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Full text of "Physics And Philosophy"


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Call No. 3-^V / Accession No. 

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Title JIL^CA cf 

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First edition October 1942 

Reprinted January 1943 

July 1943 



Preface page vii 

I What are Physics and Philosophy? i 

II How do we know ? 32 

(Descartes to Kant; Eddington) 

III The two voices of Science and Philosophy 82 

(Plato to the present) 

IV The Passing of the Mechanical Age 105 

(Newton to Einstein) 

V The New Physics 126 

(Planck, Rutherford, Bohr) 

VI From Appearance to Reality 15^ 

(Bohr, Heisenberg, de Broglie, Schrodinger, Dirac) 

VII Some Problems of Philosophy 174. 
Index 219 



Science usually advances by a succession of small steps, through a 
fogjn which even the most keen-sighted explorer can seldom see 
more than a few paces ahead. Occasionally the fog lifts, an eminence 
is gained, and a wider stretch of territory can be surveyed some- 
times with startling results. A whole science may then seem to 
undergo a kaleidoscopic "rearrangement, fragments of "knowledge 
being found to fit together in a hitherto unsuspected manner. 
Sometimes the shock of readjustment may spread to other sciences ; 
sometimes it may divert the whole current of human thought. 

Events of this last kind are rare, but instances come readily to 
mind. We are likely to think first of tfofr reanlte of replacing the 
egocentric astronomy of mediaeval times by the Copernican system 
*-~man^saw J:hat his home was not the majestic fixed centre of *he 
universe round which all else had to revolve, but one of many 
fragments of matter which were themselves revolving round a very 
ordinary one of the myriads of stars in the sj$;y . Or_we may think of 
the implications of the Darwinian biologfofeman saw that his bo3y 
had not been specially designed for himself, the lord of creation, 
but was an adaptation and development of the bodies of animals 
which had preceded him on earth, and were in fact his own ancestry ; 
all terrestrial creatures, even the meanest, proved to be his blood- 
relations, and if he had dominion over them it was only because he 
happened to have been born into the clever branch of the big 

?) A third such rearrangement of ideas occurred when Newton's 
system of mechanics and law of gravitation gained general accept- 
ance men saw that the heavenly bodies were no longer to be 
feared or even consulted as influences in human affairs; they were 


only chunks of inert matter moving as they were driven by uni- 
versal laws. The Newtonian scheme of things seemed further to 
suggest although it was never quite able to prove that all bodies, 
even the smallest, were subject to the same scheme of universal law, 
so that all change and motion were mechanical in their nature, the 
future following from the past with the inevitability of the motions 
of a machine. If this scheme controlled animate as well as in- 
animate matter, then clearly man's imagined freedom to choose 
between good and evil or select his own path through life was a 
pitiable illusion; the ball could only go where the player sent it. 

A fourth such revolution has occurred in physics in recent years. 
Its consequences extend far beyond physics, and in particular they 
affect our general view of the world in which our lives are cast in 
a word, they affect philosophy. The philosophy of any period is 
always largely interwoven with the science of the period, so that 
any fundamental change in science must produce reactions in 
philosophy. This is especially so in the present case, where the 
changes in physics itself are of a distinctly philosophical hue; a 
direct questioning of nature by experiment has shown the philoso- 
phical background hitherto assumed by physics to have been faulty. 
The necessary emendations have naturally affected the scientific 
basis of philosophy and, through it, our approach to the philo- 
sophical problems of everyday life. Are we, for instance, automata 
or are we free agents capable of influencing the course of events 
by our volitions? Is the world material or mental in its ultimate 
nature? Or is it both? If so, is matter or mind the more funda- 
mental is mind a creation of matter or matter a creation of mind? 
Is the world we perceive in space and time the world of ultimate 
reality, or is it only a curtain veiling a deeper reality beyond? 

The primary aim of the present book is to discuss the inter- 
relation between physics and philosophy. While the discussion is in 
general terms, it naturally has very special reference to the changes 
of recent years, and their bearing on philosophical questions such 
as those just mentioned. But as a preliminary let us consider the 
general questions: What is physics and what is philosophy? 



Both physics and philosophy had their beginnings in those dim 
ages in which man was first differentiating himself from his brute 
ancestry, acquiring new emotional and mental characteristics which 
were henceforth to be his distinguishing marks. Foremost among 
these were an intellectual curiosity out of which philosophy has 
grown, and a practicaljcuriosity which was ultimately to develop 
into, science. 

For primitive man, thrown into a world which he did not under- 
stand, soon found that his comfort, his well-being, and even his 
life were jeopardized by this want of understanding. Inanimate 
nature seemed helpful and friendly to him at times, but could 
become hostile when the life-giving sunshine and gentle rain gave 
place to the thunderbolt and whirlwind ; these inspired in him the 
same feelings of awe and fear as the wild beasts and human foes 
which threatened his life. His first reaction was to project his own 
human motives and passions on to the inanimate objects around 
him ; he peopled his world with spirits and demons, with gods and 
goddesses great and small until, as Andrew Lang has said, 'all 
nature was a cpngeries of animated personalities '. Such imaginings 
were not confined to cave-men and savages ; even Thales of Miletus 
(640-546 B.C.), astronomer, geometer and philosopher, maintained 
that all things were 'full of gods'. 

Primitive man endowed these personalities with characteristics 
and qualities almost as definite as those of his real friends and foes! 
In so doing he was not altogether wrong, for they seemed to be 
creatures of habit; what they had done once they were likely to do 
again. Even the animals understand this ; they avoid a place where 
they have suffered pain in the past, suspecting that what hurt them 
once may hurt them again, and they return to a place where they 
have once found food, considering it a hopeful place in which to 
look for more. What were mere associations of ideas injthe brains 
ofjmimals readily became translated into natural laws in the minds 
of thinking men, and leJ1to""te dlsc^ 
uniformlty~oF nature wliat TaasTEappened once will, in similar 

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circumstances, happen again; the events of nature do not occur at 



randgm^butjafter an unvarying pattern. Oncg tWs^discoveiy had 
been made, physical science becapie possible. Its primary aim is to 

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discover this patternjgf eventg T injsojar as it governs the happenings 
of the inanimate world. 


The primitive stage of human development which we have just 
depicted is that which Auguste Comte (1798-1857) described as 
the stage offetichism, although we now usually call it animism. In 
this stage man believed he could modify the course of events by 
his own volition and to his own advantage, by influencing the gods 
and spirits with which he had filled his world sometimes through 
a policy of appeasement, as by worship and sacrifice, and some- 
times through prayers, spells and incantations. 

Comte says that in time this stage of animism gave place to a 
second stage of metaphysics, in which the spirits and gods of the 
animistic stage become depersonified, and are replaced by vaguely 
conceived forces, activities or essences. In this stage the world is 
depicted as being controlled by * vital forces', 'chemical activities', 
a 'principle of gravity', and the like. These finally amalgamate into 
a single activity which is usually referred to as 'nature', although 
we still occasionally personify it and spell it with a capital N. The 
sequence of events has now passed beyond human control. 

Comte considers that this second or metaphysical stage must 
in due course give place to yet a third stage the positive stage. 
The 'forces' which expelled the spirits and gods now suffer 
expulsion in their turn. Nothing is left in the world but hap- 
penings for which no explanation or interpretation is offered or 
even attempted, and science has now for its single aim the discovery of 
the laws to which these happenings conform the pattern of events 

Thus to primitive man the sun was a life-giving god to the 
Greeks the horse-drawn chariot of a god while a later and less 
Dagan age supposed that angels had been entrusted with the task 
>f pushing along the sun, moon and planets, and of maintaining 
:he motion of the celestial spheres to which the more distant stars 
vere supposed to be affixed. This animistic stage ended when the 
jod, his horses and his chariot, the angels and their celestial 


spheres, were eliminated by the progress of science. To be more 
explicit, it ended when Copernicus, in accordance with the earlier 
teaching of Pythagoras, Aristarchus and others, showed how the 
apparent motion of the sun, moon and stars across the sky resulted 
from a daily rotation of the earth, while the motions of the planets! 
through the stars could be explained by their revolutions round! 
a fixed sun. Even when Kepler discovered the true shapes of these 
planetary orbits sixty years later, he still postulated a 'power' or 
influence to keep the planets moving; he thought they would all 
stop dead if a material emanation from the sun did not con- 
tinually urge them on. The science of planetary movements had 
attained to its second stage. 

Newton retameHlP force * of gravitation, but was fully conscious 
of the pMiosogjhdca^ When Leibniz attacked 

fum for introducing occult qualities and miracles into his philo- 
sophy, he replied that * to understand the motions of the planets 
under the influence of gravity, without knowing" the cause of 
gravity, is as good a progress in philosophy as to understand the 
frame of a clock, and the dependence of the wheels upon one 
another, without knowing the cause of the gravity of the weight 
which moves the machine, is in the philosophy of clockwork*. 
Astronomy was beginning to move into the third stage,. to which it 
has only recently fully attained. ThF^stronomeFoTto^day makes 
no claim to understand why the planets move as they do; he is 
content to know that the pattern of events can be described very 
neatly and concisely by picturing planetary motions as taking place, 
in a curved space. 

Comte believed that every science must inevitably go through 
these three stages in turn this is his famous 'law of the three 
stages'. He further claimed that the abstract sciences could be 
arranged in a hierarchy, in the order 

mathematics, astronomy, physics, chemistry, biology, sociology, 
in which each science is 

(#) historically older, 

(b) logically simpler, 

(c) more widely applicable, 


than any of the sciences which come after it on the list. Certain 
sciences which loom large in present-day knowledge, as for in- 
stance geology and psychology, are absent from the list and do 
not fit at all naturally into the hierarchy. If, however, we merge the 
minor sciences into the greater, the hierarchy assumes the simpler 

mathematics, physics, biology, sociology, 

and now possesses all the virtues claimed for it by its author. 

Comte further claimed that each science in the hierarchy is 
independent of all that follow it, and also must reach the final or 
positive stage before them. Since mathematics must have been in 
the positive stage from its first beginnings, the claim for physics 
is that* it depends only on mathematics, and must be the first 
experimental science to attain to the positive stage. We shall in- 
vestigate these claims in due course, but first let us examine the true 
nature of physical knowledge. 

Physical Knowledge 

We each live our mental life in a prison-house from which there is 
no escape. It is our body; and its only communication with the 
outer world is through our sense-organs eyes, ears, etc. These 
form windows through which we can look out on to the outer 
world and acquire knowledge of it. A man lacking all five senses 
could know nothing of this outer world, because he would have no 
means of contact with it; the whole content of his mind would be 
,an expansion of what had been in it at birth. 

The sense-organs of a normal man receive stimuli rays of light, 
waves of sound, etc. from the outer world, and these produce 
electric changes which are propagated over his nerves to his brain. 
Here they produce further changes, as the result of which after 
a series of processes we do not in the least understand his mind 
acquires perceptions to use Hume's terminology of the outer 
world. These give rise to impressions and ideas in turn, an impression 
denoting a sensation, emotion or feeling at the moment when a per- 
ception first makes its appearance in the mind, and an idea denoting 


what is left of an impression when its first vigour is spent, including 
for instance the memory of an impression or the repetition of it in 
a dream. 

Thus the whole content of a man's mind can consist of three 
parts at most a part that was in his mind at birth, a part that has 
entered through his sense-organs, and a part which has been de- 
veloped out of these two parts by processes of reflection and 
ratiocination. Some have denied that the first part exists at all, 
holding with Hobbes (1588-1679) that 'there is no conception in a 
man's mind which hath not at first been begotten upon the organs 
of sense', or, in the earlier phrase of the Scholastics, nihil est in 
intellectu quod nonfuerit in sensu. Others have thought with Leibniz 
(1646-1716) that this should be amended by the addition of the 
words nisi intellect ipse there is nothing in the understanding 
that has not come through the senses, except the understanding 
itself. We shall discuss these questions more fully as the need arises. 

Whenever a man increases the content of his mind he gains new 
knowledge, and this occurs each time a new relation is established 
between the worlds on the two sides of the sense-organs the 
world of ideas in an individual mind, and the world of objects 
existing outside individual minds which is common to us all. 

The study of science provides us with such new knowledge. 
Physics gives us exact knowledge because it is based on exact 
measurements. A physicist may announce, for instance, that the 
density of gold is 19-32, by which he means that the ratio of the 
weight of any piece of gold to that of a volume of water of equal size 
is 19*32; or that the wave-length of the line Ha in the spectrum of 
atomic hydrogen is 0-000065628 centimetre, by which he means 
that the ratio of the length of a wave of Ha light to that of a centi- 
metre is 0*000065628, a centimetre being defined as a certain 
fraction of the diameter of the earth, or of the length of a specified 
bar of platinum, or as a certain multiple of the wave-length of a line 
in the spectrum of cadmium. 

These statements import real knowledge into our minds, since 
each identifies a specific number, the idea of which is already in 
our minds, with the value of a ratio which has an existence in the 
world outside ; this idea of a ratio is again something with which 


our minds are familiar. Thus the statements tell us something new 
in a language we can understand. 

Each ratio expresses a relation between two things neither of 
which we understand separately, such as gold and water. Our minds 
can never step out of their prison-houses to investigate the real 
nature of the things gold, water, atomic hydrogen, centimetres or 
wave-lengths which inhabit that mysterious world out beyond 
our sense-organs. We are acquainted with such things only through 
the messages we receive from them through the windows of our 
senses, and these tell us nothing as to the essential nature of their 
origins. But our minds can understand and know ratios which 
are pure numbers even of quantities which are themselves in- 
comprehensible. We can, then, acquire real knowledge of the 
external world of physics, but this must always consist of ratios, or, 
in other words, of numbers. 

The raw material of every science must always be an accumula- 
tion of facts ; the values of ratios of which we have just been speaking 
constitute the raw material of physics. But, as Poincar remarked, 
ah accumulation of facts is no more a science than a heap of stones 
is a house. When we set to work to build our house i.e. to create a 
science we must first coordinate and synthesize the accumulated 
piles of facts. It is then usually found that a great number of 
separate facts can be summed up in a much smaller number of 
general laws. This indeed is the most fundamental and also the 
most general fact disclosed by the experimental study of science 
the stones fit together and combine, out of their intrinsic nature, to 
make a house. In brief, nature is rational. The house, being a 
rational structure and not a shapeless pile of stones, will show 
certain marked features. These express the pattern of events for 
which we are searching. 

In physics the separate stones are numbers the ratios just 
described and the features of the house are relations between 
large groups of numbers. Clearly these relations will be most 
easily recorded and explained by embodying them in mathematical 
formulae, so that our scientific house will consist of a collection of 
mathematical formulae; in this way, and this alone, can we express 
the pattern of events. To take a simple illustration, the physicist 
finds that the spectrum of atomic hydrogen contains the line Ha 


which we have already mentioned, and also a very great number of 
other lines which are usually designated as H/J, Hy, HS, etc. The 
wave-lengths of these lines can be measured, and are found to be 
related with one another in a very simple way which can be ex- 
pressed by a quite simple mathematical formula. This is typical of 
the way in which the particular scientific house of physics is built 
up ; a great number of separate facts of observation are all subsumed 
in a single mathematical formula, and our knowledge of the physical 
world is expressed by a number of such formulae. 

Pictorial Representations 

But now the complication intervenes that our minds do not take 
kindly to knowledge expressed in abstract mathematical form. Our 
mental faculties have come to us, through a long line of ancestry, 
from fishes and apes. At each stage the primary concern of our 
ancestors was not to understand the ultimate processes of physics, 
but to survive in the struggle for existence, to kill other animals 
without themselves being killed. They did not do this by pondering 
over mathematical formulae, but by adapting themselves to the 
hard facts of nature and the concrete problems of everyday life. 
Those who could not do this disappeared, while those who could 
survived, and have transmitted to us minds which are more suited 
to deal with concrete facts than with abstract concepts, with par- 
ticulars rather than with universals ; minds which are more at home 
in thinking of material objects, rest and motion, pushes, pulls and 
impacts, than in trying to digest symbols and formulae. The child 
who is beginning to learn algebra never takes kindly to #, y and z ; 
he is only satisfied when he is told that they are numbers of apples 
or pears or something such. 

In the same way, the physicists of a generation ago could not rest 
content with the x, y and z which were used to describe the 
pattern of events, but were for ever trying to interpret them in 
terms of something concrete. If, they thought, there is a pattern, 
there must be a loom for ever weaving it. They wanted to know 
what this loom was, how it worked, and why it worked thus rather 
than otherwise. And they assumed, or at least hoped, that it would 
prove possible to liken its ultimate constituents to such familiar 


mechanical objects as occur in looms, or perhaps to billiard-balls, 
jellies and spinning-tops, the workings of which they thought they 
understood. In time they hoped to devise a model which would 
reproduce all the phenomena of physics, and so make it possible to 
predict them all. 

Such a model would, they thought, in some way correspond to 
the reality underlying the phenomena. No one seems to have con- 
sidered the situation which would arise if two different models 
were found, each being perfect in this respect. 

Yet this situation is of some interest. If it arose, there would be 
no means of choosing between the two models, since each would be 
perfect in the only property by which it could be tested, namely the 
power of predicting phenomena. Neither model could, then, claim 
to represent reality, whence it follows that we must never associate 
any model with reality, since even if it accounted for all the pheno- 
mena, a second model might appear at any moment with exactly 
the same qualifications to represent reality. 

To-day we not only have no perfect model, but we know that it is 
of no use to search for one it could have no intelligible meaning 
for us. For we have found out that nature does not function in a 
way that can be made comprehensible to the human mind through 
models or pictures. 

If we are to explain the workings of an organization or a machine 
in a comprehensible way, we must speak to our listeners in a 
language they understand, and in terftis of ideas with which they 
are familiar otherwise our explanation will mean nothing to them. 
It is no good telling a crowd of savages that the time-differential of 
the electric displacement is the rotation of the magnetic forge 
multiplied by the velocity of light^ In the same way, if an inter- 
pretation ot the workings of nature is to mean anything to us, it 
must be in terms of ideas which are already in our minds other- 
wise it will be incomprehensible to us, and cannot add to our 
knowledge. We have already seen what types of ideas can be in our 
minds ideas which have been in our minds from birth, ideas 
which have entered our minds as perceptions, and ideas which 
have been developed out of these primitive ideas by processes of 
reflection and ratiocination. 


Such ideas as originated in perceptions, and so entered our 
minds through one or more of the five senses, may be classified by 
the sense or senses through which they entered. Thus the content 
of a mind will consist of visual ideas, auditory ideas, tactile ideas, 
and so on, as well as more fundamental ideas such as those of 
number and quantity which may be inborn or may have entered 
through several senses, and more complex ideas resulting from 
combinations and aggregations of simpler ideas, such as ideas of 
aesthetic beauty, moral perfection, maximum happiness, checkmate 
or free trade. It is useless to try to understand the workings of 
nature except in terms of ideas belonging to one or other of these 

For instance, the pitch, intensity and timbre of a musical sound 
are auditory ideas ; we can explain the functioning of an orchestra 
in terms of them, but only to a person who is himself possessed of 
auditory ideas, and not to one who has been deaf all his life. Colour 
and illumination are visual ideas, but we could not explain a land- 
scape or a portrait in such terms to a blind man, because he would 
have no visual ideas. 

Clearly complex ideas of the kind exemplified above can give no 
help towards an understanding of the functioning of inanimate 
nature. The same is true of ideas which have entered through the 
senses of hearing, taste and smell as for instance the memories of 
a symphony or of a good dinner. If for no other reason, none of 
these enter into direct relation with our perceptions of extension in 
space, which is one of the most fundamental of the things to be 
explained. We are left only with fundamental ideas such as number 
and quantity, and ideas which have entered our minds through 
the two senses of sight and touch. Of these sight provides more 
vivid and also more important ideas than touch we learn more 
about the world by looking at it than by touching it. Besides 
number and quantity, our visual ideas include size or extension in 
space, position in space, shape and movement. Tactile ideas com- 
prise all of these, although in a less vivid form, as well as ideas 
which are wholly tactile, such as hardness, pressure, impact and 
force. For an explanation of nature to be intelligible it must 
depend only on such ideas as these. 


Geometrical Explanations, of Nature 

Various attempts have been made to explain the workings of 
nature in terms of visual ideas alone, these depending mainly on 
the ideas of shape (geometrical figures) and motion. Three examples 
drawn from ancient, mediaeval and modern times respectively are : 

(1) The Greek explanation that nature favours circular motion 
because the circle is the perfect figure geometrically, an explanation 
which remained in vogue at least until the fifteenth century (p. 107, 
below), notwithstanding its being contrary to the facts. 

(2) The system of Descartes, which tried to explain nature in 
terms of motion, vortices, etc. (p. 107, below). This also was con- 
trapy to the facts. 

'(3) Einstein's relativity theory of gravitation, which is purely 
geometrical in form. This, so far as is known, is in complete, 
agreement with the facts. 

We shall discuss this last theory in some detail (p. 117, below). 
In brief, it tells us that a moving object or a ray of light moves along 
a geodesic^which means that it takes the shortest route from place 
to place, or again, roughly speaking, that it goes as nearly in a 
straight line as circumstances permit. This geodesic is not in ordi- 
nary space, but in an ideal composite space of four dimensions, 
which results from blending space and time. This space is not only 
four-dimensional but is also curved; it is this curvature that pre- 
sents a geodesic being an ordinary straight line. Efforts have been 
made to explain the whole of electric and magnetic phenomena in a 
similar way, but so far without success. 

It is perhaps doubtful whether such a curved four-dimensional 
space ought to be described as a visual ideajwhich is already in our 
minds. It may be only ordinary space generalized, but if so it is 
generalized out of all recognition. The highly trained mathemati- 
cian can visualize it partially and vaguely, others not at all. Unless 
^e are willing to concede that the plain man has the idea of such a 
jpace in his mind, we must say that no appreciable fraction of the 
Yorld has been really 'explained' in terms of visual ideas. 

Even if it had, such an explanation would hardly carry any con- 
dction of finality or completeness to our modern minds. To the 


Greek mind the supposed fact that the stars or planets moved in 
perfect geometrical figures provided a completely satisfying ex- 
planation of their motion the world was a perfection waiting only 
to be elucidated, and here was a bit of the elucidation. Our minds 
work differently. Optimism has given place to pessimism, at least 
to the extent that we no longer feel any confidence in an overruling 
tendency to perfection, and if we are told that a planet moves in a 
perfect circle, or in a still more perfect gfeodesic, we merely go on to 
inquire: Why? When Giotto drew his perfect circle, his pencil was 
not guided by any abstract compulsion to perfection if it were, 
we should all be able to draw perfect circles but by the skill of his 
muscles. We want to know what provides the corresponding 
guidance to the planets, and this requires that the purely visual 
ideas of geometrical form shall be supplemented by the addition of 
tactile ideas. 

Mechanical Explanations of Nature 

Explanations which introduce tactile ideas forces, pressures 
and tensions are of course dynamical or mechanical in their 
nature. It is not surprising that such explanations also should have 
been attempted from Greek times on, for, after all, our hairy an- 
cestors had to think more about muscular force than about perfect 
circles or geodesies. Plato tells us that Anaxagoras claimed to be 
able to explain the workings of nature as a machine. In more 
recent times Newton, Huyghens and others thoughtfthat the only 
possible explanations of ^nature were mechanical. Thus in 1690 
Huyghens wrote: 'In true philosophy, the causes of all natural 
phenomena are conceived in mechanical terms. We must do this, 
in my opinion, or else give up all hope of ever understanding any 
thing in physics/ 

To-day the average man probably holds very similar opinions. 
An explanation in any other than mechanical terms would seem in- 
comprehensible to him, as it did to Newton and Huyghens, through 
the necessary ideas the language in which the explanation was con- 
veyed not being in his mind. When he wants to move an object, 
he pulls or pushes .it through the activity of his muscles, and cannot 
imagine that Nature does not effect her movements in a similar way. 


Among attempted explanations in mechanical terms, the New- 
tonian system of mechanics stands first. This was supplemented in 
due course by various mechanical representations of the electro- 
magnetic theories of Maxwell and Faraday (p. 120, below). All 
envisaged the world as a collection of particles moving under the 
pushes and pulls of other particles, these pushes and pulls being 
of the same general nature as those we exert with our muscles on 
the objects we touch. 

We shall s^jp later in the present book how these and other 
ittempted mechanical explanations have all failed. Indeed the 
progress of science has disclosed in detail the reasons why all 
ailed, and all must fail. Two of the simpler of these reasons may be 
nentioned here. 

'The first is provided by the theory of relativity. The essence of a 
nechanical explanation is that each particle of a mechanism ex- 
periences a real and definite push or pull. This must be objective as 
regards both quantity and quality, so that its measure will always 
be the same, whatever means of measurement are employed to 
measure it just as a real object must always weigh the same whether 
it is weighed on a spring balance or on a weighing-beam. But the 
theory of relativity shows that if motions are attributed to v T6rces, 
these forces will be differently estimated, as * regards both quantity 
and 'quality, by observers who happen to be moving at different 
speeds, and furthermore that all their estimates have an equal claim 
to be considered right. Thus the supposed forces cannot have a 
real objective existence; they are seen to be mere mental con- 
structs which we make for ourselves in our efforts to understand 
the workings of nature. A simple specific example of this general 
argument will be found below (p. 121). 

A second reason is provided by the theory of quanta. A mechan- 
ical explanation implies not only that the particles of the universe 
move in space and time, but also that their motion is governed by 
agencies which operate in space and time. But the quantum theory 
finds, as we shall see later, that the fundamental activities of nature 
cannot be represented as occurring in space and time ; tfyey cannot, 
then, be mechanical in the ordinary sense of the word. 
In any case, no mechanical explanation could ever be satisfying and 


final ; it could at best only postpone the demand for an explanation. 

For suppose to imagine a simple although not very likely 
sibility that it had been found that the pattern of events could be 
fully explained by assuming that matter consisted of hard spherical 
atoms; and that each of these behaved like a minute billiard-tjall. 
At first this may look like a perfect mechanical explanation, but we 
soon find that it has only introduced us to a vicious circle; it first 
explains billiard-balls in terms of atoms, and then proceeds to 
explain atoms in terms of billiard-balls, so that we have not ad- 
vanced a step towards a true understanding of the ultimate nature 
of either billiard-balls or atoms. All mechanical explanations are 
open to a similar criticism, since all are of the form A is like B y and 
B is like A '. Nothing is gained by saying that the loom of nature 
works like our muscles if we cannot explain how our muscles work. 
We come, then, to the position that nothing but a mechanical 
explanation can be satisfying to our minds, and that such an ex-' 
planation would be valueless if we attained it. We see that we can 
never understand the true nature of reality. 

The Mathematical Description of Nature 

In these and similar ways, the progress of science has itself shown 
that there can be no pictorial representation of the workings of 
nature of a kind which would be intelligible to our limited minds. 
The study of physics has driven us to the positivist conception of 
physics. We can never understand what events are, but must 
limit ourselves to describing the pattern of events in mathematical 
terms; no other aim is possible at least until man becomes en- 
dowed with more senses than he at present possesses. Physicists 
who are trying to understand nature may work in manjT3merenf 
fields and by many different methods ; one may dig, one may sow, 
one may reap. But the final harvest will always be a sheaf of 
mathematical formulae. J ihese will never describe nature itself. Eut 
only ourobservations on nature. Our studies can never put us into 
contact with reality; we can never penetrate beyond the impressions 
that reality implants in our minds. 

Although we can never devise a pictorial representation which 
shall be both true to nature and intelligible to our minds, we may 


still be able to make partial aspects of the truth comprehensible 
through pictorial representations or parables. As the whole truth 
does not admit of intelligible representation, every such pictorial 
representation or parable must fail somewhere. The physicist of 
the last generation was continually making pictorial representations 
and parables, and also making the mistake of treating the half- 
truths of pictorial representations and parables as literal truths. 
He did not see that all the concrete details of his picture his 
luminiferous ether, his electric and magnetic forces, and possibly 
his atoms and electrons as well were mere articles of clothing that 
he had himself draped over the mathematical symbols; they did not 
belong to the world of reality, but to the parables by which he had 
tried to make reality comprehensible. For instance, when observa- 
tion was found to suggest that light was of the nature of waves, it 
became customary to describe it as undulations in a rigid homo- 
geneous ether which filled the whole of space. The only ascertained 
fact in this description is contained in the one word * undulations', 
and even this must be understood in the narrowest mathematical 
sense; all the rest is pictorial detail, introduced to help out the 
limitations of our minds. Kronecker is quoted as saying that in 
arithmetic God made the integers and man made the rest; in the 
same spirit we may perhaps say that in physics God made the 
mathematics and man made the rest. 

To sum up, physics tries to discover the pattern of events which 
controls the phenomena w& observe. But we can never know what 
this pattern means or how it originates ; and even if some superior 
intelligence were to tell us, we should find the explanation un- 
intelligible. Our studies can never put us into contact with 
reality, and its true meaning and nature must be for ever hidden 
from us. 


Such is physics, but it is less easy to say what philosophy is. While 
most philosophers seem to have had their private and differing 
views on the question, few have been willing to venture on a 
definition. Hobbes (1588-1679) defined it as 'a knowledge of 


effects from their causes and of causes from their effects' in other 
words the philosopher differs from the physicist only in that he 
tries to discover the pattern of events in the world at large, and not 
only in inanimate nature. Hegel (1770-1831) took a different view, 
defining philosophy as 'die denkende Betrachtung der Gegenstande', 
the investigation of things by thought and contemplation, again 
suggesting a relation although a different one to science, which 
is the investigation of things by experiment and direct inquiry. 
While the workshop of the scientist is his laboratory, or perhaps 
the open field or the star-lit sky, that of the philosopher is his own 

In whatever ways we define science and philosophy, their terri- 
tories are contiguous; wherever -science leaves off and in many 
places its boundary is ill-defined there philosophy begins. Just 
as there are many departments of science, so there are many de- 
partments of philosophy. Contiguous to the department of physics 
on the scientific side of the boundary lies the department of meta- 
physics on the philosophical side. The boundary here is clearly 
defined, at least if we accept the positivist view of physics explained 
above. For then we must agree with Comte that the task of 
physics is to discover and formulate laws, while that of philosophy 
is to interpret and discuss. But the physicist can warn the 
philosopher in advance that no intelligible interpretation of the 
workings of nature is to be expected. 

In view of this contiguity, it is not surprising that many philo- 
sophers have been physicists also. Indeed from the beginnings of 
recorded history down to the end of the seventeenth century from 
the times of Thales, Epicurus, Heraclitus and Aristotle down to 
those of Descartes and Leibniz the great names in philosophy 
were often great names in science as well. 

It is, however, hardly possible to understand the true relation 
between physics and philosophy until we have glanced at some of 
the many forms which philosophy has assumed in the course of its 
long history. Without attempting anything like a sketch of the 
general history of philosophy (which would lie quite outside the 
scope of the present book), we may perhaps trace certain threads 
which run clearly through this history. 

JP 2 


Ancient Philosophy 

Ancient European philosophy was almost exclusively Greek, and 
to the Greeks philosophy was simply what its name implies the 
love of wisdom. Yet the Greek idea of wisdom was not quite the 
same as our own; their wisdom was based more on speculation, 
conjecture and contemplation, and less on firm knowledge or bed- 
rock facts, which they had but little capacity for acquiring. In 
brief, it was less scientific than ours. Nevertheless it entered into 
some relation with science, for it comprised some real knowledge of 
mathematics, physics and astronomy, as well as a great mass of 
speculation as to cosmology, the fundamental structure of the 
world, and the principles governing the order of events. 

But it was more especially concerned with 'the conduct of life, 
public and private', taking as its main topics for discussion such 
problems as the aim and meaning of life, the ethical principles of 
conduct, the most effective organization of human society, the best 
forms of government, education and so forth; as well as more 
abstract, but not entirely irrelevant, questions such as the meaning 
of justice, truth and beauty. In common language the philosopher 
was the man who could look beyond the narrow groove in which his 
daily work lay, and steer his way through life by availing himself of 
the accumulated wisdom of the race a little knowledge mixed 
copiously with speculative conclusions drawn from this knowledge 
by contemplation, abstract reasoning and discussion. 

Mediaeval Philosophy 

Then came those darker ages in which the bright light of Greek 
culture suffered eclipse, and European philosophy with it. During 
this period Christianity appeared and conquered a large part of the 
earth, introducing a new moral code and reshaping men's views as 
to the meaning and purpose of life. In so doing, it took over a large 
part of what had hitherto been the province of philosophy, since it 
provided dogmatic and professedly infallible answers to problems 
that had so far been topics for philosophic debate ; guides to human 
conduct were no longer to be sought through the, study of philo- 
sophy or the exercise of reason, but in the precepts of religion. 


If philosophy retained any existence during this period, it was 
mainly through the Church trying to graft the dogmas of religion on 
to the older doctrines of Greek philosophy. It was studied almost 
exclusively by ecclesiastics, usually monks, and its language was 
Latin the language of the Church, but not of any living people. 
Greek philosophy had been primarily concerned with problems 
of citizenship, ethics, and the search for the good and the 
beautiful ; mediaeval philosophy with the subtleties and casuistries 
of theological doctrine. Greek philosophy had tried to advance by 
the exercise of reason and by controlled speculation; mediaeval 
philosophy by the barren methods of the syllogism and of logic- 
chopping. Greek philosophy had ever aimed at progress to higher 
things; mediaeval philosophy tried to instil an unquestioning ac- 
ceptance of established authority and resignation to an unchanging 
order; the watchword was no longer excelsior but semper eadem. 

And if science retained any existence through this period, it was 
mainly a useless science which concerned itself with, as we now 
know, wholly unprofitable quests such as the search for the philo- 
sopher's stone and the elixir of life, with alchemy and astrology, 
with magic and the black arts ; its aims were almost wholly utilitarian 
and mostly unwortKy. 

^he Philosophy of the Renaissance 

In the middle years of the fifteenth century, glimmers ot a new 
light were seen; a dawn began to break, and the darkness of these 
dismal ages gradually gave place to a brighter period of intellectual 
and spiritual activity. For the first 150 years or so, the interest was 
preponderatingly humanistic, its inspiration being drawn from 
classical literature. But with the coming of the seventeenth century, 
a new scientific interest also began to emerge, of an intellectual 
rather than of a utilitarian type ; the foundations of modern science 
were being laid. 

It began with astronomy. The world of mediaeval cosmology had 
consisted of a central earth equipped with a hell beneath and a 
heaven above in which God sat for ever on a throne at the point 
vertically above Jerusalem; the sun, moon and the star-bespangled 
sphere of heaven, which angels continually pushed round the earth, 


figured as mere adjuncts designed to secure the greater comfort of 
the earth's inhabitants. The writings of Copernicus, the specula- 
tions of Bruno and the observations of Galileo had shattered this 
old world beyond repair, and a new one was being built by the 
scientific astronomy of Galileo, of Kepler and, later, of Newton. 

Physics soon experienced a similar change. The heathen gods and 
goddesses had long since passed into oblivion, so that nature could 
no longer be interpreted as the congeries of animated personalities 
who contended with one another and occasionally interfered capri- 
ciously in human affairs. Men now began to ask what it was, and 
how it functioned. In time it came to be interpreted as a vast 
macliine a network of cogs, shafts and thrust-bars, each of which 
could only transmit the motion it received from other parts of the 
mechanism and then wait for a new impulse to arrive. 

This brought a beautiful simplicity into inanimate nature, but it 
also threatened to bring a most unwelcome simplicity into human 
life. For out of this view of nature there grew a philosophy of 
materialism, with Hobbes as its principal exponent and advocate. 
[ts central doctrines were that the whole world could be con- 
structed out of matter and motion; matter was the only reality; 
5 vents of every kind were simply the motion of matter; man was 
only arTammal with a material body, his thoughts and emotions 
aliEeTresulting from mechanical motions of the atoms of this body. 

If, then, the world of atoms worked with the inevitability of a 
machine, the whole race of men seemed to be reduced to cogs in the 
machine; they could not initiate but only transmit. Exhorting a 
man to be moral or useful was like exhorting a clock to keep good 
time; even if it had a mind, its hands would not move as its mind 
wished, but as the already fixed arrangement of its weight and 
pendulum directed. We could not choose our paths for ourselves ; 
these were already chosen for usjby the arrangement of the atoms 
in our bodies, and the imagined freedom of our wills was illusory. 

Yet on this imagined freedom man had built his social system and 
tiis ethical code ; it alone gave a meaning to his ideas of right and 
wrong, of purpose and moral responsibility; it formed the corner- 
stone of the religions in which his nobler aspirations and emotions 
ay crystallized ; on it he had built his hopes of heaven and his fears 


of hell. Through the sufferings and trials of this world, he had 
consoled and sustained himself with the vision of the rich reward 
he would reap in a world to come, a reward which was to reimburse 
him a thousand times for the sacrifices and struggles he had so 
willingly made here unless perchance, like Dante, he found his 
consolation in picturing the torments awaiting his enemies. But if 
human conduct was only a matter of the push and pull of atoms, all 
this became meaningless; it was in vain that he had starved his 
appetites, lacerated his body, and renounced all normal human 
pleasures ; he was no more worthy of reward than the man who had 
wholeheartedly grasped at pleasure. 

Never had a train of ideas seemed to touch human interests and 
everyday human life more closely; nothing could be of more tre- 
mendous import to the question of man's significance in the general 
scheme of things, and we might have expected that it would produce 
a turmoil at least comparable with those produced by the scientific 
findings of Copernicus and Darwin. And there were some, it is 
true, who showed great interest in the new doctrine. Bentley, 
Master of Trinity College, Cambridge, wrote that 'the taverns and 
coffee-houses, nay Westminster Hall and the very churches, are 
full of it', and added that from his own observation ninety-nine 
per cent of English infidels were Hobbists. 

Yet the average man, who was no infidel, gave no countenance" 
to the new doctrine partly perhaps because he was not prepareq 
to face its religious implications, but even more, we may conjecture* 
because it made no appeal to his common sense. He was perfectly 
clear in his mind that his will was free, no matter what abstruse 
arguments might be adduced to the contrary was he not conscious 
of choosing freely at almost every moment of his life? Even though 
he might conceivably be mistaken in this, the world around him 
was so obviously a world of purposeful activity men tried and 
they succeeded. The whole intricate fabric of civilized life was a 
standing record of achievement, not by atoms pushed and pulled 
by blind purposeless forces, but by resolute minds working to 
pre-selected ends. 

Not only so, but the new doctrines of science merely restated, in 
rather more exact language, ideas which had long formed part of 


the common stock of philosophy and theology. We have already 
seen how Anaxagoras had explained the world as a machine in 
which every part moved only as diredted by some other part. 
Seneca, again, had maintained that God {has determined all thjpgs 
by an inexorable law of destiny which He nas decreed and Himself 
obeys *1 Some fifteen hundred years later, the Archbishops, Bishops 
and clergy of the Anglican Church, assembled in Convocation in 
London in the year 1562, agreed on very similar ideas which they 
incorporated in their Articles of Religion, and ordered to be printed 
in every Book of Common Prayer. After another eighty years 
Descartes, who certainly tried hard not to say anything that was 
not entirely orthodox, wrote: 'It is certain thaL God has fore- 
ordained all things \ and 'Thepower of the will consists only Jn 
this, that we so act that we are not conscious of being determined to 
a~particiiiar action by any external force." 

In other words, the great machine follows its foreordained 
course, and we small cogs are compelled unwittingly to acquiesce 
in its motion which is just about what science was beginning to 
say on 4;he subject. 


Although the conclusions of science^accorded well enough with 
^cologjcaljdogni_a on the questions of free-will and predestination, 
they entered into no relation at all with the teachings of pastoral 
greadier did n^i^^his fioc^il^^GQd, JiacT fore- 

ordained all things, buTexhorted them tq^^J^accomplish jhings 
of theirjwn ipEriontriVe'Ster virtue and ri^ojusness and in 

brief to attempt precisely those things which their Articles, _of 

... ,. . , .,_, i ,. _,,,i.Ai- ---- .~. ^j*^, ---- ... j _ u ._ _ _ ^_ _ _^Q -- ' -- ~ ~~ - 

Religion pronounced to be impossible.^Jle^id^jiot teU them they 
w^^urmble to^^6ose7^uTrafEer that an eternitjMof bliss or tor- 
ment depende3^nli^^cli^e^n^y made. 

rm^ place himself and his thoughts 

unreservedly in the hands of his spiritual teachers, but others saw 
that there was a case for investigation. It seemed to be a case for 
philosophy to decide and yet, if philosophy was to sit in judgment, 
its verdict might well seem to be a foregone conclusion. It is said 


that a man's philosophy is determined by his personality, or, in 
Fichte's words : ' Tell me of what sort a man is, and I will tell you 
what philosophy he will choose/ and the history of human thought 
supplies many confirmations of the truth of this remark. As Prof. 
W. K. Wright has said: 'No one in the seventeenth century but a 
lonely excommunicated Jew like Spinoza would have snatched at 
the mechanistic side of Descartes and Hobbes and given it a 
spiritual interpretation that could afford peace and serenity to his 
own tortured soul. Only enthusiastic lovers of the strenuous life 
like Leibniz and Fichte could have found ground for unqualified 
optimism in the prospect of an immortal life of unceasing activity. 
No one but a neurotic and selfish lover of success, with a distaste 
for having to work for it, such as Schopenhauer, would have seen in 
such a prospect the justification for a philosophy of unqualified 
pessimism and world renunciation. The philosophy of every great 
thinker is the most important part of his biography.' To which we 
may surely add that the biography of every great thinker is the most 
important part of his philosophy. 

Now most of the great thinkers of this period had rather similar 
biographies. They lived in a highly religious age in which serious 
men had been educated to be, and mostly were, devout Christians. 
Thus most of the philosophers of the period, while ostensibly 
searching objectively and impartially for truth, and following the 
path of reason wherever this might lead, were nevertheless con- 
vinced in their own minds that their journeys could only end in a 
triumphant vindication of Christian doctrines, and a laying of the 
doubts which had been raised by science. Also, whatever their 
personal convictions may have been, religious feeling was so strong, 
and religious authority so dominant, that every writer felt himself 
under pressure to arrive at conclusions which conformed with the 
teaching of the Church ; he arrived at others at his peril, as Gior- 
dano Bruno and Gajfileo had discovered. Further, it was an age in 
which consistency did not rank very high among the virtues. This 
is not necessarily a condemnation; it may be that we rate con- 
sistency too high to-day. Anyone whose mind is not completely 
petrified must find his opinions continually changing under the 
pressure of new experience and further consideration. And if, 


even at the same instant, he sees two possible solutions to a problem, 
no matter how inconsistent these may be with one another, there 
can be no reason why he should not marshal the arguments for 
both; he will do this in a more valuable way than two men each 
of whom can only see one side to the question. However this may 
be, even the foremost thinkers of the age we are now considering 
seem to have felt no embarrassment in propounding entirely in- 
consistent doctrines; there was even a convenient doctrine of the 
twofold truth, which proclaimed a sort of relativity of truth a 
conclusion might be true in philosophy but false in theology, or 
vice versa. 

Considerations such as these must have influenced the courses 
which, consciously or unconsciously, the philosophers set them- 
selves; indeed some openly admitted their ultimate aims. For 
instance, in his Critique of Pure Reason, Kant asserted that 'The 
science of metaphysics has for the proper object of its inquiries 
only three grand ideas, GOD, FREEDOM [of the will] and IMMOR- 
TALITY, and it aims at showing that the second conception, 
conjoined with the first, must lead to the third as a necessary 
conclusion. All other subjects with which it occupies itself are 
merely means for the attainment and realization of these ideas/ 

In the preface to the same book, Kant had explained that he had 
to abolish knowledge to make room for belief. * I cannot even make 
the assumption as the practical interests of morality require of 
God, Freedom and Immortality, if I do not deprive speculative 
reason of its pretensions to transcendental insight/ 

In such terms as these philosophy declared itself the handmaiden 
of theology. 

In brief, philosophy awakened from its long mediaeval slumber 
to find itself confronted, among many others, with a special task. 
Just as the task of mediaeval philosophy had been to remove all 
ground for conflict between philosophy and religion, so that of the 
newly awakened renaissance philosophy was to avoid conflict be- 
tween science and religion. 



The foremost philosopher of this period was Descartes (1596-1650). 
Undeterred by his having written the sentences quoted above 
(p. 22), he wished above all things to maintain the freedom of the 
human will against the scientific considerations which seemed to be 
abolishing it. Apparently the crux of the whole matter, as he saw it, 
was the supposition that the brain consisted of ordinary matter; 
discredit this, and science would become harmless. 

When he had written as a physiologist, he had speculated that 
the brain contained a fluid which he called animal spirits. This was 
neither mind nor matter, but formed a sort of intermediary be- 
tween the two ; mind could act on it to the extent of changing the 
direction, but not the amount, of its motion for Descartes be- 
lieved that the amount of motion of a material system must remain 
constant (p. in)*. This fluid could in turn act on matter. To this 
Leibniz subsequently raised the objection that not only the total 
amount of motion must remain constant, but also the amount in 
each separate direction in space, and that any change in the direc- 
tions of motion of the animal spirits would obviously change the 
amounts of motion in these separate directions. 

When, however, Descartes wrote as a philosopher apd Christian 
apologist, he maintained that mind was of a completely different 
nature from matter, and could have no contact with it. The two had 
entirely different functions to perform mind to think and matter 
to occupy space and they were so completely divorced that 
neither could affect the other to the slightest degree. In this way 
thejwill was set free, but only at thfi cost of creating a new problem 
which was to dominate philosophy for generations if my will has 
ncT contact^of anysort with the matter of my body, how can it 
compel this body to turn to the right or to the left as it pleases? 

* By motion Descartes meant what we now call the momentum, mv. He 
believed that ^mv retained a constant value, where S indicates summation over 
all the moving bodies. Leibniz introduced the concept of energy at a later date, 
describing it as force (vis viva, equal to mv 2 ), and found that Era*; 2 retained a 
constant value. He also discovered the constancy of the momenta lmv x , etc. in 
the separate directions in space. Descartes wanted his animal spirits to change 
the direction of motion while keeping Smv constant. Leibniz's objection was 
that this would change ^mvy, ; energy did not come into the question at all. 


Descartes left this problem unsolved, but we find certain of his 
followers Malebranche, Geulincx, Mersenne and others, now 
known as the Occasiohalists solving it to 

supposingjhat the volitions of our minds are only the * occasional ' 
causes of the movements of our bodies, the real, ultimate, or 
'efficient' cause beingT God. Mind and matter never interact 
directly, but rather run on parallel never-intersecting tracks. The 
good God has so arranged things that the activities of mind and 
matter correspond exactly to one another, and keep in such perfect 
step that each seems to influence the other without actually doing 
so. In the same way they might have said, had they known of 
such things the makers of a cinematograph film arrange that the 
voices and action shall correspond and synchronize through the 
whole length of the film ; we see a soldier move smartly at the word 
of command, and his movement seems to be a direct consequence 
of the command, but actually it is the result of a pre-arranged 



Leibniz (1646-1716) went further in the same direction, describing 
Descartes' doctrine of the distinctness of mind and matter as 'the 
ante-room of truth, but only the ante-room'. 

Giordano Bruno had already supposed the world to consist of a 
number of ultimate indivisible units which he called ' monads'; 
these were at the same time spiritualand. material m their nature. 
Every human being and every living thingfwas such a monad. The 
monads were all distinct and different, and could not be resolved 
into anything simpler. 

Leibniz also supposed the world to consist of a great number of 
simple units, which he too described as monads whether he 
borrowed the name from Bruno is not known. These monads, he 
says, are the true atoms of the universe, the ultimate constituents of 
everything, and they possess neither shape nor size nor divisibility. 
Now, as Plato had argued in the Phaedo, dissolution and decay 
appertain only to complex, and above all to divisible, structures. 
Thus their very simplicity shields the monads from dissolution and 
decay, so that they are necessarily eternal and immortal. Each 
man's soul is a single monad, and his body a collection of monads 


of various kinds. All substances are of the nature of force , and 
consist of individuaTcentres of force, which must^thus be monads, 
and 'in imitation of the notion which we have of souls' miisFcon- 
tam something of the nature of feeling and appetite. These monads, 
then, are more or less spiritual in their nature. The lowest monads 
of all, Leibniz writes, resemble animals in a swoon, higher monads 
have clearer perceptions and are endowed with memory, while 
God is the highest monad of all. Since all monads are spiritual in 
their nature, matter can have no real existence, and must come 
from seeing monads in a confused way. 

The monads have no windows on to the outer world through 
which anything could come in or go out, so that each lives its 
utterly secluded life, uninfluenced by its fellow-monads. Its 
changes are determined only by its own internal state; it can 
come into existence only through ^ggtive^actofGod t and can go 
outof existence only through annihilation Iby God* Yet God, Ihe 
supreme monad, keeps all the other monads in step on a series of 


Leibniz calls this the System of pre-established Harmony. * Under 
this system', he wrote, 'bodies act as though there were no souls, 
and souls act as though there were no bodies, and both act as 
though each influenced the other.' 

Leibniz explained this further by comparing the soul and body 
(or mind and matter as we should now say) to two clocks which 
always show the same time, a comparison which the Occasionalists 
had used before him. There are, he says, three ways in which two 
clocks can be made always to show the same time. One and here 
he refers to the experiments of Huyghens is by putting them in 
close physical contact, so that each clock transmits its vibrations to 
the other, and the two clocks advance in unison ; this is the solution 
of ordinary philosophy, but must, Leibniz thinks, be rejected be- 
cause we cannot imagine anything being transmitted between mind 
and matter. The second way is to have a clockmaker continually 
putting the clocks in agreement; this also Leibniz rejects because it 
requires the incessant intervention of a deus ex machina 'for a 
natural and ordinary thing'. The third and only other way, says 
* By ' force * Leibniz here means energy or vis viva. 


Leibniz, is to construct the two clocks so perfectly at the outset 
that they will agree through all time. 

This last is the way of the system of pre-established Harmony. 
In the beginning God created mind and matter in such a way that 
each can follow its own laws, and yet the two move in the same 
perfect agreement as would prevail * if God were for ever putting in 
his hand to set them right'. 

To use Leibniz' own illustration, we, with our puny abilities, 
can make an alarm clock and set it to sound an alarm at any hour 
we require. Obviously, then, so great a craftsman as God could 
make Caesar's body and pre-arrange its atoms so that it should go 
to the Senate House at such or such an hour on the Ides of March, 
should utter such and such words, and so on. The same great 
Craftsman could also create the soul of Caesar in such a way that 
it should experience certain emotions in a pre-arranged order and 
at pre-arranged moments of time, and could, if He so wished, plan 
that these should exactly correspond to, and synchronize with, 
Caesar's bodily movements. According to Leibniz, He had so 

The wheel had now come round full circle. In his eagerness to 
establish the freedom of the will, Descartes had divided the uni- 
verse into two ingredients, mind and matter, which could not 
interact ; this raised the problem of how mind and matter could keep 
in step without interacting. Leibmz T trying to explain this, had to 
suppose that neither had more freedom than a machine which, 
having once been set in motion, was compelled to execute a pre- 
destined series of mechanical movements. In this way, every mind 
became an automaton, which is precisely the conclusion that Des- 
cartes had been drying to escape, and one that Leibniz would 
presumably have liked to avoid if he could. 


So the question stood when Kant brought his mind to it. He saw 
that the circle of arguments of Descartes and Leibniz could lead 
nowhere except to the very conclusion that both were, like himself, 
eager to avoid. He was as much concerned as his predecessors to 
establish the freedom of the will, but he had a clearer conception of 


the difficulties in the way. 'As the complete and unbroken con- 
nection of phenomena is an unalterable law of nature/ he wrote^ 
' freedom isimpossible on the supposition that phenomgaa~are 
abgoluteTy reaL Hence those philosophers who adhere to the 
common opinion on this subject can never succeed in reconciling 
thejdeas of nature and freedom/ 

By 'the common opinion 1 Kant meant what would now be 
described as Naive Realism, or Common-sense Realism. This re- 
jects all metaphysical subtleties, and maintains that the phenomena 
we observecorrespond fairly closely to the realities of the world 
outside us; whenwe think weTee a brick at some point of space, 
there really is something 'there', which is_much_Hke^ wKaF~we 
imagine a brickto be. Thus the world is just about whatjtjeems to 
be, consisting simply of the ^particles and _objegteJgiiich arejfound, 
by observation and experiment, to obey a causal law. If, says Kant, 
this is all there is to the world, then obviously the will cannot be 

On the other hand, many philosophers have found it difficult to 
accept the hypothesis that an object is just about what it appears to 
be, and so is like the mental picture it produces in our minds. For 
an object and a mental picture are of entirely different natures a 
brick and the mental picture of a brick can at best no more resemble 
one another than an orchestra and a symphony. In any case, there 
is no compelling reason why phenomena the mental visions that a 
mind constructs out of electric currents in a brain should re- 
semble the objects that produced these currents in the first instance. 
If I touch a live wire, I may see stars, but the stars I see will not in 
the least resemble the dynamo which produced the current in the 
wire I touched. In this instance, the current produces a vision in 
my mind which differs utterly from the object which created the 
current. May it not be the same with all the phenomena of nature ? 

When we perceive an object, v^e pergejvp x\ fnnst a few of its 
qualities. Having perceived these few qualities, vye frequently 
jump to the conclusion that the object belongs to some familiar 
class of object possessing these qualities. We see a kittenish patch 
of colour behaving in a kittenish way, and conclude that we are 
seeing a kitten. But our identification may be wrong; the little 


creature may be a skunk. Again, when a tiny meteor smaller than a 
pea is falling through the air, it will send the same electric currents 
to our brains as will a giant star millions of times larger than the sun 
and millions of times more distant. Primitive man jumped to the 
conclusion that the tiny meteor was really a star, and we still 
describe it as a shooting-star. This and innumerable other in- 
stances show that two objects may differ widely in their intrinsic 
natures and yet produce similar, and even identical, phenomena. 
And as the two objects of such a pair cannot both be like their 
mental images, there is no longer any sufficient reason for thinking 
that either of them must be. 

Thus we can no longer hold that objects in general are pretty 
*nuch like their mental images. The images need not resemble the 
objects in which they originate, and our perception of the outer 
world may consist only of representations which are constructed by 
our minds out of the activities flowing into our brains, and bear 
little or no resemblance to the realities outside. They may be like 
the code signals which the signalman sends over the wires to say 
what kind of train is coming next; these bear no resemblance to the 
train. Or, as Boltzmann suggests, they may be merely symbols 
(which are related to the objects as letters are to sounds, or as notes 
are to musical tones. 

Kant r holding that phenomena are 'only representations, argues 

that they must ctngiost^^ 

that even tJiQughthephgnomena may be bound to otEgrjghenomena 

need^notjbe If we limit our attention to 
the phenomena, our observations suggest that causality governs 
everything, but if we could make contact with the reality under- 
lying the phenomena we might see that this is not so. 

A few pages later, he explains that his remarks were not intended 
to prove the actual existence of freedom, or even to demonstrate the 
possibilities of freedom; 'that nature and freedom are at least not 
opposed this was the only thing in our power to prove, and the 
question which it was our task to solve*. 

Still, it seems difficult to accept this as providing even a possible 
solution to the problem of human free-will,. The average man is 
not interested in the origins underlying phenomena; the freedom of 


which he wants to assure himself, and instinctively believes himself 
4x> possess, is a freedom to control, or at least to influence, the pheno- 
mena, or, according to Kant, the representations. Imagine two men 
who are similar down to the last atoms of their bodies, placed in en- 
vironments which again are similar down to the last atoms. If free- 
will is to""be cAplaincd hrthe Way Kant suggestsTwe caiTimagine 
one" exercising his freedom and deciding on a saintly life, while 
the other may decide at the same moment that he is more hedo- 
nistically inclined. Uj^to the moment of making these choices, the 
phenomena have been the same for both, so that if causality pre- 
vails in the world of phenomena, as Kant supposes, the subsequent 
phenomena must also be the same" for both; the twojnen must 
mutter tficTsame prayers and drink similar drinks with similar 
results. So far as the phenomena go, their two lives will be identical, 
andlo their acquaintances will be indistinguishable. It follows that 
tHe men can have no moral responsibility for their actions- onlyjat 
most tor their intentions and desires. Clearly this is not what the 
plain man means by freedom ^f the will, and neither is it what 
Kant wanted to establish. But the question is no longer of more 
than academic interest, since, as we shall soon see, science now 
finds that even the phenomena are not governed by causal laws. 
' On other questions besides that of human free-will at which we 
have just glanced it was obvious that tEe methods of ^ienne cpyld 
lead only to the conclusions of science; if philosophy was to reach 
otHeFconcIusion^she must employ~other methods, furthermore, 
if sK<f~wished her conclusions to take precedence over those of 
science, she must be able to claim that her methods were in some 
way more trustworthy than the methods of science. This led to a 
critical examination of the methods by which scientific knowledge 
was obtained, and to an intensive study of certain problems of what 
is now called epistemology the science of knowledge. This will 
form the subject of our next chapter. 





We have already noticed how knowledge is gained by establishing 
relations between an inner process of understanding in our private 
minds ^nd the facts of that publuTouter world which is common 
to us all. As Plato pointed out, the use bFa common language 
is based on the supposition that such relations can be established 
by all of us. 

In the period we have been considering, science claimed only one 
source "of knowledge of the facts and , objects ot the outer world, 
namely the impressions they make on the mind througlithe 
medium of the senses. Yet the untrustworthiness of the senses had 
been one of the commonplaces of philosophy from Greek times on, 
and if the same facts and objects of the outer world made different 
impressions on different minds, wherejdM science stand? If we 
trusted to individual sense-impressions, we could never get beyond 
the position described by Protagoras (c. 481-411 B.C.): 'What seems 
to me is so to me, what seems to you is so to you ' ; each individual 
would become his own final arbiter of truth, and there could be no 
body of objective knowledge. Six centuries before Christ, in the 
earliest days of Greek philosophy, Thales of Miletus had urged 
the importance of gaining a substratum of facts, independent of 
the judgment of individuals, on which a body of objective knowledge 
could be built. 

These difficulties are non-existent to the modern physicist, who 
can trust his instruments to give absolutely objective and unbiased 
information, but they loomed large when there were no instruments 
beyond the unaided human sepses. To avoid them, Plato argued in 
the Theaetetus (c. 368 B.C.), we must distinguish between what the 
mind perceives through the senses and what it apprehends of itself 


bythinking. Such concepts as number and quantity, sameness and 
difference, likeness and unlikeness, good and bad, right and wrong 
do not^nter our minds through our senses, but reside permanently 
in ourlnmds. And as concepts such as these provide the formal 
element in all true knowledge, it follows that this does not come 
from our sensations, but rather from the judgments our minds pass 
on our sensations. 

Plato elaborated this into an argument that the human mind is 
equipped from birth with a set of forms or ideas which exist in it 
independently of the objects of the outer world. These latter serve 
as a sort of raw material for the impress of the forms, so that each 
object becomes a sort of meeting-place for a number of forms. A 
red square brick, for instance, is a lump of this raw material 
stamped with the impresses of the forms of redness, squareness and 
brickiness. When we declare that a particular object is a red square 
brick, we mean that in our judgment this particular piece of matter 
fits into these three forms. We may of course be mistaken; seen in a 
different light, the object may appear of some colour other than 
red, measured against a set-square it may prove to be far from 
square, and hit with a trowel it may prove not to be a brick at all. 

On such grounds as these, Plato maintained that we have sure 
and certain knowledge only of the forms and their relations; our 
knowledge of the objects of the outer world consists at best of 
fleeting impressions and shifting opinions. In the matter of reality 
and certainty, the ideas which reside permanently in our minds, 
namely the forms, may claim precedence over ideas put there 
temporarily by objects we perceive with our senses: it is in this 
world of permanent ideas which exists outside space and time, the 
world sub specie aeternitatis, that truth alone can dwell. 

This train of thought retained existence of a sort through the dark 
ages of philosophy; it figured prominently, although in a modified 
form, in the philosophies of St Thomas and the scholastics, and 
finally reappeared, still further modified, in the philosophy of 

The ideas of Plato, the forms, had been ideas of qualities or 
properties; he supposed that these were inborn in our minds, as 
though for instance they were memories carried over from a 


previous existence. The ideas of Descartes, on the other hand, were 
ideas of facts, or propositions as we should now call them. He 
thought they were innate in a sense rather different from that of 
Plato; the mind was not born with these ideas inside it, but with a 
predisposition to acquire them as soon as it came into contact with 
the world. I called them innate in the same sense in which we say 
that generosity is innate in certain families and certain diseases 
such as gout or gravel in others not that the infants of those 
families labour under those diseases in the womb of the mother, 
but because they are born with a certain disposition or faculty of 
contracting them.' 

Leibniz subsequently challenged this, arguing that all ideas are 
innate in this sense, but that they only mature into actual thought 
when they have been developed by the growth of knowledge. The 
mind at birth is not a clean sheet of paper, but rather an unworked 
block of marble, in which there is already a latent structure of 
veins; this will to some extent determine the form the marble will 
assume when the sculptor chisels it into shape. 

Others differed still more widely from Descartes, and in the 
period we now have under consideration we find the philosophers 
divided, broadly speaking, into two camps the rationalists, who 
maintained that the highest truth resides in our own minds and so 
is to be discovered by reason, and the empiricists, who thought that 
truth resides outside our minds and so is only to be discovered by 
observation and experiment on the world outside. 

The Rationalists 

The rationalists, headed by Descartes, argued that all knowledge 
obtained through direct observation of nature was suspect because 
it comes through the senses, and such knowledge can notoriously 
be deceptive, as all kinds of hallucinations and dreams show. 
Descartes added that even knowledge obtained by mathematical 
proof may be deceptive first, because mathematicians have often 
been wrong, and second, because we can never be certain that an 
omnipotent God^may not have decreed that we should be deceived 
even in the things we think we know best. In this way the ration- 
alists discredited, even if they did not dispose of, practically the- 


whole of scientific knowledge it came from tainted sources. They 
proposed replacing it by the store of knowledge which, as they 
believed, was to be derived from pure contemplation. 

Descartes claimed that his innate ideas, representing knowledge 
which came from 'the clear vision of the intellect', must necessarily 
be true. The fact that he could clearly and distinctly conceive 
something in his mind as, for instance, the existence of God was 
for him a sufficient proof of its truth. Others claimed that, inborn 
in the human mind, there are a number of ready-made principles or 
faculties by the recognition and skilful use of which it must be 
possible to discover truths about the universe, just as Euclid was 
able to discover geometrical truths from a few axioms, the truth 
of which was obvious. Kant went so far as to claim that it ought 
to be possible in this way to construct a 'pure science of nature', 
which should be independent of all experience of the world, and 
therefore uncontaminated by the errors and illusions of observa- 
tion. A very similar claim has again been put forward in recent 
years by Eddington (p. 72, below). 

Kant attempted a reasoned discussion of this question in his 
famous Critique of Pure Reason. He reminds us of Plato when he 
says that a phenomenon, or object of perception, contains both 
substance and form ; the substance produces the effect in the mind 
of the percipient, while the form enables us to allocate the pheno- 
menon to a wider class. The substance of a phenomenon comes to 
us as the result of an experience of the world, or, in Kant's termino- 
logy, a posteriori^ but the form, which is already in our minds 
lying in wait for the substance, comes to us a priori j&^ previously 
to, and independently of, all actual experience of the y/ nr ^ 

Relations between a priori concepts which are such that they 
can be known without any appeal to experience will constitute a 
body of knowledge altogether independent of experi^nre, ..and 
even of all sensuous impressions ' . (Such knowledge Kant described" 
as a priori knowledge, in contradistinction toempirical or a postenozL 
iSTPrtfledge, which has its' sources in experience. A priori know- 
le^geTTEen, came direct from heaven through the gates of _ horn, 
ancTso was in every way superior to knowledge discovered through 
expeninehtr bv observation, or even (according to Descartes^ b v 



mathematical demonstration, all of which came only through the 
fgatesTof ivory. A priori knowledge was necessarily applicable to 
every possioie experience, whereas empirical knowledge, which 
was known only as the result of limited experience or observation, 
could make nqsuch claims. 

Also a priori knowledge was applicable to every possible uni- 
verse, and not only to this one for we can distinguish this universe 
from other possible universes only by observation, and once we do 
this our knowledge ceases to be a priori. Thus in claiming a priori 
knowledge, we claim to know enough of the ultimate nature of 
things to be able to say what kinds of universe a Creator could have 
created, and what kinds He could not have created. Kant's claim 
that a 'pure science of nature' is possible in principle involves just 
this claim. Like every other claim to a priori knowledge, it not only 
denies the omnipotence of God, but also claims to have detailed 
knowledge of His limitations. It is a high claim to make for the 

human intellect. 

The Empiricists 

In opposition to this, the empiricists held that in general knowledge 
comes from experience alone, so that the only way to discover the 
facts about the universe is to go out into the world and search for 
them. Most empiricists were nevertheless willing to concedfiJJiat 
certain tmflis^could b(Tknown by intuition or through demongtr^ 
tipns based on intuitions. 

Locke and Hume, the two most prominent of the empiricists, 
were in agreement that the truths of pure mathematics could be 
known in this way, as also are most modern philosophers, as for 
instance Whitehead and Russell. But J. S. Mill held the opposite 
view, maintaining that the laws of arithmetic embodied generaliza- 
tions derived from observations of actual objects, while geometry 
dealt merely with idealizations of objects of experience we could 
not imagine a mathematical point, line, or triangle unless we had 
first made the acquaintance of their imperfect representations in 
the outer world. Locke thought that not onlyjibe truths of pure 
mathematics but also tfie existences of God and ourselvesj&id^ the 
truths of morality ought to be admitted to the class 
Irnfrtia. ~~ " ~ 


The whole question is obviously largely one of words. As re- 
gards the truths of morality, for instance, the question at issue is 
whether God could have made a world in which a different morality 
would have been 'true'. And surely the answer depends at least as 
much on what we mean by morality and truth as on what we know 
about morality and truth. 

In general, however, the empiricists held firmly to the principle 
that knowledge about the outer world must come from the outer 
world, and so can be acquired only by observation and experiment.! 
As this is precisely the method of science, it might have been 
expected that those philosophers who were also scientists, or were 
of a scientific turn of mind, would be found in the camp of the 
empiricists, while those of a mystical or religious turn of mind > 
would be found among the rationalists. 

"~Ttctually almost the exact opposite was the case. I suppose the 
four most distinguished advocates of rationalism were (in chrono- 
logical order) Descartes (1596-1650), Spinoza (1632-1677), Leibniz 
(1646-1716) and Kant (1724-1804). Two of these four names are 
among the very greatest in mathematics. Descartes was not only 
i-hftjgtfrp.r nf mQfa philosophy but also of modern mathematics*, 
being, amongst other things, thejnventor of analytical geometry, 
while Leibniz shares with Newton the honour of having created 
the differential calculus, and incidentally anticipated Einstein by 
maintaining that sgace and time consist only ot relations^m 
opposition to the Newtonian view that they are absolute. 

jgnt anTmake no claims comparable with these, and yet we 
should remember that astronomy and physics had interested him 
more than philosophy in his earlier years ; according to Helmholtz, 
he only turned from science to philosophy, at the age of thirty-one, 
because there were no facilities for scientific research in his Univer- 
sity of Konigsberg. And he gave scientific lectures regularly to the 
end of his academic career, and wrote on a variety of scientific 
subjects, such as earthquakes, lunar mountains and the possibility 
of changes in the revolution of the earth. Most^ of his jscientific 
work has long been forgotten, but he was the first to suggest the 
true nature of tKe external galaxies clusters of myriads of stars 
ahcThelias tKe ftotincotisideraBIe distinction of having propounded 


one of the first theories of the evolution of the solar system. Besides 
fivEoHucmg these evolutionary ideas Into astronomy^ he' was one 
of the earliest of biological evolutionists. In his Anthropology he 
declares in favour of all animals being descended from a common 
ancestor, althojigh he doesnot mclude humanity in this statement 
possTBTyEecause o7 its dangerous religious implications. Still," he 
suggests that man must have changed fundamentally in the course 
of time, ad^ding^h^^ revolution orang- 

outangs might acquire not only liuman form, but also the organs 
^T speech ^andlhFu^ He once wrote that he was 

'thinking many things, with "the clearest conviction and to his 
great satisfaction, which he would never have the courage to say'. 
Prof. Paneth has suggested that one of these things may well 
have been that what could happen to orang-outangs and chim- 
panzees in the future might also have already happened in the past. 
Engraved on his tomb at Konigsberg are words from the end of his 
Critique of Practical Reason 'Two things fill the mind with ever 
new and increasing admiration and awe, the oftener and more 
steadily we reflect on them; the starry heavens above and the moral 
law within.' The order is significant. 

Spinoza can advance no claims to scientific distinction, although 
his thought is obviously often guided by mathematical and 
scientific knowledge. 

Against this, none of the more prominent of the empiricists 
Francis Bacon (1561-1626), Locke (1632-1704), Berkeley (1685- 
1753) and Hume (1711-1776) had any special scientific attain- 
ments; Berkeley wrote an 'Essay towards a new theory of Vision ', 
but its scientific value is not great. 

The reason for this rather strange division of forces may have 
been in part that those who understood science best were also most 
acutely conscious of its anti-religious implications. But the true 
line of demarcation between the two schools of thought was geo- 
graphical. The Continentals, with their love of abstract ideas, can 
claim all the rationalists, while the British, with their love of 
practical investigation, claim the empiricists, the four just men- 
tioned being English, English, Irish and Scottish respectively. 


A Priori Knowledge 

The debate as to whether genuine a priori knowledge exists need 
hardly concern us; the question which matters for our present 
discussion is not whether such knowledge exists, but the much 
simpler question of whether, if it exists, it is important. To this it 
seems possible to give a negative answer without appealing to any- 
thing more recondite than the well-known principle that the proof 
of the pudding is in the eating. Of course we must ourselves be 
judge and jury, since it is an obvious impossibility for a man who 
does not claim infallibility to convince one who does that he is 
wrong. But even if I cannot persuade my cook that her puddings 
are bad, I can still dismiss her from my service. 

The main reason which seems to call for an adverse judgment on 
alleged a priori knowledge is that it has often been proved false by 
the subsequent advances of science. 

As examples of the kind of knowledge of which the truth was 
claimed to be obvious a priori may be taken : 

'The same thing cannot at once be and not be.' 

'Nothing cannot be the efficient cause of anything.' 

'The liberty of our will is self-evident/ 

* ^X^EYl^ 1 ' 11 ^^ happens is predetermined by causes according 
to fixed law.' """"" - 

"Descartes gives the first three of these, describing the last of 
them as 'a truth which must be reckoned as among the first and 
most common notions which are born with us'. With any reasonable 
use of language, it is obviously in contradiction with the fourth, 
which is taken from Kant, so that a priori knowledge begins to 
discredit itself by its contradictions even before the evidence of 
science has been called. 

Nothing would be gained by trying to analyse these statements in 
detail, but one general remark at once suggests itself. It is surely 
improbable, on principle, that these or any similar statements can 
express absolute truths when stated without qualification in the 
crude bald forms permitted by common language. Such words as 
thing) cause y liberty and predetermined mean nothing definite until 
they have been defined. If we are free to supply our own defini- 


tions, we shall probably be able to find a sense in which all the 
propositions will be true, and a sense in which all will be untrue; 
or we may be able to find a group of cases in which they are true 
and a group in which they are untrue. Thus they do not present 
universal truths so much as topics for debate, the question at issue 
being the limits or conditions within which each is true. Stated in 
the uncompromising terms permitted by common language, the 
propositions merely prejudge questions on which philosophy has 
broken its teeth through the ages. 

, Other pieces of alleged a priori knowledge were of a more 
{scientific kind, and these are of more interest to our present dis- 
jcussion. We may take two examples from Descartes: 

(a) the sum of the three angles of a triangle is 180, 

(b) divisibility is comprised in the nature of substance, or of an 
extended thing, 

and three from Kant: 

(c) space has three dimensions, 

^(d) between two points there can be only one straight line, 
^(e) in all changes of phenomena, substance is permanent, and 
the quantity thereof in nature can be neither increased nor dimi- 

Kant describes (c) and (d) as principles * which are generated in 
the mind entirely a priori* y and (e) as a piece of knowledge which 
' deserves to stand at the head of the pure and entirely a priori laws 
of nature'. 

As soon as we try to discuss these propositions in the light of 
modern science, we again feel the need of precise definitions of the 
terms used. Thus (a) and (d), which are geometrical in their nature, 
are true in the kind of space which is defined by the so-called 
* axioms ' of Euclid Euclidean space, as it is usually called but 
not in the curved space in which the planets are now usually 
pictured as moving. Did then Descartes and Kant intend their 
propositions to refer to Euclidean space, or to this possibly more 
real curved space? The answer is almost certainly that they were 
thinking of Euclidean space. In Descartes' time no other kind of 
space was contemplated. In Kant's time, other kinds may have been 


contemplated, but Kant held that Euclidean geometry was 'true' 
in a sense in which other geometries were not, although admitting 
that he could not prove this because the axioms of Euclid could 
be denied without any inconsistency or contradiction. Thus we can 
see now, although Descartes and Kant could not, that their sup- 
posed a priori knowledge cannot claim to be applicable to any 
objective space of the outer world, but only to private worlds of 
their own. In so far as they thought that their a priori know- 
ledge applied to the real world, they were more wrong than right. 

Kant's proposition (c) that space has three dimensions is in a 
different class ; it is hard to see how it can claim to be a priori 
knowledge. For every mathematician knows that it is just as easy, 
as an abstract exercise, to imagine a space of one, two or four 
dimensions as one of three. If, then, a new-born baby knows that 
the space of the outer world has three dimensions, this must be 
because he has already been peeping at the outer world, or has 
otherwise made its acquaintance ; his knowledge is empirical and 
not a priori. 

It is much the same with the two remaining propositions, which 
are of a more physical nature. In (b) Descartes tells us that divisi- 
bility is a property of substance or of an extended thing, but fails to 
tell us what he means by substance or thing. Actually of course 
divisibility is a property of an elephant or a sandstorm, but not of a 
photon or an electron; but Descartes does not give any definition of 
a thing which will include elephants but exclude electrons. In (e) 
Kant tells us that substance is permanent, but fails to define sub- 
stance. He does, however, say that his statement is tautological, 
which seems to imply that he would define substance as that which 
is permanent, in which case the statement tells us something about 
Kant's use of words, but still nothing about the objective world. 
Since Kant's time physicists have found that substantial electrons 
and other material particles may dissolve into, and also be created . 
out of, insubstantial radiation. Even if these phenomena had not 
been observed, we now know that there is, in principle, no per- 
manence in substance; it is mere bottled energy, and possesses no 
more inherent permanence than bottled beer, although it is of 
course true that under the physical conditions prevailing on our 


particular planet, rpatter may be regarded as very approximately 


The Three Worlds of Modern Science 

It iji natural transition from this to a reflection of a very general 
kind, which provesTto bcToF tfie utmosf Importance ^fof pur dis- 
cussion of the bearings of science on^jpKilosopE^ The human race 
first became acquainted with the properties of matter in the special 
forms they assume under the physical conditions prevailing on our 
planet. In the same way, the laws of nature first became known to 
our race in the restricted form of laws applicable to the behaviour 
of objects comparable in size with human bodies, the reason of 
course being that only objects of these sizes could be studied 
without elaborate instrumental aid. In such studies time was 
usually measured in seconds or minutes and length in inches or 
yards, while nothing ever moved much faster than a galloping horse. 
But with the instrumental aid now at its disposal, science can 
study phenomena in which times are measured, maybe in fractions 
of a millionth of ji millionth of a second, maybe in thousands oi 
f years ; the lengthTlnvoIveH' may Be 'small fractions of 

a milfionfE of an~ mcE7 15FTKey^ 
of milHom^oilmiles, while Hie objects concerned may move at a 
millionth part of a snail's pace or at a million times the speed of an 

" Surveying these immense ranges as a whole, we find that ordi- 
nary human activities occupy a fairly central position in the scheme 
of the universe ; the world of man lies just 

the world of the electron andjthejvprld of the 
occupies only an excessively mipite fraction of the whole range 
between electrons and(^Bu|S^rThe smallest piece of matter we 
can feel, see or handle without instrumental aid still contains 
millions of millions of millions of atoms and electrons, while even 
the smallest of the planets stands in about the same relation to the 
largest piece of matter we can move with our unaided bodies. 

Elaborate studies made with instrumental aid have shown that 
the phenomena of the world of the electron do not in any way form 
a replica on a minute scale of the phenomena of the man-sized 


world, and neither are these latter a replica on a minute scale of the 
phenomena of the world of the nebulae. As we leave the man- 
sized world behind us, and proceed either towards the infinitely 
great in one direction or towards the infinitely small in the other, 
the laws of nature seem at first sight to change, not only in detail 
but in their whole essence. 

Mor^cajeful_scrutjny discloses that the apparent change is 
illusory; actually the same laws prevail thf^^ourtTiTrailg^^W 
diSefenTlealDares of these laws become of pl^onHeraHng impor- 
tance in different parts of the range. A soap-bibbre obeys precisely 
ttie same law of gravitation as a cannon-ball, and also^ 

same law of air-resistance. Clearly, then, we can combine these 
two laws into a single law which must govern the motion of soap- 
bubble and cannon-ball equally. But if we let the two objects fall 
together from the leaning tower of Pisa, their motion will seem to 
be governed by entirely different laws. The reason is that gravita- 
tion is all important for the cannon-ball, while air-resistance is all 
important for the soap-bubble. 

In the same way, all objects are governed by the universal laws o] 
physics, but one aspect of the*Jaws is all important for the 
electron, another for man-sized xfojficls, and yet a third for the 

^fft-J " ' "**** J-.1M..H i mini --------- *-*^^*.>-***lf~* f ~~- ' J -,*,, - ~*- - 

Inovements of the . n^hulae. These three departments of the uni- 
versaTscheme of law are so different that we are justified in thinking 
of them as constituting three distinct and separate sets of laws wit! 
a different pattern of events in each. 

This is a fact of tremendous importance to philosophy as a whole 
Its immediate importance to our present disctission Is that it opens 
up two new worlds in which to test the alleged a priori knowledge 
of the rationalists. If this knowledge is found to be true in the twc 
new worlds, the question of whether it is genuine a priori know- 
ledge must still remain unanswered. If, however, it is found to be 
untrue in either or both of the new worlds, its claim to be genuine 
a priori knowledge is obviously discredited the a priorists have 
told us that the Creator could not make a world in such and such a 
way; we study the world of the electron or nebula and find that He 
has done so already. Thus the alleged a priory knowledge can only 
be empirical knowledge of the man-sized world 


Now when the actual intuitions of the rationalists are tested in 
these two new worlds, we find that those which were of a scientific 
nature are frequently not true for the two new worlds which science 
has just opened for us; they are only true for the man-sized world 
which was familiar to the rationalists because it did not need 
elaborate instrumental aid for its exploration. For instance, three 
of the examples of a priori knowledge just given ought, as a pre- 
liminary step towards the truth, to be amended to read: 

'The sum of the three angles of a triangle is 180, so long as the 
triangle is not of astronomical size.' 

'Divisibility is comprised in the nature of substance, so long as 
the object in question is not of the smallness dealt with in atomic 

1 Substance is permanent, so long as we experiment only to the 
degree of accuracy possible for eighteenth century physics.' 

No philosopher seems to have had an inkling, either a priori or 
otherwise, of the need for these or any similar reservations until 
modern physics arrived to point it out. The plain fact seems to be 
that when a rationalist, guided by his experience of the world, but 
subject to the scientific limitations of his day, could only imagine 
tilings being one way, he confidently announced that they were 
that way and had to be that way, describing his knowledge as 
a priori. Now that recent scientific investigations and discussion 
have opened up new worlds to the imagination, we can think 
soberly of possibilities that would have seemed sheer absurdities to 
Descartes and Kant. Not only can we imagine them, but we know 
that many of them find their counterparts in the actual world, and 
tell us that the supposed a priori knowledge of the rationalists was 
erroneous. Kant tells us that there are two infallible tests for 
true a priori knowledge necessity and strict universality. The 
supposed scientific knowledge of the a priorists fails conspicuously 
under both tests, and this failure of their scientific intuitions naturally 
discredits their non-scientific intuitions. But knowledge of a 
mathematical kind requires further investigation. 


Mathematical Knowledge 

,v? /V? V ,?" < / v 6 

While ghilosophers. may have differed as to the possibility of -ob- 
taining jz^norzUuw>wkdg& about the world of physics, they have 
been in very general agreement apart from Descartes (p. 34) and 
J. S. Mill (p. 36) that abstract knowledge of a mathematical kind 
could be obtained through purely mental processes, without any 
appeal to experience of the world, so that such knowledge can be 
truly a priori. They would have claimed this knowledge to be true 
in all possible worlds ; it would be a knowledge of facts which it was 
beyond the powers of the Creator to vary. Thus it could tell us 
nothing about the properties of our particular world, as distin- 
guished from those of other possible worlds which might have been 

We have cited three instances of supposed a priori knowledge of 
this kind, all three being geometrical in their nature, but the progress 
of science has shown that they all three fail to qualify as true 
knowledge of the physical world. 

Now that science is actively concerned with non-Euclidean 
geometries, philosophers have become chary of finding examples 
of a priori knowledge in geometry, and are more inclined to look to 
arithmetic or simple algebra. The proposition that two and two 
make four is frequently cited in this connection, although its pre- 
cise content is seldom stated, so that we feel that the first need is for 
definitions and explanations. The simple question is: Could Goc 
have made a world in which two and two did not make four ? anc 
however much or little we may claim to know about the Creator, ii 
is obvious that, before we can discuss this, we must know what the 
two and two are which form the subject of the proposition. Are 
they things which exist in reality or in our minds? Are they 
numbers or objects? And in the latter event, what kind of 

If the two and two refer to mere numbers, then the proposition is 
concerned with simple counting, and its content would seem to be a 
definition of the term four. We count two and then another two, 
and this brings us to a number to which we must give some sort of a 
name. The proposition tells us to call it four, although we might 


equally call it something else, such as quatre or vier; as indeed 
many people do. Clearly there can be no question of a priori 
knowledge here. 

Obviously then, the proposition must be interpreted as referring 
to real physical objects. It tells us that if we take two objects of any, 
but the same, kind, and add to them two more objects of still the 
same kind, we shall then have a collection of four objects in all 
not that we shall have taken four in all, for this would bring us 
back to mere counting, but shall have four objects under our 
observation as the result of doing something other than counting. 
The child is shown that when two apples are placed in juxtaposition 
with two other apples the result is a collection of four apples ; he 
sees that the same is true of fingers or counters or pennies, and then 
jumps to the conclusion that it is true of everything we can imagine, 
as for instance bananas or sea-serpents or unicorns. The knowledge 
about the apples or fingers is admittedly empirical, but this merely 
serves to pull a trigger; what is claimed as a priori knowledge is that 
we may generalize from apples and fingers to sea-serpents and 

If this is the true content of the proposition, does it not merely 
provide another instance of incomplete or ill-considered knowledge 
being labelled a priori ? For the generalization (which is the essence 
of the proposition) proves to be permissible for some classes of 
objects and for some circumstances, but only for some. It is 
impossible to say whether it is true in any particular case without 
detailed knowledge of the case, and such knowledge from its nature 
can never be a priori. We cannot say what two sea-serpents and two 
sea-serpents make until we know what a sea-serpent is, and this 
cannot be a priori knowledge. A sea-serpent is often said to be a 
cloud of birds; do then two sea-serpents when placed in juxta- 
position with two more make four sea-serpents, or do they make 
one big sea-serpent, or perchance two or three? And what about 
two raindrops meeting two more on the window-pane? If two 
negatives make a positive, while two positives also make a positive, 
what results from adding two negatives to two more? Clearly the 
proposition is applicable only to objects which retain their identity 
through the process of physical addition, and we cannot know 


a priori whether any particular class of objects possesses this property 
or not. Qfjrccen^e^ algebras in 

which L two ancj .two make numbers other than Jour, perhaps two qr^ 

algebras , do not. of course apply to mere 
^Sft% Before we can 
[assert that two objects plus two objects make four objects, we must 
find a definition of object that will exclude such things, and clearly 
this cannot be inborn in us as a priori knowledge. 

Kant did not discuss the proposition that 2 + 2 = 4, but the 
proposition that 7 + 5 = 12. He described this as a synthetical a priori 
proposition (p. 49), meaning that a special addition with fingers was 
needed to pull the trigger in his mind, and suggest the truth of the 
general proposition. But he does not define 12, or specify the 5 and 
7, other than fingers, to which the proposition is supposed to apply. 

A better example would perhaps have been the proposition that 
5 x 7 = 7x5, for this at least does not require a definition of 12, 
or even of 5 and 7, since it is equally true if 5 and 7 are re- 
placed by undefined numbers or numerical quantities p and q. The 
proposition then states that the product pq is equal to the product 
qp\ in other words, when we multiply^) and q together, the order 
in which we take them is a matter of indifference. This is obviously 
so if p and q denote pure numbers, but before we can assent to the 
general proposition, p and q must be defined with some care. 
Mathematicians now employ algebras^ which they describe as non- 
commutative, in which pq is not the same thing as qp ; these are 
found to be specially applicable to the sub-atomic world. In most 
of the problems which arise for discussion in the man-sized world, 
p and q have such meanings thatpq is equal to qp, but in the world 
of the electron this is ^ not so. We may conjecture that a denizen of 
tFe world of the electron might vigorously challenge the general 
proposition that pq = qp, insisting that it was true only under very 
special conditions (p. 157, below). 

Thus a large part of our mathematical knowledge proves on 
examination to be more empirical, at least in its application, than 
is evident at first sight, or than appears to have been suspected by 
the a priorists. We may say that a general proposition, such as that 
2 + 2 = 4 can be true in either of two ways either a posteriori or 


a priori. It is not true for objects in the outer world unless these 
conform to certain conditions. These conditions cannot even be 
stated, still less applied, without some knowledge of the outer 
world, so that when the proposition is applied to real objects, it 
obviously represents a posteriori knowledge ; we first test whether 
the proposition is true for the class of objects under consideration, 
and the proposition then merely gives back to us the knowledge we 
have previously put into it. But the proposition can also be applied 
to classes of objects we imagine in our minds in such a way that 
they satisfy the conditions necessary for the proposition to be true. 
When used in this way, the proposition contains pure a priori 
knowledge, but it can never tell us anything about the outer world 
only about the imaginings of our own minds. 

For instance the proposition 2 + 2 = 4 as applied to apples is 
a posteriori because we call on our experience of the world to assure 
us that apples retain their individual identities through the process 
of adding. But as applied to unicorns, it is a priori because the 
unicorn is a creature of our imagination, which we imagine to 
retain its identity through the process of adding. 

We see that when mathematical propositions are applied to 
objects in the a posteriori manner, they can supply no knowledge 
about the outer world beyond that we have previously put into 
them, while when they are applied in the a priori manner, they 
can give us no knowledge at all about the outer world ex nihilo 

There is, nevertheless, a wide range of abstract mathematical 
knowledge which can be derived by purely mental processes, without 
introducing any knowledge of the world outside. The clearest 
instance of such knowledge is to be found in the properties of pure 
numbers or numerical quantities, as expressed in arithmetic or 
ordinary algebra, but we must notice that even here we have 
to assume that numbers and measurable quantities exist. For 
example, we can show by purely mental processes, and without 
calling on our experience of the outer world at all, that if a is a pure 
number, then (a+ 1) x (a i) is always less than a 2 for example, 
8 x 6 is less than y 2 . In the same way, it can be discovered that 
8, 9 and 10 are composite numbers (i.e. numbers obtained by 


multiplying smaller numbers together), while 7 and 1 1 are primes 

(i.e. non-composite numbers). 

Such facts involve no knowledge or experience about the par- 
ticular world in which we live (unless we regard the existence of 
measurable quantity as an empirical fact), but, in so far as they 
have to do with worlds at all, are true of all worlds which could be 
either built or imagined. In whatever way this or any other world 
is constructed, 7 must be a prime, and, just because of this, the 
primeness of 7 can never tell us anything about the special structure 
of our particular world ; no bridge can be built between the two. 
The same is true of all the discoveries of the pure mathematician ; 
they are universal in the sense that they would be true in any 
world, and so cannot tell us anything about the special properties of 
this particular world. 

Indeed any knowledge which is truly a priori must, as Kant says, 
be universal, and so can tell us nothing about our particular world. 
Let us imagine a totally uneducated man being told he was going 
to be sent to Procyon. He would not know whether Procyon was 
a prison or a gin-palace, an island or a star. But he would know 
just as much about Procyon as our unaided a priori knowledge can 
tell us about the universe we live in, and if he tried to construct 
a 'pure science of Procyon', his efforts would be no more futile 
or misguided than those of Kant to construct a 'pure science of 
nature '. In such jways we^ see thatjthere can be only one possible 
source ofT^wledge as to the special properties ^^5|^uFjown 
wofiar namel^ ^ .00L 

method of acquiring such knowledge z namely the method of scieng^ 

Synthetic Knowledge 

As the admission of this obvious truth would have undermined 
Kant's whole position, he made two attempts to evade it; they are 
quite distinct, although he does not seem to have realized this. 

In the first he claimed to be in possession of a special kind of 
a priori knowledge synthetic a priori knowledge as he called it 
which conveyed knowledge about our particular world. 

In the second he claimed in effect that our physical knowledge 


is not knowledge about the world, but about the workings of our 
own minds not knowledge of the world we perceive, but of our 
mode of perception of the world. Let us consider,, these two 
tttempts/at escape in turn^J 

We have already Ea^Tan example of Kant's synthetic a priori 
.mowledge in the proposition that 7 + 5 = 12. A more characteristic 
instance is the proposition that 'all bodies are heavy'. In dis- 
cussing this, Kant first cites the proposition that 'all bodies are 
extended ' as a typical piece of a priori knowledge which was, in 
his judgment, obvious apart from all experience of the world. He 
then says that, after encountering extended bodies in the actual 
world, we find that they are heavy as well as extended. Adding 
this new fact to his previous knowledge he arrives at the pro- 
position that 'all bodies are heavy'. 

; He considers that all the propositions of arithmetic, and many of 
the principles of physics, are of the synthetic a priori type. As 
instances he selects the conservation of matter (pp. 40, 41) and 
Newton's third law of motion (p. 108), expressing them in the 
words ' In all changes of the material world, the quantity of matter 
remains unchanged ' and ' In all communications of motion, action 
and reaction must always be equal'. 

"" Science can of course have nothing favourable to say about this. 
On Kant's own admission, he only knows of heaviness through 
observing it in the actual world, and this immediately removes it 
from the category of a priori knowledge synthetic a priori is seen 
to be merely a new, and question-begging, name for a posteriori. 
In the instance just quoted, Kant's claim is in effect to know of 
the existence of gravitation, but if he could know of this, why did 
he not also know of electric attractions and repulsions? Would 
he have known a priori that two objects similarly electrified would 
not attract but repel one another? 

In such ways Kant persuaded himself that this supposed a priori 
knowledge provided definite and certain information about the 
actual universe. Claims of this kind at once raise questions such 
as the following: 

(i) If a priori knowledge does not come from our experience of 
the world, whence does it come? The rationalists claimed to have 


a priori knowledge that everything must have a cause; what, then, 
is the cause of a priori knowledge itself? 

(2) If a priori knowledge does not come from our knowledge of 
the world, how can it tell us anything about the world? How does 
it happen that, when we step out into the world, we find this world 
conforming to our a priori knowledge? If Kant or Eddington 
succeeded in constructing a whole universe out of such knowledge, 
on what grounds would he expect the actual universe to conform 
to his predictions? 


Kant saw very vividly the difficulties presented by this and similar 
questions, and this led him to fall back on his second line of 
defence, developing a set of ideas as to the precise meaning of 
which philosophers themselves are not altogether in agreement. 
Indeed there is every justification for wondering whether Kant alto- 
gether understood them himself. Sixteen years after the publication 
of the Critique of Pure Reason, Kant's doctrines were making a great 
turmoil in Germany ; university professors were being forbidden to 
lecture on them and one at least was forced to resign for venturing 
to disagree with him. This moment was chosen for asking Kant to 
say which of his commentators had best grasped his meaning. In 
reply, Kant indicated a certain Schultze, the author of an elemen- 
tary explanation which seems to have over-elaborated the easier 
parts of Kant's philosophy at painful length, while dismissing the 
more difficult parts in a few words which were demonstrably 
wrong. Thus the problem of discovering what Kant had been 
trying to say remained unsolved, as it still is to-day. James Ward 
tells us that no fewer than six different formulations of Kant's 
philosophy were current in the years 1865 to 1878. 

Although no one can say precisely what Kant meant to convey, I 
hope what follows will be found to express an average view as to 
his meaning, in so far as it affects the problems before us. 

To the first of the two questions stated above If a priori know- 
ledge~tt5B not come from our experience of the ^worTeT, ..wKence 
does it come? Kant's answer seems to be that a priori knowledge 



comes from the inherent constitution of the human mind. Just as the 
human body is .built in a certain way, with two eyes and two ears 
and other sp^ftic organs which perform specific functions, so the 
human mincTis built in a certain way, with specific faculties which 
perform 'specific functions. It is to these faculties that we must 
look for the sources of our a ^norTKiilowTeSge. They sift out the 
seffSe-tfaia ^vith which^tif Senses continually deluge our minds, 
allowing some to slip through unheeded while retaining others. 

Out of such as are retained, the mind creates its own picture of 
the external world. As a result of the sifting action of the mind, 
certain laws and regularities emerge to which all our perceptions 
conform. If we run a miscellaneous collection of potatoes over a 
sieve of one-inch mesh, we know that any potato-pattern left on the 
sieve will conform to at least one law every one of its ingredients 
will be more than one inch in diameter. This law is not obeyed by 
potatoes in general, nor by the miscellaneous collection of potatoes 
that were passed over the sieve ; it is a law thrust on to the potatoes 
by the selective action of the sieve, and expresses a property of the 
sieve rather than of potatoes. Kant suggests that those laws of 
nature which we know (as he thought) a priori are thrust on to the 
perceived world by a sefective action of the human mind, which 
thus facts as a lawgiver to nature;] a priori knowledge merely 
specifies the conditions to which phenomena must conform if they 
are to be perceived. 

~ "Possible modes of selection can perhaps be illustrated by two 
simple analogies. Light is a blend of constituents of different 
wave-lengths. If we pass the light through a spectroscope, the 
different constituents are separated out, and we observe a spectrum 
of colours ranging from red, through orange, yellow, green to blue 
and violet the colours of the rainbow. Outside the limits of this 
spectrum all looks dark, yet if a thermometer is placed in the dark 
region beyond the red, the mercury begins to rise, showing that 
beyond the reddest of visible radiation there is an invisible radia- 
tion ; it is in fact the infra-red heat-radiation. Beyond the violet at 
the other end of the spectrum there is another region in which our 
syes can see nothing, but in which certain salts phosphoresce, 
showing that here too there is radiation which is invisible to our 


eyes. This is the ultra-violet radiation; out beyond this we come to 
X-radiation, and further still to the y-radiation emitted by radio- 
active substances. 

Our instruments reveal a continuous spectrum of rays ranging 
from long radio waves to short y-rays, the wave-lengths of these 
extremes standing in a ratio of about twenty thousand million 
million to one. By contrast, the extremes of radiation that our eyes 
can.see have wave-lengths standing in a ratio of only about two to 
one. Thus of the whole range of radiation known to us through 
our instruments, only one part in ten thousand million million is 
perceptible to our eyes an infinitesimal fraction of the whole. 
/r The restriction of our vision to so minute a part of the whole 
spectrum acts as a sieve to our perceptions. All sorts of radiation 
fall on the retina, but this is sensitive only to a small part of what it 
receives; it forwards such radiation, and only such, to the mind for 
its attention. The mind might draw the inference that all radiation 
lies between the red and the violet. On Kant's view this would 
correspond to the a priori knowledge claimed by the rationalists^ 
and it may be noticed that, in so far as the|palogy iS*s6urTd^the only 
inference to be drawn is that a priori knowledge is wholly untrust- 

Itjs the same with sound. Our ears are sensitive only to sounds 
the pitch of which lies within about ten octaves, out of the infinite 
range which can occur in nature. If we took the data provided 
by our unaided sense-organs at their face-value, we might claim 
to know that all sounds lay within a range of ten octaves. 

Such is the way in which the physical sieves of our sense-organs 
work^f A simple analogy may explain how our mental sieves may 
worly^ielilgTit: sky exhibits a confused mass of stars which migKt 
Be sbrted into constellations in many ways. The Greeks, with their 
minds accustomed to run on legend and romance, sorted the stars 
out into figures of heroes and their accompanying animals; the more 
prosaic Chinese saw the same groups of stars as quite commonplace 
animals. But there are also stars in the southern sky which the 
Greeks had never been able to see, because their travels were 
confined to the northern hemisphere. When the navigators of a 
later age explored the southern seas, and first saw these stars, they 


did not see them as groups of new heroes and animals. The age of 
such fancies had passed, and the explorers left it to their prosaic 
astronomers to group the new stars in the forms of triangles, clocks, 
telescopes, and so on; they chose these because their practical minds 
were accustomed to thinking of such things. The division of the 
stars into constellations tells us very little about the stars, but a 
great deal about the minds of the earliest civilizations and of the 
mediaeval astronomers. 

Kant thinks that it is in such ways as these that our minds sort 
out the phenomena of nature. The outer world provides us with a 
confused mass of impressions which our minds might sort out in 
many ways. They choose one particular way because they are con- 
stituted in one particular way; other types of mind might choose 
other ways. The laws we deduce from our a priori knowledge or 
reasoning merely represent habits of thought embedded in our own 
minds. These habits of thought form blinkers, restricting the free 
vision of our minds. But the mind, not recognizing its own limita- 
tions, proceeds to attribute these limitations to nature itself. Thus, 
in Kant's own jvords l5 * reason only perceives that which it produces 
after its own design', * objects conform to the nature of our faculty 
of perception' and 'we know a priori of things only what we our- 
selves put into them*. 

Kant described this as his Copernican revolution. When no 
further progress seemed possible to an astronomy which supposed 
that the sun revolved round the astronomer, Copernicus cleared up 
the situation by supposing that the astronomer revolved round the 
sun. Kant thought that he had removed the difficulties of a priori 
knowledge in a similar way if our minds conformed to the pheno- 
mena they perceived, our knowledge could not be a priori; we must 
therefore (so Kant thought) make the phenomena conform to our 

If this were the true significance of a priori knowledge, it would 
of course tell very little about nature only something about our 
own minds. Our knowledge would not be of the structure of the 
universe without, but of the structure of our minds within. Here, 
then, we have the answer to our second question If a priori 
knowledge does not come from our knowledge of the world, how 


can it tell us anything about the world? The answer is that it 
cannot; it can only tell us about the structure of our own minds. 

All this throws a vivid light on the different methods of science 
and philosophy. Kant proposed in effect that we should base our 
knowledge of things on something that 'we ourselves put into 
them'; the scientist is anxious to eradicate just this something, 
knowing that it is not knowledge of the outer world at all. 

The * sieves' which Kant attributed to the human mind are 
fourteen in number. First of all come two which he describes as 
* Forms of Perception' these are merely space and time. Then 
come twelve others, which could well be described as * Forms of 
Understanding', although Kant preferred to describe them as 'Pure 
Conceptions of the Understanding' or 'Categories', this latter 
term being borrowed from Aristotle. 

We want ultimately to bring Kant's views on space and time into 
relation with present-day science; for this reason we may con- 
veniently proceed at once to discuss space and time in rather 
general terms. ^ v . 


As present-day science knows, the words space and time admit of 
mlny^nte^ distinct meanings may be discerned^ 

for each, those for space being approximately as follows : 

Conceptual space is primarily the space of abstract geometry. It 
has no existence of any kind except in the mind of the man who is 
creating it by thinking of it, and he may make it Euclidean or non- 
Euclidean, three-dimensional or multi-dimensional as he pleases. 
It goes out of existence when its creator stops thinking about it 
unless of course he perpetuates it in a text-book. 

Perceptual space is primarily the space of a conscious being who 
is experiencing or recording sensations. We feel an object, and our 
sense of touch suggests to us that it is of a certain shape and size ; 
we see a collection of objects, and our vision suggests to us that 
these objects stand in certain relations to one another. We find that 
we can reconcile th^se. and all other suggestions of, our senses by 
'imagining all objects arranged in a threefold ordered aggregate 
which we then call space. This is perceptual space, created for 


himself by a man experiencing sensations, and it goes out of 
existence as soon as his sensations cease. For a one-eyed man, or 
one viewing objects so remote that his binocular vision conveys no 
idea of distance, perceptual space is two-dimensional at least so 
long as no sense other than seeing is involved. Thus the ancients 
located the fixed stars on the two-dimensional surface of a sphere. 
As soon as near objects are viewed by a normal man, so that 
binocular vision is employed, or as soon as objects are seen to 
move one behind another, or as soon as senses other than seeing 
are employed, a third dimension of perceptual space instantly 
springs into being. 

"Physical space is the space of physics and astronomy. Concep- 
tual space and perceptual space are both private spaces, the one 
being private to a thinker, and the other to a percipient. Science 
finds, however, that the pattern of events in the outer world is 
consistent with, and can be explained by, the supposition that 
material objects are permanently located in, and move about in, a 
public space which is the same for all observers, apart from a 
complication introduced by the theory of relativity to which we 
shall return later (p. 63). Disregarding this complication for the 
moment, we may say that this public space is physical space. 

Absolute space is the particular type of physical space which 
Newton introduced to form the basis of his system of mechanics 
(p. 108, below), and remained in general scientific use throughout 
the period between Newton and Einstein. When we say that a train 
has moved 10 miles nearer to King's Cross, we mean that it has 
moved a distance of 10 miles along the pair of rails along which it is 
running to King's Cross, as, for instance, frorn milestone 105 to 
milestone 95. In the same interval of time, the earth carrying 
this pair of rails wth it may have carried the train 100 miles to 
the east by its daily rotation around its axis, and may have moved 
10,000 miles in its yearly orbit round the sun, while the sun, 
dragging the earth along with it, may have moved 100,000 miles 
rifearer to the nearest star and i ,ood^oob miles farther away from a 

distant nebula. All these motions are equally real and equally true, 

t r - lf -fHj^^^^^MH,^^,^. ,, J? ,* J . .,-.-'-, _,x^,*- ':**** 

but all are relative only to some other moving body. 

V-^a|i^^*w^>fcy^^**^ V'T* 

Tne sequence might go on indefinitely, but Newton imagined 


that it did not. He thought that the remotest parts of the universe 
were occupied by vast masses which might provide fixed points of 
reference from which to measure motion, while themselves pro- 
viding standards of absolute rest, although he qualified this by 
remarking ' it may be there is no body really at rest, to which the 
places and motions of others can be referred*. At a later period, 
space was supposed to be filled with a jelly-like ether, and this 
again was thought to provide a standard of absolute rest until it 
was abolished by the coming of the theory of relativity. Assuming 
that such standards existed, Newton described the space in which 
measurements were made from them as absolute space; this, he 
said, * in its own nature and without regard to anything external, 
always remains similar and unmoveable'. He contrasted it with 
perceptual space, which he described as relative space 'some 
moveable dimension or measure of * absolute space which our 
senses determine'. _ * ' *i ^ - * \ } > >r 

In a precisely similar way we mayjdjscern four distinct meanings 
for time ; there are a conceptual time, a perceptual time, a physical 
time and an absolute time. 

Conceptual time is the time of theoretical dynamics, and of all ab- 
stract attempts to study change and motion. Like conceptual space 
it exists only in the mind of a thinker. He usually makes it one- 
dimensional, but not always. Dirac, for instance, found it convenient 
to measure time by a g-number, which amounts to supposing that 
time has as many dimensions as we please to assign to it. 

Perceptual time records the flow of time for any single percipient. 
Thus it is related to the consciousness of a particular individual, 
and goes out of existence as soon as this individual loses con- 
sciousness. Experience shows that the acts of perception of every 
percipient lie on a single linear series in other words, they come 
one after another. Thus perceptual time is one-dimensional. 

Physical time is the time of the active world of physics and 
astronomy. Like physical space it is public, in contrast with con- 
ceptual and perceptual time, which are private. Sg^ain disregarding 
complications introduced by the theory of relativity, science finds 
that the pattern of events is consistent with the supposition that all 
events can be arranged uniquely in a single linear sequence, the 

5 8 H0\y DO WE KNOW? 

position on this sequence determining the time. Thfe still permits 
of an infinite number of ways of measuring the time, so that a 
convention must be introduced as to how the actual measure is 
made. We agree to select jspme motion which repeats itself regu- 
larly, such as that of the earth in its or6It,To^fofm a 'clock', and let 
each repetition oftEis motion count as a unit of time in this case 
a year. But as this unit is too large for most practical purposes, 
other regularly repeating motions must be found, such as the 
oscillations of a pendulum or the vibrations of a crystal, which 
repeat many times in a year, and these provide the units needed 
for ordinary life and for scientific investigations in which time is 

Absolute time is the counterpart of absolute space. We have just 
seen how a * clock* can be devised to give a consistent measure of 
time at any one point of space. The problem of synchronizing 
clocks in different parts of space is a different problem, to which 
we shall return later. If light travelled with infinite speed, it would 
be as simple, in principle, to synchronize distant clocks as it is to 
set our watches by Big Ben. Newton, disregarding the finite speed 
of travel of light, assumed that this could be done, and that a 
universal time * flows equably, and without regard to anything ex- 
ternal' throughout the universe. This we describe as absolute time. 

What are Space and Time? 

There can be no serious difficulty in understanding the meaning of 
conceptual and perceptual space and time, for they are our own 
creations. They exist in our individual consciousnesses, and go out 
of existence when these consciousnesses cease to function. But a 
variety of views can be held, and have been held, as to the true 
significance of physical space and time. 

Science has usually adopted a realist view of the world of 
nature, assuming that our perceptions originate in a stratum of 
real objects stars, bricks, atoms, etc. which exist outside, and 
independently of, our minds. If our minds go out of existence or 
cease to function, the stars, bricks and atoms continue to exist, and 
are still capable of producing perceptions in other minds. On this 
view, space and time have just as real existences as these material 


objects ; they existed before mind appeared in the world, and will 
continue to exist after all mind has gone. 

But philosophy has pointed out that other views are possible. 
We can have no knowledge except self-knowledge ; what is in our 
minds we know, but what is outside we can only conjecture. And 
our conjectures may be erroneous. The mentalist or idealist philo- 
sophies suppose that there is no stratum outside all mind having an 
existence of its own in the way the realists suppose ; consciousness 
is fundamental in the world, and the supposed real objects which 
produce our perceptions are creations either of our own or of some 
other minds (p. 196). There is no reason to attribute a higher 
degree of reality to space and time than to the objects we locate in 
space and time, so that these also become mental creations. Con- 
ceptual and perceptual space and time are now as real as anything 
there is, fahile physical space and time become attempted mental 
generalizations of these realities in strong contrast to the realist 
view which makes physical space and time the realities, while con- 
ceptual and perceptual space and time are mere reflections of, 
and abstractions from, these realities. 

The first modern to discuss the nature of space and time was 
Nicholas of Cusa (1401-1464). He held that space and time are 
products of the miriH7 and" so are inferior in reality to the mind 
which has created them. In contrast with this purely philosophical 
conclusion, Giordano Bruno (1548-1600), discussing space and 
time in their astronomical aspects, argued that the words 'above' 
and 'below', 'at rest' and 'in motion' become meaningless in the 
world of eternally revolving suns and planets which know of no 
fixed centre. Thus all motion is relative as Emsteinjsubsequently 
convinced thejggrld aruJj^D^ 

Leibniz (1646-1716) held very similar 

opinions, believing that space and time exist only relative to objects 
and not in their own right; space is merely the arrangement of 
things that co-exist, and time the arrangement of things that suc- 
ceed one another. All these thinkers, then, reduced space and time 
simply to conceptual and perceptual space and time; physical 
space and time had no real existence, and absolute space and time 
did not come into the picture at all. 


In opposition to them all came Isaac Newjon (1642-1727), 
tacitly assuming that jspace and time were no mere dependents on 
consciousness but existed in their own rights, and introducing the 
hypothesis that absolute measures of space and time were possible, 
at least in principle. 

Kant's Discussion of Space and Time 

Kant began his discussion of space and time by asking the 
questions: What are space and time ? Are they real existences? Or 
are they merely relations between things ? And in this case, would 
these relations belong to the things even though the things should 
never be perceived, or do they belong only to things when these 
are perceived i.e. are they contributions of the perceiving mind? 
He made no distinction between the different kinds of space and 
time that we have mentioned, identifying them all with perceptual 
space and time. His general view was that space has no real 
existence of its own, but is supplied by our minds as a framework 
for thejtrrangements of objects, so thatjtJsjDnly from the Jiuman 
point of view that we can speak oTsjgace, thejsxtension of objects 
and so forth. ' Space is not a conception which has been derived 
from outward experience ; it is a necessary representation a priori, 
which serves for the foundation of all external perceptions/ Time 
again is not an empirical conception and has no real existence of its 
own, but whereas space serves for the representations of external 
perceptions, time serves for the representations of internal per- 
ceptions 'the perception of self and of our internal states'. 

Kant tries to justify these views in the discussion of his ' first 
antinomy'. By an antinomy Kant means a pair of more or less contra- 
dictory assertions, each of which seems to be proved by disproving 
the other. In his own words, we originate a conflict of assertions 
'not for the purpose of finally deciding in favour of either side, but 
to discover whether the object of the struggle is not a mere illusion, 
which each strives in vain to reach but which would be no gain 
even when reached'. 'Perhaps after [the combatants] have wearied 
more than injured each other, they will discover the nothingness of 
their cause of quarrel, and part good friends.' A new set of ideas 
which reconciles the combatants is described as a solution of the 


antinomy. It may or may not be true; its truth is established only 
if it can be shown to provide a unique solution of the antinomy, 
but not otherwise a point which Kant overlooks. 

Kant's first antinomy consists in brief of the assertions that it is 

(a) the world had a beginning in time, and is also limited in 

or (b) the world had no beginning in time, and has no limits in 

The reasons he gives for dismissing both alternatives as absurd 
seem entirely unconvincing to a modern scientific mind. There is, 
of course, no justification for tying up an infinity of space with an 
infinity of time in the way that Kant does. Mathematicians have 
investigated the properties of universes in which space is finite 
but time infinite^ and no logical inconsistency has so far been 
cTetected In the concept. It is, however, quite simple to discuss 
time and space separately. 

In opposition to alternative (6), Kant argues that any quantity 
must be regarded as the synthesis of a succession of separate unit 
quantities. For example, a mile must be regarded as the length 
of 1760 yardsticks put end to end. If, then, the quantity is infinite, 
the synthesis fcan never be comoletedj this, he says, is the true 
definition of infinity^ "Hence 'itvfollows, without possibility of 
mistake, that an eternity of actual ^uccessive states up, to a given 
(the present) moment cannot have \elapsed, and that the world 
must therefore have a .beginning/ *-J .. _,.,, 

In this argument the words 'lean never be ^cpmpjetedi' are 
obviously ambiguous. We want to know who or what can never 
complete them, why he should want to, and whether he wants to 
complete them in his imagination or in some sort of reality; until 
this information is given us, the argument is simply a meaningless 
collection of words. 

Apart from this, the argument fails because a quantity can be 
regarded in other ways than as a succession of units. Must we 
always think of a mile as 1760 yards? Why this rather than as a 
succession of eight furlongs? And why either rather than as just 
one mile? Yet as soon as we concede the last possibility, the 


bottom drops out of Kant's argument, since we need only increase 
our unit pan passu with the length of space or time to be measured. 
Even though our finite lives may be too short to imagine eternity 
as a succession of hours or years, we can still think of it as one 

In opposition to (#), on the other side of the antinomy, Kant 
argues that if the world had a beginning in time, there must have 
been a previous void time in which there was no world. But there 
can be no reason for anything beginning in a void time, since 'no 
part of any such time contains a distinctive condition of being, 
in preference to that of non-being '. Thus the world cannot have 
had a beginning. 

This argument fails through assuming that time would neces- 
sarily go further back than the beginning of the world. This has 
not been the usual view of philosophy. Plato, for instance, said 
that time and the heavens came into being at the same instant; 
Augustine wrote, * Non in tempore, sed cum tempore, finxit Deus 
mundum', while Kant himself tells us that time does not subsist 
of itself, but is 'the form of the internal state, that is, of the per- 
ceptions of ourselves and of our state*. But if time is in ourselves, 
and we in the world, then time must be in the world, and it is a 
petitio principii to argue as though the world were in time. 

After adducing arguments of somewhat similar type for space, 
Kant proposes the solution that space and time have no real 
existences, but are only forms of human perception. As they are, 
then, creations only of the human mind, we are free to imagine 
alternative (a) at one moment and (b) at the next, if we so wish; the 
two assertions of the antinomy become no more contradictory than 
the uses of a Mercator Projection and a stereographic projection in 
map-making, and we are free to use whichever serves our purpose 
best. But even if Kant's arguments were sound, we should be 
under no obligation to accept his proposed 'solution' of the 
antinomy, for he does not even attempt to prove that it is the only 
possible solution. 

Three general reflections on the problem of space and time will 
perhaps not be out of place here in view of their bearing on Kant's 
doctrines of space and time. 


The Finite Speed of Light 

Light takes time to travel through space, a fact which does not 
appear to have been known to Kant, although it had been dis- 
covered by the Danish astronomer Roemer as far back as 1675. 
Jupiter has a number of moons which circle round it with the same 
regularity as that of the moon round the earth. When the precise 
period of revolution of any moon of Jupiter has been found, it 
would seem to be a simple matter to draw up a time-table of its 
future movements. Roemer made such a time-table, and dis- 
covered that tliQ moons did not keep to it ; they seemed to get late 
and run behind their scheduled time whenever Jupiter was at more 
than its average distance from the earth, and to be ahead of time 
when Jupiter was at less than its average distance. He found, 
however, that all the observations could be explained by supposing 
that light travelled through space at a uniform finite speed; the 
apparent irregularities of Jupiter were then explained by the varia- 
tions in the time which light took to travel from the planet to the 
earth. The truth of this explanation was established beyond all 
doubt when Bradley discovered the phenomenon of aberration in 

This shows that space and time are not totally independent of 
one another as Kant and many others seem to have imagined; 
on the contrary there must be a fairly intimate connection be- 
tween them. 

The Space-Time Unity 

The theory of relativity has revealed the nature of this connection. 
Newton "supposed mat all objects could be located in his absolute 
space, and that all events, wherever they occurred, could be as- 
signed positions uniquely and objectively on an ever-flowing stream 
of absolute time. These assumptions provided him with an ap- 
proximation which was good enough for his purpose, and fitted 
in with the scientific knowledge of the seventeenth century. 
Subsequent investigation has shown that they are inadequate to 
explain the passage of light and the behaviour of objects moving 
at a speed comparable with that of light. The physical theory of 
relativity suggests, altfaemgh without absolutely conclusive proof. 


that physical space land physical time have no separate and inde- 
pendent existences^theyseem more likely to be abstractions or 
selections from'^omethm^more coirv^lex, namely a blend of space 
an3^IIjSejwhich comprises both. 

It is of course always possible to take any two things of not too 
dissimilar nature, and blend them into a single unity which shall 
comprise both. Before the advent of the theory of relativity, no one 
could have imagined that space and time were sufficiently similar in 
their natures for the result of blending them together to be of any 
special interest. Yet such a blend has proved to be of outstanding 
importance for thTunderstandinfi of physics. 
"" Any ordinary three-dimensional space may be regarded as hung 
iround a framework of three perpendicular lines, these indicating 
three perpendicular directions in the space, as for instance East- 
West, North-South and up-down. The surveyor is accustomed to 
treat his perceptual space In this way, and the mathematician treats 
.his conceptual space in the same way, except that he replaces the 
three perpendicular directions of the surveyor by purely mental 
abstractions which he usually denotes by Ox, Oy and Oz. Now let 
us imagine the surveyor's perceptual space sliced into horizontal 
layers of infinite thinness much as a skilled chef will cut a round of 
beef into infinitely thin slices Any single slice, contemplated by 
itself, forms a mere horizontal plane which possesses extension in 
the East- West and North-South directions, but ijpt in the up-down 
direction. If we imagine these various slices now to be laid back, 
one above the other, in their original positions and then welded 
together, we shall have reconstituted the original three-dimensional 
space. We may say that, in performing this last operation, we have 
welded verticality on to horizontally and obtained something 
different from either, namely a three-dimensional space. 

Let us now imagine these two-dimensional slices replaced by 
the perceptual three-dimensional spaces of some individual A at 
successive instants of his experience. Let us take all these perceptual 
spaces, and place them contiguous to one another in their proper 
order. As they are to be contiguous and not overlapping, we must 
imagine them all placed in a four-dimensional space before we 
can do this. If we now imagine tkeiii welded together, they will form 


a four-dimensional continuum which we may describe as the space- 
time unity for the individual A. It is a conceptual space of four 
dimensions, and as it is constructed out of the perceptual three- 
dimensional spaces of a single individual A, we might reasonably 
expect it to be private and subjective to this individual. 

We can create a second space-time unity out of the perceptual 
spaces of some second individual B, which we might expect to be 
private and subjective to the individual B. The theory of relativity 
shows that the two space-time unities we have constructed in this 
way will be identical for A and J8, and so also of course for any 
other percipients C, D y E, . . . as well. In other words the space- 
time unity which we build up out of private perceptual spaces 
of a single individual proves to be public, and so objective. 
Space and time separately are private, but the blend of..,the two is 
public. " \:-.\ ^ * >x -< ; <' ^ 

We cannot speak of /right-handj and fleii-Mikh^ in 

ordma^spac^ right hand andTeft hand do noTEelong 

to space, but to an observer in space ; the division of space into 
right-hand and left-hand is meaningless except relative to a par- 
ticular observer. In the "same way,' we' cannot speak of space and 
time in the space-time unity space and time dpjioj belong to the 
space-time unity, but^o an observer in it. But it is the body of the 
observer that we want, and not his mind; a laboratory equipped 
with cameras and various instruments of measurement would serve 
our purpose just as well. 

Two observers who always keep close together will have the 
same perceptual space, but if they are moving at different speeds, 
and so changing their relative positions, they will have different 
perceptual spaces. The theory shows that these different perceptual 
spaces are to be obtained by taking cross-sections of the sjiace-time 
unity in different directions. In other words, each percipient divides 
up the public space-time unity into space and time in his own 
individual way, the mode of division depending on his speed of 

In the same way, to use a rather imperfect analogy, a cannon-ball 
may be conceived as having any number of different diameters, all 
pointing in different directions. It would be inaccurate to speak of 

J* 5 


any one of these as the height of the ball, to the exclusion of the 
others. Each one has an equal claim to be regarded as the height, 
and can indeed be made the height by turning the cannon-ball the 
right way up. But so long as the cannon-ball enters into no kind of 
relation with other objects, such terms as height, width and length 
are meaningless. In the same way, time and space are meaningless 
when applied to the four-dimensional continuum in the abstract. 
But, just as, when the ball is placed on a horizontal floor, one 
particular diameter immediately becomes the height of the ball, so, 
when we put a particular scientist or observer inside the four- 
dimensional continuum to measure or explore, one direction 
immediately becomes identifiable with his time; which particular 
direction it will be is determined by the precise speed at which 
this observer is moving. 

The questionjaow arises as to whether it is possible to dividq up 
thislinlty^mto space and time separately in a wav which shall jiot 
depend on the circumstancg of individual percipients. If such a 
way can be found, we mightidentify the space and time so obtained 
with Newton's absolute space and time. If such a way is not found, 
it will not prove that no such way exists, and still less that absolute 
space and time do not exist; the most we could say would be that 
they had not so far disclosed themselves. Actually in so far as 
ordinary physics i.e. physics on the man-sized scale is con- 
cerned, no such way has so far been found, and it seems highly 
improbable that it ever can be. For the pattern of events is known 
with tolerable completeness, and must, it is found, be described in 
terms of the space-time unity as a whole and not in terms of its 
separate dimensions, which do not enter into the description at all. 
This might have been anticipated from the circumstance that 
nothing less than the unity as a whole is completely objective. 

Although physics on the man-sized scale may be unable to 

^entangle space from time, it is still possible that atomic physics 
T * astronomy i.e. physics on the scale of the nebulae may have a 
Different story to tell. Once again, an analogy may help to explain 
Hfie possibilities. 

Let us imagine a race of deep-sea fish, living so far below the 
irface of the ocean that no ray of sunlight ever reaches them; let 


them be of precisely the same density as the water in which they 
Jiive> so that it is just as easy for them to swim up as down; let their 
semi-circular canals, and any other mechanism they may have for 
distinguishing directions, be abolished. Such a race of beings will 
nave no means within themselves of distinguishing directions, and 
frf .they study physical phenomena, they will fi n dthat the laws of 
bptics, electricity, magnetism, etc. make no distinction between 
the different directions in space. They may then announce that 
nature treats all directions of spacg^e^ually. Having no means of 
disentangling thehorizontal fronTthe vertical, they will describe 
different directions in a purely subjective way. Up and down will 
not refer tb^3irectTons determined relative to the earth's centre, but 
relative to" their own backs and bellies. They will know nothing of 
an'bbjec5yg_north, south, east andjsvestj only of subjective direc- 
tions, to describe which they may use such words as fore and aft, 
right and left. 

In this analogy, the race of fish represents physicists who study 
physics on the man-sized scale. The three-dimensional space in 
which the fishes swim corresponds to the four-dimensional space- 
time unity of the theory of relativity in which we exist. Man-sized 
nature has provided no means of dividing this into space and time 
separately, just as the fishes had found no means of dividing their 
watery space into horizontal and vertical. 

Now suppose that one fish has the enterprise to swim as far as 
the surface of the sea. He no longer studies nature on the fish- 
sized scale, but on a wgrld-sized scale. When he does this, he finds 
a whole range of new phenomena, and amongst them a surface, 
objective and fixed by nature, which at once determines up-down 
and horizontal directions in space in a wholly objective way. 

There is still a possibility that when we leave man-sized physics 
for astronomical physics, we may have experiences similar to those 
of the enterprising fish. The hypothesis that absolute time and 
space do not exist brings order into man-sized physics, but seems 
so far to have brought something very like chaos into astronomy. 
Thus there is some chance that the hypothesis may not be true. 
Newton thought that the vast masses which occupy the remotest 
parts ^toe^imivgrse might provide a framework from which to 



measure absolute rest and motion, and something of the kind seems 
to be needed if the pattern of events recently revealed by nebular 
astronomy is to make sense. It may be that before it can make 
sense, the new astronomy must find a way of determining an 
absolute time f which it will then describe as cosmical time. The 

'**'*! .!> ii-riiifc fmittf^m ' " 'W "'' " ' _ ii^ni* 1 '' " J ~ ll * L*^"''''" K" ^ 

space-time unity will menbe divided into space anotime separately 
by nature itself. Apart from this possibility, all observers stand on 
the same footing, each dividing the space-time unity into his own 
perceptual space and his own perceptual time. 

The Theory of Relativity 

The foregoing remarks embody the main conclusions of the re- 
stricted, or physical, theory of relativity which Einstein put for- 
ward in 1905. We i^^a^aysjre^mber that this theory is a 
deduction from the observed pattern of events.^Ss tEe J^tern can 
Dnly be expressed in mathematical terms, the theory of relativity 
also oarTlilT) terms. It deals with 

SeaiuFes of things, and not with things themselves, and so_can 
Sever tell us anything about the nature of the things with the 
tneasures ^jfjyhich it is concerned. In particular, it can tell, ^us 
nothing as to the nature of space and time. 

-- f-.air.jjm * -- - """ - -"- --- " -"-"- - *-~ --- ._ --- __ ^ , 

Nevertheless, as it shows the mathematical measures of space and 
time to be so intimately interwoven, it seems reasonable to suppose 
that space and time themselves must at least be of the same general 
nature. The distinction, which many philosophers besides Kant 
tiave drawn, between space and time as forms of perception of 
external and internal experience is one which can no longer be 
maintained in respect of physical space and time, although it can 
For perceptual space and time. 

This space-time unity of the theory of relativity figures yery 
prominently in^e-pliilosopill^ii system of Alexander (1859-1938), 
for he supposes tKat it is the primordial reality out of which all 
tEings have evolved. He conjectures that the most primitive^ as 
dso the simplest, land prstuff in the world is pure space-timejjaiit 
jfthis various kinds of matter emerge and, gradually risinglidgher, 
develop into life, consciousness and Deity in turn. All thereon- 


tinental thinkers whpm we have mentioned 

time as creations of mind, but for Alexander mind is a creation of 
sgace andjime. 

Representation in Space and Time 

As a final remark, science and the various materialistic philo- 
sophies have proceeded for centuries on the supposition that all 
objects and all events, and indeed the whole universe, can be 
arranged in space and time. Quite recently science has found that 
such an arrangement is inadequate. The rays of light, waves of 
sound, and the various other messengers that bring us information 
as to the happenings of the outer world may quite properly be 
regarded as travelling in space and time ; such a representation is 
self-consistent, makes sense, and gives a rational account of our 
perceptions. But we shall see below that we are hardly free to 
depict the events which despatch these messengers as happenings 
in space and time; such an interpretation does not make sense or 
lead to a rational view of the universe. We find there is something 
in reality which does not permit of representation in space and 
time. Thus space and time cannot contain the whole of reality, 
but only the messengers from reality to our senses. 


Besides the two forms of perception space and time which we 
have just discussed, Kant's fourteen mental sieves comprised twelve 
categories, or ' forms of* understanding'. There is no need to enter 
into any detailed discussion of these categories, for while eleven 
of the twelve may or may not be of interest to logic, they are of 
no interest to science. One only makes any kind of contact with 
science, and this is the category of Causality; Kant thinks that 
our minds are so constituted that we see all sequences of events in 
terms of the cause-effect relation. 

Categories figure in a somewhat different manner in other philo- 
sophical systems. Aristotle regarded them as forms of structure, 
not of the mind but of the world. For Hegel they are forms of 
thought in the Absolute mind, while Alexander returns to the 
Aristotelian conception of categories as forms of the world itself. 


Up to the present, the conclusions of philosophy have all been 
reached by minds which have all been of one type the human 
type contemplating their perceptions of one and the same world. 
So long as there is only one type of mind contemplating one world, 
there can obviously be no means of deciding whether Kant's forms 
of perception and understanding result from the structure of the 
world or from the structure of the mind which perceives the 

But we have seen (p. 43) how science has just presented us with 
two hew worlds. The world of modern science can be divided into 
three fairly distinct 'divisions a man-sized world in the centre 
Hanked By the minute world of atomic physics on the one side, and 
the vast-scaled world of astronomy on the other. The same laws of 
nature prevail in all three divisions, but different aspects oTthem 
assume prominence in each, to th^almogt complete raclujdon of all 
other aspects, so that we~may almost regard the three divisions as 
tGeejdifferent worlds, with different setsjof Jawsjn^each. But the 
human minds wlilch study them are "the same in each case, and so 
must contribute the same modes of thought to the study of each. 

The two new worlds have already provided us with a testing 
ground for a priori knowledge. If this really represented some 
inborn quality of the mind, we should have found its assertions 
true in all worlds; actually most of them prove to be true only in 
that world which we can see and study without instrumental aid. 
We accordingly concluded that such knowledge was found in the 
human mind, not because it was born there, but as a sort of sedi- 
ment left by the flow of experience of the man-sized world through 
our minds. Residence in the worlds of the electron or of the 
nebulae would have left quite a different sediment in our minds, 
which those of us who were rationalists would then have announced 
as a priori knowledge. 

A test of a similar kind can be applied to Kant's theory of 
knowledge. For, as we shall see in subsequent chapters, those 
forms of perception and understanding which are of scientific 
interest namely causality and the possibility of representation 
in space and time prevail in the man-sized world, but not 
in the small-scale world of atomic physics which we know only 


through instrumental study. If they really were contributions of 
the human mind to nature, they would be contributed to all three 
worlds equally. But as they are not contributed to all three worlds, 
we may conclude that they are not inborn modes of human thought. 
Again they would seem to be ingrained rather than inborn, not so 
much laws that we thrust on nature as laws that we with our 
limited knowledge of the world have allowed nature to thrust on 
us. We think everything can be located in space and time because 
the world that we perceive with our unaided senses seenis to 
admit of location in space and time; the reason is not that things 
are so located, but that the messengers from them to our sense- 
organs travel through space and time (p. 139). In the same way 
we think we see cause and effect running through everything, 
because the phenomena of the man-sized world seem to conform 
to a law of causality; here again the reason is not that they do 
so conform, but that they obey statistical laws which produce an 
impression of causality on our coarse-grained organs of per- 
ception (p. 131). Our experiences of our man-sized world create 
in our minds habits of thought which take causality and space- 
time representation for granted. We cannot imagine anything else 
because we have never experienced anything else. 

If this is so, Kant's forms of perception and understanding are not 
so much blinkers which restrict our knowledge of the outer world 
as lenses which condense our knowledge. But the knowledge they 
condense is knowledge only of the man-sized world, being crystal- 
lized experience of this world alone ; denizens of the underworld of 
atoms and electrons would have had other experiences, and a Kant 
of this underworld, even though endowed with a mental consti- 
tution just like our own, would have produced other categories and 
other forms of intuition. 

In any case, it is probably fair to say that all that modern 
philosophy retains of Kant's theories on this subject is the possi- 
bility of certain forms of thought whether inborn or ingrained 
hardly matters causing our particular type of mind to select what 
it does rather than something else. Our own minds contribute 
something to the nature they study a view, incidentally, which 
dates back to Nicholas of Cusa and the fifteenth century. 


r,ven this remnant means little, unless we concede the possibility 
of a priori knowledge about the external universe. Kant's whole 
theory was an ad hoc structure designed to remove an obvious 
difficulty about a priori knowledge, and if a priori knowledge passes 
away, the need for, and to some extent the importance of, this 
theory passes away with it. 

At the same time a priori knowledge was itself, in a sense, an ad 
hoc structure designed to help metaphysics in its self-appointed 
task of championing the doctrines of theology. It can hardly have 
mattered much to Descartes or Kant whether they knew that 
the sum of the three angles of a triangle was 180 by having proved 
it in their minds, or having measured it with their instruments, or 
having seen it by the clear light of reason. Their primary interest 
was in the question of principle ; they wanted to be able to claim 
that they were possessed of knowledge which was unchallengeable 
because it had not reached them through the deceitful gates of the 
senses. And the kind of knowledge they wanted to claim was not 
knowledge about triangles, but about GOD, FREEDOM and IM- 
MORTALITY. They wanted for instance to be able to say that, 
science or no science, the will was free because they saw it to be so 
by the clear vision of their intellects. 

With the passing of this special phase of philosophy, a priori 
knowledge lost its special significance, and, apart from mathematical 
knowledge, few philosophers have much to say in its favour to-day; 
at least it is generally conceded that it is of little consequence. Yet, 
just when a priori knowledge has become discredited in philosophy, 
an attempt has been made to revive it in physics. 


We hare seen how Kant thought that we ought to be able to build 
up a 'pure-eeience of nature^ solely by the use of the a priori 
knowledge inborn in our minds. This amounted to claiming that 
the world could only be of one kind or rather could only appear 
in one way to us, with our minds constituted as they are. Keeping 
our minds as they are, the Creator could not have made the world 
appear different to us from what it does. 


Sir Arthur 'Eddingf oft also thinkS that we ought to be able to 
build up what we may Describe as a pure science of nature from 
a priori knowledge^ but he thinks of this a priori knowledge as 
epistemologje^Trather than as inboM^In other words, he thinks we 
should find logical inconsistencies in reaching any other conclu- 
sions about the physical world than those which the physicists have 
actually reached from centuries of toil in their laboratories. It 
should be explained that this claim applies only to the general laws 
of nature and not to individual objects in nature, and also 'that 
when Eddington speaks of nature, he is concerned only with nature 
as it appears to us, and not with an objective nature outside our- 

The general point of view will best be understood in terms of a 
specific example. 

We have already seen that, if light travelled with infjriite velocity, 
it would be a simple matter in principle to synchronize all the clocks 
in the universe. The method would be as simple as that of setting 
our watch by Big Ben, and we could call in the help of telescopes as 
needed. But, as light does not travel with infinite velocity, we 
cannot synchronize distant clocks in this way; we must allow for 
the time light takes to travel from one clock to another, and the 
theory of relativity has made it clear that the synchronization of 
distant clocks, if it coufd be achieved at all, would call for a far more 
elaborate technique than looking through telescopes at distant 

~Th the years 1887-1^05 a great number of experiments were 
performecT for another purpose, any one of which might have re- 
sfffted in the discovery of such a technique. But none of themj^ 
'^^ the ' s^i^ronization of distant 

clocks is an impossiBifify. It is not impossible in the sense in which 
iris" impossible to By an aeroplane at 1000 miles an hour i.e. 
because our technical skill is not yet sufficiently advanced -but 

is" impblsiH^^ 

the moon i.el because, ew otisiwatwnKas sfiown, nature provides 
us*witK nothing on which we can get a hold, no resistant medium 
to support our aeroplane. Tlxe main result of the physical theory 
of relativity is usually expressed in the form that it is impossible to 


determine an absolute velocity in space, but might almostj[npt 
quile)l>qu3^ in the form that it is impossible to 

synchronize distant clocts. 

As a matter of historical fact, this conclusion was reached as a 
generalization from a very large number of experiments. Let 
us, however, imagine a race of beings who know without experi- 
menting that it is impossible ever to synchronise distant clocks to 
avoid cumbersome repetition, let us agree to describe them as 
Tiese beings would not dream of performing the 

'wEoFeset oiexperiments just mentioned, because their innate con- 
victions would tell them the results without. If they had a Kant, he 
would describe this knowledge as a priori knowledge. If they had a 
Descartes, he would point out that this knowledge, being indepen- 
dent of all experience, could claim a higher degree of certainty than 
if it had been derived from a finite number of experiments, any 
generalization from which might be negatived by further experi- 

Now Eddington claims, in brief, that we are ourselves asynchro- 
nists, that we have knowledge in our minds as to the impossibility 
of synchronizing distant clocks. Like Kant he describes this know- 
ledge as a priori * knowledge we have of the physical universe 
prior to actual observation of it'; like Descartes he claims for it a 
higher degree of certainty than can be possessed by knowledge 
derived from experiment 'generalizations that can be reached 
epistemologically have a security which is denied to those that can 
only be reached empirically'. This a priori, or epistemological, 
knowledge is not confined to asynchronism; this is merely a some- 
what trivial example. Again like Kant (p. 35), Eddington believes 
that ' all the laws of nature that are usually classed as fundamental 
can be foreseen wholly from epistemological considerations' and 
further that ' not only the laws of nature but also the constants of 
nature can be deduced from epistemological considerations, so that 
we can have a priori knowledge of them'. It follows that 'an 
intelligence unacquainted with our universe, but acquainted with 
the system of thought by which the human mind interprets to 
itself the content of its sensory experience, should be able to attain 
all the knowledge of physics that we have attained by experiment. 


He would not deduce the particular events and objects of our 
experience, but he would deduce the generalizations we have based 
on them/ 

Thus for Eddington knowledge of this fundamental kind results 
from the constitution of our minds, which are thus once again 
rehabilitated as law-givers to nature in the Kantian sense. We need 
never have built physical laboratories, except to study matters of 
detail; it would have been better to have delved into our own 
minds, where we should have found the results of all the funda- 
mental experiments of physics, together with the values of the 
fundamental constants of physics. Eddington goes on to remind 
us that ' whatever is accounted for epistemologically is ipso facto 
subjective ; it is demolished as part of the objective world \ Funda- 
mental physics, then, tells us something about our own minds, but 
nothing about the outer world. To use one of Eddington's own 
metaphors: 'When science has progressed the furthest, the mind 
has but regained from nature what the mind has put into nature. 
We have found a strange footprint on the shores of the unknown. 
We have devised profound theories, one after another, to account 
for its origin. At last, we have .succeeded in reconstructing the 
creature that made the footprint, and lo ! it is our own/ 

Eddington's claim that the fundamental laws of physics can be 
foreseen epistemologically would carry more conviction if he could 
himself establish any one of them, even the simplest, epistemo- 
logically in other words, if he could show that there would be a 
logical inconsistency in believing the laws to be different from what 
they are. This he never does. 

It seems improbable that he ever could, for surely to speak of 
establishing any fact of science by epistemology alone involves a con- 
tradiction in terms. Epistemology has only one tool in its armoury. 
This is pure logic, and before it can be applied to a scientific fact, 
we must define the scientific objects about' which the fact is stated. 
We can only do this by calling upon knowledge which has been 
obtained empirically. In so doing we pass beyond the realm of 
a priori knowledge, and our discussion ceases to be purely epis- 

To illustrate by a concrete case, Eddington believes it is possible 


to establish epistemologically that the mass of the proton must be 
1847 time^ that of the electron. Clearly, though, he must be care- 
ful to avoid proving at the same time that the mass of the apple is 
1847 times that of the orange; if his argument proves this, we shall 
feel suspicious of it. He can escape this danger by defining his 
electrons and protons in a. way which makes it clear that they are 
not apples and oranges. Actually he neglects to do this, with the 
result that, in so far as his proof of the 1847 ratio is epistemological, 
it is equally applicable to apples and oranges. 

Of course Eddington is entirely justified in assuming that we know 
what he means by electrons and protons, but what about the 
visiting intelligence from another universe? Will he not be in the 
position of the lecturee who said the lecturer had explained beauti- 
fully how the astronomers discovered the sizes and temperatures 
and masses of the stars, but had forgotten to explain how they 
found out their names? He will not know the difference between an 
apple and an electron until we tell him, and before we could do 
this, we should have to acquaint him with whole masses of labora- 
tory knowledge, and epistemology would be left far behind. For 
the visitor is supposed to be acquainted only with our system of 
thought, and can it be seriously maintained that this includes the 
knowledge that the world is made up of similar fundamental par- 
ticles of two and only two kinds? So far from being an innate part 
of our mental equipment, this is a hypothesis that did not even 
enter science until a few years ago (and incidentally left it again, 
very hurriedly, a few years later). 

It is in fact necessary to build a bridge between the abstractions 
of epistemology and the actualities of observed phenomena; without 
this epistemology is left up in the air, and cannot know what it 
is talking about. Kant did this by introducing his synthetic a priori 
knowledge; Eddington does it by withdrawing his claim that his 
a priori knowledge is ""knowledge that we have of the physical 
universe prior to actual observation of it', and writing instead that 
'to the question whether it can be regarded as independent of 
observational experience altogether, we must, I think, answer: No'. 
But this admission obviously weakens his position enormously; his 
natural laws are no longer foreseen 'wholly from episteftiological 


considerations', but only from a mixture in unknown and un- 
knowable proportions of these and observation, which means 
simply observation combined with sound reasoning. And surely 
this is just the ordinary procedure of all science. Eddington's laws, 
being no longer reached by pure epistemology, must renounce their 
claims to pure subjectivity, and to ' a security that is denied to those 
[laws] that can only be reached empirically*. They become ordinary 
scientific laws, obtained in the ordinary scientific way, and the only 
question is whether the mathematics is right or wrong. 

A simple test case is provided by the finite velocity of light. 
We introduced Eddington's philosophy, as he himself has done, 
by considering the impossibility of synchronizing distant clocks. 
The reason why such synchronization is impossible is that light 
does not pass instantaneously from place to place. Those, then, 
who believe it is possible to prove all the fundamental laws of 
nature from epistemological considerations, ought to find it possible 
to prove in this way that the velocity of light is finite i.e. they 
ought to be able to point to some logical inconsistency involved 
in the idea of light travelling with an infinite velocity. Eddington, 
however, merely dismisses the question with the statement that it 
is absurd to think of the speed of light as infinite as absurd, he 
says, as to think of it as hexagonal or blue or totalitarian. 

So long as we look at the question from the purely epistemo- 
logical point of view forgetting all that experience has taught us 
about space, time and propagation it is hard to find anything 
absurd in the idea of instantaneous propagation. Prof. A. Wolf 
writes that ' down to the seventeenth century [the velocity of light] 
had usually been regarded as infinite, and Kepler, and perhaps 
also Descartes, seem to have held this view. Descartes. . .believed 
that light was not a moving substance, nor a motion at all, but 
a tendency to motion, or a thrust exerted by the luminous body: 
and he supposed that this thrust, being incorporeal, required no 
time for its propagation/ Ift the same way, most people still 
think of the thrust of an iron bar as an example of instantaneous 
propagation. Newton and his contemporaries took it for granted 
that gravitation was propagated instantaneously; it was over a 
century later that the alternative possibility of a finite speed 


of transmission was first considered by Laplace not because it 
seemed inherently probable to him, but because he wished to leave 
no avenue unexplored which might solve the mystery of the moon's 
acceleration. And when the first observational evidence (p. 63) of 
the finite speed of light was produced by Roemer, it was hailed 
as a sensational new discovery not as confirmation of something 
that had been known all the time as a matter of course. Indeed, 
for a time it was rejected by many of Roemer's contemporaries 
who continued to believe in the infinite velocity of light. 

All this seems to show that there can be nothing epistemo- 
logically absurd in the idea of an infinite velocity of propagation. 

Even if it could be conceded that we have a priori knowledge 
that light travels with only finite velocity, it would still be a long 
step further to the fundamental postulates of the theory of relati- 
vity, and Eddington claims these also as a priori knowledge. Sixty 
years ago physicists were almost unanimous in imagining space to 
be filled with an ether through which waves travelled at the finite 
speed of 186,000 miles a second. This constituted a perfectly self- 
consistent scheme, it made sense, and explained all the phenomena 
as then known, so that, so far as epistemological considerations 
went, it was entirely eligible as a possible explanation of the pheno- 
mena; it had to be abandoned only because experiment decided 
against it. If these experiments had turned out otherwise than as 
they did and we can easily imagine them doing so this scheme 
would probably still have prevailed. This of itself gives a sufficient 
proof that no epistemological arguments compel the abandonment 
of this scheme, whence it follows that none can require the accept- 
ance of the opposite scheme, which is that of the theory of 
relativity. Indeed as this latter scheme is purely a generalization 
from the results of a large number of experiments, there is still 
a possibility in principle although not much probability that 
further experiments may still be found to compel its abandonment* 

An Alternative View 

There is an alternative way of regarding the matter which would 
seem to be more true to the facts. 

Borrowing a simile from Poincar, we have already compared 


the construction of a science to the building of a house. Our stones 
are a collection of facts of observation. Just because nature is 
rational, we find that these can be made into something other than 
a mere shapeless pile; they show definite regularities, and so can be 
fitted together to form a house with definite characteristic features. 

It will be possible to describe these characteristic features in 
simple terms which will evoke a ready response in our minds; 
indeed we can describe them in terms of ideas which are already in 
our minds and familiar to our minds. They are familiar, not be- 
cause we are familiar with the general laws of physics, but because 
we are familiar with special and restricted instances of them; it is 
of such that our daily lives are made up. We may, for instance, say 
that the house shows no unnecessary ornamentation (Occam's 
razor) or no cracks (conservation laws). The ideas of ornamentation 
and cracks are not innate in our minds, but have been acquired 
from experience in very special small corners of the world. 

Now the design of this house is nothing other than the pattern of 
events which it is the aim of physics to discover. The physicist 
finds after sweat and toil in the laboratory that this pattern of 
events shows features like those we have attributed to our house. 
There is no doubt that a great part (and perhaps all) of the funda- 
mental facts of physics can, when once they have been discovered 
empirically, be summed up in general statements which seem very 
simple and intelligible to us because we are familiar with detailed 
instances of them. These can often (perhaps always) be expressed 
in the form of what IJ. T. Whittaker has called 'jPostulatesjof^ 
impotence ', these asserting 'the impossibility of achieving some- 
thing, even though there may be an infinite number of ways of 
trying to achieve it'. It is, for instance, impossible to get mecha- 
nical work out of matter which is at a lower temperature than the 
surrounding objects, and impossible ever to measure an absolute 
velocity in space. These two postulates of impotence contain prac- 
tically the whole contents of thermodynamics and of the physical 
theory of relativity respectively. 

Hence, as Whittaker has remarked, * It seems possible that, while 
physics must continue to progress by building on experiments, 
any branch of it which is in a highly developed state may be ex- 


hibited as a set of logical deductions from postulates of impotence, 
as has already happened to thermodynamics. We may therefore 
conjecturally look forward to a time in the future when a treatise 
on any branch of physics could, if so desired, be written in the same 
style as Euclid's Elements of Geometry, beginning with some a priori 
principles, namely postulates of impotence, and then deriving 
everything else from them by syllogistic reasoning/ 

These principles would not of course be a priori in Kant's sense 
of 'pre-observation'; they would be very much a posteriori, being 
the highly concentrated extracts of immense masses of observations. 
But we can imagine a scientist pondering over their simplicity until 
they became endowed in his eyes with a quality of 'inevitableness', 
and he would begin to regard them as laws of thought. In a sense 
they would have become laws of thought for him. 

This, we may conjecture, is what Eddington has done. And, 
just as the true nature of Kant's supposed categories of thought is 
disclosed by experiments on the atomic world, which show that 
causality and space-time representation no longer prevail there, so 
at any time a new experiment may show that Eddington's supposed 
a priori principles are mere mental sediments left over from actual 
experience of the world. Indeed to some extent the discovery of 
positrons has done this already. 


Our discussion seems to bring us back to the age-old conclusion 
that if we wish to discover the truth about nature the pattern of 
events in the universe we inhabit the only sound method is to go 
out into the world and question nature directly, and this is the long- 
established and well-tried method of science. Questioning our own 
minds is of no use ; just as questioning nature can tell us truths 
only about nature, so questioning our own minds will tell us only 
truths about our own minds. 

The general recognition of this has brought (bhilofcophy into 
closer relations with science, and this approach has coincided with 
a change of view as to the proper aims of philosophy. The ancient 
jgMo^phgrg^pursued their studies in the hope of finding a lantern 
which should guide their feet along the best path in their journey 


and eighteenth 

centuriesjnji fixed^dgtermination to find evidencejEatlhjsJoutlley 
ejidei33nIaJieJ:o come. This humanistic tinge has taken a long time 
to disappear, but has almost done so in recent years; philosoph 
has become less concerned with ourselves and more concerned wltn 
the universe outside ourselves. It is now recognized that, in Ber- 
trand Russell's words: 'Man on his own account is not the true 
subject-matter of philosophy. What concerns philosophy is the 
universe as a whole; man demands consideration solely as the 
instrument by means of which we acquire knowledge of the uni- 
verse . . . .We are not in a mood proper to philosophy so long as we 
are interested in the world only as it affects human beings; the 
philosophic spirit demands an interest in the world for its own 

This may seem to suggest that philosophy should have not only 
the same methods but also the same aims and also, broadly speaking, 
the same field of work as science. But the distinction mentioned at 
the beginning of the present chapter still holds good. The tools of 
science are observation and experiment; the tools of philosophy are 
discussion and contemplation. It is still for science tojtry to dis- 
cover thgjjttern of events, and for pjiilosophy^o try "tolhteroret it 
when found. 





We have seen that knowledge of the external world can come only 
through observation and experiment. These tell us that the world 
is rational its events follow one another according to definite laws, 
and so form a regular pattern, ^he primary aim of physicsJsJhe 
disco veryrgf this pattern; we have seen that it can be described only 
ii\ mathematical language/ 

We have seen that physics cannot clothe the mathematical 
symbols of this description with their true physical meaning, but 
physics and philosophy may properly engage in joint discussion as 
to their possible meanings, and the most provable interpretation 
Df the pattern of events. Yet there are many hindrances to such 
discussion. In the present chapter we shall try to unearth some of 
these and eliminate them with a view to clearing the ground for the 
discussions which are to follow. 


Foremost among these hindrances are differences of language and 
of terminology; when science and philosophy are not speaking 
entirely different languages, they often seem at least to employ 
different idioms. 

More than three hundred years have elapsed since Francis Bacon 
wrote of the ' Idols ' which beset men's minds when they try to 
discover truth. The most troublesome of these, he said, are the 
idols of the market-place, the place where men meet to talk with 
sne another. For words are unsuited to the expression of accurate 
3r scientific thought, and apparent differences of opinion often 
result from inadequate definition of the terms ertiployed in the 


In the intervening period science has constructed its own lan- 
guage, or jargon as some may prefer to call it. Unbeautiful though 
it may be at times, it has the great merit of exactitude ; generally 
speaking, its terms are clearly and unambiguously defined, so that 
each word means the same thing to every scientist, and this thing 
is perfectly precise. When a physicist reads a sentence of Newton 
or Einstein, he may rar may not understand thelrneaning^ of the 
sentence, but JuT is ill no douHt as to the meaning of the 

As science advances, new accessions to knowledge are continually 
being interwoven into its terminology, with the result that this 
contftiually gains in richness and precision. Here a group of new 
words will be necessitated by a group of new facts; there a modifi- 
cation in the usage of old words is called for by new knowledge 
of old facts. For instance the new knowledge introduced by the 
theory of relativity compelled us to modify our use of the words 
'motion', 'velocity', 'simultaneity', 'interval of time', and so on. 

There is nothing to correspond to this in philosophy, which still 
has no precise or agreed terminology. A great number of common 
words, as well as more technical terms, are used in a variety of 
different senses, often by the same writer. And even when philo- 
sophy uses a word in a precise and unique sense, this sense is often 
different from that of science. 

This not only constitutes a serious hindrance to discussion be- 
tween science and philosophy, but may even obscure the issue in 
purely philosophical problems. Indeed it is hardly too much to 
say that a large proportion of the puzzles and problems of the 
philosophy of the past owe their very existence to the imperfections 
of language. Many of these old problems look very different when 
translated into the idiom or language of science, while some vanish 
away in the process of translation. 

There seem to be three principal causes for these differences of 
language and usage; it may be well to enumerate these first, and 
discuss them in detail, with examples, afterwards. 

I. Philosophy seems to have no agreed or precise terminology 
because there is no agreed body of fundamental knowledge for a 
precise terminology to describe. 



II. The language ot^)hiio$Ogy differs from foat of science 
largely because philosophy tendsto use words in subjective, and 
science in objective, senses. 

III. The language of philosophy further differs from that of 
science because philosophy tends to think in terms of facts as they 
are revealed by our primitive senses, while science thinks of them 
as they are revealed by instruments of precision. 

As a preliminary to considering the first suggested cause, let us 
notice that science also had no agreed or precise terminology until 
it had something agreed and precise to describe. We have had 
an illustration of this on p. 25, where we saw the word Motion* 
used in a very indefinite sense. Indeed three centuries ago 
there was a general confusion of thought between the three dis- 
tinct measures that are now described as velocity, momentum and 
energy, and the same word 'motion' was often used to denote all 
three. It is the same now in those departments of science in which 
the fundamental facts are still under discussion; for instance, 
Eddington remarks that 'the terminology of the quantum theory is 
now in such utter confusion that it is well nigh impossible to make 
clear statements in it*. 

A large part of philosophical terminology has always been in a 
corresponding state, and it may perhaps be argued that such a 
state is inevitable now, and will remain inevitable until philosophers 
can agree on their fundamental facts. Still, there can be other 
opinions about this. For fifty years, off and on, Leibniz was trying 
to devise a precise technical language and r to construct a calculus 
for philosophy. He hoped to find that all the fundamental ideas of 
reasoning could be reduced to a very small number of primitive 
elements or 'root-notions', each of which could then be desig- 
nated by a universal character or symbol like the symbols of 
algebra. If once this could be done, it ought to be possible to 
construct a calculus for the operation of these symbols. Leibniz 
considered that such a calculus would settle disputes between 
philosophers as easily as arithmetic settles disputes between ac- 
countants; if two disagreed, they would simply say, 'let us reckon 
it out', and sit down with their pens. But his efforts failed, and 
more recent attempts of the same kind have been applicable at most 


to small regions of the whole province of thought. The result is that 
philosophy still struggles to express itself in the inadequate words 
of common speech. It is still true, as Anatole France said, that 
*un metaphysicien n'a, pour constituer le systme du monde, que 
le cri perfectionne des singes et des chiens*. 

Yet the major problems of philosophy are for the most part very 
difficult; many of them tax the human mind to the utmost limits of 
its capacity, and have baffled the most acute intellects of our race 
for thousands of years indeed it is hardly too much to say that not 
one of them has been solved yet. In discussing these problems we 
have to deal with subtle and delicate shades of meaning, and to 
travel in fields of thought which are far removed from those of our 
everyday life; this would seem to demand a perfectly precise, 
perfectly flexible and perfectly refined instrument. Ordinary lan- 
guage is none of these things; it is a rough practical tool which the 
common man, or the unthinking savage before him, has developed 
from his first rough contacts with the world to express the ideas 
which arose out of those contacts. It would surely be an amazing 
coincidence if such a tool should be found suited for abstract 
discussions which have but little to do with the world of everyday 
experience. We might as well expect a surgeon to perform a delicate 
surgical operation with carpenters' tools spokeshaves, chisels and 

The inadequacy of popular language to express the subtleties of 
philosophic thought is well illustrated by the famous proposition 
of Descartes cqgito ergo sum. Descartes, believing this proposition 
to be true beyond all shadow of doubt, proposed basing the whole 
of philosophy on it. A later generation of philosophers has pointed 
out the inadequacy of the proposition, and their criticism is based 
mainly on Descartes' use of common language. For this compelled 
the subject of the proposition to fall into one of three clear-cut 
categories cogito, cogitas, cogitat or their plurals; if the thinking 
does not fit into one of these moulds, common language cannot 
express it. Anything of the nature of telepathy, for instance, is 
ruled out from the outset, not on the grounds that it cannot or does 
not occur, but simply because common language cannot cope with 
it; this makes thinking the prerogative of detached personalities. 


But even detached personalities change with every experience; I, 
who have thought, am different from that other I who existed 
before the thought came to me. And again the tenses of language 
sum, fuij eram, ero are totally inadequate to express the infinite 
gradations of change. 

Bertrand Russell says that ' grammar and ordinary language are 
bad guides to metaphysics. A great book might be written showing 
the influence of syntax on philosophy/ In illustration he mentions 
Descartes, who 'thought that there could not be motion unless 
something moves, nor thinking unless someone thought. No doubt 
most people would still hold this view; but in fact it springs from 
a notion usually unconscious that the categories of grammar 
are also the categories of reality. 1 We can find a more modern 
illustration of the same tendency in the physics of the eighteenth 
and nineteenth centuries. When it had become clear that light was 
of an undulatory nature, physicists argued that if there were un- 
dulations, there must be something to undulate one cannot have 
a verb without a noun. And so the luminiferous ether became 
established in scientific thought as 'the nominative of the verb to 
undulate ', and misled physics for over a century. 

Even when philosophical writers all use a word in the same 
sense, their usage is often different from that of science, and this 
brings us to the second of our suggested causes. Until recently 
science has taken it for granted that there exists an objective world 
entirely apart from and outside our minds, and has designed its 
terminology for the description of such an'objective world. Philo- 
sophy has never taken such a world for granted, although individual 
philosophers may have argued for it; on the contrary it has realized 
that its primary concern must be with the sensations and ideas in 
our minds, which suggest to us that such a world exists. Hence an 
obvious tendency for science to use words in an objective, and 
philosophy in a subjective, sense. 

As examples of this difference of usage let us consider the verb 
see and the adjective red. 

The scientist's use of the word see is quite definite; when he says 
that he sees Sinus, he means that he believes that Sirius exists 
outside his mind, and that rays of light which have come from 


Sirius are forming an image of Sirius on his retina and thereby 
affecting his brain. If a drunkard says he sees purple snakes, the 
scientist objects that he cannot see purple snakes because there are 
none to see to the scientist the essence of seeing is the passage of 
rays of light from the object seen to the retina of him who sees. 

Many philosophers object to this. They point out that, when I 
say I am seeing Sirius, I-am claiming to see something which may 
no longer exist, since it may have disappeared in the eight years 
which have elapsed since the light left Sirius. Bertrand Russell 
considers it as incorrect to say you see a star when you only see the 
light from it as to say that you see New Zealand when you see a 
New Zealander in London, He treats the case of a physiologist 
examining the brain of his patient in the same way ; most people, 
he maintains, would say that what the physiologist sees is in the 
brain of the patient, but the philosopher must insist that actually it 
is in the brain of the physiologist. On this view, the drunkard can 
really see purple snakes in his bedroom, but the sober man can 
never see green snakes in the grass, because they may have gone out 
of existence while their light was travelling to his eyes. In brief, 
the philosophers consider we can only see things which are inside 
our heads, while the scientists, following the more ordinary use of 
language, consider we can only see things which are outside our 

The adjective red is used in science to describe light which 
possesses quite definite objective properties; these can be specified 
by mentioning a number of complete waves to the inch or of 
complete oscillations to the second the two definitions are exactly 
equivalent. When light so specified falls on a normal human eye, it 
produces what we describe as a sensation of redness. 

The mechanism by which it does this is still imperfectly under- 
stood, but appears to be somewhat as follows. The optic nerve of 
the human eye is a bundle of nerve fibres which terminate in the 
retina in the form of rods and cones. When light falls on these 
nerve-endings, chemical changes occur in them which send certain 
electric activities along the nerve fibres to the brain ; these produce 
sensations of light or colour in the mind. The rods are stimulated 
by light of any colour, even though it be very faint it is through 


these that we see at night or in dim light but they produce 
sensations only of light and shade, and not of colour. Stimulation 
of the cones, on the other hand, produces definite sensations of 
colour. If the rods are in an unsatisfactory state, we suffer from 
night-blindness; if the cones, from colour-blindness. 

The development of the cones is determined by certain heredi- 
tary elements which are believed to reside in a special chromosome 
(the X-chromosome), of which every man has one, and every 
woman two, in each cell of their bodies. In western Europe, about 
one man in forty started life at his conception with this hereditary 
element defective, and so is permanently and unalterably colour- 
blind; a woman is only colour-blind if she has two hereditary 
elements defective, so that only one woman in several hundreds is 

Apart from man, it is believed that very few of the larger animals 
are endowed with colour vision; most of them see the world only 
as a series of contrasts of light and darkness somewhat as we see 
it by moonlight. The human sensation of redness is the origin 
of our conception of redness as a quality, but provides only a rough 
test for redness ; the true test is by a set of completely inanimate 
instruments spectroscope, camera and photographic plate. 

When a scientist says that a flower or a motor bus is red, he 
means that any light that they reflect is red in the scientific sense as 
defined above. When sunlight, which is a blend of light of many 
colours, falls on a red flower, the petals of the flower reflect the red 
constituent of the light, and this constituent alone, into my eyes so 
that I see the flower by red light. If I have normal vision, this 
produces a sensation of redness in my mind, and I say that the 
flower is red. If I have not normal vision, but am colour-blind to 
red, I shall still see the flower by light which is red in the scientific 
sense of the word, although my colour-blindness may result in the 
light appearing of a different hue, or making very little impression 
on my retina; I may see it as a dull, instead of a vivid, red. 

But when a philosopher says an object is red, he usually means 
that it produces a sensation of redness in his own, or in someone 
else's, eye. As with the word see we previously discussed, the 
scientist applies the adjective red to something objective outside his 


head primarily to light while the philosopher applies it to some- 
thing inside his head primarily to a colour-sensation. Thus 
colour-blindness can alter colours in the philosophic sense, but not 
in the scientific. 


In addition to such crude and rudimentary difficulties of pure 
language, further difficulties originate in the different idioms em- 
ployed by the philosopher and the scientist. Not only do they 
express their thoughts in different languages, but the thoughts 
themselves tend to run on different lines of rails. This seems to 
result, at least in part, from the third and last of our suggested 
causes. The philosophers still think in a way which dates back to 
the earliest days of their subject, to times when no instruments of 
measurement were available of greater precision than the five 
human senses; they still describe things in terms of the effects they 
produce on these senses, while the scientist describes them in 
terms of the effects they produce on his sensitive instruments of 
measurement. The philosopher not only speaks but thinks in 
subjective, and the scientist in objective, terms. 

Quantities and Qualities 

^ "' - "" . - ^ / (/ 

One of the more obviou^ 

usually thinks in terms ^gualkjes, the scientist in terms^of quan- 
titles. The philosophical lectureFmay be telling his audience that a 
lump of sugar possesses the qualities of hardness, whiteness and 
sweetness, while his colleague in the science room next door may 
be explaining coefficients of rigidity, of reflection oTTight and 
hydrogen-ion concentration measures ofme degree to whichjthe 
qualities^QJhar^ness, whiteness anS^weetness^^possessed. While 
the philosophical lecturer argues on the supposition that hot and 
cold are incompatibles, so that no object can be hot and cold at the 
same time, the science lecturer discourses on temperature, which 
not only measures the infinite gradations of what his philosophical 
colleague describes as hotness or coldness, but also bridges a gulf 
which the latter still treats as unbridgeable. 

The consequences of this can be illustrated in terms of a simple 


philosophical argument which had a very long innings in different 
suits of clothes lasting the 2000 years from Plato through Berkeley 
to Bradley. It runs somewhat as follows: 

We are in a comfortable room when a man A comes in from a 
snowstorm outside and says, 'It is warm in here'. Another man B 
then comes in from a Turkish bath and says, 'It is cold in here*. 
The argument proceeds to Assert that as the room cannot be warm 
and cold at the same time, the heat and cold cannot be real qualities 
of the room, but can only be ideas in the minds of A and B: Two 
other men C and D now come in, the one from a palace and the 
other from an Anderson shelter, and remark respectively that the 
room is small and large. As the room itself cannot be large and 
small at the same time so the argument runs the largeness and 
smallness can only exist in the minds of C and D ; the room cannot 
have any quality of size in itself. By continued repetition . of 
this argument the room can be stripped of all its qualities in turn, 
and as it is nothing more than the sum of its qualities (so this 
particular argument runs), it disappears entirely except in so far as 
it exists in the minds of A, B y C and D. 

The argument looks very different when it is translated into the 
idiom of science. When A comes in he will say, c It is warmer here 
than outside ', while B will say, * It is colder here than in the Turkish 
bath*. The argument would have to proceed that a room cannot be 
both hotter than a snowstorm and colder than a Turkish bath and 
we see at once that the attempted inference fails entirely. 

Of course we cannot dispose of an argument merely by trans- 
lating it into another idiom, any more than we could disprove the 
propositions of Euclid by translating them into French. There 
must obviously be more in it all than this. 

The argument fails through disregarding the distinction between 
subjective estimates and objective measures of temperature. When 
it says that a room may be deemed hot and cold at the same time, it 
is dealing with subjective heat and cold ; these, it goes on to prove, 
can only be ideas in the minds of A and B. But here it suddenly 
swings over and erroneously identifies them with objective tem- 
peratures. The subjective room may be the sum of its subjective 
qualities, and the objective room the sum of its objective qualities, 


but abolishing all the subjective qualities of the room cannot 
abolish the objective room. Before his argument can stand, the 
philosopher must show that there is no difference between the 
subjective and objective temperatures of a room, and every time he 
tries to do this the thermometer on the mantelpiece will prove him 

The psychologist may put in a word here, since he can tell us 
that our senses are not very good at estimating absolute heat and 
cold; we do not judge that an object is hot or cold, so much as that 
it is hotter than or colder than something else, the comparison 
usually being with the warmth of our own bodies or with our last 
experience with heat or cold. Thus common language speaks of 
marble as cold and of woollen blankets at the same temperature as 
warm, because touching marble makes our hand colder than it has 
been and wrapping it in a blanket makes it hotter than it has been; 
the ultimate reason for this is that marbles are good, and woollen 
materials are bad, conductors of heat. The psychologist knows 
from his laboratory experiments that considerations such as this 
are important, while the philosopher of the old-fashioned type 
apparently did not. Science knows from its observations that its 
own idiom is the correct one to employ. 

Since the time of Aristotle, philosophers have been 'inclined to 
regard substance as something that is wrapped up in a number of 
qualities, much as a package may be wrapped up in a number of 
layers of paper, and have speculated as to what, if anything, will be 
found when all the wrappers are removed. 

Galileo, Descartes, Locke and others imagined that qualities 
could be divided into an outer layer of what Locke described as 
secondary qualities those perceived by the senses, such as redness 
and coldness and an inner layer of primary qualities which a 
substance or an object possesses in its own right and by virtue of its 
mere existence, independently of whether it is perceived or not 
such as solidity and extension in space; these, in Locke's words, 
'are utterly inseparable from the body in what state soever it be'. 

Looked at from the objective viewpoint of science, such a dis- 
tinction appears highly artificial. Redness indicates a capacity for 
reflecting red light, solidity and extension in space a capacity for 


* reflecting 1 any other body which tries to trespass upon the space 
of the body in question. It is not clear why one of these capacities 
should be classified as primary and the other as secondary, one as 
fundamental and the other as superficial. 

The philosopher may protest that to him redness has nothing to 
do with the reflection of light, but means simply a capacity for 
producing a mental sensation of redness. This will not do, since it 
makes the distinction between primary and secondary qualities 
purely subjective. Redness must now be classified as a secondary 
quality for a normal man, but as a primary quality for a blind man, 
who cannot see at all, as also for a dog, who has no colour vision. 
Locke and his fellow philosophers may argue that redness is a 
secondary quality, but a canine philosopher would argue, with 
precisely equal validity, that it was a primary quality. 

The problem is sometimes approached by imagining an object 
to be stripped, one by one, of all the qualities which we can imagine 
being stripped from it. The qualities which we can imagine re- 
moved are of course secondary, the unremovable residuum primary. 
The philosophic lump of sugar, for instance, is pictured as wrapped 
up in its qualities of whiteness, sweetness, hardness and so forth. 
If we strip these away, one after the other, what unstrippable 
residue is finally left? Or is nothing left? Is it true, as was assumed 
in the argument just quoted, that an object is nothing but the sum 
of its qualities? 

Science finds that the qualities of a substance or object depend in 
part on the intrinsic nature of its constituent parts and in part on 
the way in which these constituent parts are arranged in space, its 
physical qualities depending on the mode of arrangement of its 
molecules, and its chemical qualities on the mode of arrangement 
of the atoms of which its molecules are formed. This being so, it is 
meaningless to speak of * stripping* anything of its qualities. The 
most we can do is to rearrange its constituent units, and in so doing 
replace one quality by another for instance, the hardness of ice by 
the liquidity of water or the compressibility of steam, the brilliance 
of diamond by the heavy dullness of graphite or by the deep 
blackness of lampblack. To the scientist all qualities are primary in 
the sense that they 'are utterly inseparable from the body in what 


state soever it be* ; a red tulip is not made less red by being looked 
at in a blue light. 

Again it is not to the point for the philosopher to protest that the 
scientist insists on looking at things objectively, while he, the 
philosopher, is accustomed to keep his thoughts on the subjective 
plane. If he insists that he can easily imagine things stripped of 
their qualities, the reply is that philosophy, just as much as physids, 
is out to obtain knowledge about the real world, and not about an 
imaginable but wholly unreal world in which qualities can be 
stripped away and nothing left in their place ; it is only in Wonder- 
[and that a cat can be stripped of everything but a grin. 


A. second difference of idiom, closely connected with that we have 
just discussed, arises out of the philosophical practice of depicting 
the world entirely in black and white, and so ignoring all the half- 
;ones, gradualness and vagueness which figure so prominently in 
>ur experience of the actual world. The obvious example of this is 
provided by the law of the excluded micjdle, which has dominated 
formal logic, with devastating results, from the time of Aristotls on. 
The law asserts that everything must be either A or not-^4, what- 
ever A may be. The scientist, on the other hand, knowing that 
everything will generally possess some ^4-ness, and some not-^4- 
aess, is very little concerned as to whether an object is classed as A 
3r not-^4 ; what he wants to know is how much ^4-ness it possesses. 
For example, the law asserts that every quantity must be either 
inite or not-finite. If this is so, the half of a finite quantity must 
ilways be finite; it cannot be not-finite, or the sum of two not-finite 
juantities would be finite, which is absurd. Thus in the series of 

I > 2> 4> 4> WT> sfe> ! 

n which each is half of the preceding, every member of the series 
nust be finite no matter how far the series extends. If it continues 
ndefinitely, we have an infinite sequence of quantities each of which 
s finite. The sum of all the members of the series is now the sum 
>f an infinite number of finite quantities, and so must, according 


to the law, be infinite. Yet very simple arithmetic will show that 
the sum is actually finite, being 2. 

This is the fallacy underlying Zeno's well-known paradox of the 
hare and the tortoise. For simplicity, let us suppose that the hare 
goes only twice as fast as the tortoise. Let it give the tortoise a 
start of a minute, during which the tortoise travels from the 
starting-point A to a point B. The hare now starts, and takes half a 
minute to reach B. During this time the tortoise travels a distance 
BC, which is of course half of the distance AB. The hare accord- 
ingly takes a quarter of a minute to traverse the stretch BC. And 
so it goes on, the total time of the race, in minutes, being 

i + i + J + i+... adinf. 

Obviously the series can never end, and as, according to the law, 
it consists of an infinite number of finite terms, the total time of the 
race must be infinite the hare can never catch the tortoise. As 
before, the fallacy lies in the supposition that quantities can be 
sharply divided into finite and not-finite in other words, in the 
law of the excluded middle. 

To turn to a more serious example, the same fallacy lies at the 
root of the so-called pntological proof of the existence of God. 
In the form in which St Anselm originated it, this assumed that a 
being, like an object, must either possess or not possess every con- 
ceivable quality. Thus a Perfect Being must either possess or not 
possess the quality of existence; He must, in fact, possess it, since 
the non-possession of this quality would be an obvious imperfec- 
tion. Hence, runs the argument, a Perfect feeing must really exist. 
The detailed argument, in the form given by the usually clear- 
minded Descartes, ran as follows: 'To say that an attribute is 
contained in the nature or in the concept of a thing is the same as to 
say that this attribute is true of this thing, and that it may be 
affirmed to be in it. But necessary existence is contained, in the 
naturejpr in the concepfat Uod. Hence it may with truth bfi-said 
that .necessary ftrifttfence is m God, or that God exists. ' 
-^H"can almost see the rabbit being put into the hat, and it seems 
strange that such a transparent piece of logical legerdemain could 
impose not only on the confused logicians of the Middle Ages, 


but also on later thinkers of the calibre of Descartes and Leibniz, 
until Kant finally pointed out its logical inadequacy : ' that unfor- 
tunate ontological argument, which neither satisfies the healthy 
common-sense of humanity nor sustains the scientific examination 
of the philosopher*. 

The explanation seems to be that only two different degrees 
of existence were recognized existence and non-existence. The 
argument proves that if we set out to think of a Being endowed 
with every perfection, we must think of Him as really existing 
nothing more. It can never assign a higher degree of existence 
to such a Being than existence in our thoughts ex nihilo nihilfit. 

As soon as the argument is translated into the scientific idiom, 
we are no longer concerned with mere qualities, but with degrees of 
qualities, and if the being is to be identified with the Supreme Being, 
the degree of each can only be infinite. But, as Leibniz pointed out, 
there are pairs of qualities which become incompatible when taken 
in infinite amounts, as infinite justice and infinite mercy. Thus, so 
far as this argument goes, we have no right to imagine such a 
Supreme Being, even in our thoughts. 

The law of the excluded middle entails other disconcerting con- 
sequences of a more practical kind. From it we learn that at every 
moment of his life a man must be either young or not-young, so 
that the transition from young to not-young must occur at a single 
moment of his life. Thus youth passes away in the twinkling of an 
eye, and it is the same with the beauty of a woman and the health of 
an invalid. We reach strange conclusions by following the strait 
path of formal logic. 

In practical affairs all life is a compromise, and most things 
reside in precisely that middle region which the law attempts to 
abolish. This does not in the least interfere with the popularity of 
the law for dialectical purposes: 'Gentlemen, it is surely obvious 
that there is either a shortage of feeding stuffs for pigs, or there is 
no shortage/ 



A natural transition of thought brings us to a third difference of 
idiom, or perhaps rather of method, which has somewhat more 
serious consequences than any so far considered. The metier of the 
philosopher is to synthesize and explain facts already known; that 
of the scientist is in large part to discover new facts. When the 
philosopher finds himself called upon to explain a very complex 
and very unintelligible world, he is tempted to reduce every pro- 
blem to its crudest and barest skeleton by discarding everything 
which does not seem to him to be essential. The scientist, on the 
other hand, ever looking for something new, naturally preserves all 
complications ; indeed he welcomes them, since they may show him 
the way to new fields of knowledge. The point of interest to us at 
the moment is that the philosopher is in danger of over-simplifying 
his problem, and leaving out essentials through not seeing that they 

are essential. 

Over- Simplification 

To take a simple illustration, the philosopher may set out to 
inquire why a flower looks red in the philosophic sense wherein 
does its philosophic redness reside? Like so many of the funda- 
mental problems of philosophy, this dates back to Plato; in the 
Theaetetus, Socrates reaches the conclusion that colour resides 
neither in our eye nor in the perceived external object. The modern 
philosopher usually follows the lead of Plato to the extent of 
eliminating all factors from the discussion except the flower and 
the mind which perceives it, for surely these and these alone (so he 
will say) are the essentials of the problem. He can now argue that 
to one mind the flower may appear crimson and to another scarlet; 
hence the colour cannot reside in the flower; hence it must reside 
in the perceiving mind; and so on, as on p. 90. 

The scientist knows how many other factors are involved. In 
particular, the light by which the flower is illuminated must be 
important, since if there were no illumination, the flower could not 
look red at all it would look black. Actually it cannot look red 
unless there is some red light to be reflected, so that there must be a 
red constituent in the light by which the flower is illuminated. And 


even if there is red light to be reflected, a man will not see this 
unless his retina is sensitive to red light, so that he must not be 
colour-blind to red. Thus we see that for a flower to look red, three 
conditions must be fulfilled : 

(a) The illumination of the flower must contain some red light. 

(b) The surface of the flower must have the power of reflecting 
red light. 

(c) The man who looks at the flower must not be colour-blind 
to red. 

The question as to where the philosophic redness of the flower 
resides no longer seems to be expressed in the best form, but if an 
answer must be given, it should clearly be that the redness resides 

(a) The sun or some other illuminant which emits red light. 

(b) The surface of the flower which reflects red light. 

(c) The retina of the percipient which perceives red light. 
This brief discussion will have shown that the perception of 

redness is far more complicated than the simple treatment of the 
philosopher usually assumes, and even so it is still far from covering 
the whole ground. 

If, instead of asking why a flower looks red, we ask why the 
setting sun looks red, the answer just given fails entirely. The new 
answer is that the earth's atmosphere abstracts certain constituents 
from the sunlight as this passes through it; that it abstracts more 
blue light than red, making the blueness of the sky therewith; that 
this abstraction increases the proportion of red in the remaining 
light, so that the sun always looks redder than it really is. But at 
sunrise or sunset the sunlight makes a longer journey than usual 
through the atmosphere, so that more than the average amount of 
blue light is abstracted, and the sun looks even redder than usual; 
comparing it with its ordinary appearance, we say the sun looks red. 

To put it in another way, a long process of evolutionary develop- 
ment has given our race eyes which are sensitive only to those 
wave-lengths of radiation with which the sun mainly lights the 
earth, and are most sensitive to those which arrive in greatest 
profusion. At sunset the normal balance of these colours is dis- 
turbed in the way just explained, and sunlight looks red. 


If again we ask why the most distant objects in space look red, as 
they all do, we come into contact with one of the big outstanding 
problems of present-day astronomy. The objects in question are 
the great extragalactic nebulae, and they do not reflect, light as a 
flower does, but emit light of their own. The more distant a 
nebula, the redder its light. It may be that the light would appear 
yellow, green or blue to an inhabitant of the nebula, and that it 
looks red to us only because we are receding from the nebula (or 
the nebula from us, which comes to the same thing) at a speed 
comparable with that of light. This would result in the light- waves 
entering our eyes at less frequent intervals, and this in turn would 
cause the light to appear redder to us than to an inhabitant of the 
nebula. But there are other possibilities, too technical for discus- 
sion here. 

Other colour-problems, with entirely different answers, are pro- 
vided by the redness of a fire, the blueness of the electric arc, the 
blueness of the sky (partially explained above), the blueness of 
moonlight and of shadows on the snow, and by the varied colours 
of the rainbow, of the butterfly's wing and of the patch of dirty oil 
on the road. But whether we discuss the colours of the rose or of 
the butterfly, of the nebula or of the rainbow, the philosophers 
must concede that there are more things in heaven and earth than 
are dreamed of in their philosophy; the world is not as simple as 
they try to make it. 

Atomistic Modes of Thought 

Another difference of method is that the philosopher is much more 
* atomistic* in his thought than the scientist. He is inclined to see 
the world as a collection of separate objects, nature as a collection of 
detached events, time as a collection of moments each of finite 
duration, and space as a collection of regions each of finite extent. 
The scientist, on the other hand, thinks mainly in terms of con- 
tinuity. He sees nature as a theatre of continuous change rather 
than as a succession of jerks, as a cinematograph show rather than 
as a series of magic-lantern slides. While the philosopher thinks of 
time as a succession of finite moments, the scientist represents it as 
an ever-flowing stream ; if he divides it into moments, each is infini- 


tesimal in size, so that the time interval between two successive 
moments is nil. It is the same with space; the philosopher divides 
this up into small finite regions, but the scientist into infinitesimals 
or points, the distance between two again being nil. In brief, the 
philosopher tends to think in terms of what the mathematician calls 
finite differences^ whereas the scientist thinks in terms of infini- 

Possibly this last remark not only summarizes the difference, but 
also explains its origin, which seems, at least in part, to be historical. 
The modes of philosophical thought had become crystallized be- 
fore Leibniz invented the differential calculus or Newton the theory 
of fluxions. As science progressed to ever new types of problems, 
the scientist had perforce to acquaint himself with the newer and 
more accurate modes of thought or fail in his attack, whereas the 
philosopher, still concerned with the same old problems, ex- 
perienced no such need. There are of course exceptions. Leibniz, 
as was to be expected from the inventor of the differential cal- 
culus, always insisted strongly on the continuity of all change in 
nature, as has also Bergson in more recent times. 

The question is one of more than mere form. There is a common 
belief that discontinuous change inevitably passes over into con- 
tinuous change if the intervals of the discontinuities are made 
vanishingly small. In some respects this is true, but in others it is 
not. No matter how small we make its steps, a staircase will never 
become the same thing as an inclined plane. A sufficiently small 
particle can always stand at rest on the staircase, but will roll down 
the inclined plane; more paint is needed to paint the staircase than 
to paint the inclined plane 41 per cent more if the angle is 45, 
regardless of how large or small the steps may be. Again, a saw is 
not turned into a knife by making its teeth infinitely small ; the two 
will cut their way through matter by quite different processes. 

An example of this atomic mode of thought and its consequences 
is provided by another of the famous paradoxes of Zeno. Imagine 
that a moving arrow has some position P in space at some moment 
A > and some other position Q at the next moment B. If we regard 
time as a succession of separate moments A, B, C, . . . , there must 
be some instant of time at which the moment A gives place to the 



moment J5, and this instant is common to the moments A and J3. 
Because it belongs to A, the arrow must be at P when it occurs, 
and, because it belongs to J5, the arrow must be at Q. But it is 
impossible for the arrow to be at two different places P and Q at 
the same instant of time, so that P and Q must be the same, which 
means that in the time-interval from A to B the arrow cannot have 
moved at all. In this way, Zeno claimed to prove, although per- 
haps with his tongue in his cheek, that all motion was impossible 
and all change an illusion. Reality must then be changeless, the 
doctrine which Parmenides had set up in opposition to the nama 
pec, Kal ouSev p,Vi of Heraclitus. 

When this argument of Zeno is translated into the scientific 
idiom nothing of it is left. As the interval between two successive 
moments is now nil, there is no significance in the motion of the 
arrow in this interval also being nil. To come to grips with the 
problem, we must consider the motion of the arrow throughout an 
infinite number of moments, since nothing less than this will give 
us a finite interval of time. The distance through which the arrow 
moves in an infinite number of these infinitesimal moments is of 

course . r . 

infinity x zero, 

which, as every schoolboy knows, may be zero or finite or infinite. 
Thus the possibility of motion is re-established, and the universe 
again becomes free to change. 

When the philosophers oa later age came to study problems of 
motion and change, a large part of their arguments was vitiated by 
their habit of still dividing time into detached moments and change 
into detached events; it was as though they could see nothing in 
the Great North Road except a succession of milestones. Neither 
Kant nor Berkeley seems ever to have grasped the general principle 
of infinitesimals, the latter protesting that it had been 'contrived on 
purpose to humour the laziness of the mind, which had rather 
acquiesce in an indolent scepticism than be at the pains to go 
through with a severe examination of those principles it hath ever 
embraced for true'. Maintaining, as he ever did, that existence 
consisted in being perceived, he indignantly refused to admit that 
infinitesimals could exist which were too small to be perceived, or 


that mathematicians could stand to gain by imagining them to 
exist when they did not. He was especially severe on those who 
* assert that there are infinitesimals of infinitesimals of infinitesimals, 
without ever coming to an end. So that according to them an inch 
doth not barely contain an infinite number of parts, but an infinity 
of an infinity of an infinity ad infinitum of parts'.. . /Whatever 
mathematicians may think of fluxions or the differential calculus or 
the like, a little reflection will show them that, in working by those 
methods, they do not conceive or imagine lines or surfaces less than 
what are perceivable to sense. They may indeed call those little and 
almost insensible quantities infinitesmah or infinitesimals of inftni- 
tesimals y if they please : but at bottom this is all, they being in truth 
finite, nor does the solution of problems require the supposing any 



The results were particularly disastrous in discussions of the 
problems of causality. Many philosophers imagined that the hap- 
penings in nature could be broken up into isolated events, and 
that these could be grouped in pairs, in such a way that the 
events of each pair were related through the cause-effect relation. 

On this fallacious basis, Kant argues that 'the greater part of 
operating causes in nature are simultaneous with their effects', for 
the reason that ' if the cause has but a moment before ceased to be, 
the effect could not have arisen '. He instances a warm room which, 
he says, is warm because a fire is burning in it, although, as every 
housemaid knows, the reason is that a fire has been burning in it. 

Kant sees that, if cause and effect really are simultaneous, it 
becomes difficult to say which of a pair of related events is cause 
and which is effect, but claims to be able to distinguish between 
the two 'through the relation of time of the dynamical connection 
of both'. To take his own illustration, a leaden ball lying on a 
cushion is invariably accompanied by a Jiollow in the cushion. ' If 
I lay the ball upon the cushion, then the hollow follows upon the 
previously smooth surface ; but if the cushion has for some reason 
or other a hollow, there does not follow thereupon a leaden ball.' 

Hume subsequently propounded a different view of causality, 
holding that all effects are contiguous in space with their causes, 


and also successive in time. But contiguity and succession are not 
enough of themselves to proclaim two objects or events to be cause 
and effect; there must also be constant conjunction. In other words 
we must have noticed the contiguity and succession repeated in 
a great number of instances. 'We remember to have seen that 
species of object we call flame, and to have felt that species of 
sensation we call heat. We likewise call to mind their constant 
conjunction in all past instances. Without any further ceremony, 
we call the one cause, and the other effect, and infer the existence of 
the one from that of the other/ This also is very unconvincing 
scientifically, partly because heat is frequently experienced without 
flame, and flame without sensible heat; partly because there is no 
means of deciding which is cause and which effect. In actual fact 
heat often produces flame and flame usually produces heat, but 
when we come upon a house on fire, it is not easy to say whether 
the ultimate origin of the conflagration was heat or flame or some- 
thing different from both. 

Obviously, too, the constant conjunction of two events does not 
entitle us to ascribe the cause-effect relation to them at all. I may 
remember having repeatedly seen the Scotch express pass through 
my station when the hands of my watch pointed to 12 o'clock, but 
this does not show that either event is the cause of the other. We 
may have repeatedly seen the full moon when the sky is clear, and 
never when the sky is clouded over, but must not conclude that the 
full moon makes the sky clear (although there is a popular super- 
stition to this effect), or that a clear sky makes the moon full. 

Typical of a more modern and more scientific attitude to causality 
is the definition recently proposed by Bertrand Russell : * Given an 
event Ei, there is an event 2 and a time-interval T, such that 
whenever Ei occurs, Z?2 follows after an interval TV Yet scientific 
study shows that even this is not true, to the exactness which 
philosophy ought to aspire to, except in the one special case in 
which Ei is the state of the whole world at one instant of time, and 
2 is the state after a time-interval T. 

The scientific fact is that it is not permissible to treat the causal 
relation in any of these ways. All are based on unwarranted 
simplifications of the complexities of the actual world; they are 
abstractions which can at best provide approximations to the truth. 


There is no scientific justification for dividing the happenings of 
the world into detached events, and still less for supposing that 
they are strung in pairs, like a row of dominoes, each being the 
cause of the event which follows and at the same time the effect of 
that which precedes. The changes in the world are too continuous 
in their nature, and also too closely interwoven, for any such pro- 
cedure to be valid. We shall see this more clearly when we discuss 
the scientific view of causality in the next two chapters, but it may 
be of service to illustrate it by a simple example here and now. 

Suppose I shoot a bird and it falls to the ground. The falling to 
the ground may obviously be regarded as an effect, but where are 
we to look for the cause? In spite of Kant's argument to the con- 
trary which has just been mentioned, most men would say that it 
was my having previously pulled the trigger of my gun. Yet this is 
an obvious over-simplification of the situation ; to my pulling the 
trigger must be added my having previously loaded the gun with a 
cartridge in which someone had previously put powder and shot in 
the right places and in the right amounts, that I further pointed 
the gun in the right direction, and pulled the trigger at the right 
moment, having previously made a correct allowance for the speed 
and direction of flight of the bird, for the strength and direction of 
the wind, and for the effects of air resistance and gravitation. That 
the shot found its mark when I aimed in this particular direction 
was perhaps because a depression which had been centred over 
Iceland three days ago had moved eastward and caused strong 
south-west winds ; this was because there had been a hurricane in 
the West Indies a week before, and so on ad infinitum. Any effect 
is seen to be connected to previous events by an endless succession 
of strings of events all of which meet in the effect. 

We see how excessively naive it is to suppose that all the events 
in the world can be arranged in pairs with the cause-effect relation 
obtaining in each. This would imply that each effect has only one 
cause and each cause only one effect. If we suppose that the hap- 
penings of nature are governed by a causal law, we must suppose that 
the cause of any effect is the whole previous state of the world, so 
that every effect has an infinite number of causes. Some of these 
may of course exert an influence so slight as to be negligible. For 
instance, my success in hitting my bird will not depend to any 


appreciable extent on whether Sirius is in the ascendant, or whether 
I have just broken a mirror or spilt the salt, although it may depend 
on how late I sat up the night before. 

Yet in considering any event, it is not necessary for all previous 
events in the history of the world to be considered as separate 
causes. The effects of the earlier of them are already taken into 
account in the later, and they need not b allowed for twice over. 
It is enough to consider a cross-section at one particular instant of 
time. The state of the world at this instant any instant I choose 
will provide the adequate cause of the effect under consideration. 
If for example I select the instant at which I pulled the trigger to 
shoot the bird, then the state of the world at this instant already 
comprised a cartridge in my gun and a strong south-westerly wind ; 
there is no need for us to bother about who loaded the gun or what 
caused the wind. 

The cross-section we select need not extend over the whole of 
space; the more distant regions may be left out of consideration 
altogether. For no influence can travel faster than light, and some 
parts of the universe will always be so distant that light which left 
them at the instant of the cross-section would not have reached us 
yet; happenings at such places obviously cannot affect the present 
course of events here. 

Two particular cases of cross-section are of special interest. 
First, the cross-section may be taken at the beginning of time, or, 
if we prefer so to $all it, at the creation of the world ; we then see 
how everything that occurs now is a direct consequence of the way 
in which the atoms of the world were arranged at their creation. 
Second, we may bring our cross-section forward through time 
until its instant differs only infinitesimally from the present. All 
those parts of the universe which are not in our immediate vicinity 
may now be disregarded, and we find that the state of things here 
and now depends only on the state of things which prevailed in our 
immediate vicinity an infinitesimal moment ago. This brings us 
back to the very restricted view of causality adopted by Kant, but 
science sees no reason for confining itself to this view. Neither 
does the common man, who will continue to insist that his dog 
died to-day because it had eaten poison yesterday. 





The earliest attempts to discover the pattern of events were limited, 
naturally enough, to the visible movements of objects either on 
what we have called the man-sized scale or on the far grander scale 
of astronomy these were the only movements which could be 
studied without instrumental aid. 

The movements of the astronomical bodies were treated only in 
their geometrical aspect. The * fixed stars' hardly came under 
discussion at all, since they appeared to have no motion beyond 
their diurnal rotation round the pole. This was of course a conse- 
quence of their great distance from the earth, but it was explained 
by supposing them to be immovably attached to a sphere which 
rotated round the earth as centre. 

There remained the sun, moon, and planets. A whole succession 
of astronomers from Aristarchus through Ptolemy to Copernicus 
and Kepler had investigated the paths in which these bodies 
moved, but had shown very little concern as to why they moved in 
these particular paths rather than in others. Aristotle'j* pronounce- 
ment that a circular motion was natural to all bodies, because 
the circle was the perfect geometrical figure, seems to have stifled 
curiosity fairly thoroughly for nearly two thousand years; it was un- 
critically accepted by Copernicus, and even at one time by Galileo. 

It was different with terrestrial bodies; there had been many 
attempts to explain their movements in what we should now de- 
scribe as dynamical terms. The earliest Greek thinkers had imagined 
that the motion of every object was controlled by a tendency, 
Inherent in the object, to find its * natural place* in the world. A 
stone sank in water because the natural place for stones was the 
Dottom of a stream ; flames ascended in air because their natural 


place was up in the sky, and so on. Aristotle explained this by the 
supposition that ^bodies j^ssessed jurying degrees of heaviness and 
lightness, and that the natural arrangement of the world was one in 
order of heaviness, the heavier bodies taking their places below and 
the lighter above like layers of oil and water. This view prevailed 
until it was challenged by Giordano Bruno (1548-1600), who 
pointed out that as heaviness and lightness were merely relative 
terms, substances could have no natural places in the universe. 

It was of course obvious that many objects were not in their 
natural places, and some explanation had to be found for this. 
Aristotle had thought that a body could only be kept away from its 
natural place through continued contact with some other body, 
such as the hand which held it or the table on which it lay; it could 
only be moved by the pressure of some other body, and this contact 
had to persist throughout the motion. When a stone was thrown 
upwards, the air surrounding it was also set into motion and pressed 
on the projectile through its flight, thus keeping it from returning 
to its natural place which was on the ground. 

A different view was held by Hipparchus (c. 140 B.C.), who 
thought that a body was set into motion by receiving an * impulse ' 
from some other body; this stayed in the moving body for a time, 
but then gradually weakened and finally disappeared, with the result 
that the moving body first slowed down and finally came to rest. 

It was natural that such beliefs should be held, for they seemed 
to be confirmed by the actual behaviour of moving bodies at the 
surface of the earth. Here every moving 'body obviously did slow 
down and finally come to rest; had it done anything else it would 
have formed a perpetual-motion machine, and this was generally 
agreed to be an impossibility. Indeed Aristotle had branded it as 
an absurdity, using it thus in an argument which ended in a sup- 
posed reductio ad absurdum. But the true reason for the slowing 
down was not that conjectured by Hipparchus; it was the action of 
air resistance, friction and other * dissipative ' forces. 

A first glimmering of the truth seems to have been seen by 
Plutarch, who wrote (c. A.D. 100): ' Everything is carried along by 
the motion natural to it, if it is not deflected by something else/ 
Apart from this, none of the ancients seems ever to have conjectured 


that a body set in motion in empty space, or in any region in which 
dissipative forces did not operate, would not slow down at all, but 
would really act as a perpetual-motion machine and continue to 
move, either for ever or until something extraneous brought it to 

The idea that such motion could occur is, however, definitely 
found in the writings of Nicholas of Cusa (1401-1464). He believed 
that the earth continually moved through space without our being 
conscious of its motion just as a boat may drift down a rivet 
without its occupants knowing they are moving until they notice 
the banks sliding past them and also accepted the Pythagorean 
doctringJLhat : thejearthjurns steadiR^onjts axis olnice^every twenty 
fouiiio^rs. lie f urtherjrema^edlEat a smooth bairwhich "has been 
v set moving on a smooth floor will continue to move until something 
checks its motion. Herejhis facts were right, but his explanation 
wrong; he supposed that the motion continued because every par- 
ticle of the ball tended to retain its natural circular motion round 
the centre of the ball, remarking that a ball which was-not perfectly 
round woukLnot persist in its motion. 

TKen came Galileo, who saw that the primary effect of outside 
influences acting oh a body is to produce a change in the motion of 
the body, changes of position being only secondary effects. Thus a 
body which is not acted on by any outside influence at all can 
experience no change of motion, and so must move on for ever at 
the same uniform speed, as Nicholas of Cusa had said. 

Descartes was probably the first to enunciate this principle 
clearly and unambiguously, writing: * A body when it is at rest has 
the power of remaining at rest, and of resisting everything that 
could make it change. Similarly when it is in motion it has the 
power of continuing in motion with the same velocity and in the 
same direction/ 

Descartes was also the first, at least since the era of Greek 
speculation, to attempt to bring all the phenomena of physics 
within the scope of a single system of laws. His system was not 
dynamical but kinematical ; he tried to explain phenomena in terms 
of motiq^aji^n^t.aLfQrces; ^T do not 'aECept lariy other principles 
in physics than there are in geometry and abstract mathematics, 


because all the phenomena of nature may be explained by their 
means/ But the system was mostly erroneous. 

By contrast, the system which Newton published in the year 
1687 under the title Philosophiae Naturalis Principia Mathematica 
was purely dynamical in its nature. If it was not perfectly true to 
nature, it was at least so true that two hundred years were to elapse 
before its imperfections began to show themselves. 


Newton regarded the material world as a collection of particles or 
pieces of matter, each of which could be either at rest in space or 
moving through space. If a particle was at rest it stayed at rest, 
and if it was in motion it continued in motion at the same speed 
and in the same direction unless 'forces' intervened to change 
this state of rest or motion (First Law). Thus perpetual motion 
became the normal state of things Tor a moving body unless some- 
thing checked it. 

Forces were explained only by their effects, which were to change 
motion; a force was measured by the change it produced in the 
velocity of the body on which it acted, multiplied by the mass of 
the body (Second Law). Here the word 'velocity* must be under- 
stood 10 specify nut only the speed but also the direction of the 
motion. Thus a change of velocity must be supposed to occur 
when a body changes the direction of its motion, even though it 
continues to move at the same speed as with the moon's motion 
round the earth ; the force which causes this change of velocity is of 
course the gravitational pull of the earth. 

Newton added that when any body A exerts a force on a second 
body 5, then B must exert on A a force which is equal in amount 
but opposite in direction (Third Law). 

For two reasons Newton's system of mechanics was incom- 
parably better than anything that had preceded it. In the first 
place, it was based on the experimental results of Galileo and 
others, whereas previous systems had been based on conjecture 
and speculation. And, in the second place, it was free from any 
special concern with the local conditions prevailing at the earth's 


surface, and so was able to provide a sound basis for the vast 
superstructure of dynamical astronomy which was subsequently to 
be reared upon it it was a dynamics for the sky as well as for the 
earth. Yet it represented only one step, although an important one, 
towards final truth. Forunderlying it was the assumption that 
bodies moved against a ^background of absolute time and^ space; 
twoTiunxlred and thirty years later the theory of relativity was to 
disclose that nature provides no such background. And after 
another ten years the theory of quanta was jto show that JNewton's 
lawsTare valid onljrfor ^ be- 

yond tHeseHes a whole world oF atomic and sub -atomic processes 
which do noFbbey JNewton'slaws at alL_ "*~ 

Mechanistic Determinism 

This system of mechanics threw into perfectly sharp focus the 
problem of determinism upon which we touched at the end of the 
preceding chapter. According to Newton's laws, any particle A in 
the world will be subject to forces from any or all of the other par- 
ticles By C, Z), . . . in the world. These forces may come from 
contiguous particles as when two billiard-balls collide or from 
distant particles through gravitational attraction as when the sun 
and moon raise tides on the ocean. In either case the amount of 
force exerted at any instant depends only on the positions which 
the various particles of the world occupy in space at that instant. 

It follows that the changes of the world at any instant depend 
only on the state of the world at that instant, the state being_dg : 
finedby the positions and velocities of the particles ; changes in 
position are determined by the velocities, and changes in velooities 
by theTofces, which in turn are determine3nby the positions. 

If, Tfteii7 we know the state of the world atlmylme instant, we 
can in principle calculate to the last detail the manner and rate at 
which this state will change. Knowing this, we can calculate the 
state at the succeeding instant, and then, using this as a stepping- 
stone, the state at the instant next after, and so on indefinitely. 
Thus, as Laplace pointed out in his Essay on Probability (1812), the 
present state of the world may be regarded as the effect of its 
antecedent state, and also as the cause of the state that is to follow. 


He went on to say that if the state of the world at its creation were 
specified in its minutest details to an infinitely capable and infinitely 
industrious mathematician, such a being would be able to deduce 
the whole of its subsequent history. * Nothing would be uncertain 
for him; the future as well as the past would be pre'sent to his eyes/ 

Even though no such mathematician exists, the whole future 
history of the world must have been implicit in its configuration at 
its creation; its so-called evolution is a mere unrolling of what is 
already there, and we have as little power to affect the pattern of 
things to come as a man who weaves a carpet on a loom which is 
already set, or indeed as a man who unrolls an already woven 
carpet for our inspection. 

When once this evolutionary point of view has been gained, it 
becomes a mere question of words whether we speak of ' causation ' 
with Kant or of 'constant conjunction' with Hume. If the pattern 
of the world is such that after A always comes JB, who shall care 
whether we say that A is the unvarying cause of B, or that B is the 
unvarying concomitant of At The true and indisputable cause of 
everything was the arrangement of the particles of the world at the 
beginning of time, so that it is true to say, in the language of 
orthodox theology, that all things were fore-ordained by God at 
the creation of the world. But itj equally true to say, in the 
language of science, that the cause of everything is to bgjound jn 
tfie arrangement of the particles ot the world at anylpast instant in 
Its history that we may choose; every past instant ma^ equally well 
be treated, tor our present purpose, asjthe .moment of the world's 
creation. And what is essential is the arrangement of the particles, 
and not the God Who arranged them. 

General Principles 

Although it would need Laplace's infinitely industrious and in- 
finitely talented mathematician to trace out the future of every 
particle in the universe, yet quite ordinary mathematicians have 
been able to obtain a good deal of simple but important knowledge 
flf particles in generaL 

The kinetic energy of a moving particle is defined to be half of the 
mass of the particle multiplied by the square of its speed of motion 


(iww 2 ), this be^ the j-mouaLof work that must be done to set the 
particle iflFp. motion at a speed v._ When two or more particles 
affect one another's motion by contact or impact, it is easily shown 
that any increase in the kinetic energy of one is exactly offset by an 
equal decrease in the kinetic energy of the others, so that the total 
kinetic energy of all the oarticles j-emains constaafc^i^kuphoiit the 

Again the momentum^tot a moving particle is^dpfed to be the 
mass of the particle multiplied by its speed of motion (mv\. When 
two particlesact on one another T the momentum of both is changed. 
IfThelnotrpn is confined to one direcjtionIm.SB^^it is easily shown 
that any gain ofmomentum by one particle j 

losjTto the Second, l$ol;Iia^ 

If the motionls"not confined to one direction in space, the situation 
is~more complicated. We must now select any three directions in 
s^acewfilch^are mutuallylit right angles to one another, asjSouth- 
North, West-East, and down-up. The motion of each particle must 
now be separated"mto its constituent motions in these three direc- 
tions. This of course divides the momentum into three parts, one 
in eacIToFthe three chosen directions. The West-East momentum 
of the particle is now defined as the mass of the particle multi]3lied 
by the speed at which it moves from West to East and so on. It can 
noW"15e" showiLJJiaLJLJie__total momentum in each of the three 
directions separately must remain unaltered, and the same is of 
course true of any other direction in space that we may select. 

In whatever way a nymber of particles may move, their motion 
must always conform to the general principles just stated. If a 
problem is of a sufficiently simple nature, these principles may 
suffice of themselves to provide a complete solution, without our 
troubling about the motions of the individual particles at all. 

Suppose, for instance, that in a shunting yard an empty truck 
weighing 5 tons runs at 5 miles an hour into a loaded truck weighing 
ao tons, wlilcins standing at rest. Suppose that the^ trucks are 
fitted with an automatic coupling of the American type, so^that 
tEey become locked together after their impact, and tEen havcTto 
mon^elorward at the same speed; we wish to know what this speed 

will h*v 


We need only notice that the forward momentum of the coupled 
tmclcsjifter impact must be exactly equal to their forward mo- 
mentum before, so that the amount of momentum \yhich originally 
resided in theonlT g -torx^ truck must now be distributed over 
actons.. This 2$ tons will accordingly ifrpve forward at one-fifth 
of ^he,sgeed at which the 5 -ton truck originally moved t^e two 
trucks move forwkrd together at i mile an hour. 

If there is no automatic coupling, the problem becomes slightly 
more complicated, because the trucks can then rebound after im- 
pact and move at different speeds afterwards. As we have now to 
find the values of two different quantities the two speeds after 
impact we need two relations from which to find them. A second 
relation is supplied by the fact that the total kinetic energy must 
be the same after the impact as it was before. Using these two 
relations, we find that the loaded truck will now move forward at 
2 "miles arfTiour, whiletire Jjght truck rebounds and moves Back- 
wards^ aT~3 miles ariTiour. 

Equations of Motion 

More complicated problems cannot be solved in this simple way, 
but other and somewhat similar methods are available; let us try 
to illustrate them by the simplest of examples. 

In the game of billiards, three balls roll about on a rough surface 
bounded by resilient cushions; they move as they are impelled by 
the impact of extraneous objects, the cues. It would be possible to 
follow out their motion by treating each baU as an infinite number 
of minute particles, first reckoning out how each particle pulled or 
pushed its neighbour, and then calculating the resulting motion of 
the balls as a whole. This, indeed, is what we should have to do, if 
we were limited to using Newton's laws in the crude form in which 
they were originally enunciated. But such a problem would be one 
for Laplace's infinitely patient mathematician, and not for ordi- 
nary mortals, whose life is too short; they need other methods. 

The position of any ball on the table can be specified by two 
measurements, namely the distances of its centre from each of two 
cushions, one on a long and one on a short side of the table. Such 
measurements are called coordinates. Thus the oosition of all 


three balls can be specified by mentioning the values of six co- 

This takes no account of any spins or rotations the balls may 
have. Now the orientation of any ball can be specified by men- 
tioning the values of three angles, and these may also be regarded 
as coordinates, although of a slightly different kind. Thus we see 
that the positions, not only of the balls as a whole but of every 
particle in the balls, can be specified by the values of fifteen co- 
ordinates,, six of which measure position and nine orientations. If 
we are further told the rate at which each of these coordinates is 
increasing, these fifteen new quantities give us a complete know- 
ledge of the motion of every particle in the balls. These thirty 
quantities specify the state of the three balls completely. 

Thus all the knowledge that Laplace's mathematician would 
demand for a prediction of the whole future motion of the infinite 
number of particles in the three balls is contained in the values of 
only thirty quantities fifteen coordinates and their fifteen rates of 
change and all the information he could give us as to the state 
of the balls at any future instant would be comprised in the values 
of these same thirty quantities at that future instant. 

Short cuts have been found by which we can pass from the 
values of the thirty quantities at one instant to those at another 
instant without troubling about the movements of individual par- 
ticles, and there are similar methods for tracing out the motions 
of any mechanical system whatever; the rules for doing so ap- 
pear in mathematical fojm, and are known as equations of motion. 
Such sets of equations have been given, in various forms, by a 
number of mathematicians, especially by Maupertuis, Lagrange 
and Hamilton. 

Hamilton's equations are perhaps the most interesting. They 
occur in pairs, one pair for each coordinate, and the form of each 
pair is always the same, regardless of whether the coordinate 
represents a distance, an angle, or something else. This form of 
equations of motion is described as the canonical form. 

We can discover something of the inner meaning of these equa- 
tions by discussing a very simple case the motion of a particle 
moving in a straight line. Here we define the momentum of the 

JP 8 


moving particle as its speed of motion multiplied by its mass; 
Newton's second law then tells us that this momentum increases at 
a rate equal to the force acting on the particle. These statements 
may be put in the form of equations, thus : 

mass x velocity = momentum, 
rate of increase of momentum = force. 

Now every pair of Hamiltonian equations is simply a generaliza- 
tion of this pair; the first member of the pair tells us the relation 
between the velocities of bodies (or, more generally, the rates of 
increase of coordinates) and certain quantities described as mo- 
menta, while the second member tells us the rate at which these 
momenta increase in terms of the forces, these often including 
what are usually called centrifugal forces. This second equation is 
thllS 1 d generalizationof Newton's seconcT law of motion. 


So far we havp imagined .^11 thp fTlfirgy anc ^ a ^ ^e momentum of 
the world to reside in the motion of material particles. When it 
does, we can show, from Newton's laws, that the total kinetic 
energy of any group of particles will retain a constant value through- 
out all changes in the motion of individual particles, provided only 
that no forces act on the group of particles from outside. This is 
the law of Conservation of Energy in its simplest form. The same is 
true of the total niblnentunrin any direction in space. This is the 
law of Conservation of Momentum. 

But when gravitation, chemical action, radiation, electricity and 
magnetism are taken into account, neither the total energy nor the 
total momentum of the material particles remains constant. We 
can, for instance, increase the energy of motion of a motor-car, 
either by letting it run downhill or by burning some of the petrol 
in the tank. We cannot of course go on doing this indefinitely, since 
after a time the car will have dropped to sea-level or the tank will 
be empty. This leads us to picture both the height above sea-level 
and the petrol in the tank as representing stores of energy upon 
which we may draw to increase the energy of the car until these 
stores are exhausted, but no longer. 


To make a consistent picture, we have to suppose that energy 
can be stored in a great variety of forms, as for example in the 
raised weight of a clock, in the coiled spring of a watch, in the 
chemicals used in the cells of an electric accumulator, in the coal 
we burn in our boilers and in the petrol we burn in our cars. By 
attributing certain specific amounts of energy and momentum to 
gravitation, chemical energy, electricity, magnetism and radiation, 
nineteenth-century physics found it possible to define both energy 
and momentum in such ways that both were conserved, or at least 
appeared to be. It was found possible to extend the Newtonian 
mechanics in this and similar ways until it was able to account for a 
great range of physical phenomena, and hopes were entertained 
that in time it would explain all hopes which, as we shall soon 
see, were not to be fulfilled. 

This extension of the Newtonian mechanics is generally de- 
scribed as the 'classical mechanics'. We are only concerned here 
with such of its features as are of general philosophical interest. 

One of these may be mentioned at once ; let us again avail our- 
selves of a specific example. 

Suppose we return to the billiard-table we discussed three pages 
back, and find that it has been made more complicated in our 
absence. The original table was suited to the illustration of the 
Newtonian mechanics, the new to the illustration of the more 
complicated classical mechanics. Someone has put magnets inside 
the balls and also inside the cushions of the table, has laid electric 
wires through the be4 of the table and installed batteries and 
switches to create and control electric currents. To describe the 
state of this system completely, we shall certainly need more than 
our original fifteen coordinates, but the classical mechanics assures 
us that some finite number will suffice, and further provides us 
with equations of motion for the new coordinates. 

It is surprising and significant that these new equations of motion 
are precisely similar in form to the simple canonical equations of 
the Newtonian mechanics. That is to say, the same sort of symbols 
occur in the new equations as in the old, and enter in precisely the 
same way although of course they have different meanings. The 
new equations accordingly admit of the same sort of general in- 



terpretation as the old; in each canonical pair, one equation tells 
us, as before, that the momentum associated with one coordinate 
increases at a rate equal to the force which operates to increase this 
coordinate; the other specifies the rate of change of this coordinate 
in terms of the various momenta. This similarity of interpretation 
shows that the classical mechanics is still fundamentally Newtonian 
in conception; nature can still be pictured as consisting of particles 
which are pushed and pulled about by forces. 

Action at a Distance 

Difficulties occur as soon as we try to picture these pushes and pulls 
in detail. 

When one billiard-ball strikes another and sets it into motion, it 
is easy to imagine pairs of molecules, one in each ball, pushing each 
other and so transmitting force from the one ball to the other; 
Newton's concept of force makes it possible to form a perfectly 
definite picture of what happens in such a case. But it is not so 
easy to picture what happens when the moon raises tides on the 
ocean, or the sun holds the earth in its orbit. Newton's law of 
gravitation specified the amount^ 5O^ ^?.^J:^i5g_k^ een two 
bodies such as the sun and the earth, but made no attempt to 
explain the nature of the force, or how the ForceT couT3[ operate 
across stretches of apparently empty space. How can the moon 
move the waters 61 ouFoceaSs limlessTEere is some chain of con- 
tinuous contact between moon and earth such as might be 
provided, for instance, by ajrtieaf of strjggs or elastics, or {perhaps 
by ajjquid transmitting a conrfnuousj>ressure or tension? What, it 
may be asked, plays the part in jreality of such a system of strings, 
elastics or liquids? 

Newton and his contempuiancs a&*cu &uun questions as these, 
and it was generally felt that an answer must be found before 
Newton's theory of gravitation could be accepted. In a famous 
letter to Bentley, Newton wrote: 'It is inconceivable that inani- 
mate brute matter should, without the mediation of something else 
which is not material, operate upon and affect other matter without 
mutual contact. . . . That gravity should be innate, inherent and 
essential to matter, so that one body may act upon another at a 


distance, through a vacuum, without the mediation of anything 
else by and through which their action may be conveyed from one 
to another, is to me so great an absurdity that I believe no man, 
who has in philosophical matters a competent faculty of thinking, 
can ever fall into it/ 

The question remained unanswered until Einstein's generalized 
Jieognqf relativity came in 1915 and showed that there is probably 
neither any answer to it, nor need for an answer. ^^ j^, 

We havealready seen (p. 63) that a three-dimensional: space 
does nofof itself provide a suitable framework against which to 
represent the motions of objects. When a number of objects stand 
at rest7~their spatial relations may be represented in a three- 
dimensional continuum, and such an arrangement, if properly 
made, will be consistent with itself and will 'make sense' we 
shall be able to represent not only some but all of the spatial 
relations of the objects in a single arrangement. But such an 
arrangement is found to be inadequate when the objects are in 
rapid motion; no such arrangement can then represent all the 
observable facts. A fourth dimension, of the general nature of 
time, must be adctexl to the three ^imensions~of simple space, 
forming tKf four-dimensional continuum that we have described 
as the spa^-timejmjty (p. 63). We cannot say that any one par- 
ticular dimension in this represents -time, while the other three 
represent space. The four-dimensional continuum forms an indis- 
soluble unity, and must always be regarded as a whole. In it are 
any number of different, directions, any one of which may be taken 
to represent time, and will adequately, represent it to an observer 
who is moving through space at the right speed. 

This four-dimensional continuum, formed by the indissoluble 
welding of space and time to form something different from either, 
is found to provide by far the most suitable framework for the 
discussion and explanation of the^benQia^nQn of ftravitflfr'nn .. JV 
pdmfhl the comimramT^^ of space at an instant of 

time. Thus the fact that a gravitating mass such as the sun occupied 
a particular point of space at a particular instant of time can be 
represented by the position of one single point P in the continuum, 
while the position of the same mass in space at other instants will 


be represented by the positions of other points Q, R 9 S, . . . in the 
continuum. The line obtained by joining up the points PQRS. . . 
constitutes a record of the various positions occupied by the mass 
through an interval of time, or through the whole of time if we 
wish. Such a line is called the 'world-line' of the mass in question. 

With this framework before us, we find that a concise, complete 
and perfect picture of the pattern of events can be obtained in the 
following way. 

We first suppose that the presence of a gravitating mass at the 
place and time represented by the point P of the continuum im- 
presses a curvature on the continuum in the proximity of the point 
P, just as the presence of a lead ball on a cushion at a certain place 
and time impresses a curvature on the cushion in the proximity of 
these points of space and time. Thus the continued existence of the 
sun will impress a curvature on the continuum in the region 
surrounding the world-line of the sun. 

Having introduced us to a curved continuum in this way, the 
theory of relativity now tells us that the world-lines of small bodies 
moving in the neighbourhood of the mass as, for example, planets, 
comets or meteorites moving round the sun are either straight 
lines, or are the straightest lines that are consistent with the cur- 
vature of the continuum. 

This simple statement describes the whole pattern of events, 
except that it must obviously be put in a slightly different form 
when more than one gravitating mass is involved. If there are no 
gravitating masses present, the continuum an have no curvature*. 
Thus the world-line of a particle is a straight line i.e. the particle 
continually moves in the same direction and at the same speed. 
This is Newton's first law, which now appears as a simple inference 
from the theory of relativity. If gravitating masses are present, a 
particle appears to move in a curved path, but the apparent curva- 
ture of path merely reflects the curvature of the continuum. 
Newton thought that a planet followed a curved path in a straight 
(flat) space; the theory of relativity pictures it as following a 
straight path in a curved space. 

* We need not discuss the possibility of space having inherent in itself a 
cuivature on a universe-wide scale; such curvature, if it exists, is unimportant 
for our present discussion. 


We notice that all reference to force has disappeared, so that the 
motibns of the planets and of other gravitating bodies present 
problems in geometry, but not in dynamics. Also the question of 
action at a distance has dropped out altogether. Nature has dodged 
it by the simple manoeuvre of making gravitation act on space 
instead of across or through space, although, in a sense this only 
postpones the difficulty; it provides a new description, but not a 
satisfying explanation, of the facts. 

At the same time, the question of causality has assumed a new 
aspect. We can no longer say that the past creates the present; 
past and present no longer have any objective meanings, since the 
four- dimensional continuum can no longer be sharply divided into 
past, present and future. All we can say is that the world-lines of all 
objects in the universe follow the simple pattern already described. 
If these world-lines have a real existence in a real continuum, the 
whole history of the universe, future as well as past, is already 
irrevocably fixed. If on the other hand the world-lines are merely 
constructions we draw for ourselves, to help us visualize the 
pattern of events, then it is as easy to extend these world-lines 
from our already completed past into our future as it is to carry 
on the weaving of a fabric when the pattern is already set in the 
loom. In either case the future is unalterable, and inescapable 
determinism reigns. 

Electric and Magnetic Forces 

Superficially at least the forces of electricity and magnetism seem > 
to present the same kind of problem as the forces of gravitation. 
Experiment shows that two electrically charged bodies attract one 
another (or repel if their charges are of the same kind) with a force 
which conforms to the same mathematical law as the force of 
gravitation both forces fall off as the inverse square ofjLhe 
distance. The same is true of magnetic force also; two magnetic 
poJTes attract or repel one another with a force which again follows 
t^eTavToT the inverse square of the distance. 
~\TEis being so, we might well expect these forces to admit of a 
pictorial interpretation of the same kind as that for gravitational 
force,) But no such, interpretation has yet been found, and the 
prospect of finding orifexnow looks very remote. Electric and mag- 


netic forces in general present a far more intricate problem than 
gravitational forces. Gravitational force is simple, and a thing by 
itself, as also are electric and magnetic forces so long as the electric 
charges and magnetic poles stand at rest. But as soon as motion 
comes into the picture, the whole situation is changed. Forces of 
new kinds come into play, for moving electric charges exert mag- 
netic forces in addition to the electric forces they exert when at 
rest, while moving magnets exert electric forces in addition to the 
magnetic forces they exert when at rest. When the exact laws 
governing these intricate laws had been discovered by a great 
number of experimenters, Clerk Maxwell succeeded in expressing 
them in a mathematical form which was both simple and elegant. 

At this time, space was supposed to be filled with an ether, a 
substahc^ which might well serve, among other functions, to trans- 
mit forces across space. So long as such an ether^could^be^aHed 
7m, the traifemission of force to a distance ^vas easy to understand ; 
it was like ringing a distant bell by pulling a bell- rope. 

The pattern of electrical events being known with complete 
precision in mathematical terms, it was natural to try to discover 
the properties of the ether from this pattern. It was taken for 
granted that these properties would prove to be mechanical 
either the particles of the ether would be found capable of motion 
in the Newtonian sense and in accordance with the Newtonian 
laws, or else they would conform to some more general principle, 
such as * least action' (p. 187), which formed a sort of generalization 
of the Newtonian laws; they would in either case be pushed or 
pulled about by forces, Faraday, Maxwell*, Larmor and a great 
number of others all tried to explain electromagnetic action on 
these lines, but all their attempts failed, and it began to seem im- 
possible that any properties of the ether could explain the observed 
pattern of events. 

Then the theory of relativity came and explained the causes of 
failure. Electric action requires time to travel from one point of 
space to another, the simplest instance of this being the finite speed 
of travel of light (p. 63). Thus electromagnetic action may be said 
to travel through space and time jointly. But by filling space and 
space alone with an ether, the pictorial representations had all 


presupposed a clear-cut distinction between spaqe and time. Clearly, 
if such a distinction existed, it ought to be possible to separate the 
two out by experiment. Yet when the experiment was attempted 
by Michelson and Morley it failed, thus showing that the space and 
time assumed in the picture were not true to the facts of nature. 

On this failure the theory of relativity was built. It provided a 
clue to the solution of the puzzle by showing that the pattern of 
events could not be altered by making the whole electric structure 
move through the supposed ether at any speed whatever. This, 
indeed, was the fundamental postulate of the theory, which every 
experiment so far made has confirmed the pattern of events can- 
not " 3e a ^ tere d by altering the speed of motion. Ingthgr ; words, the 
pattern of events was the same whether the world stood at rest in 
th^ supposed ether, or had an ether wind blowing through itjitj* 
mflfioli miles^an Tiour^,It began to look as though the supposed 
etHer was not very important in the scheme of things, and further 
discussion showed that it could not serve any useful purpose and 
so might as well be abandoned. But if the bell-rope has to be dis- 
carded, what is to ring the bell? 

Clearly, if electric action is to be explained in mechanical terms, 
the mechanism must be supposed to be attached to the electric 
charges, and to move through space with them. It must extend 
through the whole of space, because the attraction and repulsion of 
an electron extend through the whole of space, and it must be the 
same for all directions in space, since an electron at rest exerts a 
force which is the same for all directions in space. Further, as the 
pattern of events is unaltered by motion, the mechanism must be 
the same when the electron is in motion as when it is at rest. But 
experiment shows that an electron in motion exerts additional forces 
which are not the same for all directions in space; if we picture this 
electron as moving head-foremost through space, these forces sur- 
round it like a belt round its waist. 

Thus direct experimental evidence shows that the forces exerted 
by an electron (or of course by any other charged body) can neither 
be attributed to any mechanism attached to the body, nor to action 
transmitted through an ether or any medium surrounding the 
body. We have a perfect specification of the pattern of events 


written, as it necessarily must be, in the language of mathematics, 
but this does not admit of interpretation in mechanical terms, or 
indeed in any terms other than those of mathematics. 

This is true also of the greater part of the classical mechanics. 
The only part that we understand pictorially is the Newtonian part 
which deals with mechanical phenomena on the man-sized scale; 
we can understand this because the phenomena directly affect our 
senses ; the pictorial explanation is in terms of forces such as we 
exert with the muscles of our bodies, and the idea of such forces is 
familiar to our minds, 

If we wish to visualize other processes pictorially, no single 
perfect picture is available, an0I the best we can do is to construct a 
number of imperfect pictures, each representing one, but only one, 
aspect of the complete range of phenomena. For instance, if a 
shower of electrons is shot on to a zinc sulphide screen, a number 
of flashes are produced one for each electron and we may picture 
the electrons as bullet-like projectiles hitting a target. But if the 
same shower is made to pass near to a suspended magnet, this is 
found to be deflected as the electrons go by. The electrons may now 
be pictured as octopus-like structures with tentacles or 'tubes of 
force' sticking out from it in every direction. 

It would, however, be wrong to think of an electron as a bullet- 
like structure with tentacles sticking out from its surface. We can 
calculate the mass of the bullet, and also the mass of the tentacles. 
The two masses are found to be identical, each agreeing with the 

known mass of the electron. Thus we cannot take the electron to be 


bullet plus tentacles this would give us twice too great a mass 
we must take it to be bullet or tentacles. The two pictures do not 
depict two different parts of the electron, but two different aspects 
of the electron. They are not additive but alternative; as one comes 
into play, the other must disappear. 

Actually the situation is even more complicated, since a separate 
tentacle picture is needed for each speed of motion of the electron, 
the speed being measured relative to the suspended magnet or 
other object on which the moving electron is to act. The reason is 
that already explained. When the electron is at rest, the tentacles 
stick out equally in all directions. But an electron which is at rest 


relative to one magnet may be in motion relative to another, and to 
discuss the action of the electron on this second magnet we must 
picture it as having a belt of tentacles round its waist. This shows 
that we must have a different picture for every speed of relative 
motion, so that the total number of pictures is infinite, and we 
cannot form the picture we need until we know the speed of the 
electron relative to the object it is about to meet. 


By the end of the nineteenth century the classical mechanics 
might almost be said to have met with complete success in 
explaining and predicting the phenomena of what we have called 
the man-sized world. It had also been very successful with the still 
larger scaled problems of astronomy, although missing complete 
success in a comparatively small group of problems which are now, 
we hope, in process of being cleared up by the gravitational theory 
of relativity. But at the other end of the scale there was no success 
at all ; experimental physics was particularly interested in the pro- 
cesses taking place inside the atom, and in this field the classical 
mechanics was failing conspicuously and completely. Perhaps its 
most spectacular failure was with the fundamental problem of the 
structure of the atom. 

Atomic Structure 

Experimental physics had provided strong reasons for thinking 
that an atom consists, of a collection of electrons negatively 
charged particles together with something which carries just 
enough positive electricity to counteract the total negative charge 
of them all for the total charge on a normal atom is always zero. 

Now there is no mechanism within the framework of the clas- 
sical mechanics for endowing such a structure with a permanent 
unchanging size. Its charges cannot stand at rest, or they begin to 
fall into one another, and they cannot be in motion or they become 
a perpetual-motion machine of the kind not permitted by the 
classical mechanics. Thus the mere permanence of the atom 
showed the need for a revision of the classical mechanics. 

And whatever system of mechanics we finally adopt, we should 


expect that the fixed and unchanging sizes of atoms could be 
calculated by combining the known constants of nature in some 
way or other. But the constants known to the classical mechanics 
cannot be combined to form a length of the requisite order of 
magnitude, and this seemed to suggest that some other fundamental 
constant of nature still remained to be discovered. 

The Problem of Radiation 

Another conspicuous failure of the classical mechanics was with 
one aspect of the problem of radiation. Here it predicted very 
general and particularly clear-cut results, which observation was 
found to negative completely. A simple illustration will explain 
the nature of the conflict. 

Imagine a crowd of steel balls set rolling about on a steel floor. 
If two balls bump into one another, their individual speeds and 
directions of motion will change, but the incident will not alter the 
total energy of motion of the balls. There must, however, be a 
steady leakage of energy from other causes, such as air resistance 
and the friction of the floor, so that the "balls continually lose energy 
and, after no great length of time, will be found standing at rest on 
the floor. The energy of their motion seems to have been lost, 
although we know that actually most of it has been transformed 
into heat. The classical mechanics predicts that this must happen; 
it shows that all energy of motion, except possibly a minute fraction 
of the whole, must be transformed into heat whenever such a 
transformation is physically possible. It is because of this that 
perpetual-motion machines are a practical impossibility. 

Precisely similar ideas are applicable to the molecules which 
form the air of a room. These also move about independently, and 
frequently bump into one another. The classical mechanics now 
predicts that the whole energy of motion will be changed into 
radiation, so that the molecules will shortly be found lying at rest 
on the floor as the steel balls were. In actual fact they continue 
to move with undiminished energy, forming a perpetual-motion 
machine in defiance of the classical mechanics. 

Why does the classical mechanics meet with such different de- 
grees of success in these two cases? Why does it fail so conspicuously 


for molecules of air, when it gave the right results for steel balls? 
The short answer is that we have passed from one to another 
of the three worlds we discussed on p. 42 from the man-sized 
world to the world of the electron. 

We can go further than this. It seems fairly clear although no 
absolutely compelling proof can be provided that if any system of 
bodies whatever is moving continuously in time and space under any 
system of laws whatever, provided only that there is a causal law so 
that one state is followed uniquely by another, then the final upshot 
of the motion must be that predicted by the classical mechanics 
all the energy of the bodies must be transferred from matter to 
radiation. This fallacious result is not, then, a peculiarity of the 
classical mechanics; it is given also by a very wide class of possible 
systems of mechanics. This being so, no minor modification of the 
classical mechanics can possibly put things right. Something far 
more drastic will be needed ; we are called upon to surrender either 
the continuity or the causality of the classical mechanics, or else the 
possibility of representing changes by motions in time and space. 

Motions in Time and Space 

Now these three concepts formed the foundation-stones of the 
philosophy of materialism and determinism to which the physics 
of the nineteenth century seemed to lead. Thus as soon as any 
one of the three has to be rejected, the philosophical implications 
of physics undergo a great change; the mechanical age has passed, 
both in physics and philosophy, and materialism and determinism 
again become open questions at least until we have seen what 
the new physics has to say about them. We shall discuss this new 
physics in the next two chapters, and its philosophical implica- 
tions in our final Chapter vn. 




With the coming of the twentieth century, there came into being a 
new physics which was especially concerned with phenomena on 
the atomic and sub-atomic scale. It brought with it a new way of 
interpreting the phenomena of inanimate nature, which was destined 
in time to sweep away all the difficulties besetting the old classical 
mechanics. A preliminary glance over the vast territory of this new 
physics reveals three outstanding landmarks. 

First we notice an investigation which Prof. Planck of Berlin 
published in 1899. His aim was so to amend the classical mechanics 
that it should fit the observed facts of radiation, and show why the 
energy of bodies was not wholly transformed into radiation. We 
have already seen that this was likely to involve giving up either 
continuity or causality or the representation of phenomena as 
changes taking place in space and time. Actually his investigation 
seemed to show that continuity had to be given up, suggesting that 
in the last resort changes in the universe do not consist of con- 
tinuous motions in space and time, but are in some v^ay discon- 

The classical mechanics had envisaged a world constructed of 
matter and radiation, the matter consisting of atoms and the radia- 
tion of waves. Planck's theory called for an atomicity of radiation 
similar to that which was so well established for matter. It sup- 
posed that radiation was not discharged from matter in a steady 
stream like water from a hose, but rather like lead from a machine- 
gun ; it came off in separate chunks which Planck called quanta. 
This, as we shall shortly see, carried tremendous philosophical 
consequences with it. 

An extension of Planck's ideas, due to Prof. Niels Bohr of 


Copenhagen, went on to suggest that, viewed through a microscope 
of sufficient power (this being far beyond anything attainable in 
practice), the ultimate particles of matter would be seen to move, 
not like railway trains running smoothly on tracks, but like kan- 
garoos hopping about in a field. 

A second conspicuous landmark in the field of the new physics 
is the enunciation of the fundamental law of radioactive disintegra- 
tion by Rutherford and Soddy in 1903. This law was in no sense 
a consequence or development of Planck's theories ; indeed fourteen 
pears were to elapse before any connection was noticed between 
the two. The new law asserted that the atoms of radioactive sub- 
stances broke up spontaneously, and not because of any particular 
conditions or special happenings. This seemed to involve an even 
more startling break with classical theory than the new laws of 
Planck; radioactive break-up appeared to be an effect without a 
cause, and suggested that the ultimate laws of nature were not even 

A theoretical investigation which Einstein published in 1917 pro- 
vides a third conspicuous landmark. It connected up the two great 
landmarks already mentioned by showing that the disintegration 
of radioactive substances is governed by the same laws as the 
jumps of the kangaroo electrons in the theory of Bohr. In fact 
radioactive atoms were now seen merely to contain a special breed 
Df kangaroos, much more energetic and ferocious than any that 
had hitherto been encountered. * 

The laws which governed the spontaneous jumps of kangaroos 
were shown to be of the simplest; out of any number of kangaroos 
i certain proportion always jumped within a specified time, and 
nothing seemed able to change this number. Also, before the 
jumps took place, there was nothing in the world of phenomena 
to distinguish those kangaroos that were about to jump from those 
that were not neither good nor bad treatment could make a 
kangaroo jump until it hopped out, apparently of its own accord, 
to help fill the quota demanded by the statistical law. As discon- 
tinuity marched into the world of phenomena through one door, 
causality walked out through another. We shall see later why this 
had to be. 



After this preliminary glance, let us turn to a more detailed survey 
of the situation. Planck's theory asserted that radiation was as 
atomic in its structure as matter, but with one essential difference. 
There are only ninety-two different kinds of atoms of matter or 
somewhat more when isotopic differences are taken into account 
but there are an infinite number of different kinds of radiation, 
these being distinguished by the different lengths of their waves. 
Planck found it necessary to postulate an infinite number of kinds 
of quanta or atoms of radiation, one for every length of wave. The 
energy contained in an atom, or quantum, of radiation is large 
when the wave-length is small, and vice versa. The precise relation 
is that the energy is equal to h times the frequency of the radiation, 
this being the number of complete wave-oscillations which occur 
at any specified point in a second, or again the number of com- 
plete waves which pass over the point in a second the two defini- 
tions are equivalent. The factor of proportionality h is found to 
be a universal constant of nature. It is generally known as Planck's 
constant, and incidentally has dominated atomic physics since its 
discovery. We have already seen (p. 124) that some such constant 
was much needed to give a definite size to the atom; here it was. 

The Photo-electric Effect 

Not only was Planck's theory immediately successful with those 
particular problems of radiation for which it had been especially 
designed, but further confirmation of its truth was soon forth- 
coming from entirely different quarters. Much of the evidence had 
been known for some time, but it needed an Einstein to point out 
its significance (1905). 

The evidence in its simplest form was provided by a phenomenon 
known as the ' Photo-electric Effect'. When ultra-violet radiation 
(p. 53) falls on a metal surface, a stream of electrons is found to 
be ejected from the metal. If radiation is pictured as waves, there 
is no difficulty in seeing in a general way why this should be; the 
incidence of the radiation may well shake the electrons about in 
the atoms of the metal, and under very intense radiation they might 


break loose altogether like boats breaking loose from their moor- 
ings in a stormy sea. Yet if this were the true explanation, weakening 
the radiation ought to result in the electrons being ejected with less 
energy, or perhaps not coming off at all. Actually a weakening of 
the radiation leaves the energy of each electron unimpaired, 
although reducing the number of electrons shot off. This number 
is proportional to the intensity of the radiation, so that even the 
feeblest stream of radiation produces a minute trickle of electrons 
in which each individual moves just as vigorously as in the bigger 
flow produced by more intense radiation; it is as though the 
radiation was a hail of projectiles, hitting some electrons out of 
their atoms, but leaving the rest untouched. 

Further, when an electron is ejected, the total energy it has 
absorbed from the radiation is found always to be exactly one 
whole quantum of the radiation. Not all of this energy will appear 
as energy of motion, since the electron must expend some of it 
in breaking loose from its atom, and more in fighting its way out 
through the other atoms to outer space. 

We have seen that radiation of low frequency has quanta of low 
energy and conversely. Radiation may be of such low frequency 
that the absorption of a quantum by an atom will not liberate an 
electron; the limiting frequency at which this change occurs is 
called the thresholdjjreguency . Thus radiation only liberates electrons 
when its "frequency is above the threshold-frequency. 

As the amount of energy required to set an electron free naturally 
depends on the properties of the atom to which the electron belongs, 
different substances have different threshold-frequencies. Those of 
most substances are well above the frequencies of visible light, so 
that the quanta of sunlight and of ordinary room lighting are too 
feeble to tear electrons off common objects. Even so, they may 
still carry enough energy to cause some rearrangement of the 
molecules of the substance on which they fall. Such rearrange- 
ment is known as photo-chemical action, and it is found that the 
absorption of a single quantum never affects more than one mole- 
cule this is known as Einstein's law of photo-chemical action. 
This chemical action of photons explains why bright sunlight 
causes our curtains and furnishings to fade, and why certain 


chemicals such as peroxide of hydrogen must be kept away from 
bright light if the molecules are not to change their composition. 
It explains too why blue and violet lights the lights of highest 
frequencies affect photographic plates more than lights of other 

When the frequency of radiation is above the threshold-fre- 
quency, electrons are torn off, and the energy of their motion 
obviously ought to be proportional to the excess of the radiation- 
frequency above the threshold-frequency; experiment confirms 
this law completely. 

The process we have been considering is the transfer of energy 
by radiation from matter at one place to matter at another place ; 
the experiments just mentioned show that this transfer always takes 
place by complete quanta. Confirmation is provided by Heisen- 
berg's contributions to the subject, which are discussed in the next 
chapter. Heisenberg finds that facts of observation lead uniquely 
and inevitably to the theoretical structure known as matrix 
mechanics. This shows that the total radiation in any region of 
empty space can change only by a single complete quantum at a 
time. Thus not only in the photo-electric phenomenon, but in all 
other transfers of energy through space, energy is always trans- 
ferred by complete quanta ; fractions of a quantum can never occur. 
This brings atomicity into our picture of radiation just as definitely 
as the discovery of the electron and its standard charge brought 
atomicity into our picture of matter and of electricity. 

The Atomicity of Radiation 

In 190^ Einstein proposed a pictorial representation of all this, 
which was in many ways reminiscent of the corpuscular theory 
by which Newton had tried to explain light two centuries earlier. 
Planck had supposed that an atom could only emit radiation, by 
complete units or quanta. Einstein now pictured each emitted 
quantum as travelling through space in the form of a compact and 
indivisible unit an unbreakable packet of radiation. Such a packet 
he called a light-arrow, although the more non-committal term 
photon is more usual to-day. 


According to this picture, a stream of radiation may be visualized 
as a shower., of photons. WEen this falls on a material surface, like 
a hail of arrows hitting a target, each photon will hit one of the 
electrons in the surface, and will do damage which is confined to 
the point of impact. This picture explains at once why weakening 
the stream of radiation does not stop electrons coming off, why 
doubling the intensity of radiation doubles the number of electrons 
and, more generally, why the two are proportional. 

Simple considerations of a general kind show that a free electron 
i.e. one which is not attached to an atom can never absorb a 
quantum of radiation. If, then, a light-arrow should strike such 
an electron, we must picture the two as colliding like two billiard- 
balls, and the collision will change the directions of motion of 
both.*; In 102? Compton and Simon were able to photograph the 

\ '-^.,__ * ...^V -IT- ... A. -is.. - 4. ,,. ..^Q -V - r rinmm, 

paths d electrons both before and after such * collisions ^and found 
that the light-arrows of Einstein's picture must be supposed to 
carry precisely^ tEe amounts of energy and momentum that the 
quantum theory demanded. 

The Undulatory Nature of Radiation 

While there is convincing experimental evidence that^ radiation is 
both emitted and absorbed in complete quanta, tEere is none to 
showtEat these quanta travel through . 3'page 

units suggested bv jffipsftein's picture, Indeed, there cannot be; 
it is only at the beginning of its journey, when it is emitted by 
-natter, and again at its, end, when it again interacts with matter, 
Jiat radiation can make its presen^Jsjaos^ 

But there is a great deal of evidence that light does not travel 
through space in the form of these unbreakable units ; there is in 
fact the evidence of the whole undulatory theory of light. It will 
be enough to illustrate this, by a single example, which shows the 
evidence in a particularly clear form. 

Suppose that light of pure colour, and so of uniform wave- 
length, is emitted from the source of light S in fig. i. Let us 
further imagine a screen AB> punctured by two movable pinholes 
at A and B, to be set up as shown, and let a second screen be 




placed behind it, the lines SA and SB meeting this second screen 
at the points P and Q. 

When the source S emits light, we should expect to find the 
points P and Q illuminated, while the rest of the screen remained 
dark. And so we do, so long as we do not examine the screen too 
closely ; at a cursory glance we might well think that photons had 
passed like arrows through the holes A and B. But a more careful 
examination shows that the illumination at P and Q consists of 
something more than the small circular patches of light which the 


Fig. i 

arrow picture of radiation would lead us to expect ; at each of the 
two points we find a complicated pattern consisting of concentric 
circles of light alternating with concentric circles of darkness. 

Before discussing this, let us extend our experiment by moving 
the pinholes A and B gradually nearer to one another. At first the 
patterns at P and Q simply approach one another in the way we 
should expect, but when they have come quite near to one another 
a new phenomenon occurs. The pattern we now observe can no 
longer be obtained by the mere addition of the two circular patterns 
at P and Q. These patterns have begun to interact with one another, 
and for certain positions of A and B, the points P and Q become 


completely dark. Keeping A and B in such positions, let us stop 
up the hole B. We find that the point P immediately changes from 
being dark to being light. If we unstop B y P becomes dark again. 
Thus to all appearances a decrease in the illumination adds to the 
light at P, while an increase subtracts from it. 

Such results obviously cannot be explained in terms of photons 
travelling like arrows through holes. The undulatory theory, on 
the other hand, explains them at once. It tells us that the illumina- 
tion at each point is produced by the combined action of two sets 
of waves, one coming through A and one through 5, and it is one 
of the commonplaces of physics that two such sets of waves can 
neutralize one another. The process consists in the crests of one 
set of waves coinciding exactly with the troughs of the other set, 
so that the effects of the two sets of waves just cancel out, and is 
known as interference. This not only provides a general explanation 
of the phenomenon, but also enables us to predict the pattern 

The Particle- and Wave-pictures 

We now have two distinct pictures of the nature of radiation. on 
depicting it as particles and thej)ther_as waves. The particle- 
picture is obviously the more suitable when the radiation is falling 
on matter, and the wave-picture when it is travelling through space. 
For a time there was a disposition to conclude that light niust con- 
sist of two parts, a wave part and a particle part, but it is now clear 
that this is not so. The t wave-picture and the particle-picture do 
not show two different things, but two aspectsjof the same thin|y 
They are simply partial pictures which are appropriate to different 
sets of circumstances like the two pictures of the electron which 
we introduced on p. 122 and so are complementary but not 
additive. As soon as light shows the properties of particles, its 
wave properties disappear, and vice versa; the two sets of pro- 
perties are never in evidence at the same time. Thus as we follow 
a beam of light, or even a single quantum, in its course, we must 
imagine the wave- and particle-pictures taking control of the situa- 
tion alternately. 
The wave-oicture explains much in its own proper province, 


(but it brings its own difficulties with it. In particular, it is not 
easy to pass back from the wave-picture to the particle-picture. 
For all waves scatter as they travel through space, and it is difficult 
to imagine how waves which have once scattered as the undulatory 
theory directs can recombine and concentrate their attack on 
single molecules or electrons in the way they are observed to do 
as soon as they encounter matter. 

Suppose, for instance, that the source S (p. 132) emits only a 
single quantum of light. If this travels through space in the form 
of the waves required by the undulatory theory, some of it must 
pass through the hole A and some through the hole JB, while the 
greater part will be absorbed or reflected by the screen AB. We 
cannot imagine all these various parts recombining and directing 
the whole of their energy upon a single molecule, either on the 
near or on the far side of the screen AB y so that our picture seems to 
fail entirely. We must always remember^that the actual physical 
processes are essentially unpicturable, but obviously their results can- 
not be obtained from any activities which we can imagine operating 
in time and space, so that we here obtain our first intimation that 
the space-time framework of the classical mechanics is inadequate 
for the complete representation of natural phenomena. 

The undulatory theory of light attained its most precise, and 
(as many then thought) its final, form in Maxwell's electro- 
ma^e^theory^of^ light. This interpreted theWaVes^f the ^n3ula- 
fory~theory as oscillating ele"HHc*an3 magnetic f qrceg. Jia^sllillg 
tErougK janjether. At each instant of time, there was at every point 
of this ether a definite electric force (which Maxwell tried to repre- 
sent as a Misplacement' of the ether), and a definite magnetic 
force just as, at any point on the surface of a stormy sea there is 
a definite elevation above, or depression below, the mean level of 
the sea. 

With the passing of absolute space, these ideas became un- 
tenable. The theory of relativity washed away the ether, and not 
onlyshoweT^ different measures 

to the forces at the same point and at the same instant of time, but 

ja _ B ,i "* ......la, .....-..* in* mi it iiiiiii - J *~ "~*^**" w "" "* ii. nim. , ^^^^^^^ JJU||[|)M , ^.-^iu..i ' " 

also that the^ could all b^equally, right. T^^qH?alKdL^ec^iji^nd 
magnetic forces, then, are not physical realities, as. for instance. 


displacements of the ether would have been; the^jgejot evgn_ 
objective^ but are subi^twe^eiHarc^onstructs which we have 
made for ourselves in our efforts to interpret th^jwaves ,of the 
ui^ulatoiyTh'eofyr In3eed, as they were created in an attempt to 
a mechanical expIanatioK""6rthe" 

come under the same l^tmitefmral^ 

forces with which we tried to explain the action of an electric charge 
(p. 119) and, mutatis mutandis, for the same reasons. Clearly we 
must search for a better interpretation of the waves of the undula- 

tory theory. r f 7 T . T . 

Waves of Probability 

Let us return to the imaginary experiment of p. 134, in which a 
single quantum of radiation is emitted from a source of light to fall 
on one point or another of a system of distant screens. We know 
that the whole energy of the quantum will concentrate on a single 
point of the screens, but which point will it be ? 

The obvious answer is that sometimes it will be one point, and 
sometimes another. It cannot always be the same point, or else 
when quanta were being emitted in millions, this one specially 
favoured point would be intensely bright and all others completely 
dark. Actually when quanta are being emitted in millions, there 
are some places on the screens at which the illumination is very 
bright, these indicating regions in which many photons have struck, 
and also places of less illumination, these indicating regions in 
which few photons have struck. Even the most faintly illuminated 
parts of the screen must have been struck by some photons. 

If we now fix our attention on a single quantum of radiation of 
which we know nothing except that it belonged to the original 
beam, we may say that the extent to which either screen is illu- 
minated at any point gives a measure of the probability that the 
quantum shall condense into a photon at this point. In this way 
we may interpret the waves of the undulatory theory as waves 
of probability ; the extension of the wave system in space marks 
out the region within which a photon may be supposed to be 
travelling, while the intensity of the waves at each point within 
this region gives a measure of the probability that a photon will 
occur at that point if matter is placed there. 


When half a million babies are born in England in a year, we 
may say that 20 per cent of them are born in London, 2 per cent 
in Manchester, i per cent in Bristol, and so on. But when we 
think of the one baby born in a single minute of time, we cannot 
say that 20 per cent of it is born in London, 2 per cent in Man- 
chester, and so on. We can only say that there is a 20 per cent 
probability of its being born in London, a 2 per cent probability 
of its being born in Manchester, and so on. If we disregard varia- 
tions of birth-rate with locality, a map exhibiting the density of 
population in different parts of England will also act as a chart 
showing the number of births per annum ; but with reference to 
the birth occurring at any one instant, it merely shows the relative 
probabilities of the baby appearing in different areas. As soon as 
the waves of the undulatory theory fall on matter, they provide a 
precisely similar chart for the probability of photons appearing in 
the different areas of the matter. The waves, then, are again mental 
constructs not enabling us to see what will happen, but what may 

Waves of Knowledge 

The waves may equally well be interpreted as representations of 
our knowledge. In the experiment with the single photon, we do 
not know where the photon is, but the wave-picture gives a sort 
of diagrammatic representation of what we do know. We know* 
that the photon must be within a certain region of space, this being 
the region mapped out by the waves at each instant. We may know 
that it is much more likely to be in a region A than in some other 
region B ; if so, the waves represent this knowledge by being much 
more intense in the region A than in the region , and so on. 

These two interpretations of waves as representations of prob- 
ability and of knowledge are well illustrated in an idealized 
experiment imagined by Einstein and Ehrenfest. 

An ordinary glass mirror functions because a thin coat of 
silvering on its back reflects all light falling on it. The silvering 
can be made so thin that the mirror will reflect only a part of the 
light falling on it for simplicity let us suppose half while the rest 
goes through to the further side and continues on its way as 


though the mirror had not been in its path. When a beam of 
radiation falls on such a mirror, we must imagine that half of its 
quanta are reflected and half transmitted. 

But suppose that only a single quantum falls on the mirror. As 
quanta are indivisible, we must picture the whole of the radiation 
as going either one way or the other ; the most we can say is that 
there is a 50 per cent chance that it will be reflected, and a 50 per 
cent chance that it will be transmitted. 

So far the waves have been figuring as representations of prob- 
ability, telling us the relative probabilities of the quantum being in 
one or other of the two paths. Suppose, however, that we now 
place a screen across the path of reflection, and allow a solitary 
quantum to fall on the mirror as before. If the quantum happens 
to be reflected, the screen will be struck by a photon, and we can 
detect its presence (in principle) in a variety of ways, mechanical 
or photographic. If the photon shows itself in the path of reflec- 
tion, the intensity of the waves in the transmitted beam is imme- 
diately reduced to zero. We may either say that this is because the 
probability that the photon is following this path has been reduced 
to zero, or that it is because we now know that the photon is not 
on this path. If, on the other hand, no photon is seen to strike 
the screen, the transmitted beam is immediately doubled in 
strength, while the reflected beam is annihilated, and the same two 
interpretations are available as before. It may seem odd that we 
can annihilate a beam of light by conducting an experiment at an 
unlimited distance away, but this becomes obvious when we con- 
sider that the beam is a representation of our knowledge, so that 
if our knowledge changes abruptly, the beam must also change 
abruptly. A simple analogy may clear up the matter and show that 
there is nothing mysterious or mystical about it. 

Imagine a ship crossing the Atlantic from New York to Southr 
ampton. The first day out, the ship's position would normally 
be determined by taking readings of the sun's altitude ; the navi- 
gating officer would then mark this position on the ship's chart. 
If the sky was cloudy, it would be necessary to fall back* on an 
approximate position calculated by " dead reckoning " ; the officer 
would know the approximate speed of the ship, or the distance it 


had travelled through the water as recorded by the log, and could 
make a rough allowance for the motion superposed by currents in 
the sea. He might in this way be able to fix his position to within, 
say, 5 miles. He could not mark a cross on his chart to fix his 
position, but might draw a circle 5 miles in diameter; this, like 
the waves of the undulatory theory, would represent his know- 
ledge of his position. As the ship progressed on its journey, we can 
picture this circle travelling over the chart, like a wave travelling 
through space, at a speed representing the speed of the ship. As new 
uncertainties accumulated, the circle would continually increase in 
size. If the sun was still invisible on the next day, it might be 
necessary to indicate the ship's position by a circle 10 miles in 
diameter. If the sun could not be seen throughout the voyage, the 
uncertainty as to the ship's position would continue to increase, 
until, by the time the ship was close to land, it might have to be 
represented by a circle 50 miles in diameter. Suppose, that when 
such a circle had been marked on the chart, half of it was found 
to lie over the Cornish coast. As the ship could not be on land, 
this half of the circle could at once be ruled out ; this bit of know- 
ledge would at once reduce the extent of the uncertainty to half 
just as happened in the experiment with the half-silvered mirror. 
If the Lizard was sighted a few moments later, the further know- 
ledge thus provided would reduce the uncertainty practically to 
zero, and the ship's position could now be marked by a point. 

This analogy clears up the physical situation in other respects. 
We know how in practical life one uncertainty leads to another; 
foij instance, the uncertainty which prevailed as to the ship's posi- 
tion when it was one day out continually increased ; this uncertainty 
made it impossible to allow exactly for the currents encountered 
on the second day's run, and as the voyage proceeded uncertainty 
was piled on uncertainty. The wave-picture of radiation faithfully 
reproduces this cumulative property of uncertainty in knowledge, 
because it is an inherent property of a group of waves always to 
spread out, and so occupy more space. 

In this analogy the ship represents a photon, the sea represents 
the space in which the photon moves, and the land represents 
barriers, such as the screen on p. 132, which prevents the photon 


moving through the whole of space. The sea, land, ship and photons 
all exist and move in the ordinary space of everyday life ; indeed 
this is what we mean by ordinary space the space in which we 
see things through the impact of photons on our retina, and travel 
by ship. But the waves which represent the navigator's knowledge 
of his ship's position do not travel through ordinary space, but 
over a nautical chart, which is a sort of diagrammatical representa- 
tion of ordinary space. In precisely the same way, the space tra- 
versed by those waves which represent our knowledge of photons 
is not ordinary space but a mathematical representation of ordinary 
space. If it contains barriers, these are representations of barriers 
in ordinary space like the coastline on a nautical chart. In brief, 
the space of photons is ordinary physical space, while the space 
traversed by the waves of the undulatory theory is a conceptual 
space. Indeed, it must be, sjnc.lh^waves, as we have seen^jure 
mere mentajjconstructs and possess no physical existence. 

It may be thought thafTF we are concerned only with mathe- 
matical representations, it is a matter of indifference whether we 
imagine them set up in ordinary space or in some conceptual space 
of our own construction. This is so, provided the two spaces have 
the same number of dimensions. And, as the waves of the undula- 
tory theory of light need a conceptual space of three dimensions 
for their representation, generations of physicists have identified 
this with ordinary physical space, and thought of light as waves 
travelling through the space of everyday life in which we travel 
by car or train. This is now seen to be a little irrational rather like 
marking out the time-table of a railway along the tracks. It can, 
however, find justification in the fact that an ordinary beam of light 
contains so many photons that probabilities may be replaced by 
actualities. When we take this step, the space through which prob- 
abilities of photons travel becomes identical with the space through 
which the photons themselves travel, and this is the space of 
everyday life the space in which we see things. In this way we 
come back to the view of light propagation which all physicists 
held as a matter of course before the quantum theory came to 
trouble them. 


The Uniformity oj l\atun 

Before the quantum theory appeared, the principle of the uni- 
formity of nature that like causesproduce like effects had been 
accepted as a universal and^ indisputable fact of science, AS soon 
asjhejttpmicity of radiation became established, this principle had 
to bejiiscarded. ~~ ~ ~ - 

In the experiment described on^p. 135, the uniformity of nature 
would have required that every -photon should hit the screen at 
the same point. Actually we have seen that they hit at different 
points, so that if a single quantum is discharged from the source 
several times in succession, different experiments will be found to 
give different results, and this although the conditions before the 
experiments were, so far as we could tell, precisely identical. 

The same thing is shown, even more convincingly, by the experi- 
ment with the half-silvered mirror. If we shoot solitary photons, 
one after another at the same point of the mirror, half of them will 
get through and half will not, so that again a succession of similar 
experiments will not give similar results. 

It may perhaps be objected that if the results of two experiments 
differ, this must be because either the conditions before the experi- 
ments, or else the conditions during the experiments, were not 
absolutely identical. If we shoot peas at a piece of wire-netting, 
~we~may find that half of them get through, while half of them hit 
the wires of the netting and fall back. If we only shoot a single 
pea, there is a 50 per cent chance that it will get through. If we 
shoot a second pea, aiming it so that it meets the netting at pre- 
cisely the same point as the first, and so making the conditions of 
the experiments absolutely identical, we may be sure that the 
experiments will have the same result; if the first pea gets through, 
the second will also get through. If the two peas were observed 
to meet different fates, we should conclude that the conditions of 
the two experiments had not been absolutely identical. It may be 
objected that similar considerations apply also to the experiments 
just described, and that if the two quanta of radiation had different 
experiences, the conditions of the experiments cannot have been 
absolutely identical. 


The conditions of the two sets of experiments are, however, not 
parallel. In the experiment with the wire-netting, all those peas 
which failed to get through, as well as many of those that got 
through, hit the wires of the netting at some point or other, and 
the exact point at which they hit decided both their fate in the 
matter of getting through, and also the angle at which their paths 
lay after impact. Both sets of peas moved at all sorts of angles. 
But in the experiment with the mirror, all the radiation which gets 
through moves along exactly the same path, and the same is true 
of that which is reflected. It follows that the angles of these paths 
are not determined by the positions of individual molecules, but 
by the direction of the surface as a whole, and this is sufficient to 
show that the phenomenon is not molecular or atomic. 

In thisjway we find that the atomicity of radiation destroys the 
principle of the uniformity of nature, and the phenomena of nature 
are no longer governed by a causal law or at least if they are so 
governed, the causes lie beyond the series of phenomena as known 
to us. If, then, we wish to picture the happenings of nature as 
stifl governed by causal laws, we must suppose that there is a sub- 
stratum, lying beyond the phenomena and so also beyond our 
access, in which the happenings in the phenomenal world are 
somehow determined. 

It is natural to wonder why the| atomicity of radiation carries 

nore far-reaching consequences man the similar atomicity of 

^ratter. But we shall soon see that the atomicity of matter entails 

recisely similar consequences, the only difference being that these 

have not been recognized for so long. 

The Principle of Uncertainty 

Of the further consequences which follow from the atomicity of 
radiation, one is of the utmost importance to physics as a whole, 
and especially to those aspects of it that are under discussion in 
the present book. Physics sets before itself the task of coordinating 
the various sense-data which reach us from the world beyond our 
sense-organs. If our senses could receive and measure infinitely 
delicate sense-data, we should be able in principle to form a per- 
fectly precise picture of this outer world. Our senses have limita- 


tions of their own, but these can to a large extent be obviated by 
instrumental aid ; telescopes, microscopes, etc. exist to make good the 
deficiencies of our eyes. But there is a further limitation which no 
instrumental aid can make good ; it arises from the circumstances 
that we can receive no message from the outer world smaller than 
that conveyed by the arrival of a complete photon. As these 
photons are finite chunks of energy, infinite refinement is denied 
us ; we have clumsy tools at best, and these can only make a blurred 
picture. It is like the picture a child might make by sticking 
indivisible wafers of colour on to a canvas. We might think we 
could avoid this complication by using radiation of infinite wave- 
length. For the quanta of this radiation have zero energy, and so 
might be expected to provide infinitely sensitive probes with which 
to explore the outer world. And so they do, so long as we only want to 
measure energy, but a true picture of the outer world will depend 
also on the exact measurement of lengths and positions. For this, 
long-wave quanta are useless. To measure a length accurately to 
within a millionth of an inch, we must have a measure graduated 
to millionths of an inch ; a yard-stick graduated only to inches is 
useless. Now quanta of one inch wave-length are, in a sense, 
graduated only to inches, while quanta of infinite wave-length are 
not graduated at all. Passing from quanta of short wave-length to 
quanta of long wave-length only shifts, but does not remove, the 

A rough analogy is to be found in the problem of photographing 
i rapidly moving object. A sensitized film can record no detail 
3n a scale which is smaller than the grain of the film, so that if we 
ase a large-grained film, all the fine detail of our picture will be 
blurred. If we try to escape this difficulty by using a film of veiy 
small grain, we merely cross over from Scylla to Charybdis; the 
speed of the film is now reduced so much that the picture is blurred 
:hrough its subject having moved appreciably during the time 
accessary for exposure. 

We shall return later to a more detailed discussion of the physical 
consequences of this. For the moment we pass to yet another 
consequence of the general fact that our knowledge of the outer 
vorld comes to us only through the impact of complete quanta. 


Subject and Object 

It used to be supposed that in making an observation on nature, 
as also in the more general activities of our everyday life, the uni- 
verse could be supposed divided into two detached and distinct 
parts, a perceiving subject and a perceived object. Psychology 
provided an obvious exception, because the perceiver and per- 
ceived might be the same; subject and object might be identical, 
or might at least overlap. But in the exact sciences, and above 
all inphysics, subject and object were supposed to be entirely 
distinct, so that a description of any selected part of the universe 
could hepreared whiqlLj^Quld be entirely independent jaLthe 
observer as \vcij_as of the special circumstances surTouiading him, 

The theory of relativity (1905) first showed that this cannot be. 
entirely so ; the picture which each observer makes of the world is 
in some degree subjective. Even if the different observers all make 
their pictures at the same instant of time and from the same point 
of space, these pictures will be different unless the observers are 
all moving together at the same speed ; then, and then only, they 
will be identical. Otherwise, the picture depends both on what 
an observer sees, and on how fast he is moving when he sees it 

The theory of quanta carries us further along the same road. 
For every observation involves the passage of a complete quantum 
from the observed object to the observing subject, and a complete 
quantum constitutes a not negligible coupling between the observer 
and the observed. We can no longer make a sharp division between 
the two ; to try to do so would involve making an arbitrary decision 
as to the exact point at which the division should be made. Com- 
plete objectivity can only be regained by treating observer and 
observed as parts of a single system ; these must now be supposed 
to constitute an indivisible whole, which we must now identify 
with nature, the object of our studies. It now appears that this 
does not consist of something we perceive, but of our percep- 
tions; it is not the object of the subject-object relation, but the 
relation itself. But it is only in the small-scale world of atoms and 
electrons that this new development makes any appreciable dif- 
ference ; our study of the man-sized world can go on as before. 


For instance, when an astronomer is observing the motion of a 
planet injth^jolar^ystem, it is emitting millions of quanta every 
second, some of which pass through tKe telescope of the astronomer 
and into his eye. B^npting tlie directions from which these arrive, 
he canjKSTqw and describe the motion of the planet across the sky. 
With the departure of each quantum, the planet suffers a recoil 
which changes its motion, but the changes are so minute that they 
may properly be disregarded. But it is different when a physicist 
tries to follow the motion of an electron inside an atom. He can 
only obtain knowledge of the internal state of the atom by causing 
it to discharge a full quantum of radiation, and we shall soon see 
(p. 146) that the emission of a quantum of radiation is so atom- 
shaking an event that the whole motion of the atom is changed, and 
the result is practically a new atom. A succession of quanta may 
give scraps of information about various stages of the atom, but can 
give no record of continuous change. In fact there can be no con- 
tinuous change to record, since every departure of a quantum breaks 
the continuity. 

For this^ reason it is futile to discuss whether the motion of th^ 
atom conforms to a causal law or not. The mere i'ormularion"of the 
law ot causality presupposes the~existence of an isolated objective 
system which an isolated observer can observe without disturbing 
it. The question is whether he, noticing that such a system is in a 
certain state at one instant, can or cannot foretell that it will be, 
in some other specifiable state at some future instant. But if there 
is no sharp distinction between observer and observed, this be- 
comes meaningless since any observation he makes must influence 
the future course of the system. 

In more general terms, we may say that the law of causality 
acquires a meaning for us only if we have infinitesimals at our 
disposal with which to observe the system without disturbing it. 
When the smallest instruments at our dispoaaLare photons^ and 
electrons, tfle law oi: causality becomes meaningless for usTexcept 
with jgference to systems containing immense numbers of photons 
and^ls^rons. For such systems the classical mechanics Eas already 
told us t^j^s^i^ tor other "System^ CSusatify becomes 

me^ungless so farjis our knowledge of the system is concerned ; 
if it controls the pattern of events, we can 


We have now seen that six important consequences follow from 
the mere fact of the atomicity of radiation, coupled with those well- 
established facts of the iindiilatgryj^ory of light that have been 
mentioned jrhese^are : 

(1) So far as the phenomena are concerned, the uniformity of 
nature disappears, 

(2) Precise knowledge of the outer world becomes impossible 
for us. 

(3) The processes of nature cannot be adequately represented 
within a framework of space and time. 

(4) The division between subject and object is no longer definite 
or precise; complete precision can only be regained by uniting 
subject and object into a single whole. t 

(5) So far as our knowledge is concerned, causality becomes 

(6) If we still wish to think of the happenings in the phenomenal 
world as governed by a causal law, we must suppose that these 
happenings are determined in some substratum of the world which 
lies beyond the world of phenomena, and so also beyond our access, 

Bohr's Theory of Atomic Spectra 

Let us now pass from the general inferences to be drawn from the 
quantum theory to particular developments of it. Perhaps the most 
striking of these appeared in 1913, when Bohr suggested that it 
would provide a solution of the long-standing puzzle of atomic 

In 1911 Rutherford Mdpictured the atom as a miniature solar 
system a crowd of electrons revolving round a massive Central 
nucleu^ tHe electron had to be in orbital motion round the nucleus 
to esca P e falling into it. We have already seen (p. 123) that such a 
picture was incompatiblewitE the classical mechanics; according 
to tHis, the electron would continually'radiate energy as the result 
of its^orbitar'motion, and so would gradually spiral down into- 
the nucleus, which would finally absorb it. Thus atomswould be 
temporary structures~oFvaned and ever-varying sizes. 

Bohr planned to remedy these defects by introducing an atomi- 
city of energy into the atom itself. We can explain this sufficiently 

JP 10 


by considering the simplest kind of atom-^-the hydrogen atom, in 
which only one solitary electron revolves round the nucleus. Bohr 
assumed that the atom could not be of any size whatever, but only 
of the sizes in which it contained i, 2, 3, . *. or some other integral 
number of quanta of energy. Hitherto the energy of a quantum 
had always been h times the frequency of the radiation to which 
the quantum belonged, but there was now no radiation to provide 
a measure of frequency. Bohr accordingly measured his quanta 
against the frequency with which the electron described its orbit. 

In this way he avoided both the continual diminution of size 
of the atom, and its continuous leakage of energy, but there was no 
opportunity left for the atom to radiate at all. Yet hydrogen atoms 
certainly could both emit and absorb radiation. He accordingly 
supposed that tEe electron did nor permanently"remain in the same 
orblFin the electron, but oc^asionalljMimpedfrom one of the 
permittee! orfiits to another-~thegJre the kangaroo jumps of 
wWch~W6^ha^^TfeaHy spoken (p. 127); again the process is un- 
picliTfable"ifilfs ulfimat^ails. Whenever 


Qrbrt7'ffielntfmsic energy of*tEe"Hom changed, so that energyjwas 
either liberated or absorbed. Bohr supposed that, in eSigr case, 

orabsorbed formed preciselyone quantum 

ofradiation. This of course fixed the frequency of the radiation. 
In every previousjaj^licafi^ law, 

that the ~ energy isjk times the_frec[ueBcy, had been usei^to 
deduce the energy of & quantum when the frequency JQ the 
radiation was already known. In the present case the formula was 
used the other way; the energy~oF the emitted ^EHoJnTwas^ known 
tcrtregifl :with7^3TEe7ormuIa^was utilized to deducelts frequency. 
I'he^'equencigs^calculated in this way are 


completely and exactly with those observed ii^hjpggtrum of 
hydrogen. ~ ~~ 

This" spectrum is of the type known in spectroscopy as a line- 
spectrum. Its appearance is that of a group of bright lines on a 
dark background, indicating that the radiation divides itself be- 
tween a number of clearly defined- frequencies, and that there is no 
radiation in between. Before Bohr's explanation appeared, these 
frequencies had been supposed to belong to some sort of vibration 


taking place in the hydrogen atom like the frequencies of the 
musical note which is heard when a bell or piano-wire is made to 
vibrate. It now became clear that they had an entirely different 
origin. The energy exhibited in the spectrum was not liberated 
by a vibration, or by any kind of continuous motion, but by the 
sudden jump of an electron to an orbit of lower energy, and its 
frequency was determined by the compulsion put upon it to forrrr 
a smgle quantum. "" 

Tin ther same year in which Bohr produced this revolutionary 
theory, Franck and Hertz passed a beam of slowly moving electrons 
through a gas, and measured the amounts of energy that individual 
electrons yielded up to the molecules of the gas at collisions. The 
various amounts which the electrons were found to have lost proved 
always to be one or other of the various amounts needed to raise 
the atoms from one of the states permitted by Bohr's theory to 
another. This showed that these states had a real existence, and 
that transitions between them actually occurred. 

To sum up, the success of Bohr's theory suggested that an atom 
was not a continuously varying structure, from which radiation 
trickled 'away 7 like gas from a ; leaky balloon, but a structure which. 
emitted and__absorbed radiation in definite packets at definiteJn- 
stants l)f Jtime. Thusjiie energy of the atom did not vary con- 
tinuously, butfiumped suddenly at these instants from one value 
to another. Only certain definite calculable values were permissible 
for IKeTenergy ; these formed a chain of * energy-levels', arranged 
like the steps of a laddei;, and the energy of an atom could step 
from any one to any other but could not stand suspended in mid- 
air between two steps. When an atom stepped to a lower energy- 
level, its constituents rearranged themselves suddenly like the 
collapsing of a house of cards. 


The second great lamTmarE in the new physics j|s the discovery 
of the fundamental laws of radioactivity by Rutherford and Soddy. 
In 1898 and the immediately succeeding years, Becquerel and 
the Curies had discovered a group of substances, subsequently 
known as 'radioactive substances', which possessed very unusual 



properties, such as a capacity to fog photographic plates kept in 
their vicinity, and to stand permanently at a slightly higher tempera- 
ture than the objects surrounding them. In due course the explana- 
tion was found ; the newly discovered substances not only emitted 
the normal radiation appropriate to their temperature, but other 
and additional radiation as well radioactive radiation as we may 
call it from sources which seemed to be internal to the atom. 
This radiation was finally traced to its origin or rather origins, 
for there proved to be three, all of the nature of internal explosions. 

Each atom of a radioactive substance can be pictured, likejrther 
atoms, as a central nucleus with its crowd of surrounding electrons. 
The central nucleus must not be pictured as a structureless particle, 
but aS"^n05inpt^^fr^geihent oFmany constituents. These con- 
stituents, it was found, may suddenly rearrange themselves, and 
in so doing may eject either a massive particle (known as an 
a-particle) or a very rapidly moving electron (known as a j8-particle) 
or a quantum of very high frequency radiation (known as a y-ray). 

These three processes may all be included under the common 
term * radioactive transformation '^ince each transforms the original 
radioactive atom into something different. It was soon found that 
most radioactive substances had their own characteristic type of 
radiation, each atom of a substance A transforming into an atom 
of some other substance fi, this into an atom of C, and so on. Thus, 
apart from unimportant exceptions, radioactive transformation 
follows a single one-way track with no branches. 

The next step was to investigate the speed with which an atom 
travels ~*aiong this path. Ordinary radiation is emitted at a rate 
which is determined by the temperature of the emitting substance, 
hot matter emitting radiation profusely and cold matter meagrely. 
It might not unreasonably have been expected that the same would 
be true of radiqactive emission, but experiment showed that it is 
not. Two similar masses of radioactive substance may be taken, 
one of them heated to the highest, and the other cooled to the 
lowest, of the temperatures available in the laboratory, and both 
will still emit their radioactive radiation at exactly the same rate 
as before. 

The same is found to be true of all other physical chancres. Tn a 


milligram of radium, about 500 million atoms disintegrate every 
second, each giving out its characteristic radiation, and nothing 
that can be done to the radium or to its environment will change 
either the number of atoms which disintegrate, or the quality of 
the resulting radiation. The radiation may be described as spon- 
taneous in the sense that its Amount and quality are determmeH 
fr6m inside and not from outside. 

Such Is^^fundaniental law of all radioactive disintegration, 
whfch Rutherford jand Soddy enunciated in iQQ.^. it was entirely 
different in^clwacteFfirom any natural law hithertCKlmown, and 
made it clear that nature proceeded on a plan,. which was entirely 
different from anything hitherto suspected^ 

Interesting but-'difficult questions arise when we discuss which 
atoms will disintegrate first, and which will survive longest without 
disintegration. In the particular instance just given, 500 million 
atoms are due to disintegrate in the next second. What, we may 
inquire, determines which particular atoms will fill the quota? 

It cannot be anything in the present physical condition or en- 
vironment of individual atoms, for if it were, we could make more 
or fewer atoms disintegrate by modifying the physical state of the 
radium as a whole, and so altering the states of individual atoms. 
Neither can it be anything in the past histories of the atoms, for 
if it were, assemblies of atoms with different past histories would 
show different rates of disintegration, and this again is contrary 
to the facts; the rate of disintegration is found to be precisely the 
same for young atoms of, radium which have just been formed by 
the disintegration of heavier elements as it is for old veterans which 
are the sole survivors of a stock of radium many thousands of years 
old. Clearly, then, it is not a case of the young surviving and the 
old falling. We must rather picture the atoms of radium as drawing 
lots, young and old on the same footing like shipwrecked mariners 
on a raft drawing lots to determine which is to survive. But there 
is no drawing of lots in nature, so that the choice of one particular 
atom rather than another appears, from our present point of view, 
to be an event without a cause. 

While the interest of all this to physics was immense and far- 
reaching, the interest to philosophy was, if possible, even greater, 


since it seemed to remove causality from a large part of our picture 
of the physical world. We have, let us say, half a million atoms of 
radium in this room. If we are told the position and the speed of 
motion of every one of them at any moment, we might expect 
that Laplace's super-mathematician would be able to predict the 
future of every atom. And so he would if their motion had con- 
formed to the classical mechanics. But the new laws merely tell 
him that one of his atoms is destined to disintegrate to-day, another 
to-morrow, and so on. No amount of calculation will tell him 
which atoms will do this ; we must rather picture Fate as picking 
out her atom, by methods undiscoverable by us. This will then 
eject its oc-particle, which will proceed to mix with the other atoms 
and disorganize their motions in one wjay if it is atom A that 
disintegrates, but in some quite other way if it is atom J3. From 
the state of the matter at one instant, it is impossible in principle 
to discover what the state will be at a future instant. 

Einstein's Synthesis 

A third landmark was reached in 1917, when Einstein linked up 
these surprising (as they then seemed) laws of radioactive trans- 
formation with the equally surprising laws of Planck's quantum 

We have seen how the electrons in an atom can rearrange them- 
selves in new positions of higher or lower energy, and we hav 
compared falls into positions of lower energy with the collapsing 
of a house of cards. A cannon-ball at,_say, 1000 F. consists of 
atoms of iron, and while the majority of these are in the collapsed 
state, some are in states of higher energy, like standing houses of 
cards. A wind blowing over a town of card-houses may blow some 
down, but it may also cause some which have already been blown 
down to stand up again or so we may imagine for purposes of 
illustration. It is much the same inside the cannon-ball. Every 
small particle of it is emitting radiation in all directions, and as this 
radiation falls on the atoms it may change their condition, causing 
some of the standing houses of cards to collapse and some of the 
collapsed houses to stand up again. If this were all, it would be 
easy to discover how many houses of cards would be standing at 


any assigned temperature, how many would be fallen down, and 
what the constitution of the radiation would be. But the results 
obtained on this hypothesis do not agree with the facts of observa- 

Einstein obtained agreement, brilliantly and completely, by the 
introduction of a single additional supposition. He supposed that 
the standing houses of cards could not only be knocked down by 
the impact of radiation, but that they could sometimes collapse 
of themselves in the same way, and according to the same laws, 
as atomic nuclei collapse in radioactive disintegration, the rate of 
collapse being entirely independent of environment and physical 

In its new appearance, the law is not concerned with the rather 
recondite phenomena of radioactivity, but with familiar everyday 
radiation: it governs the radiation which the sun showers on the 
earth by day, as well as the light of the electric torch which lights 
our footsteps at night. Every atom in the universe is not only 
liable to spontaneous collapse, but also does collapse at frequent 
intervals. Thus the abdication of determinism appears to be com- 
plete, not only from the domain of radioactivity, but from the 
whole realm of physics. 

Determinism in Nature 

had hithertobeen^ based on the ^assumption _ofjhe 
uniformity of nature -like causes produce like effects^r an d if_thifi 
fails, the whole of^science_^ould seem to be left hanging in the air, 
for its existence, and no explanation of its 

success. Yet the success is indisputable, and explanation there 
must be. 

"The explanation is twofold. In the first place, the indeterminism 
disclosed by the quantum theory is confined to the small-scale 
processes of nature, and in the second place even these indeter- 
minate events are governed by statistical laws. In all man-sized 
phenomena, billions of electrons and atoms are involved, and for 
the discussion of such phenomena as are perceptible to us, these 
may be treated statistically as a crowd. But these crowds obey 
statistical laws which now take control of the situation, with the 


result that the phenomena can be predicted with almost the same 
precision as though the future motion of each particle were known. 
In the same way, the statistician, knowing the birth-rate, death- 
rate, etc. of a population, can predict the future changes in the 
population as a whole, without being able to predict what each 
separate individual will do in the matter of births and deaths. On 
the man-sized scale, and indeed far below down to pieces of 
matter far too small to be seen in any microscope nature is, 
to all appearances, strictly deterministic; like causes produce like 
effects. Thus the uniformity of nature is re-established except in 
the realm of the infinitesimal, and science can justify the funda- 
mental assumption on which her existence rests. We see why 
determinism has become ingrained in our modes of thought, and 
how Descartes and his followers came to announce it as a priori 
knowledge which they saw by the clear vision of their intellects. 
Nevertheless, it may not be true for those ranges of nature which 
were not accessible to them. 




The new physics just described was still based largely on New- 
tonian ideas. Indeed, in its theoretical aspects, it might not un- 
fairly be described as a final attempt to explain the world in 
materialistic terms as particles being pushed and pulled about in 
space and time. Nevertheless, the new physics had found it neces- 
sary to abolish most of the forces of pushing and pulling, replacing 
the gradual changes of motion of the particles under these forces 
by sudden and unpredictable jumps. These appeared to involve 
violations of the law of causality, both in the disintegration of radio- 
active atoms and also in the internal changes of ordinary atoms. 
We seemed to see Fate defying this law as she picked out certain 
atoms for disintegration or collapse and, by her apparently capri- 
cious acts, sent the universe along one path or another according 
to her whim. 

On such lines the new physics had explained many phenomena 
which had hitherto seemed inexplicable, but it had by no means 
met with complete success. For instance, while it gave a perfect 
interpretation of the simplest spectrum of all, namely that of the 
hydrogen atom, it failed with more complex spectra. This was not 
necessarily a fatal objection; a few emendations and possibly a few 
new ad hoc assumptions might have effected a complete reconcilia- 
tion, although this seems improbable. What seemed far more 
serious to many was that success had been achieved only at the 
price of ejecting continuity and causality from the scheme of 
nature, and replacing the exact laws of the classical mechanics by 
an assemblage of statistical laws and this without disclosing any 
reason why these statistical laws should be obeyed. 

Perhaps this ought not to have been a matter for surprise. We 
have already seen that the erroneous predictions of the classical 
mechanics are probably inevitable in any scheme which pictures 


physical processes as happenings in space and time, and also 
assumes causality and continuity in these happenings. Planck's 
original quantum theory attempted to remedy these shortcomings 
by postulating processes of a very novel kind, but these were still 
supposed to occur in space and time. This being so, it was almost 
a foregone conclusion that either causality or continuity would 
have to be renounced, and there was no special reason for surprise 
when it was found necessary to renounce both. These general con- 
siderations were not, however, widely appreciated at the time, so 
that few scientists and perhaps even fewer philosophers were pre- 
pared to accept the discontinuities and indeterminism of the old 
quantum theory as final. 


In 1925 Heisenberg made a new attempt, on entirely novel lines, 
to obtain an explanation of atomic spectra. Working in collabora- 
tion with Bohr, he had come to the conclusion that the imper- 
fections of Bohr's earlier theory had been the consequence of 
assuming too simple a model for the atom. For Bohr had not only 
assumed that the atom consisted of particles moving in space and 
time, but also that the particles inside atoms were of the same kind 
as the electrons outside atoms. 

Now the electron, can never be seen directly. The nearest ap- 
proach to this is in the Wilson cloud-chamber, where we may see 
the condensation trail which an electron leaves behind it as it 
pushes its way through the molecules of gas, much as we see the 
condensation trail left by an aeroplane high up in the sky when 
we cannot see the aeroplane itself. There is much more evidence 
of a similar kind, but all of it refers only to electrons outside atoms ; 
the electron inside the atom remains unobserved and unobservable, 
and there is no solid justification for supposing that it resembles 
the electrons we see (or so nearly see) outside. We may watch the 
sparks fly as the blacksmith hammers a piece of iron into a horse- 
shoe, but we must not infer that the piece of iron is an accumula- 
tion of sparks, each having the properties of those we see flying 
through the air. 


Bohr's investigation had typified what had become a standard 
procedure in problems of theoretical physics. The first step was to 
discover the mathematical laws governing certain groups of phe- 
nomena ; the second was to devise hypothetical models or pictures 
to interpret these laws in terms of motion or mechanism; the 
third was to examine in what way these models would behave in 
other respects, and this would lead to the prediction of other 
phenomena predictions which might or might not be confirmed 
when put to the test of experiment. For instance, Newton had 
explained the phenomena of gravitation in terms of a force of 
gravitation ; a later age had seen the luminiferous ether introduced 
to explain the propagation of light and, subsequently, the general 
phenomena of electricity and magnetism ; finally Bohr had intro- 
duced electronic jumps in an attempt to explain atomic spectra. 
In each case the models had fulfilled their primary purpose, but 
had failed to predict further phenomena with accuracy. 

Heisenberg now approached the problem from a new philo- 
sophical angle. He discarded all models, pictures and parables, 
and made a clear distinction between the sure knowledge we gain 
from observation of nature and the conjectural knowledge we 
introduce when we use models, pictures and parables. Sure know- 
ledge, as we have already seen, can only be numerical, so that 
Heisenberg's results were inevitably mathematical in form, and 
could not disclose anything about the true nature of physical pro- 
cesses or entities. 

As Heisenberg was concerned primarily with the problem of 
atomic spectra, he found his main observational material in a mass 
of measurements on the frequencies of the light emitted by the 
atoms of the chemical elements. 

A great deal of regularity had already been detected in these 
numbers. In 1908 Ritz had noticed that they were the differences 
of a set of even more fundamental frequencies, being of the form 
a b,b c,a c, etc. where a, 6, c, . . . were the more fundamental 
frequencies. These fundamental frequencies were further known 
to fall into groups, the numbers in any one group being associated 
with the series of integers i, 2, 3, 4, .... Bohr had further dis- 
covered that the frequencies corresponding to very large integers 


could be calculated accurately from the classical mechanics; they 
were simply the number of times that an ordinary electron would 
complete the circuit of its orbit in one second when it was at a 
very great distance from the nucleus of th6 atom to which it 
belonged. This could only mean that when an electron receded 
to a great distance from the nucleus of its atom, it not only assumed 
the properties of an ordinary electron, but also behaved as directed 
by the classical mechanics. Yet the classical mechanics failed com- 
pletely for the calculation of frequencies corresponding to small 

A similar situation had occurred in astronomy, where the New- 
tonian law of gravitation had been found to predict the orbits of 
the outer planets with great accuracy, but had failed with the orbits 
of Mercury and Venus. The relativity theory of gravitation had 
provided the needed modification of Newton's law, and in working 
out the details of the new theory, Einstein had utilized the fact 
that the Newtonian law gave_the fipht result at jjreat distances 
from the sun^ Heisenberg, confronted with a similar prbBIem, was 
able to avail himself of the fact that the classical mechanics gave 
the right result at great distances from the atomic nucleus. Here, 
and here alone, Heisenberg's theory made contact with the world 
of the older physics. For the classical mechanics was based on 
the conception of particles moving in space, so that through it 
Heisenberg's theory entered into relation with space, motion and 
material particles. 

Thus in the outer regions of the atom, Heisenberg's theory 
coincided both with the classical mechanics and with the newer 
theory of Bohr. In the interior of the atom, Bohr had tried the 
plan of retaining the particle-electron and modifying the classical 
mechanics. Heisenberg took the opposite course, his procedure 
amounting in effect to retaining the classical mechanics, at least 
in form, and modifying the electron. Actually, the electron dropped 
out altogether: it had to, because it exists only as a matter of 
inference and not of direct observation. For the same reason, the 
new theory contains no "mention of atoms, nuclei, protons, or of 
electricity in any shape or form. The existences of all these are 
matters of inference, and Heisenberg's purely mathematical theory 


could no more make contact with them than with the efficiency of 
a turbine or with the price of wheat. 

Developing these ideas mathematically, Born and Jordan 
showed that the classical mechanics can account for all spectral 
phenomena, provided entirely new meanings are given to such 
symbols as the p and q which had hitherto been taken to describe 
the position and motion of an electron. The things represented 
by these symbols acquire new properties which make it impossible 
that they should any longer represent the simple momentum and 
distance of a moving particle. In fact, they cease to be mere 
quantities of any kind, each becoming a whole group of quantities. 

The most significant of the new properties is that the product 
pq is no longer the same thing as the product qp in other words, 
the order in which the two factors are multiplied together is no 
longer a matter of indifference. The difference between pq and qp 
is found to be always the same, being Planck's constant h multiplied 
by a numerical multiplier. 

This last relation in combination with the canonical equations, 
which are taken over complete from the classical mechanics/ pro- 
vide sufficient mathematical relations for the solution of any 
problem of quantum mechanics, and, so far as is at present known, 
invariably lead to the correct solution. Here, then, so far as we 
can at present see, the true description of the pattern of events 
must lie. 

It may be thought that there is one relation more in the quantum 
than in the classical mechanics, namely that just mentioned, which 
gives the value of pq qp- But this is not so ; pq qp has one value 
in the quantum mechanics and a different value, namely zero, in 
the classical mechanics. The real difference is that the value of 
pq qp is mentioned explicitly in the quantum mechanics, but not 
in the classical mechanics, where p and q are tacitly assumed to be 
of such a nature that pq must be equal to qp. 

Even when this is agreed, it may still seem that the quantum 
mechanics must represent a complete break with the classical 
mechanics, since pq qp has entirely different values in the two 
systems. But again this is not so. Suppose we use the quantum 
mechanics to solve a problem on the man-sized scale ; p and q are 


now so large that pq is an immense multiple of h, and so also of 
pqqp- But this is only to say that, to a very close approximation, 
pq may be taken equal to qp, and we are brought back to the classical 

Thus in problems in which pq is a large multiple of h, the 
quantum mechanics necessarily gives the same result as the 
classical mechanics, while in problems in which pq is not a large 
multiple of h, it provides a genuine extension of the classical 
mechanics. Heisenberg's quantum mechanics is universally true, 
and the classical mechanics is merely a special case of it. 

When a problem is solved by the classical mechanics, the solu- 
tion we obtain depicts continuous motion and change ; when it is 
solved by the quantum mechanics, the solution tells us of jumpy 
motions and changes of the kind we have already met in Bohr's 
theory of the hydrogen atom if the solutions of the classical 
mechanics describe a ball rolling down an inclined plane, those of 
the quantum theory describe it as bumping down a staircase. The 
amount of each jump is proportional to h y so that in problems in 
which pq is a large multiple of h, each jump is so small compared 
with the main motion that the succession of jumps becomes in- 
distinguishable from continuous motion. In this way the jumps 
of the quantum theory merge into the continuous motion of the 
Newtonian mechanics. 

Pictorial Representations 

If, as now appears fairly certain, Heisenberg's system describes 
the true pattern of events, it is natural to inquire whether any 
pictorial representation of the system can be obtained. 

The simplest course is to try to imagine that p and q still specify 
the position and momentum of a moving something, this unknown 
something becoming identical with the familiar electron when it is 
at great distances from the atomic nucleus, but this is of no real 
value, since our minds cannot imagine any kind of structure for 
which pq would be different from qp. If we wish to obtain a really 
helpful representation, our primary problem must be to find some 
interpretation of p and q, such that the order in which- p and q 
are compounded shall not be a matter of indifference. The simplest 


procedure is to picture p and q as some sort of operators, since the 
order in which operations are performed is not usually a matter of 
indifference. Fining a man 100 and then confiscating half his 
fortune is not the same thing as confiscating half his fortune and 
then fining him 100. The difference to the victim is 50, and this 
corresponds to the value of pqqp on Heisenberg's theory. 

At an early stage in the development of the theory, Born and 
Wiener found very simple operators which satisfied the require- 
ment that pq qp should be equal to a constant quantity. But 
before this had occurred, other attempts to improve on Bohr's 
theory had resulted in yet another form of the quantum theory, 
the form which is jasually^, described as the wave_mechanics. This 
was of a much more physicaljriature than the ^^tx^j^^^E^^cal 
theo^^F^lsenE^rg, and" led to a picture of atomic j^rocesses 
which was not altogether unlike^ tjiatj^r^nteTT>y Bohr's earlier 

Now the replacement of p and q in Heisenberg's theory by the 
operators just mentioned was found to lead exactly to the equations 
which had already been found to express the wave mechanics. The 
wave mechanics accordingly falls naturally into place as a pictorial 
representation of the more general quantum mechanics of Heisen- 
berg. In its mathematical implications, it can be shown to be 
completely equivalent to Heisenberg's quantum mechanics, and 
has shown itself able, in principle, to aolve every problem so far 
solved by the quantum mechanics. *But we must be on our 
guard against supposing that the two are exactly equivalent; it 
must always be remembered that the quantum mechanics consists 
of a statement of facts in abstract mathematical form, whereas the 
wave mechanics consists of a pictorial representation of these 
facts in which the pictorial details may or may not correspond 
truly to the realities of nature.} irf 

Before proceeding to describe this wave mechanks, it will be 
convenient to mention some experimental results which are of 
importance for its understanding. 


As s^jmcejdiJ^ 

atoms and electrons were discovered in turn. The last of these 

a PB?H!JJ^J^^~^^ I the 

electron or of the _electrqnic^iaige. 

A current of electricity, such as carries our telephone messages 
or rings our electric bells, consists of ^^r^nL^^dectrons all 
moving in the same direction. SucE currents can not only be passed 
throjjgh^ solids,Jiquids^and gases, but also through empty space. 
Irfthis last case, it can be arranged that the electrons shalTall move 
in parallel paths and at the same speed ; they may then be described 
as a shower rather than as a current. 

If a thin layer of metal is placed in the path of such a shower, 
some at least of the electrons of the shower must strike the nuclei 
and electrons of the atoms of the metal. As they will strike at all 
sorts of angles, we might expect to find that their courses would be 
deflected rtmch as bagatelle-balls are deflected by the pins of the 
bagatelle-board, so that they will emerge at the far side of the 
metal film as a disordered mob of electrons. 

The actual course of events is very different. Part of it was dis- 
covered, almost by accident, by two American physicists, Davisson 
and Germer. They were intending to study the law of scattering 
of electrons at metal surfaces, and were projecting a shower of 
parallel-moving electrons on to a sheet of nickel, when their 
apparatus broke. In the process of mending it, they made their 
nickel surface so hot that it Crystallized. 

Now crystal surfaces possess very special properties. The atoms 
of a non-crystalline substance are ,not arranged in any regular 
formation, but are thrown together as though at random, like the 
grains in a pile of sand. But the atoms of a crystalline substance 
are arranged in perfect regularity, forming a repeating geometrical 
pattern of squares, triangles and so forth a property which has 
been of great value to experimental physics. 

The properties of light are often studied by using a piece of 
apparatus called a diffraction-grating a metal plate having parallel 
lines scratched on its surface with the utmost regularity and pre- 


cision at the rate of 15,000 to 40,000 to the inch. When a beam of 
light is reflected from such a surface, it is sorted out into its different 
spectral colours, much as though it had been passed through a 
spectroscope. The closer the lines are drawn on its surface, the 
shorter the wave-lengths of the light with which the apparatus 
can deal, because the grating becomes ineffective if the distance 
between successive lines is much greater than the wave-lengths 
of the light. Red light has about 30,000 waves to the inch, violet 
light about 60,000. It is easy to rule the lines on a grating close 
enough to deal with such radiation as this. 

On the other hand X-radiation has hundreds of millions of waves 
to the inch, so that a grating could only cope with this if its lines 
were ruled at only atomic distances apart. It is obviously impossible 
to rule lines as close as this by mechanical methods, but some experi- 
ments by Laue (1912) showed that it is also unnecessary, since 
quite perfect gratings of this kind already exist in the surface of 
crystals, in which the atoms are arranged in perfectly regular 

Innumerable experiments have shown that the ridges and depres- 
sions formed by these regular chains of atoms cause the surface 
to act as a natural diffraction-grating for radiation having the 
wave-length of X-rays. This has opened up new fields of scientific 
investigation. Sir W. H. Bragg and Sir W. L. Bragg, together with 
an army of other investigators, have studiea the arrangement of 
atoms in solids by noticing how X-radiation is treated when it falls 
on the solids, while Siegbahn and others, measuring the wave- 
lengths of the X-rays emitted by atoms of the various chemical 
elements, have gained valuable information as to the internal 
structure of these atoms. 

We can now understand what happened when Davisson and 
Germer shot electrons on to the surface of their crystallized nickel. 
They found that the reflected electrons were not scattered at 
random, but showed marked preferences for certain directions in 
space. They saw that this must result from the regular spacing of 
the atoms in the nickel surface, but unhappily the electrons they 
were using moved too slowly for their investigation to be carried 
to its proper conclusion. 


Shortly afterwards Prof. G. P. Thomson performed similar 
experiments, using faster electrons and improved methods. He 
made thin films, only about 100 atoms in thickness, out of metals 
which were naturally crystalline ; these were strong enough to hang 
together, and yet so thin as to be almost transparent. Electrons 
which moved at about 50,000 miles a second were found to pene- 
trate through these films, instead of being reflected back at their 
surfaces, and could be made to record their positions after penetra- 
tion on a photographic plate. The imprints were found to show 
extreme orderliness in their arrangement; they formed a pattern 
of concentric circles, light and dark circles alternating, round the 
point at which the shower of electrons would have struck the plate 
had the film of metal not been in their way. This showed that the 
film does not throw the electron formation into disorder, but 
spreads it in a very regular way. The pattern was found to be the 
same as would have been formed by X-rays of a certain definite 
wave-length passing through the same film of metal. If the film 
is replaced by one of some other substance, the pattern is replaced 
by the pattern which the same X-rays would form if they passed 
through the new substance. 

We may be tempted at this stage to imagine that the regular 
pattern is simply impressed on the shower of electrons by the 
regular arrangement of the atoms in the crystal. But this cannot 
be the whole cause of the scattering ; if it were, passing the shower 
of electrons through two plates of the solid in succession would 
produce twice as much scattering as passing it through one. 
Instead of doing this, it merely lessens the intensity of the pattern, 
thus proving that the pattern must be produced by some property 
inherent in the electrons, which is brought to light by their passage 
through the metal film. This is further shown by the fact that the 
electrons can be reflected from the metal surface, and still show 
the same sort of regular pattern. 

In each case, the pattern is the same as would be produced by 
X-rays, so that the shower of electrons must have something in 
common with X-radiation, and so must possess some quality of 
an undulatory kind. It is of course a mere numerical accident that 
in all the experiments the electrons behave like one special type of 


radiation, namely X-radiation ; this results from X-radiation being 
the only type of radiation with a wave-length comparable with 
interatomic distances. 

If the speed of the electron shower is changed, the pattern 
changes to one that would be produced by X-radiation of a dif- 
ferent wave-length the slower the electrons, the longer the waves 
of the equivalent radiation. This wave-length is found to be 
inversely proportional to the speed of the electron shower, the 
product of wave-length and speed being equal to h (Planck's 
constant) divided by m (the mass of the electron). The appearance 
of Planck's constant here clearly suggests that the wave properties 
of the electron must in some way be connected with the quantum 
theory ; indeed de Broglie had predicted the relation we have just 
mentioned from pure quantum considerations, before the wave- 
pattern had ever been observed. 

Such are the purely experimental results. In the preceding 
chapter we saw how radiation, which was once thought to be 
wholly undulatory, can be pictured as possessing some of the 
properties of particles a beam of radiation falling on a material 
surface may be pictured as a shower of photons, each located at 
a definite point of space and possessing mass and energy. We now 
find that a shower of electrons, which was once thought to consist 
wholly of particles, may be pictured as possessing some of the 
properties of waves, at least to the extent of having a definite 
wave-length associated with it. 


These waves form the subject-matter of the wave mechanics, and 
at the same time, as we have seen, provide a pictorial representa- 
tion of Heisenberg's quantum mechanics. The fact that the mathe- 
matical wave-lengths (although never physical waves) show their 
presence experimentally provides confirmation both of the truth 
of the quantum mechanics and of the validity of the wave mechanics 
as a pictorial representation of it. 

When we study the properties of these waves further, we find 
that they are very similar to the waves of the undulatory theory of 


light. We have already seen that these latter may be described 
as waves of probability, the intensity of the waves at any point 
giving a measure of the probability of a photon occurring at 
the point. Electron waves may be interpreted in a precisely 
similar way. 

To see this, we need only imagine that in the experiments just 
described, the strength of the shower of electrons is reduced until 
it consists of one solitary electron. This, being indivisible, must 
strike the photographic plate at one and only one point. This point 
must be one which was darkened in the original experiment, other- 
wise we should have to suppose that one electron could do what 
millions failed to do. The darker the plate was then made at any 
point, the more electrons struck the plate here, and so the greater 
the chance that the single electron shall strike here now. Thus 
electron waves, just like waves of radiation, may be interpreted as 
waves of probability, their intensity at any point giving a measure 
of the probability of an electron being found at the point. 

According to Planck's original theory, a photon is a store of 
energy of amount equal to h times the frequency of its waves. 
Now an^lectron is also known to be a_store of energy of amount 
mc*y where m is the mass of the electron and c is the velocity of 
light, llie general principles of the quantum theory sugggstjtKat 
here also the energy will bejfr jdmes the_ frequency of the vsiay.es, 
so that the frequency of the electron waves will be mc*/h. 

This means that mc*\h complete waves pass over any assigned 
point in a second, and as each wave is of length h/mu, where u 
is the speed of the electron, the total length of waves passing 
over any assigned point in a second will be mc 2 /h x h/mu, or c 2 /u. 
Thus the electron waves travel at a speed c 2 /u. 

This result seems at first very surprising. It is a well-established 
resulFpf physics that nothing material can travel fasteiJLhan_light> 
Thus^w, the speed of the material electron, must be less thian c, 
the jspeed _of. light, so that c*/u, the speed of the electron waves, 
niustj^jsjaffj^^ the speed of light. This would sufficiently show, 

if we did not know it already, that thjgggjffaYfiS transport nothing 
Probability is of course not material, being 
with neither mass nor energy. 


Since the electron waves travel faster than light, it looks at first 
ias though they would rapidly run away from their electrons. Yet 
Jiis would involve an obvious absurdity. For if electrons travel 
through space at a speed u, the regions where we are likely to find 
them i.e. the regions defined by the presence of waves must 
obviously travel at the same speed u. Actually, as we shall now 
see, these regions actually do travel only at the speed u ; the proof 
bf this turns on a somewhat technical point in the general theory 

01 mathematical discussion, thg_simplest wave 
system is a train of perfectly regular waves extending for^an infinite 
distance in every direction,, Each wave is of precisely the same 
shape and length, its contour being that of a ripple on still water. 
Out of a combination of such units, we can build up any formation 
of waves, no matter how complicated. Conversely of course any 
wave-formation as, for instance, a storm at sea can be analysed 
into a number of these simple units. The storm may be confined 
within a circle of 100 miles radius, but each unit must be supposed 
to extend to infinity in every direction. Outside the storm circle, 
the waves of the various units still exist in a mathematical sense, 
but destroy one another by interference, a point which is a crest on 
one set of waves being a trough on another and so on, in such a 
way that the total elevation of the surface of the water at every 
point is nil, and the sea is calm. 

When the original cause of the storm the whipping-up of 
waves by the friction of wind on water has subsided, each wave- 
unit pursues its natural motion over the sea as though the other 
wave-units did not exist. When the motion is traced out mathe- 
matically, two distinctive featifres emerge. In the first place, the 
interference of waves outside the storm circle becomes less com- 
plete as the motion progresses, so that the roughness of the sea 
gradually extends to areas outside the circle. Also the shorter waves 
are destroyed more rapidly than the longer by the action of dissipa- 
tive forces, so that finally only the long waves are left, and we have 
rollers or a long swell prevailing over the whole ocean. 

A somewhat different application of the theory is of special 
interest to our present problems. By combining a number of units 


having wave-lengths nearly equal to any assigned wave-length A, 
a wave-formation can be built up which will consist entirely of 
waves having the precise wave-length A, and will extend only over 
a small region of space. As before, the waves outside this small 
region of space destroy one another by interference. A short 
sequence of waves of this kind is called a wave-packet. 

Let us now imagine each of the constituent units of a wave- 
packet travelling through space in the way appropriate to its 
wave-length. It is common in nature for waves to travel at a speed 
depending on their wave-length, and in the present case each train 
of waves will travel nearly but not quite at the speed appropriate 
to the wave-length A. We might expect that the whole of the 
wave-packet would travel at approximately the same speed, but 
mathematical analysis shows that it does not. In front of the 
wave-packet, the waves are continually destroying one another by 
interference, while at the back the reverse process is taking place. 
This results in a slowing down of the speed of the wave-packet 
as a whole, so that it advances more slowly than the individual 
waves of which it is constituted. Detailed analysis shows that, 
although each individual wave travels at a speed c 2 /u, the packet 
as a whole travels only at a speed u, which is precisely the speed 
of the electron. Thus the waves as a whole do not run away from 
the electron. 

We MWj^t^adiatipn canno^suitably bejgictured asjparticles 
when^it is travelling tlSough empty space. There is a corre- 
sponding property for electrons; these should not be pictured as 
waves so long as they are travelling through empty space. The 
reason is that the quantities h/mu and c^ju which specify the waves 
Kave no meaning until u can be defined, and, as the theory of 
relativity shows, it is meaningless to define u as the speed at which 
electrons are travelling through empty space ; it can only be defined 
in connection with some frame of reference, as for instance some 
material surface on which the electrons are falling or about to fall. 
Thus we must think of the electron waves as springing into 
existence when a current of electricity enters into relation with a 
material surface, just as we think of photons as springing into 
existence when radiation meets a material surface. 


All this shows that the waves cannot have any material or real 
existence apart from ourselves. They are not constituents of nature, 
but only ot our efforts to understand nature, being only the in- 
gredients of a mental picture that we draw for ourselves in the 
hope of rendering intelligible the mathematical formulae of the 
quantum mechanics. The mathematical specification of the waves 
is unalterably fixed, being the equivalent of the formulae of the 
quantum mechanics. But the details of the physical picture are 
not unalterably fixed. If this picture were perfect, it woilld enable 
us to comprehend the incomprehensible, so that we cannot expect 
it to be very perfect; it may well show some want of precision, 
and may even be adjusted to meet the special circumstances of 
a particular problem. Often, for instance,it is a convenience to 
imagine the electron waves as existing m emptyjspace, just jasjt 
may occasionally be^onvenient to imagine photon waves existing 
in matter! ~ ~~ ~ " 

^Nevertheless, all waves tend to spread like the waves of our 
storm at sea, or ripples on the surface of a pond. Whether a wave- 
packet is large or small, it must continually increase in size, and 
the smaller it is to begin with, the more rapidly it grows. This shows 
that no wave-packet can permanently represent a single electron; 
an electron is a permanent structure, while a wave-packet is not. 
Indeed the wave mechanics has no concern with single electrons. 
But we may import the concept of the atomicity of electricity into 
it from experimental physics, and we then notice that if any 
wave-packet represented an electron at one moment it would have 
ceased to do so by the next, because the wave-packet would have 
changed, while the electron had not. 

Wfe might perhaps conjecture that different wave-packets repre- 
sent electrons in different circumstances. If so, let us see whether 
we can discover the circumstances of the electron corresponding 
to a few simple types of wave-packet. 

Suppose first that the wave-packet is of only infinitesimal length, 
a mere point in space. Such a wave-packet might seem specially 
suited to represent an electron under most circumstances. But it 
is a mathematical fact that a wave-packet of only infinitesimal 
length cannot have, or even be associated with, any definite wave- 


length; there is not room, so to speak, for the wave qualities to 
develop. We have seen that a packet of wave-length A represents 
an electron moving with a speed h\m\ so that, if we cannot form 
any idea of the value of A, we are equally unable to form any idea 
of the speed of the electron. 

If the length of the wave-packet is gradually increased, definite 
wave qualities gradually emerge. Finally the packet becomes an 
endless train of waves, each of wave-length equal to the wave- 
length of the packet. If an electron is represented by such an 
infinite train of waves, we can of course determine its speed of 
motion with absolute precision; it is simply h/mX, and there is 
now no uncertainty about the value to be assigned to A. But now 
we are totally unable to say where the electron is. The wave-packet 
has become an endless and featureless train of waves, and there 
can be no reason for assigning the electron to any one point of it 
rather than to another. Thus we see that a short train of waves 
would fix the position of the electron in space, but would fail to 
fix its speed of motion, while a long train of waves would tell us 
the speed of motion, but could not fix the position of the electron 
in space. No conceivable wave-packet could indicate both the 
speed of motion and the position of an electron with absolute 

This immediately reminds us of a result we obtained in Chapter 
iv (p. 142). We there saw that our experimental explorations *of 
nature do not admit of absolute precision, owing to the fact that 
nothing less than a complete photon can b^ received from the outer 
world. Regarding the electron asa moving particle, we saw that 
no experiTnent could fix both its speed of motion and its position 
ifTspace with complete accuracy, it quanta of low frequency are 
used in an experiment, the determination of the electron's position 
will necessarily be very uncertain, while if quanta of high frequency 
are used, the uncertainty is merely transferred to the determination 
of the electron's momentum, since the energetic photon gives the 
electron a big kick in leaving it. No possible experimental arrange- 
ment can make these two uncertainties both vanish simultaneously, 
so that the product of the two can never be zero. A detailed study 
by Heisenberg has shown that the product can never be less than 


Planck's constant h. This is known as Heisenberg's principle of 
uncertainty (or indeterminacy). 

We have just seen that the wave-packet of an electron shows 
a precisely similar lack of precision. Again a detailed mathematical 
discussion shows that whatever kind of wave-packet we select to 
represent an electron, the product of the two uncertainties of posi- 
tion and momentum can never be less than h, which is precisely 
what Heisenberg found about the experimental investigation. 

When an electron is depicted as a particle in space, it has an 
exact speed of motion and also an exact position in space, both of 
which can be specified by numerical quantities; the trouble re- 
vealed by the uncertainty principle is not that these quantities do 
not exist, but that we have no practical means of measuring them ; 
they can exist in the electron, but not in our knowledge of the 
electron. But when we depict the electron as a wave-packet, these 
quantities do not even exist in the wave-packet. 

As Bohr was the first to point out, this gives us the clue to the 
whole situation and lets the secret out different kinds of wave- 
packet must not be supposed to represent different kinds of elec- 
trons, or electrons in different states, or electrons under different 
conditions, but the different kinds of knowledge we can have about 
electrons. Indeed, just as the waves of the undulatory theory of 
light were found to represent our knowledge about photons (p. 136), 
so the waves of the wave mechanics are now seen to represent our 
knowledge about electrons. Both sets of waves are mental con- 
structs of our own; boj:h are propagated in conceptual spaces. 

There is complete parallelism except in one respect. The waves 
of the undulatory theory need a space of only three dimensions 
for their representation, so that we may conveniently and legiti- 
mately represent them in ordinary physical space. The waves of 
a single electron can also be represented in a space of three dimen- 
sions, but the waves of two electrons need a space of six dimensions, 
three for each electron, while the waves of a million electrons need a 
space of three million dimensions. Thus the wave-picture of even 
the simplest group of electrons, or of other particles, cannot be 
drawn in ordinary space. 
The wave-picture just described is due to de Broglie, Schrodinger, 


Bohr ana Heisenberg. It is subjective in the sense that it may 
depend on experiments that we have recently performed on elec- 
trons, but it is also objective in the sense that it shows a capacity 
at least equal to that of the particle-picture to interpret objective 
reality, giving correct solutions to many problems in which the 
particle-picture fails. Indeed, in its mathematical formulation the 
wave-picture is exactly equivalent to Heisenberg's scheme which, 
from the mode of its derivation, is necessarily true to reality. 

It should, however, be added at once that those cases in which 
the wave-picture meets with more success than the particle- 
picture are not those in which it represents the knowledge of any 
particular individual. The majority have to do with the spectra of 
atoms, and so are concerned with the motion of electrons round 
nuclei, and not in free space. The wave-packet still represents our 
knowledge of the electron, but it is now knowledge as to the 
possible or probable positions of the electron inside the atom, 
knowledge which is independent of any particular observation or 
observer. The wave-packet of a free electron represents private 
knowledge, individual to a particular observer who has recently 
made an observation on the electron, but the wave-packet of an 
electron inside an atom represents public knowledge, accessible to 
all without an experiment. An observer could of course find out 
more about the electron inside the atom by a new ad hoc experi- 
ment as for instance by bombarding the atom with oc-particles 
and noting the Wilson-chamber condensation trail of the electron 
as it was shot out of the atom but he woiild destroy the atom in 
so doing. The wave-packet of the electron would be concentrated 
into a smaller region, and would become the wave-packet of a free 
electron starting off on a new journey. 

Thus there is a standard wave-packet for an electron inside an 
atom, or rather there are several distinct standard packets one 
for each state of steady motion which can take place inside the 
atom but there is no standard wave-packet for an electron travel- 
ling freely through space. This reminds us of what we found in 
discussing the pictures of an electron provided' by the classical 
mechanics. We found a bullet-picture which corresponded to out 
present wave-picture, and a * tentacle '-picture which corresponded 


to our present wave-picture, but there was no standard tentacle- 
picture suited for all circumstances. The appropriate picture de- 
pended on the motion, not only of the electron but of other bodies 
as well. 

If the waves of a free electron or photon represent human know- 
ledge, what happens to the waves when there is no human know- 
ledge to represent? For we must suppose that electrons were in 
existence while there was still no human consciousness to observe 
them, and that there are free electrons in Sirius where there are 
no physicists to observe them. 

The simple but surprising answer would seem to be that when 
there is no human knowledge there are no waves ; we must always 
remember that the waves are not a part of nature, but of our efforts 
to understand nature. Whether we are thinking about electrons or 
not, and whether we are experimenting with them or not, their 
motion is determined by the equations of the Heisenberg dynamics. 
When an electron joins an atom or is knocked out of an atom, its 
motion undergoes just the same changes whether we are presiding 
over the experiment or not ; if a photon is emitted, it makes no 
difference to the electron whether this photon ends up in a 
human eye or elsewhere. 

Similar remarks may be made about the waves of the undulatory 
theory of light, and about the electric and magnetic forces of which 
we have hitherto imagined them to be constituted. Energy may be 
transferred from place to place, but the waves and the electric and 
magnetic forces are not P ar ^ of the mechanism of transfer ; they are 
part simply of our efforts to understand this mechanism and 
picture it to ourselves. Before man appeared on the scene, there 
were neither waves nor elecfric nor magnetic forces ; these were 
not made by God, but by Huyghens, Fresnel, Faraday and 


The third form of quantum mechanics, that of Dirac, must be 
dismissed very briefly, not because it is in any way unimportant, 
but because it is so intensely mathematical in form as to lie entirely 
beyond the scope of the present book. Dirac's ambition was to 


put the whole of quantum mechanics in a perfectly consistent form, 
deducing all its conclusions from a few simple assumptions much 
as Euclid deduced the whole of geometry from a few simple axioms. 

Dirac remarks that the classical mechanics had tried to explain 
physical phenomena in terms of particles and radiation moving in 
space and time ; it made a few simple assumptions about the factors 
governing the bodies which figure in the phenomena, and then 
tried to account for their behaviour in terms of these assumptions. 
In brief, it tried to explain the phenomena without going beyond 
the phenomena, as though the world of phenomena formed a 
closed whole. This attempt failed, and it became clear that nature 
works on a different plan. Exhaustive studies by many investigators 
have shown that the fundamental laws of nature do not control 
the phenomena directly. We must picture them as operating in 
a substratum of which we can form no mental picture unless 
we are willing to introduce a number of irrelevant and therefore 
unjustifiable suppositions. 

Events in this substratum are accompanied by events in the 
world of phenomena which we represent in space and time, but the 
substratum and the phenomenal world together do not form a 
complete world in itself which we can observe objectively without 
disturbing it. The complete closed world consists of three parts 
substratum, phenomenal world, and observer. By our experiments 
we drag up activities from the substratum into the phenomenal 
world of space and time. But there is no clear line of demarcation 
between subject and object, and by performing observations on 
the world we alter it, much as a fisherman, dragging up a fish 
from the depths of the sea, disturbs the waters and also damages 
the fish. 

Dirac introduces operators of an abstract mathematical kind, to 
represent the effect of dragging an activity up to the surface i.e. 
of observing it. He finds it necessary to assume that the series 
of observable types of activity #, 6, , . . . is more restricted than 
the corresponding series of types in the substratum. The latter 
series consists of certain pure types -4, B, C,. . . which appear as 
a, bj c, . . . in the world of phenomena, and also of certain composite 
types, which we may denote by AB, BC, AC,. . . and have no 


direct counterparts in the phenomenal world. AB may give rise 
to a or to b, but never to both, and there is an assignable probability 
as to whether a or b will appear. Thus the substratum of reality is 
in some way richer and more varied than the world of phenomena. 

After elaborate mathematical discussion, Dirac reaches a formal 
theory of a very complete kind. The matrix mechanics of Heisen- 
berg and the wave mechanics of de Broglie and Schrodinger are 
then shown to be included in the theory as special cases. 

It will be seen from this that the pattern of events implied in 
Dirac's theory is necessarily the same as the pattern implied in the 
theories of Heisenberg, and of de Broglie and Schrodinger, and 
so agrees entirely with that observed in nature. It is an essential 
feature of Dirac's theory that events in the phenomenal world are 
not uniquely associated with events in the substratum; different 
events in the substratum may result in phenomena which are pre- 
cisely similar, at least to our observation. Thus the same pheno- 
menon in the space-time world may be associated with a number 
of different states in the substratum, and so may be followed by 
different events. Because of this, experiments which are similar so 
far as our observation goes need not, and usually will not, lead to 
identical results. Thus the uniformity of nature is jettisoned at the 
outset in so far as the phenomena are concerned, and causality 
disappears from the world we see. 

It does not entirely disappear from the world which is hidden 
from our view. The mathematical equations of both forms of the 
new quantum theory the wave mechanics and the matrix me- 
chanics are completely deterministic in form. So far as these 
equations go, the future of the world appears to be a mere unrolling 
so that the future follows uniquely and inexorably from the past. 
But this unrolling is not, as we have already seen, of the course 
of events, but of our knowledge of events. Causality disappears 
from the events themselves to reappear in our knowledge of events. 
But, since we can never pass behind our knowledge of events to 
the events themselves, we can never know whether causality 
governs the events or not. Indeed the considerations mentioned 
on p. 144 suggest that even to discuss the question is meaningless. 


We have now concluded our summary of the findings of modern 
pKysrcgpmcTmay turn to consider how these findings affect Hie 
practical problems of philosophy and of everyday life. But let us 
first recapitulate the conclusions we have reached in our scientific 


Because we are human beings and not mere animals, we try to 
discover as much as we can about the world in which our lives 
are cast. We have seen that there is only one metfrpd of gaining 
such knowledge the method of science, which consists in a direct 
questioning of natiire^^By^observation and experiment. 
- The lirst thing we learn from such questioning is that the world 
is rational ; its happenings are not determined by caprice but by. 
Jaw. There exists what we have called a 'pattern of events', and 
the primary aim of physical science is the discovery of this pattern. 
This, as we have seen, will be capable ojf^description only in 
[liathematical terms? " ~~ 

The new quantum theory explained in the preceding chapter 
has provided a mathematical description of the pattern of events 
which is believed to be complete and perfect. For it enables us 
n principle at least to predict every possible phenomenon of 
shysics, and not one of its predictions has so far proved to be 
wrong. In a sense, then, we might say that theoretical physics 
las achieved the main purpose of its being, and that nothing 
emains but to work out the details. 

But we not only wish to predict phenomena, but also to under- 
stand them. Thus it is not surprising that philosophy and science 
lave alike found this mathematical description unsatisfying, 
ind have tried to attach concrete meanings to the mathematical 
;ymbols involved to replace unintelligible universals by in- 


telligible particulars. We may argue that if there is a pattern, 
there must be some sortjgfjo^ ever weaving it; we want to 

know ^^^^ffi^loom is, how it works, and why_it works thus 
rather than otherwise. 

The physicists of the last century thought that one of the primary 
concerns of science should be to devise models or draw pictures to 
illustrate the workings of this loom. It was supposed that a model 
which reproduced all the phenomena of a science, and^sojrnadeJt 
possible to predict them all, mustjn some way correspond to the 
reaTi^underlying the phenomena. But obviously this cannot be 
so. After one perfect model had been found, a second of equal 
perfection might appear, and as both models could not correspond 
to reality, we should have at least one perfect model which did not 
correspond to reality. Thus we could never be sure that any model 
corresponded to reality. In brief, we can never have certain know- 
ledge as to the nature of reality. 

We know now that there is no danger of even one perfect model 
appearing at least of a kind which is intelligible to our minds. 
For a model or picture "will only be intelligible to us if it is made 
up of ideas which are already in our minds. Of such ideas some, 
as for instance the ideas u of abstract mathematics, have no special 
relation to our particular world ; all those which have must, as we 
have seen, have entered our minds through the gateways of the 
senses. These are restricted by our having only five senses of which 
only two are at all important for our present purpose. 

A detailed investigation of the sources of our ideas has shown 
that there is only on type of model or picture which could be 
intelligible to our restricted minds, namely one in mechanical 
terms. Yet a review of receipt physics has shown that all attempts 
at mechanical models or pictures have failed and must fail. For a 
mechanical model or picture must represent things"asliappening 
in space and time, while it~has recently become^ clear that tfie 
ultimate processes of nature neither occur in, nor admit of repre- 
sgntation in, space and time. Thus an understanding oTthe ultimate 
piy cesses of nature is for ever beyond our reach: we shall never 
be able even in imagination to open the ca^je of our watch aad 
^ go round. The true object of scientific study 


can never be the realities of nature, but only our own observations 
on nature. 

The Particle-picture and the Wave-picture 

Although there can be no complete picture of the workings of 
nature which will be intelligible to our minds, yet we can still 
draw pictures to represent partial aspects of the truth in an in- 
telligible way. The new physics places two such partial pictures 
before us one in terms of particles, and one in terms of waves. 
Neither of these can of course tell the whole truth. 

In the same way, an atlas may contain two maps of North 
America drawn on different projections: neither of them will 
represent the whole truth, but each will faithfully represent some 
aspect of it. An equal-area projection, for instance, represents me 
relative areas of any two regions accurately, but their shapes 
wrongly, while a Mercator projection represents the shapes rightly, 
but the areas wrongly. So long as we can only draw our maps on 
flat pieces of paper, such imperfections are inevitable ; they are the 
price we pay for limiting our maps to the kind that can be bound 
up in an atlas. 

The pictures we draw of nature show similar limitations ; these 
are the price we pay for limiting our pictures of nature to the kinds 
that can be understood by our minds. As we cannot draw one 
perfect picture, we make two imperfect pictures and turn to one 
or the other according as we want one property or another to be 
accurately delineated. Our observations tell us which is the right 
picture to use for each particular purpose for instance, we know 
we must use the particle-picture for the photo-electric effect, the 
wave-picture for illumination effects, and so on. 

Yet some properties of nature are so far-reaching and general 
that neither picture can depict them properly of itself. In such 
cases we must appeal to both pictures, and these sometimes give 
us different and inconsistent information. Where, then, shall we 
find the truth? 

For instance, is nature governed by causal laws or not? The 
particle-picture answers : No, the motions of my particles can only 
be compared to the random jumps of kangaroos, with no causal 


laws controlling the jumps. But the wave-picture says: Yes, at 
every instant my waves follow uniquely, and so inevitably, from 
those of the preceding instant. 

Or again, is reality ultimately atomic or is it not? The particle- 
picture tells us of a material world in which matter, electricity and 
radiation occur only in indivisible units ; the wave-picture merely 
tells us that it knows of none of these things. 

The two pictures seem to tell different stories, but we must 
remember that they are not equally trustworthy. The particle- 
picture embodies the findings of the old quantum theory which we 
discussed in Chapter v. This proved to be both inaccurate and in- 
complete, so that the new quantum theory was brought into being 
to remedy its deficiencies which it has successfully done. The 
wave-picture is not only a pictorial representation of the new 
quantum theory, but also, as regards the mathematical facts in- 
volved, is its exact equivalent. Thus the predictions of the wave- 
picture cannot be other than true, whereas those of the particle- 
picture may or may not be true. When there is a conflict, the 
evidence of the wave-picture must be accepted, while we may be 
sure that the conflict results from some imperfection of the particle- 
picture. In the instances just given, it is not difficult to trace out 
a possible origin for the conflict. 

The mathematical laws of the quantum theory show that radiant 
energy is transferred by complete quanta. But in depicting a 
beam of light as a hail oi bullet-like photons, the particle-picture 
is clearly going further than the facts warrant. A man's balance at 
the bank always changes'by an integral number of pence, but this 
does not justify him in picturing its changes as caused by a flight 
of broilze pennies. If he does, Ijis child may ask him what decides 
which particular pennies shall be sent to pay the rent. The father 
may reply: Mere chance a foolish answer but no more foolish 
than the question. In the same way, if we make the initial mistake 
of depicting radiation as identifiable photons, we shall have to call 
on mere chance to get us out of our difficulties and here is the 
origin of the indeterminacy of the particle-picture. 

For instance, when a beam of light falls on a half-silvered mirror 
(p. 137), the particle-picture shows half the photons being turned 

JP 12 


back by the silvering of the mirror, while the other half pass on 
, their way undisturbed. We ask at once : What singles but the lucky 
photons? It is a question which had confronted Newton's cor- 
puscular theory of light, and he had answered it by a vague wave 
of the hand towards Fortune's wheel his corpuscles, he had said, 
were 'subject to alternate fits of easy transmission and easy 
reflection*. In the same way, if we depict radiation as identifiable 
photons, we can find nothing but the finger of Fate to separate the 
sheep from the goats. But the finger of Fate, like the sheep and 
the goats, is mere pictorial detail. As soon as we turn to the 
more trustworthy wave-picture, all this pictorial drapery drops out 
of the picture, and we find a complete determinism. Yet this 
determinism, as we have seen, does not control events, but our 
knowledge of events. The wave-picture does no show the future 
following inexorably from the present, but the imperfections of 
our future knowledge following inexorably from the imperfections 
of our present knowledge. 

What is true of radiation is true also of electricity. We know that 
electricity is always transferred from place to place by complete 
electron-units, but this does not justify us in replacing a current 
of electricity by a shower of identifiable particles. Indeed, the 
quantum theory definitely tells us that we must not do so. When 
two balls A, B collide on a billiard- table, A may go to the right and 
B to the left. When two electrons A, B collide, we might also expect 
to be able to say that A would go to the right and B to the left ; 
actually we cannot, because we have no right to identify the two 
electrons which went into the collision tvith the two which come 
out; we must rather think of the two electrons A and B which 
entered into collision as combining into a drop of electric fluid, 
which then breaks up again to form two new electrons C and Z). 
If we ask which way A will go after collision, the true answer is 
that A no longer exists. The superficial answer is that it is an even 
chance whether A goes to the right or to the left, for it is a toss-up 
whether we identify A with C or D. But the toss-up is not in 
nature ; it is in our minds. 

We see, then, that the particle-picture goes wrong in attributing 
indeterminism to nature; it is not a property of nature, but of our 


way of looking at nature. The particle-picture further goes wrong 
in attributing atomicity to the ingredients of the material world, 
whether matter or radiation ; the atomicity does not reside in these 
ingredients but in the events which affect them. To return to our 
former analogy, all payments into and out of a bank account are 
by complete mathematical pence, but they do not consist of bronze 
pennies flying hither and thither. But we can now carry this train of 
ideas a little further; we know matter only through the energy or 
particles it emits, but this provides no warrant for assuming that 
matter itself consists of atoms either of substance or of energy 
this would be like assuming that our balance at the bank must 
consist of a pile of bronze pennies. 


We have seen that efforts to discover the true nature of reality are 
necessarily doomed to failure, so that if we are to progress further 
it must be by taking some other objective and utilizing some new 
philosophical principles of* which we have not so far made use. 
Two such suggest themselves. The first is the principle of what 
Leibniz described as probable reasoning ; we give up the quest for 
certain knowledge, and concentrate on that one of the various 
alternatives before us which seems to be most probably true. But 
how are we to decide which of the alternatives is most likely to be 
true? This question has been much discussed of late, particularly 
by H. Jeffreys. For our purpose it is sufficient to rely on what 
may be described as the simplicity postulate ; this asserts that of 
two alternatives, the simpler is likely to be the nearer to the truth. 

Let us try to illustrate these new principles by considering a 
simple, although very artificial, analogy. 

Let us imagine that in the centre of Europe there lives a peasant 
who has never seen or heard of the sea, and cannot even read about 
it, but is in possession of a super-perfect radio-set which can pick 
up messages from every ship in the world. Suppose further that 
every ship is continually sending out its position in a standard 
form, such as 

'Queen Mary', +41 10', -72 26', 



this meaning that, at the moment of speaking, the ship ' Queen 
Mary* is in latitude 41 10' north and longitude 72 26' west. 

At first he may merely amuse himself by listening to the various 
messages, but after a time he may take to recording them and, 
if he is of an inquiring turn of mind, he may try to discover some 
method or order in them. He will soon notice that all latitudes 
lie between +90 and -90, and all longitudes between +180 
and 1 80. If he then tries plotting out these numbers on squared 
paper, he will find that successive positions of any one ship form 
a continuous chain, and may begin to construct a mental picture 
for himself by thinking of the senders of the messages as moving 
objects. He will then find that each supposed object moves at an 
approximately uniform rate on his chart, although this law is not 
exact or universal. A ship may move from longitude +170 to 
+ 174 in one day, and on to + 178 the next, but the third day 
may take it to -178, an apparent journey of 356. Further, a 
ship may move at a regular 4 a day when its latitude is near to o, 
but this daily motion will increase as the latitude increases, and 
may shoot up almost beyond limits if ever the latitude approaches 
to 90. 

If, notwithstanding their peculiar nature, our listener succeeds 
in formulating exact laws, he will then be able to predict the 
motions of the ships. Or, to be more precise, he will be able, 
without assuming that he is dealing with either motions or ships, 
to predict what he will hear when he turns on his radio. He can 
predict the result of every experiment he can perform, since the 
only experiment within his power is to turn a knob and listen. 

Those who are content with a positivist conception of the aims 
of science will feel that he is in an entirely satisfactory position ; 
he has discovered the pattern of events, and so can predict ac- 
curately; what more can he want? A mental picture would be an 
added luxury, but also a useless luxury. For if the picture did not 
bear any resemblance at all to the reality it would be valueless, and 
if it did it would be unintelligible, since we are supposing that 
our listener cannot imagine either sea or ships. 


Probable Reasoning 

At this point, let us notice that the supposition that the signals 
came from moving objects was hypothetical in the sense that 
nothing in the observations compelled it from the nature of the 
case the observer is debarred from knowing whether the signals 
come from moving objects or not. It expresses a possibility and 
not certain knowledge, and can never be proved true. In real 
science also a hypothesis can never be proved true. If it is negatived 
by future observations we shall know it is wrong, but if future 
observations confirm it we shall never be able to say it is right, 
since it will always be at the mercy of still further observations. 
A science which confines itself to correlating the phenomena can 
never learn anything about the reality underlying the pTienQmena r 
wiule a^cience which goes further than this, and introduces hypo- 
theses about reality ,/cafl ftpver acquire ceitam knowledge pi a 
positive^ kind about^ reality; ia-^erfiatev^ 

Certain knowledge is, however, equally beyond our reach in 
most departments of life. Oftener than not, we cannot wait for 
certain knowledge, but order our affairs in the light of probabilities. 
There is no reason why we should not do the same in our efforts 
to understand the universe, provided we always bear in mind that 
we are discussing probabilities and not certainties. 

The philosopher does it as much as the rest of us. I am conscious 
only of my own thoughts and sensations, so that, for aught I know 
to the contrary, I may be the only conscious being in the universe. 
If I choose on these grounds to become a solipsist i.e. one who 
supposes that he is the only conscious being in the whole universe 
nothing can definitely prove me wrong. But my sensations inform 
me of other objects that look like my body, and seem to experience 
sensations and thoughts like my own. I assume, although only on 
grounds of probable reasoning, that these other objects are beings 
essentially similar to myself. If we refused to admit probability 
considerations, we ought all to be solipsists ; with things as they 
are, any genuine solipsists there may be are kept safely shut up. 

The physicist also relies on probability considerations every day 


of his life. He measures the wave-lengths of spectral lines in the 
light emitted by Sirius, and finds they are identical with those in 
the light emitted by hydrogen at a temperature of 10,000 C. He 
concludes without more ado that there are atoms of hydrogen at 
10,000 in Sirius. There is no proof of this and never can be, for 
we shall never be able to go to Sirius to find out. But the probabili- 
ties against the agreement being a mere coincidence are so over- 
whelming that the physicist feels justified in disregarding this possi- 
bility, and announces that this part of the light of Sirius comes from 
hydrogen at a temperature of 10,000. 

In these two instances, the philosopher and physicist are both 
guided by probable reasoning rather than by certain deductions. 
If our radio listener allows himself to be guided by similar con- 
siderations, he may decide provisionally that his signals come from 
moving objects. This idea may lead him to think of pasting together 
his +180 and 180 lines, thus transforming his plane diagram 
into a cylinder. This simplifies the situation enormously, for it 
now seems the most natural thing in the world that a sequence of 
readings equidistant in time should read 170, 174, 178, 178, 
etc. But he is still faced with the peculiarity that his moving 
objects traverse more degrees of longitude per day in high latitudes 
than in low. With a little ingenuity, he may further think of 
crumpling in the two ends of his cylinder, and so making the degrees 
of longitude smaller in higher latitudes. If he finally tries the 
experiment of replacing his cylinder by a sphere, he will find that 
his laws assume an exceedingly simple form from which all oddity 
has disappeared. Each ship takes the shortest course from point 
to point, and performs its journey at a uniform speed. 

Even the original laws were true laws, since they enabled the 
listener to predict accurately. But they were not simple, because 
their discoverer had expressed them against a bad background. As 
soon as he changed from one background to another from a 
rectangular projection to a spherical surface the laws changed from 
being strange but true to being simple and true. Precisely for this 
reason, most men will consider that the second set of laws was 
preferable. Without assigning any special attributes to the Designer 
of the universe, we probably feel that the simpler laws are likely 


to be in some way closer to that reality which we can never under- 
stand, than complicated and odd laws in brief, that artificiality 
comes from man, and not from nature. In the example just con- 
sidered, it is certainly more true to say that the earth's surface is 
spherical than to picture it as plane. 

And in the real problems of science also, it is true, as Einstein 
has remarked, that ' In every important advance the physicist finds 
that the fundamental laws are simplified more and more as experi- 
mental research advances. He is astonished to notice how sublime 
order emerges from what appeared to be chaos. And this cannot 
be traced back to the workings of his own mind but is due to a 
quality that is inherent in the world of perception.' 

This not only shows that our minds are in some way in harmony 
with the workings of nature a harmony which Einstein compares 
with the pre-established harmony of Leibniz (p. 27) but also 
that our investigations of nature are proceeding on the right lines ; 
;it further shows that the simplicity which is inherent in nature is 
of the kind which our minds adjudge to be simple. Indeed any 
other kind of simplicity would probably escape our notice. 

The Simplicity Postulate 

This suggests the introduction of a further principle, if not into 
the technique of scientific investigation at least into the practice 
of philosophical discussion the principle of simplicity. When two 
hypotheses are possible, we .provisionally choose that which our 
mlMraajudp^^lI^ ^EPl?!L^?L^ J*upPsitiqn that IKIs Is 
the mor^^ lead in_the jdirection ofj^trujh. It includes 

as a special case tEe principle of Occam's razor entia non multi- 
plicanda praeter necessitatem. 

There can of course be no absolute criterion as to which of two 
hypotheses is the simpler ; in the last resort this must be a matter 
of private judgment. In the fictitious example we have just been 
discussing there could be no room for doubt, but in actual scien- 
tific practice there have been cases in which two investigators have 
differed as to which of two hypotheses was the simpler, as for 
example with the one-fluid and two-fluid theories of electricity. 


The history of science provides many instances of situations such 
as we have been discussing. To begin with the most obvious, 
Ptolemy and his Arabian successors built up the famous system 
of cycles and epicycles which enabled them to predict the future 
positions of the planets with almost perfect precision. At first, the 
sun, moon and stars were supposed to revolve round the fixed 
central earth, while the planets revolved about other centres which 
themselves revolved about the earth. It was soon found that this 
did not quite fit the facts, and the orbits had to be changed to 
slightly eccentric circles neither the earth nor the moving centres 
were any longer at the exact centres of the circles which were 
described around them. Finally, as the planetary motions came to 
be known to a still higher degree of accuracy, epicycle was piled on 
epicycle until the system became exceedingly complex. 

Many, indeed, felt that it was too complex to correspond to the 
ultimate facts. In the thirteenth century, Alphonso X of Castile 
16 reported to have said that if the heavens were really like that, 
*J could have given the Deity good advice, had He consulted me 
at their creation.' At a later date Copernicus also thought the 
Ptolemaic system too complex to be true and, after years of thought 
and labour, showed that the planetary motions could be described 
much more simply if the background of the motions were changed : 
Ptolemy had assumed a fixed earth ; Copernicus substituted a fixed 
sun. We now know that the sun can* no more be said to be at rest, 
in any absolute sense, than the earth; iMs one of the thousands 
of millions of stars which together fornT^he galactic system, ami 
it moves round the centre of this system just as the^earth moves 
round the centre of the solar system! And even this centre of 

the galactic system cannot be said ' t j'bg3L rest ^ ^ or m ^li Qns f 

galactic systems can Ue seen in the sky, all pretty much like our 
own, and all in motion relative to our own galaxy and to one 
another. No one of all these galaxies has a better claim than any 
other to constitute a standard of 'rest' from which the 'motions' 
on the others can be measured. Nevertheless, many complications 
ire avoided by imagining that the sun and not the earth is at rest. 
Neither the sun nor the earth is at rest in any absolute sense, and 
pet it is, in a sense, nearer to the truth to say that the earth 


moves round a fixed sun than to say that the sun moves round a 
fixed earth. 

Copernicus had still to retain a few minor epicycles to make his 
system gree with the facts of observation. This, as we now know, 
was the inevitable consequence of his assumption that the planetary 
orbits were circular : neither he nor anyone else had so far dared 
to challenge Aristotle's dictum that the planets must necessarily 
move in circular orbits, because the circle was the only perfect curve. 
As soon as Kepler substituted ellipses for the Copernican circles, 
epicycles were seen to be unnecessary, and the theory of planetary 
motions assumed an exceedingly simple form the form it was to 
retain for more than three centuries, until an even greater sim- 
plicity was imparted to it by the relativity theory of Einstein, to 
which we shall come in a moment. 

The restricted (or physical) theory of relativity provides a second 
illustration of the same thing. The Newtonian mechanics, with its 
background of absolute space and time, had explained the motion 
of objects well enough so long as their speeds of motion were not 
comparable with that of light. But, as experiment ultimately 
showed, it could only explain the motion of rapidly moving objects 
at the price of introducing extreme complications. Objects in rapid 
motion had to contract and assume new shapes, while no one could 
ever quite say what happened to objects in rapid rotation. The 
theory of relativity introduced a tremendous simplification into the 
whole subject when it discarded Newton's absolute space and time 
as a background, and substituted the new space-time unity, as 
explained on p. 63. 

The generalized (or gravitational) theory of relativity provides 
an even more striking instance of the same thing. The Newtonian 
theory of gravitation, which required the planets to move round 
the sun in elliptical orbits, gavejm excellent account of the move- 
ments of theouter planets, but failed with thejnner. Attempts were 
madil:oremedythis by slightlyaltenng theNewtonian law of gravi- 
tation, fay suppOfginglKe^sunTo be surrounded beclouds of gas or 
dust which impeded the free motion of the inner planets, and in a 
variety of other ways. The relativity theory of gravitation then, 
cleared up the whole situation at one strokeJpy reiectine Newton's 

1 86 


force of gravitation altogether, and^hnpressing a curvature on the 
fime unity in which the motions oiF^^plan^sjwer 
e change was fn^ 

Orice a 

ngiable ^ 

grouncT. l^he^wEoIe^notjgn of planets and otherjbodies, as weffas 
ofHrays of jight, could now J)e described by the simple statement 
that they all described geodesies i.e. took the shortest possible 
course from point tojpoint in the new curved space-time unity. 

The simplification which this change introduced was not only 
tremendous in itself, but was in line with a number of earlier 
simplifications, all based on the idea of a length of path or some 
similar quantity assuming the smallest value which was possible 
for it. 

Fig. 2 


The principle made its first appearance in optics. If a candle is 
burning at C, and my eye at E looks at a mirror MM', I shall 
seem to see the candle at some point A in the mirror. This shows 
that rays of light are travelling along the path CAE from the 
candle to my eye, and along no others ; for if they travelled along 
any other path CBE as well, I should seem to see candles at 
both A and B y which I do not. Hero of Alexandria set himself 
the problem of finding what it was that specially distinguished 
the path CAE which the light actually took from every other 


possible path such as CBE which it might have taken, but did 
not. He found that CAE was the shortest path from C to E 
which touched the mirror on its way. Even though the light is 
reflected from hundreds of mirrors, the path is still determined 
by the same principle; it is the shortest path that can be found, 
subject to the condition of its touching all the mirrors in turn. 
Alternatively the path may be described as the quickest from C 
to E ; the light chooses its path on the principle of wasting as little 
time as possible on the way. 

Fermat (1601-1665) showed that this latter principle still deter- 
mines the path when the light travels through water, glass, or 
other refracting substances of any kind whatever. Thus it is true 
under all circumstances that light always travels by the quickest 
route; this provides another instance of the tremendous simpli- 
fications to which Einstein refers (p. 183). 

Maupertuis (1698-1759) subsequently conjectured that the 
motions of tangible objects must conform to some similar prin- 
ciple, arguing that Divine perfection would be opposed to any 
expenditure of energy by moving bodies, beyond the absolute 
minimum necessary to get from one place to another. In time 
such a principle was found to govern the motion of all bodies of 
tangible size the principle of 'Least Action'. This principle in- 
cludes the Newtonian mechanics and the classical mechanics as 
special cases, so that it covers not only mechanical activities but 
those of electricity and magnetism as well. It can best be under- 
stood through a simple analogy. 

When I hire a taxicab, the taximeter piles up the charges against 
me at a rate which depends both on where I am, and on how fast 
I am travelling. I have to pay one sum per five minutes when I am 
at rest in a city, some other sum per five minutes when I travel at 
1 5 miles an hour in the city, twice as much when I travel at 30 miles 
an hour in the city, and so on, and on an entirely different tariff 
when I am outside the city limits. Now let us imagine a taximeter 
attached to every moving object in the universe, piling up charges 
at a rate which depends on both the speed of motion and the position 
of the object. Let all the objects move for some specified time, such 
as an hour, and at the end of the motion let all the charges shown 
on the various taximeters be added up. The principle of Least 


Action tells us that the actual objects in nature will have chosen 
their paths so as to make the total charge shown by all the taxi- 
meters a minimum Nature, setting her face against unnecessary 
expenditure on taxicabs, always chooses the cheapest route. 

Suppose, for instance, that a single particle has to be transferred, 
within a specified time, from one point A to another point B, 
through a region in which conditions are absolutely uniform, so 
that the taxicab tariff is of course uniform also. The cheapest way 
of making the journey will be to travel in a perfectly straight line 
at a perfectly uniform speed, which is what Newton's law of 
motion tells the particle to do. Or again, suppose that a planet 
has to be transported from its preselit position to the corresponding 
position at the other sI3e of the sun. The shortest route would be 
gliaiglil diiougli llie~centre ot the sun, but, as the tanlt in intense 
gravitational fields is exorbitant, the charges by this route would 
be prohibitive. We find we can avoid these excessive charges by 
taking a curved path round the sun, even though this lengthens 
the journey somewhat. If part of tFe route still goes near tolEEe 
sunlit j^fo ea p est tQ p er fo rm this part of thejourney at high speed, 
so as to spend as little time as possible in tlie region of exorbitant 
tariffs. Exact mathematical analysis is needed to find exactly what 
combination of path and speed reduces the total charge to an 
absolute minimum; it tells us that the path must be an ellipse 
having the sun in one of its foci. This is precisely the path demanded 
by the Newtonian mechanics, but we notice that it is no longer 
mapped out by the action of * forces' of the Newtonian kind. 

Logically, and to some extent chronologically also, the principle 
of Least Action forms a direct successor to the principle of Least 
Time of Hero and Fermat. The principle of Least Distance, or 
geodesies, in the curved space-^ime .jpLlglativity is clearlyJaJhe 
sam^lilif^ot succession^! t introduces a great simplicity by changing 
to the new background of a curved space like the change of back- 
ground of our radio listener when he changed from a rectangular 
projection to a curved spherical surface. Like the principles of 
Least Time and of Least Action, this principle of Least Distance 
shows an extreme simplicity which suggests that we are keeping 
in close touch with the true significance of natural processes. 


The old quantum theory did not show any such simplicity. We 
need not concern ourselves with it any further since it has now 
become clear that it was only an unsatisfactory hybrid between 
the classical mechanics and the new quantum theory, being, in 
fact, a last desperate effort to represent nature against a background 
of time and space. 

In the new quantum theory the same simplicity reappears in 
full strength and almost in the same form. So far as its formal 
mathematical description goes, the theory is a genuine extension 
of the old Newtonian mechanics, so much so that the same mathe- 
matical equations will serve for the description of both, namely the 
canonical equations of which we spoke on p. 113, these in turn 
being an expression of the principle of Least Action. 

But the pictorial representations that must be given to these 
equations differ widely in the two cases. The classical mechanics 
came into existence as an effort to describe the continuous motions 
of objects under pushes and pulls ; it is in this way that it is usually 
interpreted. But the new quantum mechanics must be interpreted 
rather as a description of steady states in which either there is no 
motion or else the state of motion does not change. Now and then, 
as we have seen, a jump occurs from one of these steady states to 
another, and it is with jumps of this kind rather than with gradual 
changes that the new mechanics is concerned. Are these jumps 
final, or will they ultimately be resolved into some kind of rapid 
continuous motions of which we have so far no knowledge, either 
observational or theoretical ? We simply cannot form a judgment. 

The main difference between the old mechanics and the new is, 
however, once again a difference of background. The classical 
mechanics and the old quantum theory had both assumed that the 
whole world existed in time and space ; the new mechanics is most 
simply expressed in terms of symbols which are best interpreted 
by passing beyond space and time. In transcending space and time, 
the new quantum mechanics finds a new background which makes 
for far greater simplicity and so probably comes nearer to ultimate 
truth. In passing from the old mechanics to the new, the mathe- 
matical description of the pattern of events stands almost unaltered, 
while the interpretation we put upon the symbols is utterly changed. 


The history of theoretical physics is a record of the clothing of 
mathematical formulae which were right, or very nearly right, with 
physical interpretations which were often very badly wrong. When 
Newton had found laws of motion of a mechanical system which 
were true (apart from the minor refinements of the theory of 
relativity), he put science on a wrong track for two centuries by 
interpreting them in terms of forces and absolute space and time. 
It was much the same with his supposed force of gravitation. 
Again, when the true laws of the propagation of light had been dis- 
covered, they were interpreted as applying to the propagation of 
waves in an ether which was supposed to fill all space, and again 
science was started along a wrong road which it was to follow for 
nearly two centuries. 

Now when philosophy has availed itself of the results of science, 
it has not been by borrowing the abstract mathematical description 
of the pattern of events, but by borrowing the then current pictorial 
description of this pattern; thus it has not appropriated certain 
knowledge but conjectures. These conjectures were often good 
enough for the man-sized world, but not, as we now know, for 
those ultimate processes of nature which control the happenings of 
the man-sized world, and bring us nearest to the true nature of 

One consequence of this is that the standard philosophical dis- 
cussions of many problems, such as those of causality and free-will 
or of materialism or mentalism, are based on an interpretation of 
the pattern of events which is no longer tenable. The scientific 
basis of these older discussions has been washed away, and with 
their disappearance have gone all the arguments, such as they were, 
that seemed to require the acceptance of materialism and deter- 
minism and the renunciation of human free-will. This does not 
mean that the conclusions previously reached were necessarily 
wrong, for a bad argument may lead to a good conclusion. But 
it does mean that the situation must be reviewed afresh. Every- 
thing is back in the melting-pot, and we must start anew and try 
to discover truth on the basis of the new physics. Apart from our 
knowledge of the pattern of events, our tools can only be probable 
reasoning and the principle of simplicity. 


A posteriori concepts, 35 

A posteriori knowledge, 35, 47, 48, 50, 


A priori concepts, 35, 60 
A priori knowledge, 35, 39, 40, 43 ff., 

54, 70, 72, 79, 152 
Absolute mind, 69, 203 
Absolute space, 36, 37, 57, 59, 63, 66, 

67, 109, 134, 190, 199 
Absolute time, 37, 58, 59, 63, 66, 67, 

190, 199 

Action at a distance, 116 
Alchemy, 19 

Alexander, S., 68, 69, 205, 206, 213 
Alphonso X, 184 
Anaxagoras, 13, 22 
Ancient philosophy, 18 
Animal spirits (of Descartes), 25 
Animism, 3, 4 
Anselm, St, 94 
Antinomy (Kant), 60 
Appearance and reality, 192!!. 
Arabian astronomy, 184 
Aristarchus of Samos, 5, 165 
Aristotle, 17, 69, 91, 93, 105, 106, 185 
Arithmetical knowledge, 35, 45, 50 
Articles of religion (Anglican), 22, 209 
Atomicity of radiation, 126, 130 ' 
Atomistic modes of thought, 98 
Auditory ideas, 1 1 
Augustine, St, 62 
Automata, 2, 28 
Axioms of Euclid, 40, 41 

Bacon, Francis, 38, 82 

Becquerel, 147 

Bentley, 21, 116 

Bergson, 99 

Berkeley, Bishop, 38, 100, 196, 198, 

199, 203 
Bertrand Russell, 36, 80, 86, 87, 102, 


Bohr, N., 126, 127, 145, 154, 155, 170 
Bohr's theory of spectra, 145, 154 
Born and Jordan, 157 
Born and Wiener, 159 

Bragg, Sir W. H. and Sir W. L., 161 
Bruno, Giordano, 20, 23, 26, 59, 106 

Canonical equations of motion, 113 

Categories, 69 

Causality, 69, 101, 119, 144. 173, 190, 


Cave, the parable of the, 193 
Circular motion, 12 
Classical mechanics, 114, 123, 126 
Clocks, 58 
Cogito ergo sum, 85 
Colour-blindness, 88, 89 
Colour vision, 87 
Common-sense realism, 29 
Compton and Simon, 131 
Coirlte, Auguste, 4 
Conceptual space, 55, 59 
Conceptual time, 57, 59 
Conservation of energy, 114 
Conservation of momentum, 114 
Continuity of change, 99 
Coordinates, 112, 113 
Copernican revolution of Kant, 54 
Copernican system of astronomy, i, 5, 

20, 21, 105, 184 
Corpuscular theory of light (Newton's), 

130, 178, 202 
Cosmology, mediaeval, 19 
Critique of Practical Reason, 38 
Critique of Pure Reason, 24, 35, 51 
Crystalline structure, 160 
Curie, P. and M., 147 
Curvature of space, 118 
Curved space, 5, 12 

Dante, 21 

Darwinian biology, i, 21 

Davisson and Germer, 160, 161 

de Broglie, 169, 173 

Descartes, 12, 17, 22, 23, 25, 26, 28, 

33, 34, 37, 39, 4, 4*, 45, 74. 85, 
86, 91, 94, 95, 107, 152, 195, 196, 
198, 204, 205, 211, 214 

Determinism, 109, 151, 205, 216 

Diffraction grating, 160 

220 INDEX 

rfirac, P. A. M., 57, 171 ff. 
Discontinuity, 99 
Dissipative forces, 106 
Divisibility of substance, 40, 41, 44 
Dynamical explanations of nature, 13 

Eddington, Sir A., 35, 51, 72 ff., 84 
Ehrenfest, P., 136 

Einstein, A., 12, 37, 56, 59, 117, 127, 
129, 136, 150, 151, 183, 187, 200, 


Electric forces, 119, 134, 164 
Electromagnetic theory of light, 1 34 
Electron, structure of, 122 
wave-packet of, 167, 170 
Electron- waves, i6off. 
Empiricists, 34, 36 
Energy, 25, 27, no, 114 
Epicurus, 17 

Epistemology, 31, 32ff., 73, 75 
Equations of motion, 113 
Ether, luminiferous, 119 
Euclid, axioms of, 40, 41 
Euclidean space, 40 
Excluded middle, law of the, 93 
Extension in space, 1 1 

Fara4ay, Michael, 14, 120 

Fermat, 187, 188 

Fetichism, 4 

Fichte, 23 

Finiteness of space and time, 61 

Force, 27 

Force of gravitation, 5, 120, 190 

Forms, of perception (Kant), 55 

of Plato, 33, 195 
Four-dimensional space, 12 
France, Anatole, 85 
Franck and Hertz, 187 
Freedom of will, 2, 20, 21, 24, 29, 
30, 31, 190, 2osff. 

Galileo, 20, 23, 91, 105, 107, 108, 196 

Geodesies, 12 

Geulincx, 26 

God, existence of, 35, 94, 198 

Gravitation, force of, 5, 120, 190 

law of, 1 1 6, 120, 185, 190 

propagation of, 77 

relativity theory of, 12, 185 
Greek philosophy, 12, 13, 18, 19 (and 
also under individual names) 

Half-silvered mirror, 136, 177 
Hamilton, Sir W. R., 113 

equations of, 113, 114 
Hegel, 17, 69, 203, 204, 205, 206 
Heisenberg, W., 130, 154, 155, 168, 
170, 173 

uncertainty principle of, 142, 169 
Helmholtz, W. von, 37 
Heraclitus, 17, 100 
Hero of Alexandria, 186, 188 
Hipparchus, 106 
Hobbes, 7, 16, 20, 23 
Hume, 6, 36, 38, 101, no, 205, 206 
Huyghens, 13, 27 

Idealist philosophies, 59, 196 
Ideas, 6, 10, 33, 34, 198 
Immortality, 24 
Impressions, 6 

Indeterminacy principle of Heisen- 
berg, 142, 169 
Indeterminists, 210 
Infinitesimals, 100 
Innate ideas, 34, 35 

James, William, 210 
Jeffreys, H., 179, 199 

Kant, Immanuel, 24, 28, 35, 37, 39, 
40, 41, 51, 60, 62, 63, 74, 76, 95, 
100, 101, 103, no, 205, 206, 214 
theory of knowledge of, 51, 70, 71, 


Kepler, 5, 20, 105, 185 
Kinetic energy, no, 114 
Knowledge, physical, 6, 7, 8 
the nature of, 7 
waves of, 136, 169, 195 
Kronecker, 16 

Lagrange, 113 

Lang, Andrew, 3 

Language, 32, 82, 83, 85 

Laplace, 109 

Larmor, 120 

Laue, 161 

Least Action, principle of, 120, 187, 

T88, 189 " 

Leibniz, 5, 7, 17, 23, 25, 27, 28, 34, 
37, 59, 84, 95, 99, 179, 198, 204, 



Light, finite speed of, 58, 63 

physical nature of, 52 
Locke, 36, 38, 91, 92, 196, 205, 206 
Lotze, 210 

Magnetic forces, 119, 134 
Malebranche, 26 

Materialism, 2, 20, 190, 192, 195 ff., 216 
Mathematical description of Nature, 
15, 155, 201 

knowledge, 34, 36, 45 
Maupertuis, 113, 187 
Maxwell, J. Clerk, 14, 120, 134, 211 
Mechanical explanations of Nature, 

13, 14, 28 
Mechanics, classical, 114, 123, 126 

Newtonian, i, 2, 14, 108, 109, 115, 

of Descartes, 12, 107 
Mediaeval cosmology, 19 

philosophy, 18 
Mentalism, 2, 59, 190, I95ff. 
Mersenne, 26 
Metaphysics, 4, 17 
Michelson-Morley experiment, 121 
Mill, J. S., 36, 45, 205, 206 
Momentum, 25, in, 112, 114 
Monads, 26 
Morality, 37, 38 
Motion, 25, 84, 85, 86 

Naive realism, 29 

Nature, 4, 8, 12 

New quantum theory, I54ff. 

New worlds of modern science, 42, 

43, 7 
Newton, Sir Isaac, 13, 57, 56, 57, 58, 

60, 77, 108, 116, 178/217 
Newtonian astronomy, 5, 20 
Newtonian laws of motion, 108, 118, 

Newtonian mechanics, i, 2, 14, 108, 

109, 115, 194 

Nicholas of Cusa, 59, 71, 107 
Non-commutative algebra, 47 
Number, ideas of, n, 33 

Objective idealism, 204 
Occam's razor, 79, 183 
Occasionalists, 26, 27 
Ontological proof of the existence of 
God, 94 

Paneth, 38 

Paradoxes of Zeno, 94, 99 

Parmenides, 100 

Particle- and wave-pictures of radia- 
tion, 133, i76ff., 201 

Peirce, C. S., 184 

Perceptions, 6, n, 37 

Perceptual space, 55, 59 

Perceptual time, 57, 59 

Permanence of substance, 40, 41, 44 

Phenomena, 10, 30, 193 

Philosophy, definitions of, 16 

Photo-chemical action, 129 

Photo-electric effect, 128 

Photons, 130, 136, 164, 191 

Physical knowledge, 6, 7 

Physical space, 56, 59 

Physical time, 57, 59 

Pictorial representations, 9, 15, 122, 175 

Planck, Max, 126, 128, 212 

Planetary movements, 4, 5, 12 

Plato, 13, 26, 32, 33, 62, 96, 193, 195 

Plutarch, 106 

Pluto (planet), 196 

Poincare*, 8, 78 

Positivism, 4, 15, 184 

Postulate of simplicity, 179, 183 

Postulates of impotence, 79 

Pre-established harmony (Leibniz), 
27, 28, 183 

Pre-Newtonian mechanics, 105 

Primary and secondary qualities, 91, 

Principia of Newton, 108 

Probability, waves of, 135, 136, 137, 

Probable reasoning (Leibniz), 179, 181 

Protagoras, 32 

Ptolemy, 105, 184 

Pythagoras, 5 

Qualities (primary and secondary), 91, 


Quanta, 14, 126 
Quantity, ideas of, 1 1 
Quantum theory, 128, 154, 201 

Radioactivity, 127, 147 
Rationalists, 35, 37 
Rationality of Nature, 8 
Realism, 29, 58 
Reality, 10, 15, 16, 193 



Redness, meaning of word, 87 
origin of sensation, 96, 197 

Relativity, 14, 37, 83, 117, 120, 200 
Einstein's theory of, 68, 83, 117, 185, 

of motion, 37, 56, 59 

Renaissance philosophy, 19 

Representation in space and time, 69, 
134, 172, 103 

Representationalism, 30 

Ritz, 155 

Roemer, 63, 77 

Russell, Bertrand, 36, 80, 86, 87, 102, 

Rutherford, 127, 147 

St Anselm, 94 

St Augustine, 62 

St Thomas, 33 

Scholastics, 33 

Schopenhauer, 23, 213 

Schrftdinger, 170, 173 

Schultze, 51 

Secondary and primary qualities, 91, 

196, 197 
Seneca^ 22 
Sense-organs, 6, 7, 8 
Sidgwick, H., 207, 209 
Siegbahn, 161 
Simultaneity, 83 
Soddy, 127, 147 
Solipsism, 181 
Space, 40, 41, 55> S&, 63 
Space-time unity, 63 ff., 68, 117, 192 
Spinoza, 23, 37, 38, 205, 206, 213 
Subject-object relation, 143 
Substance, divisibility of, 40, 41, 44 
permanence of, 40, 41, 44 

Synchronization of clocks, 58, 73, 74 
Synthetic a priori knowledge (Kant), 
47, 49, 50, 76 

Tactile ideas, n, 13 
Teleological causation, 210 
Thales of Miletus, 3, 17, 32 
Thomson, G. P., 162 
Time, 55, 58, 63 
Tychism, 210 

Uncertainty principle of Heisenberg, 

141, 169 

Undulatory theory of light, 1 3 1 ff . 
Uniformity of nature, 3, 140 
Universal law, 2 

Velocity of light, 63, 77 
Vision, mechanism of, 87 
Visual ideas, u, 13 
Vortices (Descartes), 12 

Ward, James, 51 
Wave mechanics, 1630. 
Wave-packet, i66ff. 

of electron, 167, 170 
Wave-picture of radiation, 133, 176, 


Whitehead, A. N., 36 
Whittaker, E. T., 79 
Wilson cloud-chamber, 154 
World-line, 118 
Wright, W. K., 23 

X-radiation, 161, 163 


Zeno, paradoxes of, 94, 99