
*** START OF THE PROJECT GUTENBERG EBOOK 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
1929
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 especially 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 experience, including religion. The general spirit
of the inquiry followed in the lectures is stated in the concluding
paragraph of the Introduction (p. xvii).
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 introduction
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 greatest philosophical significance are to be found.
Whilst much of the book should prove fairly easy reading, arguments of

considerable difficulty have to be taken in their turn.
My principal aim has been to show that these scientific developments
provide new material for the philosopher. 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 mathematical
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 appearance 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 “on the side of the angels” and judge
its trustworthiness accordingly. During 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 generally considered
rather unsuitable for a long book, but I decided not to modify it. A
scientific writer, in forgoing 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 1928



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 substantiality 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 scientific 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 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 satisfactorily 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 abnormal circumstances befall, then my scientific table shows to
advantage. If the house catches fire my scientific table will dissolve
quite naturally into scientific smoke, whereas my familiar table
undergoes a metamorphosis 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, 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 substitute 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 apparatus of
measurement nor the mode of using it suggests that there is anything
essentially different in these problems. 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 scientific scrutiny. Science has at last revolted
against attaching the exact knowledge contained in these measurements
to a traditional picture-gallery of conceptions which convey no
authentic information of the background 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 mostly empty and the table of everyday conception
which we regard as the type of solid reality—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 supported because there
is substance below it, it being the intrinsic nature of substance to
occupy space to the exclusion 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 delicate test and
remorseless logic assured me that my second 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 external 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 according 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 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
scrupulously careful to guard these from contamination by conceptions
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 premature 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
experience or even something explicable in terms of common experience.
The man in the street is always making 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 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
prematurely, before the letters are strung into words and the words
into sentences. The symbol  is not the counterpart of anything
in familiar life. To the child the letter  would seem horribly
abstract; so we give him a familiar conception along with it. “
was an Archer who shot at a frog.” This tides over his immediate
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
fundamental 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 concern 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 determination 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 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, suffering and evil.
The frank realisation that physical science is concerned with a
world of shadows is one of the most significant of recent advances.
I do not mean that physicists are to any extent preoccupied with the
philosophical implications of this. From their point of view it is not
so much a withdrawal of untenable claims as an assertion of freedom 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 symbolism it is hard to avoid dressing our
symbols in deceitful clothing. When I think of an electron there rises
to my mind a hard, red, tiny ball; the proton similarly 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 Hamiltonian 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
universe 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 cannot but have undergone change,
since our whole conception 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 transcending 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 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
 
v

Introduction
 
ix

Chapter I.
The Downfall of Classical Physics
1

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 Einstein and
Minkowski introduced fundamental changes in our ideas of time and
space. In 1911 Rutherford introduced 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 matter underwent the ordinary experience of scientific
discovery; 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 Bolshevism of modern science and regrets for the old-established
order, I am inclined to think that Rutherford, not Einstein, 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 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 material with the characteristics of air should have
relatively little substance in it, and “airy nothing” is a common
phrase for the insubstantial. In solids the atoms are packed tightly in
contact, so that the old atomic theory agreed with our preconceptions
in regarding 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
electricity, 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 1911 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 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 elementary 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
hydrogen, which seems to be the most primitive form of matter, 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 balance 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 ¹⁄₅₀₀₀₀ 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 question 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 perplexing 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 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 (1) 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 developments 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 corpuscular theory; heat was
changed from substance (caloric) 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 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 contraction, 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 2½ 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 experiments 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 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. Moreover 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 FitzGerald 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 an entirely natural property of a swarm of particles held

in delicate balance by electromagnetic forces, and occupying space by
buffeting away anything that tries to enter. Or you may look at it
this way: your expectation that the rod will keep its original length
presupposes, 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 contraction 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.

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 electromagnetic
laws universally accepted since the time of Maxwell. 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 relativity, 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 measuring a
length—takes a scale and turns it in the direction 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 measuring 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 FitzGerald contraction.
First take what may seem to be a rather fantastic case. Imagine you are
on a planet moving very fast indeed, say 161,000 miles a second. For
this speed the contraction is one-half. Any solid contracts 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 a

right angle. The inhabitants copy Alice in Wonderland; 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
unprofessionally) 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 appreciable in measurements of scientific or
even of engineering accuracy. One of the most fundamental procedures
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 insignificant 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 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 overlooked this
correction!”
There is no means of deciding which is right—to which of us the
observed relative velocity of 1000 miles a second really
belongs. Astronomically the galaxy 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 unnoticed. 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 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 experiments—but there is
really no contradiction. They were all null experiments, just
as your experiment of watching 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 contraction 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 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 supplementary 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 observed or capable of observation. The compensation
does not apply to the intermediate part of our deduction—that system
of inference from observation which forms the classical physical theory
of the universe.

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 direction; 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 physicists 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 recognise 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, absurd 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 compared 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 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 misunderstanding 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″ x 1″ 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 objection 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 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
naïve 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 measurements 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 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 commonsense. 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 scientific 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 suggest
all sorts of ways in which the apparatus might have misbehaved; but
if the knowledge of location is 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 likewise 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
position 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 behaviour is a constant property of all
kinds of matter and is even shared by optical and electrical measuring
devices. Thus it is not betrayed by any kind of discrepancy in

applying the usual methods of measurement. The discrepancy is revealed
when we change the standard motion of the measuring appliances, e.g.
when we compare lengths and distances as measured by terrestrial
observers with those which would be measured by observers on a planet
with different velocity. Provisionally 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 invalidated if our methods of measuring lengths were fundamentally
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 indistinguishable 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 suggestion—
"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
to 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.

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 observer, another frame
relative to the nebular observers, others relative
to other stars. Frames of space are relative. 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 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 1000 miles a second. A
moving charge constitutes an electric current which in accordance with
the laws of electromagnetism gives rise to a magnetic 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 instruments 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 absolute specification.
It is not quite true to say that all the physical quantities 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, 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 quantities. 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 precautions
necessary to obtain an accurate measurement of distance. They bring me
results which differ. I ask 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 important
body to regulate the motion of the scale, and so he had given it a
motion agreeing with that of the enormous star Betelgeuse. Naturally
the FitzGerald contraction 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 Betelgeuse 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, “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 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—something 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 something 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 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 Cambridge relative to Edinburgh and
another direction relative 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 undiscoverable) 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 difference 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 unfortunately 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 convinced
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 statements—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, merely recognising

that the pound is relative and therefore 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
consequences may be more sensational—a mark-exchange soaring to
billions, a high-speed  particle contracting to a third of its
radius. But it is not these casual manifestations 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
understanding 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 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 compute 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 perhaps this
particular tribe was not enthusiastic about astronomy.
Like the critic I do not think Nature has been particularly 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 we 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 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 conceptions, 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 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 something
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 significant 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 disappears 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-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 characters of the particles. Perhaps a philosopher might
question 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 have no ground for any preconception,

the experimental conclusions can be accepted without surprise or
misgiving. 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 particular 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 material 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 Dickens 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  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 distinction 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 generally 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-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 purposes 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.[1] 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 someone 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 four dimensions.

We have been confronted with something not contemplated in classical
physics—a multiplicity 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 confronted 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 quantities 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 quantities
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 travelled 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
absolute scheme underlying it, so that we may see the world in a truer
perspective.

[1]
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.




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 nonsensical,
Prof. Bergson would probably end the discussion 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 jure, the Astronomer Royal’s time is
time de facto. His time permeates every corner of physics.
It stands in no need of logical defence; 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 space must be

of entirely different nature. They cannot possibly be mixed up.” The
Astronomer Royal complacently retorts, “It is not impossible. I
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 investigation 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 lessened if it should prove
to be only imperfectly representative 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, unrecognised 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 representation 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 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 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 retardation 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 digestion and fatigue; the rate of muscular response
to stimulus; 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 emotions; 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 traveller 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 reckoning, confirms
this. Judging by the time which consciousness 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’ll 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 subjective
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 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” quite scientifically,
e.g. by a watch travelling 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 universal 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 private 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 external events are located

in a frame of space and time. We have seen that there is an infinite
choice of alternative frames; so, to be quite explicit, I will tell you
how I locate events in my frame.



Fig. 1


Location of Events. In Fig. 1 you see a collection of events,
indicated by circles. They are not at present in 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 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).



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 conclusion that Now is not confined
to Here-Now. I therefore 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 consistently. 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 accommodate these events. We had been mixing
up two distinct 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 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 provided 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 I locate events; how about you? We must first
put You into the picture (Fig. 3).



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 will cause different Doppler effects, FitzGerald

contractions, 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, 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.

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 FitzGerald 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; because you take
this to be the velocity relative to your planet and I take it to be the
velocity relative to mine.[2] 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 I drew the picture.
You, of course, would redraw it with your Now lines at right angles to
yourself.

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 longitude 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 convenient to adopt.
We have been accustomed to regard the world—the enduring world—as
stratified into a succession of instantaneous 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 cone

or hour-glass figure (Fig. 4).



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 absolute 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.
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
infinitely narrow present. It suggests itself that the 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 prearranged—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 instantaneous 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.

As the speed of matter approaches the speed of light its mass increases
to infinity, and therefore it is impossible 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 dividing 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 elsewhere), but not in both. The
temporal relations radiate into the past and future cones and the
spatial relations into the neutral wedge; they are kept absolutely

separated 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 artificial Now lines.
I would direct your attention to an important difference 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
experience 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 mysterious 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 nature of time and so it baffles our comprehension. It is the same

paradox which makes us believe we understand 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 mistaken 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 supposed

planes of structure represented by 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.[3]
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 unfamiliar 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 relativity 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.


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 reference body I
have in mind. No one will understand anything 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? Relative 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 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.

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 unwitting compliment to its
universal importance.[4] Turning 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 “universal” 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 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 considered in this chapter.[5]
The velocity of light in matter can under certain circumstances (in the
phenomenon of anomalous dispersion) 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 signalling.
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-day! The limit to the velocity of signals
is our bulwark 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 convinced, 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 “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 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  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 quantities length and time.
Let us look at a  particle from its own point of view. It
is an ordinary electron in no wise different from any other. But is
it 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  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, . But
mass and radius are relative quantities, and in this case the frame

to which they are referred is evidently the frame appropriate 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 problem were wrong; an important
compensating factor escaped notice. Its conclusions as to the second
problem were (after some false starts) entirely correct numerically.
But since the result was deduced from the electromagnetic 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 increase 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 egocentric 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
connected with the denial of absolute velocity; knowledge of absolute
velocity would enable us to assert that 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 distinction 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
absolute 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 introduced 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.

[2]
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.


[3]
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.


[4]
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.


[5]
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.




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 will never come back however long you shuffle.
Something has been done which cannot be undone, namely, the
introduction 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 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 happened 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.




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 motion,
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, astronomers 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 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 illustration; 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 controlling 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—because 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 , right is ; past is ,
future is . 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 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 subject-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 backwards. If space is “looking-glassed” the world
continues 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—the second

law of thermodynamics—to put some sense into the world. It opens up a
new province of knowledge, namely, the study of organisation; and it
is in connection 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 sometimes 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 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-dimensional 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
contention 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
arrangements 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 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
direction; 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 unavoidable 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.

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 coincidences;
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
of 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 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 compartment 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
 is the number of molecules (roughly a quadrillion) the chance
of this happening is . 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 favourable 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.

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.[6] 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. 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 accidental
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 coincidence. 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 experimentalists
do bungle things sometimes. But if your theory is found to be against
the second law of thermodynamics 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 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 connection 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 problems 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 conviction of nearly all physicists[7] that at the root of

everything there is a complete scheme of primary law governing 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 backwards, 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 insufficient to exclude any result as impossible.” Secondary law
replies plainly, “It will boil because it is too improbable that it
should do anything else.” Secondary law is not in conflict with primary
law, nor can we regard it as essential to complete a scheme of law
already 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 thermodynamics 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 “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.

Thermodynamical Equilibrium. Progress of time introduces 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-morrow. 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 expressing 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 antithesis
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 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 boundary 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 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?[8] The
question is answered by another important law of Nature which runs—
Nothing in the statistics of an assemblage can distinguish 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.[9]

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; we
had to be content to admit in the physical world an inconceivable
conception—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 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 force are distributed. Both are a matter for

experiment.
If we go a very long way to a point  in one direction through
the universe and a very long way to a point  in the opposite
direction, it is believed that between  and  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 contemplated 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 conception.
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 compared with ordinary standards. On
rather insecure evidence it has been estimated to be not many times

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 dimension. There is a bending round by which
East ultimately 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.  will not happen until we reach A.D. , and
presumably in order to reach A.D.  the difficulty must
first have been surmounted. It should also be noted that according
to the second law of thermodynamics 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.

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 postulates 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. Moreover, this organisation is
admittedly the antithesis of chance. It is something which could not
occur fortuitously.
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 somewhat crude the naïve
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 as the working-hypothesis

of thermodynamics rather than its declaration of faith. It is one of
those conclusions 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 connection 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 counter-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 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 possibility of re-formation of matter
from radiation is not crucial and we can await conclusions with some
indifference.
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.

[6]
If the gas in expanding had been made to move a piston,
the organisation would have passed into the motion of the piston.


[7]
There are, however, others beside myself who have recently
begun to question it.


[8]
In a kaleidoscope the shuffling is soon complete and all
the patterns are equal as regards random element, but they differ
greatly in elegance.


[9]
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 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.




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 intended 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
neighbour who has his good and bad ends the other way round, I cannot
imagine.”
A problem here rises before us concerning the 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 interpretation 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 “becoming” into it and
not an artificial symbolic substitute.
The linkage of entropy-change to “becoming” presents features unlike
every other problem of parallelism 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 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.

I am afraid that the average reader will feel impatient 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 conveniently 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

meaning. The view here advocated is tantamount to an admission that
consciousness, looking out through a private door, can learn by
direct insight an underlying character 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 nature, Time occupies the
key position. I have already referred 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 private
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 on private doors, through which all
forms 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 structure 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 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 consciousness 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
phenomena. 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 elementary 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 (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.
corresponding 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 unambiguous
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 inorganic causes, but sorting is the
prerogative of mind or instinct. We cannot believe that it is merely an
orientation 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 interchanged, 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 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 unbecoming
shuffled—and, if we wish to pursue this aspect further, we must
discuss not the causes but the uncauses. 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
objective 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 coloured 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

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
organisation; 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 constellation; 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 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 indefinables 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 entropy-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,

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 symbolic knowledge as on an inferior plane. If
I grasp the notion of existence because I myself exist, I grasp the
notion of becoming because I myself become. It is the innermost Ego of
all which is and becomes.
The incongruity of symbolising this fundamental intuition by a property
of arrangement of the microscopic constituents of the world, is
evident. What this difficulty portends is still very obscure. But it
is not 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.[10] I believe that
Nature is honest at heart, and that she only resorts to these apparent
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

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 arrangement for measuring the
disorganisation of energy (e.g. a thermometer) in order to discover
which event is the later.[11] 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 measurer of time, the more completely does it conceal
time’s arrow.

This paradox seems to be explained by the fact pointed out in chapter
III 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 experiences 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 messages 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 better analogy

would be an entropy-clock, i.e. an instrument designed primarily for
measuring the rate of disorganisation of energy, and only very roughly
keeping 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 ordinary
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 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 Scientific Reaction from Microscopic Analysis. From the
point of view of philosophy of science the conception 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 accordingly 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 microscopic 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 particular fraction of the organisation is
located in a particular 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 completed our study of one we

know all about two, because “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
compartments. 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 unfounded 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 extension—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).

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 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 aesthetic 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 discrimination 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.

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
environment means to us much that is not warranted by anything found
in the scientific inventory of its structure. 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.]

You are given the instantaneous position and velocity[12] 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 motion 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 between my fingers. I grant that you can
do all that without 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 unhappening? There is no need to alter a word of your
description 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 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
conceptions 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 physical world of atoms, aether and electrons has any
significance 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 eternal 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 consciousness. 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 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 arrow), 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.

[10]
See p. 221.


[11]
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 difference 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.


[12]
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.)




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 1915 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 experiments. 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 repetition 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 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 we 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 usual 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. Therefore 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 firma? 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 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 scientific 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,[13] let us enter the runaway lift
again. Here nothing need be supported; our bodies, our galvanometers,

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 converge 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 altogether
in the frame of the man in the lift and we ought to disregard it in
any attempt to form an absolute picture of gravitation. But there
always remains something 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 Newton’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 including 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
question 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. Newton
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 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 gravitation.
The apple looks down and sees Newton approaching it; but this time it
cannot attribute Newton’s acceleration 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
hammering. 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.[14] The essence
or absolute part of gravitation cannot be a force on a body, because
we are entirely vague as to the body to which it is applied. We must
picture it differently.

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 would 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 represented
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.

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 simplicity. Objects at
rest remain at rest; objects in uniform 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
activity 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 simply 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.

Do not imagine that this preliminary change of conception 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 picture of
gravitation, we require a new law of gravitation; for the Newtonian law
told us the amount 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. two-dimensional

manifolds embedded in a three-dimensional space. The absolute curvature
at any point is measured by a single quantity called the radius of
spherical 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 discovering 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 curvature. Ten of the
coefficients stand out more prominently than the other ten.
Einstein’s law of gravitation asserts that the ten principal
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 without 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 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 gravitating 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 gravitation 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
hundreds 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
astronomical phenomena in which the difference is large enough to be
observed. In these phenomena the observations support Einstein’s law
against Newton’s.[15]
It is essential to our faith in a theory that its predictions 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 Newton’s law just failed. But our main reason for rejecting
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. 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 appearances
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 trying 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 success 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 waves by tugging at them, and

clearly another representation 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 reduced 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 altogether, 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 he blandly invents a new force to account for

the deviation. 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 absolute interferences
which can change the motion of a body. I have nothing to say against
them. The molecular 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 interfering 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 gravitational 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.[16]
Einstein’s Law of Motion does not recognise this dissection. There
are certain curves which can be defined on a curved surface without

reference to any frame or system of partitions, viz. the geodesics or
shortest routes from one point to another. The geodesics 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
“elsewhere”. 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.[17] This may be different from the time recorded

by a clock which has taken some other route between the same terminal
points. On p. 39 we considered 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 calculate the curved form of
space-time by using Einstein’s law of gravitation, viz. that the ten
principal curvatures 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 Newton’s law of gravitation and Newton’s law
of motion. But there is a remarkable addendum which applies only

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.



Fig. 5


Two “particles” in the four-dimensional world are shown in Fig.
5, namely yourself and myself. We are not empty
space so there is no limit to the kind of curvature entering into
our composition; 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
“singularities”. These two non-empty ridges are joined by empty space,
which must be free from those kinds of curvature 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 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 cannot occur alone. But if they bend a little
towards one another the connecting region can lie smoothly and satisfy
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 submit 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-dimensional 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 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 reasonable enough. If left alone the earth must
pursue its proper curved orbit; but if some kind of stress or pressure
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, Newtonians and Einsteinians alike would infer
that there existed a stress causing this behaviour. It is true that
causation has apparently been turned topsy-turvy; according 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
arising 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 Einstein’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 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 accidents, 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 indifferently as an acceleration of
the train (relative to the station) 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 Cambridge 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 difference 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 injure anyone,

even when it is (according to the Newtonian 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 injuring
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 acceleration of the station—and in fact of
the earth and the rest of the universe? To put it mildly, relative
acceleration 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 bombardment) 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 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 ourselves relative to the
stone. Which end of the relation must we choose? The one indicated by
molecular bombardment? 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
suggestion 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 “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 unfortunate 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. 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 interlocked 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 subject-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 considered 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.[18] 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 possible 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 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 meaning 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 discussion 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 proposition then expresses that
“If you measure with a scale from  to  and from  to
 the sum of your readings will be greater than the reading
obtained by measuring with a scale from  to .”
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 to  and from  to 
the sum of your readings will be less than the reading obtained
by measuring with a clock from  to .”
In order to measure from an event  to an event  with a clock
you must make an adjustment of the clock analogous to orienting a scale
along the line . What is this analogous adjustment? The purpose
in either case is to bring both  and  into the immediate
neighbourhood of the scale or clock. For the clock that means that
after experiencing the event  it must travel with the appropriate
velocity needed to reach the locality of  just at the moment
that  happens. Thus the velocity of the clock is prescribed. One
further point should be noticed. After measuring with a scale from

 to  you can turn your scale round and measure from  to
, 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
impossible 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
sluggish, 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 difference
between time geometry and space geometry is not very profound, and the
mathematician easily glides over it by a discrete use of the symbol
. We still 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-dimension 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 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 anyone 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 previously explained it has the
other meaning of distance, volume, etc., quantities expressing physical
measurement just as much as force is a quantity expressing physical
measurement. Thus the (rather crude) statement 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.

[13]
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.


[14]
It will probably be objected that since the phenomena
here discussed 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 symmetrical ones “right” and unsymmetrical
ones “wrong”.


[15]
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.


[16]
The reader will verify that this is the doctrine the
teacher would have to inculcate if he went as a missionary to the men
in the lift.


[17]
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 surmounted in the same way. The usual theory of “rectification
of curves” applies to these time-tracks as well as to space-curves.


[18]
This would be an instantaneous space-triangle. An
enduring triangle is a kind of four-dimensional prism.




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 giving 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 hummocky 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 precautions were
taken to exclude it. If the curvature were entirely arbitrary this
would be the end of the explanation; 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 curvature which differentiate the
world from a flat world, but by the vanishing everywhere of the ten
principal coefficients.

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[19] curvature of every
three-dimensional 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 equivalent to the first if for
“same constant length” we read “infinite length”. Apart from very
speculative estimates we do not know the constant length referred to,
but it must certainly be greater than the distance of the furthest
nebula, say  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.

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 (1) 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,  trillion
miles.” I cannot tell you what number to put for  because that is
still a secret of the firm.

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
adaptation 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.[20] 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 unthinkable as action at a distance;
more so, because comparison 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 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 characteristics of its surroundings,
present and past.[21] We cannot say that it was chosen to keep the same
absolute length since there is no such thing known; but it was chosen

so that it might not be prevented by casual influences 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 territory 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.

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.
Extension which is not relative to something in the surroundings 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 belonging 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 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-comparison of space, it is perhaps surprising that
such a self-comparison should be able to show up any heterogeneity at
all. It can in fact be proved that the metric of a two-dimensional or
a three-dimensional world surveyed 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.”[22] But we now see that
Einstein’s theory asserts a casual heterogeneity of only one set of

ten coefficients and complete uniformity of the other ten. It therefore
does not leave us without the basis of uniformity of which Whitehead in
his own way perceived the necessity. Moreover, this uniformity 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 condition 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 important 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
 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 rigorous 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 behaves in the simple way

which we might at least hope to find.[23]

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 on 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 taking 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 discreditable 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.”[24]
This sounds particularly foolish even for a philosopher; but I believe
that there is a sense in which it is true.

We have already had three versions of what the earth is trying to do
when it describes its elliptic orbit around the sun.
(1) 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, may 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).

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.



Fig. 6

We must take measurements of its position, for example, measurements
of its distance from the sun. In Fig. 6,  shows the ridge in
the world which we recognise as the sun; I have drawn the earth’s ridge
in duplicate  because I imagine it as still undecided
which track it will take. If it takes  we lay our measuring
rods end to end down the ridges and across the valley from  to
, 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 large and the radius of

curvature small. The rods therefore are small, and there will be more
of them in  than the picture would lead you to expect. If the
earth chooses to go to  the curvature is less sharp; the greater
radius of curvature implies greater length of the rods. The number
needed to stretch from  to  will not be so great as the
diagram at first suggests; it will not be increased in anything like
the proportion of  to  in the figure. We should not
be surprised 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  or . 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 mathematical 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 interplanetary 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 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 curvature 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  (full line) and  (dotted line) is the
same. There are two dimensions of space-time omitted in the picture
besides the extra dimensions 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, expressed
mathematically by , 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  so we shall
find the same position of the earth, if we assume nothing more than
that the practical unit of length involved in measurements 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  or
, and it would convey no information as to any observable
phenomena if we knew the representation.

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 measurements 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;[25] 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.


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 (sufficiently
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 components. 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 considered 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 exploring
the world with a measuring scale and a magnetic compass respectively
ought to be welded together into 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 determines the symmetrical properties
and the other the antisymmetrical properties of the underlying
world-structure.[26]
Objection has often been taken, especially by philosophical 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 gravitation cannot be regarded as
subsisting apart from the experimental procedure by which we have
ascertained its truth.

The conception of frames of space and time, and of the non-emptiness
of the world described as energy, momentum, etc., is bound up with
the survey by gross appliances. When they can no longer be supported
by such a survey, the conceptions melt away into meaninglessness. 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 distance,
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 philosophic interest.
The ten principal coefficients of curvature 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 electromagnetic 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

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 properties 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 complete 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 radiating 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 superfluous as to ask
how light gets a grip on the electromagnetic 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 unusual 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
commemorates it can have caused so much carnage?
The philosophic outcome of this identification will 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 synonymously; 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. Distances for the
second man obey a Euclidean geometry of three dimensions; distances for
the first man obey 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 describing 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 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 intuitively, 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 experiment 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 adopt any geometry we choose.[27] 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
assertion 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 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 gateway 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 curvature 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 material
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 measures; a potential theory
unhampered by any question as to how gravitational and electrical
potentials really 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 mathematics. 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 overweigh 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
development 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.

[19]
Cylindrical curvature of the world has nothing to do with
gravitation, 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.


[20]
This relativity with respect to a standard unit is, of
course, additional to and independent of the relativity with respect to
the observer’s motion treated in chapter II.


[21]
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 corrections for casual influences,
regard that as license to adopt any arbitrary system of corrections—a
procedure which would merely have the effect of concealing what the
measures can teach us about essential characteristics.


[22]
A. N. Whitehead, The Principle of Relativity,
Preface.


[23]
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 .


[24]
Hegel, Werke (1842 Ed.), Bd. 7, Abt. 1, p. 97.


[25]
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.


[26]
See p. 236.


[27]
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.




Chapter VIII
MAN’S PLACE IN THE UNIVERSE


The Sidereal Universe. The largest telescopes reveal about a
thousand million stars. Each increase in telescopic 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 uniformly 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 sometimes quoted. But the difficulty is that the part of
the stellar universe which we mainly survey is a local condensation 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 directions 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 a real extent. It is

partly broken up into subordinate condensations, 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 Sagittarius; 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 flattened
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
extension. Its size is such that light takes at least 2000 years to
cross from one side to the other; this measurement 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 ¹⁄₁₀₀₀₀ of its light. But those of inferior
light 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 phenomena 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 discounted 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 universes” 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 arrangement 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 variables) it
is possible to judge the distance. The nearest spiral nebula is 850,000
light years away.

From the small amount of data yet collected it would seem that our
own nebula or galactic system is exceptionally large; it is even
suggested that if the spiral nebulae are “islands” the galactic system
is a “continent”. 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 100 million stars.
Again the question raises itself, How far does this distribution
extend? Not the stars this time but universes 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 signifying 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 1000 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 has really been observed is the

slowing down of vibrations 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.[28]

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 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 evolution 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 abandoned speculation
is that lucid stars may be the exception, 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 substances seems to leave no escape from
the conclusion that the older (Archaean) rocks in the earth’s crust
were 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
() 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 billions of years, and we may speculate
on processes of rejuvenescence which might prolong the existence 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 backwards—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 to the habitability of other worlds. The
popular idea that an answer to this question is one of the main aims
of the study of celestial objects is rather disconcerting to the
astronomer. Anything that he has to contribute is of the nature of
fragmentary hints picked up in the course of investigations with
more practicable and commonplace purposes. Nevertheless, the mind is
irresistibly drawn to play with 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 conditions
differing from those of our planet. If I have rightly understood the
view of palaeontologists, mammalian life is the third terrestrial
dynasty—Nature’s third attempt to evolve an order of life sufficiently
flexible 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
before 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 conditions 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 observation 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
inference. If transplanted to Venus we might perhaps continue to live
without much derangement of habit—except that I personally would
have to find a new profession, since Venus is not a good place for
astronomers. 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. Evidently this cavity fulfils an
important function in draining 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 satellite. 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 circumstances must have occurred in the history of the earth to
have led to the breaking away of so unusual a fraction of the mass. The
explanation proposed by Sir George Darwin, which is still regarded as
most probable, is that a resonance in period occurred between the solar
tides and the natural free period of vibration of the globe of the

earth. The tidal deformation of the earth thus grew to large amplitude,
ending in a cataclysm 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 possibility 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 oxygen. There is no ocean; the surface
markings represent, not sea and land, but red desert and darker ground
which is perhaps moist and fertile. A conspicuous feature 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 different 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 more easily.[29] Still clearer detail is

obtained by photographing with the long infra-red waves.
Great attention has lately been paid to the determination of the
temperature of the surface of Mars; it is possible to find this by
direct measurement of the heat radiated to us from different parts of
the surface. The results, though in many respects informative, are
scarcely accurate and accordant enough to give a definite idea of the
climatology. Naturally the temperature 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 temperature 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 conditions.
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 attentive 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 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 maintained 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 magnificent
contribution to our knowledge of Mars; but few would follow him all
the way on the more picturesque side of his conclusions.[30] 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 contemporaneously.

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 surprised 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
spectroscope 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
frequently the development has taken another turn and resulted in two
suns closely associated. We may probably rule out the possibility
of planets in double stars. Not only is there a difficulty in

ascribing to them permanent orbits under the more complicated field
of gravitation, 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 rotation. 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. According 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 theoretical study of rotating masses of
gas; the work presents 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.

By elimination of alternatives it appears that a configuration
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
distortion 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 occurrence, 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 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 million 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 expedition we shall find it necessary to reject many
apparently eligible mansions on points of detail. Trivial circumstances
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? Assuming 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.

[28]
A very much larger radius of space ( light
years) has recently been proposed by Hubble; but the basis of his
calculation, though concerned 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.


[29]
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.


[30]
Mars is not seen under favourable conditions except
from low latitudes and high altitudes. Astronomers who have not
these advantages are reluctant to form a decided opinion on the many
controversial points that have arisen.




Chapter IX
THE QUANTUM THEORY


The Origin of the Trouble. Nowadays whenever enthusiasts 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 attitude is often one of naïve 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 insistent. There is something radically wrong with
the present 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  which
crops up continually in a wide range of experiments. In one sense
we know just what  is, because there are a variety of ways of
measuring it;  is

That will (rightly) suggest to you that  is something very small;
but the most important information is contained 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  is of the nature of energy
multiplied by time.
Now in practical life it does not often occur to us to 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 multiplies 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 1915 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 class of physical conceptions)

all the quantities prominent in pre-relativity physics refer to the
three-dimensional 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 developments of dynamics have been made on this basis. I
need only refer to the standard treatise on Analytical Dynamics by your
own (Edinburgh) Professor[31], 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 developments 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 1917, it is necessary to have plunged
rather deeply into the Hamiltonian 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 headway.
But neither the absolute importance of action in the four-dimensional
world, nor its earlier prominence in Hamiltonian dynamics, prepares

us for the discovery that a particular lump of it can have a
special importance. And yet a lump of standard size

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 fundamental 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 deadlock 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 constitution 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 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
proceeding 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 discontinuously. It sends
out a long train of waves and then stops. It has to be restarted by
some kind of stimulation 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
. 
This energy is, as we have seen, marked by a distinctive
period . We have thus the two
ingredients necessary for a natural lump of action. Multiply them
together, and we obtain .
That is the quantity .
The remarkable law of Nature is that we are continually 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 forms of radiation. It applies to light

absorbed by an atom as well as to light emitted, the absorption being
discontinuous also. Evidently  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 hydrogen
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 remarked that this is only possible if indivisible
units are being shuffled. If the cards can be torn into smaller
and smaller pieces without limit there is no end to the process of

shuffling. The indivisible units in the shuffling of energy are the
quanta. By radiation absorption 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 definiteness
of thermodynamical equilibrium which first put Prof. Max Planck on
the track of the quantum; and the magnitude of  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 ; then we turned our attention to a
case in which the energy was obviously dissolving away through space,
viz. light-waves, and immediately  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 -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 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 . Just
 
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 whatever
comes into it shall be just . 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 amount in the box reaches a whole

quantum, it enters the atom. In the second the atom uses the small
fraction 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 provides the energy of

these explosions, but the phenomenon 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 ejections 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 -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 exposure 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 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 conceptions.
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 photoelectric 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 interference and diffraction it
seems impossible to deny this uniformity, 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.

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 experimental
knowledge of quanta. The mysterious quantity  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 . The intermediate orbits are excluded because they would
involve fractions of , and  cannot be divided.
But there is one relaxation. When wave-energy is sent out from or

taken into the atom, the amount and period must correspond exactly
to . But as regards its internal arrangements the atom has no
objection to , , , etc.; it only insists that
fractions shall be excluded. That is why there are many alternative
orbits for the electron corresponding to different integral multipliers
of . We call these multipliers quantum numbers, 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
; 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

periods of its electronic circulation are ignored and the period of
the aether-waves is settled not by any picturable mechanism but by the
seemingly artificial -rule. 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 . If this
unmechanical process of emission 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  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 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 , so that there is no
fear of a vicious circle when we use the measured energies to test the
 rule.[32] Incidentally this experiment provides another argument
against the collection-box theory. Small contributions of energy are
not thankfully received, and electrons which offer anything 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 follow 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, magnetism, X-rays, radioactivity, and so on. We must leave this
and return to a general consideration of the relation between classical
laws and quantum laws. For at least fifteen years we have used
classical laws and quantum 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 inconsistent with those laws. The energies of the orbits

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 condenses the light of
a star on a film of sodium in a photoelectric 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
electrons 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 classical theory on Mondays, Wednesday and Fridays, and the quantum
theory on Tuesdays, Thursdays and Saturdays. Perhaps that ought to make
us feel a little sympathetic towards the man whose philosophy of the
universe 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 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 uniform 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 especially 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.

This is the famous Correspondence Principle enunciated 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 converge to the
classical laws, and predict just what the classical laws would predict.
For an example, take a hydrogen atom with its electron 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 continuously, 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  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 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 continuous
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 disagreement 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 conceptions, 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 conceptions 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 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—something
ready to merge into space and time when the state numbers are infinite.
And simultaneously the interactions 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 quantum numbers; consequently our first survey of the world has
stumbled across the classical laws and our present conceptions of the
world consist of those entities 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 survive 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
describing a circular or elliptic orbit. This is only a model; the real
atom contains nothing of the sort. The real atom contains something

which it has not entered into the mind of man to conceive, which has,
however, been described symbolically by Schrödinger. This “something”
is spread about in a manner by no means comparable 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 Schrödinger’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 Schrödinger’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 Schrödinger’s
mist like a genie emerging from his bottle.

[31]
Prof. E. T. Whittaker.


[32]
Since the  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.




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 conflict 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
100-inch reflector at Mt. Wilson. The diffraction pattern shows that
each emission from each atom must be filling the whole mirror. 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 100-inch mirror.
But if this same star-light without any artificial concentration 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 the 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 quantum; 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 difficult 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 spreading 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
compromise 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

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 commonly 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 experimentally 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 aperture—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 circumstances 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 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 produced 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 phenomena 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.[33]
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 happens the change from the classical mechanics, though
far-reaching in principle, gave only insignificant corrections 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 universe. 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 material

breaks up (ionises) the outer electron systems 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 classical 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![34]

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 particular

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 spectroscopists 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
spectroscopists 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.[35]
The time was evidently ripe for the birth of a new theory. The
situation then prevailing may be summarised 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 must be made equal to , or sometimes to an integral

multiple of .
(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 governed by classical electrodynamics whilst its
radiation would be governed by the  rule. But in classical
electrodynamics the acceleration and the radiation are indissolubly
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
complete system of more general laws must not be hampered 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
spectroscopically. Thus a reconstruction was required not only to
remove logical objections but to meet the urgent demands 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

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, Schrödinger. 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 reaction
towards classical ideas, but that was probably a false impression.
You will realise the anarchy of this branch of physics when three
successive pretenders 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 contemplate; 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 physical
world lies the mystic formula

We do not yet understand that; probably if we could understand it we
should not think it so fundamental. 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
 rule, but to many related phenomena which the older formulation could
not treat.
On the right-hand side, besides  (the atom of action) and the
merely numerical factor , there appears  (the square root
of -1) 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  in their formulae was a kind of
signal to look out for waves or oscillations. The right-hand side
contains nothing unusual, but the left-hand side baffles imagination.
We call  and  co-ordinates and momenta, 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  is so ill-behaved as to be unequal to .
It is here that the three theories differ most essentially. Obviously
 and  cannot represent simple numerical measures, for then
 would be zero. For Schrödinger  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  is a matrix—not one quantity, nor several
quantities, but an infinite number of quantities arranged in systematic
array. For Dirac  is a symbol without any kind of numerical
interpretation; 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 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 experience, 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 physicist 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 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 necessary 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  and  individually have
no arithmetical interpretation, the combination  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 Schrödinger has rent much
of the mystery from the ’s and ’s 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.
Schrödinger’s theory is now enjoying the full tide of popularity,
partly because of intrinsic merit, but also, 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 admittance except on business”, and
particularly to warn the doorkeeper to keep out prying philosophers.
I will, however, content myself with the protest that, whilst
Schrödinger’s theory is guiding us to sound and rapid progress in many
of the mathematical problems confronting us and is indispensable in its
practical utility, I do not see the least likelihood that his ideas
will survive long in their present form.

Outline of Schrödinger’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 Brobdingnagian point of view. Such a disturbed area is
recognised 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. Moreover the speed may be modified by local conditions. This
modification is the counterpart in Schrödinger’s theory of a field
of force in classical physics. It will readily be understood that

if we are to reduce all phenomena to a propagation of waves, then
the influence of a body on phenomena in its neighbourhood (commonly
described as the field of force caused by its presence) must consist in
a modification of the propagation 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 experience. 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 constant relation between period and
energy which we have called the  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 , the equation expressing the law of propagation
of the ripples will contain a term in . 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 , since it emerges into our gross
experience by the same method that  does. If  produces
those phenomena which make us recognise it as energy, the spurious
 will produce similar phenomena corresponding to a spurious kind
of energy. Clearly the latter will 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  and the spurious or
potential  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 particle 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
observed 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 mechanics on this rather
fantastic basis. Its distinctive merits begin to be apparent when
we deal with phenomena 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 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 concentrated particle. That
is not Schrödinger’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 Schrödinger 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 representing 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  which it provides in the
equation of wave-propagation for the electron. The waves travelling
in accordance with this equation constitute Schrödinger’s equivalent
for the electron; and any solution 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 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 possible 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 these energies by a wave-theory instead of by an
inexplicable mathematical rule. Further, when applied to more complex
atoms Schrödinger’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)
simultaneously; but there is nothing to prevent waves of two different
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 oscillations too rapid to affect our gross senses;
but their beats 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.
Heterodyning 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 different lines in the spectra, but it also
predicts their intensities—a problem which the older quantum theory
had no means of tackling. It should, however, be understood 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 oscillating 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 ,
and properly speaking we should regard it as an elementary indefinable
of the wave-theory. But can we give it a classical interpretation 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  in that region. So that
if  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 -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.[36]

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 therefore that a particle definitely localised as a moving point
is strictly 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 terminated 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 0.1 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-lengths the range is

increased to 10 per cent.; 10 of the longest and 11 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 Schrödinger’s picture of the hydrogen 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 permissive, 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 1 per cent., a series of energies
ranging over 1 per cent, can be treated as one definite energy.
Hitherto I have only considered the waves corresponding to one
electron; now suppose that we have a problem involving two electrons.
How shall they be represented? “Surely, that is simple enough! We
have only to take two stormy areas instead of one.” I am afraid 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 Schrödinger waves there represent a probability belonging to a
second electron. Each electron wants the whole of three-dimensional
space for its waves; so Schrödinger 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 Schrödinger has given us what
seemed to be a comprehensible 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 configuration space had
a close correspondence with physical space, suggesting some degree of
objective reality of the waves. Schrödinger’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
Schrödinger waves, the frequency of the waves represents the number of
systems of a given kind. The wave-equation is formulated 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 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 1, 2, 3, 4, ..., but can never be 2¾
for 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 fulfilled; 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.

Suppose for example that we wish to know the position 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 ¹⁄₁₀₀₀₀ 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 unexpected
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 something 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 systematic, 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 discoveries are made; and new entities, coming
out of the unknown, 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
position and velocity of an electron beyond a limited number of places
of decimals is an attempt to describe something 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 conceptions 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
determination 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  is
a co-ordinate and  the corresponding momentum, the necessary

uncertainty of our knowledge of  multiplied by the uncertainty of
 is of the order of magnitude of the quantum constant .
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 universe 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 receives 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 ascertaining the position is that we disturb the
electron in an incalculable way which will prevent our subsequently
ascertaining how much momentum it had. However, we shall be able to
ascertain the momentum with an uncertainty 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 small 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 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
proportional 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 quantum 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. Schrödinger’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 position 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 Schrödinger’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 Schrödinger’s theory;
it is the other way about. The principle of indeterminacy, like the
principle of relativity, represents 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.

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
astronomy has not been based on this spasmodic kind of moon. In the
scientific world (which has to fulfil functions 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 experience. On the other hand we cannot limit the
description 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 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 challenged. Similarly,
when we say that the distance of Sirius is 50 billion miles we are not
giving a merely conventional interpretation to its measured parallax;
we intend to give it the same status in knowledge as if someone 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
measurers, 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 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 prediction of the future will include the unknowable elements of
the past. I think it was Heisenberg who said, “The question whether
from a complete knowledge of the past we can predict the future,
does not arise because a complete knowledge of the past involves a

self-contradiction.”
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 expense of an addition to
ignorance. It is hard to empty the well of Truth with a leaky bucket.

[33]
The evidence is much stronger now than when the lectures
were delivered.


[34]
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.


[35]
Each orbit or state of the atom requires three (or, for
later refinements, 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.


[36]
The probability is often stated to be proportional
to instead of , as assumed above. The whole
interpretation is very obscure, but it 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 
is obtained by introducing two symmetrical systems of -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 necessarily 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.




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; doubtless 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 remember that as an
impecunious schoolboy I used to read attractive articles on how
to construct wonderful contrivances 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
relations and relata. The relations unite the relata; the
relata are the meeting points of the relations. The one 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 transactions, 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 numerical 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 relations between the relata
but some kind of relation of likeness between some of the relations.
The slightest 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 genealogist, 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 with 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

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  and a relation  radiating from
it. Now step to a contiguous relatum  and pick out the “like”
relation . Go on to another contiguous relatum  and pick
out the relation  which is like . (Note that since 
is farther from  than from , the relation at  which is
like  is not so definite as the relation which is like .)
Step by step we may make the comparison round a route  which
returns to the starting-point. There is nothing to ensure that the
final relation  which has, so to speak, been carried round the

circuit will be the relation  with which we originally started.
We have now two relations ,  radiating from the first
relatum, their difference being connected with a certain circuit in
the world . The loose ends of the relations  and 
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 
round the circuit. As we vary the circuit and the original relation, so
the change  varies; and the next step is to find a mathematical
formula expressing 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 combinations,
the required mathematical formula contains  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 degrees and kinds of unlikeness; but we have found a means of

specifying the kind of unlikeness of  and  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 monomarks used for the
identification of relata. This, however, 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 accordance 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
duplicates 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 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 structure in a
symmetrical manner and by ignoring others we reduce the really
important measures to 16.[37] 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 (). We derive from them space, time, and the
world-curvatures representing the mechanical properties of matter, viz.
momentum, energy, stress, etc.
Antisymmetrical coefficients (6). Out of these we construct
Electromagnetism. They are the three components 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 feature; the
field laws—conservation of energy, mass, momentum and of electric

charge, the law of gravitation, Maxwell’s equations—are not controlling
laws.[38] 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 conservation 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. For every

item on the credit side an equal item appeared somewhere else on the
debit side. “Ha!” said the Bursar, “I have discovered one of the great
laws controlling 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 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. Commonly
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 conservation 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
infallibility 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.

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 distribution 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 properties 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 combination;
the significance of this process is not in inorganic nature. Its
significance is that it corresponds 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 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.[39] Ultimately 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 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 , we could presumably
construct from the same lumber an appliance for measuring . 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 
reads 5 units no human mind is aware of it. Neither  nor the
appliance for measuring  have any interaction with consciousness.
Thus the responsibility for the fact that the scheme of the scientific
world includes mass but excludes  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 qualitative discrimination of likeness and
unlikeness.
Secondly, the laws which we have hitherto regarded 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
different 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: (1) identical laws,
(2) statistical laws, (3) transcendental 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 uniformity 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 mathematical 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 probability; but we
do not yet seem able to find a place for this in any of the current

conceptions of the world substratum.
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 implied in the scheme of world-building. They are concerned
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 making 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
indirectly 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 transcendental 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 intrinsic system of government. But I scarcely know what to think.
We must not assume that the possible developments 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 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 imposed as a frame for natural phenomena.
Perhaps our final conclusion as to the world of physics will resemble
Kronecker’s view of pure mathematics.
“God made the integers, all else is the work of man.”[40]

[37]
Mathematically we contract the original tensor of the
fourth rank to one of the second rank.


[38]
One law commonly grouped with these, viz. the law of
ponderomotive force of the electric field, is not included. It seems
to be impossible to get at the origin of this law without tackling
electron structure which is beyond the scope of our present exercise in
world-building.


[39]
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.


[40]
Die ganzen Zahlen hat Gott gemacht; alles anderes ist
Menschenwerk.




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 conception. Other advances of knowledge
can be accommodated by that very useful aid to comprehension—“like
this only more so”. For example, if you think of something 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”; but 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 conglomeration 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 beginning of the alphabet. We have suffered, and we
still suffer, from expectations that electrons and quanta must 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
philosophy 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. Whitehead’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 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 synonymous. 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 complex 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 chemical 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

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 60°.” Now the hillside fades out of
the problem and an angle of 60° takes its place. What is 60°? There

is no need to struggle with mystical conceptions of direction; 60° 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 directly 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 problem, 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 observation of any kind

of coincidence—or, as it is usually expressed in the language of
the general relativity 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 measurement.
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 described 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 apprehension, 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 calipers. Similarly the greyish black appearance in our
mental impression is replaced in exact science by the readings 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 correspondence—
(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;
(c) 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”; 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 particular 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. It is this connectivity
of pointer readings, expressed by physical laws, which supplies the
continuous background that any realistic problem demands.

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 experiment. How can this identity be expressed in a
description 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 continually 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
continuity; the elephant must be watched all the way from the scales
to the hillside. The elephant, we must remember, 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 outline of the image, the observer
satisfied himself that he was following 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.[41] 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
meaningless in physics. We use instead the law of conservation 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 constancy of certain properties of the elephant is not
assumed as self-evident from its sameness, but is an inference from
experimental and theoretical laws relating 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 comprehensive. A knowledge of the response of all kinds
of objects—weighing-machines and other indicators—would determine
completely its relation to its environment, leaving only its inner
un-get-atable nature undetermined. 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 mathematics is so closely

adapted to the physical conceptions. It is not so in all subjects. For
example, we may admire the triumph of patience of the mathematician
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 different methods of finding the lunar orbit. But
by the use of tensors the mathematical physicist precisely describes
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 external 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
on 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
discovers 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 down thought as an illusion—some perverse interpretation of the

interplay of the physical entities that he has found. Or if he sees
the folly of calling the most undoubted element of our experience an
illusion, he will have to face the tremendous question, How can this
collection 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 “concrete”
nature inconsistent with thought, and then to wonder where the thought
comes from. We have dismissed 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 insight 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 more specialised attributes of consciousness.[42] 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 expounded 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 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
mathematical 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 relations 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 matter which.... (MEM. Explain “matter”.) On second
thoughts I will leave the rest of the description as “an exercise to
the reader” since it 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 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 achievements 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 explanation 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 pursue the

question, What is Mr. X? It is not disposed to admit that its elaborate
structure of a physical universe is “The House that Mr. X built”.



Fig. 8


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 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 justified. That matter
in some indirect way comes within the purview of Mr. X’s mind is not
a fact of any utility 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 something 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[43] its

subject-matter has undergone, and indeed retains evidences 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 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 properties when 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 certified 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 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 inorganic 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; 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 inquiry
must take place in the domain of my own consciousness. I find there
certain data purporting to relate to this unknown X; and I can (by
using powers 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 inference 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 scientific 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 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 investigation 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.[44] Thus there is no
opportunity for acquiring by any physical 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
mathematical equations forming a deterministic scheme. This knowledge
has all been inferred and it was bound therefore 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 environment 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
developing 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 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?

[41]
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.


[42]
For example, we should most of us assume (hypothetically)
that the dynamical quality of the world referred to in chapter V
is characteristic 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 “becoming” 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.


[43]
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 experiment.


[44]
The solitary exception is, I believe, Dirac’s
generalisation which introduces -numbers (p. 210). There is as yet
no approach to a general system of inference on a non-numerical basis.




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 outwards 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. 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 formulate 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 chapter we are taking 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 scientific. 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 superciliously on the comparative
disorganisation of our knowledge 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.


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 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 knowledge of the physical world. It is necessary to
keep reminding ourselves that all knowledge of our environment 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.[45] In the central clearing station the incoming
messages are sorted and decoded, partly by instinctive image-building
inherited 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 configuration of brain-matter. Again Bertrand Russell
writes—[46]

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, 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 admitting a
more abstract kind of dissection.
This might give rise to some difficulty if we were postulating complete
identity of mind-stuff with consciousness. 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 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
description. The term “mind-stuff” might well be amended; but neutral
stuff seems to be the wrong kind of amendment. 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.

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 deliberate. 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 omnipresent spirit to whom we trace the sacred things
of consciousness, would there not be even graver apprehension? 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 controlling 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
misinterpretation. It is, of course, possible to obtain consistent use
of the word “reality” by adopting a conventional 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 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 element 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 conscious 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 untrustworthy. The impression of externality is
equally strong in the world that comes to me in dreams; the dreamworld
is less rational, but that might be used as an argument in favour of
its externality as showing its dissociation 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 knowledge 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 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 hallucination 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 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 rôle, 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 significance
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 competition rather than on other problems which
our experience might suggest to us. Just what kind of 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 scientific
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 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
responsibilities 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

plane, are just as much facing the hard facts of experience as those
who start from consciousness as a device for reading the indications of
spectroscopes and micrometers.

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 gravitation; 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 indistinguishable by any
possible experimental test. Starting on a journey round the spherical
world we come across a group , and then after going half round we
come to an exactly similar group  indistinguishable by any test;
another half circle again brings us to an exactly similar group, which,
however, we decide is the original group . 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 conclusion
that this happened when we reached ; everything 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 “spherical”. But which is

the real truth? Disregard the fact that I introduced  and 
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  and
 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, sometimes with a rush, caught
in a cul-de-sac of metastability, hesitating before “forbidden
passages”. Behind it all the quantum  regulates each change with
mathematical 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 conventional way of depicting a particular change of
state 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 particularly 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 belonging 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 fundamental entities of physics to translate it into
“Jabberwocky”; provided all numbers—all metrical attributes—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”.

[45]
I mean, resemble in intrinsic nature. It is true (as
Bertrand Russell has emphasised) that the symbolic description of
structure will be identical for the 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 indifferent as to which of the two we
imagine ourselves to be discussing.


[46]
Analysis of Matter, p. 320.




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 extravagant 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 configurations 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 contrary 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

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. Determinism has dropped out altogether in the
latest formulations 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 determinism. 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.[47]
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
hurried 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
elements—unpredictable not merely because it is impracticable 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 freewill. 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 experimental 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 necessary 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
inconsistent with a strictly causal scheme. How can I cause an event in
the absolute future, if the future was predetermined 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 alterations in its
future state. Thus in primary physics, which knows nothing of time’s
arrow, there is no discrimination of cause and effect; but events are
connected by a 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 confident 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 secondary 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 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 ultimately on a scheme of completely
deterministic laws? I think there are two reasons—
(1) 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 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 involves 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 messages 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.

(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 progress 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 11 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 VII 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  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 excluded.
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 arrangements.
It cannot be refined to an accuracy beyond the 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
exceedingly 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 phenomena 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 company. The average law is
so trustworthy that it may be considered predestined that half the
children now born will survive the age of  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 argument 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 predictable because they are averages, irrespective of the type of
government of the phenomena underlying them.

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 quantum 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 determining 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
predetermined. 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 of a particular throw is not marked in the

dice; nevertheless it is strictly causal (apart perhaps from the human
element involved in throwing the dice) being determined by the external
influences which are concerned. Our own position at this stage is that
future developments 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
appeal 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.[48] 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 microscopic elements of
the physical world—individual atoms, electrons, quanta—do not make
definite predictions as to what the individual will do next. I am

here speaking of the laws that have been actually discovered 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
successful predictions hitherto attributed to causality are traceable
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 doctrine, 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 investigation does
not lead to knowledge of the intrinsic nature of things. “Whenever we
state the properties of a body in terms of physical quantities we are

imparting knowledge 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 indicators, 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 anything 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
substitution of probability for certainty makes little difference;
it adds only a negligible haziness to the world. But 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 expressed 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 Schrödinger tries to picture the atom—as a
wave centre of his probability entity .
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 fundamental point in Schrödinger’s theory that his probabilities
are not to be replaced in that way. When his  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  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 think this means that the spread of  is
not a symbol for uncertainty arising 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 would 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. Schrödinger’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-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 indeterminacy 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 determination 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. It 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 becomes knowable.
The classical view that a particle necessarily has a definite (but not
necessarily knowable) velocity now, 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 phenomenon 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 momentum 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 contemplated 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 eliminated 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 anticipated.
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. 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 Newtonian 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 admitted 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 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 determinism 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 atmosphere;
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 appears to be no longer any

obstacle to this emancipation.
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 phenomenon occurring simultaneously with
the physical phenomena. But this assumes that the result of applying
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 undetermined. 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 consequences 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. 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 background. 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.
something 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 behaviour
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 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 relativity? 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 interference 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-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 improbable 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
undetermined 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 of “probability in the light of certain given information”, and

the probability alters according to the extent of the information. It
is, I think, one of the most unsatisfactory 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 behaviour 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 based
on the hypothesis of random disconnection. 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.

[47]
A few days after the course of lectures was completed,
Einstein wrote in his message on the Newton Centenary, “It is only
in the quantum 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.)


[48]
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.




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 ,  be given multiples of
, where  and  are prescribed, the
equations in question determine  and , and thence, by (9)
the value of . Thus we find


where  has been written for  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 investigations give considerable insight
into the incipient stages of wave-formation”.
On another occasion the same subject of “Generation 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

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
storehouse grew bolder. Perhaps we recalled the phrase “rippling
laughter”. Waves—ripples—laughter—gladness—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 single-minded purpose to ensuring that 

shall be equal to .
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 untruth “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 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 surrender 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 materialist does not advocate, or at any rate
does not practice, the first alternative; therefore I assume the second
alternative, 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.

But I cannot think there is much amiss with our appreciation 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 essential 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 knowledge 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
consideration 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 knowledge 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 experience 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 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 consciousness 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 inquiry 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. But if some

prophetic voice had warned us that it was an illusion and therefore
we had not troubled to investigate 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 environment 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 everyday 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 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 professes 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 carrying 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;







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 scientists
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 afterwards 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 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
kickable, 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 ′s and
′s 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-ordinates.
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 this sanction, for the pursuit of science springs from a

striving which the mind is impelled to follow, a questioning that will
not be suppressed. Whether in the intellectual pursuits of science or
in the mystical pursuits 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. Experience 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

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 ourselves 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 attributing 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 dependent 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 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 we 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 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 alternatives.
Either there are no absolute values, so that the sanctions of the
inward monitor in our consciousness are the final court of appeal
beyond which it is idle to inquire. 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 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 indications 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 fundamental position instead of representing it
as an inessential 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. Through fourteen chapters you have followed 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 repudiate 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 fundamental, we can proceed to discuss the
various criticisms 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 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 significance 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.[49]
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 misunderstood 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 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 misrepresentations 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 biological
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 consultation 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 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 conviction 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 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 mathematical 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 convictions 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 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
asserting 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
represented in this inadequate way. One of the greatest puzzles of
science is to discover why protons and electrons 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 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 basis 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 consciousness, 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
temperament 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 importance 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 experiences 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 concerned 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.

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 relationships, 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 incorporated 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 conception that was unfolded nearly two thousand years
ago.

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 violently 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
direction 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.

[49]
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.




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 foolishness. 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
unnecessarily 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 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 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 65 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 reasoning would
probably run somewhat as follows. By reasoning we are sometimes able
to predict events afterwards confirmed by observation; the mental
processes follow a sequence ending in a conception which anticipates
a subsequent perception. We may call such 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 constitute “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 observational 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.

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 theologian 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 theologians 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 superstition. 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 rôle
allowed 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 consciousness 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 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 interpretation 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 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 philosophy. 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 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 nowhere 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 it will require a skilful drawing of the

boundary line to frustrate the development of a conflict here.[50]
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 enthusiastically 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 proceeds.

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 but 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.

[50]
This difficulty is evidently connected with the dual
entry of time into our experience to which I have so often referred.




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), 111, 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 () 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, 341
Commonsense knowledge, 16
Companion of Sirius, 203
Comparability of relations, 232
Compensation of errors, 12

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 survey,
154
Curvature of space-time, 119, 127,
157; coefficients of, 120, 155
Cyclic method of physics, 260, 277,
348
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; inscrutable
nature of, 81; macroscopic
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, 111, 185, 203
Einstein’s law of gravitation, 120,
139, 151, 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 mathematician,
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

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; reality
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, 111,
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; explanation
of, 138, 145
Greenland, 117
Gross appliances, survey with, 154
Growth, idea of, 87
Group velocity, 213
, 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
 (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

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 properties,
275
Michelson-Morley experiment, 5, 11
Microscopic analysis, reaction from,
103
Milky Way, 163
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, 11
Models, 198, 209, 344
Molecular bombardment, 113, 131
Momentum, 153, 208, 223, 239, 262
Monomarks, 231
Moon, origin of, 171
Morley, E. W., 5
Motion, law of, 123
Multiplicationist, 86
Multiplicity of space and time
frames, 20, 35, 61
Myself, 42, 53
Mysticism, defence of, 323; religious,
338
Nautical Almanac, 150
Nebulae, 165
Nebular observers, 9, 12
Neptune, 49
Neutral stuff, 280
Neutral wedge, 48
New quantum theory, 206

Newton, 111, 122, 201; quotation
from, 111
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 ’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; insufficiency
of, 107
Primary scheme of physics, 76, 129,
295
Principal curvature, 120, 139
Principia, 4
Principle of 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 (), 216, 305
Pure mathematician, 161, 337, 347
Purpose, 105
-numbers, 208, 270
Quantum, 184; size of, 200
Quantum laws, 193
Quantum numbers, 191, 205
Quest of the absolute, 26, 122; of
science, 110, 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, 111
Nursery Rhymes, 64, 70, 262
Omar Khayyam, 64, 293
O’Shaughnessy, A., 325
Russell, Bertrand, 160, 278

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 magnetic
field, 22; of distance, 25;
of pound sterling, 26; of Now
(simultaneity), 46, 61; of acceleration,
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
Schrödinger’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 lines, 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; displacement
of, 121, 166
Spherical curvature, radius of,
140
Spherical space, 82, 166, 289; radius
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 interference
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

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 recognition
of, 51, 100; time’s arrow,
69; infinity of, 83; summary of
conclusions, 101; time-triangles,
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




TRANSCRIBER’S NOTES
 On page 32, the reference to Einstein
has been replaced by Dickens as shown in the Index. This quote
appears in “Martin Chuzzlewit” published in 1843.



*** END OF THE PROJECT GUTENBERG EBOOK THE NATURE OF THE PHYSICAL WORLD ***


Updated editions will replace the previous one—the old editions will
be renamed.

Creating the works from print editions not protected by U.S. copyright
law means that no one owns a United States copyright in these works,
so the Foundation (and you!) can copy and distribute it in the United
States without permission and without paying copyright
royalties. Special rules, set forth in the General Terms of Use part
of this license, apply to copying and distributing Project
Gutenberg™ electronic works to protect the PROJECT GUTENBERG™
concept and trademark. Project Gutenberg is a registered trademark,
and may not be used if you charge for an eBook, except by following
the terms of the trademark license, including paying royalties for use
of the Project Gutenberg trademark. If you do not charge anything for
copies of this eBook, complying with the trademark license is very
easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project
Gutenberg eBooks may be modified and printed and given away—you may
do practically ANYTHING in the United States with eBooks not protected
by U.S. copyright law. Redistribution is subject to the trademark
license, especially commercial redistribution.

START: FULL LICENSE
THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase “Project
Gutenberg”), you agree to comply with all the terms of the Full
Project Gutenberg™ License available with this file or online at
www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing Project Gutenberg™
electronic works

1.A. By reading or using any part of this Project Gutenberg™
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/copyright) agreement. If you do not agree to abide by all
the terms of this agreement, you must cease using and return or
destroy all copies of Project Gutenberg™ electronic works in your
possession. If you paid a fee for obtaining a copy of or access to a
Project Gutenberg™ electronic work and you do not agree to be bound
by the terms of this agreement, you may obtain a refund from the person
or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement. There are a few
things that you can do with most Project Gutenberg™ electronic works
even without complying with the full terms of this agreement. See
paragraph 1.C below. There are a lot of things you can do with Project
Gutenberg™ electronic works if you follow the terms of this
agreement and help preserve free future access to Project Gutenberg™
electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the
Foundation” or PGLAF), owns a compilation copyright in the collection
of Project Gutenberg™ electronic works. Nearly all the individual
works in the collection are in the public domain in the United
States. If an individual work is unprotected by copyright law in the
United States and you are located in the United States, we do not
claim a right to prevent you from copying, distributing, performing,
displaying or creating derivative works based on the work as long as
all references to Project Gutenberg are removed. Of course, we hope
that you will support the Project Gutenberg™ mission of promoting
free access to electronic works by freely sharing Project Gutenberg™
works in compliance with the terms of this agreement for keeping the
Project Gutenberg™ name associated with the work. You can easily
comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg™ License when
you share it without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are
in a constant state of change. If you are outside the United States,
check the laws of your country in addition to the terms of this
agreement before downloading, copying, displaying, performing,
distributing or creating derivative works based on this work or any
other Project Gutenberg™ work. The Foundation makes no
representations concerning the copyright status of any work in any
country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other
immediate access to, the full Project Gutenberg™ License must appear
prominently whenever any copy of a Project Gutenberg™ work (any work
on which the phrase “Project Gutenberg” appears, or with which the
phrase “Project Gutenberg” is associated) is accessed, displayed,
performed, viewed, copied or distributed:

    This eBook is for the use of anyone anywhere in the United States and most
    other parts of the world at no cost and with almost no restrictions
    whatsoever. You may copy it, give it away or re-use it under the terms
    of the Project Gutenberg License included with this eBook or online
    at www.gutenberg.org. If you
    are not located in the United States, you will have to check the laws
    of the country where you are located before using this eBook.
  
1.E.2. If an individual Project Gutenberg™ electronic work is
derived from texts not protected by U.S. copyright law (does not
contain a notice indicating that it is posted with permission of the
copyright holder), the work can be copied and distributed to anyone in
the United States without paying any fees or charges. If you are
redistributing or providing access to a work with the phrase “Project
Gutenberg” associated with or appearing on the work, you must comply
either with the requirements of paragraphs 1.E.1 through 1.E.7 or
obtain permission for the use of the work and the Project Gutenberg™
trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg™ electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any
additional terms imposed by the copyright holder. Additional terms
will be linked to the Project Gutenberg™ License for all works
posted with the permission of the copyright holder found at the
beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg™
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg™.

1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg™ License.

1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including
any word processing or hypertext form. However, if you provide access
to or distribute copies of a Project Gutenberg™ work in a format
other than “Plain Vanilla ASCII” or other format used in the official
version posted on the official Project Gutenberg™ website
(www.gutenberg.org), you must, at no additional cost, fee or expense
to the user, provide a copy, a means of exporting a copy, or a means
of obtaining a copy upon request, of the work in its original “Plain
Vanilla ASCII” or other form. Any alternate format must include the
full Project Gutenberg™ License as specified in paragraph 1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg™ works
unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg™ electronic works
provided that:

• You pay a royalty fee of 20% of the gross profits you derive from
        the use of Project Gutenberg™ works calculated using the method
        you already use to calculate your applicable taxes. The fee is owed
        to the owner of the Project Gutenberg™ trademark, but he has
        agreed to donate royalties under this paragraph to the Project
        Gutenberg Literary Archive Foundation. Royalty payments must be paid
        within 60 days following each date on which you prepare (or are
        legally required to prepare) your periodic tax returns. Royalty
        payments should be clearly marked as such and sent to the Project
        Gutenberg Literary Archive Foundation at the address specified in
        Section 4, “Information about donations to the Project Gutenberg
        Literary Archive Foundation.”
    
• You provide a full refund of any money paid by a user who notifies
        you in writing (or by e-mail) within 30 days of receipt that s/he
        does not agree to the terms of the full Project Gutenberg™
        License. You must require such a user to return or destroy all
        copies of the works possessed in a physical medium and discontinue
        all use of and all access to other copies of Project Gutenberg™
        works.
    
• You provide, in accordance with paragraph 1.F.3, a full refund of
        any money paid for a work or a replacement copy, if a defect in the
        electronic work is discovered and reported to you within 90 days of
        receipt of the work.
    
• You comply with all other terms of this agreement for free
        distribution of Project Gutenberg™ works.
    

1.E.9. If you wish to charge a fee or distribute a Project
Gutenberg™ electronic work or group of works on different terms than
are set forth in this agreement, you must obtain permission in writing
from the Project Gutenberg Literary Archive Foundation, the manager of
the Project Gutenberg™ trademark. Contact the Foundation as set
forth in Section 3 below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
works not protected by U.S. copyright law in creating the Project
Gutenberg™ collection. Despite these efforts, Project Gutenberg™
electronic works, and the medium on which they may be stored, may
contain “Defects,” such as, but not limited to, incomplete, inaccurate
or corrupt data, transcription errors, a copyright or other
intellectual property infringement, a defective or damaged disk or
other medium, a computer virus, or computer codes that damage or
cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right
of Replacement or Refund” described in paragraph 1.F.3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg™ trademark, and any other party distributing a Project
Gutenberg™ electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium
with your written explanation. The person or entity that provided you
with the defective work may elect to provide a replacement copy in
lieu of a refund. If you received the work electronically, the person
or entity providing it to you may choose to give you a second
opportunity to receive the work electronically in lieu of a refund. If
the second copy is also defective, you may demand a refund in writing
without further opportunities to fix the problem.

1.F.4. Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO
OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of
damages. If any disclaimer or limitation set forth in this agreement
violates the law of the state applicable to this agreement, the
agreement shall be interpreted to make the maximum disclaimer or
limitation permitted by the applicable state law. The invalidity or
unenforceability of any provision of this agreement shall not void the
remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg™ electronic works in
accordance with this agreement, and any volunteers associated with the
production, promotion and distribution of Project Gutenberg™
electronic works, harmless from all liability, costs and expenses,
including legal fees, that arise directly or indirectly from any of
the following which you do or cause to occur: (a) distribution of this
or any Project Gutenberg™ work, (b) alteration, modification, or
additions or deletions to any Project Gutenberg™ work, and (c) any
Defect you cause.

Section 2. Information about the Mission of Project Gutenberg™

Project Gutenberg™ is synonymous with the free distribution of
electronic works in formats readable by the widest variety of
computers including obsolete, old, middle-aged and new computers. It
exists because of the efforts of hundreds of volunteers and donations
from people in all walks of life.

Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg™’s
goals and ensuring that the Project Gutenberg™ collection will
remain freely available for generations to come. In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg™ and future
generations. To learn more about the Project Gutenberg Literary
Archive Foundation and how your efforts and donations can help, see
Sections 3 and 4 and the Foundation information page at www.gutenberg.org.

Section 3. Information about the Project Gutenberg Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service. The Foundation’s EIN or federal tax identification
number is 64-6221541. Contributions to the Project Gutenberg Literary
Archive Foundation are tax deductible to the full extent permitted by
U.S. federal laws and your state’s laws.

The Foundation’s business office is located at 809 North 1500 West,
Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up
to date contact information can be found at the Foundation’s website
and official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project Gutenberg
Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread
public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine-readable form accessible by the widest
array of equipment including outdated equipment. Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States. Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements. We do not solicit donations in locations
where we have not received written confirmation of compliance. To SEND
DONATIONS or determine the status of compliance for any particular state
visit www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including checks, online payments and credit card donations. To
donate, please visit: www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg™ electronic works

Professor Michael S. Hart was the originator of the Project
Gutenberg™ concept of a library of electronic works that could be
freely shared with anyone. For forty years, he produced and
distributed Project Gutenberg™ eBooks with only a loose network of
volunteer support.

Project Gutenberg™ eBooks are often created from several printed
editions, all of which are confirmed as not protected by copyright in
the U.S. unless a copyright notice is included. Thus, we do not
necessarily keep eBooks in compliance with any particular paper
edition.

Most people start at our website which has the main PG search
facility: www.gutenberg.org.

This website includes information about Project Gutenberg™,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.


