Skip to main content

Full text of "The Serial Universe"

See other formats



gj<OU 160265 >m 

>o: - 7} 

CD -< 


Call NQ. S3 Accession 

A U th.r 



T*h book ariotfKJ bo id; u ,.>i on u> batoia lit* u 
last marked below. 


by the same author 


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 lady who typed this book 



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 


Preface page 9 

Introduction 13 

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 


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 



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 


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 

Conclusion 235 

Appendix 239 


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 



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 



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 



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 



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 



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 



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 



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 


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. 








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 


a parent 

who was 

child of 

a parent 

who was 

child of 

a parent 

who was 

child of 

etc., etc., 



the relation between the terms becomes readily 

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 



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 



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 



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: 



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. 


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 

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 . 


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? 



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). 


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 



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. 


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 


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 



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 


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- 



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 


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 



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 

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. 



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 

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- 



intentionally, he was claiming for 'self-conscious- 
ness 5 the status of given, undeniable know- 

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- 

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 

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- 



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 



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 



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. 






In the next row I put the great-grandfather's de- 
scriptions of the child, the father and the grand- 












In the next row we include the great-grandfather, 
and give the great-great-grandfather's descriptions 
of all his descendants. 















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 



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- 

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 



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. 



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 


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, 


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, 




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, 



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. 


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. 




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 

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- 



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- 

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 knowledge involved in a scientific experi- 
ment may be classified, then, as follows. 

Observed by 
(abstracted by) B 


Known to ourselves 
and regarded as ex- 
isting independently 
of each other 



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. 



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 

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 



number of A 2 entities affecting B, we may call 
'World as observed by B\ 

World as observed by B 





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 


World as observed by 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: 



World as observed by B l 


World as observed by C 

A t 





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 


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 


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: 



World as observed by B 


World as observed by C^ 


B l 

World as observed by D 

A 3 







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- 

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 

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 



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 



can do so. And that, of course, is insufficient for our 

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 



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. 



* * 



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 

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- 



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- 

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 


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 

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 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 



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 


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 

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*. 



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, 



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. 



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. 


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. 



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 



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 



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 



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: 



L (A 

f) 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? 


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 



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, 




L (^ 

f) 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 



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 



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 



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 

Next, we were particularly careful not to say 

FSU 8l 6 


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 



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 





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 



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 


L @ 







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 



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 

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- 


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 


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 

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. 



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 


Present in the more compre- 
hensive system known to us 


The observer's 

In the second-term system, L, M and JV are being 
treated as: 


(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. 



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 


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. 




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 



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 




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 

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, 



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 



O 1 2 3 4 X 


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 

4 - 





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 - 



f 1 






I I 


9 1 2 

* 4 



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 


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. 









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 



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- 



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. 




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 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. 



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 



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 



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, 



... IJL 

m 71 




. b I 




. 11 

. I 

. 6 I 



m U rr 

e < 

f) " 



1 06 


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 












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 


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. 



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. 





G 1 


P 1 



(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 



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 

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 



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 



World as observed by D 





Nevertheless, we can prepare from FIGURE 11 a 
very similar table showing the 'now-marks 5 as 



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 . 








? H" 

B-2 o" 





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 



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. 




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 

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 


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- 



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 






G 1 





P 1 

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 



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. 



Is it possible, now, for us to regard this observer 2 
as that ultimate source of the impulse the ex- 


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 


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 


(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 



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 










Q- - 








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- 



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 



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 

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. 



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 


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 

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 



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- 



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 



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 


* * * 



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. 



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 



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 

- = i, i.e., /! = t 2 (i). 


One what? To give this velocity a meaning we 
must realise that the seconds marked off on the T 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 


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. 

We will call this constant, k. Then 

~ = k, a velocity. 

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. 



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 


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. 



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 



"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. 



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 



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. 


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 



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 


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? 


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, 



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 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. 



(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 




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, 


t second of Tj 

= 300,000 kilo-metres 


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 



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. 


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- 



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 





G" ] H f 









.P 1 



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 



writ of T, 

=J 4 

unitof T 2 *k 

^axis of TI 

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. 



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" 



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- 



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- 




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 

traversed per interval of time, or =., by V meaning 


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 



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 



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 

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 



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 












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 


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 


T 2 




V S 





P? 2 

icPT 2 - PS 4 


PST t 

icPSTt - PSS< 




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 

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, 



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 


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 

Our table 



1 PS 

1 PS 





PS x eT t (i) 


PSxicT, (i) 


-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 



(the GH of that diagram). So our table would 




PS x icT t 
PS x S t 



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 


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 


PS x icT 3 


=PS x S t 


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 

1 66 


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, 

- 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 


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 



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. 



If, instead of regarding Tas period, we regard S 
as wave-length, it is clear that 


,, Action 

or Momentum = ll7 -, r , 


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. 



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 



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 

The discovery of the photo-electric effect equal- 
ised matters. If particles could not account for 



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- 

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 

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'. 



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 



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- 



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 

(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 



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- 

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 



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. 


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 

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, 

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 


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 


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, 



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 

That we shall do in the next chapter. 



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 

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- 



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 





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 



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 

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 



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 

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 



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 


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 




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 


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 

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.) 



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 



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 

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 



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 

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 

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 


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- 



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 

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 


193 '3 


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 


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 



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 


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 



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 

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 



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 

* The Physical Principles of the Quantum Theory. (Cambridge 
University Press.) 



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 = * . 


The rest is simple. We can write the above 
equation in the form 

d (sin 0'- sin 0) =m x -. 

But, in the ordinary wave theory, 

d (sin 0' sin 0) = wA; 

therefore -=A. 


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, 


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- 

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 



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 



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, 



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 





-Axis of q 


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, 



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- 

It is to be noted, of course, that, while C will 



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 



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, 



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 



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 

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 



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 


209 14 


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. 



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 


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 



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 



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 

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 



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? 



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 

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 



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. 



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 

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 

Physical Interactor with Brain A^ 



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 



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 



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 



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- 

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 


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 

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 



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. 


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 If 


(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 




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, 


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 


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 



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 



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 



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- 



* * * * 




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 

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 



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. 



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 



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 



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 

The middle horizontal plane-section of this 
block-figure, the plane G'G"H"H', is our instan- 



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- 









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 



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 


(and O"} 


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 



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 .