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Title: The Nature of the Physical World
Author: Eddington, Arthur Stanley (1882-1944)
Date of first publication: 1928
Edition used as base for this ebook:
   Cambridge: Cambridge University Press, 1929
Date first posted: 27 July 2013
Date last updated: 28 July 2013
Project Gutenberg Canada ebook #1097

This ebook was produced by
Iona Vaughan, Paul Ereaut, Mark Akrigg
& the Online Distributed Proofreading Canada Team
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  THE NATURE

  OF THE

  PHYSICAL WORLD


  Cambridge University Press
  Fetter Lane, London

  _Bombay, Calcutta, Madras_
  _Toronto_
  Macmillan
  _Tokyo_
  Maruzen-Kabushiki-Kaisha

  _Copyrighted in the U.S.A._
  _by_
  The Macmillan Company

  All rights reserved




  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_


  GIFFORD LECTURES

  1927


  CAMBRIDGE

  AT THE UNIVERSITY PRESS

  1929


  _First Edition_ 1928

  _Reprinted_ 1929

  PRINTED IN GREAT BRITAIN



  (Transcriber's note:- An explanation of mathematical symbols can be
  found in the main Transcriber's Note at the end of the text)


                         CONTENTS


  _Preface_                                          vii

  _Introduction_                                      xi

  _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                   178

            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




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. xviii).

I hope that the scientific chapters may be read with interest apart from
the later applications in the book; but they are not written quite on
the lines that would have been adopted had they been wholly independent.
It would not serve my purpose to give an easy 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 _A_ is not the counterpart of anything in familiar life. To
the child the letter _A_ would seem horribly abstract; so we give him a
familiar conception along with it. "_A_ was an Archer who shot at a
frog." This tides over his 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?




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 1/50,000 of the diameter of an atom; that of the nucleus is not
very much larger; an isolated proton is supposed to be much smaller
still.

Thirty years ago there was much debate over the 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" × 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 _β_
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--

                                   0=0.

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.[A] 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.




_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.

[Illustration: 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). 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.

[Illustration: Fig. 2]

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). 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.

[Illustration: 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.[B] 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). 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.

[Illustration: Fig. 4]

Since the hour-glass is absolute its two cones provide respectively an
Absolute Future and an Absolute Past for the event Here-Now. They are
separated by a wedge-shaped neutral zone which (absolutely) is neither
past nor future. The common impression that relativity turns past and
future altogether topsy-turvy is quite false. But, unlike the relative
past and future, the absolute past and future are not separated by an
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.[C]

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 a 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 a 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.[D] 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 a 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 a second, provided
that the velocity is referred to one of the frames of space and time
considered in this chapter.[E]

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 forwards 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 _β_ 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 _β_ 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 _β_ particle, smugly thinking itself at rest, pays no
attention to our goings on, and arranges itself with the usual mass,
radius and charge. It has just the standard mass of an electron,
9 . 10^{-28} grams. But mass and radius are relative quantities, and
in this case the frame 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.




_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
      Shall 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
-x, right is +x; past is -t, future is +t. 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
_n_ is the number of molecules (roughly a quadrillion) the chance of
this happening is (½)^{n}. The reason why we ignore this chance may be
seen by a rather classical illustration. If I let my fingers wander idly
over the keys of a typewriter it _might_ happen that my screed made an
intelligible sentence. If an army of monkeys were strumming on
typewriters they _might_ write all the books in the British Museum. The
chance of their doing so is decidedly more 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.[F] 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[G] 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?[H] 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.[I]

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 _A_ in one direction through the
universe and a very long way to a point _B_ in the opposite direction,
it is believed that between _A_ and _B_ there exists a linkage of the
kind indicated by a very small code number; in other words these points
reached by travelling vast distances in opposite directions would be
found experimentally to be close together. Why not? This happens when we
travel east and west on the earth. It is true that our traditional
inflexible conception of space refuses to admit it; but there was once a
traditional conception of the earth which refused to admit
circumnavigation. In our approach to the conception of spherical space
the difficult part was to destroy the inside of the hypersphere leaving
only its three-dimensional surface existing. I do not think that is so
difficult when we conceive space as a network of distances. The network
over the surface constitutes a self-supporting system of linkage which
can be 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.[oo] will
not happen until we reach A.D.[oo], and presumably in order to reach
A.D.[oo] 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 draw 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.




_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.[J] 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.[K] 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 perhaps 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--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 paints,
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[L] 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.




_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,[M] 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.[N] 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.[O]

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.[P]

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.[Q] 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 principle
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.

[Illustration: Fig. 5]

I should like to show you in a general way how it is possible for a law
controlling the curvature of empty space to determine the tracks of
particles without being supplemented by any other conditions. Two
"particles" in the four-dimensional world are shown in Fig. 5, namely
_yourself_ and _myself_. We are not empty space so there is 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.[R] 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 _A_ to _B_ and from _B_ to _C_ the sum
of your readings will be _greater_ than the reading obtained by
measuring with a scale from _A_ to _C_."

For a time-triangle the measurements must be made with an instrument
which can measure time, and the proposition then expresses that

"If you measure with a clock from _A_ to _B_ and from _B_ to _C_ the sum
of your readings will be _less_ than the reading obtained by measuring
with a clock from _A_ to _C_."

In order to measure from an event _A_ to an event _B_ with a clock you
must make an adjustment of the clock analogous to orienting a scale
along the line _AB_. What is this analogous adjustment? The purpose in
either case is to bring both _A_ and _B_ into the immediate
neighbourhood of the scale or clock. For the clock that means that after
experiencing the event _A_ it must travel with the appropriate velocity
needed to reach the locality of _B_ just at the moment that _B_ happens.
Thus the velocity of the clock is prescribed. One further point should
be noticed. After measuring with a scale from _A_ to _B_ you can turn
your scale round and measure from _B_ to _A_, obtaining the same result.
But you cannot turn a clock round, i.e. make it go backwards in time.
That is important because it decides _which_ two sides are less than the
third side. If you choose the wrong pair the enunciation of the time
proposition refers to an 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
[sr]-1. 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.




_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[S] 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
10^{20} miles. A distinction between so great a length and infinite
length is unnecessary in most of our arguments and investigations, but
it is necessary in the present chapter.

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, x trillion miles." I cannot
tell you what number to put for x 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.[T] 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.[U] 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."[V] 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 [sr]-1 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.[W]


_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;
their's is a free motion. They go on their way, as the ancients said,
like the blessed gods."[X]

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 round 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).

[Illustration: Fig. 6]

You will say that the earth must certainly get into the right position
for the eclipse next June (1927); so it cannot be free to go anywhere it
pleases. I can put that right. I hold to it that the earth goes anywhere
it pleases. The next thing is that _we_ must find out where it has been
pleased to go. The important question for us is not where the earth has
got to in the inscrutable absolute behind the phenomena, but where we
shall locate it in our conventional background of space and time. We
must take measurements of its position, for example, measurements of its
distance from the sun. In Fig. 6, SS_{1} shows the ridge in the world
which we recognise as the sun; I have drawn the earth's ridge in
duplicate (EE_{1}, EE_{2}) because I imagine it as still undecided
which track it will take. If it takes EE_{1} we lay our measuring rods
end to end down the ridges and across the valley from S_{1} to E_{1},
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
S_{1}E_{1} than the picture would lead you to expect. If the earth
chooses to go to E_{2} the curvature is less sharp; the greater radius
of curvature implies greater length of the rods. The number needed to
stretch from S_{1} to E_{2} will not be so great as the diagram at
first suggests; it will not be increased in anything like the proportion
of S_{1}E_{2} to S_{1}E_{1} 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 EE_{1} or EE_{2}. And the Superintendent of the
Nautical Almanac who published this same distance some years in advance
will claim that he correctly predicted where the earth would go.

And so you see that the earth can play truant to any extent but our
measurements will still report it in the place assigned to it by the
Nautical Almanac. The predictions of that authority pay no attention to
the vagaries of the god-like earth; they are based on what will happen
when we come to measure up the path that it has chosen. We shall measure
it with rods that adjust themselves to the curvature of the world. The
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 S_{1}E_{1} (full line) and S_{1}E_{2} (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 G_{μν}=λg_{μν}, means
nothing more nor less than that the unit of length everywhere is a
constant fraction of the directed radius of the world at that point.
And as the astronomer who predicts the future position of the earth
does not assume anything more about what the earth will choose to do
than is expressed in the law G_{μν}=λg_{μν}, 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 EE_{1} or EE_{2}, 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;[Y] 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.[Z]

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.[AA] 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
investigation. 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.




_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 were
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 areal 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 1/10,000 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 starlight. 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.[AB]


_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 (5~·~10^{12}) years
ago. The theory which assigns this date cannot be trusted confidently,
but it seems a reasonably safe conclusion that the sun's age does not
exceed this limit. The future is not so restricted and the sun may
continue as a star of increasing feebleness for 50 or 500 billion years.
The theory of sub-atomic energy has prolonged the life of a star from
millions to 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.[AC] 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.[AD] 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 faster 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.




_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 _h_ which crops
up continually in a wide range of experiments. In one sense we know just
what _h_ is, because there are a variety of ways of measuring it; _h_ is

           ·00000000000000000000000000655 erg-seconds.

That will (rightly) suggest to you that _h_ 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 _h_ 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[AE], 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 6·55 . 10^{-27}
erg-seconds is continually turning up experimentally. It is all very
well to say that we must think of action as atomic and regard this lump
as the atom of action. We cannot do it. We have been trying hard for the
last ten years. Our present picture of the world shows action in a form
quite incompatible with this kind of atomic structure, and the picture
will have to be redrawn. There must in fact be a radical change in the
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 3·4 . 10^{-12} ergs. This
energy is, as we have seen, marked by a distinctive period
1·9 . 10^{-15} secs. We have thus the two ingredients necessary for a
natural lump of action. Multiply them together, and we obtain
6·55 . 10^{-27} erg-seconds. That is the quantity _h_.

The remarkable law of Nature is that we are continually getting the same
numerical result. 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 _h_ is a kind of atom--something which
coheres as one unit in the processes of radiation; it is not an atom of
matter but an atom or, as we usually call it, a _quantum_ of the more
elusive entity action. Whereas there are 92 different kinds of material
atoms there is only one quantum of action--the same whatever the
material it is associated with. I say the _same_ without reservation.
You might perhaps think that there must be some qualitative difference
between the quantum of red light and the quantum of blue light, although
both contain the same number of erg-seconds; but the apparent difference
is only relative to a frame of space and time and does not concern the
absolute lump of action. By approaching the light-source at high speed
we change the red light to blue light in accordance with Doppler's
principle; the energy of the waves is also changed by being referred to
a new frame of reference. A sodium flame and a 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 _h_ was first calculated by analysis of
the observed composition of the radiation in the final state of
randomness. Progress of the theory in its adolescent stage was largely
due to Einstein so far as concerns the general principles and to Bohr as
regards its connection with atomic structure.

The paradoxical nature of the quantum is that although it is indivisible
it does not hang together. We examined first a case in which a quantity
of energy was obviously cohering together, viz. an electron, but we did
not find _h_; then we turned our attention to a case in which the energy
was obviously dissolving away through space, viz. light-waves, and
immediately _h_ appeared. The atom of action seems to have no coherence
in space; it has a unity which overleaps space. How can such a unity be
made to appear in our picture of a world extended through space and
time?

_Conflict with the Wave-Theory of Light._ The pursuit of the quantum
leads to many surprises; but probably none is more outrageous to our
preconceptions than the regathering of light and other radiant energy
into _h_-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 _h_. The period is carried by the waves without change, but the
energy spreads out in an ever-widening circle. Eight years and nine
months after the emission the wave-front is due to reach the earth. A
few minutes before the arrival some person takes it into his head to go
out and admire the glories of the heavens and--in short--to stick his
eye in the way. The light-waves when they started could have had no
notion what they were going to hit; for all they knew they were bound on
a journey through endless space, as most of their colleagues were. Their
energy would seem to be dissipated beyond recovery over a sphere of 50
billion miles' radius. And yet if that energy is ever to enter matter
again, if it is to work those chemical changes in the retina which give
rise to the sensation of light, it must enter as a single quantum of
action _h_. Just 6·55 . 10^{-27} erg-seconds must enter or none at
all. Just as the emitting atom regardless of all laws of classical
physics is determined that whatever goes out of it shall be just _h_, so
the receiving atom is determined that whatever comes into it shall be
just _h_. Not all the light-waves pass by without entering the eye; for
somehow we are able to see Sirius. How is it managed? Do the ripples
striking the eye send a message round to the back part of the wave,
saying, "We have found an eye. Let's all crowd into it!"

Attempts to account for this phenomenon follow two main devices which we
may describe as the "collection-box" theory and the "sweepstake" theory,
respectively. Making no effort to translate them into scientific
language, they amount to this: In the first the atom holds a
collection-box into which each arriving group of waves pays a very small
contribution; when the 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 _h_-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 _h_ crops up inside the atom as well as
outside it. Let us take the simplest of all atoms, namely, the hydrogen
atom. This consists of a proton and an electron, that is to say a unit
charge of positive electricity and a unit charge of negative
electricity. The proton carries nearly all the mass of the atom and
remains rock-like at the centre, whilst the nimble electron moves round
in a circular or elliptic orbit under the inverse-square law of
attraction between them. The system is thus very like a sun and a
planet. But whereas in the solar system the planet's orbit may be of any
size and any eccentricity, the electron's orbit is restricted to a
definite series of sizes and shapes. There is nothing in the classical
theory of electromagnetism to impose such a restriction; but the
restriction exists, and the law imposing it has been discovered. It
arises because the atom is arranging to make something in its interior
equal to _h_. The intermediate orbits are excluded because they would
involve fractions of _h_, and _h_ 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 _h_. But
as regards its internal arrangements the atom has no objection to 2_h_,
3_h_, 4_h_, 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 _h_. 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 _h_; but it is something which, viewed in the
four-dimensional world, is at once seen to be action though this may not
be so apparent when we view it in the ordinary way in three-dimensional
sections. Also several features of the atom are regulated independently
by this rule, and accordingly there are several quantum numbers--one for
each feature; but to avoid technical complication I shall refer only to
the quantum numbers belonging to one leading feature.

According to this picture of the atom, which is due to Niels Bohr, the
only possible change of state is the transfer of an electron from one
quantum orbit to another. Such a jump must occur whenever light is
absorbed or emitted. Suppose then that an electron which has been
travelling in one of the higher orbits jumps down into an orbit of less
energy. The atom will then have a certain amount of surplus energy that
must be got rid of. The lump of energy is fixed, and it remains to
settle the period of vibration that it shall have when it changes into
aether-waves. It seems incredible that the atom should get hold of the
aether and shake it in any other period than one of those in which it is
itself vibrating. Yet it is the experimental fact that, when the atom by
radiating sets the aether in vibration, the 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
_h_-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 _h_. 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
_h_ is perhaps the most striking evidence of the dominance of the
quantum, it will be worth while to explain how the energy of an orbit
jump in an atom can be measured. It is possible to impart to a single
electron a known amount of energy by making it travel along an electric
field with a measured drop of potential. If this projectile hits an atom
it may cause one of the electrons circulating in the atom to jump to an
upper orbit, but, of course, only if its energy is sufficient to supply
that required for the jump; if the electron has too little energy it can
do nothing and must pass on with its energy intact. Let us fire a stream
of electrons all endowed with the same known energy into the midst of a
group of atoms. If the energy is below that corresponding to an orbit
jump, the stream will pass through without interference other than
ordinary scattering. Now gradually increase the energy of the electrons;
quite suddenly we find that the electrons are leaving a great deal of
their energy behind. That means that the critical energy has been
reached and orbit jumps are being excited. Thus we have a means of
measuring the critical energy which is just that of the jump--the
difference of energy of the two states of the atom. This method of
measurement has the advantage that it does not involve any knowledge of
the constant _h_, so that there is no fear of a vicious circle when we
use the measured energies to test the _h_ rule.[AF] 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 _h_, 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, Wednesdays 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 _h_ rule.

"Preposterous! You cannot seriously mean that the electron does
different things on different days of the week!"

But did I say that it does different things? I used different words to
describe its doings. I run down the stairs on Tuesday and slide down the
banisters on Wednesday; but if the staircase consists of innumerable
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.




_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 starlight 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.[AG]

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![AH]

_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
electrons 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.[AI]

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 _h_, or sometimes to an integral
multiple of _h_.

(2) The proviso often led to a self-contradictory use of the classical
theory. Thus in the Bohr atom the acceleration of the electron in its
orbit would be governed by classical electrodynamics whilst its
radiation would be governed by the _h_ 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

                           qp-pq=ih/2π.

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 _h_
rule, but to many related phenomena which the older formulation could
not treat.

On the right-hand side, besides _h_ (the atom of action) and the merely
numerical factor _2π_, there appears _i_ (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 [sr]-1 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 _q_ and _p_ 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 _qp_ is so ill-behaved as to be
unequal to _pq_.

It is here that the three theories differ most essentially. Obviously
_q_ and _p_ cannot represent simple numerical measures, for then
_qp-pq_ would be zero. For Schrödinger _p_ is an _operator_. His
"momentum" is not a quantity but a signal to us to perform a certain
mathematical operation on any quantities which may follow. For Born and
Jordan _p_ is a _matrix_--not one quantity, nor several quantities, but
an infinite number of quantities arranged in systematic array. For Dirac
_p_ is a symbol without any kind of numerical interpretation; he calls
it a _q_-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 _p_ and _q_ individually have no arithmetical interpretation,
the combination _qp-pq_ has the arithmetical interpretation expressed
by the formula above quoted. By furnishing numbers, though itself
non-numerical, such a theory can well be the basis for the
measure-numbers studied in exact science. The measure-numbers, which are
all that we glean from a physical survey of the world, cannot be the
whole world; they may not even be so much of it as to constitute a
self-governing unit. This seems the natural interpretation of Dirac's
procedure in seeking the governing laws of exact science in a
non-arithmetical calculus.

I am afraid it is a long shot to predict anything like this emerging
from Dirac's beginning; and for the moment Schrödinger has rent much of
the mystery from the _p_'s and _q_'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 _h_ rule.

Generally the oscillations in the sub-aether are too rapid for us to
detect directly; their frequency reaches the plane of ordinary
experience by affecting the speed of propagation, because the speed
depends (as already stated) on the wave-length or frequency. Calling the
frequency _ν_, the equation expressing the law of propagation of the
ripples will contain a term in _ν_. 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 _ν_, since it emerges into our gross experience by the
same method that _ν_ does. If _ν_ produces those phenomena which make us
recognise it as energy, the spurious _ν_ 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 _ν_ and the spurious or potential
_ν_ 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 _ν_ 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 _ψ_, 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 _ψ_ in that region. So that if _ψ_ 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 _ψ_-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.[AJ]

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 at 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 1/10,000 of a millimetre? Certainly; but in that case it
will only be possible to fix the velocity with an error of 10 kilometres
per second.

The conditions of our exploration of the secrets of Nature are such that
the more we bring to light the secret of position the more the secret of
velocity is hidden. They are like the old man and woman in the
weather-glass; as one comes out of one door, the other retires behind
the other door. When we encounter 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 _q_ is a co-ordinate and _p_ the corresponding momentum,
the necessary uncertainty of our knowledge of _q_ multiplied by the
uncertainty of _p_ is of the order of magnitude of the quantum constant
_h_.

A general kind of reason for this can be seen without much difficulty.
Suppose it is a question of knowing the position and momentum of an
electron. So long as the electron is not interacting with the rest of
the 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 succeeded 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.




_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.

[Illustration: Fig. 7]

Let us start with a relatum _A_ and a relation _AP_ radiating from it.
Now step to a contiguous relatum _B_ and pick out the "like" relation
_BQ_. Go on to another contiguous relatum _C_ and pick out the relation
_CR_ which is like _BQ_. (Note that since _C_ is farther from _A_ than
from _B_, the relation at _C_ which is like _AP_ is not so definite as
the relation which is like _BQ_.) Step by step we may make the
comparison round a route _AEFA_ which returns to the starting-point.
There is nothing to ensure that the final relation _AP´_ which has, so
to speak, been carried round the circuit will be the relation _AP_ with
which we originally started.

We have now two relations _AP_, _AP´_ radiating from the first relatum,
their difference being connected with a certain circuit in the world
_AEFA_. The loose ends of the relations _P_ and _P´_ have their
monomarks, and we can take the difference of the monomarks (i.e. the
difference of the identification numbers comprised in them) as the code
expression for the change introduced by carrying _AP_ round the circuit.
As we vary the circuit and the original relation, so the change _PP´_
varies; and the next step is to find a mathematical formula 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 4^{4} or 256 numerical coefficients.
_These coefficients give a numerical measure of the structure
surrounding the initial relatum._

This completes the first part of our task to introduce numerical measure
of structure into the basal material. The method is not so artificial as
it appears at first sight. Unless we shirk the problem by putting the
desired physical properties of the world directly into the original
relations and relata, we must derive them from the structural
interlocking of the relations; and such interlocking is naturally traced
by following circuits among the relations. The axiom of comparability of
contiguous relations only discriminates between like and unlike, and
does not initially afford any means of classifying various degrees and
kinds of unlikeness; but we have found a means of specifying the kind
of unlikeness of _AP_ and _AP´_ by reference to a circuit which
"transforms" one into the other. Thus we have built a quantitative study
of diversity on a definition of similarity.

The numerical measures of structure will be dependent on, and vary
according to, the arbitrary code of 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.[AK] These can be divided into 10 forming a symmetrical scheme and 6
forming an antisymmetrical scheme. This is the great point of
bifurcation of the world.

_Symmetrical coefficients_ (10). Out of these we find it possible to
construct Geometry and Mechanics. They are the ten potentials of
Einstein (g_{μν}). 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.[AL] 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.[AM] 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 _x_, we could presumably
construct from the same lumber an appliance for measuring _x_. The
difference is this--if the pointer of the weighing machine is reading 5
lbs. a human consciousness is in a mysterious way (not yet completely
traced) aware of the fact, whereas if the measuring appliance for _x_
reads 5 units no human mind is aware of it. Neither _x_ nor the
appliance for measuring _x_ have any interaction with consciousness.
Thus the responsibility for the fact that the scheme of the scientific
world includes mass but excludes _x_ rests ultimately with the phenomena
of consciousness.

Perhaps a better way of expressing this selective influence of mind on
the laws of Nature is to say that _values_ are created by the mind. All
the "light and shade" in our conception of the world of physics comes in
this way from the mind, and cannot be explained without reference to the
characteristics of consciousness.

The world which we have built from the relation-structure is no doubt
doomed to be pulled about a good deal as our knowledge progresses. The
quantum theory shows that some radical change is impending. But I think
that our building exercise has at any rate widened our minds to the
possibilities and has given us a different orientation towards the idea
of physical law. The points which I stress are:

Firstly, a strictly quantitative science can arise from a basis which is
purely qualitative. The comparability that has to be assumed
axiomatically is a merely 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."[AN]




_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.[AO] 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.[AP] 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". 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.

[Illustration: Fig. 8]

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[AQ] 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 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.[AR] 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?




_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.[AS] 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--[AT]

     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 dream-world 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, or 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 newspaper. 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 must 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 _A_, and then after going half round we come to an exactly similar
group _A´_ indistinguishable by any test; another half circle again
brings us to an exactly similar group, which, however, we decide is the
original group _A_. Now let us ponder a little. We realise that in any
case by going on far enough we come back to the same group. Why do we
not accept the obvious conclusion that this happened when we reached
_A´_; 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 _A_
and _A´_ to you as though they were not the same particles, because that
begs the question; imagine that you have actually had this adventure in
a world you had not been told about. You cannot find out the answer. Can
you conceive what the question means? I cannot. All that turns on the
answer is whether we shall provide two separate haloes for _A_ and _A´_
or whether one will suffice.

Descriptions of the phenomena of atomic physics have an extraordinary
vividness. We see the atoms with their girdles of circulating electrons
darting hither and thither, colliding and rebounding. Free electrons
torn from the girdles hurry away a hundred times faster, curving sharply
round the atoms with slide 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 _h_ 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".




_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.[AU]

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 2+2 will be equal to 4 even in 1999; but if this should prove
correct it will not help to convince anyone that the universe (or, if
you like, the human mind) is governed by laws of deterministic type. I
suppose that in the most erratically governed world _something_ can be
predicted if truisms are not 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 _x_ years. But that does not tell us
whether the span of life of young A. McB. is already written in the book
of fate, or whether there is still time to alter it by teaching him not
to run in front of motor-buses. The eclipse in 1999 is as safe as the
balance of a life-insurance company; the next quantum jump of an atom is
as uncertain as your life and mine.

We are thus in a position to answer the main 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.[AV] 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 _ψ_.

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 _ψ_ 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 _ψ_ itself which acts as the source of the light emitted from the
atom, the period of the light being that of the beats of _ψ_. I think
this means that the spread of _ψ_ 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. Thereappears 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.




_Chapter XV_

SCIENCE AND MYSTICISM


One day I happened to be occupied with the subject of "Generation of
Waves by Wind". I took down the standard treatise on hydrodynamics, and
under that heading I read--

     The equations (12) and (13) of the preceding Art. enable us to
     examine a related question of some interest, viz. the generation
     and maintenance of waves against viscosity, by suitable forces
     applied to the surface.

     If the external forces p´_{yy}, p´_{xy} be given multiples of
     e^{ikx+αt}, where k and α are prescribed, the equations in
     question determine A and C, and thence, by (9) the value of
     η. Thus we find


     p´_{yy}   (α^{2} + 2νk^{2}α + σ^{2}) A - i (σ^{2} + 2νkmα)C
     ------- = ----------------------------------------------------,
       gρη                      gk (A - iC)

     p´_{xy}    α     2iνk^{2} A + (α + 2νk^{2}) C
     ------- = --- . ----------------------------,
       gρη      gk             (A - iC)

     where σ^{2} has been written for gk + T´k^{3} 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 qp-pq shall be equal to
ih/2π.

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 p's and q'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
artificial electrical 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 consciousnesses, 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.[AW]

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
pertinacy. 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 the
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.




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.[AX]

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.

FOOTNOTES:

[Footnote A: 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.]

[Footnote B: 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.]

[Footnote C: 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.]

[Footnote D: 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.]

[Footnote E: 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.]

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

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

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

[Footnote I: 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.]

[Footnote J: See p. 221.]

[Footnote K: 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.]

[Footnote L: 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.)]

[Footnote M: 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.]

[Footnote N: 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".]

[Footnote O: 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_.]

[Footnote P: 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.]

[Footnote Q: 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.]

[Footnote R: This would be an _instantaneous_ space-triangle. An
enduring triangle is a kind of four-dimensional prism.]

[Footnote S: 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.]

[Footnote T: 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.]

[Footnote U: 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.]

[Footnote V: A. N. Whitehead, _The Principle of Relativity_, Preface.]

[Footnote W: 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 [sr][-1].]

[Footnote X: Hegel, _Werke_ (1842 Ed.), Bd. 7, Abt. 1, p. 97.]

[Footnote Y: 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.]

[Footnote Z: See p. 236.]

[Footnote AA: 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.]

[Footnote AB: A very much larger radius of space (10^{11} 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.]

[Footnote AC: 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.]

[Footnote AD: 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.]

[Footnote AE: Prof. E. T. Whittaker.]

[Footnote AF: Since the _h_ rule is now well established the energies of
different states of the atoms are usually calculated by its aid; to use
these to test the rule would be a vicious circle.]

[Footnote AG: The evidence is much stronger now than when the lectures
were delivered.]

[Footnote AH: 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.]

[Footnote AI: 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.]

[Footnote AJ: The probability is often stated to be proportional to
ψ^{2}, instead of ψ, 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 ψ^{2} is obtained by introducing two
symmetrical systems of ψ-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.]

[Footnote AK: Mathematically we contract the original tensor of the
fourth rank to one of the second rank.]

[Footnote AL: 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.]

[Footnote AM: 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.]

[Footnote AN: Die ganzen Zahlen hat Gott gemacht; alles anderes ist
Menschenwerk.]

[Footnote AO: 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.]

[Footnote AP: 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.]

[Footnote AQ: 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.]

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

[Footnote AS: 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.]

[Footnote AT: _Analysis of Matter_, p. 320.]

[Footnote AU: 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.)]

[Footnote AV: 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.]

[Footnote AW: 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.]

[Footnote AX: 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 (_β_) 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


_h_, 179, 183, 223

Halo of reality, 282, 285, 290

Hamilton, W. R., 181

Hamiltonian differentiation, 240

Heaven, 351

Hegel, 147

Heisenberg, W., 206, 220, 228, 306

Heredity, 250

Here-Now, 41

Heterodyning, 216

Hour-glass figures, 48

House that Jack Built, 262

Hubble, E. P., 167

Humour, 322, 335

Humpty Dumpty, 64

Huxley, T. H., 173

Hydrodynamics, 242, 316

Hydrogen, 3

Hyperbolic geometry, 136

Hypersphere, 81, 157


_i_ (square root of [-]1), 135, 146, 208

Identical laws, 237

Identity replacing causation, 156

Illusion, 320

Impossibility and improbability, 75

Impressionist scheme of physics, 103

Indeterminacy, principle of, 220, 306

Inertia, 124

Inference, chain of, 270, 298

Infinity, 80

Infra-red photography, 173

Inner Light, 327

Insight, 89, 91, 268, 277, 311, 339

Instants, world-wide, 43

Integers, 220, 246

Interval, 37, 261

Intimate and symbolic knowledge, 321

Introspection, 321

Invariants, 23

Inventory method, 103, 106, 280, 341

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


_p_'s and _q_'s, 208, 223, 327

Pacific Ocean, 171

Particle, 202, 211, 218

Past, relative and absolute, 48

Pedantry, 340, 342

Permanence, 241

Personal aspect of spiritual world, 337

Phoenix complex, 85

Photoelectric effect, 187

Photon, 190

Physical time, 40

Picture and paint, 106

Picture of gravitation, 115, 138, 157

Plan, Nature's, 27

Planck, M., 185

Plurality of worlds, 169

Pointer readings, 251

Ponderomotive force, 237

Porosity of matter, 1

Potential (gravitational), 261

Potential energy, 213

Potential gradient, 96

Pound sterling, relativity of, 26

Predestination, 293, 303

Predictability of events, 147, 228, 300, 307

Primary law, 66, 75, 98;
  insufficiency of, 107

Primary scheme of physics, 76, 129, 295

Principal curvature, 120, 139

_Principia_, 4

Principle, Correspondence, 196

Principle of detailed balancing, 80

Principle of indeterminacy, 220, 306

Probability, 216, 315

Proof and plausibility, 337

Proper-distance, 25

Proper-time, 37

Proportion, sense of, 341

Proton, 3

Psi (_ψ_), 216, 305

Pure mathematician, 161, 337, 347

Purpose, 105


_q_-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
  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




PRINTED
BY

WALTER LEWIS, M.A.

AT
THE CAMBRIDGE
UNIVERSITY
PRESS


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  | Transcriber's note:                                                |
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  | The symbol _{} in an equation represents a subscript.              |
  | The symbol ^{} in an equation represents a superscript.            |
  | The symbol [oo] represents infinity.                               |
  | The symbol [sr] represents square root.                            |
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  | In the text version where underscores represent italicised matter, |
  | some underscores have been omitted to simplify the presentation,   |
  | e.g. variables in equations etc.                                   |
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  | Capital letters in square brackets are anchors                     |
  | for footnotes e.g. [O]                                             |
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  | Punctuation errors have been corrected.                            |
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  | The following suspected printer's errors have been addressed.      |
  |                                                                    |
  | 'Contents' 'Chap IX,  p 178' to 'Chap IX,  p 179'                  |
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  | Page 32 'Einstein' to 'Dickens'                                    |
  | 'and Dickens has inspired us' (confirmed from a quote              |
  |  in the paragraph)                                                 |
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  | Page 59 'it is' to 'is it'                                         |
  | 'But is it travelling with unusualy high speed'                    |
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  | Page 190 'inverse square-law' to 'inverse-square law'              |
  | 'under the inverse-square law'                                     |
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  | Page 251 'it' to 'is'                                              |
  | 'What exactly is this two tons'                                    |
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[End of The Nature of the Physical World, by A. S. Eddington]
