conviction
which is rarely if ever disputed — that surrender to the
mystic influence of a scene of natural beauty is right and
proper for a human spirit, although it would have been
deemed an unpardonable eccentricity in the "observer"
contemplated in earlier chapters. Religious conviction
is often described in somewhat analogous terms as a
surrender; it is not to be enforced by argument on those
who do not feel its claim in their own nature.
I think it is inevitable that these convictions should
emphasise a personal aspect of what we are trying to
grasp. We have to build the spiritual world out of
symbols taken from our own personality, as we build
338 SCIENCE AND MYSTICISM
the scientific world out of the metrical symbols of the
mathematician. If not, it can only be left ungraspable —
an environment dimly felt in moments of exaltation but
lost to us in the sordid routine of life. To turn it into
more continuous channels we must be able to approach
the World-Spirit in the midst of our cares and duties in
that simpler relation of spirit to spirit in which all true
religion finds ｅxpression.
Mystical Religion. We have seen that the cyclic scheme
of physics presupposes a background outside the scope
of its investigations. In this background we must find,
first, our own personality, and then perhaps a greater
personality. The idea of a universal Mind or Logos
would be, I think, a fairly plausible inference from the
present state of scientific theory; at least it is in harmony
with it. But if so, all that our inquiry justifies us in assert-
ing is a purely colourless pantheism. Science cannot tell
whether the world-spirit is good or evil, and its halting
argument for the existence of a God might equally well
be turned into an argument for the existence of a Devil.
I think that that is an example of the limitation of
physical schemes that has troubled us before — namely,
that in all such schemes opposites are represented by
+ and — . Past and future, cause and effect, are repre-
sented in this inadequate way. one of the greatest
puzzles of science is to discover why protons and elec-
trons are not simply the opposites of one another,
although our whole conception of electric charge
requires that positive and negative electricity should be
related like + and — . The direction of time's arrow
could only be determined by that incongruous mixture
of theology and statistics known as the second law of
thermodynamics; or, to be more explicit, the direction
MYSTICAL RELIGION 339
of the arrow could be determined by statistical rules,
but its significance as a governing fact "making sense
of the world'* could only be deduced on teleological
assumptions. If physics cannot determine which way
up its own world ought to be regarded, there is not much
hope of guidance from it as to ethical orientation. We
trust to some inward sense of fitness when we orient the
physical world with the future on top, and likewise we
must trust to some inner monitor when we orient the
spiritual world with the good on top.
Granted that physical science has limited its scope
so as to leave a background which we are at liberty to,
or even invited to, fill with a reality of spiritual import,
we have yet to face the most difficult criticism from
science. "Here", says science, "I have left a domain
in which I shall not interfere. I grant that you have
some kind of avenue to it through the self-knowledge
of consciousness, so that it is not necessarily a domain
of pure agnosticism. But how are you going to deal with
this domain? Have you any system of inference from
mystic experience comparable to the system by which
science develops a knowledge of the outside world?
I do not insist on your employing my method, which
I acknowledge is inapplicable; but you ought to have
some defensible method. The alleged b4sis of experience
may possibly be valid; but have I any reason to regard
the religious interpretation currently given to it as
anything more than muddle-headed romancing?"
The question is almost beyond my scope. I can only
acknowledge its pertinency. Although I have chosen the
lightest task by considering only mystical religion —
and I have no impulse to defend any other — I am not
competent to give an answer which shall be anything
like complete. It is obvious that the insight of con-
340 SCIENCE AND MYSTICISM
sciousness, although the only avenue to what I have
called intimate knowledge of the reality behind the
symbols of science, is not to be trusted implicitly
without control. In history religious mysticism has
often been associated with extravagances that cannot
be approved. I suppose too that oversensitiveness to
aesthetic influences may be a sign of a neurotic tem-
perament unhealthy to the individual. We must allow
something for the pathological condition of the brain
in what appear to be moments of exalted insight. one
begins to fear that after all our faults have been detected
and removed there will not be any "us" left. But in
the study of the physical world we have ultimately to
rely on our sense-organs, although they are capable of
betraying us by gross illusions; similarly the avenue of
consciousness into the spiritual world may be beset with
pitfalls, but that does not necessarily imply that no
advance is possible.
A point that must be insisted on is that religion or
contact with spiritual power if it has any general im-
portance at all must be a commonplace matter of
ordinary life, and it should be treated as such in any
discussion. I hope that you have not interpreted my
references to mysticism as referring to abnormal experi-
ences and revelations. I am not qualified to discuss
what evidential value (if any) may be attached to the
stranger forms of experience and insight. But in any
case to suppose that mystical religion is mainly con-
cerned with these is like supposing that Einstein's
theory is mainly concerned with the perihelion of
Mercury and a few other exceptional observations.
For a matter belonging to daily affairs the tone of
current discussions often seems quite inappropriately
pedantic.
MYSTICAL RELIGION 341
As scientists we realise that colour is merely a question
of the wave-lengths of aethereal vibrations; but that does
not seem to have dispelled the feeling that eyes which
reflect light near wave-length 4800 are a subject for
rhapsody whilst those which reflect wave-length 5300
are left unsung. We have not yet reached the practice of
the Laputans, who, "if they would, for example, praise the
beauty of a woman, or any other animal, they describe
it by rhombs, circles, parallelograms, ellipses, and other
geometrical terms". The materialist who is convinced
that all phenomena arise from electrons and quanta and
the like controlled by mathematical formulae, must
presumably hold the belief that his wife is a rather
elaborate differential equation; but he is probably
tactful enough not to obtrude this opinion in domestic
life. If this kind of scientific dissection is felt to be
inadequate and irrelevant in ordinary personal relation-
ships, it is surely out of place in the most personal
relationship of all — that of the human soul to a divine
spirit.
We are anxious for perfect truth, but it is hard to say
in what form perfect truth is to be found. I cannot
quite believe that it has the form typified by an inventory.
Somehow as part of its perfection there should be in-
corporated in it that which we esteem as a "sense of
proportion". The physicist is not conscious of any
disloyalty to truth on occasions when his sense of
proportion tells him to regard a plank as continuous
material, well knowing that it is "really" empty space
containing sparsely scattered electric charges. And the
deepest philosophical researches as to the nature of the
Deity may give a conception equally out of proportion
for daily life; so that we should rather employ a concep-
tion that was unfolded nearly two thousand years ago.
342 SCIENCE AND MYSTICISM
I am standing on the threshold about to enter a room.
It is a complicated business. In the first place I must
shove against an atmosphere pressing with a force of
fourteen pounds on every square inch of my body.
I must make sure of landing on a plank travelling at
twenty miles a second round the sun — a fraction of a
second too early or too late, the plank would be miles
away. I must do this whilst hanging from a round
planet head outward into space, and with a wind of
aether blowing at no one knows how many miles a
second through every interstice of my body. The plank
has no solidity of substance. To step on it is like stepping
on a swarm of flies. Shall I not slip through? No, if
I make the venture one of the flies hits me and gives a
boost up again; I fall again and am knocked upwards
by another fly; and so on. I may hope that the net result
will be that I remain about steady; but if unfortunately
I should slip through the floor or be boosted too vio-
lently up to the ceiling, the occurrence would be, not
a violation of the laws of Nature, but a rare coincidence.
These are some of the minor difficulties. I ought really
to look at the problem four-dimensionally as concerning
the intersection of my world-line with that of the plank.
Then again it is necessary to determine in which direc-
tion the entropy of the world is increasing in order to
make sure that my passage over the threshold is an
entrance, not an exit.
Verily, it is easier for a camel to pass through the eye
of a needle than for a scientific man to pass through a
door. And whether the door be barn door or church
door it might be wiser that he should consent to be an
ordinary man and walk in rather than wait till all the
difficulties involved in a really scientific ingress are
resolved.
"1
CONCLUSION
A tide of indignation has been surging in the breast of
the matter-of-fact scientist and is about to be unloosed
upon us. Let us broadly survey the defence we can set
up.
I suppose the most sweeping charge will be that I
have been talking what at the back of my mind I must
know is only a well-meaning kind of nonsense. I can
assure you that there is a scientific part of me that has
often brought that criticism during some of the later
chapters. I will not say that I have been half-convinced,
but at least I have felt a homesickness for the paths of
physical science where there are more or less discernible
handrails to keep us from the worst morasses of foolish-
ness. But however much I may have felt inclined to
tear up this part of the discussion and confine myself to
my proper profession of juggling with pointer readings,
I find myself holding to the main principles. Starting
from aether, electrons and other physical machinery we
cannot reach conscious man and render count of what
is apprehended in his consciousness. Conceivably we
might reach a human machine interacting by reflexes
with its environment; but we cannot reach rational man
morally responsible to pursue the truth as to aether and
electrons or to religion. Perhaps it may seem unneces-
sarily portentous to invoke the latest developments of
the relativity and quantum theories merely to tell you
this; but that is scarcely the point. We have followed
these theories because they contain the conceptions of
modern science; and it is not a question of asserting a
faith that science must ultimately be reconcilable with
an idealistic view, but of examining how at the moment
343
344 CONCLUSION
it actually stands in regard to it. I might sacrifice the
detailed arguments of the last four chapters (perhaps
marred by dialectic entanglement) if I could otherwise
convey the significance of the recent change which has
overtaken scientific ideals. The physicist now regards
his own external world in a way which I can only describe
as more mystical, though not less exact and practical,
than that which prevailed some years ago, when it was
taken for granted that nothing could be true unless an
engineer could make a model of it. There was a time
when the whole combination of self and environment
which makes up experience seemed likely to pass under
the dominion of a physics much more iron-bound than
it is now. That overweening phase, when it was almost
necessary to ask the permission of physics to call one's
soul one's own, is past. The change gives rise to
thoughts which ought to be developed. Even if we
cannot attain to much clarity of constructive thought
we can discern that certain assumptions, expectations
or fears are no longer applicable.
Is it merely a well-meaning kind of nonsense for a
physicist to affirm this necessity for an outlook beyond
physics? It is worse nonsense to deny it. Or as that
ardent relativist the Red Queen puts it, "You call that
nonsense, but I've heard nonsense compared with which
that would be as sensible as a dictionary".
For if those who hold that there must be a physical
basis for everything hold that these mystical views are
nonsense, we may ask — What then is the physical basis
of nonsense? The "problem of nonsense" touches the
scientist more nearly than any other moral problem.
He may regard the distinction of good and evil as too
remote to bother about; but the distinction of sense
and nonsense, of valid and invalid reasoning, must be
CONCLUSION 345
accepted at the beginning of every scientific inquiry.
Therefore it may well be chosen for examination as a
test case.
If the brain contains a physical basis for the nonsense
which it thinks, this must be some kind of configuration
of the entities of physics — not precisely a chemical
secretion, but not essentially different from that kind
of product. It is as though when my brain says 7 times
8 are 56 its machinery is manufacturing sugar, but
when it says 7 times 8 are 6$ the machinery has gone
wrong and produced chalk. But who says the machinery
has gone wrong? As a physical machine the brain has
acted according to the unbreakable laws of physics;
so why stigmatise its action? This discrimination of
chemical products as good or evil has no parallel in
chemistry. We cannot assimilate laws of thought to
natural laws; they are laws which ought to be obeyed,
not laws which must be obeyed; and the physicist must
accept laws of thought before he accepts natural law.
"Ought" takes us outside chemistry and physics. It
concerns something which wants or esteems sugar, not
chalk, sense, not nonsense. A physical machine cannot
esteem or want anything; whatever is fed into it it will
chaw up according to the laws of its physical machinery.
That which in the physical world shadows the nonsense
in the mind affords no ground for its condemnation. In a
world of aether and electrons we might perhaps encounter
nonsense; we could not encounter damned nonsense.
The most plausible physical theory of correct rea-
soning would probably run somewhat as follows. By
reasoning we are sometimes able to predict events
afterwards confirmed by observation; the mental pro-
cesses follow a sequence ending in a conception which
anticipates a subsequent perception. We may call such
346 CONCLUSION
a chain of mental states "successful reasoning" —
intended as a technical classification without any moral
implications involving the awkward word "ought". We
can examine what are the common characteristics of
various pieces of successful reasoning. If we apply this
analysis to the mental aspects of the reasoning we obtain
laws of logic; but presumably the analysis could also
be applied to the physical constituents of the brain. It
is not unlikely that a distinctive characteristic would be
found in the physical processes in the brain-cells which
accompany successful reasoning, and this would con-
stitute "the physical basis of success."
But we do not use reasoning power solely to predict
observational events, and the question of success (as
above defined) does not always arise. Nevertheless if
such reasoning were accompanied by the product which
I have called "the physical basis of success" we should
naturally assimilate it to successful reasoning.
And so if I persuade my materialist opponent to
withdraw the epithet "damned nonsense" as inconsistent
with his own principles he is still entitled to allege that
my brain in evolving these ideas did not contain the
physical basis of success. As there is some danger of
our respective points of view becoming mixed up, I
must make clear my contention:
(a) If I thought like my opponent I should not worry
about the alleged absence of a physical basis of success
in my reasoning, since it is not obvious why this should
be demanded when we are not dealing with observa-
tional predictions.
(b) As I do not think like him, I am deeply perturbed
by the allegation; because I should consider it to be the
outward sign that the stronger epithet (which is not
inconsistent with my principles) is applicable.
CONCLUSION 347
I think that the "success" theory of reasoning will
not be much appreciated by the pure mathematician.
For him reasoning is a heaven-sent faculty to be enjoyed
remote from the fuss of external Nature. It is heresy
to suggest that the status of his demonstrations depends
on the fact that a physicist now and then succeeds in
predicting results which accord with observation. Let
the external world behave as irrationally as it will, there
will remain undisturbed a corner of knowledge where
he may happily hunt for the roots of the Riemann-
Zeta function. The "success" theory naturally justifies
itself to the physicist. He employs this type of activity
of the brain because it leads him to what he wants — a
verifiable prediction as to the external world — and for
that reason he esteems it. Why should not the theo-
logian employ and esteem one of the mental processes
of unreason which leads to what he wants — an assurance
of future bliss, or a Hell to frighten us into better
behaviour? Understand that I do not encourage theo-
logians to despise reason; my point is that they might
well do so if it had no better justification than the
"success" theory.
And so my own concern lest I should have been
talking nonsense ends in persuading me that I have to
reckon with something that could not possibly be
found in the physical world.
Another charge launched against these lectures may
be that of admitting some degree of supernaturalism,
which in the eyes of many is the same thing as super-
stition. In so far as supernaturalism is associated with
the denial of strict causality (p. 309) I can only answer
that that is what the modern scientific development of
the quantum theory brings us to. But probably the
more provocative part of our scheme is the role allowed
348 CONCLUSION
to mind and consciousness. Yet I suppose that our
adversary admits consciousness as a fact and he is aware
that but for knowledge by consciousness scientific
investigation could not begin. Does he regard con-
sciousness as supernatural? Then it is he who is
admitting the supernatural. Or does he regard it as
part of Nature? So do we. We treat it in what seems
to be its obvious position as the avenue of approach to
the reality and significance of the world, as it is the
avenue of approach to all scientific knowledge of the
world. Or does he regard consciousness as something
which unfortunately has to be admitted but which it is
scarcely polite to mention? Even so we humour him.
We have associated consciousness with a background
untouched in the physical survey of the world and have
given the physicist a domain where he can go round in
cycles without ever encountering anything to bring a
blush to his cheek. Here a realm of natural law is
secured to him covering all that he has ever effectively
occupied. And indeed it has been quite as much the
purpose of our discussion to secure such a realm where
scientific method may work unhindered, as to deal with
the nature of that part of our experience which lies
beyond it. This defence of scientific method may not
be superfluous. The accusation is often made that, by
its neglect of aspects of human experience evident to
a wider culture, physical science has been overtaken
by a kind of madness leading it sadly astray. It is
part of our contention that there exists a wide field
of research for which the methods of physics suffice,
into which the introduction of these other aspects would
be entirely mischievous.
A besetting temptation of the scientific apologist for
religion is to take some of its current ｅxpressions and
CONCLUSION 349
after clearing away crudities of thought (which must
necessarily be associated with anything adapted to the
everyday needs of humanity) to water down the meaning
until little is left that could possibly be in opposition
to science or to anything else. If the revised interpre-
tation had first been presented no one would have raised
vigorous criticism; on the other hand no one would have
been stirred to any great spiritual enthusiasm. It is the
less easy to steer clear of this temptation because it is
necessarily a question of degree. Clearly if we are to
extract from the tenets of a hundred different sects any
coherent view to be defended some at least of them must
be submitted to a watering-down process. I do not
know if the reader will acquit me of having succumbed
to this temptation in the passages where I have touched
upon religion; but I have tried to make a fight against it.
Any apparent failure has probably arisen in the following
way. We have been concerned with the borderland of
the material and spiritual worlds as approached from the
side of the former. From this side all that we could
assert of the spiritual world would be insufficient to
justify even the palest brand of theology that is not too
emaciated to have any practical influence on man's
outlook. But the spiritual world as understood in any
serious religion is by no means a colourless domain.
Thus by calling this hinterland of science a spiritual
world I may seem to have begged a vital question,
whereas I intended only a provisional identification. To
make it more than provisional an approach must be made
from the other side. I am unwilling to play the amateur
theologian, and examine this approach in detail. I have,
however, pointed out that the attribution of religious
colour to the domain must rest on inner conviction; and
I think we should not deny validity to certain inner
35o CONCLUSION
convictions, which seem parallel with the unreasoning
trust in reason which is at the basis of mathematics, with
an innate sense of the fitness of things which is at the
basis of the science of the physical world, and with an
irresistible sense of incongruity which is at the basis of
the justification of humour. Or perhaps it is not so much
a question of asserting the validity of these convictions
as of recognising their function as an essential part of
our nature. We do not defend the validity of seeing
beauty in a natural landscape; we accept with gratitude
the fact that we are so endowed as to see it that way.
It will perhaps be said that the conclusion to be
drawn from these arguments from modern science, is
that religion first became possible for a reasonable
scientific man about the year 1927. If we must consider
that tiresome person, the consistently reasonable man,
we may point out that not merely religion but most of
the ordinary aspects of life first became possible for him
in that year. Certain common activities (e.g. falling in
love) are, I fancy, still forbidden him. If our expectation
should prove well founded that 1927 has seen the final
overthrow of strict causality by Heisenberg, Bohr, Born
and others, the year will certainly rank as one of the
greatest epochs in the development of scientific philo-
sophy. But seeing that before this enlightened era men
managed to persuade themselves that they had to mould
their own material future notwithstanding the yoke of
strict causality, they might well use the same modus
vivendi in religion.
This brings us to consider the view often pontifically
asserted that there can be no conflict between science
and religion because they belong to altogether different
realms of thought. The implication is that discussions
such as we have been pursuing are superfluous. But it
CONCLUSION 351
seems to me rather that the assertion challenges this kind
of discussion — to see how both realms of thought can
be associated independently with our existence. Having
seen something of the way in which the scientific realm
of thought has constituted itself out of a self-closed
cyclic scheme we are able to give a guarded assent. The
conflict will not be averted unless both sides confine
themselves to their proper domain; and a discussion
which enables us to reach a better understanding as to
the boundary should be a contribution towards a state
of peace. There is still plenty of opportunity for frontier
difficulties; a particular illustration will show this.
A belief not by any means confined to the more
dogmatic adherents of religion is that there is a future
non-material existence in store for us. Heaven is no-
where in space, but it is in time. (All the meaning of
the belief is bound up with the word future; there is no
comfort in an assurance of bliss in some former state of
existence.) on the other hand the scientist declares that
time and space are a single continuum, and the modern
idea of a Heaven in time but not in space is in this
respect more at variance with science than the pre-
Copernican idea of a Heaven above our heads. The
question I am now putting is not whether the theologian
or the scientist is right, but which is trespassing on the
domain of the other? Cannot theology dispose of the
destinies of the human soul in a non-material way
without trespassing on the realm of science? Cannot
science assert its conclusions as to the geometry of the
space-time continuum without trespassing on the realm
of theology? According to the assertion above science
and theology can make what mistakes they please
provided that they make them in their own territory ; they
cannot quarrel if they keep to their own realms. But
352 CONCLUSION
it will require a skilful drawing of the boundary line
to frustrate the development of a conflict here.*
The philosophic trend of modern scientific thought
differs markedly from the views of thirty years ago.
Can we guarantee that the next thirty years will not see
another revolution, perhaps even a complete reaction?
We may certainly expect great changes, and by that
time many things will appear in a new aspect. That is
one of the difficulties in the relations of science and
philosophy; that is why the scientist as a rule pays so
little heed to the philosophical implications of his own
discoveries. By dogged endeavour he is slowly and
tortuously advancing to purer and purer truth; but his
ideas seem to zigzag in a manner most disconcerting
to the onlooker. Scientific discovery is like the fitting
together of the pieces of a great jig-saw puzzle; a
revolution of science does not mean that the pieces
already arranged and interlocked have to be dispersed;
it means that in fitting on fresh pieces we have had to
revise our impression of what the puzzle-picture is
going to be like. one day you ask the scientist how he is
getting on; he replies, "Finely. I have very nearly
finished this piece of blue sky." Another day you ask
how the sky is progressing and are told, "I have added a
lot more, but it was sea, not sky; there's a boat floating on
the top of it". Perhaps next time it will have turned out
to be a parasol upside down ; but our friend is still enthusi-
astically delighted with the progress he is making. The
scientist has his guesses as to how the finished picture will
work out; he depends largely on these in his search for
other pieces to fit; but his guesses are modified from time
to time by unexpected developments as the fitting pro-
*This difficulty is evidently connected with the dual entry of time
into our experience to which I have so often referred.
CONCLUSION 353
ceeds. These revolutions of thought as to the final
picture do not cause the scientist to lose faith in his
handiwork, for he is aware that the completed portion
is growing steadily. Those who look over his shoulder
and use the present partially developed picture for
purposes outside science, do so at their own risk.
The lack of finality of scientific theories would be a
very serious limitation of our argument, if we had staked
much on their permanence. The religious reader may
well be content that I have not offered him a God
revealed by the quantum theory, and therefore liable
to be swept away in the next scientific revolution. It is
not so much the particular form that scientific theories
have now taken — the conclusions which we believe we
have proved — as the movement of thought behind them
that concerns the philosopher. Our eyes once opened,
we may pass on to a yet newer outlook on the world,
but we can never go back to the old outlook.
If the scheme of philosophy which we now rear on
the scientific advances of Einstein, Bohr, Rutherford
and others is doomed to fall in the next thirty years, it
is not to be laid to their charge that we have gone astray.
Like the systems of Euclid, of Ptolemy, of Newton,
which have served their turn, so the systems of Einstein
and Heisenberg may give way to some fuller realisation
of the world. But in each revolution of scientific thought
new words are set to the old music, and that which has
gone before is not destroyed T^ut refocussed. Amid all
our faulty attempts at ｅxpression the kernel of scientific
truth steadily grows; and of this truth it may be said —
The more it changes, the more it remains the same
thing.
INDEX
A B C of physics, xiv, 88
A priori probability, 77, 244, 305
Absolute, 23, 56; past and future,
48, 57, 295; elsewhere, 49, 50;
values, 288, 331; future perfect,
307
Absorption of light, 184, 186
Abstractions, 53
Accelerated frames of reference, 113
Acceleration, relativity of, 129
Action, 180, 241; atom of, 182
Actuality, 266, 319
Aether, nature of, 31
Aether-drag, 3
Age of the sun, 169
And, study of, 104
Anthropomorphic conception of
deity, 282, 337, 341
Antisymmetrical properties of
world, 236
Ape-like ancestors, 16, 81, 273
Apple (Newton's), hi, 115
Arrow, Time's, 69, 79, 88, 295
Astronomer Royal's time, 36, 40
Atom, structure of, 1, 190, 199, 224
Atom of action, 182. See Quantum
Atomicity, laws of, 236, 245
Averages, 300
Awareness, 16, 334
Background of pointer readings,
137, 255, 259, 268, 330, 339
Balance sheet, 33
Beats, 216
Beauty, 105, 267, 350
Becoming, 68, 87
Beginning of time, 83
Berkeley, Bishop, xii, 326
Beta (3) particle, 59
Bifurcation of the world, 236
Billiard ball atoms, 2, 259
Blessed gods (Hegel), 147, 155
Bohr, N., 2, 185, 191, 196, 220, 306
Boltzmann, L., 63
Bombardment, molecular, 113, 131
Born, M., 208
Bose, S. N., 203
Bragg, W. H., 194
Brain, 260, 268, 279, 311, 323
Broad, C. D., 160
de Broglie, L., 201, 202
Building material, 230
Bursar, 237
Casual and essential characteristics,
142
Categories, xi, 105
Causality, 297
Cause and effect, 295
Cepheid variables, 165
Chalk, calculation of motion of, 107
Chance, 72, 77, 189
Classical laws and quantum laws,
193, 195, 308
Classical physics, 4
Clifford, W. K., 278
Clock, 99, 134, 154
Code-numbers, 55, 81, 235, 277
Coincidences, 71
Collection-box theory, 187, 193
Colour and wave-length, 88, 94,
329, 34i
Commonsense knowledge, 16
Companion of Sirius, 203
Comparability of relations, 232
Compensation of errors, 12
355
356
INDEX
Concrete, 273
Configuration space, 219
Conservation, laws of, 236, 241
Constellations, subjectivity of, 95,
106, 241
Contiguous relations, 233
Contraction, FitzGerald, 5, 24;
reality of, 32, 53
Controlling laws, 151, 245
Conversion, 336
Conviction, 333, 350
Co-ordinates, 208, 231
Copenhagen school, 195
Correspondence principle, 196
Counts of stars, 163
Crudeness of scale and clock sur-
vey, 154
Curvature of space-time, 119, 127,
157; coefficients of, 120, 155
Cyclic method of physics, 260, 277,
Cylindrical curvature, 139
Darwin, G. H., 171
Deflection of light by gravity, 122
Demon (gravitation), 118, 309
Dense matter, 203
Design, 77
Detailed balancing, principle of, 80
Determinism, 228, 271, 294, 303, 310
Differential equations, 282, 329, 341
Diffraction of electrons, 202
Dimension, fourth, 52; beyond
fourth, 120, 158, 219
Dirac, P. A. M., 208, 219, 270
Directed radius, 140
Direction, relativity of, 26
Distance, relativity of, 25; inscru-
table nature of, 81; macroscop-
ic character, 155, 201
Door, scientific ingress through, 342
Doppler effect, 45, 184
Double stars, 175
Dual recognition of time, 51, 91, 99,
334, 352
Duration and becoming, 79, 99
Dynamic quality of time, 68, 90, 92,
260
Eclipses, prediction of, 149, 299
Ego, 97, 282, 315
Egocentric attitude of observer, 15,
61, 112
Einstein, A., 1, 53, in, 185, 203
Einstein's law of gravitation, 120,
J 39» I 5 I > 260; law of motion,
124
Einstein's theory, 20, 111
Electrical theory of matter, 2, 6
Electromagnetism, 236
Electron, 3 ; mass of, 59 ; extension
in time, 146; in the atom, 188,
199, 224; nature of, 279, 290
Elephant, problem of, 251
Elliptical space, 289
Elsewhere, 42
Emission of light, 183, 191, 216
Encounters of stars, 177
Engineer, superseded by mathe-
matician, 104, 209
Entropy, 74, 105
Entropy-change and Becoming, 88
Entropy-clock, 101
Environment, 288, 328
Epistemology, 225, 304
Erg-seconds, 179
Essential characteristics, 142
Euclidean geometry, 159
Events, location of, 41 ; point-
events, 49
Evolution, irreversibility of, 91 ; in
stellar system, 167, 176
Exact science, 250
Existence, 286
Experience, 288, 328
Explanation, scientific ideal of, xiii,
138, 209, 248
Extensive abstraction, method of,
249
External world, 284
INDEX
357
Familiar and scientific worlds, xiii,
247, 324
Fictitious lengths, 19
Field, 153
Field-physics, 236
Finite but unbounded space, 80, 139,
166, 289
FitzGerald contraction, 5, 24; real-
ity of, 32, 53
Flat world, 118, 138
Flatness of galaxy, 164
Force, 124
Formality of taking place, 68
Fortuitous concourse of atoms, 77,
251
Fourth dimension, 52, 231
Fowler, R. H., 204
Frames of space and time, 14, 20,
35, 61, 112, 155
Freak (solar system), 176
Freewill, 295
Fullness of space, measures of, 153
Future, relative and absolute, 48 ;
see Predictability
Future life, 351
Future perfect tense, 307
Galactic system, 163
Geloeology, 335
General theory of relativity, in,
129
Generation of Waves by Wind,
316
Geodesic, 125
Geometrisation of physics, 136
Geometry, 133, 157, 161
Grain of the world, 48, 55, 56, 90
Gravitation, relative and absolute
features, 114; as curvature,
118; law of, 120, 139; explana-
tion of, 138, 145
Greenland, 117
Gross appliances, survey with, 154
Growth, idea of, 87
Group velocity, 213
h, 179, 183, 223
Halo of reality, 282, 285, 290
Hamilton, W. R., 181
Hamiltonian differentiation, 240
Heaven, 351
Hegel, 147
Heisenberg, W., 206, 220, 228, 306
Heredity, 250
Here-Now, 41
Heterodyning, 216
Hour-glass figures, 48
House that Jack Built, 262
Hubble, E. P., 167
Humour, 322, 335
Humpty Dumpty, 64
Huxley, T. H., 173
Hydrodynamics, 242, 316
Hydrogen, 3
Hyperbolic geometry, 136
Hypersphere, 81, 157
i (square root of — 1), 135, 146,
208
Identical laws, 237
Identity replacing causation, 156
Illusion, 320
Impossibility and improbability, 75
Impressionist scheme of physics, 103
Indeterminacy, principle of, 220,
306
Inertia, 124
Inference, chain of, 270, 298
Infinity, 80
Infra-red photography, 173
Inner Light, 327
Insight, 89, 91, 268, 277, 311, 339
Instants, world-wide, 43
Integers, 220, 246
Interval, 37, 261
Intimate and symbolic knowledge,
321
Introspection, 321
Invariants, 23
Inventory method, 103, 106, 280,
341
358
INDEX
Inverse-square law, 29
Island universes, 165
Isotropic directed curvature, 144
Jabberwocky, 291
Jeans, J. H., 176, 187
Johnson, Dr., 326
Jordan, P., 208
Knowable to mind, 264
Knowledge, nature of physical, 257,
304; complete, 226
Laplace, 176
Laputans, 341
Larmor, J., 7
Laws of Nature, 237, 244
Laws of thought, 345
Lenard, P., 130
Length, 6, 160. See Distance
Life on other planets, 170
Life-insurance, 300
Lift, man in the, 111
Light, velocity of, 46, 54; emission
of, 183, 191, 216
Likeness between relations, 232
Limitations of physical knowledge,
257
Linkage of scientific and familiar
worlds, xiii, 88, 156, 239, 249
Location, frames of, 14, 41
Logos, 338
Longest track, law of, 125, 135,
148
Lorentz, H. A., 7
Lowell, P., 174
Luck, rays of, 190
Lumber (in world building), 235,
243
Macroscopic survey, 154, 227, 299,
304
Man, 169, 178
Man-years, 180
Mars, 172
Mass, increase with velocity, 39, 50,
59
Mathematician, 161, 209, 337, 347
Matrix, 208
Matter, 1, 31, 156, 203, 248, 262
Maxwell, J. C, 8, 60, 156, 237
Measures of structure, 234, 268
Mechanical models, 209
Mechanics and Geometry, 137
Mendelian theory, 250
Mental state, 279
Metric, 142, 153
Metrical and non-metrical proper-
ties, 275
Michelson-Morley experiment, 5, zi
Microscopic analysis, reaction from,
103
Milky Way, Z63
Miller, D. C, 5
Mind and matter, 259, 268, 278;
selection by mind, 239, 243, 264
Mind-stuff, 276
Minkowski, H., 34, 53
Mirror, distortion by moving, it
Models, 198, 209, 344
Molecular bombardment, 113, Z3Z
Momentum, 153, 208, 223, 239, 262
Monomarks, 23 z
Moon, origin of, Z7Z
Morley, E. W., 5
Motion, law of, 123
Multiplicationist, 86
Multiplicity of space and time
frames, 20, 35, 6z
Myself, 42, 53
Mysticism, defence of, 323 ; reli-
gious, 338
Nautical Almanac, Z50
Nebulae, 165
Nebular observers, 9, Z2
Neptune, 49
Neutral stuff, 280
Neutral wedge, 48
New quantum theory, 206
INDEX
359
Newton, in, 122, 201; quotation
from, in
Newtonian scheme, 4, 18, 125
Non-empty space, 127, 153, 238
Non-Euclidean geometry, 157
Nonsense, problem of, 344
Now-lines, 42, 47, 49, 184
Nucleus of atom, 3
Objectivity of "becoming", 94; of
a picture, 107
Observer, attributes of, 15, 337
Odds, 301, 303
Official scientific attitude, 286, 334
Operator, 208
Orbit jumps of electron, 191, 196,
205, 215, 300, 312
Organisation, 68, 70, 104
Ought, 345
Oxygen and vegetation, 174
/»'s and q's, 208, 223, 327
Pacific Ocean, 171
Particle, 202, 211, 218
Past, relative and absolute, 48
Pedantry, 340, 342
Permanence, 241
Personal aspect of spiritual world,
337
Phoenix complex, 85
Photoelectric effect, 187
Photon, 190
Physical time, 40
Picture and paint, 106
Picture of gravitation, 115, 138, 157
Plan, Nature's, 27
Planck, M., 185
Plurality of worlds, 169
Pointer readings, 251
Ponderomotive force, 237
Porosity of matter, 1
Potential (gravitational), 261
Potential energy, 213
Potential gradient, 96
Pound sterling, relativity of, 26
Predestination, 293, 303
Predictability of events, 147, 228,
300, 307
Primary law, 66, 75, 98 ; insuffi-
ciency of, 107
Primary scheme of physics, 76, 129,
295
Principal curvature, 120, 139
Principia, 4
Principle, Correspondence, 196
Principle of detailed balancing, 80
Principle of indeterminacy, 220, 306
Probability, 216, 315
Proof and plausibility, 337
Proper-distance, 25
Proper-time, 37
Proportion, sense of, 341
Proton, 3
Psi {\p), 216, 305
Pure mathematician, 161, 337, 347
Purpose, 105
g-numbers, 208, 270
Quantum, 184; size of, 200
Quantum laws, 193
Quantum numbers, 191, 205
Quest of the absolute, 26, 122; of
science, no, 287; of reality, 328
Quotations from
Boswell, 326
Brooke, Rupert, 317
Clifford, W. K., 278
Dickens, 32
Einstein, A., 294
Hegel, 147
Huxley, T. H., 173
Kronecker, L., 246
Lamb, H., 316
Lewis Carroll, 28, 291, 344
Milton, 167
Newton, in
Nursery Rhymes, 64, 70, 262
Omar Khayyam, 64, 293
O'Shaughnessy, A., 325
Russell, Bertrand, 160, 278
360
INDEX
Quotations from (cont.)
Shakespeare, 21, 39, 83, 292, 330
Swift, 341
Whitehead, A. N., 145
Radiation pressure, 58
Random element, 64; measurement
of, 74
Reality, meaning of, 282, 326
Really true, 34
Rectification of curves, 125
Rejuvenescence, theories of, 85, 169
Relata and relations, 230
Relativity of velocity, 10, 54, 59, 61 ;
of space-frames, 21 ; of mag-
netic field, 22 ; of distance, 25 ;
of pound sterling, 26; of Now
(simultaneity), 46, 61; of ac-
celeration, 129; of standard of
length, 143
Religion, 194, 281, 288, 322, 324,
326, 333, 349
Retrospective symbols, 307, 308
Revolutions of scientific thought, 4,
352
Right frames of space, 18, 20
Roemer, O., 43
Rotating masses, break-up of, 176
Running down of universe, 63, 84
Russell, B., 160, 277, 278
Rutherford, E., 2, 327
Scale (measuring), 12, 18, 24, 134,
141
Schrodinger's theory, 199, 210, 225,
305
Scientific and familiar worlds, xiii,
247, 324
Second law of thermodynamics, 74,
86
Secondary law, 75, 79, 98
Seen-now line9, 44, 47
Selection by mind, 239, 243, 264,
330
Self-comparison of space, 145
Sense-organs, 51, 96, 266, 329
Shadows, world of, xiv, 109
Shuffling, 63, 92, 184
Sidereal universe, 163
Signals, speed of, 57
Significances, 108, 329
Simultaneity, 49, 61
Singularities, 127
Sirius, Companion of, 203
de Sitter, W., 167
Slithy toves, 291
Solar system, origin of, 176
Solar system type of atom, 2, 190
Sorting, 93
Space, 14, 16, 51, 81, 137
Spasmodic moon, 226
Spatial relations, 50
Spectral lines, 205, 216; displace-
ment of, 121, 166
Spherical curvature, radius of,
14c
Spherical space, 82, 166, 289; ra-
dius of, 167
Spiral nebulae, 165
Spiritual world, 281, 288, 324, 349
Standard metre, 141
Stars, number of, 163 ; double, 175 ;
evolution of, 176; white
dwarfs, 203
States, 197, 301
Statistical laws, 244; mind's inter-
ference with, 313
Statistics, 201, 300, 303
Stratification, 47
Stress, 129, 155, 262
Structure, 234, 277
Sub-aether, 211, 219
Subjective element in physics, 95,
241
Substance, ix, 273, 318
Success, physical basis of, 346
Sun, as a star, 164; age of, 169
Supernatural, 309, 348
Survey from within, 145, 321, 330
Sweepstake theory, 189
INDEX
361
Symbolism in science, xiii, 209, 247,
269, 324
Synthetic method of physics, 249
Temperature, 71
Temporal relations, 50
Tensor, 257
Tensor calculus, 181
Thermodynamical equilibrium, 77
Thermodynamics, second law of,
66, 74, 86
Thermometer as entropy-clock, 99,
101
Thinking machine, 259
Thought, 258 ; laws of, 345
Time in physics, 36; time lived
(proper-time), 40; dual recog-
nition of, 51, 100; time's arrow,
69; infinity of, 83; summary of
conclusions, 101 ; time-triangies,
133 ; reality of, 275
Time-scale in astronomy, 167
Touch, sense of, 273
Track, longest, 125, 135, 148
Trade Union of matter, 126
Transcendental laws, 245
Traveller, time lived by, 39, 126,
135
Triangles in space and time, 133
Tug of gravitation, 115, 122
Undoing, 65
Unhappening, 94, 108
Uniformity, basis of, 145
Unknowable entities, 221, 308
Utopia, 265
Values, 243, 330
Vegetation on Mars, 173
Velocity, relativity of, 10; upper
limit to, 56
Velocity through aether, 30, 32
Velocity of light, 46, 54
Venus, 170
Victorian physicist, ideals of, 209,
259
View-point, 92, 283
Void, 13, 137
Volition, 310
Watertight compartments, 194
Wave-group, 213, 217, 225
Wave-length, measurement of, 24
Wave-mechanics, 211
Wave-theory of matter, 202
Wavicle, 201
Wells, H. G., 67
White dwarfs, 203
Whitehead, A. N., 145, 249
Whittaker, E. T., 181
Winding up of universe, 83
World building, 230
World-lines, 253
Worm, four-dimensional, 42, 87, 92
Wright, W. H., 172
Wrong frames of reference, 116
X (Mr.), 262, 268
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