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THE SERIAL UNIVERSE
by the same author
*
AN EXPERIMENT WITH TIME
C.T.R. Wilson
Track of an 'alpha' particle, in air
G.P. Thomson
Effect produced on a photographic plate by electrons which have passed
previously through a thin metal film
THE
SERIAL UNIVERSE
BY
J. W. DUNNE
LONDON
FABER & FABER LIMITED
24 RUSSELL SQUARE
FIRST PUBLISHED IN NOVEMBER^CMXXXIV
BY FABER AND FABER LIMITED
24 RUSSELL SQUARE LONDON W.C,
PRINTED IN GREAT BRITAIN AT
THE UNIVERSITY PRESS CAMBRIDGE
ALL RIGHTS RESERVED
DEDICATION
To
the lady who typed this book
MY WIFE
PREFACE
In this book I have tried to give the reader a
bird's-eye view of the territory covered by the
theory called 'Serialism'. Some of the chapters,
greatly condensed, have been delivered in lecture
form to the Royal College of Science (Mathe-^
matical Society and Physical Society). But the
main outline of the subject is, I believe, clear
enough to be appreciated by those who have no
special technical knowledge.
Where all is fog, a blind man with a stick is not
entirely at a disadvantage. In my case, Fortune
presented me with a stick; and I have used this
with considerable temerity. Certainly, it has led
me somewhere possibly only into the roadway,
where I shall be run over by a motor-bus full of
scientific critics. But, if I have crossed safely to
the other side, then I should like to express my
gratitude to Mr J. A. Lauwerys of the University
of London, whose continuous encouragement has
been the chief factor which has kept me tapping
along.
CONTENTS
Preface page 9
Introduction 13
PART I. THE THEORY OF SERIALISM
Chap. I. Meaning of a 'Regress' 23
II. Artist and Picture 29
III. Tabular Analysis of a Regress 38
IV. Regress of Self-consciousness 46
V. Meaning of ' Observation 5 in
Physics 50
VI. Regress of a Self-conscious Ob-
server 53
PART II. GENERAL TEST OF THE THEORY
Chap. VII. 'Now' 63
VIII. Regress of Time 72
IX. Regress of 'Reality'. Regress
of Physics. Spatial Repre-
sentation of Time 84
X. Dimensions, Magnitudes and
Mesh-systems 94
XI . Graphical Analysis of the Time
Regress 103
XII. The Immortal Observer and
his Functions 115
CONTENTS
PART III. SPECIAL TESTS OF THE THEORT
Chap. XIII. An Approach to Relativity 133
XIV. Velocity of the ' Now 5 138
XV. The Regress in Relativity 150
XVI. The Physical Outlook of Ob-
server 2 157
9
XVII. Quanta, Waves, Particles and
the Uncertainty Principle 169
XVIII. The Regress of Uncertainty 183
XIX. The Wave Effects 196
XX. Introducing the Real Ob-
server 208
XXI. The Place of Brain 216
XXII. 'A' 222
XXIII. Chronaxy 228
PART IV
Conclusion 235
Appendix 239
Frontispiece
Track of an 'alpha 3 particle, in air
Effect produced on a photographic
plate by electrons which have passed
previously through a thin metal film
12
INTRODUCTION
The men who little guessing the magnitude of
their adventure set out upon the earliest attempts
to understand the world in which we live were
rewarded by three surprising discoveries.
They had opened a door closed till then in the
human mind ; and they saw, in a first, dazzling vista,
the tremendous powers of abstract reasoning with
which Man, all unsuspecting, had been equipped.
They had peered behind Nature's mask of happy
anarchy; and they stared upon Order portentous
and unassailable. But the strangest discovery was
that this orderliness in Nature, and this intelligence
in Man, seemed to have been specially created to
play partners in a kind of cosmic cotillion of
rationality. Mind made laws of reason: Nature
obeyed them.
They discovered these early philosophers that
they were wonderful people in a wonderful world.
To many, the first of these marvels seemed the
more admirable of the two. But there were others
of a different temperament. In this respect, indeed,
the entire company might have been divided, very
early, into two parties. On the one side were those
who loved above all things to present abstract
problems to that fascinating new toy, the human
intellect : on the other were those who found their
13
INTRODUCTION
greatest happiness in the discovery of a new fact to
be fitted to facts of nature already ascertained.
Friction between these two divisions must have
arisen very soon. For one of the commonest cha-
racteristics of a newly-discovered fact is that it
appears, at first sight, to be unintelligible. Con-
sequently, every advance of this kind serves to
bring into prominence the difference between the
pure 'empiricist' (the man who would put facts before
reason) and the pure 'rationalist (the man who would
put reason before facts). The former is willing to
accept the new fact simply because it seems to be a
fact: the latter would prefer to withhold recogni-
tion until the alleged discovery has proved itself to
be reasonable. In the early days of the research,
new facts were both plentiful and marvellous; and
the cumulative effect of all the little hesitations on
the part of the reason- worshippers was, sometimes,
considerable. But, always, they caught up again;
for the empiricist's structure of facts proved, in-
variably, in a little while, to be entirely reasonable.
Nevertheless, these delays in admitting new dis-
coveries were harmful to the prestige of the ration-
alists; for every such lagging-behind meant that
the empiricists had obtained knowledge (admitted,
later on, to be true) which had been established
upon a basis other than that of pure reason.
All this, however, was merely first-line skirmish-
ing. In their main position, the rationalists had
dug themselves in so deeply that none, save a few
14
INTRODUCTION
complete sceptics, dreamed of trying to dislodge
them. Their cardinal tenet that reason, unaided,
could discover the great fundamental truths which
facts of experience served merely to illustrate had
been adopted by the metaphysicians as the basis
of an energetic inquiry into the constitution of the
universe. And the empiricists, although they may
have doubted the expediency of the metaphysi-
cian's methods, never supposed for one momenj
that such facts of nature as remained to be dis<
covered would prove to be, at bottom, otherwise
than wholly reasonable.
Now, nobody had disputed that reasoning is a
machine which deals faithfully with all the material
offered to it, provided its owner does not attempt
to alter its method of working. But it is a machine
which needs feeding with 'premisses', i.e., asser-
tions presumed to be true. The rationalists claimed
to have discovered the most fundamental premisses
of all basic truths which could not be denied, bul
which, because they were basic, could not be
proved. Knowledge which satisfies that descrip-
tion is said to be 'given', and the supposed given
knowledge which the rationalists selected as the
base of their edifice consisted of a set of axioms
asserting what could or could not exist without
self-contradiction. The empiricists, however, were
able to point to given knowledge of an apparently
different kind. The evidence of the senses is
notoriously unreliable, but what none can deny
15
INTRODUCTION
is the existence of the evidence. We may doubt what
a sensory experience seems to assert; we may be a
little vague even regarding the precise character of
the experience itself: but we reach, through our
senses, a limit to what it is possible for us to deny
we arrive at what is (for us) an undeniable
residuum which we call the 'sensation 5 , or, in less
popular language, the c sense-0fofaw'.
The fact that the sense-rf0fa of the empiricists
happened to obey the axioms of the rationalists,
and were never self-contradictory, shed no light on
the main problem. Was the universe the product of
Mind, so that it, and experience of it, must illus-
trate Mind's axioms? Or did the universe exist
independently; and were our infrangible axioms
no more, at bottom, than our recognitions of the
special kind of order which we happened to have
discovered pervading that universe, and so, no
more than illustrations of our inability to grasp the
possibility of any other kind of order?
That question was never answered. An inter-
ruption occurred. In the height of the discussions,
an Irishman, Bishop Berkeley, threw into the
philosophic duck-pond a boulder of such magni-
tude that the resulting commotion endures in
ripples to this very day. He asked an entirely
different question. If sensations such as those of
colour, form and feeling, plus their derivatives of
memory-images, associated ' ideas', concepts and
the like, were the sole bases of our knowledge, the
16
INTRODUCTION
only objects with which we were, or could be,
directly acquainted, what evidence had we that there
existed any substantial, non-mental world at all?
You may imagine the joyous rallying of ration-
alists which followed the appearance of this c Ideal-
ism' (as Berkeley's theories were called). No
physical universe! Nothing but a vast, collective
hallucination! Then Mind was Lord of All.
Philosophy, split horizontally by the division
between rationalists and empiricists, was riven ver-
tically by the far fiercer dispute which arose be-
tween the idealists and the realists. Peacemakers
suggested an 'intuitive' knowledge of objective
reality. Voluntarists argued that this intuitive
knowledge was knowledge of opposition to 'Will'.
But the rationalists wished to limit the intuitive bases
of their structure to cognition of the three 'Laws of
Thought'; while intuition, if it existed, would be
a process beyond reach of the empiricist's tests.
But the idealists were not only assailed from
without: they were betrayed from within. There
arose very quickly a critic who said, in effect,
'What is all this talk about a "collective" hallu-
cination? /fall that I can know directly are my
sensations, and no external universe can be in-
ferred from these; then I have no reason to sup-
pose that there exists any mind other than my
own. / am the only experient, and the hallu-
cinatory external world is my world, and mine
alone.' The logic of the argument seemed to be
FSU
INTRODUCTION
unassailable. No answer could be found then: none
was found later.
Most of the idealists were unable to face this
unescapable consequence of their thesis. c Solipsism '
(as this completed theory was called) proved too
indigestible for any but the absolute purists. The
rationalist quarter, moreover, had been worried
considerably by the logical discoveries of Hume,
who proved that, if the world of sense-data were all
that existed, a Mind controlling this display would
be as hallucinatory as an external world. In the
end, so far as the majorities were concerned, the
rationalists abandoned their rationalism, the em-
piricists discarded their empiricism, and both
agreed to accept the external world as 'given 5 by
some concealed process which (it was hoped) would
prove some day to be both rational and empirical,
but which, till then, could not be classified as any-
thing beyond that irrational and intangible thing
intuition. And so, on a basis of intuition, Science
came into its own.
Progress was now rapid. Rationalists and em-
piricists hurried hand in hand towards a goal which
showed ever clearer and more brilliant. It was
discovered, with profound relief, that the real uni-
verse consisted of conglomerations of little round
things like billiard balls, called ' atoms '. Electricity
was found to be a modification of an all-pervading
elastic solid called 'aether'. There were laggards
who pointed out that the primary sense-data such
18
INTRODUCTION
as colour could not be composed of, or accounted
for by, either billiard balls or waves ; but the gleam of
the Absolutely Reasonable shining just ahead blinded
nearly all to the mists of irrationality gathering
on either side. They reached that gleam and it
vanished at that moment. The solid atoms fled
away. In their places lay voids tenanted by
minute specks too unreal to possess both precise
position and precise velocity. Did I say c specks ' ?
They were not specks, but waves filling all space.
Photographs proved it. Worse, each of these wave-
entities needed a whole three-dimensional world
to itself, so that no two could be together in the
same ordinary space. Did I say 'waves'? I am
sorry, they were specks in one and the same space.
Experiments proved it, and they could be even
counted by a specially designed apparatus. They
were not mixtures of specks and waves : each was,
definitely, both. A strange phantasmagoria. It
was founded upon the indubitable existence of
a tiny, irreducible, four-dimensional magnitude
called the c Quantum ' itself the very acme of ir-
rationality. And the behaviour of this irrational
universe could be calculated only by the aid of a
specially invented 'irrational 5 algebra.
On another side they were faced by the world
of Relativity. Here the aether had either disap-
peared, or it survived merely as a purely personal
appendage as subjective as any Solipsist could
desire. Space and time had not vanished : they had
19 2-2
INTRODUCTION
done worse: they had become interchangeable.
And the c space-time 5 world of the relativists
appeared to be governed throughout its expanse by
the square root of minus one famous in mathe-
matics as the basic 'imaginary 5 number.
Now, reasoning must start from c given 5 know-
ledge, and that knowledge is, consequently, not
rational. No science, therefore, proposes to explain,
or expects to explain, the existence of whatever it ac-
cepts as the fundamental realities. But its object is to
employ those elementary indefinables as characters
in a narrative of rational happenings. And there is
a fairly general feeling that, in the tale which our
science offers us to-day, the irrationalities are far
too numerous. It is a true story; but it looks as if,
somewhere, somehow, it had been made into
'printer's pie 5 . The right words are there, but they
seem to be in the wrong places ; and there is more
than a suggestion that paragraphs which ought to
have been consecutive have become superimposed.
Waves, particles, space-time, quanta and evensense-
data must, we feel, fit together in some simpler
fashion. And we suspect that, if only we could
discover that scheme, all these surplus irration-
alities would vanish, leaving us with nothing that
was not obvious and expectable to the most
ordinary intelligence, and with nothing more
obstreperous than the two basic indefinables of
Mind and Matter.
20
THE SERIAL UNIVERSE
*
PART I
THE THEORY OF
SERIALISM
CHAPTER I
MEANING OF A 'REGRESS 5
A ' series' is a collection of items linked together,
chain-fashion, by some recurrent relation. The
notion of series has reference, always, to some
underlying unity; this is implicit in the fact that
the separated items are related to one another.
The distinctive items of a series are called its
c terms'. For example, if we regard a child as a
creature who had a parent who had a parent who
had a parent, etc., etc.; the child is the first term,
his parent the second term, and his grandparent
the third term of a receding series. And, if we
tabulate that series thus:
ist term
2nd term
3rd term
4th term
A child
of
a parent
who was
child of
a parent
who was
child of
a parent
who was
child of
etc., etc.,
23
THE SERIAL UNIVERSE
the relation between the terms becomes readily
apparent.
We know, from various biological indications,
that this particular sequence stretches back to
before the dawn of history. But the old-time philo-
sophers thought that it must either recede to a
time infinitely remote, or have been started by
some magical act of creation. And it is rather
interesting to consider what were their grounds for
that assumption.
If we look at the first term in the table, we find
there an individual to whom we have allotted only
one character the character of being a child. Now
the fact that every child has or had a parent is
merely a truism ; it is asserted already in the mean-
ing attaching to the word 'child'. And, taken by
itself, it does not compel us to entertain the notion
of remoter ancestors. But suppose we go on to the
second term. We come to a person who is declared
to possess a double character a person who is both
parent and child. As a parent, he is related to the
first-term individual already examined; and, as a
child, he must be related to some ancestor not yet
taken into account. Now, the early philosophers
supposed, wrongly, that it was a matter of logical
necessity for every parent to be also a child. If that
had been true, the series, obviously, would have
been bound to extend backward to infinity.
The point the point which is so often over-
looked is this : The extension of a simple series to
24
MEANING OF A 'REGRESS*
infinity involves some necessarily dual character in
its terms. But, to discover that dual character, we
must trace the series as far as its second term. A
study of the first term (such as the child in the
above example) with its single character, will yield
us only half the required information. And it may
be noted that the third and remaining terms do np
more than repeat the information already asserted
by the second term. All the remoter individuals in
the purely imaginary example we have taken would
have possessed the double character of being both
parent and child; but we could have discovered
that from an examination of the second term alone.
In brief: Every simple series to infinity is the ex-
pression of some logical fact which is asserted in the
second term but not in thejirst.
And, as we shall see later, it may be impossible
to exaggerate the importance, to the human race,
of this very simple characteristic of a simple in-
finite series.
Now, a series may be brought to light as the
result of a question. Someone might enquire,
'What was the origin of this man?', or a child
learning arithmetic might set to work to discover
what is the largest possible whole number. The
answer to the first question has not yet been as-
certained : the answer to the second can never be
given. It will be seen, however, that the reply in
each case must develop as a series of answers to a
series of questions. In the first instance, we reply
25
THE SERIAL UNIVERSE
that the man is descended from his father; but that
only raises the further and similar question, c What
was the origin of his father? 5 . In the second case,
the child will discover that 2 is a greater number
than i ; but he is compelled to consider then
whether there is not a number greater than 2 and
so on to infinity. A question which can be answered
only at the cost of asking another and similar
question in this annoying fashion was called, by
the early philosophers, 'regressive', and the ma-
jority of them regarded such a 'regress to infinity 5
with absolute abhorrence.
Their attitude is easy to understand. They wished
to regard the universe as something completely
explicable. To admit that there were questions
with answers which receded as a rainbow recedes,
was, in their opinion, to admit, before they started,
that their task of explaining everything was fore-
doomed to failure. Then, again, a considerable
number of the early philosophers supposed that the
universe must be, at bottom, something extremely,
even childishly, simple; a naive theory which in-
volved that to every question there must be a
simple and straightforward answer. This provided
another reason for the ancient dislike of regressions.
And we must add to the list that very numerous
class which wished, and still wishes, from motives
of policy, to divide the world sharply into things
which are comprehensible and things which are
incomprehensible. To such persons, a question
26
MEANING OF A 'REGRESS*
which is answered by an ' infinite regress' is
anathema, because it provides, very obviously, a
class between the two division? ,
In brief, it was universally recognised that a
regress might be logically incontrovertible; men
moulded their lives and their sciences upon the
immense stock of reliable information provided by
the study of these incompleted series of questions
and answers; and yet the regress to infinity was
looked upon as being, in some fashion apparent
only to intuition, not actually untrue, but not
precisely that aspect of the truth which it was the
business of philosophy to discover.
They were quite unable to put this feeling into
words. They wandered off into loose talk of 'com-
plexities', which was a dubious charge, and of
'contradictions', which was a libel unjustified in
anyone with any pretensions to intelligence for a
contradiction produces no regress at all, and the
whole trouble about the infinite regress is its damn-
able logicality. If the truth of the premiss (i.e., the
double character of the second term) is acknow-
ledged, the regress becomes mathematically in-
evitable. Yet the feeling has persisted to this day:
it crops up afresh whenever some new regression,
to the sight of which we have not grown accus-
tomed, is discovered. And Bradley, perhaps, gave
it its nearest approach to verbal expression when he
said, 'Reality cannot be an infinite regress'.
The answer, I think, is this:
27
THE SERIAL UNIVERSE
The truth or falsity of Bradley 5 s dictum depends
upon the meaning it attaches to the word c reality 3 .
If it refers to reality pure and undefiled by any
attempt at translation into terms of human com-
prehension, his statement, probably, is true (though
you must not ask me to give reasons for that belief) .
But if the word means reality in the scientific sense,
rational cum empirical reality, then the asser-
tion is, definitely, wrong. The difference is that
which lies between 'things as they are 5 and
'things as they seem to be 5 . Of 'things as they
are 5 we know nothing rational ; and, if we suspect
Bradley to be right, it is merely because of the
feeling of dissatisfaction aroused in us by any re-
gress. But of ' things as they seem to be 5 things as
they affect an observer we can say a great deal. As
I hope to show in this book, we can say, with
absolute assurance, that 'reality 5 as it appears to
human science must needs be an infinite regress. And
it is only when it is expressed in that form that we
can treat it as the reality upon which we can rely.
CHAPTER II
ARTIST AND PICTURE
A certain artist, having escaped from the lunatic
asylum in which, rightly or wrongly, he had been
confined, purchased the materials of his craft and
set to work to make a complete picture of the
universe.
He began by drawing, in the centre of a huge
canvas, a very small but very finely executed re-
presentation of the landscape as he saw it. The
result (except for the execution) was like the sketch
labelled X in FIGURE i .
FIGURE I.
On examining this, however, he was not satis-
fied. Something was missing. And, after a
moment's reflection, he realised what that some-
thing was. He was part of the universe, and this
fact had not yet been indicated. So the question
arose : How was he to add to the picture a repre-
sentation of himself?
29
THE SERIAL UNIVERSE
Now, this artist may have been insane, but he was
not mad enough to imagine that he could paint
himself as standing in the ground which he had
already portrayed as lying in front of him. So he
shifted his easel a little way back, engaged a pass-
ing yokel to stand as a model, and enlarged his
picture into the sketch shown as X 2 (FIGURE 2).
FIGURE 2.
But still he was dissatisfied. With the remorseless
logic of a lunatic (or genius you may take your
choice) he argued thus :
This picture is perfectly correct as far as it goes.
X 2 represents the real world as I the real artist
suppose it to be, and X l represents that world as an
artist who was unaware of his own existence would
suppose it to be. No fault can be found in the
30
ARTIST AND PICTURE
pictured world X 2 or in the pictured artist, or in
that pictured artist's picture X l . But I the real
artist am aware of my own existence, and am
trying to portray myself as part of the real world.
The pictured artist is, thus, an incomplete de-
scription of me, and of my relation to the universe.
FIGURE 3.
So saying, he shifted his easel again, seized his
brush and palate, and, with a few masterly strokes,
expanded his picture into X 3 (FIGURE 3).
Of course, he was still dissatisfied. The artist
THE SERIAL UNIVERSE
pictured in X 3 is shown as an artist who, though
aware of something which he calls himself, and
which he portrays in JSf 2 , is not possessed of the know-
ledge which would enable him to realise the necessity of
painting X 3 the knowledge which is troubling the
real artist. He does not know, as the real artist
knows, that he is self-conscious, and, consequently,
he pictures himself, in X^ as a gentleman unaware
of his own existence in the universe.
The interpretation of this parable is sufficiently
obvious. The artist is trying to describe in his
picture a creature equipped with all the know-
ledge which he himself possesses, symbolising that
knowledge by the picture which the pictured
creature would draw. And it becomes abundantly
evident that the knowledge thus pictured must
always be less than the knowledge employed in
making the picture. In other words, the mind which
any human science can describe can never be an adequate
representation of the mind which can make that science.
And the process of correcting that inadequacy
must follow the serial steps of an infinite regress.
This pictorial symbol does not lend itself very
readily to detailed analysis, and we shall make
little further use of it. It provides, however, an
excellent illustration of the differences which under-
lay the views of ( i ) the old-fashioned man of science,
(2) the materialist, and (3) the average philoso-
pher. The classical physicist held (wrongly, as we
shall see) that the picture X l9 which contains no
32
ARTIST AND PICTURE
reference to an artist, ought to prove self-consistent
and self-sufficient. The materialist held (wrongly, as
we have seen) that the second picture, X 2 (q-v.),
would describe closely enough for practical pur-
poses the relation between man and his universe.
He omitted to note that the artist shown in that
picture is only the first term of a regressive con-
ception, and that, to get at the practical infor-
mation which is expressed in such a series, we must
study the second-term individual. The average philo-
sopher found himself in a quandary. He could
see that the materialist was at fault, but he was
unable to point to the error without pointing to a
regress which he did not know how to handle.
Consequently, he hesitated while the error gained
adherents. And thus there became established that
picture, so popular to-day, which exhibits the
universe as nothing more or less than an indiffer-
ently gilded execution chamber, replenished con-
tinually with new victims. The materialist was
scarcely to blame: he was honestly myopic. But
the philosopher was a politician.
The regressive picture of our symbol contains,
not only a series of artists of increasing capacity,
but also a series of the landscapes which such
imagined individuals would draw. One might sus-
pect that the details of those landscapes the hills
and trees and houses ought to bear some witness
FSU 33 3
THE SERIAL UNIVERSE
to the increasing skill of the draughtsmen and
exhibit a serial progress towards a regressive per-
fection. Now, we shall discover, in the course of
this book, that the entire symbol, with this addi-
tional interpretation, is absolutely correct. This
means that, whatever the universe may c be 5 in
itself, all sciences thereof must be regressive, so that
we are faced with what is, for all empirical pur-
poses, a serial world. And, when we recall that the
relation of such a world to ourselves the repetitive
relation which makes the regress is given by the
second term and not by the first, it will become
evident that the theory of the 'execution chamber 5
was a particularly ludicrous blunder.
Omitting the arguments, the conclusions of the
theory I call 'Serialism 5 are, briefly, as follows.
We are self-conscious creatures aware of some-
thing which we are able to regard as other than
ourselves. That is a condition of affairs which it is
impossible to treat as rational (i.e., systematic)
except by exhibiting it in the form of an infinite
regress. Consequently, the first essential for any
science which can satisfy us as fitting the facts of
experience is that it shall employ some method of
description which is suitably regressive. It turns out
that the possibility of viewing all experience in
terms of 'time 5 provides us with just the method of
description required. The notion of absolute time
is a pure regress. Its employment results in ex-
hibiting us as self-conscious observers. It intro-
34
ARTIST AND PICTURE
duces the notion of 'change 5 , allotting to us the
ability to initiate changes in a change-resisting
'not-self. It treats the self-conscious observer as
regressive, and it describes the external world as it
would appear to such a regressive individual. Thus
it fulfils all the requirements of the situation. But
time does more than that. By conferring on the
observer the ability to interfere with what he
observes and to watch the subsequent results, it
introduces the possibility of experimental science.
The notion of experiment implies always an inter-
ference with the observed system by an observer
outside that system. This is the cardinal method of
physics, which postulates, thus, from the outset
the possibility of interference with every system by
an observer who, in relation to that system, is
'free'. The essential point here, however, is that
physics, as a science of experiment, of alter it and
see ', is based upon the notion of time. So, for that
matter, are all our systems of practical politics,
ethical or otherwise. In that way only by the
employment of this flagrantly regressive method
of description have we been able to convert our
otherwise irrational knowledge into a systematic
and serviceable scheme.
But is this regressive way the proper way to de-
scribe the universe? That question has little, if any,
meaning. Is 'decimal point three recurring' the
* proper ' way to describe ' one- third ' ? The regress of
the recurring decimal and the regress of time both
35 3-2
THE SERIAL UNIVERSE
rank as series to infinity; and, though the former
series is 'convergent 5 and the latter 'parallel 5 ,
the underlying principle in each is the same.
There is probably another way of describing the
universe, just as there is another way of describing
one-third. We use the decimal method because it is
convenient for our purpose and just as valid as the
other. We use the time regress because it gives us
a valid account of the universe in its relation to our-
selves, that is, in its reaction to experiment. It is
the proper method for its purpose, and I know of
no profounder meaning in the word c proper 5 . J|ut
this I do know : It is impossible to imagine a more
effective way of losing knowledge than that of
expressing it in the form of an infinite regress and
then restricting attention to the first term alone.
And that is what mankind has been doing.
All talk about 'death 5 or ' immortality 5 has re-
ference to time, and is meaningless in any other
connection. But a time-system is a regressive
system, and it is only in the lop-sided first term of
that regress that death makes its appearance. It
will become clear in the course of this book that,
in second-term time (which gives the key to the
whole series) we individuals have curious very
curious beginnings, but no ends. Is that a hor-
rible thought? Perhaps. But I do not think so. The
present-day terror of immortality is based, almost
entirely, upon an imperfect appreciation of what
that immortality means. We try to imagine it as
36
ARTIST AND PICTURE
fitted somehow into the first-term world, (where, of
course, it won't go), and so plague ourselves with a
lugubrious picture of bored individuals dragging
memory's ever-lengthening chains, desperately sick
of themselves and the world and all that therein is,
craving an extinction which they cannot find. We
imagine, in fact, our present kind of daily life con-
tinued for ever. If that were true, there could be
no act more cruel than the act of giving birth to
a child. But, fortunately, our immortality is in
multi-dimensional time, and is of a very different
character.
And now for the proofs. These must develop, so
to say, backward. We must take the world of our
present-day knowledge, show that it is regressive,
show that it is described as if it were viewed by
a regressive observer, and show that this imagined
regressive individual would constitute a self-con-
scious human being. That will be conclusive
evidence that we are self-conscious creatures who
are using that regressive method of defining our-
selves and our surroundings.
37
CHAPTER III
TABULAR ANALYSIS OF A REGRESS
1 he French philosopher Descartes, while engaged
in subjecting all so-called knowledge to the acid
test of doubt (in the hope of discovering something
indubitable), was seized by a sudden inspiration.
'I am thinking! 5 he exclaimed, 'Therefore I
exist.'
Critics have declared that this saying embodied
two assertions concerning two empirical discoveries
and that these findings should have been an-
nounced in the following order:
(1) c There is thinking going on 5 (an undeniable
fact, c given' to introspective observation).
(2) 'This thinking is my thinking.'
For awareness of activities, and awareness that
there is a 'self which is active, are two very dif-
ferent matters.
Be that as it may, the initial fact which Descartes
announced (before he brought in his unnecessary
'therefore') was: / am (thinking). And it is im-
portant to bear in mind that he was seeking, at the
time, for something which he could regard as in-
dubitable. So that he was regarding it as 'given'
to him, without necessity of argument, that there
was an 'I' thinking. Thus, intentionally or un-
38
TABULAR ANALYSIS OF A REGRESS
intentionally, he was claiming for 'self-conscious-
ness 5 the status of given, undeniable know-
ledge.
We are, all of us, aware of our thoughts. We can
watch, critically, the sequence of mental operations
we are performing in any reasoned argument, so
that an error is detected and arrested before the
next step is made. We can retrace any train of
ideas we may happen to have followed in mind-
wandering. Indeed, it was only because a great
part of our thinking processes remembering
and associating are observable to introspection
that the science of psychology came into ex-
istence.
But, if it is, for you the present reader an ex-
perimentally ascertainable fact that JWM can observe
such thinking processes, this involves, not only
your direct knowledge of the processes but also
your direct knowledge of the something called
or miscalled c yourself which thus observes
them.
Now, if there be such a 'self, it is not readily
discoverable by introspection. We seem to know of
it, in fact, from the presented verdict of mental
processes which we have been unable to follow.
Yet the knowledge thereof is, certainly, 'given 5 , in
the sense that we cannot rid ourselves of it by any
means whatsoever not even by reflections on the
obscurity of its origin.
Most people are prepared to accept self-con-
39
THE SERIAL UNIVERSE
sciousness as a fact; even though they regard it
(wrongly) as a fact which plays no part in our in-
terpretation of the physical world. But everyone
finds it unsatisfactory to be confronted with some-
thing which claims the status of existence while
declining to submit to examination. I suggest,
therefore, that we make one more attempt to track
down this elusive 'self; and, since our powers of
conscious introspection seem to be too feeble for
this purpose, I propose that we set about our task
in an entirely different fashion.
We shall begin by imagining that there exists a
'self-conscious' observer. He is to be aware of his
'self 5 as something observed. He is to distinguish
that 'self from an antithesis a 'not-self also
observed. And he is to be aware of his 'self 5 as an
intermediary entity an instrument which he can
employ in observing the ' not-self 5 . In other words,
he is to be aware, by observation, of what is called
'the subject-object relation 5 .
Then we shall ask ourselves what sort of a thing
such a creature would need to be in a rational
world a world which science could handle.
When we have ascertained those requirements,
we shall look around to see whether there is, or is
not, in nature as we know it to-day, anything
which meets that bill.
We shall find that our bill of requirements con-
stitutes an infinite series which we shall need to
draft in the form of a table. The table will be
40
TABULAR ANALYSIS OF A REGRESS
triangular; consisting of an arrangement of com-
partments like this.
which looks, at first sight, as if I proposed asking
the reader to examine something much more com-
plicated than the simple series of ancestors, or of
whole numbers, we glanced at in Chapter i. That,
however, is not the case.
This tabular construction is only a convenient
way of exhibiting the relations between all the
c terms' of any simple series. Let us glance at an
example. We can realise, quite easily, that every
schoolboy is the child of the child of the child of
the child of the remainder of an extremely
long series of ancestors. But, if I were to ask you
what was the relation between the second and fifth
individuals in that series, you would have to think
for a moment or two before you could reply that
the one was the great-grandchild of the other. You
would have to think much longer, if I asked you
the relationship between the ninth and the thirty-
second terms. But I could prepare for you a tri-
angular table which would save you any trouble of
that kind. And I should construct it as follows.
In the top compartment of the table I put the
41
THE SERIAL UNIVERSE
first person of the series, the schoolboy, as de-
scribed by the second person, the father.
In the next (horizontal) pair of compartments I
put the grandfather's descriptions of the first and
second persons, the child and the father.
ist
person
2nd
person
child
grand-
child
child
In the next row I put the great-grandfather's de-
scriptions of the child, the father and the grand-
father.
ist
person
2nd
person
3rd
person
child
grand-
child
child
great-
grand-
child
grand-
child
child
42
TABULAR ANALYSIS OF A REGRESS
In the next row we include the great-grandfather,
and give the great-great-grandfather's descriptions
of all his descendants.
1st
person
and
person
3rd
person
4th
person
child
grand-
child
child
great-
grand-
child
grand-
child
child
great-
great-
grand-
child
great-
grand-
child
grand-
child
child
And so on for as far as you like.
Please note that,
(1) Each row gives the relations which all the
persons considered therein bear to the person on
the extreme right of the line below. The last row
gives, of course, the relations of the persons to the
individual who comes next in the series.
(2) Since each row describes the persons con-
cerned as these would be described by the person next to be
considered, the descriptions change in each row.
For example, the second person of the series
(counted from left to right) is child in the opinion
43
THE SERIAL UNIVERSE
of the third person, grandchild in the opinion of
the fourth person, great-grandchild in the opinion
of the fifth person (not yet entered) , and so on.
(3) The descriptions given of each person are
only characters pertaining to them on account of
their different relations to the different individuals
of the series. We are trying, throughout this table,
to arrive at a description of each individual as the
descendant of the ultimate ancestor. When we arrive
at the stage where we discover the great-great-
grandfather, we declare that the person with whom
we started is to be described, properly, as the great-
great-grandchild of that ancestor. That definition
is given in the left-hand compartment of the fourth
row. This child's other descriptions (in ascending
order up the first vertical column) are regarded then
as merely characters which, we have discovered, are
bound to pertain to any great-great-grandchild.
Unfortunately, we cannot reach, in the space at our
disposal, the ultimate ancestor; but we shall find
that a great-great-grandchild, in turn, is only a
character which must be possessed by a great-
great-great-grandchild.
The reader need not trouble, here, to learn the
ins and outs of this table by heart. He will have
plenty of opportunity to familiarise himself with
these as we go along. The essential thing now is for
him to realise that the table is quite comprehen-
sible, and that it deals with various aspects of only
one simple series. Also, that the descriptions given
44
TABULAR ANALYSIS OF A REGRESS
are all relative the table does not tell us what
anything is in itself. For instance, our first entry
tells us nothing about the schoolboy except the
way in which he is related to his father; it
describes him simply as 'child 3 . The other entries
follow the same rule.
45
CHAPTER IV
REGRESS OF SELF-CONSCIOUSNESS
When we are trying to describe what we mean
by self-consciousness, we say that you are aware of
'jwwrself 5 , that I am aware of ' myself, that she is
aware of 'A^rself 5 , but that he is aware of 'Azwself 5 .
This last is a bad error, for the possessive pronoun
is all-important. There could be nothing rational
in a Jones who was aware of Jones, and science
could have no dealings with such an individual.
You are speaking quite properly when you say
that you are aware of C jy0#rself 5 and not of
The only 'self 5 that you could be aware of, in
a rational world, would be something which was
an object to the ultimate, real you. But your self-
consciousness does not lie merely in your being
aware of such an object it involves the recognition
of that object as yours. Suppose you decide (rightly
or wrongly) that your body is 'yourself 5 ; you do
not do so because you are aware of a body a body
belonging to, say, Smith but because you are
aware of the body in question as yours. And so it is
with any subtler object you may designate by that
title of 'self 5 . A man who was aware that c he 5 was
observing would be aware of an observing thing
which was an object to the ultimate him; but, to be
REGRESS OF SELF-CONSCIOUSNESS
self-conscious, that man would have to be aware of
that observing thing, not as an object apparent to
the human race in general, but as an entity per-
taining strictly to him. He would need to be aware
of it as his observing self.
It is easy to see, now, thatjany rational self-
consciousness would involve an infinite regress.
For, whatever were observable to a man as a
proper 'self' would need to be observable to him
as his self, involving awareness of something owning
the self first considered. Let us suppose, for
example, that B is recognised by the self-conscious
individual as his observing self and A as the object
(the c not-self 5 ) observed an arrangement which
we can tabulate thus,
B
putting (for future convenience) the observing en-
tity to the right of, and below, the entity observed.
Then, since the self-conscious creature regards B
as his self, he must be aware of a self C which owns
B. So that the table must be extended thus,
B
47
THE SERIAL UNIVERSE
indicating that C observes B while B observes A.
But, since our friend is aware of C as a c self 5 owning
B, he must be aware of that C as his self, and so be
aware of a self D owning C, thus,
B
D
where D is observing Cs observations of B's obser-
vations of A.
D, of course, must be a c self observed by an
owner E, and so on ad infinitum.
It looks rather fantastical, as do all regressions
when we first encounter them. But there is no
getting away from it. Unless D is aware of C, he
cannot regard B as his self not, at least, in a
rational world.
The reader, however, studying this table, will
ask the following question : * If C observes B while B
observes A, how can C be aware of A as distinct
from B? Surely he would observe B's response to
A as merely a modification in B\ This criticism is
quite justified. It is, indeed, the basis of the philo-
sophy called Idealism the theory which denies the
separate existence of A.
REGRESS OF SELF-CONSCIOUSNESS
We must recognise, then, that our table, though
correct, is incomplete. There is a great deal missing.
And what that great deal is we shall discover in the
next two chapters.
49
CHAPTER V
MEANING OF 'OBSERVATION'
IN PHYSICS
Let A and B be two entities existing independently
of each other. Let A be affecting (I am choosing the
word with the broadest meaning) B. And let us sup-
pose that we are studying the effect of A upon B. In
making that investigation we are, actually, em-
ploying B as an instrument for discovering some-
thing about A.
Now, it is clear enough that the knowledge of
A provided for us by B can be knowledge of only a
single character possessed by A the character of
being-able-to-affect-B '. This character is said to be
'relative 5 to 5; since, by our definition thereof, it
does not exist except with reference to B. But it
cannot be the only character which A possesses;
because, if that were the case, the complete A
would be merely relative to B and have no inde-
pendent existence such as we hypothecated at the
outset.
Suppose we designate the fully charactered A
by A 2 , and represent the character of being-able-to-
affect-B by A l . Then what , the instrument, is
said to c observe 5 is simply this A l for characters of
AZ which do not affect B are, obviously, not dis-
50
MEANING OF * OBSERVATION 5 IN PHYSICS
covered for us by B. The instrument B is referred
to, in science, as 'the observer'.
Thus, in science, to 'observe 5 is to abstract a
character from some entity existing independently
of the observer. And the character abstracted
must be one which, in some way, affects that ob-
server.
We see, then, that an 'observing instrument 5 is
not, in strict scientific parlance, a mere measuring
appliance (though it may have a scale attached to it
as a refinement) . As examples of observation by an
instrument, I may cite : A dynamometer abstract-
ing Force from Impulse; a metal film abstracting
Energy from Light; a moving body with its
motion restricted by the proximity of another
body and which, thus, abstracts that other body's
character of Attraction or Repulsion. All these
abstractions could be made without the use of any
scale to give a merely numerical magnitude to the
character abstracted.
It is to be noted that, if our knowledge were con-
fined solely to knowledge of J3, we should have no
grounds for supposing that J5 5 s behaviour was due
to anything beyond its own intrinsic nature. Our
science would consist then of a mere classified
catalogue of the incidents in jB 5 s career, and we
should have no right to speak of B as an c instru-
ment 5 . The use of that term implies that we have
some previous knowledge of A 2 as an entity other
than the known B.
51 4-2
THE SERIAL UNIVERSE
The knowledge involved in a scientific experi-
ment may be classified, then, as follows.
Observed by
(abstracted by) B
A
Known to ourselves
and regarded as ex-
isting independently
of each other
A*
B
It will be perceived that, from the outset, we
credit B with a reality which we deny to A. For
AI$ existence is merely relative to B. It will be
realised, moreover, that it is impossible for us to
regard an instrument B as something which we can
add to a system consisting of entities (such as A^
which have been described solely by the way in
which they affect B.
CHAPTER VI
REGRESS OF A SELF-CONSCIOUS
OBSERVER
We are now in a position to tackle the individual
to whom it is a 'given 5 fact that 'he 5 is observing
something which is not his observing 'self.
Let A be the object observed, B the observing
'self, and C something which knows that B is ob-
serving A. These we can tabulate as before (vide
Chapter m).
The question was : How can C be aware of A as
anything but a modification in the B which he is
observing?
We know from the last chapter that A, being
something observed by B, is merely a character
abstracted from some entity in the world which
contains B. We can describe A, therefore, as an A l
abstracted from an A% , and can amplify our table in
the fashion shown below. Since there may be any
53
THE SERIAL UNIVERSE
number of A 2 entities affecting B, we may call
'World as observed by B\
World as observed by B
4
^
B
C
Now, since it is to be, for C, an unavoidable
judgment that B is observing some character of A 2 ,
he must have a knowledge of A 2 as much c given' as
is his knowledge of 5, that is to say, it must be
knowledge by observation. So we can fill in a little
more of our table ; thus :
World as observed by B
4
World as observed by C
^
B
C
Now, since A% and B are observed by C, they
must be characters abstracted by C from corre-
sponding entities in some more fundamental world
containing C the observer. So we can change B
into B l and can tabulate the two more fundamental
entities as A 3 and J5 2 ; thus:
54
REGRESS OF A SELF-CONSCIOUS OBSERVER
World as observed by B l
A,
World as observed by C
A t
BI
A
B,
C
Here, Cis aware of an objective A 2 , and of B l as
an object which is being modified by the character
AI -
We know that, since B 2 is having its character B l
modified by A l9 it is recording the presence of A.
But to record the presence of A l the character of
A 2 is not to record the presence of A 2 as a whole.
AD as a whole, is not being observed by B 2 , and B 2
is not abstracting A 2 from A 3 . It is C who is
doing that, i.e., A 2 is that character of A 3 which
is relative to C, but it is not in any way relative
to B 2 .
But the regress of self-consciousness, which we
studied in Chapter iv, declares that C itself is only
a 'self observed by a remoter owner, Z>, who is
the real, ultimate observer of the series, as far as
we have considered this.
Now, by our hypothesis, this (so-far) ultimate
observer D has to know that A 2 is an object existing
independently of his self B l . Of course, C records,
as we have seen, the separate existences of A 2 (con-
taining A^) and B l . But these recordings are only
modifications of, or changes in C. The question is,
55 /7ZJJ
THE SERIAL UNIVERSE
again, how can this ultimate observer D know
that A 2 (containing A^ and B l are existing in-
dependently of, and being observed by, C, and
are not merely modifications in the structure of C.
D cannot discover that by merely observing C.
The answer is that to discover that A 2 and B l are
observed by C is to perceive that C abstracts them
from some more fundamental entities. The en-
tities from which C does abstract them are, as we
have seen, A 3 and B 2 . Z), therefore, must perceive
that A 2 and B l are abstracted from A 3 and J5 2 by C.
But, as a preliminary to observing this function of
C, he must be able to observe A^ and B%.
So we can amplify our table by labelling the
third row, 'World as observed by D\
World as observed by B l
World as observed by C
World as observed by D
A,
Then, again, since A 3 and B 2 and C are observed
by D, they must be characters abstracted from
more fundamental entities, A^ B 3 and C 2 , in the
same world as D. So we can change C into C x and
extend our table thus:
56
REGRESS OF A SELF-CONSCIOUS OBSERVER
World as observed by B
A,
World as observed by C^
A*
B l
World as observed by D
A 3
B*
Ci
A*
B*
c*
D
But the regress of self-consciousness insists that
Z), itself, is only a 'self observed by a remoter
owner , and so on ad infinitum.
Clearly, then, if we wish to complete our
analysis of an individual to whom it is c given 5
that his 'self is observing something, we shall
have to extend our table to infinity, repeating
the old arguments for each new entity intro-
duced.
It is to be noted again that the abstractions are
all performed by the series of observers B l) C l9
Z), etc., along the diagonal edge, and not by any
other entities shown in the table. We saw, before,
that B 2 does not abstract A 2 from A^ and similar
arguments will show that B 3 does not abstract A 3
from AD and that C 2 does not abstract B 2 from jB 3 .
This rule must hold good throughout the infinite
regress.
It is evident that, in the four-world table shown,
there is only one world adjudged as being real
the world of the bottom row. The 'worlds' tabu-
lated in the other and upper rows are merely lists of
57
THE SERIAL UNIVERSE
characters abstracted from that more real world
by D employing the primary observing instrument
C x and the secondary instrument B l .
The character of the regress is clear enough. We
have a horizontal series of entities, indicated by the
alphabetical sequence A, B, C, etc., and a vertical
series of characters of those entities, indicated by
the numerals i, 2, 3, etc. The regress of the self-
conscious observer who is aware of an object A l
other than his 'self lies along the diagonal edge
B 19 C 15 A etc.
That the ultimate observer should be able to
treat the series of entities A ly B ly C l9 etc., as in-
dependently existing systems is a condition essen-
tial to his possession of any knowledge of a ' self
situated in an external world. But that is only the
half of our trouble. In order to fulfil our require-
ments, the observer in question must be able to
recognise, not only that A 2 exists independently of
B l9 but also that A l is being observed by B^ which
means that he must be able to perceive that the
modification in B l is caused by the nature of A 2 .
And, similarly, throughout the regress, he must be
able to perceive, not only the separate existences
of the observing instruments and the systems from
which those instruments are abstracting, but also
the fact that the instruments are being affected by
characters of those systems. Now, our present
table does not show how the ultimate observer is
enabled to perceive this : it merely assumes that he
58
REGRESS OF A SELF-CONSCIOUS OBSERVER
can do so. And that, of course, is insufficient for our
purpose.
It will be realised that our test is very drastic.
We have to discover, in our everyday, scientific
methods of describing the universe, some unnoticed
assumption which actually takes into account all that
infinite series of different entities indicated in the
horizontal extension of the table. In addition, this
commonplace method of description has to make it
clear that the ultimate observer will perceive the
observing entities as observing and the observed
entities as observed. And not till we have dis-
covered this immensely significant assumption, and
have shown that all our empirical sciences are
founded upon it, shall we be in a position to assert
that we are self-conscious individuals, aware of
an external world, and employing the regressive
method of the artist and the picture because it
shows in a reliable and useful fashion the otherwise
incomprehensible relation between ourselves and
our universe.
That descriptive convenience exists. We put it to
everyday use. And, if you like to say, in view of the
enormous difficulty of the problem, that any such
device would need to be the product of a master
Mind, I, for one, shall not attempt to contradict
you. But the greater marvel, I think, lies in the
fact that the device which solves the tremendous
problem of rendering systematic an otherwise in-
comprehensible world proves to be, at the same
59
THE SERIAL UNIVERSE
time, of such a character that the veriest half-wit,
lacking all clear understanding of its nature, is
compelled to employ it. The Mind which devises
the method devises it for the advantage of both the
genius and the fool.
60
THE SERIAL UNIVERSE
* *
PART II
GENERAL TEST OF THE THEORY
CHAPTER VII
'NOW
Let M represent a particular configuration of the
external world as this last is described by you from
observation, experiment and calculation. The parti-
cular configuration which M is to represent is the
one which is open to your observation at the
present moment. Let L represent, similarly, a past
configuration remembered. From your knowledge
of L and M you calculate, let us suppose, what will
be the character of a future configuration JV". Your
descriptions are made in the language of classical
science.
If, now, you examine your three descriptions,
you will discover that these amount to no more than
descriptions of three separate worlds. For there is
nothing to show that one description refers to any-
thing more or less real than does another. Equally,
the descriptions give no indication that any of the
configurations are past or present or future.
Further examination brings to light that the
three worlds described differ from one another in
the condition known to science as 'entropy', and
that the nature of this difference is such as to allow
you to consider these worlds as arrangeable in
order of their amount of entropy (an arrangement
which will correspond nicely with our alpha-
63
THE SERIAL UNIVERSE
betical order LMN}. This entropy order we may
hope to describe, presently, (though we are not
yet entitled to do this), as time order. So far, how-
ever, the descriptions fail to show,
(1) That they refer to successive states of one and
the same world, or
(2) That those states have any relation to a
'now 5 .
As we shall see shortly, these two requirements
are merely different ways of expressing the same
thing. We cannot assume condition (i) without as-
suming condition (2). But we need not enter into
that question here. It is sufficient, for the moment,
to note that our descriptions do not fulfil con-
dition (2).
Examining condition (2), we remember that M
was to represent the configuration which is open to
your observation 'now 5 . A doubt assails us here.
For a great many people have supposed that the
notions of a c now 5 and of ' happening in succes-
sion 5 are references to a psychological observer
which ought not to be made. The order exhibited
in our present descriptions Z,, M and JV, provides, it
has been said, all that is needful for scientific pur-
poses.
Very well, suppose we ignore the fact that the
actual starting point of your description was your
observational knowledge of M and your remem-
bered knowledge of L. We have no shadow of right,
of course, to do any such thing; but we are trying
'NOW'
to put ourselves into the position of these objectors.
Let us say that the reference to yourself as the
observer the reference which was implicit in the
demand that M was to represent the configuration
open to your observation at the present moment
was a reference which ought not to have been
made. Let us say, if you like, that the 'now 3 is
psychological though classical psychology was as
'now '-less as classical physics. Let us say, even,
(since we have lapsed into nonsense, and may as
well be hung for a sheep as for a lamb), that the
'now' is an 'illusion'. Good. Our present de-
scription of L, M and JV has been made by your-
self from your memory, observation and calcu-
lation we cannot avoid that but it contains nc
reference to the observer and describer, and nc
unique 'now'. It is, in fact, the description which,
according to these people, describes three tem-
poral 'states', and which they assert to be entirely
sufficient for the practical purposes of any man of
science.
We must agree that it is very satisfactory to have
arrived, by this drastic process of elimination, at a
reliable account of the universe around us. But
how can we be sure that it is reliable? Ah ! that is
the beauty of science as distinguished from mere
philosophy. We can test the truth of its assertions
by actual experiment. Splendid. Let us test the
accuracy of our present descriptions, Z,, M and JV.
Let us make an experiment and see.
FSU 65 5
THE SERIAL UNIVERSE
The best configuration for us to employ for this
purpose will be, I suggest, the one we have de-
scribed as L\ because, by experimenting upon
(altering) that one, we shall be able to note
whether configuration M is changed accord-
ing to the calculated result, and to see, also,
whether the change carries through to configura-
tion JV.
What's that you say? We cannot alter L! Why not?
Because L is past! But we have just agreed that the
world which we have described as L, M and JV" is
devoid of such mystical characteristics as 'past' or
' present ' or ' future ', and that this is the world with
which experimental science has to deal. What, then,
is wrong with my proposal that we should experi-
ment with the state L? Something was omitted from
that description! Well, perhaps you are right. But
what did we omit?
It needs no pointing out that any system which
can be classified as an object to be experimented
upon must be distinguishable arbitrarily or other-
wise from the apparatus which is regarded as
interfering therewith for the purposes of the ex-
periment. The two systems must be treated as
extraneous to each other. Now, the essence of a
scientific description has been, always, that the
validity of the description must be experimentally
verifiable by everyone, including the describer.
This limits the universe which can be described.
It must be one which the describer can regard as
66
'NOW'
extraneous to his instruments and as subject to inter-
ference by these.
But, if the objective universe which is thus
described is regarded by the describer as a series of
'states 5 possessing time order, it is, as we have just
discovered, an essential condition that he regards
his experimental apparatus (the excluded system
which interferes) as operative at only one 'state* in that
apparent temporal series the 'state 5 he calls c now 5 .
And anyone who delegates to him the task of
verification must agree with his verdict concerning
which is that unique, assailable 'state 5 .
But how does the describer know which is this
critical 'state 5 ? What marks the 'now 5 for him? Is
it physical as well as 'psychological 5 ?
Consider this 'now-mark 5 . We know that it has
a reference to the experimenter system. We know
that it is a finger-post reading: 'This way to the
interfering system which we left outside 5 . And,
once we have perceived this, we realise that the
excluded system must include every instrument,
large or small, which exerts pressure upon the object
system, and which thereby experiences recoil.
Consider, again, that we must regard this finger-
post (whatever it may be) as changing from associa-
tion with one configuration of the object series to
association with the configuration which the de-
scriber regards as next in time order. Thus only
can the mark indicate an important aspect of the
problem, viz., that, if the experimenter system
67 5-2
THE SERIAL UNIVERSE
postpones its interference, it will find that its
chance of altering the configuration which was
'now' has gone. The interfering-and-observing
system follows, of course, these changes of the
finger-post.
But, in these circumstances, the excluded instru-
ments of the experimenter system, following the
changes of the 'now', must mark that 'now 5 !
Quite so. And they constitute a physical 'now-
mark' which the observer has made for himself.
For, when he extrapolates the observed system in
time, he leaves his instruments, automatically, at
the psychological 'now 5 .
When we have taken into account this behaviour
of the c now-mark 5 (the observer's instruments)
a behaviour indicating clearly that the series of
configurations in entropy order, pertaining to the
observed system, is being presented to the observer's
instruments in succession we shall be entitled to
say that these configurations have been described,
quite properly, as states successive in time to
those instruments.
And that is the truth about the time device as
employed by all experimental science. It separates
the observed and observing systems in the most
effective fashion possible by providing them with
what are (as easily may be proved) two different
time systems interacting at a 'now*.
68
'NOW'
Now, this simple fact about scientific analysis in
terms of time that a system which is accepted
as obtaining information by experiment must be
treated as an interactor which is (to use simple
metaphors) 'travelling through 5 any 'time map 5
which that acceptance allows us to draft was not
appreciated by the classical employers of the de-
vice. The fact itself is, evidently, a special example
of the general law to which we directed our
attention at the end of Chapter v, viz., that it is
mathematically impossible to treat B (a thing
which is affected) as an additional part of any
system A l which is being described by the way it
affects that B. The materialist, for example, would
have argued that it is possible to add to the se-
quence of material states LMN three correspond-
ing states of a system of material instruments, Imn,
thus > LMN
I m n
and to regard Imn as the system of instruments
which provides the information from which the
description of LMN as material is compiled. And
the reply would be: (i) (On general grounds)
That this would be to commit the mathematical
fallacy of trying to put the observer into a tem-
poral system which has been described by the
temporal features it presents to that observer; and
(2) That as an empirical fact which is merely
illustrative of (i) the experimenting, interfering,
69
THE SERIAL UNIVERSE
pressure-exerting instruments which provide the
information from which the material description
is compiled must, of necessity, be treated by the
describer as confined to a 'now 5 : a state .of affairs
which he must represent thus,
L M N
where is the instrumental system in question. If
we ask: What, then, is represented by Imn in the
materialist's picture? the answer is: The successive
states of some piece of mechanism designed for use
as an instrument but which is not being employed by the
describer as a source of information or as a means of inter-
ference. In that picture both LMN and Imn are being
described by the way they affect the describer's
instruments, which last have not been shown.
In their actual work, all the men of science,
guided by sound intuition, avoided the mate-
rialist's fallacy. They had no clear notion that
they were relegating observer and observed to two
different time systems, or that they were entertain-
ing the idea of a material c now-mark' changing
from association with one state of the system
observed to association with the next. But they did
this, unconsciously, whenever they separated the experi-
menter and his instruments from the system to be experi-
mented upon, and accepted that experimenter's view of the
object system as a series of states in time order. And they
did that in every experiment they made.
70
'NOW'
Before we go on, there is one rather remarkable
fact to which we should direct attention. All this
means that 'determinism' is 'non-suited'. Not
only has it no case to present: it never had a case.
Classical science involves, employs and asserts the
contrary view the view of every observer as an
external potential interferer with an otherwise de-
terminate universe. We need no microscopic 'Un-
certainty Principle' to assist us there. The deter-
minist bogey that alleged offspring of classical
science was never even conceived, and the birth
certificate signed by the materialist was a fake.
CHAPTER VIII
REGRESS OF TIME
We have seen that time is an analytical device
which effects the sharpest possible distinction be-
tween subject and object. We can see, also, that
each person will apply it differently. Jones will
regard the system upon which he is experimenting
(which may include Smith) as a series of states in
an objective time order, while he treats his instru-
ments of observation and interference as confined
to a 'now' which changes from association with
one state of the system observed to association with
the next. Smith will regard Jones (and Jones's
instruments) as pertaining to the objective time
series, while considering that it is his own instru-
ments which are excluded and confined to a c now'.
Thus, Jones's instruments may be considered, in
some cases, as belonging to a series of objective
states, and, in other cases, as confined to a changing
'now'; according to whether we are employing
Smith or Jones as our source of information.
Obviously, then, analysis in terms of c time' is
merely a mathematical convenience. And it is one
which gives the maximum prominence to the sub-
ject-object relation. We need not be surprised,
therefore, if we discover, presently, that its mathe-
matical character is regressive.
72
REGRESS OF TIME
In the last chapter, we represented the three
distinctive entropy configurations by three letters,
Z,, M and Ji. This was in anticipation of the later
stage where we should be able to regard those con-
figurations as successive in time to the observer's
instruments. The alphabetical sequence of the letters
would serve, at that stage, to indicate the order of
succession of the states of the observed system. Now,
although we may, for convenience, write the letters
in a row, it must be understood that this positional
arrangement is not essential to the argument. We
could, if you preferred it, write the letters on
counters and shake these up in a bag. The entropy
order which indicated the time order would still be
indicated, quite adequately, by the alphabetical
order of the letters.
We have seen that the 'now-mark' which indi-
cates the presence of our experimental instruments
must be thought of as changing from association
with one state of the system observed to association
with the next in whatever represents, to those instru-
ments, the order of objective events. In the state
of affairs we have been imagining as confronting us,
we assumed the ' now-mark ' to be at Af, thus,
L @ JV
the mark being represented here by a circle en-
closing the significant letter. In this state of affairs,
M is present, L is past, and JV" is future to the
instruments in question. But we may not think of
73
THE SERIAL UNIVERSE
the mark as remaining indefinitely at Af, allowing
us as much time as we desire to prepare for an
experiment on the basis of that present state of
affairs. A little later on we shall find that the mark
is associated no longer with M. We may have to
represent that future state of affairs thus,
L M
where JV is present and L and M are both past to
the instruments concerned.
Again, we have realised that we cannot experi-
ment with Z,, because L is past (to the instruments) .
But we have to recognise that there was a past state
of affairs where L was present and M and JV* were
both future (to the instruments), a state which we
may represent thus:
(L) M N
Now, what precisely did we mean when we said
that (Z)AfjV represents a 'past' state of affairs, that
L(M)N represents the 'present 3 state of affairs, and
that LM(N) represents a 'future 5 state of affairs?
Let us label these three states of affairs i , 2 and 3,
and let us place them (for convenience) one above
the other, thus:
3. L M (N)
2. L @ N
I. (L) M N
74
REGRESS OF TIME
We know that M represents that entropy con-
figuration of the observed system which we re-
garded originally as 'present 5 , and that we ac-
cepted L as 'past' and N as 'future'. For that
reason we placed the c now-mark 3 at M. But we
are realising now that this mark has changed from
association with L to its present association with M,
and is going to change to association with N. We
intend, therefore, that (5f) shall indicate the pre-
sent state of the 'now-mark* (i.e., of the instruments) .
Similarly, we intend that (Z) shall indicate a past
state, and (#) a future state of the c now-mark' '. But
these intended past, present and future states of
the now-mark, (/f), (M) and (N) are being regarded
as successive in a time order which can be repre-
sented only by our numerals i, 2 and 3 !
Clearly, in that time order, the three states of
affairs i. (~L)MN> 2. L(M)N and 3. LM(N) repre-
sent successive states of a more comprehensive
system a system which includes the three object
states L, M and N plus the changing 'now-mark 5 .
Now, if M is to be present to the instruments, (M)
must represent (as we have just said) the present
state of the c now-mark', and this means, in turn,
that 2. L(M)N must represent that state of the
more comprehensive system which is present in the
more fundamental time order indicated by the
numerical sequence i, 2 and 3. But our descrip-
tions do not indicate this! For all they tell us,
i. (l?)MN or 3. LM(N) might indicate, equally
75
THE SERIAL UNIVERSE
well, the present state of this circular mark. Clearly,
then, we must add to the states i , 2 and 3 of our
more comprehensive system a new 'now-mark 3
indicating that 2. L\M)N is present in the more
fundamental time concerned. We can do this by
enclosing 2. Z(M)JV within an oblong, thus:
M
2.
L (A
f) N
1,
M
N
It is clear enough that the time order indicated
by i, 2 and 3 is more fundamental than the merely
apparent time order which we indicated by the
alphabetical sequence Z, M and JV*. For it is the as-
sociation of the oblong 'now-mark 5 with 2. Z,(M)jV
which makes (M) the present state of the circular
mark Q> an d which, thereby, indicates M as the
' present 5 state of the originally considered system.
If the oblong ' now-mark 3 were to enclose 3. LM(N)
then (#) would be the present state of the 'now-
mark 5 , and jVthe 'present 5 state of the originally
considered system despite the fact that M in 2 is
enclosed also by a circular mark.
It will be asked: Since we are trying to regard
real time order as represented by the succession of
the more comprehensive states i, 2 and 3, what
was indicated by the entropy order of the original,
less comprehensive configurations Z, M and JV?
REGRESS OF TIME
Are we to try to imagine the more comprehensive
system as embracing two kinds of time?
Certainly not : the more comprehensive system
possesses only one time order, viz., that indicated
by i, 2 and 3. It contains, also, all present, the
items of the order indicated by JL, M and JV; but
that order is not time order in the more compre-
hensive state of affairs. Then what sort of order is it?
Well, I am going to answer that question in the
next chapter; but I have a particular reason for not
wishing to do so here. In this chapter I am con-
cerned to show only that real time order is the re-
ceding element in an infinite regress. As such, we
shall be coming continually upon orders which
serve the purpose of time order for the particular
stage we happen to have reached in the regress,
but which turn out to be something different from
time order when we get on to the next stage, just
as each c child' in the fictitious ancestry regress
turns out to be c grandchild' in the next stage. But
What that 'something different' is in the case of
time, we need not consider at this moment.
Before we go any further, we had better note that
the placing of our three second-term states of
affairs i, 2 and 3 one above the other on the page
is in no way essential to the arguments we have
used. These would proceed in precisely the same
way if Z, M and JV had been written on counters
shaken up in a bag. We should have required three
such bags to represent the three distinctive states
77
THE SERIAL UNIVERSE
of affairs where the circular c now-mark 5 surrounds,
respectively, L and M and JV. And, to make the
bag containing (M) represent the present state of
affairs, we should have had to label the three bags,
i, 2 and 3, and then add another label, represent-
ing a second-term c now-mark 5 , to the bag marked
2 and containing (Af).
To prove that real time order recedes in an in-
finite regress, we have to show that the arguments
which led us from first-term time to second-term
time are bound to repeat themselves thereafter.
We have arrived at a system containing three
second-term states, of which states, number 2 is
surrounded by an oblong 'now-mark 5 . We repre-
sented the total system thus,
3.
M
2.
L (^
f) N
1,
M
N
and we noted that it is the presence of the second-
term, oblong, 'now-mark 5 which makes (A?) in 2
(instead of (Z) in i or (N) in 3) the first-term
4 present 5 configuration with which we started.
State i is, thus, past, and state 3 is future. But we
have agreed that (N) will become, in a little while,
the first-term configuration which is thus uniquely
'now'. But, for (N) in 3 to become thus uniquely
'now 5 , the seconcTterm, oblong c now-mark 5 must
78
REGRESS OF TIME
change from association with 2 to association with
3. States i and 2 will be then both past. Again,
(Z) was once 'now', and the oblong c now-mark 3
must then have enclosed i . (Z)MJV. States 2 and 3
were then both future. Consequently, we are con-
fronted with three different states of the whole col-
lection of letters and c now-marks 3 so far dealt with
three states each containing i , 2 and 3 plus an
oblong 'now-mark 3 , but with this mark associated
respectively with i, 2 and 3. And one of those
third-term states (the one where the oblong c now-
mark 3 encloses 2) will have to be enclosed in a
third-term c now-mark 3 . Or (to employ the other
method) we shall need three sacks, each containing
three bags with counters, with a c now-label ' on one
sack, a ' now-label 3 on one bag in each sack, and a
c now-mark 3 on one counter in each bag, in order
to show that one unique counter of all the lot
represents the first- term c present 3 state with which
we started. And so it must go on ad infinitum.
It is to be noted, particularly, that nowhere in
the analysis of this regress have we introduced a
new hypothesis. We do not state that the first-term
series LMN may be the present state of a more
comprehensive system : we show that it must be so.
We show, in brief, that the entire regress was
implicit in our opening statement that M is the
'present 3 state of three states of the observed
system. That, of course, is a characteristic of all
regressions: they do not proceed by adding new
79
THE SERIAL UNIVERSE
terms, but by showing that the existence of one
term with a dual character involves the existence
of all the remainder.
We are going to abandon, in a little while, the
method of representing our series of states by letters
of the alphabet or by numbers. We shall represent
the original states by dots, and their intended time
order by the space order in which those dots are
placed in the page. That, of course, is the conven-
tional scientific way of picturing time. We shall
represent the changes of the 'now-mark' by
changes in its position on the page ; that is to say, we
shall imagine it as moving over the row of dots re-
presenting first-term temporal states. This is a far
easier method of studying our present problems.
But it begins by what a few people would regard as
begging a question. Is it legitimate to use space
order for our first attempt at representing time
order? Actually, the answer is, yes; but the point
is a very subtle one, and many people who have not
gone deep enough into the matter would answer,
no. Such persons might then proceed to the further
error of supposing that the entire regress arose from
our having begun by trying to represent time in
an erroneous fashion. It is to avoid that objection
that the present chapter has been written. The
Bergsonians (the people with whom we are argu-
ing) admit that states of time are distinctive and
80
REGRESS OF TIME
successive, but deny that they can be regarded as
separated in the way that points of space are
separated. Very well, our original descriptions of
the three distinctive configurations Z,, M and Ji do
not indicate that these are separated. For all that
the descriptions tell us, M might contain L, and N
might contain L and M . Again, the configurations
are imagined; and, for all that their three descrip-
tions tell us, we might be dealing with three con-
figurations pertaining to three different worlds
imagined at three different times by three different
people. We intended, of course, that our descrip-
tions should convey more than this; but we found
that they failed to do so. They indicated nothing
but the existence of an entropy order in the three
imagined configurations. They do not suggest that
there is, or that there is not, any connection what-
ever between the three configurations: they in-
dicate merely the differences between these the
fact that they are distinctive.
Next, we note that nowhere have we used space
order to represent time order. It is true that the
counters in our bags are spatially separated, but
their space orders in the bags may be changed as
often as we please (by shaking the bags) without
this affecting the alphabetical sequence corre-
sponding to that entropy order which we hope to
be able to regard as sequential successive in
time.
Next, we were particularly careful not to say
FSU 8l 6
THE SERIAL UNIVERSE
that the 'now-marks 5 moved from one state to
another; for to do this would have been to declare
that the states were being presumed to be spatially
separated. We said, instead, that the marks
'changed from association with 5 the next in what-
ever series we had been hoping, previously, to
regard as real time order. That change, again, was
not presupposed; it was discovered to be an em-
pirical fact that our chance of interfering with any
particular 'state 5 of the object system would van-
ish, and would be replaced by a chance of inter-
fering with the 'state 5 which came next in what we
.were trying to regard as time order. It may be
urged that the admission of this behaviour of the
c now-mark 5 is an admission that the states are
separated in the same way that points in space are
separated. Quite so. But this new view of the re-
lationship between the states is a development of our
original, less explicit view a development forced
upon us by the logical development of the regress.
The new view is one which we have endeavoured to
avoid, and had successfully avoided up till that
moment. It is a consequence of the regress, and not
a primary supposition causing the regress.
Finally, suppose we think of the distinctive state
L as changing into the distinctive state Af, while
thinking of the observing entity outside the system
as changeless (except when observing) . Is that the
same thing as thinking of the observer as changing
from association with the state L to association with
82
REGRESS OF TIME
the state M ? Of course it is. We are thinking of
the states L and M as being successively associated
with the unchanging observer; and it comes to
precisely the same thing whether we say that L
and Af are successively associated with the observer,
or that the observer is successively associated with
L and M.
83 (HI
CHAPTER IX
REGRESS OF 'REALITY'. REGRESS OF
PHYSICS. SPATIAL REPRESENTATION
OF TIME
You will remember that we began by saying that
M was to be our description of the state which is
open to your observation at the present moment,
and that L and JV were to be described from
memory and calculation respectively. According
to popular notions, those descriptions should have
shown M as real and L and JV" as unreal. They
did not do so. They exhibited only three differing
conditions of entropy with no reality distinction
between them. Equally, the descriptions gave no
indication that any of those conditions were pre-
sent or past or future. Putting in the 'now-mark 5
at M rectified the latter deficiency. But did this
addition reduce L and N to descriptions of the
unreal states contemplated in the popular view?
We had best, I think, call upon one of the ex-
ponents of this common opinion and ask him what,
precisely, is he trying to assert. His answer is as
follows. M is a state which exists 'now 5 . L is a
state which did exist once but does not exist 'now 5 .
JV is a state which will exist but does not exist
'now 5 . To say that states do not exist 'now 5 is to
say that they are 'now 5 unreal.
We reply to this by asking him to which 'now 5 is
84
REGRESS OF 'REALITY*
he referring. Of course L and JVdo not exist at the
first-term 'now 5 we have been at pains to show
that. But, certainly, they exist all right in the
second-term c now 5 .
This does not satisfy him. He suspects that our
arrival at the second-term 'now 5 depended in
some way on a presumption that L and N were
existing states (which, of course, would have been
begging the question). If, he thinks, we had been
quite clear about the non-existence of these states
when we referred them to the first and only 'now 5
we recognised at that stage, the regress would not
have developed.
He is quite wrong. Let us suppose that the first-
term 'now-mark 5 is, as he wishes, a mark con-
ferring reality on the state described. Good: M
describes a state which is real; L and JV are de-
scriptions of unreal states unreal simply because
they are not existing now. But, by his own ac-
count, JV will be in a little interval of absolute
time the description of a real state existing 'now 5 ,
and L was once the description of such a real state.
Analysing this conception, we find that it is simply
the concept of our second-term state of affairs,
8. L M
2.
L @
N
1.
M
N
85
THE SERIAL UNIVERSE
where second-term time is real time, and first-term
time is only a pseudo-time. Here, i contains de-
scriptions of three states which are all past, while 3
contains descriptions of three states which are all
future in absolute time. Consequently, none of
those six states is real.
But i and 3 each contain the original c now-
mark 5 which was regarded by our friend as con-
ferring reality. So this mark has lost its supposed
potency. It does not represent anything beyond a
description of the observer's three-dimensional instruments
and it gives three descriptions of these; viz., as
past or unreal, as present or real and as future or
unreal in real time. But the recognition of them
as present or real (in 2) is not due to anything dis-
tinctive in their description : it is due solely to the
fact that everything in 2 is defined as a description
of some state present in real time.
So this popular definition of reality regresses.
And that means that it is only a definition of re-
lative reality. It means that the state M seems real
to the instrument simply because it is the state
Which is being observed by that instrument. But
that we regard it as real depends, obviously, upon
whether we are regarding the instrument as real.
And the nature of the regress is such that, when we
are regarding the instrument as real, we are regarding
as equally real all states which are past or future in fast-
term time.
It may seem strange that an attempt to regard
86
REGRESS OF ^REALITY*
the past and future as unreal should break down
in this hopeless fashion. But the fact is that nobody
actually has ever thought of them as unreal. We
think of them merely as ' having been real ', and fail
to notice that this is thinking of them as real in
what we are regarding as real time.
We have arrived at a satisfactory account of the
man-in- the-street's views ; but we must attend now
to an interruption by a physicist of the old school.
'You admit', he says, 'that this first-term reality
of yours is relative to the instrument. Well, that is
the only kind of reality in which I am interested.
I do not even consider whether my instruments are
real they are outside my picture. That picture
is concerned only with what it is that affects the
instruments.'
We will allow him to maintain this view, but only
on one condition. He must agree that this 'real 5
world which he is examining with his instruments
is one which he has never tested by experiment
has never altered. If he has altered it, it is a world
in which his instruments have played a part other
than that of mere observation; and to account for
the present state of that world is to take into con-
sideration the extent of the interference by the
instruments to consider, that is to say, the quan-
tity of energy they have supplied. I think our
classical physicist will prefer to bide his time and
look for some weaker point of attack. Meanwhile,
we may ask him to consider whether, if he con-
THE SERIAL UNIVERSE
templates any further experiments, he is regarding
the future of his world as stable or unstable. (I pre-
fer those words to ' certain' and 'uncertain 5 , which
do not mean precisely the same thing.)
But here is a modern physicist with a perfectly
legitimate question. 'Do you 5 , he asks, 'regard
this second- term "now-mark 55 of yours as physical
or merely as psychological? If the latter, it has
nothing to do with my science and I am not com-
pelled to take it into account. I can see, of course,
that if I have to recognise it, I am launched, past
all saving, upon an infinite regress. But you must
not expect me to take this critical step except under
dire compulsion. 5
I am afraid that compulsion is there. Glance
back at the last diagram. The circles enclosing L
in i, M in 2 and N in 3 represent past, present
and future states of the interfering instrument. To
make your experiment, you must, at some time or
other, do something to your instrument ; you must
move, at least, some of its parts. But you cannot
alter a past state of the instrument : you can act
upon it only when it is in the state which you re-
gard as present. Consider what that means. You
can alter the instrument in 2. Z,(M)jV, associated
with the second-term c now-mark 5 ; but you cannot
alter it in 3. LM(N) until (in absolute time) the
second-term 'now-mark 5 has changed to associa-
tion with this state of affairs. Thus, the second-
term 'now-mark 5 represents to you a facility for
REGRESS OF PHYSICS
moving the instrument. The increase in the instru-
ment's momentum results, in the course of the
experiment, in an increased momentum of the
original object system. So, the second-term c now-
mark 3 is a facility for adding momentum to the
original object. Such a facility must be physical;
and the physicist is obliged to take it into account
for the same reason that compels him to take into
consideration the instrument viz., because it is a
cause of the observed behaviour of the external
world.
Before, however, we attempt to elucidate in
greater detail the physical aspects of the time re-
gress, it would be advisable for us to see whether our
present analytical method is not open to simpli-
fication. Our treatment of states and 'now-marks 5
has been, so far, entirely algebraical a matter of
the manipulation of five signs, viz., L, Af, JV, Q
an d I "1- The spatial order in which we
have distributed these signs upon the pages of the
book has had no significance of any kind. But
most algebra is amenable to simple pictorial illus-
tration, and we may as well make use of this fact
in the present case. Readers who do not like
diagrams may, however, continue to employ our
past method of treating these problems: our dia-
grams will introduce nothing that cannot be ex-
pressed by continued combinations of algebraical
signs. But those combinations would become
immensely complicated.
89
THE SERIAL UNIVERSE
Our three original states of entropy Z, M and JV
exhibit what is called ' betweenness ' order. M
comes ' between 5 L and JV; and this holds good
even though M be merely a more broken-up Z, and
JV be merely L in a greater condition of internal
mixture. Now, we can think of intermediate con-
ditions between L and M and between M and JV",
and of intermediate conditions again between the
five states thus considered. And we can continue
this process indefinitely. We do not need, however,
to carry it so far as to produce an infinite number of
states. Before we reach such a condition of affairs
we shall have arrived at a curious mathematical
phase in our process; we shall have unearthed the
notion of what mathematicians call a 'limit 3 . Then
we shall be able to regard our immense number
of states as constituting what we can recognise as
a 'Continuum'.
Consider, now, the first and second terms of our
series of more and more comprehensive systems.
We can tabulate these as follows:
Present in the system apparent
to the observer's instruments
M
Present in the more compre-
hensive system known to us
LMN
The observer's
instruments
In the second-term system, L, M and JV are being
treated as:
SPATIAL REPRESENTATION OF TIME
(1) Of the same class (entropy configuration).
(2) Equally real.
(3) Parts of a continuum.
(4) Equally present.
(5) Associated with an independently existing
thing which changes from one to the other.
And, if you wished to describe three configurations
as separated in space, you could say no more about
them than we have said of Z, M and JV.*
So, although we begin by using the entropy
order of the three configurations to represent their
time order, the result is the discovery that, in the
second term of an inevitable regress, this entropy
order represents order in an unsuspected dimension
of space. And it is clear that whatever we may select,
at the outset," to represent time order must repre-
sent, at the next stage, nothing but space order.
In other words, time order must change to space
order at each stage of the infinite regress of real time.
I shall raise no objection if you prefer to speak
of this new dimension as * configuration space',
meaning thereby a mathematical device to be dis-
tinguished from the 'ordinary 5 three-dimensional
space of the first-term system M. It is part of our
argument that analysis in terms of time is a purely
mathematical device. The essential thing is to
recognise that this space, ' configuration ' or other-
wise, is space and not time in the second term of the
regress. And, as such, the ' betweenness 5 order of
* The relativity aspects of this matter are considered later.
91
THE SERIAL UNIVERSE
LM 'N therein may be adequately represented by
the positions of three dots on a sheet of paper,
while the c now-mark' may be represented by a
fourth dot, superposed on Af, and with its presence
indicated by a letter O^thus:
L M N
This represents the c betweenness 5 order of
LMN\ but that is not enough. We have to indicate
also that the arrangement will appear to the
observer's instruments as time order. In other
words, we must show that will regard LMJV as
a sequence in which L comes first and JV last. That
condition, however, is satisfied if we add an arrow
to show the direction in which is moving along
the newly-discovered dimension, thus:
L M M
We have now an excellent graphical representa-
tion of first-term time order. But we have not yet
shown that the three configurations Z,, M and JV
represent three successive states of one and the same
world external to 0. We have to introduce the
> notion of continuity. This we can effect by drawing
a continuous line from L to JV, thus:
L M jy
SPATIAL REPRESENTATION OF TIME
Then any point in that line will represent one
particular configuration of the world external to 0,
and the whole line will represent the endurance of
that world in first-term time.
Since the time sequence of these states is in-
dicated by the arrow, we can abandon the alpha-
betical sequence of the letters LA/JVas redundant.
A line labelled, say, GH 9 with an somewhere
between G and H to indicate the position of the
c now-mark 3 , and an arrow to show the direction of
its travel, thus,
G H
will be ample for our purpose.
And that is the method which was adopted when
the time regress first was analysed. This was
effected in a book called An Experiment with Time,
published in March 1927. The method has great
advantages of simplicity, and we shall employ it
for the remainder of our present demonstration.
93
CHAPTER X
DIMENSIONS, MAGNITUDES
AND MESH-SYSTEMS
I must ask permission to make a digression. The
present reader, no doubt, is well acquainted with
the meaning of the word ' dimension'. But I have
in mind a potential peruser of these pages who
happens to be a little hazy in his ideas on this sub-
ject. The digression is intended for his benefit.
A dimension is neither a line nor strictly speak-
ing a magnitude. It is a manner in which some-
thing may be measured. For example, 'momentum*
consists of c mass ' multiplied by the velocity with
which that mass moves. Consequently, it has to be
measured in two totally distinct ways one dealing
with the mass and the other with the velocity. It
possesses, therefore, two dimensions. We could say
that mass and velocity are the two dimensions of
a momentum, even though we did not know the
amount of mass or the amount of velocity possessed
by the particular momentum we are considering.
Those amounts would be the magnitudes, and would
need to be indicated by numerical figures ; whereas
the two dimensions can be represented simply by the
symbols M (meaning mass) and V (meaning
velocity).
94
DIMENSIONS, MAGNITUDES Sf MESH-SYSTEMS
Spatial dimensions provide us with a very con-
venient way of representing other dimensions. For
example, we can employ the up-and-down dimen-
sion of this page to represent mass, and the side-to-
side dimension to represent velocity. To indicate
the amount of mass, we need a line OT laid down
somewhere in the up-and-down dimension and
marked off with a scale representing units of mass.
Similarly, to indicate the amount of velocity, we
need a line OX laid somewhere in the horizontal
dimension and marked with a scale indicating units
of velocity. But to employ the two dimensions of
the paper to indicate the amount of momentum,
we must place the two scales so that they meet at a
common point 0, and start the scale measurements
from that point; thus:
O 1 2
FIGURE 4.
95
THE SERIAL UNIVERSE
The two scales OX and OTare called c axes', and
the point at which they meet is called the
'origin'. You will notice that I have made the
divisions on one scale quite different from -those on
the other. It is often a matter of pure convenience
what sized scale you choose to employ in each
case.
Now, consider any point #, placed in the angle
between the two lines. The height of that point
above OX, that is to say, its distance from OX
in the up-and-down dimension, will give you a
measurement of mass. You discover the amount of
this by drawing a line through #, parallel to OX,
to cut the scale on OT. In the present case, the
mass magnitude thus indicated is 6. Again, the
horizontal distance of a from OT will give you a
measurement of velocity, the value of which you
ascertain by dropping a perpendicular line from a
to cut the scale on OX. The velocity magnitude
indicated in this case is 2. Thus, the position of a
with regard to the two axes indicates a mass magni-
tude of 6 and* a velocity magnitude of 2, that is to
say, a momentum magnitude of 6 x 2 = 12. The
two magnitudes of mass and velocity (viz., 6 and 2)
are called the 'coordinates 5 of the momentum.
The trouble about this dodge of using the dimen-
sion of surfaces to represent dimensions of other
kinds is that the surface has only two dimensions
available for the purpose. We can use a drawing in
perspective to indicate a third you can imagine,
96
DIMENSIONS, MAGNITUDES & MESH-SYSTEMS
that is to say, a third axis sticking out from the
page towards your eye but this is a rather cum-
bersome device; and, when the dimensions with
which we have to deal exceed two in number, it is
more convenient to choose the two of these which
you wish most to represent diagrammatically, and
to refer to the others by letters of the alphabet. The
treatment of those others is, of course, algebraical,
while the treatment of the chosen two is pictorial;
but this combination of treatments is quite easy and
quite legitimate, since the diagrams are, really,
only pictorial algebra. The point to be grasped
here, however, is that, if you have to deal with
something possessing a hundred dimensions, you
can select any pair of these for pictorial treatment
the twenty-first and the seventy-fifth, for ex-
ample, if it suits you sticking to algebraical treat-
ment for the remainder.
Let us return now to FIGURE 4, and let us draw
through each of the divisions of the scale on OT a
line parallel to OX. If we draw then through each
of the divisions on OX a line parallel to OT, we
shall have a network of crossing lines, as below.
This arrangement is called a c mesh-system'.
You will notice that the two lines we drew from
a in FIGURE 4 (a horizontal line to 6 in T, and a ver-
tical line to 2 in OX) were really two of the crossing
lines of the mesh-system shown in FIGURE 5. In fact,
FIGURE 4 was simply FIGURE 5 with a lot of the lines
of the mesh-system left out for purposes of clarity.
FSU 97 7
THE SERIAL UNIVERSE
a
O 1 2 3 4 X
FIGURE 5.
Now, suppose that our axes of X and T meeting
at were used to indicate, not measurements of
momentum and velocity, but measurements of dis-
tance in space. Distance from what? Well, look at
FIGURE 6. Y
4 -
23
FIGURE 6.
98
DIMENSIONS, MAGNITUDES fif MESH-SYSTEMS
Clearly, the scale on OT shows that the point b is
3 space-units distant from the axis OJf, while the
scale on OX shows that the point in question is
4 space-units from the axis OT. Conversely, if we
were told that the coordinates of some other point
C were 3 in the horizontal dimension and 2 in the
vertical dimension, we could place that point on
the paper by drawing a vertical line upward from
3 on the X scale, and intersecting this by a hori-
zontal line drawn from 2 on the T scale; thus:
4 -
1
1
f 1
1
1
1
1
1
1
I I
1
9 1 2
* 4
FIGURE 7.
X
With the aid of the readings on the two scales,
and a little knowledge of elementary Euclid, we
can calculate the direct distance in space-units be-
tween the two points c and b. But, if we propose to
make the calculation, we must make the divisions
representing inches on the T scale equal to those
99 7-2
THE SERIAL UNIVERSE
representing inches on the X scale. Consequently,
the meshes of the mesh-system supposing that we
fill this in will consist of four-sided figures with
all the sides equal.
Let us consider, next, a diagram which is fairly
common in this era of influenza, viz., a tempera-
ture chart. Here we are using the scale on the
vertical axis to indicate the height of the mercury
in the thermometer (a space measure), so we may
call this axis, the axis of S (S standing for space).
The scale on the horizontal axis indicates periods of
time as told by some clock, and we may label this
axis, T. Here is one such chart.
S
103
102
101
100
99*
98
97
3
FIGURE 8.
It seems to indicate malaria rather than 'flu, but
that is immaterial to you and me. The point I want
you to notice is that I have made the vertical
100
DIMENSIONS, MAGNITUDES & MESH-SYSTEMS
spaces in the mesh-system much smaller than the
horizontal spaces, and that this is immaterial. That
is because the doctor is not interested in deter-
mining the lengths of the lines joining the points, but
wishes to know only what was the height of the
thermometer at certain instants of time. Any mesh-
system will serve to inform him of this.
The nurse shows by round blobs the points
where the patient's temperature was actually taken,
and the lines joining the blobs are largely matters
of guesswork. It is precisely such a line, however,
which is called, in Relativity parlance, a 'world-
line'. Now, the relativist is particularly interested
in determining the lengths of portions of such a
world-line by methods which bear some analogy to
the Euclidean calculation referred to earlier. Con-
sequently, the nurse's mesh-system is not the sort
of thing he likes. He prefers to make the space
divisions of his mesh-system equal to the time
divisions. How he contrives to make a period of
time equal to a length of space is a matter we may
discuss later.
Before proceeding with our analysis, it will be
advisable to remind ourselves of a fact which was
recognised by physics and philosophy long before
Einstein embodied it in his greater 'relativity'
the fact that all measurements of velocity are
relative to something. Now, the observer's instru-
101
THE SERIAL UNIVERSE
ment for determining the velocity of anything in
the system observed can record only such velocities
as are relative to that instrument. Suppose, then, that
the observer, employing such an instrument as his
source of information, prepares a space and time
diagram exhibiting the spatial positions of the
various parts of the observed system at different
instants of time. The world-lines thus constructed
will show, of course, by their inclinations to the
axis which indicates time, whether the objects to
which they refer are being regarded as moving in
space or as at rest. And the time axis will represent
the track, along the time dimension, of the observ-
ing instrument. The instrument itself is not shown
the diagram is a space and time map of the
entities of the observed system only.
102
CHAPTER XI
GRAPHICAL ANALYSIS OF THE
TIME REGRESS
Let us return now to our graphical representation
of that second-term, more comprehensive system
which includes the successive states of the first-
term system plus the observer's instruments. We
had discovered that the series of states of the
original observed system could be treated as cross-
sections of a continuous line GH representing the
endurance, in first-term time, of that system, as in
the figure below,
G H
FIGURE Q.
(FIGURE 8 of An Experiment with Time, first edition.)
and that the observer's instruments the physical
c now-mark' could be represented by a point
superposed on that line somewhere between G and H. .
We had ascertained that the actual world which we
represent by GH must be thought of as extended in a
hitherto unconsidered dimension of space* (a fourth
dimension), that the observer's instruments repre-
sented by must be regarded as travelling in that
dimension, (the direction of travel being indicated
* We shall see, later on, that this involves no contradiction
of relativity.
103
THE SERIAL UNIVERSE
by the arrow in the diagram), and that the points
on GH must be considered, consequently, to be ap-
pearing to the observer's instruments as the succes-
sive states of an ordinary, three-dimensional world.
It will be remembered that these instruments are
interfering instruments exerting force upon the
object system and, so, are observing, by reaction,
that inertia which is the characteristic of mass.
The states represented by the points in GH are
supposed to be described by us from information
obtained by use of the instruments at 0. The pro-
cess is somewhat analogous to that by which a
man, having thrown, through a narrow vertical
slit, a searchlight beam upon a dark external
world, has prepared, from the glimpses thus
obtained, a map of a countryside through which,
he judges on other grounds, the searchlight and
the slit, contained in a railway carriage, are pass-
ing. The analogy assumes that the man can
estimate, from what has been seen, the probable
character of the country to which he is coming;
but, that much being allowed, it is obvious that he
could both prepare his map and mark upon it the
present position of the travelling searchlight.
The successive states of our second-term world
will consist of a series of pictures like FIGURE 9,
with the c now-mark 5 at different places in each
picture. FIGURE 9 the whole of it will be the
present state of this more comprehensive world.
States where is nearer to G will be past states and
104
GRAPHICAL ANALYSIS OF TIME REGRESS
states where is nearer to //will be future states
in what we are regarding now as real time. Here
the arguments of Chapter ix repeat themselves.
The states of second-term time, showing the suc-
cessive positions of as this travels along GH,
possess ' betweenness J order; and may be exhibited
as a continuum (which is, of course, only a way of
showing that the motion of along GH is being
regarded as continuous). Then, since the second-
term c now-mark 3 represents, as we saw earlier,
something to which we are obliged to give a
physical significance, and since this physical thing
is changing from association with one part of the
new continuum to the next part in order of con-
tinuity, we may represent second-term time by
a dimension of space over which the second-term
c now-mark 5 is travelling. We have to consider,
however, that three dimensions of space are re-
served for c ordinary' space in which the parts of
the object system have different positions at dif-
ferent instants of first-term time, and we are con-
sidering that a fourth dimension of space is being
employed to represent first-term time order. Con-
sequently, the new continuum in which we indicate
second-term time order will necessitate our em-
ploying a fifth dimension of space. The surface of
our paper will allow us to represent this very
nicely; the side-to-side dimension representing, as
before, the fourth dimension, the up-and-down
dimension representing the fifth dimension, while
105
THE SERIAL UNIVERSE
the three dimensions of 'ordinary 5 space are left
out of it for algebraical treatment.
Here, however, a difficulty confronts the printer
of the book. Strictly speaking, we should begin
with the representation of our second-term world
by the line GH and the point as in FIGURE 9.
That would indicate the present position of the first-
term c now-mark' 0. We should then draw similar
horizontal lines below this line to represent past
states of this second-term world, (with nearer to
G), and we should draw another set of lines above
GH to represent future states (with nearer to H] .
But, to get continuity in second-term time, we
should have to draw these lines so close together
that no gaps could be noticed between them. The
result would be a completely black block on which
we should be unable to indicate the varying
positions of 0. There are two ways of dodging that
difficulty. We can separate the horizontal lines thus,
tjr
G-
... IJL
m 71
IJ[
G.
77
. b I
ri
"
tr
. 11
. I
. 6 I
ri
n
m U rr
e <
f) "
nQ
77
1 06
GRAPHICAL ANALYSIS OF TIME REGRESS
or we may consider only a few points in GH 9 thus,
G d *
and draw the past and future states of this row of
dots above and below. Then, when we have drawn
the vertical lines connecting the selected points in
GH with the corresponding points above and below,
and have represented the continuity of the experi-
menter's instruments by a diagonal line linking
together all the O's, we shall have a picture like
this:
G
T.2
T.1-
O'
G'
-T.1
H"
0"
H'
T.2
FIGURE IO.
107
THE SERIAL UNIVERSE
The chief advantage of this diagram is that it
throws into relief the world-lines which pertain to
second-term time. I had best, perhaps, explain
this at some length. The two little crossed lines
drawn by the left-hand bottom corner of the figure
serve very much the same purpose as does the
little compass one finds printed in the corners of
some maps. The compass shows which dimension
of the map represents North and South, and which
represents East and West. Our present little cross
shows which dimension of our paper represents
first-term time, and which represents second-term
time. First- term time we shall refer to in future as
' Time i ' : second-term time we shall speak of as
c Time 2 ' . Time i , which we had hoped, originally,
to be able to treat as real, absolute time, has
turned out to be merely a fourth dimension of
space in which the original observed system is ex-
tended. Time 2, which takes into account the
motion of the first-term instruments along the
fourth dimension, we are regarding as absolute
time; but we are representing it by the up-and-
down dimension of the paper in anticipation of the
step where we shall have to regard it, not as real
time, but as a fifth dimension of space which will
happen when we take the motion of the second-
term 'now-mark 5 into consideration. The line
O'O" shows the positions which the experimenter's
instruments (the first-term c now-mark') occupy in
the fourth dimension (the side-to-side dimension of
1 08
GRAPHICAL ANALYSIS OF TIME REGRESS
the paper) at different instants of time 2. It is,
thus, the world-line of those instruments in a time
and space diagram where space is the fourth
dimension and time is the fifth. (The three
'ordinary 5 dimensions of space are not indicated;
but can be dealt with algebraically, if we wish to
enlarge, unnecessarily, the task we have set our-
selves in this chapter.)
GH if we had filled in all the points along its
length would have extended into the past and
future parts of time 2 as a 'world-plane 9 thus
making the black block on the paper which we are
trying to avoid. The left-hand edge of that world-
plane would have coincided with our present line
G'G". So G'G" represents the world-line, in fifth-
dimensional time, of that point in GH which is G.
Similarly H'H" represents the time 2 world-line of
whatever configuration is represented by the point
H. And the intermediate vertical lines in our
figure represent the time 2 world-lines of those few
points along GH which we have decided to take
into consideration. All those points in GH represent
cross-sections of a line (not drawn) which stretches
along the fourth dimension (time i). The positions
of these cross-sections in that fourth dimension are
fixed, and do not like the position of change
in the successive states of that line (unless the ex-
perimenter interferes). Consequently, the time 2
world-lines of these sections run straight up the
paper parallel to the axis of time 2.
109
THE SERIAL UNIVERSE
We have still to represent the time 2 'now-mark',
which is the ultimate physical thing that we are
considering so far. We can do this by ruling a
horizontal line PP' from G to H across the middle
of the figure, and by adding an arrowhead to the
time 2 line of the little dimension indicator.
T.2
T.1-
-T.J
P
G
G 1
/o"
P 1
H
T.2
FIGURE II.
(FIGURE 9 in An Experiment with Time, first edition.)
It must be grasped that this diagram (repre-
senting the third-term world) consists of three
parts. First, there is the original system which was
objective to the experimenter's instruments. This
no
GRAPHICAL ANALYSIS OF TIME REGRESS
was a three-dimensional world; but, in the analysis
of the regress, it has expanded into a four-dimen-
sional and, afterwards, a five-dimensional world.
It ought to be represented by a plane G'G"H"H',
but for convenience we have substituted for that
plane a grid of vertical lines. This grid represents
a time 2 map of the original object system. That
system, no matter to how many dimensions it may
prove to extend, we shall refer to, usually, as
the "substratum'. Upon the time 2 map of this sub-
stratum we have imposed a time 2 map, O'O", of
the first-term system of instruments. Then, upon
this combined map we have imposed a line PP'
representing the (so far) ultimate c now-mark'; but
we have drawn no time map of the past or future states
of that physical thing. For the thing represented by
the line PP' is travelling over the time 2 map.
Consequently, the whole diagram is a 'working
model', and real time is the time (not indicated)
which times the movement of PP. This will be
time 3. The time map of PP' showing the different
positions of PP' at different instants of this real
time, would need to be mapped out in a sixth
dimension.
At in the middle of FIGURE 1 1 there are three
entities, viz., a point in the substratum, a point in
the world-line O'O" of the first-term system of in-
struments, and a point in the (so far) ultimate c now-
mark 5 PP. It will be more convenient in future to
regard itself as the intersection point of PP' and
in
THE SERIAL UNIVERSE
O'O", rather than as one specific state of the
instrument. It indicates, in this way, the place in
the diagram which is 'now' in time i. Clearly, it
must travel up the diagonal line O'O" as PP r moves
up the diagram. In travelling up O'O" this point
travels, obviously, from left to right of the dia-
gram, coming upon the original entropy configura-
tions of the substratum (now represented by the
vertical lines) one after another in order of that
absolute time which is not yet pictured.
We are not yet in a position to describe our
FIGURE 1 1 as a pictorial representation of the regress
of observer and observed for which we are seeking.
The entities shown in that diagram cannot be
fitted yet into the table on page 56 the table
which we drafted thus:
World as observed by B
\
World as observed by C
^
Br
World as observed by D
A*
*
C
D
Nevertheless, we can prepare from FIGURE 11 a
very similar table showing the 'now-marks 5 as
112
GRAPHICAL ANALYSIS OF TIME REGRESS
geometrical determinants which abstract from a
real world a series of worlds of progressively fewer
dimensions terminating in the three-dimensional
world of c ordinary' space. Here it is: compare it
with FIGURES 9 and 1 1 .
^1
A,
G
2
H
#.
$
G'
? H"
B-2 o"
V
C
PT*f
H'
I'
We read it as follows. B l is the first-term
travelling c now-mark'. It abstracts from the four-
dimensional world A 2 (or GH), along which it is
travelling, the three-dimensional world A. But
B l (or 0) is itself abstracted from the diagonal
world-line B 2 (or O'O") by the second-term c now-
mark' C (or PP) moving up time 2. And A 2 is
abstracted from the five-dimensional world A 3 (or
G'G"H"H'} by that same C (or PP).
Note that A^ B l and C along the diagonal edge
of the table, represent, respectively, the ultimate
abstracted object, an abstractor i and an abstractor
FSU
THE SERIAL UNIVERSE
2, as required by the table on page 55. And the
curious feature which we noted in that table viz.,
that B 2 does not abstract J 2 from A 3 is borne out
in the present analysis: O'O" does not 'abstract
GH from G'G"H"H f . Clearly, we are getting
4 warm', and it may repay us to examine the nature
of these 4 abstractors ' these c now-marks ' rather
more closely.
114
CHAPTER XII
THE IMMORTAL OBSERVER AND
HIS FUNCTIONS
An experiment is made, and the object system
the world external to the experimenter's instru-
ments is disturbed. It has received an impulse,
and the physicist cannot account for its behaviour
as subsequently observed unless he takes that im-
pulse into consideration.
The problem of the origin of the impulse is
one which the older philosophies enabled him to
ignore. They assumed that it was possible to in-
clude both observer and observed in one and the
same four-dimensional system, so that the classical
laws of physical causation would suffice to account
for every kind of physical interchange between the
two parties concerned. Consequently, jhe physicist
could leave the question of the origin of the
impulse to the physiologist. The latter, however,
could not start work until the physicist had laid
down laws for his guidance. And the instructions
which the physiologist received were simple. He
was not to take into account the possibility of any intrusion
from any world outside the supposed single temporal
system.
But, if time in physics is regressive, those in-
structions no longer may be issued. And the
physiologist is brought to a standstill. He must
115 8- 2
THE SERIAL UNIVERSE
wait until the physicist can tell him whence he may
regard that impulse as coming.
Now, whatever supplies the impulse must ex-
perience a reaction, and is, thus, an observer
of that reaction. In this chapter, I propose to
deal solely with that kind of observation which
consists in recoil. With this proviso, JL shall refer to
the instrument as 'Observer i ', and shall speak of
the ultimate source of the impulse as the ' Ultimate
Observer 5 .
We saw in Chapter ix that the second-term
'now-mark' constitutes for the experimenter a
facility for moving the instrument. We have represented
that mark in FIGURE u by the line PP. (This
figure is reprinted on p. 117). It will be re-
membered that we are, for simplicity, regarding 0,
not as one specific state of the instrument, but as
a mere abstraction the intersection point between
PP f and O'O" the place in the diagram which is
'now' in time i. And it has to be thought of as
travelling in time i, that is, as moving from left to
right in the diagram.
Since 0, while travelling in time i , has to remain
in O'O", it must be considered as travelling up
time 2 (the vertical dimension of the diagram).
Consequently, the physical thing which, at the
point 0, gives the impulse to the instrument is
travelling up time 2. Since this thing is the re-
116
IMMORTAL OBSERVER AND HIS FUNCTIONS
cipient of the reaction, we may call it c Observer 2 '.
This observer 2, then, has a field of observation
travelling up time 2. But the thing which de-
termines for this entity the order of succession
T.2
j
T.7-
-T.1
P
G
G 1
T.2
O/
H"
AO"
P 1
H
in which the states of the instrument arranged
along O'O" are presented for observation is the
time 2 'now'. So, for this observer 2, time 2 is real
time. Aeain. the rate at which the observed point
r.MlVI!*<" *-* A
travels along time i is determined entirely by the
rate of travel of the time 2 'now' and the amount of
inclination of O'O". The rate of this travel of
117
THE SERIAL UNIVERSE
governs, in turn, the apparent rates of all the
apparent motions of the original object system in
three-dimensional space apparent, that is, to this
observer 2 (he observes motion as a comppnent of
impulse). Consequently, the motion of the time 2
'now 5 , (the only real motion in the diagram), and
the direction in which it moves relatively to O'O",
determine for him which parts of the object
system (the substratum) appear to be moving and
which parts appear to be at rest. And this means
that his own motion in time 2* must be parallel to
the time 2 axis.
We might say, at once, that since time 2 is time
for him, he belongs to the second-term world of
four dimensions (with time as a fifth), and, so, is a
four-dimensional entity. But we can give an addi-
tional argument for this. We have seen that his
field of observation lies in the time 2 'now 5 , and
travels straight up the diagram. That field cannot
be shorter in the fourth dimension than is the
time i interval between his first and last obser-
vations of the instrument. He has observed this
instrument at G 1 (when the time 2 'now 5 was
there) ; he will observe it at H" (when the time 2
'now' reaches that level); and his field of obser-
vation moves straight up the diagram during the
interval of time 3 between these observations. That
field must extend, therefore, in the present diagram,
the whole width of the figure. He, therefore, is the
second-term physical entity PP.
118
IMMORTAL OBSERVER AND HIS FUNCTIONS
Is it possible, now, for us to regard this observer 2
as that ultimate source of the impulse the ex-
perimenter?
No.
FIGURE 1 1 shows the present state of the five-
dimensional world; but it had past states, (when
PP f was at the bottom of the diagram), and will
have future states (when PP' will have moved to
the top) . If we drew diagrams of these states we
should be showing past, present and future states
of PP' in time 3. We should discover then that the
considerable disturbance we are visualising can be
effected by the experimenter when PP' is in its present
state, but not when it is in its past or future states.
We should discover, also, that there is nothing in the
physical characters of its past, present and future
states to provide it with a unique ability for this
interference at the time 3 'now'. So PP' does not
represent the experimenter. It is a second-term
physical instrument. And the time 3 'now' ex-
hibits itself as a facility for altering that instrument.
And so the argument goes on, ad infinitum.
The physicist introduces that multi-dimensional
world and that endless series of physical instru-
ments of more and more dimensions whenever he
thinks of the object system as a series of states (or,
for that matter, of events) in time, and as a system
which can be made the object of experiment.
The non-technical reader may be inclined to
wonder how it is that this observer 2, which is a
THE SERIAL UNIVERSE
four-dimensional thing with a four-dimensional
outlook, can observe a three-dimensional thing like
observer i the first-term instrument. There is,
however, no difficulty about that, when the thing
observed is resistance to force.*
He may wonder, also, what sort of a thing a four-
dimensional instrument can be from the physical
point of view. But that aspect of the matter does
not disturb the modern physicist, most of whose
work is concerned with four-dimensional entities.
He would say that the four-dimensional substra-
tum GH consists of a recognised physical quantity
known as Action. Its dimensions are Energy multi-
plied by Time, and we shall have a great deal to
say about it later on. For the moment, it suffices to
point out that PP' is an entity of exactly the same
physical dimensions as GH.
But that brings us to a really unexpected fact.
The regress compels us to regard PP' as a real
entity abstracting an unreal from a real O'O"
(vide the table at the end of the last chapter).
Moreover, a body which you are employing for
the observation of a second body does not become
unobservable whenever that second body is ab-
sent. Consider, then, what happens to the entity PP'
when it is not utilising the first-term instrument at
consider, for instance, that this first-term in-
strument gets broken and, subsequently, is repaired.
* Note for physicists: It must be remembered that time,
for this observer 2, is the fifth dimension.
1 2O
IMMORTAL OBSERVER AND HIS FUNCTIONS
(We should show that state of affairs in FIGURE 1 1
by making a gap in the middle of the line O'O".)
While passing over that gap, PP' would continue
to exist, ready to re-commence observing an as
soon as the gap in O'O" had been traversed. Now,
the truth of that assertion would not depend upon
whether the gap in O'O" were long or short.
Clearly, then, its truth would not be affected if
the instrument were never repaired.
Would this continued existence of PP' in time 3
be affected if PP' did not extend beyond the left
and right-hand edges of the diagram? (The sub-
stratum itself extends, of course, a long way in
both directions beyond those limits.) The answer
here, again, must be in the negative.
Turn now to the substratum. In the time i
dimension (from left to right) its character is
differentiated; i.e., each state represented by a
vertical line is different from the state next to right
or left. But there is no differentiation in the
vertical dimension, above GH. Such differentiation
would be logically impossible. For the states from
left to right are supposed to be related to one
another in the manner dictated by the laws of
classical science: they represent a causal scheme.
An interference at ? for example, (see FIGURE 12),
would change all the states between and H. The
vertical lines above OH would become then differ-
ent from all the lines below OH, but that breach in
the continuity of the lines cannot occur at a level not
121
THE SERIAL UNIVERSE
yet reached by PP. Suppose, for instance, that the
line running up from had a changed character
above a point Qa, little ahead ofPP'. All the lines
above QR at that level would have correspondingly
changed characters, so that a causal relation could
be traced from Q, to R. But below QR the lines
G"
V
P
G
T.2
T.I
T.1
7!2
G'
W
Q- -
O/
-AO"
R
P'
H
H"
H'
FIGURE 12.
would be causally related so as to agree with the
different condition of the line between and Q.
Then, as PP* moved upward, observer i , travelling
from to S, would come upon substratum states
in a certain causally related condition. (We are
ignoring microscopic physics in this chapter.) But
on arriving at S 9 he would encounter a state be-
122
IMMORTAL OBSERVER AND HIS FUNCTIONS
longing to an entirely different causal scheme
originating at Q,- He would discover that a miracle
had happened !
Now let us consider that PP' has travelled up the
diagram as far as the level 70", and that, at this
level, the instrument O'O" ends through, say,
breakage. Alternatively, let us say that PP' extends
no farther than the width of the diagram. In
either case, when PP' reaches 70", its chance of
interfering with the substratum ends. For it is our
initial supposition that the experimenter can in-
teract with the substratum at the time i 'now'
only; that is to say, can interact only via the first-
term instrument. (Interaction at any other point
would produce miracles for the observer at 0.)
Hence, after PP' has passed 70" neither the sub-
stratum nor PP' can effect changes in each other;
the lines above 70" persist unaltered in time 2 for
ever; and PP' moves over them in time 3.
What, then, is to interrupt the continued exist-
ence (in time 3) of this observer 2? Nothing save
a miracle.
Now, PP' is not the experimenter: it is one of
that individual's instruments. It, like the first-
term instrument, is one of an endless series of
' observers ' intervening between the experimenter
and the substratum. And the really interesting
thing is the way in which those observers are
related by the time device.
Everything in the diagram which runs from left
123
THE SERIAL UNIVERSE
to right is differentiated in that dimension. The
result of that differentiation is, as we all know,
a beginning and an end in time i for any entity
which depends for its identity upon a condition of
internal organisation. Let us assume, for security
in this vital question, that everything pertaining to
the experimenter is limited in this way, and let us
say that the width of the diagram indicates those
limits.
Observer 2, as we have seen, will lose touch
with observer i , leaving it behind him in the fifth
dimension. A moment's consideration shows us
that this is simply because observer I's world-line
O'O" crosses the diagram from left to right, that is
to say, from beginning to end in time i. But
observer 2 thereafter travels straight up between
those two boundaries, and there are no limits or
changes assigned to the substratum ahead of him
in time 2, and no limits assigned, as we have seen,
to his endurance in time 3. The endurance of the
substratum in time 3 would have to be shown by
arranging a series of diagrams like FIGURE 1 2 one
above the other in the fashion of the leaves of a
book, making a tall pile.* The pile would have
boundaries on the left and the right, but no boun-
daries towards the tops and bottoms of the pages. And
it would be unlimited in height. The successive posi-
tions ofPP' in that pile, each a little more towards
* See Appendix for a perspective drawing of this figure,
taken from An Experiment with Time.
124
IMMORTAL OBSERVER AND HIS FUNCTIONS
the top of the page than the one next below it,
would build up an inclined plane endlessly long in
the time 2 and time 3 directions. Observer 3 would
be represented by a horizontal level taken through
the pile. It would form a plane with boundaries on
the right and the left but none in the time 2 direction.
Its travel would be a rising motion up the tall pile,
that is to say, in the time 3 direction. A little con-
sideration shows that it would never lose contact
with observer 2 (the inclined plane). Also there
would be no limits to its endurance in time 3 (the
time which times its travel).
In brief, of the entire series of observers, the only one
which comes to an end in its own time dimension is
observer i.
But observer 2 cannot interfere with observer i,
after he (observer 2) has passed the line VO" .
What about observer 3? He can continue to inter-
fere with observer 2 ; but he cannot interfere with
observer i except via observer 2, so, when observer 2
loses touch with observer i, observer 3 is rendered
impotent to interfere with observer i. And the
same restrictions apply to all the other observers.
The first-term 'now 5 at 0, the intersection point
between PP' and O'O", represents the experi-
menter's chance of altering the substratum ahead
of in time i. Such alteration changes, as we
have seen, that part of the substratum which is
ahead of both and PP', viz., the rectangle WH"P'
in FIGURE 12. This alteration changes that corner
THE SERIAL UNIVERSE
of our imagined pile which lies ahead of and
ofPP' and of the third-term c now-mark 3 . And so
on throughout the series. Thus, interference at
alters what lies ahead in the time pertaining to
every observer in the series. But, once observer 2
has passed the point where observer i intersects
the right-hand boundary of the diagram, van-
ishes, and the experimenter has lost his last chance
of altering the future course of his originally
selected object world.
The observational powers of observer 2 in the
absence of observer i are matters of great import-
ance to mankind, and we had best look into this
question very closely. We have proved that this ob-
server, PP', is travelling parallel to the time 2 axis
and is possessed of a field of observation extending
at least from G to H. We have seen that he con-
tinues to exist in the absence of observer i, e.g.,
when observer i is inactive, or when observer 2
has passed the position VO" (FIGURE 12). But does
this mean that PP' observes the substratum, and con-
tinues to do so when he has no first-term observing
instrument to assist him? The answer is in the
affirmative.
First, I may repeat here the argument given
already in An Experiment with Time (3rd edition,
pp. 179-181).
4 The development of the series of observers
places observer i (the section of O'O" which is at
126
IMMORTAL OBSERVER AND HIS FUNCTIONS
0) between observer 2 and the substratum section
at which is, somehow, affecting that observer 2.
So that the process by which that particular state
affects observer 2 is as follows. A certain feature in
that state causes a corresponding modification in
the intervening section of O'O". It is this repro-
duced feature which affects observer 2.
'But that raises the following difficulty. Obser-
ver 2 is a four-dimensional creature, and the sec-
tion of O'O" which intervenes between him and the
substratum is only three-dimensional. His field of
observation must extend, therefore, in the fourth
dimension beyond the place where O'O" crosses
that field. In those outer parts of observer 2 5 s field
there are many other three-dimensional sections of
the substratum containing the kind of feature
which, reproduced in the intervening entity, is
affecting observer 2 . Since observer 2 is susceptible
to features of that kind, what is there to prevent him
from being affected by these other three-dimen-
sional sections of the substratum as well as by the
section of O'O" which lies in his field?
'Nothing, that I can see. So, pending the dis-
covery of some obstacle, we must assume that
observer 2 is affected by the substratum adjacent
to the section of O'O". But this collection of adjacent
sections does not affect him in the same way that he is
affected by the three-dimensional section of O'O". The
bit of the substratum beside O'O" is a four-dimen-
sional strip presented as a whole to a four-dimen-
127
THE SERIAL UNIVERSE
sional observer it has, to him, no distinguishable
three-dimensional sections. The function of obser-
ver i (i.e., the function of the only purely three-
dimensional entity within the field) is to abstract
from the substratum an aspect thereof witli which,
otherwise, observer 2 could never become ac-
quainted. 5
All of which is reasoning sound enough.
But in the present book we can arrive at the
same conclusion by a simpler route. As we saw in
the table near the end of Chapter xi, PP' (or C) is a
geometrical abstractor abstracting G//(or-4 2 ) from
G'H'H'G (or A B ). We can add to this what we
have proved earlier in the present chapter that he
is an observer with a field of observation as long as
GH. Therefore he is an observer abstracting GH
from G"H"H'G (an J 2 from an A 3 ). Clearly, then,
he is the observer C of the table of the self-
conscious observer on pp. 54, 55 of Chapter vi.
And 0, the part of O'O" which is 'now', is his first-
term 'self.
But how, the reader may ask, can this PP' ob-
serve the substratum when (in the absence of
observer i) he is not being altered by it? For, after
the disappearance of observer i, PP' simply rushes
on over a substratum which never alters and to
which he is already, so to say, * fitted 5 ! Here I
must refer the questioner to the definition of
physical observation given in the first page of
Chapter v. To 'observe' is to ' be-affected-by* and
128
IMMORTAL OBSERVER AND HIS FUNCTIONS
not necessarily to be ' alter ed-by' . Suppose that
PP"s form is adapted to the form of GH, so that in
the absence oGH, PP"s form might be otherwise.
That would mean that PP"s freedom is being
restricted by the presence of GH. And that would
amount to physical observation. Our proof that
PP' does observe GH has led us, therefore, to no
physical absurdity.
It is to be noted that this regress of time clears up
the difficulty we discerned in our general table of a
self-conscious observer. PP' can perceive perfectly
well that the substratum is altering as travels
(in PP"s field) from left to right across it. .For PP'
can see any point in the substratum ahead of in
his field, and can notice that is changed to con-
form with the new conditions when it arrives there.
FSU 129
THE SERIAL UNIVERSE
* * *
PART III
SPECIAL TESTS OF THE THEORY
CHAPTER XIII
AN APPROACH TO RELATIVITY
How fast does the 'now' travel?
At first sight this question seems either meaning-
less or impossible to answer. Very well, let us see
what a second inspection will make of it.
To begin with, we must realise that the question
is not one of deciding how fast our instrument at
(the B l of our table) is travelling over an already
marked out space and time map (A 2 in the table),
i.e., how fast in FIGURE 12 is travelling over an
already marked out GH. Our problem is to em-
ploy the knowledge of the object world provided
by our instrument B l for the purpose of construct-
ing precisely such a marked out A 2 map and this
when we have not the faintest notion of the rate at
which that instrument is travelling over the fourth-
dimensional length of the countryside to be
plotted out.
Well, let us start with the part of our task which
is easiest. Our set of instruments in the B l system
contains a scale of distances in ordinary three-
dimensional space, and the travel of that scale in
the fourth dimension will not alter its length. We
can employ that scale, therefore, to mark out a
scale on the space axis of our map.
133
THE SERIAL UNIVERSE
But trouble arises when we try to mark out a
time i scale, keeping in mind that it indicates a
fourth dimension of space along which our source
of information, B l9 is travelling. For our clock is
not something which B^ (our set of instruments)
observes : it is something which we observe without
the intervention of instruments. Its ticks are not
features in a time i world-line, but events which we
have to mark out on that time i axis to which
time i world-lines will be referred. It belongs, in
brief, to the system of present, three-dimensional
instruments which provide us with the information
from which we propose to draft a time map of the
endurances of bodies other than that clock. That
is to say, the clock belongs to the travelling B l .
Probably, this will be grasped more easily by the
employment of a diagram. Let us assume that our
clock's ticks occur at intervals of one second.
FIGURE 1 3 shows axes of time i and time 2 (indi-
cated by T! and T 2 ). The world-line of our B l
clock (a B 2 ] may be represented by any sloping line
we please, such as 00" . Its ticks will be features in
its career features which we can represent by
marks made at regular intervals along its length.
These ticks are, thus, periodic in time 2; and we
can use-them to mark off a scale of T 2 seconds, by
drawing horizontals to the T 2 axis. But the ticks
are periodic also in time i ; so they will serve to
mark off a scale of 7^ seconds, by dropping verti-
cals on to the 7^ axis. Clearly, at whatever angle
134
AN APPROACH TO RELATIVITY
we draw 00", the diagram will indicate always
that the clock is travelling along time i at the rate
of one time i second per second of time 2. In other
words,
- = i, i.e., /! = t 2 (i).
h
One what? To give this velocity a meaning we
must realise that the seconds marked off on the T t
/
/
/
/
234
FIGURE 13.
T,
axis are marked by the ticks of a clock travelling
along a fourth dimension of space, so that the 7^
'second' represents a space distance travelled by
the clock in one second of time 2. The clock is then
marking off both real seconds in time 2, and space
lengths in the fourth dimension. And the time ob-
served by us the time told by the clock will be
time 2.
Now, we are all agreed that the rate at which a
clock hand travels over its dial must be assumed to
be constant if that clock is going to be accepted as
our measure of time. Therefore, since our clock
THE SERIAL UNIVERSE
ticks out the seconds of time 2 (vide FIGURE 13) we
must not only regard its ticks as evenly spaced
along 00", but must regard also the line 00" itself
as straight, i.e., our clock is travelling along the
fourth dimension at a uniform velocity. If we call
the fourth-dimensional space-length traversed in
any period of time 2, 5 4 , then the velocity of the
clock will be s
~ = a constant.
h
We will call this constant, k. Then
~ = k, a velocity.
'2
And / 1= =A;# 2 = j 4 * (2).
Now, A:, the velocity, means
k units of space 4
one second of time 2 5
where A: is a mere number, and the distance tra-
velled in one second of time 2 is (vide FIGURE 13)
one division of our intended time i scale. So that,
in preparing our four-dimensional map, we shall
have to give each second-division of time i the
same length as we give to k units of three-dimen-
sional space.
And there we stick. What number of space units
is A:? We have not the faintest notion. And it is ob-
vious that we shall never discover its value so long
as we continue on our present lines.
Let us try another method.
* See note on the following page.
136
AN APPROACH TO RELATIVITY
Note. Readers who are unacquainted with equa-
tions of the kind we have been considering may be
momentarily puzzled by the assertion that J x = kt 2 ,
when we have seen, a little earlier, that t 1 = t 2 .
They might suppose even that k must be the num-
ber i. But t is really an abbreviation for t [T], an
expression in which [T] is the unit of time (i.e., one
second) and J is a mere number. Similarly, s is an
abbreviation for s [S], where s is a pure number
and [S] is the unit of space. The unit of velocity is
~=A 5 and the velocity k means k x ~^\ > where k is
a pure number. Hence, A; 2 means A; x rWs- x
L^2J
Since the two [7~ 2 ]'s cancel each other, the ex-
pression resolves itself into kt 2 (both pure numbers)
units of space. Multiplying t 2 by the velocity k does
not, therefore, alter its length : it merely expresses
that length as being equivalent to k units of space.
137
CHAPTER XIV
VELOCITY OF THE 'NOW
We have arrived at the facts that each second of
our time i scale will have to be made equal to a
distance of k space units, and must represent also the
distance which would be traversed by the travelling clock
while it ticked one second of time 2. But, for our simple
scale to represent these facts, it was necessary for us
to assume that the seconds of time 2 were being re-
garded as marks on an axis drawn in a fifth dimen-
sion, and that the world-line of the clock was being
represented by the inclined line in FIGURE 13.
Failing that or some alternative understanding,
our time i scale, with its divisions of k space units
apiece, would represent nothing but a space length
over which anything might be travelling at any
rate whatsoever. Now, our method of showing
that the divisions represented the distances tra-
versed during clock ticks was perfectly sound. But
there is another way of making our scale show
what is required of it; and, since we have been
brought to a standstill, we had better see what
this other way will do for us.
A quantity which can be represented diagramma-
tically by the length of a line that is to say, by
some marked-off distance on a scale is called a
138
VELOCITY OF THE C NOW*
"Scalar 9 . Ordinary space length and ordinary
time duration are examples of simple scalars.
But now we have to consider quantities of
another kind quantities which can be repre-
sented by lines of definite lengths and fitted with
arrowheads. Such lines are called ' Vectors', and
they specify several aspects of the quantity they
represent. That quantity is to be conceived as a
transportation or transference or step from one end of
the line to the other. To quote A. N. Whitehead:
'All other types of physical vectors are really re-
ducible in some way or another to this single type'.
The arrowhead gives the sense of the transporta-
tion, i.e., tells us from which end to which end of
the line the transportation is supposed to be taking
place. If we place the line within the angle made
by two axes, the slope of the line will give what we
may call 'the line 5 * of the transportation. The
ends of the vector will tell us where (as referred to
these axes) the transportation starts and ends. The
length of the line indicates the amount of the trans-
portation. This amount is a simple scalar quantity,
and it can be indicated either by a scale marked on
the line or by referring the line to scales marked on
the aforesaid rectangular axes.
If we announce that this amount of transporta-
tion is to be considered as the distance moved in a
* I should call this, ' the direction ', were it not that many
writers use the latter word, rather loosely, to signify either
the 'sense' indicated by the arrow or both the 'sense 5 and
the 'line' of the transportation.
139
THE SERIAL UNIVERSE
constant interval of time, then the length specifies
a velocity, and a long line will represent a larger
velocity than is indicated by a short one. There are
other quantities which the length of the line may
be made to represent (by suitable conventions) but
we need not stop to consider these.
The point to be borne in mind is that every vec-
tor possesses, besides its other characters, its cha-
racter as a scalar, which is the character represented
by its length. To distinguish this scalar character
of a vector from scalar quantities represented by
lines which are not vectors, we call the former, a
' Tensor \ A tensor is simply the scalar belonging
to a vector.
Let us turn now to the scale we want to mark off
along the fourth-dimensional axis of the mesh-
system pertaining to our intended four-dimen-
sional map. Each unit interval thereof will possess,
as we saw in Chapter xm, a length equal to k
space units and we do not know the value of k.
Of course, if we could ascertain the length of one
of these interspaces, we could use that length to
mark out all the remainder. But the only way in
which we can discover that length is (as we saw in
the last chapter) by discovering first the unknown
velocity k with which our clock is travelling in the
fourth dimension, and by marking thereafter the
places it has reached in that dimension at the
beginning and end of one of the seconds it is
ticking out in time 2. From the formula kt 2 = J 4
140
VELOCITY OF THE C NOW ?
we could then calculate the length of the distance
thus marked out.
But the little line arrived at in this fashion will
have a very curious character. Indeed, it will
have two distinct characters. In the first place, it
will indicate a unit distance travelled by the
moving clock. In this capacity, it is a unit vector of
transportation with its tensor (scalar value) measuring J 4 .
In the second place, it will be a pure scalar
measuring a second of time 2 since it is marked off
by the ticks of the uniformly travelling clock. It
specifies, in fact, both a distance travelled and the
time taken in travelling that distance.*
We have come upon a length of that kind in our
everyday life. An interval marked upon the cir-
cumference of a clock can have that double
character. It can specify both the amount of a
displacement of the clock hand and the time in
which that displacement is effected.
Let us remind ourselves once more of what we
are doing. We have a three-dimensional clock
which, according to serialism, is travelling along
4 and ticking out seconds of time 2 (time i is S 4 ) .
Say that at two successive ticks we observe two
objective features in the fourth-dimensional path
over which our instrument is travelling. We want
to mark upon our S 4 scale the fourth-dimensional
distance between those two features. And we
* Note that it cannot specify a velocity, since the time
increases with the distance.
THE SERIAL UNIVERSE
realise that, if we succeed in doing this, the dis-
tance marked will be both a distance moved-over
in 5*4 or 7^ and an interval upon a scale of T% .
What we had hoped to do was to make the in-
terval an 4 length only, and then to bring in time 2
as a fifth dimension at right angles to that length,
as in FIGURE 13. We had intended, thereafter, to
draw a diagonal line between the axes of 5* 4 (or Tj)
and T 2 , which line should indicate, by reference
to those axes, the rate at which the clock was
travelling. Then we should have indicated the
travel of the second-term 'now' by an arrow
pointing up time 2. (That, of course, would have
made the T 2 lengths represent amounts of trans-
portation.) We shall be able to draw something
like (though not exactly like) that picture after
we have discovered the sought-for velocity. But,
to find that velocity, we are obliged to draw, first, a
picture in which the scale of time 2 and the scale of
S (or time i) occupy one and the same position.
Now, we can show diagrammatically exactly what
it is we are doing when we draw this preliminary
picture. FIGURE 14 shows the picture we want to
arrive at, with the arrow pointing up time 2, and
with the unit of time 2 marked on the T 2 axis. The
unit of time i is shown as a space length 4 . If,
now, we were to rotate the axis of T 2 , with its arrow,
about the pivot point 0, until it lay along the axis
of 5 4 , with its T 2 divisions coinciding with the S 4
divisions, and its arrow pointing along 4, then we
142
VELOCITY OF THE NOW
should have drawn exactly the picture which is
presented to us by our method of exploring S 4 with
an instrument travelling in that dimension (vide
FIGURE 15).
For here the horizontal line is the tensor (scalar
character) of a vector of transportation with the
axis of TZ
Ws ofT,
ofT,cuidofkT 2
~s+
FIGURE 15
axis
and ofT2
necessary arrow, and is also a scalar indicating the
unit of Tg, i.e., the time recorded by the clock.
Now, how are we to shut up FIGURE 14, con-
certina-fashion, until it presents to us that FIGURE
15 which is the only picture we can draw when
we explore the four-dimensional world with our
travelling instrumental system?
THE SERIAL UNIVERSE
There is only one way to effect this, and that is
to multiply the unit of T 2 by the square root of
minus one ( written V i).
Why? I am sorry , but, to see why, the reader will
have to study a branch of mathematics known as
the * Quaternion' calculus, and invented many years
ago by the famous Sir William Rowan Hamilton.
If he does not wish to be troubled with that, then
he must take my word for it that a c quaternion 5 is
the name for any operation which changes one
vector into another. The quaternion which rotates
a vector into a new direction without changing its
length is called a c Versor'. The versor which rotates
a vector through a right angle is called a 4 Right Ver-
sor*. Multiplying a vector by a right versor turns it
through a right angle, and a second multiplication
will turn it through another right angle ; so that, at
the finish, it is pointing in the opposite direction to
that in which it started, and becomes negative in-
stead of positive. If we call the original vector, jS,
and the right versor, i, the total operation amounts
t0 i x i x j8 - |2j8 = - J8,
whence z' 2 = i. So that i, which, when multiplied
by j3, turns that vector through one right angle,
equals V~ i.
Consequently, if we multiply the vector t 2 in
FIGURE 14 by V i 5 we shall rotate it through a
right angle into its new position in FIGURE 15. But
then the value of the J 4 length (the 7^ unit) will be,
144
VELOCITY OF THE '
not k x unit T 2 , but\/ i x k x unit T 2 . Con-
sequently, while observed lengths in three-dimen-
sional space are not affected by the travel of our
instrument along the fourth dimension, all time i
lengths in our map will be equal to V i ^ 2 ,
where t% is the time recorded by the clock.
The resulting map will be rather curious, and it
is to be seen that it is a product of pure, unadul-
terated serialism.
Consider the case of a physicist who has gone so
far as to recognise the existence of a travelling
'now'. Consider even that he accepts the notion
that his instruments are situated at that 'now'. We
may take it for granted that he will perceive the
necessity of multiplying his recorded time by the
unknown k in order to get a correct map of the
temporal system in which the object world en-
dures. But there he will stop. Time 2 and time i
are the same to him there is only one time, and
when he has considered it as flowing he has done
all that is necessary from his point of view. Since
he does not intend to take a time 2 into account,
he has no time 2 axis which requires rotating, and,
therefore, he has no need to multiply recorded
time by V i. Let us suppose, then, that he has
to consider, in his four-dimensional map, an in-
clined world-line such as ab in FIGURE 1 6. Drawing
from a the line ac parallel to the space axis, and
from b the line be parallel to the time axis, he would
produce a right-angled triangle. He would express
FSU 145 10
THE SERIAL UNIVERSE
the length of ac as s, and the length of cb as kt. We
know that, in this right-angled
triangle, the square on ab is equal
to the sum of the squares on ac
and cb. For him, then, the length
of ab (let us call it, the c distance ' ) c kt b
would be given by the following FIGURE 16.
formula: ^ 2 + ^ = distance 2.
But a serialist, who recognises a distinction be-
tween time 2 and time i (or S 4 ), has to rotate his
time 2 axis by the employment of V i ; and he
would express the length of cb as V i . kt 2 . His
formula, then, would be
s* + (V- i ^ 2 ) 2 = distance 2 .
Whence s 2 k 2 t 2 2 = distance 2 (3) .
The map constructed according to this rule the
map arrived at by watching instruments which are
travelling along the fourth dimension will be a map
of the four-dimensional world of Relativity, a world'
of Euclidean character.
Let us consider now the foot-rule with which our
travelling instrument is equipped. We can repre-
sent this by the dotted line B^B^ in FIGURE 1 7. It is
travelling along the axis of S 4 at the still undis-
covered velocity k. The axis S is an axis of one of
the other three dimensions of space. B^B^ is sup-
posed to be intersected at the point by a fixed
world-line ab crossing BjJB^ at an angle of 45.
146
VELOCITY OF THE NOW
(This assumes that we draw our mesh-system with
the horizontal intervals equal to the vertical in-
tervals.) As B^BI moves with velocity k, will
travel down B^B^ towards B with a velocity equal
to k. Now, the velocity represented by the inclina-
tion of the above world-line ab at 45 in any world
where the observed time is multiplied by V i . k
will possess most extraordinary characteristics.
This was proved by Minkowski. It will be a
limiting velocity, inasmuch as nothing used for a
signal will be able to travel at a higher speed. And
any object which is travelling with that velocity
will appear, according to the measurements of the
three-dimensional observer, to shrink to nothing
in the direction in which it is moving, while re-
taining its usual magnitude in the other directions.
It is this velocity which will appear to our
travelling instrument as the velocity of down the
scale B^Bi the velocity which is equal to k.
Here is a chance to see whether our serialism is
right ! Let us examine the universe around us with
147
10-2
THE SERIAL UNIVERSE
our three-dimensional instruments and see if we
can find anywhere a velocity, in three-dimensional
space, which possesses the above paradoxical cha-
racteristics. If we are lucky enough to discover it,
it will prove that our method of assuming our in-
strument to be travelling in 4 is right. For the
magical qualities of that velocity will depend upon
the travel of our instruments. And, incidentally, its
velocity k in three-dimensional space, which velocity
we shall be able to measure with our three-dimen-
sional instruments, will be equal to the velocity of
the c now'.
We find it at once. It is the velocity of light. And
it is known to physicists as the constant, c.
Our A:, then, is this c, a velocity of 300,000 kilo-
metres per second. And we can draw the meshes of
our required mesh-system thus,
300,000
kilometres
t second of Tj
= 300,000 kilo-metres
FIGURE l8.
S 4 orTj
The relativists did not proceed as we have done.
Einstein began by assuming that light possessed
irrational properties (in order to account for the
148
VELOCITY OF THE C NOW*
results of the Michelson-Morley experiment).
Minkowski discovered thereafter that, if time in
the real world was a fourth dimension with units
equal to \/ i . c x the observed time, then the
magic would be transferred from the behaviour of
light to the behaviour of A/ i c.
We have shown that, if the regressive character of
time is taken into account, the world mapped out
by means of an instrument which is ' now 3 must be
a world in which observed velocities have an upper
limit, and where a velocity with that upper limit
will behave as light, quite rationally, does behave.
CHAPTER XV
THE REGRESS IN RELATIVITY
From now onward the reader will need to refer
continually both to the table on page 1 1 3 and to
FIGURE ii. I know from experience that it is most
troublesome to have to hunt back for these two
illustrations. Fortunately, it is just possible, I find,
to print both on one page; and I have asked
Mr Lewis to repeat the pair thus on the left-hand
page which follows next. Then, if the reader slips
a book marker in at that place, he will be able to
make his references without difficulty.
Glancing, then, at the table, he will see that the
map we have just sketched out is a picture of the
world as this would be observed by an imagined
four-dimensional observer C x the observer who
can see that the B l instrument is travelling along
A 2 (or 6*4) . Now, this imagined observer can per-
ceive that the travelling of B l along S is taking
time The question arises, therefore, why he should
not construct a map with time as a fifth dimension
a map which would show the different posi-
tions (in 5 4 ) of B l at different instants of this fifth-
dimensional time. But, as soon as we ask ourselves
this, we find that the fourth-dimensional axis we
have drawn does show the positions (in S 4 ) ofB l at
different instants of time 2, (which was our fifth-
150
THE REGRESS IN RELATIVITY
dimensional time) ; for the divisions on the scale of
that axis indicate both the distance travelled and
the time 2 taken in travelling that distance. Have
we, then, got rid of time stopped the regress?
Well, let us look at this 5 4 axis again. Time i is
marked out there, and so is time 2. Precisely, and
this means that the instrument is travelling not
only over time i (S 4 ) but over time 2. Time 2 is
timing the travel over time i . But can we say that
the time 2 divisions represent the time taken in
travelling over the time 2 divisions?
Perhaps the reader will think that this is hair-
splitting. Surely (he might argue) we can say that
time 2 and time'i have become, now, one and the
same absolute time, so that if time 2 times travel
over time i (which he agrees to) it is also timing
travel over well, travel over itself.
That argument will not survive a moment's
inspection. Our only grounds for claiming that
time 2 represents the time taken in travelling over
time i is the fact that we have rotated the axis of
T 2 so that it is superposed upon the axis of T x (or
$4). The arrow showed then What did it show?
FIGURE 14 is repeated on p. 153. Note what it
represents before the rotation takes place. Observer 2
(C x ) is travelling up the time 2 dimension, which
becomes, consequently, a fifth dimension of space
and is equipped with an arrow to show that its
lengths represent vectors of transportation. But
the time which is timing the motion of observer 2
THE SERIAL UNIVERSE
T.l-
T.Z
-H
G" ] H f
O
G"
P
G
G'
C
P-
O/
H"
'O"
.P 1
H
H'
THE REGRESS IN RELATIVITY
along that axis is not time 2. It is time 3, which
would be mapped out as a dimension of length in
the next stage of the regress. And we cannot say
that this T 2 or S 5 axis represents time 3 lengths un-
less we have multiplied, previously, the axis of
jT 3 by V i , so as to rotate it into the position of
the T 2 axis. It would require then a second multi-
plication by V i to rotate it as far as the T x or S 4
unit
o
writ of T,
=J 4
unitof T 2 *k
^axis of TI
orSf
axis. But two multiplications by V i would up-
set completely our map of the world of relativity.
The time 3 axis remains, consequently, sticking up
above the four-dimensional map that we prepared
by multiplying T 2 by V i * And it is in this fifth
* Imagine the T 3 axis as standing out at right angles to the
page on which FIGURE 1 1 is printed. Then imagine yourself
looking up the diagram with your eye level with the bottom
of the page, so that the whole figure is foreshortened into a
horizontal line with moving along this. The T* 3 axis will
be then the axis of the diagram in which you have to plot
out the successive positions of 0.
153
THE SERIAL UNIVERSE
dimension that we shall have to indicate the time
taken by B l in travelling along the T 2 axis, after
the latter has been rotated. The arrow in our four-
dimensional world has, therefore, the following
functions :
It turns not only T or S 4 but also T 2 (now
A/ i cT 2 } into the single tensor of a single vector
of transportation, leaving T 3 as the scale of time
taken by the instrument in travelling over
We have had to reach this conclusion by a
rather tortuous route; but, now that we know
where we stand, we can see that there is a simpler
way of treating the whole matter. _____
Suppose we multiply the axis of T 3 by V~ i
once. This will rotate it into a position coinciding
with that of the axis of T 2 . We will specify then, by
a constant k of unknown magnitude, the velocity
of observer 2 (C) along the axis of T 2 (or S 5 ). Then
any length on the T 2 axis will have the value
t% = s 5 .
Then we draw, in the plane defined by an axis of
ordinary space S and the axis of T 3 (lying along T 2 ) ,
a line ab calculated according to the formula
s 2 = 2 / 3 2 = square of length ab.
Next, we draw 00" from at an angle of 45.
Then, by drawing from a and b lines parallel to
T 1? we project ab on to the plane defined by 00"
154
THE REGRESS IN RELATIVITY
and the axis of S. Thereafter, by lines drawn per-
pendicular to 7^ from this projected a and pro-
jected b, we project ab on to the plane defined by
the axes of T and S. Clearly, a four-dimensional
map constructed this way will be precisely the
same as the map which we constructed formerly
by making / 1 = \/ i . kt 2 . Then k will be, to
B 19 the velocity of light, and to (7, the velocity
I have suggested the T 3 axis as the subject for
multiplication, because the map will take us then
to the crucial second term of the regress, i.e., it
will show the way in which time 2 is related to
time i and time 3. But it is clear that what,
actually, we have to do is to multiply the axis of
absolute time by V i. Then the infinitely re-
gressive map will be correct no matter how far we
carry it.
Alternatively, of course, we can regard each of
the infinite number of time axes as multiplied,
separately, by V i ; so that each is rotated into
the position occupied previously by the axis of the
term next below.
But, whichever way we choose to look at the
matter, the result will be to make the T 2 and the T
axes occupy the same position, while leaving an
axis for the further map in which the successive
positions of the travelling B l instrument have to be
plotted out. And, in all the dimensions taken into
account, the meshes of the mesh-systems will con-
155
THE SERIAL UNIVERSE
sist of squares with sides equal to V i . ct n where
t n is the ultimate time considered. This t n will be,
of course, equal to t or t 2 or t z or any further (and
really redundant) time that we may wish to con-
template.
156
CHAPTER XVI
THE PHYSICAL OUTLOOK OF
OBSERVER 2
The quantities which are considered in the prob-
lems of classical dynamics are:
Space, indicated by S 9 *
Time, indicated by T,
Mass, indicated by Af,
Force, indicated by P.
It is convenient, sometimes, to represent space
g
traversed per interval of time, or =., by V meaning
velocity.
The way in which these quantities are inter-
related is indicated in the following equation:
p _MS , .
* J~2 \4v
If we multiply both sides of this equation by T y
we get MS
PT= M = MV (5).
This quantity AfF, equal to PT, specifies the
dimensions of the 'Momentum* generated in the
moving mass in the course of the time during
which the force acts upon that mass.
* The more common practice is to denote space by Zr,
meaning length, but I regard this as liable to confuse the
reader.
157
THE SERIAL UNIVERSE
Instead, however, of multiplying the two sides
of equation (4) by T, we may choose to multiply
them by S 9 which gives us
...... (6).
This quantity, MF 2 , equal to P5, specifies the
dimensions of the ' Energy' generated in the moving
mass in the time during which the force acts.*
Finally, let us multiply both sides of equation (4)
by ST. The result is
pc ~ MS 2 , .
PST = -jr- ...... (7).
This quantity, PST or MS 2 /T, is called 'Action^
and it is a quantity of unique interest. A long time
ago, it was discovered that all the laws which
govern the paths by which a system changes from
one configuration to another could be regarded as
mere derivatives of a single general law that the
action involved in such a change must be the least
possible in the circumstances. This 'Principle of
Least Action' was said to govern everything in
physics from the path of a planet to the path of a
pulse of light.
Clearly, we can regard this curious quantity PST
as PT x S, that is to say, as momentum multiplied
by space. Or we may regard it as PS x T, which
* Numerically, mv 2 is the 'Vis Viva\ or twice the energy;
but it is, consequently, proportional to the energy, and the
numerical factor is of no importance in the present
calculations.
158
PHYSICAL OUTLOOK OF OBSERVER 2
is energy multiplied by time. This last way of re-
garding the quantity in question brings to light
very clearly the most interesting feature of action.
For energy, PS, is three-dimensional; and, when
this is multiplied by T, the result is four-dimen-
sional. Thus, action is a feature of a four-dimen-
sional world, a feature which a three-dimensional
observer divides up into components of energy and
time.
Glancing through the foregoing equations, the
reader will note that they exhibit the inter-
relations of what are, really, two systems of units.
We can express all our problems in terms of the
three dimensions P, S and T, or, equally well, in
terms of the three dimensions M, S and T. Equa-
tion (4), viz., ^
f =
which may be written also
PT 2
...... (8),
provides the connecting link between the two
systems. The first form of this expresses P in terms
of the MST system, i.e., represents force as a name
for mass x acceleration (Sj T 2 is acceleration) . The
second expresses M in terms of the PST system.
The MST system has the illusory advantage that
M, meaning 'mass 5 , may be confused with the
philosopher's 'matter 5 located at a definite place
in space. Actually, the equations tell you nothing
159
THE SERIAL UNIVERSE
about the position of the 'matter' unless you
have agreed, previously, to accept the idea that
the mass of a 'piece of matter 5 is located at the
centre of gravity of the latter. Apart from pre-
suppositions of that kind, neither system, in pure
dynamics, makes any reference to matter. In the
MS T system the M is situated at a marked point in
space: in the PST system the P is applied to a
marked point in space.
The PST system, however, has a real advantage
of simplicity, as the following table will show.
PST system
MST system
Momentum
PT
MS
T
Energy
PS
MS*
7-2
Action
PST
MS*
T
Consider now the case of a classical physicist who
is watching the behaviour of his B l instrument and
is inferring from this the character of the ultimate
object world. He would map out that world as a
four-dimensional structure with time as a merely
imagined fourth dimension (such as one sees in a
barometric chart) . But he would be quite unaware
1 60
PHYSICAL OUTLOOK OF OBSERVER 2
of the necessity of regarding his B l instrument as
travelling along that dimension over an object
system extended therein. Consequently, to him,
time would be a simple scalar quantity; and it
would appear as this in all his physical expressions.
But, in our four-dimensional continuum, some of
the physical quantities* are different. Those which
consist of P and S or combinations of P and S 9 where
S is any one of the three dimensions of c ordinary '
space, would not be affected by the travel of the
instrument; and, as regards these, the classical
physicist and ourselves would be in agreement. But
wherever he would write T 2 , (our name for the time
told by his clock), we should write icT%. Thus,
where he would enter on his map a momentum
Pt 2 , we shall have to insert a quantity icPt 2 . Making
this alteration wherever necessary, we find that
What the classical
physicist regards as
We regard as
Time
T 2
icTi~S<
Velocity
'-?.
V S
ic^St
Mass
M
-c*M
Momentum
P? 2
icPT 2 - PS 4
Action
PST t
icPSTt - PSS<
FSU
161
THE SERIAL UNIVERSE
All of which quantities, be it remembered, pertain
to the object world.
In what we may call the c original' theory of
Relativity, it was pointed out by Einstein that mass
in the four-dimensional world must be mass multi-
plied by c 2 . No satisfactory explanation was given
as to why, in that case, this Me 2 is observed by our
instruments as plain M it was inferred that Me 2
must be energy relating to some ' internal 5 tur-
bulence or what-not of the atom an internal
energy which no instrument could observe. But
the important and self-contradictory inference
which was drawn was that mass and energy were
one and the same thing. We can tell a story more
rational than that. Energy Me 2 is merely energy
along the fourth dimension due to the relative
velocity c existing between the instrument and the
substratum.
The reader may wish to know, here, whether
this relative velocity can alter. The answer is that
the formula for lengths in the four-dimensional
scale, s = ict 2 , makes those lengths dependent upon
c. If c becomes less, the distances which we mark
on the 6*4 scale whenever our travelling clock ticks
would become shorter, while the lengths of our
space units would remain unaltered. Consequently,
the inclined world-lines (the positions of which are
independent of how B regards them) would in-
dicate to B l that distances as before were being
traversed in three-dimensional space, but that, now,
162
PHYSICAL OUTLOOK OF OBSERVER 2
a larger number of seconds were being taken over
the journey. Thus, the effect of reducing the velo-
city of the c now' would be to reduce all the velo-
cities observed by the instrument, including that
limiting velocity which is always the velocity per-
taining to light. Hence, you can see whether the
velocity of your c now 5 is slowing down by seeing
whether the velocity of light is diminishing.
It is quite evident from observation that this
velocity does not vary every time B l transfers
energy to, or receives energy from, A l .
Now we can fill in our table.
A l is, to -S 15 the content of a field of three-
dimensional observation. This content appears to
B as changing ; but the history of those changes is
to be mapped out (says B^ in a time dimension, and
the field contains merely an instantaneous view of
its content. Using the PST system, A l must be PS
the resistance encountered by the instrument
multiplied by the distance S that the point of ap-
plication has moved since the last observation.
Next, we have to fill in A 2 . That is easy : A 2 is the
temporal history of the changes in A 2 . The Vic-
torian physicist would have written it PST 2 . (T 2 ,
remember, is our expression for the time told by
the BI clock.) The man who, while admitting the
travel of the instrument along time i, fails to realise
that the resulting map involves a right-angled re-
volution of all the axes of a regressive time this
man would ignore the sign of revolution, z, and
163 1 1-2
THE SERIAL UNIVERSE
would describe A 2 asPSxcT^. We, as we have seen,
substitute for that, PSxicT 2 .
B has, of course, the same dimensions as what-
ever it abstracts in A l9 viz., PS.
I show, for purposes of comparison, our table
and the table in which the revolution of thfe time
axes has been overlooked.
Revolution of time axes
overlooked
Our table
A
A,
1 PS
1 PS
A,
Bl
A,
BI
PS x eT t (i)
PS
PSxicT, (i)
PS
-PS x S t (a)
-PSxS t (2)
I have given, in each table, the two descriptions
of A 2 . It will be noticed that the only difference
between the two tables lies in the fact that we write
icT 2 in place of cT 2 in the first of these two descrip-
tions. But this makes the S 4 of our table quite
different from the 5 4 of the left-hand table.
If, now, we add to our table the proper descrip-
tion of C, we shall have carried the regress far
enough. Thereafter, there would be only repeti-
tions of the relations already discovered. Now, C
(the PP' of FIGURE n ) has the same dimensions as
164
PHYSICAL OUTLOOK OF OBSERVER 2
(the GH of that diagram). So our table would
run
AI
PS
PS x icT t
PS x S t
(i)
(2)
PS
PS x icT 2 (i)
PSxS t (a)
If we wish to fill in A 3 , that task presents no
difficulty. T 2 of FIGURE 1 1 has been multiplied by
ic, which rotates it into the position of 7^ . T 3 has
been multiplied by ic, which rotates it into the late
position of T 2 . So A 3 becomes
PS x icT 2 x icT 3 (i)
It is clear enough that C as (i) PSx icT 2 will
abstract that same quantity from A 3 as
(i) PSx icT 2 x icT 3 .
Also that C as (2) PSx S 4 will abstract that quan-
tity from A 3 as (2) PS x S 4 x S 5 .
There is, of course, no A 3 for the man who has
neglected his sign of rotation, i. His world is con-
fined to A and
with light behaving quite madly
165
THE SERIAL UNIVERSE
in the former. His B^ which he regards as travel-
ling, must not interfere; because (as we shall see
later) that would cause the most hopeless confusion.
Lacking an .4 3 , he lacks, also, a B% and more im-
portant a C. This last omission will render his
case quite desperate when he is confronted with
modern 'quantum' physics. In fact he has made a
thorough mix-up of his universe, and his multi-
plication of 7~2 by c has helped him not a whit.
Returning to the smoother pathways of the
serialist, we can fill in B 2 . Remembering that the
jT 2 of FIGURE 1 1 has been replaced by ic jf 3 , (owing
to the rotation of the axes) , we can see that B 2 must be
V
PS x icT 3
(0
=PS x S t
(2)
It is evident that C as (2) PSx S 4 will abstract PS
from B 2 as (2) PSxS%. But the reader may find it
difficult to understand why C, as (i)PSxicT 2 ,
should abstract only PS from B 2 as ( i ) PS x ic T 3 .
He might suppose that what should be abstracted
is PS x ic. He should bear in mind, therefore, that
ic has not been introduced as an independently
existing factor, but as an adjectival factor quali-
fying our time axes only. These it rotates and turns
into space. We cannot take it away from them and
attach it to anything else. PS x ic would be
meaningless.
1 66
PHYSICAL OUTLOOK OF OBSERVER 2
Where does mass enter into all this? Why, we
can always substitute MV 2 for PS. For M in our
table becomes M; 2 , and V becomes V\ic\ so,
naturally,
- Me 2 x V 2 /i 2 c 2 = ~Mc 2 x V 2 I -c 2 = MV 2 .
But mass plain mass without adjectival trim-
mings is not observed by J5 X . It is an inference,
and a very elaborate one, from observation ofMV 2 .
This we shall see when we come to deal with the
physiological aspects of the regress.
It is clear that this physical regress will proceed
on the lines sketched out for as far as we care to
carry it. But there is nothing to be gained by
analysing it beyond the second-term observer C.
The remainder will be mere repetitions exhibiting
that relation between observer, self and object
world which has been exemplified already in the
table which contains C.
******
We have finished with relativity for the moment.
Our serialism has shown us why it is that V i is
bound to enter into all relativity (and, for that
matter, all atomic) calculations. Briefly, we can-
not get Minkowski's world except by rotating the
axis of a second dimension of time so that this axis
coincides with the axis of fourth-dimensional time.
When that is done, the picture in four dimensions
appears as one which has been mapped out from
observation of a three-dimensional instrument
THE SERIAL UNIVERSE
which is travelling over the fourth-dimensional ex-
tension of the object world, and it becomes obvious
that the velocity whatever its value of that travel-
ling will produce, in the three-dimensional world
apparent to the instrument, an equal and limiting
velocity with all the remarkable attributes of the
velocity of light. But the necessary rotation of the
T 2 axis cannot be effected without previous recog-
nition of the infinite regress of time axes implicit in
the notion of a second dimension of time, and it
is a rotation which involves an equal rotation of all
those other axes.
1 68
CHAPTER XVII
QUANTA, WAVES, PARTICLES AND
THE UNCERTAINTY PRINCIPLE
On December i4th, 1900, Dr Max Planck of
Berlin announced to the German Physical Society
his discovery of a strange new constant which he
symbolised by the letter h. It became apparent
very quickly that this h was nothing less than an
atom of action an atom of PST. It is known now
universally as Planck's 'Quantum'.
Planck had been studying radiation, and what
his experiments proved may be explained quite
simply. If we make the T in PST represent the
period of the oscillation of a wave, we can say,
obviously, that DOT-
PC * 01
~ Period '
t . . Action
which means Energy = p r--? .
Planck showed that the action on the right-hand
side of the equation must consist of indivisible
atoms. Since fractions of these atoms could not
exist, the equation must take the form
where n is some whole number and h is the atom of
action the quantum.
169
THE SERIAL UNIVERSE
If, instead of regarding Tas period, we regard S
as wave-length, it is clear that
PST
Wave-length
,, Action
or Momentum = ll7 -, r ,
Wave-length
which Planck 3 s discovery compels us to write
Momentum = T - AT * ,- ( i o) .
Wave-length v '
It is to be noted that these two equations (9) and
(10) do not allow us to regard either energy PS or
momentum PT as atomic. For period in (9), and
wave-length in (10) are both variables. But the
non-atomic quantity of energy which is equal to
one atom h divided by the period of oscillation has
proved to possess an importance equal to that of
any atom. It is called, nowadays, a t photon' ; and
it is a well-established law that, in all interaction
between an observing instrument and the object
observed, what passes is energy in the form of one
or more photons. Moreover, Einstein showed,
early in the century, that each c photon 3 , A/period,
must arrive at the receiving instrument in the form
of a particle travelling like a bullet, and not in the
form of a wave. Nothing which did not possess
these bullet-like characteristics could produce what
is known as the 'photo-electric 3 effect.*
* The reader will find a very clear and simple account
of this effect in the last chapter of Sir William Bragg's
The Universe of Light.
170
THE UNCERTAINTY PRINCIPLE
But (the reader well may ask) if these photons
art particles possessed of varying amounts of energy,
what is the meaning of ' period 5 in the definition of
a photon as A/period?
Well, the trouble was that, if you exposed a
photographic plate to a direct beam of light, no-
thing but particles would arrive; but, if you passed
that beam first through what is known as a
'grating 5 , the effect produced would be exactly the
same as if that beam had consisted of nothing but
spreading light- waves. These waves would have
length and period. The photons, on the other hand,
had energy and momentum. And the law which
emerged connected the light-particle in the one ex-
periment with the light-wave in the other by the
two equations (9) and (10) amplified thus:
Energy of the) h
light-particle j Period of the light-wave '
Momentum of the) _ h
light-particle j ~ Length of the light- wave"
Now, Newton had held that light consisted of
particles shot out from the source in all directions.
His contemporary, Huygens, proposed a ' pulse 5
theory, which, when modified and extended by
Young and Fresnel, became the wave theory. This,
in the interval before the arrival of Planck, held
the field. The crucial experiment was the 'diffrac-
tion 5 of light by means of the 'grating 5 mentioned
above. A 'grating 5 may be thought of most simply
171
THE SERIAL UNIVERSE
as an obstacle which hinders the passage of the
light except through little apertures left open for
the purpose. When a wave is checked by such an
obstacle, any portion of it which arrives at a hole
passes through that hole intact, but thereafter
spreads out as a semi-circular wavelet radiating
from the hole as a centre. Spreading thus from all
the apertures in the grating, the wavelets cross one
another's paths. Now, when two waves cross, and
the crest of the one happens to coincide with the
trough of the other, the result is to cancel the wave
motion completely. If, however, the crest of one
happens to coincide with the crest of the other, the
wave effect is increased. The wavelets radiating
from the holes fall on all parts of the receiving
screen, but the part which is nearest the wavelet
starting from one side of the grating is farthest from
the wavelet starting from the other side. Thus the
screen is struck in some places by wavelets which
are in step, in others by wavelets which are com-
pletely out of step, thus cancelling one another, and
in other places by wavelets which are partly out of
step. The result is to make upon the screen a
curious pattern of concentric rings of alternate light
and darkness the ' diffraction ' pattern. It seemed
incredible that any shower of particles could pro-
duce such an effect, and the wave theory won the
day.
The discovery of the photo-electric effect equal-
ised matters. If particles could not account for
172
THE UNCERTAINTY PRINCIPLE
diffraction patterns, waves could not produce the
result we perceive after we have pressed the button
of our Kodak.
The fact that light-particles could behave as
waves suggested, of course, that particles of all
kinds might possess this curious character; and, in
due course, a wave theory of matter in general
came into being. It was produced first by de
Broglie, and presented later in an improved form
by Schrodinger (who had arrived at it quite inde-
pendently) . Dirac may be said to have completed
the work.
In the wave theories, particles are merely wave
groups, analogous to patches of rough water in a
sea. The waves of which these groups are composed
may extend, theoretically, throughout the whole
of space; but they neutralise each other every-
where except just in the region of the stormy patch.
Such a wave group will, in most cases, travel more
slowly than do the actual waves of which it is com-
posed.
It is almost impossible to analyse into distinctive
classes the philosophical attitudes adopted by
physicists towards these 'waves'. But one can
trace a hazy division between two main schools of
thought.
The first school regarded the waves as real, and
the c particle ' as being merely a name for the wave
group. Waves looked at from this point of view
might be called 'metaphysical waves'.
173
THE SERIAL UNIVERSE
The second regarded the particle as the under-
lying reality, and the waves as purely epistemo-
logical, i.e., as mathematical illustrations of the ob-
server's ignorance concerning the present p'osition
of the particle.
The objection to the first attitude was insuper-
able. Nothing could prevent these wave groups
from expanding. The expansion might be slow;
but, even at its slowest possible rate, it would be too
fast to permit of the existence of the world as we
find it to-day. To quote C. G. Darwin: 'Even if we
regarded the world as originally created in well-
defined " wave-packets 55 , they would certainly by
now have spread indefinitely. We may say that the
existence of fossils which have preserved their form
unchanged for several hundred million years dis-
proves the adequacy of the wave theory 5 .
The epistemological wave, or, as it was called,
the 'probability wave-packet 5 , was free from this
objection. If the particle was travelling at an un-
known speed in an unknown direction, our ignor-
ance as to its whereabouts would increase with in-
creasing time, and the area which might contain
it would increase as the area of a packet of real
waves would increase. Furthermore, the chances
of finding the particle at any point in that area
would be exactly equal to the 'intensity' of an
imagined expanding wave-packet at that point.
An experiment which discovered the true position
of the particle would bring the uncertainty to an
174
THE UNCERTAINTY PRINCIPLE
end, and the wave-packet of purely imagined waves
would be reduced suddenly to the tiny area occu-
pied by the real particle. The objection that the
troughs of the waves would have to represent
'negative' probabilities was an awkward one, but
it seemed less overwhelming than the objection to
the notion that the wave group was real, and
yet shrank suddenly every time an experiment
was made to ascertain whether it was, in fact,
a particle.
These questions became acute when it was found
that, just as in the case of the alleged light-particles,
electrons could produce a 'diffraction 5 pattern.
This discovery was made by Thompson.
I do not propose to drag the reader through the
technical details of the various experiments which
exhibited the apparently dual character of any
alleged particle. He will find most excellent and
lucid descriptions, abundantly illustrated, in Sir
William Bragg's The Universe of Light; while C. G.
Darwin's invaluable book, The New Conceptions of
Matter, will show him precisely how the two classes
of experiment those which discover particles, and
those which exhibit waves are interrelated. One
can summarise the empirical evidence as follows.
( i ) Alleged particles shot against a screen coated
with zinc sulphide crystals will produce tiny sparks
at the points where they strike the screen, showing,
thus, the strictly localised character of the col-
lision.
'75
THE SERIAL UNIVERSE
Alleged particles shot through a Wilson cloud
chamber cause condensations of moisture along the
tracks of the supposed tiny bodies. These tracks in-
dicate that what has passed is something very small
which is travelling in space in a perfectly normal
fashion.
(2) Showers of alleged particles falling on a
photographic plate after they have been interfered
with by a 'grating 5 produce a diffraction pattern
such as would be made by alleged waves.
(3) The two classes of experiment cannot be combined.
It is impossible to discover, at one and the same
time, both the ' particle aspect' and the 'wave
aspect 5 of whatever may be the ultimate reality.
Consequently, we cannot fall back upon the notion
of a group of real waves containing a real particle.
The whole thing boils down to this: Set a trap
to catch particles, and you will catch particles; set
a trap to catch waves, and you will catch waves.
And all the experiments appear to be crucial,
ruling out definitely either one aspect or the other.
This, to a serialist, gives rise to the suspicion that it
may be the nature of the experiment and not the nature
of the object which is really in question.
To the general cauldron of trouble we may add
a couple of ingredients. The Schrodinger waves are
not waves in space alone, but waves in space and
time. Each electron requires the whole of ordinary
three-dimensional space for its waves, and will not
permit the presence of any other electron in that
176
THE UNCERTAINTY PRINCIPLE
space. Two electrons require a space of six di-
mensions, three apiece, and so on. Which makes
the serialist, with his mild regress of time dimen-
sions, appear quite timid.
The reader must bear in mind the way in which
the quantum the atom of action is involved in
all these difficulties. The whole of the wave theory
is dotted with A's. And h appears again in what is
known as c Heisenberg's Uncertainty Principle 5
a principle which we must proceed now to con-
sider.
Every experiment (as I have pointed out ad
museum] is an interference with the object system
by something three-dimensional which is regarded
as separated from that system. Again, every ob-
servation by a three-dimensional instrument in-
volves an interchange of energy between the instru-
ment and its object and is, consequently, an
interference with that object. Now, Heisenberg
remarked that what must pass between observer
and observed in such cases cannot be less than,
and cannot be dimensionally different from,
one photon, /z/period which is the energy con-
tent of one atom of action h. Consequently,
every measurement of action PST must lack
precision to the extent of the amount contained
in h.
Such a measurement would be, for example, a
simultaneous measurement of PTand S in the case
of a particle. The total uncertainty h in the amount
12
THE SERIAL UNIVERSE
of action must appear in the separate measure-
ments of PT and S 9 so that our uncertainty about
the momentum of the particle multiplied by our
uncertainty about its position cannot be less than
h. In these calculations we write p for momentum
and q for the coordinate giving the position of the
particle at the moment of experiment. 'Uncer-
tainty' is symbolised by A. So that Heisenberg's
equation runs
(~ means, 'is of the order of magnitude of.)
This Uncertainty Principle appears to be abso-
lutely inviolable, so we had better ascertain exactly
what it means. Fortunately, the meaning is ex-
tremely clear and precise.
The impact of the apparatus for measuring
velocity alters the velocity of the supposed particle
to an unpredictable extent. The two measure-
ments of position and momentum are supposed to
be made simultaneously. Very well :
At that instant, the present position of, and the past
velocity of, the particle may be determined with
any degree of accuracy we please. The Uncertainty
Principle does not apply to these two determina-
tions. But
At that instant, the more accurately we measure
the present position of the particle the greater be-
comes the uncertainty in our knowledge of its
future velocity, so that
A present q x A future p ~ h.
THE UNCERTAINTY PRINCIPLE
All physicists, including Heisenberg himself, are
agreed upon these two facts.
Now whose is the uncertainty? It will not be dis-
puted that the observer is uncertain, so we can take
that for granted and go on to the next question. Is
there, in this Uncertainty Principle alone, the slightest
shadow of an excuse for supposing that there can be
no such thing in the universe as a particle possessing
simultaneously both definite position and definite
velocity?
I have tried to put that question plainly, but
those who suppose that there are grounds for an
affirmative answer are less explicit. ' It is the velo-
city after the measurement which alone is of im-
portance to the physicist', says Heisenberg. Why?
Is it not part of the physicist's task to explain what
has happened to show how such-and-such a situa-
tion has come about ? Sir Arthur Eddington, again,
remarks that the velocity which we ascertain by
two successive measurements c is a purely retro-
spective velocity'. But does that mean that our
acquired knowledge thereof is to be ignored? If so,
why?
The truth is that Heisenberg's Uncertainty
Principle gives a plain answer to the question as
to whether the Schrodinger 'waves' are to be
regarded as epistemological or metaphysical. And
the answer is against the metaphysicians.
For, suppose that the waves were objectively real.
Suppose that Nature knew nothing of such things
179 12-2
THE SERIAL UNIVERSE
as particles. Then we should find that our supposed
'particle 5 was a figment of our gross imaginations,
trained to the appreciation of a macroscopic (large
scale) world. And, if we were foolish enough to in-
sist that the wave-group exhibited nothing beyond
our own ignorance of what we had done to the
particle in the course of an experiment, Nature
would give us the lie.
But her verdict would be retrospective.
There is no getting round that. In such circum-
stances, we should find that the alleged particle
had never possessed, at any time, the two mutually ex-
clusive attributes of precision in position coupled
with precision in velocity. The wave-group would
not have permitted it. We should find that the pre-
cision in velocity had always varied inversely as the
precision in position.
Very well. I make six successive determinations
of the position of a supposed particle ; which de-
terminatipns, according to the Uncertainty Prin-
ciple, may be, theoretically, as accurate as I please.
Each of these determinations, after the first, in-
forms me of the velocity of the particle since the
previous measurement was made. Each deter-
mination disturbs the velocity previously ascer-
tained, but in each case, except the last, I am able
to say exactly what was the extent and direction of
that change in velocity. I have, therefore, a history
of the particle showing that it possessed definite
position and definite velocity on four occasions
1 80
THE UNCERTAINTY PRINCIPLE
according to my opponents, and on five occasions
according to myself. The admitted four occasions
are sufficient for my purpose. Nature knew no-
thing then of an Uncertainty Principle!
She has heard of it since, from the New Meta-
physicians, but is entirely unable to alter her dis-
tressing past. The most that she can do is to agree
quickly that the metaphysician's knowledge as to
what has become of the particle since the last time
he hit it is mathematically representable by the
intensity of a wave. She hopes profoundly that he
will be satisfied with this makeshift and will probe
no deeper into the matter.
He never does.
Conclusive ! Of course. But all the arguments
in this imbroglio are conclusive. If it were not so,
there would be no confusion. Here is a reply to
myself. If the waves are merely imagined, how can
they make a mark upon a photographic plate?
Note, please, that this is an instance of the way
in which the dispute is carried on. No side can re-
fute the arguments of its opponents it has to con-
tent itself with advancing another argument of a
totally different kind. In a copy of Nature which
lies open before me, I find Sir James Jeans' s an-
nouncement, to the British Association, of a sup-
posedly crucial experiment which favours the
wave; while Professor Andrade, on another page,
181
THE SERIAL UNIVERSE
is pointing out how the discovery of ' The New Ele-
mentary Particles ? furnishes a final answer to the
vexed question, and a verdict for the particle. But
the experiments in the two cases were entirely dif-
ferent. And, until we understand a little more of
what we are doing, we have no right to say, that,
in any experiment, the particle-picture and the
wave-picture have 'come into conflict 5 . In other
words, we have no right yet to presuppose that the
trap which has caught a wave was a trap for
particles, or vice versa nor shall we have that right
until we have made the trap the object of our
observation.
That we shall do in the next chapter.
182
CHAPTER XVIII
THE REGRESS OF UNCERTAINTY
It will have been obvious to the reader that, in
their interpretations of the Uncertainty Principle,
the several parties concerned have been regarding
the ' now ' as all-important, and have been treating
that 'now 5 as travelling in the fourth-dimension.
Consequently, they are drafting their pictures in
terms of an infinite regress. But to draw a picture
of a certain kind while pretending to yourself that
you are drawing something else is not the way to do
full justice to your capacity as an artist. It is not
surprising, therefore, that the picture has gone
wrong.
This is what has been drawn. The artist starts
with the state of affairs where a determination of
the position of the particle is made. Then, whether
he regards the particle as being really a wave-
group, or believes the wave-group to be a mere
abstract 'probability-packet 5 , he marks out the
future in time i as an area enclosed between two
world-lines showing the limits of the changes which
may have been made in the particle's velocity, and
these lines show the way in which the wave-group
expands in three-dimensional space. (For sim-
plicity in the diagrams, we shall show these world-
183
THE SERIAL UNIVERSE
lines as extending evenly on either side of the time
direction, vide FIGURE 19.)
Here a is (let us say) an electron. Its position is
being determined within a small space 'area (re-
presented by the thicknesses of the lines ab and ac).
This determination disturbs its velocity. The
artist's ignorance of the extent of that disturbance
is of such a magnitude that, when he makes the
Space
font
Time
FIGURE
next observation (at, say, any instant t') he may
rediscover the electron anywhere upon the line de.
He proceeds then to picture this second deter-
mination of position as being made. That is to say,
he considers the case where the 'now 3 , and, of
course, his instrument, (though he does not mention
this), has shifted to t 1 . He supposes that the elec-
tron is rediscovered at, say, a point/, and he ex-
hibits, in FIGURE 20, the resulting situation.
At this stage, the notion that the wave-packet is
184
THE REGRESS OF UNCERTAINTY
real begins to look absurd. For the new disturb-
ance given to the rediscovered electron could not
cause an expanding group of real waves to contract
instantaneously to a tiny area in the manner
shown.
How do the advocates of wave reality get over
this difficulty? I cannot tell you. At this juncture
they cease to talk about waves, and commence a
present
'w*
FIGURE 2O.
dissertation upon the inadequacy of space and time
descriptions and the folly evinced by man in sup-
posing that Nature would allow herself to be de-
scribed in terms suitable to his gross mind this
last being a theme in which they feel really at
home.
That plea, as always in the history of mankind,
proves to be inadmissible. We are crying out be-
fore we are hurt.
The idea is that the real-wave theory proves
185
THE SERIAL UNIVERSE
adequate up to a certain point, and then breaks
down. Also, that the particle theory proves work-
able for a little while, and then collapses. But, in
the picture we have shown, the particle theory
does not fail anywhere if the wave-packets are
only areas exhibiting the ignorance of the experi-
menter at the 'now 5 , an ignorance which, subse-
quently, is enlightened. There is no collapse of the
particle-picture so long as you content yourself
with seeking for the position of the particle. It is
not until you introduce an experiment which seeks
for waves that the trouble begins.
Now, it will be obvious to any serialist that
FIGURE 20, as an illustration of two successive hap-
penings at the 'now', has been wrongly drawn. It
requires the introduction of another time dimen-
sion in which to exhibit the changes in position of
that 'now' and of the instrument of discovery
which travels therewith. That we will deal with in
good time. But I want to point out that the result
is to obscure a fallacy in the picture of the past.
For the experimenter is seeking for, and discover-
ing, the particle, and is making no other kind of
experiment. He has no reason, therefore, to ex-
hibit his past wave-packets as having been anything
in the 'substratum' anything pertaining to the ob-
ject observed. They were memoranda of his own
ignorance|, an ignorance which has been en-
lightened when the experiment at t' is made. The
correct picture would have been as in FIGURE 2 1 .
1 86
THE REGRESS OF UNCERTAINTY
It represents the kind of time i map of the elec-
tron's career which could be drafted from the in-
formation provided by a series of scintillation
experiments or from observation of the track in a
Wilson chamber. Only one past position of the
electron is shown, but there is no reason theo-
retically why the past part of the picture should
not show a whole series of past positions of the
Space
Time!
FIGURE 21.
particle and the knowledge of its velocity obtained
from these, precisely as I indicated in the imagined
experiment of the last chapter.
Now, in the ordinary course of exhibiting a time
regress, the next stage is to draw a diagram which
shall include the instrument B l and map out the
successive positions of this, employing another di-
mension for ultimate time and treating the 7^ axis
of FIGURE 21 as an axis of S . But, before we can
put the instrument into any such picture, we must
THE SERIAL UNIVERSE
note what the Uncertainty Principle has to say
about that instrument.
Of Heisenberg's many illustrations, the one
quoted most frequently is the famous imagined ex-
periment with a microscope. The apparatus is sup-
posed to be an adjunct to an eye observing an
electron by means of light scattered from the latter.
Heisenberg considers the cone of rays scattered
from the electron and entering the aperture of the
instrument as yielding the necessary information
about position q. He then considers the recoil
which the electron receives from this light; and,
for that purpose, assumes that one photon of light
passes. He relates the momentum of this photon to
the wave-length of the light-waves entering the
aperture by the formula (see equation (io))p = h/ A,
where p is momentum and A is wave-length. He
has no difficulty in showing that the uncertainty in
the determination of present position is related to
the uncertainty of the future momentum by the
equation
The example is not a very good one, and I quote
it merely because of Heisenberg's concluding re-
marks, which I give in full below.*
c Objections may be raised to this consideration;
the indeterminateness of the recoil is due to the un-
certain path of the light quantum 5 (i.e., photon)
* The Physical Principles of the Quantum Theory, by Werner
Heisenberg. (Cambridge University Press.)
188
THE REGRESS OF UNCERTAINTY
6 within the bundle of rays, and we might seek to
determine the path by making the microscope
movable and measuring the recoil it receives from
the light quantum. But this does not circumvent
the uncertainty relation, for it immediately raises
the question of the position of the microscope, and
its position and momentum will also be found to be
subject to the equation
The point to be noticed in this imagined exten-
sion of the experiment is that when we put the instru-
ment into the picture, as B , and observe this from the
viewpoint ofC, we transfer the uncertainty ofp and qfrom
the original object electron A l to the instrument B . We
exhibit our uncertainty regarding A l as being due entirely
to our uncertainty concerning B , and not to anything in-
trinsic in the character ofA . We are not confronted then
with both an indeterminate electron and an indeterminate
instrument, which would give more uncertainty than the
quantum restriction h permits.
It will be perceived that, in this imagined ex-
tension, the microscope is supposed to be actually
recording the momentum received from the electron
(strictly speaking, of course, from the photon). The
C which observes the instrument's observations of
the electron (records both the light coming from
the eyepiece and the imagined motion of the eye-
piece) could be, e.g., a strip of sensitised film. But
the illustration, as said before, is not a very good
189
THE SERIAL UNIVERSE
one: the experiment is impracticable; and the
change in the momentum of the microscope would
be inappreciable, owing to the large mass of that
instrument. We will pass on, therefore, to Heisen-
berg's analysis of a real experiment, viz., the
scintillation produced by the impact of an alpha
particle upon the surface of a prepared screen.
The scintillation is produced by the 'ionisation'
of an atom in the prepared screen, that is to say,
the incident particle knocks an electron in the
screen out of the orbit in which it is circulating
within the atom. That orbit constitutes a slightly
hazy point in our mesh system, (the screen), hazy
because we do not know the position of the target
electron within that orbit. The momentum of the
incident particle is changed, of course, by the
impact.
How are we to measure that change in the
alpha particle's momentum? Clearly, whatever
momentum it loses is transferred to the electron
ejected from the atom. Now, we can measure the
momentum of the ejected electron precisely, after
it is ejected. But the trouble is that we do not know
what was its momentum before it was struck. Thus
the uncertainty in the position of the incident alpha
particle is due to the uncertainty of the position of
the instrument electron within its orbit; and the un-
certainty in the new momentum of the alpha
particle after the collision is due to the uncertainty
of the momentum of the instrument electron within
190
THE REGRESS OF UNCERTAINTY
that same orbit. Heisenberg, explaining this in
slightly more condensed language, and taking the
nature of Bohr orbits into consideration, relates
these two uncertainties by the equation
Ajb s A<? s is not less than h,
where the little s refers to the orbit of the instrument
electron.
But when the two uncertainties are regarded
thus as pertaining to the instrument, the alpha
particle is being assumed to possess a perfectly
definite track both before and after the collision;
that is to say, there is not supposed to be any in-
trinsic uncertainty in its behaviour. To assume the
contrary, while allowing for the two uncertainties
in the instrument, would give us more uncertainty
than h can provide.
So, in this experiment, again, putting the instru-
ment into the picture, as a B l observed by a (7, transfers the
uncertainty from A l to B l9
Similar considerations apply, of course, to the
ionisation of an atom in the Wilson cloud chamber
experiments.
Now we know where we stand, and we can get
on with a description of the kind of time map which
would be drawn by our imagined C.
He is a four-dimensional observer with a field of
observation extending the whole length of J 2 ,
which constitutes his c now ' in a world where time
is a fifth dimension, icT B . B l is an object at the
THE SERIAL UNIVERSE
point 0, and has just been employed by C as an
instrument for obtaining information about the
substratum at that point, i.e., information about
A lt We saw earlier (pp. 127 and 128) that C, being
a four-dimensional observer, cannot distinguish
three-dimensional sections of the substratum with-
out the assistance of B lt (B l9 since it is travelling
at the velocity of light, c, has, to C, no fourth-
dimensional extension.)
In C"s world, consisting of A% and B l9 there is no
inherent uncertainty. The particle disturbed by B l
has a perfectly definite world-line both to the left
of and to the right of (in our maps) the point of
impact. The trend of the line to the right of that
point, i.e., in the time i * future 5 , is altered by that
impact altered instantaneously in fifth-dimen-
sional time. Let us suppose that this disturbance
of the particle at has repercussions in the ob-
jective world, produces, for example, an explosion,
and alters, consequently, the general character
of that substratum to the right of 0. That change
would be apparent to our imagined four-dimen-
sional observer C. And his A% world, which is
Nature's world, would be recognised by him as
perfectly 'determinate 5 so far as the pseudo-time,
time i, is concerned. To C, the fifth-dimension
(icT^) is time, and the four-dimensional world is,
simply, 'present 5 , and equally definite everywhere.
But BI$ future does not lie in that A 2 world. B 2 is
a world-line (the 0' 0" of FIGURE 1 1) which inter-
192
THE REGRESS OF UNCERTAINTY
sects A 2 at only one point. So that J^'s future lies
outside Gs view. Now, we have just seen that,
according to Heisenberg, putting the instrument
into the picture as something observed transfers the
uncertainty from the original object particle to the
particle in the instrument. C, then, is uncertain as
to the future of B l . He does not know precisely
what has been the change in its velocity in three-
dimensional space (the space in which the impact
occurred). He cannot map out the trend of its
world-line along the four-dimensional stretch B 2 .
And his uncertainty is governed by the rule
just as in the case of 5/s uncertainty about the
future of the original object particle in A .
Note that in both cases the uncertainty is the
same. It is an uncertainty as to whereabouts in
ordinary space the instrument will encounter the particle
in a future experiment. But the correct develop-
ment of the regress shows this, first as an un-
certainty regarding the future position of the
particle as referred to B ly and then, in the all-
important second term, as an uncertainty in the
future position of B l as referred to C the time i
future of A l being certain as referred to the C
system.
It is clear that if we put C into the picture we
shall find that the uncertainty of our knowledge
concerning B l is due entirely to the uncertainty of
FSU
193 '3
THE SERIAL UNIVERSE
our knowledge concerning C. The observer who
puts C into the picture is D. The map he would
draw of space and time (time being the sixth
dimension to him) would show both JS 2 and A% as
having definite position in the 'present' five-
dimensional world, but it would show the future of
C, which is in the sixth dimension, as having the
quantum uncertainty.
Thus, the uncertainty recedes up the ladder of
the infinite regress. It is an uncertainty about the
unreachable absolute future. But, in the second
term and onward, we discover that it is an un-
certainly pertaining only to the last instrument in the
picture and never to the world which we are studying by
means of that instrument.
What alterations do we require to make now in
FIGURE II?
Well, first of all we have to change the names of
the axes, owing to the rotations which have taken
place. We must alter T l in the 'dimension indi-
cator' to icT 29 and we must change T 2 into icT$.
(This, of course, holds good throughout the regress :
Tg is altered to icT\ T 4 becomes icT 5 ; and so on.)
This has the effect of introducing i in all dimensions
except those of ordinary space. But, the re-
christening of the axes makes no other change in
the substratum. O'O", for example, does not pivot
round about 0, and lie flat along GH. The multi-
plication by i results merely in making O'O" the
world-line of an instrument which is travelling
194
THE REGRESS OF UNCERTAINTY
along icT 2 as well as along S 4 . The arrow repre-
sents still the motion of C (or PF) up the fifth
dimension, (now relabelled *VT 3 ), and this remains
the only arrow in the diagram. The motion of
along GH y indicated by an arrow in FIGURES 9
(p. 103) and 15 (p. 143), is represented now by the
method of c rectangular coordinates ' ; so that S or
icT 2 are simple scalars, and ic T 3 is the only tensor
in the figure.
As thus presented, the diagram is a picture of the
world observed by observer 3 D in the table. It is
he who observes C as a travelling instrument, and
his uncertainty is an uncertainty about the future
positions of C in three-dimensional space. That
future is not in the diagram.
If, however, we wish to make a picture of C's
world, including the future as calculated by C from
his knowledge of the present world A% (or GH), we
should need to draw 00" dotted, in order to in-
dicate C 5 s uncertainty about its future spatial
position. But O'O is a determinate line, and should
be drawn as before.
195 13-2
CHAPTER XIX
THE WAVE EFFECTS
We have to reply now to two questions, viz. :
(1) Can we prove this regress of uncertainty
prove it by actual experiment?
(2) What about those wave effects?
The answer to the first question is, c Yes 5 : the
reply to the second is that it is the wave effects
which constitute the experimental proof required.
We are going to investigate the nature of light.
A beam of light is, consequently, our A l object. For
our B l instrument, we shall employ, instead of a
scintillating screen, a complete diffraction appara-
tus comprising a ruled metallic reflecting grating,
(this diffracts just as well as a transparent plate
with opaque rulings), and a photographic plate
to receive the rays after their reflection.
The result of the experiment will be the ap-
pearance of diffraction rings on the plate. Our
business is to ascertain what must be the nature of
the rays which made those rings.
Our scintillation experiments have taught us
that the beam of light consists of a shower . of
particles. Since those experiments were more
direct and simple than the one on which we are
engaged now, we shall begin by seeing what would
happen to a shower of particles striking the grating
196
THE WAVE EFFECTS
and being scattered in all directions. On this
particle theory the diffraction effect must be due
entirely to that scattering ; for light particles do not
interfere with one another when their paths cross,
because they carry no electric charge. So what we
have to study is the nature of the interaction
between the particles and the ruled reflecting
surface.
Now, we know the position of the apparatus in
our laboratory and can regard both laboratory and
apparatus as a single spatial system. We know the
width of the beam of light relative to that system.
But we have not the remotest idea whereabouts in
that beam is any individual particle. This is a
considerable uncertainty in our knowledge of the
position of the point where that particle strikes the
grating. But position is relative, and we can ex-
press this uncertainty in two ways. We may say
either that we do not know the position of any
particle relative to the screen, or, equally well,
that we do not know the position of the screen
relative to any particle. We will interpret the un-
certainty in the second of these two ways. It is very
considerable : let us see if we can reduce it.
The demonstration which follows is Duane's,
and is one of the prettiest bits of work in the whole
of mathematical physics. But the non-mathe-
matical reader, I fear, will be unable to follow it
for more than a little way. Still, the general idea
will be apparent to him, so he should skim through
197
THE SERIAL UNIVERSE
the text. For the rest, he will have to be satisfied
with the fact that the demonstration is accepted,
and quoted with approval, by Heisenberg, who
adds interesting comments.
It turns out that we can reduce the uncertainty.
For, suppose we were to move the grating. A
movement of the whole grating to the extent of the
distance between the rulings would not affect the
diffraction; for a particle which, before the move-
ment, would have fallen on one ruling, would fall,
after the movement, on another ruling in the same
place as the first, so that the diffraction effect
would be unaltered. This critical distance between
the rulings is called the grating 'constant'. We will
symbolise it by d. The dimension in which such
movement could take place, at right angles to the
ruling, we will call x. I will continue now in
Heisenberg's own words.*
c Translation in the ^-direction may be looked
upon as a periodic motion, in so far as only the
interaction of the incident particles with the grat-
ing is considered; for the displacement of the
whole grating by an amount d will not change this
interaction. Thus we may conclude that the mo-
tion of the grating in this direction is quantized
and that its momentum p x may assume only values
nh/d (as follows at once from the earlier form of the
theory:
* The Physical Principles of the Quantum Theory. (Cambridge
University Press.)
198
THE WAVE EFFECTS
Note that this introduces the quantum as an
atom of action but not yet as a connecting link
between wave and particle. That is what has to be
proved. Heisenberg continues:
'Since the total momentum of grating and
particle must remain unchanged, the momentum
of the particle can be changed only by an amount
mh\d (m an integer) :
px'-px + d ~.
Furthermore, because of its large mass, the grating
cannot take up any appreciable amount of energy,
so that
IfO is the angle of incidence, 0' that of reflection,
we have h h /
CQS0=^, COS *'=-,
i a/ /i m h J
whence sin sm0 = * .
pd
The rest is simple. We can write the above
equation in the form
d (sin 0'- sin 0) =m x -.
P
But, in the ordinary wave theory,
d (sin 0' sin 0) = wA;
therefore -=A.
P
That is to say, from an inspection of the pattern on
the plate a length can be arrived at, really a mea-
sure of h divided by the momentum of the particle,
THE SERIAL UNIVERSE
which length would be equal to the wave-length of
the particle had the grating been treated as of fixed
position and had the particle been a veritable wave.
The following comments are, I believe, pure
Heisenberg ; but I apologise to Duane if I am mis-
taken.
c The dual characters of both matter and light
gave rise to many difficulties before the physical
principles involved were clearly comprehended,
and the following paradox was often discussed. The
forces between a part of the grating and the particle
certainly diminish very rapidly with the distance
between the two. The direction of reflection should
therefore be determined only by those parts of the
grating which are in the immediate neighborhood
of the incident particle, but none the less it is
found that the most widely separated portions of
the grating are the important factors in deter-
mining the sharpness of the diffraction maxima.
The source of this contradiction is the confusion of
two different experiments. If no experiment is
performed which would permit the determination
of the position of the particle before its reflection,
there is no contradiction with observation if the
whole of the grating does act on it. If, on the other
hand, an experiment is performed which deter-
mines that the particle will strike on a section of
length A* of the grating, it must render the know-
ledge of the particle's momentum essentially un-
certain by an amount &p~h/Ax. The direction of
200
THE WAVE EFFECTS
its reflection will therefore become correspondingly
uncertain. The numerical value of this uncertainty
in direction is precisely that which would be cal-
culated from the resolving power of a grating of
&x/d lines. If A# <d the interference maxima dis-
appear entirely; not until this case is reached can
the path of the particle properly be compared with
that expected on the classical particle theory, for
not until then can it be determined whether the
particle will impinge on a ruling or on one of the
plane parts of the surface, etc.'
We need not, in this experiment, trouble about
the uncertainty of the positions of the individual
atoms of the apparatus. We are dealing with an
uncertainty so large (the whole width of the grating
constant) that the atomic uncertainty is negligible.
Now, we have regarded the position of the in-
strument as uncertain by that large amount. The
result is to produce a diffraction pattern, provided
that the light consists of perfectly determinate particles,
behaving just as classical particles would behave. For the
momenta of the particles before impact are re-
garded as free from the restrictions of the h rule.
That they arrive at the plate in a subservient con-
dition, is due to their traffic with the atoms of
action of the grating.
If, on the other hand, we regard the position of
the grating as determinate, and not subservient to
the h rule, we shall get the same diffraction pat-
tern, provided that the light particle is a merely imagined
2OI
THE SERIAL UNIVERSE
point in what is really a wave-group governed before im-
pact by the quantum restrictions.
The illustration is clear enough. Every un-
certainty in Nature can be regarded as yoiir'bwn
uncertainty concerning your instrument. The case
here parallels on a larger scale the case of the
scintillation experiments. There we saw that, if
we assert that the uncertainty in the position and
momentum of the ionised electron follows the h
rule, then the incident particle must be deter-
minate and free from such restrictions.
The reader may be a little puzzled as to how we
can contrive to construct a science if we have to
regard our instrument as indeterminate. The
answer is : Easily enough, if you know the rule
governing that uncertainty the h rule. He may
wonder, also, whether it would not be simpler to
treat the instrument as free from h restrictions, and
to attribute these to the system under observation.
But here the rule of the regress comes in. When any
knowledge has to be expressed in the form of an
infinite regress, you must trace that regress far
enough to bring in the relation between the second
term and the third. That means, in this case, that
we must regard the universe from the point of view
of a four-dimensional observer, who would put the
instrument into his picture and regard that in-
strument as the only thing which is governed by
the h rule. And remember: it is impossible to
imagine a more effective way of losing knowledge,
202
THE WAVE EFFECTS
or a more prolific method of introducing con-
fusion, than that which consists in expressing your
knowledge in the form of an infinite regress and
then confining your study to the first term alone.
If the reader has still any doubts remaining, let
him glance at FIGURE 22. It exhibits the relations
between the atom of action and the two uncer-
Axis ofp
B D
E
M
N
F
-Axis of q
FIGURE 22,
tainties of position and momentum. I have copied
it from a sketch I made last Spring; but the de-
monstration has been published independently
since then by Professor Flint in the pages of Nature,
where it elicited no contradiction that I noticed.
So the reader may regard it as sound.
The entire area ADGJ represents action /></. The
small area jV represents an atom of that action;
and it will be seen that it is equal to A/? Ay. Thus,
203
THE SERIAL UNIVERSE
the uncertainty of an action measurement is due
to the atomicity of the action. Obviously, if you
regarded the uncertainty in the action measure-
ment as due to the difference between the area of
the whole figure ADGJ and the inner area M,
a difference, that is to say, equal to the areas
ADFC + EFGH, then the Uncertainty Principle
would not hold. So far Professor Flint goes. What
follows is my own opinion, but I do not anticipate
any disagreement from so clear-sighted a physicist.
Suppose we asserted that the instrument and the
object measured thereby were both composed of
atoms of action equal to the N of the figure. It is
clear that the uncertainty in the resulting measure-
ments of the object would be doubled. Aj&A^
would be 2JV. Can we get round this by supposing
that JVin each case = A/2, so that the sides of that
area equal A/?/\/2 and A q/\/2 respectively, instead
of the Aj& and Agr shown? No, for the total observed
uncertainty in the measurements ofp would be then
% + *t. VaAt
<V/2 V 2
instead of A/? required by quantum theory and so
with A</.
So we must have action atomicity either in the
instrument or in the external world, but not in
both. And, as already explained, the nature of the
Time picture attributes that atomicity to the in-
strument.
It is to be noted, of course, that, while C will
204
THE WAVE EFFECTS
regard JB l as indeterminate, and A l (inferred as
abstracted from A 2 ) as determinate, he will realise
that the indeterminate character of B l will make
B l observe A l as also indeterminate. Consequently,
so long as observation is confined to a single A l
only, and this is not interfered with between
observations by other entities in the external
world, no error will be perceived. But that would
be a very limited kind of science.
The correct procedure for a modern physics
which seeks to ascertain the nature of the external
world is to assume quantum uncertainty in the
instrument and no quantum uncertainty in Nature.
Then, and then only, is it possible to calculate
easily what is going on among the entities which
are not being observed at that instant. That cal-
culation having been made correctly, an experi-
ment in which, again, allowance is made for the
instrument's uncertainty will prove the accuracy
of the work. When the instrument interferes, it
passes an atom of action to the external world or
accepts an atom therefrom, but there is no need
for us to attempt the impossible picture of that
atom maintaining its integrity in that external
world. Indeed, the regress forbids us to entertain
any such notion forbids us to convert our episte-
mology into a metaphysics forbids us to attribute
to Nature an indeterminism which pertains, pro-
perly, to the observer.
The reader will appreciate now the significance
205
THE SERIAL UNIVERSE
of the warning given at the end of Chapter xvii.
We have no right to say that a trap which has
caught a wave was a trap for particles no right
to say that, in such and such a case, the wave
theory and the particle theory have been in con-
flict. Every such supposed instance requires re-
examining in the light of the knowledge that the
uncertainty which has been attributed, hitherto,
to the particle is an uncertainty which should,
rightly, have been attributed to the interfering
agents. And the result, it may be safely pro-
phesied, will be to exhibit Nature as a world of
particles obeying the laws of Relativity. For you
cannot deflect a particle at any stage of an experi-
ment without stamping upon it the trade mark of
the deflector's uncertainty.
It may be worth while, now, to glance at
FIGURE 1 1 and see in what manner the uncertainty
regresses. The t in any action measurement is
always that particular dimension of regressive
time which is being regarded, by the observer, at
the stage concerned, as absolute time. B l regards
AI& future as lying in the fourth dimension. He
regards icT 2 as, simply, time, and A^s action as,
simply, energy, PS, multiplied by this time. That
action appears to him as atomic. But C regards
the future of all objects in his field as lying in the
fifth dimension. He looks upon icT 3 as, simply,
time. According to him, the atomic action per-
tains really to JB ly which instrument, consequently,
206
THE WAVE EFFECTS
can discover no magnitude smaller than h in A^s
action an action in which the time component
must lie, also, in the fifth dimension. C cannot
measure A^s action himself, he has to let B^ do that
for him, and the atomicity offi^s action compels JB l
to report that ^'s action is atomic. But C per-
ceives no necessity to regard ^'s action as also
atomic the atomicity of B^s action is sufficient
to account for the observed facts.
C would ridicule .6/s notion that A 2 is the action
of A l . For A 2 , to C, is PSS (vide page 1 64) . There is
an atom associated with that, but it is not, to C,
an atom of action. In brief, what B l would regard
as h, C would consider to be ihc = tine 2 /a where e
is the atom of electrostatic charge, and a is the
'fine structure constant 5 .
Such is the picture of a physical universe in terms
of time. Naturally, if, in attempting that picture,
we employ time wrongly, the picture will fail.
And so, with the recognition of the regressive
character of time, the whole wave-particle mystery
vanishes. Nature regains her complete deter-
jiuuism, and her past becomes, once more, entirely
respectable. She may even smile, not unkindly, at
the observer's uncertainty concerning his own in-
struments. And he may smile back; for he, as the
proprietor of the instruments, has always the
power to interfere with Nature's determinate
scheme.
207
CHAPTER XX
INTRODUCING THE REAL OBSERVER
The reader will appreciate now the complete
artificiality of analysis in terms of time. I take two
objects, both, to me, in the A class, and hey
presto! one of them turns into a B l galloping
along the time in which the other one endures. It
is purely a matter of interpretation, and the inter-
pretation depends upon which one I choose to
select as my source of information about the other.
But the reader will have realised also, I hope,
the extraordinary way in which this device
abstracts sense out of what, otherwise, would be
nonsense.
He will guess, moreover, whither the last para-
graph is heading. I should like to hurry on to-
wards that goal. But we cannot do that yet. There
is a great host of objectors standing by a host
headed by the allied ghosts of John Locke and
Ernst Mach a host of innumerable epistemo-
logical purists.
Both Locke and Mach, I think, would have in-
sisted that our journey has been made from a
starting point which I omitted to define. For, at
the beginning of Chapter vii, I opened the time
regress in the following words :
' Let M represent a particular configuration of
208
INTRODUCING THE REAL OBSERVER
the external world as this last is described by you
from observation, experiment and calculation. The
particular configuration which M is to represent is
the one which is open to your observation at the
present moment.'
How are you to know which is this c present 3
configuration? And what is the use of my telling
you that you must put your chosen instrument at
that 'now 5 in the time map, before you have dis-
covered where that 'now 3 is? The instrument may
mark it, when found; but, since you can change
instrument and object about at will, neither of
these can make it.
So the whole analysis has been based upon the
presupposition that you, as a psychological in-
dividual, are situated at the c now ' of some time
which is apparent to you. It has been founded,
moreover, on the presupposition that you have
knowledge of a physical world as well as knowledge
of a world of phenomena. We must accept the
first assumption, otherwise the whole physical de-
monstration breaks down. We must do something
more than accept the second, if we are to construct
an edifice which philosophers will regard as other
than a phantasy.
.Note that we have not got to justify the first
hypothesis your knowledge of a psychological
'now 5 . We are trying to discover whether there is
any method of describing the universe which
would satisfy the needs of the self-conscious
FSU
209 14
THE SERIAL UNIVERSE
observer we imagined in the previous chapter. We
are proceeding by a method of trial and error.
'Here is time! Let us see if that fits. 3 So we try
what amounts to equipping you with an intuitive
knowledge of 'now'. The analysis in Part II,
'General Test of the Theory ', shows that this fits
to perfection. It shows that anyone with the
initial intuitive knowledge of a 'now' must have an
intuitive knowledge of the serial dimensions of time,
and can be a self-conscious observer.
Now, the original analysis of any self-conscious
observer showed that such a creature would re-
gard his objective world as comprehensible and as
subject to his interference. So, in Part III, we tried
equipping this psychological observer with an in-
tuitive appreciation of force, as well as of space and
of time. Possibly, you did not notice that we were
doing this; but it was implicit in the statement
that he could take P, S and T instead of Af, S and
T as elementary indefinables in terms of which the
objective world could be described. It was proved
thereafter that the world in question would be
regarded as comprehensible. But the supposition
of an intuitive knowledge of P, S and T as in-
definables suited to the description of an external
world of physics meant that, if the psychological
observer possessed that intuitive knowledge, he
could discover that physical world. This would be a
reply to Subjective Idealism. Consequently, we
must examine it rather carefully.
210
INTRODUCING THE REAL OBSERVER
There are certain phenomenal objects, e.g., a
' chair 5 , which, when you apply force to them,
move. Given the intuitive appreciation of resist-
ance and the intuitive appreciation of space, the
resistance appreciated multiplied by the appre-
ciated distance of displacement of the phenomenal
object constitutes a complete appreciation of
physical energy. The appreciation of this complex
is not elementary, it is a 'percept' and not a
'sensation 3 , but that is immaterial. External
physical energy can be discovered.
Next, let us look at the matter from the point of
view of psycho-physiology. Among the various
kinds of neurones with which your nerve endings
are equipped, there are some which can be stimu-
lated by simple pressure. These are to be found in
the skin, in the muscles and embedded in those
parts of the joints* which roll upon each other. The
pressure registered by the muscular neurones is a
measure proportionate to the strain exerted by
those muscles in moving a limb : the change in the
pressure from one neurone to another in the rolling
surfaces of the joint gives you direct information as
to the amount of rotation of the limb. Conse-
quently, when you move a limb, you can perceive
P+>, or energy.
In both cases the energy appreciated is a per-
cept, and a percept which is just as much 'pheno-
menal 3 as is that percept of the coloured sphere
which you learn to regard as an 'orange 5 . In both
211 14-2
THE SERIAL UNIVERSE
cases assimilation and association are at work to
produce the complete percept.
Now, let us add the appreciation of time, T.
Whenever you move a particular portion of your
body, a curious law comes into operation; and this
law is open to your appreciation. In all the
changes of P, S and T accompanying the change of
position of the limb there is one quantity which
remains constant, and that quantity is the force
divided by the acceleration. That quantity is the
mass of the limb. The process of learning what
force to apply in order to produce a required
acceleration of the phenomenal limb (or accelera-
tion of the rate of change of pressure from one
neurone to another in the joint) is precisely the
same thing as learning what is the mass of the
limb involved. There is, then, no reason why a
child in the pre-natal condition should not become
aware of the world of mass.
And the possibility of discoveries of this kind is
not confined to the realm of the body. The pres-
sure neurones in the skin of your finger tip will
inform you of the resistance offered by an external
object of which you have no other sensory appre-
ciation. If you move the finger, the joint neurones
inform you of the displacement of that point of
resistance. But the pressure recorded will be less
than the pressure recorded by the muscular
neurones, because the pressure in the latter case is
that needed to accelerate both the limb and the
212
INTRODUCING THE REAL OBSERVER
external mass, while the finger-tip pressure is that
which is needed to produce the same acceleration
in external mass only.
Thus, the intuitive knowledge of time and space
accepted (on trial) in Part II, plus the sensation
of pressure (demonstrable in any psycho-physio-
logical laboratory) provides any purely psycho-
logical observer with all that is necessary for the
discovery of an objective physical world.
If the reader does not like this theory, he will
have to fall back on one which is, I regret to say,
rather popular nowadays. The idea is that the
child distinguishes, after birth, phenomena appear-
ing and disappearing at certain points in space;
discovers, by consultation with his nurse or other
children, that other people perceive similar pheno-
mena; arrives at the conclusion that these other
people are real; then, by a tremendous effort of
imagination, invents something which is not the pheno-
mena to occupy that point in space; then, reading
the laws of Sir Isaac Newton, arrives at the notion
of 'mass' as the occupant; and, finally, just about
at the time he is leaving school, learns that his
limbs being composed of fixed quantities of New-
tonian ' mass ' will accelerate in proportion to the
ajnount of force he applies to them. This discovery,
made in the nick of time, enables him to perform
the motions necessary to take him to a university.
The fact that we are equipped with a special psy-
chological apparatus for discovering the physical
213
THE SERIAL UNIVERSE
world, without having to call upon any sensation
save that of pressure, came to me as a considerable
surprise. I had imagined before that the physical
universe was something which, somehow or other,
we abstracted from such sensations as light and
sound and heat and cold. But none of these is
involved. Pressure is the only sensation required.
Consequently, with the acceptance of P, S and Tas
terms for physical description, (as we have done
everywhere in Part III), we have a complete
physical universe running through from the re-
motest visible star in A l9 to the ultimate psycho-
logical observer at the unreachable end of our
table.
It is interesting to observe how this direct ac-
quaintance with the physical world, by means of
the sensation of force, is related to the remainder
of the sensations. You are constantly changing
these other psychological phenomena. Your eye-
lids tire, and you let them fall. Immediately, a
previous visual phenomena vanishes. You move
your hand; and, forthwith, a previous unpleasant
feeling of heat disappears. In such cases, you, the
psychological observer, interfere. But it is im-
portant to note that you do not interfere directly
with the sensation. You close your eyelids: you r^-
move your hand. And the eyelids are not the visual
phenomenon; the hand is not the sensation of heat.
Here you become aware of a new class of objects,
existing independently of the purely subjective
214
INTRODUCING THE REAL OBSERVER
sensory presentations the colours, lights, sounds,
etc. You may open and close your eyes in dark-
ness, when there is no visual phenomenon to be
observed. You may move your hand when it is
touching nothing. And experiment shows that, if
we classify the ordinary psychological objects as
phenomena observed, we can classify this second
class as observational facilities and observational
restrictions. It is with this world of facilities and
restrictions that we interfere when we change an
elementary phenomenon.
We may pause here to note that one value of the
physical universe seems to be that it ensures a
community of experience without which we should
be eternal strangers to one another.
We see, then, that the physical world constitutes
a thread running straight through the hitherto
separated sciences of physics and psychology. The
ultimate source of the energy transferred to the
external world in the course of an experiment is
the psychological observer himself. He is the re-
gressive physical entity. So the question arises:
How are we to bring brain into our table?
315
CHAPTER XXI
THE PLACE OF BRAIN
One method of ascertaining the connection which
exists between the world of phenomenal objects
and the observer's physical brain is to get hold of
another fellow, poke his nervous system, and listen
to what he says about it. His remarks may or may
not be instructive ; but, since he can talk, you will
gather more information by listening than merely
by watching what he does about it. The scientific
observer, however, is not really dependent upon
outside assistance, so far as regards the discovery
of the mere fact that the physical correlates of a
psychological phenomenon involve his nervous
system.
Consider again that classical illustration of a
psychological phenomenon: the globe of colour
you call an ' orange'. Interpose your hand be-
tween the phenomenal object and your eyes, and
the presentation vanishes. You have grounds then
for saying that the phenomenon has a physical
'correlate' external to your eyes. But now, press
with your finger on the corner of your eyeball. The
phenomenon alters its shape. Further, it is pos-
sible for you to sever your own optic nerve, when
the psychological object will vanish completely.
You have reason, then, for asserting that the
phenomenon possesses a neural correlate. But that
216
THE PLAGE OF BRAIN
last discovery does not permit you to assert, that
the phenomenon has no correlate external to the
brain. A stimulation of the nerve by something
external to the brain is the essential condition to the
experience of what psychologists call an c impres-
sion'. Even when you cut the nerve (an operation
which is accompanied by the impression of a flash
of light) the essential stimulus is from outside the
organism. Phenomena which involve no such
external stimulus, e.g., the memory 'image 5 of the
orange, are of an unmistakably different character.
(It may be remarked here that an 'hallucination 5 ,
according to the best authorities, involves some
external stimulation of the nerve endings and the
illusions consist of a misinterpretation of the nature
of that stimulus.)
Precisely similar considerations apply if you
trepan your enemy Smith and look at his brain.
Seeking for the physical correlates of the conse-
quent visual phenomenon, by the simple method of
exploration with your hand, you find that these
comprise a connected chain of physical objects
starting with Smith's brain and including part of
your own. The method, of course, leaves you ig-
norant of any but the most macroscopic details of
the chain, but it suffices to assure you that you
as the psychological observer B l of phenomenal
objects A l at the 'now 5 must place your own
brain in the same world as Smith's, viz., among the
physical correlates of the A l phenomena.
217
THE SERIAL UNIVERSE
Tabulating, then, the regressive observer of im-
pressional phenomena, we fill in his A l compart-
ment as follows :
Impressional Phenomena
paralleled by
Brain affected by an external object
His B l compartment will contain :
Observer of Phenomena A l
and
Physical Interactor with Brain A^
If he is merely a thinker manipulating the so-
called memory 'images', the A compartment will
contain :
Memory Phenomena
paralleled by
Internal activity of Brain
And B l will be :
Observer of Memory Phenomena A 1
and
Physical Interactor with Brain A^
218
THE PLACE OF BRAIN
What started us along the time regress, however,
was the search for the source of the energy which
makes its way into the object world in the course
of every experiment. It might be stored in the
observing instrument; but, on bringing that instru-
ment into the picture, the time regress compels us
to realise that the source of the energy which
releases the stored energy in the instrument (if such
there be) is still to seek. The result is the infinite
/ >rVr . l ^-.< ,.-. * " '
regress of a source pf eixprgy. Now, we know, all
of us, that the energy which initiates an experi-
ment with an instrument comes from the experi-
menter's brain. And I suppose most of my readers
expected (as I did myself) that brain would enter
the regress as the observer C. We see now that it
does nothing of the kind. The experimenter's
interfering brain comes into A l9 with all the rest of
the objective physical world, including the physical
instrument we employ as B i .
And this brings us back to the fact to which I
drew attention at the beginning of the last chapter.
In experimental physics, we take what is actually
an A l object selected from the external world, and
employ it as a means of observing some other
object in that same A l world. We see, from the
tables we have just worked out, that it is the psycho-
physical observer JB l who makes that selection. The
external instrument is the external object which
affects brain, in the first of the present tabulations.
But there are many such external objects and many
219
THE SERIAL UNIVERSE
corresponding affectations of the brain. B l takes
his choice. But there is a limit to what he can do
along these lines. The operator of that selected
instrument is the A l brain; and the real B l is the
psycho-physical individual who employs this A l
brain plus this A l object as a means of studying
some other suspected A^ object which may not be
affecting brain at all. He himself, the psycho-
physical B l is situated at the time i 'now', and
is travelling along the fourth dimension. He selects
an A l object as an instrument, and makes it travel
with him. That process is simple enough. The
selection of an A l for use as a B l involves merely
that you interpret it as a three-dimensional entity
of changing character instead of as the changing
contents of a mere travelling field of view passing
over A 2 . Actually, what the psycho-physical B l
observes is a travelling sectional view of the brain
A%. That view is his A. He treats that as a three-
dimensional entity which is changing its character,
and so he converts it into a companion B 1 travel-
ling with him. The external instrument which is
affecting that neural companion is being treated,
consequently, as a third party to the plot it
also is regarded as travelling along time i. But
the only entity which is really travelling, in tlje
regress of the psycho-physical observer, is the
psycho-physical B l . And, since he is three-di-
mensional, he cannot select ^four-dimensional entity
from the brain he is observing, and use that, with
220
THE PLAGE OF BRAIN
an accompanying four-dimensional entity in the
world external to brain, to play the part of an
instrumental observer C.
Thus, the real time regress in the world of physics
is the regress of the psycho-physical observer who
lies behind all nervous matter a physical creature
indeed, but one confined to the realms of biology.
It is that creature whom we imitate when we use
our clocks and measuring rods to map out an
object world in terms of time. And we can carry
that process only one stage of the regress, the stage
where an instrument is treated as a B l9 and C is
merely imagined.
But, this being the case, what about the regress of
h? It cannot regress more than one term, from
the object world to the instrument ! For there can
be no h in the uncertainties of the psycho-physical
observer : he is far too coarse a creature to respond
to anything so ultra-microscopic as a single pho-
ton. Obviously, then, h must be something which
we put into the instrument when we regard the latter
as an entity of changing character travelling along
A 2 and abstracting sectional views therefrom
something which we insert when we treat that
instrument's temporal endurance as in the fifth
instead of the fourth dimension.
But that is an investigation which deserves a new
chapter.
221
CHAPTER XXII
Let us glance back at our table of abstractions
on page 152. We see that the travelling, three-
dimensional B l9 consisting of energy PS, abstracts
energy from the four-dimensional world A 2 pos-
sessing the dimensions PS x icT 2 . We can find no
fault with that. To 'abstract' is merely to pick
out a character, as a dynamometer picks out force
P from momentum PT, or as a tape measure can
discover lengths within the area of a tennis court.
But, in the world of physics, B l does not merely
c abstract' energy : it subtracts it. Energy is actually
transferred from A 2 to B l in the course of an obser-
vation, and is passed from B 1 to A 2 in the course of
every interference with A 2 for experimental pur-
poses.
Now, A 2 is a four-dimensional quantity. And
you cannot subtract, as an independently existing
thing, a three-dimensional component from a four-
dimensional thing. If you reduce A%s energy com-
ponent, you reduce the magnitude of J 2 's content
PS x icT 2 9 just as, if you reduce the length of your
tennis court, you reduce its area. Now you can
take away from the area of your tennis court and
add what you have gained to the area of your
flower-beds. But you cannot borrow from an
222
area and say that you have utilised the borrowed
bit in increasing the length of a line. We cannot
pass PS from A 2 to B l without robbing A 2 of a
portion of PS x icT 2 and utilising it nowhere.
The most obvious thing to do seems to be to add
a little time i thickness to B 1 . Unfortunately, that
is just what we are unable to do. For B l is moving
through the four-dimensional world with the
velocity c, and, according to our regressive Re-
lativity, this velocity is as critical in four-dimen-
sional space as it is in three-dimensional space.
B l can have no thickness in the direction of its
travel.
Very well, suppose we give up all this business of
imitating the psycho-physical observer with in-
struments external to brain. All said and done, it
was we who converted an A l mass of metal,
mirrors, prisms and what-not into a B. We did
that simply by regarding it as a three-dimensional
entity of changing character, instead of as a
travelling, sectional view of a more real entity A 2 .
Let us drop that interpretation, and regard the
thing as an A l . Then it will extend in time i as an
A 2 accompanying the object A 2 . We can let the
real B l of the regressive psycho-physical experi-
menter serve to determine the 'now 3 .
That, I am afraid, will not help us. For the re-
gress we, actually, are following is the regress of
that psycho-physical individual. It is from him
that there comes the inflow of energy to the
223
THE SERIAL UNIVERSE
physical world A l . And it is the passage of energy
between his A 2 (i.e., brain) and his l which is,
really, our difficulty. If the trouble can be got
over in his case, it can be got over in the same way
in the case of the instrument and the object, where
both these are in the world external to brain.
But the fact that we can, if we please, re-convert
our JB l instrument into an A l9 merely by inter-
preting its changes in a different fashion, is of im-
mense importance in our problem. For, when we do
this, we are, as I said before, re-converting our B 2 into
an AZ, and can transfer quantities of the original
object A% to this new A 2 . Suppose we do this when-
ever we think of the instrument and object as inter-
acting. We can, immediately afterwards, treat the
instrument in the other fashion, i.e., regard it as a
BZ which has collected PS x ic T% from the object A 2 .
Now, it is important that the reader should
grasp the fact that there is no Take' in this purely
mental operation. It is absolutely legitimate for you
to regard a three-dimensional object either as (/) an
entity situated at your own travelling psychological
*now\ an entity which is changing its character, or
as (2) the view which a four-dimensional entity presents
to your travelling psychological *now\ When you are
employing that object as your source of information
about another object you are regarding it as (/): when
you cease to consider it as such a source of information,
you are regarding it as (2}. The change in your method
of interpretation involves no logical error of any kind.
224
The reader will find an illustration of view (2)
on page 69.
But, now, consider what is the result of this
change of interpretation the result in your five-
dimensional map. Your 5 2 line runs no longer
athwart that world in a continuous fashion like
the line in FIGURE 1 1. It goes, instead, like this:
G"
H"
o"
o,
G If
FIGURE 23.
(For simplicity the interior vertical lines of FIGURE 1 1 are omitted.)
The breaks between 0' and show where you,
when PP' was passing those places, were regarding
B l as an A l interacting with another A l9 that is to
say, were regarding it as part of the substratum,
with extension in the fourth dimension and en-
durance in the fifth like any other entity in that
substratum. At those places, the instrument was
being thought of as interacting with the other
objects of the physical world just as these interact
FSU
225
THE SERIAL UNIVERSE
with one another it was not being regarded as in
any way a unique determinant of the map. The
places where O'O is unbroken show where you
were examining B^ to see what had happened to
it, with the intention of drafting your map from
the information thus obtained. The dotted exten-
sion above J indicates merely your uncertainty re-
garding the change in the momentum and position
of the instrument consequent upon the last inter-
action with the substratum.
It will be noticed that the breaks the discon-
tinuities are of different lengths. Obviously, you
can leave your instrument to collect PS x icT 2 for
as long a time as you like.
The essential point is that your (purely mental)
operation makes the duration of your instrument
in fifth-dimensional time discrete. Now, in the
measurements of a B 2 quantity, PSxicT 3 , the
energy PS is already discrete. (A body may
possess definite and limited amounts of energy.)
Consequently, since both components of B 2 are
discrete, B 2 itself consists of discrete portions of
PS x ic T 3 . Now, the observer C does not regard the
fifth dimension as icT 3 : he regards it as, simply,
'time 5 . So, to him, the discrete portions of B 2 are
discrete portions of action, of varying magnitude. .
The employment of this perfectly legitimate
mental device is subject, however, to certain re-
strictions. You must not forget c, the rate of travel
of the ' now 5 . You must not interfere with P or S,
226
which are unaffected by the question as to whether
your instrument is a travelling B l or a travelling
view of an A 2 . So you must not lose energy PS in
the course of the operation. Your discrete portions
of 5 2 's action have got to be equal to correspond-
ing discrete portions of A^s PS x S 4 , and the latter
quantity involves atomic electrostatic charge, e.
Planck, finding himself faced with the necessity of
considering action as discrete, owing to the be-
haviour of c black body ' radiation, found that c and
e and the constant called the Absolute Tempera-
ture and yet another constant, Boltzmann's k,
would require to be taken into account. The last
two are connected with the 'entropy' of the ex-
ternal world, which gives the sense of the travel of
the 'now', and so must be taken into account by
ourselves. Planck did not pretend, of course, to
know why action should present itself to us as
discrete: he supposed this discontinuity to be an
inexplicable attribute of the object world. But he
discovered that the restrictions involved in the
acceptance of these four constants which are our
restrictions would limit the size of the discrete
portions. They could not be smaller than h.
And there's your quantum! perfectly logical,
ajid involving no breach of continuity in anything
save the interpretations of the ultimate observer.
And it is a quantum which pertains, as we had
expected, to the instrument and not to the object.
227 '5-2
CHAPTER XXIII
CHRONAXY
Very pretty/ says the reader, (so I hope), c but
you have forgotten one thing. Your h proves that
the physicist is describing his world as if it were
being observed by an imagined serial observer.
And he cannot obtain the "now" he requires for
that purpose unless he himself is a real serial
observer. But then, he, as this real serial observer,
is confronted by the same difficulties as confront
his imagined four-dimensional individual. He can-
not pass energy in a continuous stream between
his psycho-physical JB l and his A l brain. He, as a
four-dimensional individual, must treat the time 2
extension of his B 2 as discontinuous must accept
nothing but discrete lumps of action from his A 2
brain. Now, if he does that, the effect should be
observable in brain whenever he interferes with that
organism. And it should be a large scale effect; for
he is a macroscopic individual. I cannot accept
your h as the solution of the problem in his case.
And, remember, Nature will have a say in the
matter. He will find limits of some kind to tfye
jumps of his B^S
And so he does.
This discovery was made by Professor L.
Lapicque, and has been studied in great detail by
228
GHRONAXY
himself, Bourguignon and Haldane. Possibly
there are others who should be mentioned, for the
discovery is, now, several years old.
Suppose you apply an electrical stimulus to a
nerve. It will have to be a motor nerve, if you are
to observe a measurable effect, but nervous matter
is of the same kind everywhere, and it is with the
physical response of the nerve that we are con-
cerned. It is found that the intensity of the stimulus
necessary to produce a response from the nerve varies in-
versely as the duration of that stimulus.
That means that the nerve responds, not to
energy, but to action energy x time.
Again, it is found that there is minimal duration
necessary to produce a response. It is an extra-
ordinary fact that, if the duration is of less than
this minimal duration, there is no response, no
matter how intense the stimulus! Conversely,
there is no response unless the stimulus has a
minimal intensity, no matter how long the duration.
That seems easier to understand. But the point
is that, since (as we have just seen) the nerve is
responding to action, the minimal intensity mul-
tiplied by the minimal duration constitutes an
atom of action so far as the nerve in question is
concerned. It is true that this atom of action is un-
like the quantum, inasmuch as it is composed of an
atom of energy multiplied by an atom of time; but
that does not make the action other than atomic.
It means merely that the character of the atom of
229
THE SERIAL UNIVERSE
action in the physiological world is more restricted
than is the character of h. It is four-dimensional,
but it has to possess a certain four-dimensional
shape \ whereas that shape in h is elastic. Again,
the physiological atom of action varies with dif-
ferent nerves, but there is no reason, in our theory,
why this should not be the case. For the ultra-
microscopic world, which has to be taken into ac-
count in the ultra-microscopic experiments pos-
sible with the refined instruments of our labora-
tories, means nothing in the coarse reactions of
living matter. The minimal intensity and mini-
mal time can be, consequently, private idiosyn-
crasies of each biological structure, and even vary
at different stages of that structure's life-history.
So there is your discrete action in the case of the
world of living tissue in the psycho-physical ex-
perimenter's A\
Chronaxy in the muscles and in the sensori-
motor arcs of the spinal level must be purely
automatic. But that means nothing. Every phy-
siologist knows that a flow of nervous energy
which appears, at first, to be controlled becomes,
with constant repetition, entirely automatic. The
psycho-physical observer observer of sensations
and interactor with brain has a physical char
racter, and what becomes automatic in nerve or
muscle should become similarly automatic in
him. Since his B l must be the thing which makes
living tissue different from dead tissue, we would
230
CHRONAXY
expect to find it present, but habit-bound, in
every tissue showing automatic chronaxy.
It should be understood quite clearly that this
psycho-physical B l is not brain. For he can use
one part of the brain and body to observe another
part. When you press your finger into the corner
of your eye in order to distort a visual phenomenon,
you are discovering your eyeball with your finger,
which observes the resistance. You can use your
right hand to discover the left and then reverse
the process. In such experiments, the motor
system is an A l object employed as a B^ just as
a camera plate is an A l object, being used as a
source of information regarding another A l ob-
ject.
231
THE SERIAL UNIVERSE
* * * *
PART IV
CONCLUSION & APPENDIX
CONCLUSION
We have now completed our survey , in Part III,
of the evidence afforded by the exact sciences.
That evidence bears out completely the conclu-
sions arrived at, on purely mathematical grounds,
in Part II. The extensions of modern science : Re-
lativity ; Wave-particle effects; the Quantum itself:
these have proved to be merely examples of the
fact that a time picture is necessarily a regressive
picture, and one which could not be initiated save
by a regressive observer aware of a travelling
c now'. If we substitute, for the real observer i,
the instruments of our laboratory, and proceed to
make a time picture, we find that we are fitting
those instruments into the c now ' of the real ob-
server i we had hoped to escape, so that the object
world exhibits itself to those instruments as it
would to him, did he possess the same accuracy
of observation. And we are left, still, with the
fact that the source of certain energies which
make their way into the external world during an
experiment, and have to be accounted for, lies at
the unreachable end of the regress of the real
observer.
We find that the time picture studied in Parts II
and III fits perfectly the table of the self-conscious
observer which we worked out in Part I, and may
say, therefore, that man must be a self-conscious
235
THE SERIAL UNIVERSE
observer employing time as one of his terms of
description because its regressive character fits his
needs and gives him the only kind of picture he
could regard as both rational and empirically true.
But we discover a great deal more than that. We
find that such an observer cannot be otherwise
than immortal in his own time 2, whatever he
may be in anyone else's time 2. He survives the
destruction of his observer i, and survives with the
whole of his time i c past ' experience as his four-
dimensional equipment. It is unalterable, because
it is fitted to the unalterable past of the objective
world. This constraint this interference with his
freedom constitutes his observation of that ob-
jective world. '
Lest the reader be unduly alarmed by this
picture, I may say here that there is plenty of
evidence to show that observer 2 is essentially a
creator of imagery imagery which seems unreal
to us now, but entirely real when we glimpse it,
as we do, in our dreams. But none of this last
falls within the province of the exact sciences. All
that these can say is that, since man views the
world in terms of time, he must be immortal in
time 2. And that, I think, they may say positively.
The reader who wishes to know more about the
merely psychological aspects of this four-dimen-
sional, psycho-physical being will find a great deal
on that subject in the book called An Experiment
with Time.
236
CONCLUSION
And now we may attempt an answer to the
question we asked ourselves in the Introduction.
Is the universe rational or irrational? And the
answer isVRational in everything save the ulti-
mate observer who makes the picture. He, with
his self-consciousness and his will and his dualism
of psycho-physical outlook, is irrational; but, no
matter how far you may pursue him, you can
never discover this. For when you reach any
observer in the series, and put him into the pic-
ture, he promptly transfers the irrationality to the
observer next behind him. Thus, rationality, in
the philosophy of an epistemologist, lies in an in-
finite regress. To a metaphysician, it lies in re-
fusing to consider any subject-object relation what-
soever. And that involves the denial of all know-
ledge obtained by experiment.
The reader is at perfect liberty to become a
metaphysician and to say that the time picture is
all wrong. But he cannot then claim that the
particular metaphysical picture he may favour can
be tested by experiment. Moreover, that will not
enable him to escape his immortality. For when
he talks about c after 5 death, he is reverting to the
time picture, and in that picture he is immortal.
Do we desire this immortality, now that we may
feel reasonably assured that we possess it? Some of
us dread it, having the false notion thereof I re-
ferred to on page 37. But all of us hate, with a
hatred too deep for expression, the notion of the
237
THE SERIAL UNIVERSE
whole of Nature being, to Life, no more than
c an indifferently gilded execution chamber', 're-
plenished continually with new victims'.
But, for me, the question resolves itself very
simply. There is adventure in eternal life. There
is none in eternal death. And I am all for
adventure.
238
APPENDIX
Extract from 'An Experiment with Time'
We may, conveniently, carry the analysis one
stage further; but we need not trouble to repeat
the arguments.
We shall discover, of course, that the time and
the field and the observer, which, in stage 2, we
considered as being ultimate, were not ultimate at
all ; and we shall come upon a larger-dimensioned
lot of ultimates which, in their turn, will only re-
tain that status until the next stage is reached.
And so on to infinity.
In FIGURE 25 we exhibit three dimensions of
time as the three dimensions of a solid figure seen
in perspective. We have to draw imaginary
boundaries to this figure in order to make the
perspective clear; but, actually, there are no such
boundaries at the top or the bottom or the back or the
front. The figure has fixed sides (representing birth and
death in time /), but its extensions in the time 2 and
time 3 dimensions have no limits.
Time 3 is shown as the vertical dimension of the
block. In relation to this time the dimensions we
call time i and time 2 are akin to dimensions of
space.
The middle horizontal plane-section of this
block-figure, the plane G'G"H"H', is our instan-
239
THE SERIAL UNIVERSE
taneous photograph of FIGURE 24, shown in per-
spective. The endurances, in the new dimension
of time, of the cerebral states represented by the
time 2 extended lines in FIGURE 24 should be
shown by extending these lines in the time 3 di-
H"
0"
G'
\s
O/
H
-P'
FIGURE 24.
mension so that they form vertical planes arranged
like pieces of toast in a rack. But to fill these in
would overcrowd the diagram. Our first reagent,
O'O", will endure (extend) in time 3 as a plane
dividing the block diagonally; that is to say, the
plane ABCD.
In the c present ' condition of FIGURE 24, (shown in
240
APPENDIX
the middle of the block) , the field of presentation
GH which, be it remembered, must be marked
out by the intersection of some observing entity
with the plane of the figure is at the middle of
the plane. In the 'past' condition of FIGURE 24 (the
plane at the bottom of the block) this field this
H"
(and O"}
FIGURE 25.
line of intersection is at DE. In the 'future' con-
dition of FIGURE 24 (at the top of the block) this
field is at FB. The intersecting entity, reagent
number 2, lies, therefore, along the sloping plane
DFBE, which plane represents its endurance.
The intersection of this plane with the plane
ABCD is the line DB. The new travelling field of
presentation (field 3) is the plane G'G"H"H'. As
241
THE SERIAL UNIVERSE
this field 3 plane travels up the block, its line of
intersection with the sloping plane DFBE (the line
GH} moves over the travelling field 3 plane to-
wards G"H . That is to say, field 2 moves along
time 2. The point (where the three planes
ABCD, DFBE, and G'G"H"H' intersect) moves,
meanwhile, along the travelling line GH towards
H. That is to say, field i moves along time i .
CAMBRIDGE: PRINTED BY
WALTER LEWIS, M.A.
AT THE UNIVERSITY PRESS
