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  | Transcriber's Note: In the parts containing mathematical         |
  | notation, superscripts are denoted using a carat and subscripts  |
  | using underscores, for example x^2, e_1, e_{n+1}. Greek          |
  | letters used as symbols are written out, for example {sigma},    |
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  | Italicised words are marked using underscores _like this_, but   |
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  | font) have not been marked, to aid legibility and to avoid       |
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  | Two printer errors have been corrected. These are:               |
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  | "...relating the entity discriminated by sight with that         |
  | discriminated by [sight]..." sight has been changed to touch as  |
  | suggested by the sense.                                          |
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                            The Concept of


                                NATURE


                         THE TARNER LECTURES
                     DELIVERED IN TRINITY COLLEGE
                            NOVEMBER 1919


                        Alfred North Whitehead




PREFACE


The contents of this book were originally delivered at Trinity College
in the autumn of 1919 as the inaugural course of Tarner lectures. The
Tarner lectureship is an occasional office founded by the liberality of
Mr Edward Tarner. The duty of each of the successive holders of the post
will be to deliver a course on 'the Philosophy of the Sciences and the
Relations or Want of Relations between the different Departments of
Knowledge.' The present book embodies the endeavour of the first
lecturer of the series to fulfil his task.

The chapters retain their original lecture form and remain as delivered
with the exception of minor changes designed to remove obscurities of
expression. The lecture form has the advantage of suggesting an audience
with a definite mental background which it is the purpose of the lecture
to modify in a specific way. In the presentation of a novel outlook with
wide ramifications a single line of communications from premises to
conclusions is not sufficient for intelligibility. Your audience will
construe whatever you say into conformity with their pre-existing
outlook. For this reason the first two chapters and the last two
chapters are essential for intelligibility though they hardly add to the
formal completeness of the exposition. Their function is to prevent the
reader from bolting up side tracks in pursuit of misunderstandings. The
same reason dictates my avoidance of the existing technical terminology
of philosophy. The modern natural philosophy is shot through and
through with the fallacy of bifurcation which is discussed in the second
chapter of this work. Accordingly all its technical terms in some subtle
way presuppose a misunderstanding of my thesis. It is perhaps as well to
state explicitly that if the reader indulges in the facile vice of
bifurcation not a word of what I have here written will be intelligible.

The last two chapters do not properly belong to the special course.
Chapter VIII is a lecture delivered in the spring of 1920 before the
Chemical Society of the students of the Imperial College of Science and
Technology. It has been appended here as conveniently summing up and
applying the doctrine of the book for an audience with one definite type
of outlook.

This volume on 'the Concept of Nature' forms a companion book to my
previous work _An Enquiry concerning the Principles of Natural
Knowledge_. Either book can be read independently, but they supplement
each other. In part the present book supplies points of view which were
omitted from its predecessor; in part it traverses the same ground with
an alternative exposition. For one thing, mathematical notation has been
carefully avoided, and the results of mathematical deductions are
assumed. Some of the explanations have been improved and others have
been set in a new light. On the other hand important points of the
previous work have been omitted where I have had nothing fresh to say
about them. On the whole, whereas the former work based itself chiefly
on ideas directly drawn from mathematical physics, the present book
keeps closer to certain fields of philosophy and physics to the
exclusion of mathematics. The two works meet in their discussions of
some details of space and time.

I am not conscious that I have in any way altered my views. Some
developments have been made. Those that are capable of a
non-mathematical exposition have been incorporated in the text. The
mathematical developments are alluded to in the last two chapters. They
concern the adaptation of the principles of mathematical physics to the
form of the relativity principle which is here maintained. Einstein's
method of using the theory of tensors is adopted, but the application is
worked out on different lines and from different assumptions. Those of
his results which have been verified by experience are obtained also by
my methods. The divergence chiefly arises from the fact that I do not
accept his theory of non-uniform space or his assumption as to the
peculiar fundamental character of light-signals. I would not however be
misunderstood to be lacking in appreciation of the value of his recent
work on general relativity which has the high merit of first disclosing
the way in which mathematical physics should proceed in the light of the
principle of relativity. But in my judgment he has cramped the
development of his brilliant mathematical method in the narrow bounds of
a very doubtful philosophy.

The object of the present volume and of its predecessor is to lay the
basis of a natural philosophy which is the necessary presupposition of a
reorganised speculative physics. The general assimilation of space and
time which dominates the constructive thought can claim the independent
support of Minkowski from the side of science and also of succeeding
relativists, while on the side of philosophers it was, I believe, one
theme of Prof. Alexander's Gifford lectures delivered some few years ago
but not yet published. He also summarised his conclusions on this
question in a lecture to the Aristotelian Society in the July of 1918.
Since the publication of _An Enquiry concerning the Principles of
Natural Knowledge_ I have had the advantage of reading Mr C. D. Broad's
_Perception, Physics, and Reality_ [Camb. Univ. Press, 1914]. This
valuable book has assisted me in my discussion in Chapter II, though I
am unaware as to how far Mr Broad would assent to any of my arguments as
there stated.

It remains for me to thank the staff of the University Press, its
compositors, its proof-readers, its clerks, and its managing officials,
not only for the technical excellence of their work, but for the way
they have co-operated so as to secure my convenience.

A. N. W.

    IMPERIAL COLLEGE OF SCIENCE
        AND TECHNOLOGY.
            _April_, 1920.




CONTENTS


   CHAP.                                                  PAGE

       I NATURE AND THOUGHT                                  1

      II THEORIES OF THE BIFURCATION OF NATURE              26

     III TIME                                               49

      IV THE METHOD OF EXTENSIVE ABSTRACTION                74

       V SPACE AND MOTION                                   99

      VI CONGRUENCE                                        120

     VII OBJECTS                                           143

    VIII SUMMARY                                           164

      IX THE ULTIMATE PHYSICAL CONCEPTS                    185

         NOTE: ON THE GREEK CONCEPT OF A POINT             197

         NOTE: ON SIGNIFICANCE AND INFINITE EVENTS         197

         INDEX                                             199




THE CONCEPT OF NATURE




CHAPTER I

NATURE AND THOUGHT


The subject-matter of the Tarner lectures is defined by the founder to
be 'the Philosophy of the Sciences and the Relations or Want of
Relations between the different Departments of Knowledge.' It is fitting
at the first lecture of this new foundation to dwell for a few moments
on the intentions of the donor as expressed in this definition; and I do
so the more willingly as I shall thereby be enabled to introduce the
topics to which the present course is to be devoted.

We are justified, I think, in taking the second clause of the definition
as in part explanatory of the earlier clause. What is the philosophy of
the sciences? It is not a bad answer to say that it is the study of the
relations between the different departments of knowledge. Then with
admirable solicitude for the freedom of learning there is inserted in
the definition after the word 'relations' the phrase 'or want of
relations.' A disproof of relations between sciences would in itself
constitute a philosophy of the sciences. But we could not dispense
either with the earlier or the later clause. It is not every relation
between sciences which enters into their philosophy. For example biology
and physics are connected by the use of the microscope. Still, I may
safely assert that a technical description of the uses of the microscope
in biology is not part of the philosophy of the sciences. Again, you
cannot abandon the later clause of the definition; namely that
referring to the relations between the sciences, without abandoning the
explicit reference to an ideal in the absence of which philosophy must
languish from lack of intrinsic interest. That ideal is the attainment
of some unifying concept which will set in assigned relationships within
itself all that there is for knowledge, for feeling, and for emotion.
That far off ideal is the motive power of philosophic research; and
claims allegiance even as you expel it. The philosophic pluralist is a
strict logician; the Hegelian thrives on contradictions by the help of
his absolute; the Mohammedan divine bows before the creative will of
Allah; and the pragmatist will swallow anything so long as it 'works.'

The mention of these vast systems and of the age-long controversies from
which they spring, warns us to concentrate. Our task is the simpler one
of the philosophy of the sciences. Now a science has already a certain
unity which is the very reason why that body of knowledge has been
instinctively recognised as forming a science. The philosophy of a
science is the endeavour to express explicitly those unifying
characteristics which pervade that complex of thoughts and make it to be
a science. The philosophy of the sciences--conceived as one subject--is
the endeavour to exhibit all sciences as one science, or--in case of
defeat--the disproof of such a possibility.

Again I will make a further simplification, and confine attention to the
natural sciences, that is, to the sciences whose subject-matter is
nature. By postulating a common subject-matter for this group of
sciences a unifying philosophy of natural science has been thereby
presupposed.

What do we mean by nature? We have to discuss the philosophy of natural
science. Natural science is the science of nature. But--What is nature?

Nature is that which we observe in perception through the senses. In
this sense-perception we are aware of something which is not thought and
which is self-contained for thought. This property of being
self-contained for thought lies at the base of natural science. It means
that nature can be thought of as a closed system whose mutual relations
do not require the expression of the fact that they are thought about.

Thus in a sense nature is independent of thought. By this statement no
metaphysical pronouncement is intended. What I mean is that we can think
about nature without thinking about thought. I shall say that then we
are thinking 'homogeneously' about nature.

Of course it is possible to think of nature in conjunction with thought
about the fact that nature is thought about. In such a case I shall say
that we are thinking 'heterogeneously' about nature. In fact during the
last few minutes we have been thinking heterogeneously about nature.
Natural science is exclusively concerned with homogeneous thoughts about
nature.

But sense-perception has in it an element which is not thought. It is a
difficult psychological question whether sense-perception involves
thought; and if it does involve thought, what is the kind of thought
which it necessarily involves. Note that it has been stated above that
sense-perception is an awareness of something which is not thought.
Namely, nature is not thought. But this is a different question, namely
that the fact of sense-perception has a factor which is not thought. I
call this factor 'sense-awareness.' Accordingly the doctrine that
natural science is exclusively concerned with homogeneous thoughts about
nature does not immediately carry with it the conclusion that natural
science is not concerned with sense-awareness.

However, I do assert this further statement; namely, that though natural
science is concerned with nature which is the terminus of
sense-perception, it is not concerned with the sense-awareness itself.

I repeat the main line of this argument, and expand it in certain
directions.

Thought about nature is different from the sense-perception of nature.
Hence the fact of sense-perception has an ingredient or factor which is
not thought. I call this ingredient sense-awareness. It is indifferent
to my argument whether sense-perception has or has not thought as
another ingredient. If sense-perception does not involve thought, then
sense-awareness and sense-perception are identical. But the something
perceived is perceived as an entity which is the terminus of the
sense-awareness, something which for thought is beyond the fact of that
sense-awareness. Also the something perceived certainly does not contain
other sense-awarenesses which are different from the sense-awareness
which is an ingredient in that perception. Accordingly nature as
disclosed in sense-perception is self-contained as against
sense-awareness, in addition to being self-contained as against thought.
I will also express this self-containedness of nature by saying that
nature is closed to mind.

This closure of nature does not carry with it any metaphysical doctrine
of the disjunction of nature and mind. It means that in sense-perception
nature is disclosed as a complex of entities whose mutual relations are
expressible in thought without reference to mind, that is, without
reference either to sense-awareness or to thought. Furthermore, I do not
wish to be understood as implying that sense-awareness and thought are
the only activities which are to be ascribed to mind. Also I am not
denying that there are relations of natural entities to mind or minds
other than being the termini of the sense-awarenesses of minds.
Accordingly I will extend the meaning of the terms 'homogeneous
thoughts' and 'heterogeneous thoughts' which have already been
introduced. We are thinking 'homogeneously' about nature when we are
thinking about it without thinking about thought or about
sense-awareness, and we are thinking 'heterogeneously' about nature when
we are thinking about it in conjunction with thinking either about
thought or about sense-awareness or about both.

I also take the homogeneity of thought about nature as excluding any
reference to moral or aesthetic values whose apprehension is vivid in
proportion to self-conscious activity. The values of nature are perhaps
the key to the metaphysical synthesis of existence. But such a synthesis
is exactly what I am not attempting. I am concerned exclusively with the
generalisations of widest scope which can be effected respecting that
which is known to us as the direct deliverance of sense-awareness.

I have said that nature is disclosed in sense-perception as a complex of
entities. It is worth considering what we mean by an entity in this
connexion. 'Entity' is simply the Latin equivalent for 'thing' unless
some arbitrary distinction is drawn between the words for technical
purposes. All thought has to be about things. We can gain some idea of
this necessity of things for thought by examination of the structure of
a proposition.

Let us suppose that a proposition is being communicated by an expositor
to a recipient. Such a proposition is composed of phrases; some of these
phrases may be demonstrative and others may be descriptive.

By a demonstrative phrase I mean a phrase which makes the recipient
aware of an entity in a way which is independent of the particular
demonstrative phrase. You will understand that I am here using
'demonstration' in the non-logical sense, namely in the sense in which a
lecturer demonstrates by the aid of a frog and a microscope the
circulation of the blood for an elementary class of medical students. I
will call such demonstration 'speculative' demonstration, remembering
Hamlet's use of the word 'speculation' when he says,

    There is no speculation in those eyes.

Thus a demonstrative phrase demonstrates an entity speculatively. It may
happen that the expositor has meant some other entity--namely, the
phrase demonstrates to him an entity which is diverse from the entity
which it demonstrates to the recipient. In that case there is confusion;
for there are two diverse propositions, namely the proposition for the
expositor and the proposition for the recipient. I put this possibility
aside as irrelevant for our discussion, though in practice it may be
difficult for two persons to concur in the consideration of exactly the
same proposition, or even for one person to have determined exactly the
proposition which he is considering.

Again the demonstrative phrase may fail to demonstrate any entity. In
that case there is no proposition for the recipient. I think that we
may assume (perhaps rashly) that the expositor knows what he means.

A demonstrative phrase is a gesture. It is not itself a constituent of
the proposition, but the entity which it demonstrates is such a
constituent. You may quarrel with a demonstrative phrase as in some way
obnoxious to you; but if it demonstrates the right entity, the
proposition is unaffected though your taste may be offended. This
suggestiveness of the phraseology is part of the literary quality of the
sentence which conveys the proposition. This is because a sentence
directly conveys one proposition, while in its phraseology it suggests a
penumbra of other propositions charged with emotional value. We are now
talking of the one proposition directly conveyed in any phraseology.

This doctrine is obscured by the fact that in most cases what is in form
a mere part of the demonstrative gesture is in fact a part of the
proposition which it is desired directly to convey. In such a case we
will call the phraseology of the proposition elliptical. In ordinary
intercourse the phraseology of nearly all propositions is elliptical.

Let us take some examples. Suppose that the expositor is in London, say
in Regent's Park and in Bedford College, the great women's college which
is situated in that park. He is speaking in the college hall and he
says,

    'This college building is commodious.'

The phrase 'this college building' is a demonstrative phrase. Now
suppose the recipient answers,

    'This is not a college building, it is the lion-house in the Zoo.'

Then, provided that the expositor's original proposition has not been
couched in elliptical phraseology, the expositor sticks to his original
proposition when he replies,

    'Anyhow, _it_ is commodious.'

Note that the recipient's answer accepts the speculative demonstration
of the phrase 'This college building.' He does not say, 'What do you
mean?' He accepts the phrase as demonstrating an entity, but declares
that same entity to be the lion-house in the Zoo. In his reply, the
expositor in his turn recognises the success of his original gesture as
a speculative demonstration, and waives the question of the suitability
of its mode of suggestiveness with an 'anyhow.' But he is now in a
position to repeat the original proposition with the aid of a
demonstrative gesture robbed of any suggestiveness, suitable or
unsuitable, by saying,

    '_It_ is commodious.'

The '_it_' of this final statement presupposes that thought has seized
on the entity as a bare objective for consideration.

We confine ourselves to entities disclosed in sense-awareness. The
entity is so disclosed as a relatum in the complex which is nature. It
dawns on an observer because of its relations; but it is an objective
for thought in its own bare individuality. Thought cannot proceed
otherwise; namely, it cannot proceed without the ideal bare 'it' which
is speculatively demonstrated. This setting up of the entity as a bare
objective does not ascribe to it an existence apart from the complex in
which it has been found by sense-perception. The 'it' for thought is
essentially a relatum for sense-awareness.

The chances are that the dialogue as to the college building takes
another form. Whatever the expositor originally meant, he almost
certainly now takes his former statement as couched in elliptical
phraseology, and assumes that he was meaning,

    'This is a college building and is commodious.'

Here the demonstrative phrase or the gesture, which demonstrates the
'it' which is commodious, has now been reduced to 'this'; and the
attenuated phrase, under the circumstances in which it is uttered, is
sufficient for the purpose of correct demonstration. This brings out the
point that the verbal form is never the whole phraseology of the
proposition; this phraseology also includes the general circumstances of
its production. Thus the aim of a demonstrative phrase is to exhibit a
definite 'it' as a bare objective for thought; but the _modus operandi_
of a demonstrative phrase is to produce an awareness of the entity as a
particular relatum in an auxiliary complex, chosen merely for the sake
of the speculative demonstration and irrelevant to the proposition. For
example, in the above dialogue, colleges and buildings, as related to
the 'it' speculatively demonstrated by the phrase 'this college
building,' set that 'it' in an auxiliary complex which is irrelevant to
the proposition

    'It is commodious.'

Of course in language every phrase is invariably highly elliptical.
Accordingly the sentence

    'This college building is commodious'

means probably

    'This college building is commodious as a college building.'

But it will be found that in the above discussion we can replace
'commodious' by 'commodious as a college building' without altering our
conclusion; though we can guess that the recipient, who thought he was
in the lion-house of the Zoo, would be less likely to assent to.

    'Anyhow, it is commodious as a college building.'

A more obvious instance of elliptical phraseology arises if the
expositor should address the recipient with the remark,

    'That criminal is your friend.'

The recipient might answer,

    'He is my friend and you are insulting.'

Here the recipient assumes that the phrase 'That criminal' is elliptical
and not merely demonstrative. In fact, pure demonstration is impossible
though it is the ideal of thought. This practical impossibility of pure
demonstration is a difficulty which arises in the communication of
thought and in the retention of thought. Namely, a proposition about a
particular factor in nature can neither be expressed to others nor
retained for repeated consideration without the aid of auxiliary
complexes which are irrelevant to it.

I now pass to descriptive phrases. The expositor says,

    'A college in Regent's Park is commodious.'

The recipient knows Regent's Park well. The phrase 'A college in
Regent's Park' is descriptive for him. If its phraseology is not
elliptical, which in ordinary life it certainly will be in some way or
other, this proposition simply means,

    'There is an entity which is a college building in Regent's Park
    and is commodious.'

If the recipient rejoins,

    'The lion-house in the Zoo is the only commodious building in
    Regent's Park,'

he now contradicts the expositor, on the assumption that a lion-house in
a Zoo is not a college building.

Thus whereas in the first dialogue the recipient merely quarrelled with
the expositor without contradicting him, in this dialogue he contradicts
him. Thus a descriptive phrase is part of the proposition which it helps
to express, whereas a demonstrative phrase is not part of the
proposition which it helps to express.

Again the expositor might be standing in Green Park--where there are no
college buildings--and say,

    'This college building is commodious.'

Probably no proposition will be received by the recipient because the
demonstrative phrase,

    'This college building'

has failed to demonstrate owing to the absence of the background of
sense-awareness which it presupposes.

But if the expositor had said,

    'A college building in Green Park is commodious,'

the recipient would have received a proposition, but a false one.

Language is usually ambiguous and it is rash to make general assertions
as to its meanings. But phrases which commence with 'this' or 'that' are
usually demonstrative, whereas phrases which commence with 'the' or 'a'
are often descriptive. In studying the theory of propositional
expression it is important to remember the wide difference between the
analogous modest words 'this' and 'that' on the one hand and 'a' and
'the' on the other hand. The sentence

    'The college building in Regent's Park is commodious'

means, according to the analysis first made by Bertrand Russell, the
proposition,

   'There is an entity which (i) is a college building in Regent's
   Park and (ii) is commodious and (iii) is such that any college
   building in Regent's Park is identical with it.'

The descriptive character of the phrase 'The college building in
Regent's Park' is thus evident. Also the proposition is denied by the
denial of any one of its three component clauses or by the denial of any
combination of the component clauses. If we had substituted 'Green Park'
for 'Regent's Park' a false proposition would have resulted. Also the
erection of a second college in Regent's Park would make the proposition
false, though in ordinary life common sense would politely treat it as
merely ambiguous.

'The Iliad' for a classical scholar is usually a demonstrative phrase;
for it demonstrates to him a well-known poem. But for the majority of
mankind the phrase is descriptive, namely, it is synonymous with 'The
poem named "the Iliad".'

Names may be either demonstrative or descriptive phrases. For example
'Homer' is for us a descriptive phrase, namely, the word with some
slight difference in suggestiveness means 'The man who wrote the Iliad.'

This discussion illustrates that thought places before itself bare
objectives, entities as we call them, which the thinking clothes by
expressing their mutual relations. Sense-awareness discloses fact with
factors which are the entities for thought. The separate distinction of
an entity in thought is not a metaphysical assertion, but a method of
procedure necessary for the finite expression of individual
propositions. Apart from entities there could be no finite truths; they
are the means by which the infinitude of irrelevance is kept out of
thought.

To sum up: the termini for thought are entities, primarily with bare
individuality, secondarily with properties and relations ascribed to
them in the procedure of thought; the termini for sense-awareness are
factors in the fact of nature, primarily relata and only secondarily
discriminated as distinct individualities.

No characteristic of nature which is immediately posited for knowledge
by sense-awareness can be explained. It is impenetrable by thought, in
the sense that its peculiar essential character which enters into
experience by sense-awareness is for thought merely the guardian of its
individuality as a bare entity. Thus for thought 'red' is merely a
definite entity, though for awareness 'red' has the content of its
individuality. The transition from the 'red' of awareness to the 'red'
of thought is t is accompanied by a definite loss of content, namely by the
transition from the factor 'red' to the entity 'red.' This loss in the
transition to thought is compensated by the fact that thought is
communicable whereas sense-awareness is incommunicable.

Thus there are three components in our knowledge of nature, namely,
fact, factors, and entities. Fact is the undifferentiated terminus of
sense-awareness; factors are termini of sense-awareness, differentiated
as elements of fact; entities are factors in their function as the
termini of thought. The entities thus spoken of are natural entities.
Thought is wider than nature, so that there are entities for thought
which are not natural entities.

When we speak of nature as a complex of related entities, the 'complex'
is fact as an entity for thought, to whose bare individuality is
ascribed the property of embracing in its complexity the natural
entities. It is our business to analyse this conception and in the
course of the analysis space and time should appear. Evidently the
relations holding between natural entities are themselves natural
entities, namely they are also factors of fact, there for
sense-awareness. Accordingly the structure of the natural complex can
never be completed in thought, just as the factors of fact can never be
exhausted in sense-awareness. Unexhaustiveness is an essential character
of our knowledge of nature. Also nature does not exhaust the matter for
thought, namely there are thoughts which would not occur in any
homogeneous thinking about nature.

The question as to whether sense-perception involves thought is largely
verbal. If sense-perception involves a cognition of individuality
abstracted from the actual position of the entity as a factor in fact,
then it undoubtedly does involve thought. But if it is conceived as
sense-awareness of a factor in fact competent to evoke emotion and
purposeful action without further cognition, then it does not involve
thought. In such a case the terminus of the sense-awareness is something
for mind, but nothing for thought. The sense-perception of some lower
forms of life may be conjectured to approximate to this character
habitually. Also occasionally our own sense-perception in moments when
thought-activity has been lulled to quiescence is not far off the
attainment of this ideal limit.

The process of discrimination in sense-awareness has two distinct sides.
There is the discrimination of fact into parts, and the discrimination
of any part of fact as exhibiting relations to entities which are not
parts of fact though they are ingredients in it. Namely the immediate
fact for awareness is the whole occurrence of nature. It is nature as an
event present for sense-awareness, and essentially passing. There is no
holding nature still and looking at it. We cannot redouble our efforts
to improve our knowledge of the terminus of our present sense-awareness;
it is our subsequent opportunity in subsequent sense-awareness which
gains the benefit of our good resolution. Thus the ultimate fact for
sense-awareness is an event. This whole event is discriminated by us
into partial events. We are aware of an event which is our bodily life,
of an event which is the course of nature within this room, and of a
vaguely perceived aggregate of other partial events. This is the
discrimination in sense-awareness of fact into parts.

I shall use the term 'part' in the arbitrarily limited sense of an event
which is part of the whole fact disclosed in awareness.

Sense-awareness also yields to us other factors in nature which are not
events. For example, sky-blue is seen as situated in a certain event.
This relation of situation requires further discussion which is
postponed to a later lecture. My present point is that sky-blue is found
in nature with a definite implication in events, but is not an event
itself. Accordingly in addition to events, there are other factors in
nature directly disclosed to us in sense-awareness. The conception in
thought of all the factors in nature as distinct entities with definite
natural relations is what I have in another place[1] called the
'diversification of nature.'

[1] Cf. _Enquiry_.

There is one general conclusion to be drawn from the foregoing
discussion. It is that the first task of a philosophy of science should
be some general classification of the entities disclosed to us in
sense-perception.

Among the examples of entities in addition to 'events' which we have
used for the purpose of illustration are the buildings of Bedford
College, Homer, and sky-blue. Evidently these are very different sorts
of things; and it is likely that statements which are made about one
kind of entity will not be true about other kinds. If human thought
proceeded with the orderly method which abstract logic would suggest to
it, we might go further and say that a classification of natural
entities should be the first step in science itself. Perhaps you will be
inclined to reply that this classification has already been effected,
and that science is concerned with the adventures of material entities
in space and time.

The history of the doctrine of matter has yet to be written. It is the
history of the influence of Greek philosophy on science. That influence
has issued in one long misconception of the metaphysical status of
natural entities. The entity has been separated from the factor which is
the terminus of sense-awareness. It has become the substratum for that
factor, and the factor has been degraded into an attribute of the
entity. In this way a distinction has been imported into nature which is
in truth no distinction at all. A natural entity is merely a factor of
fact, considered in itself. Its disconnexion from the complex of fact is
a mere abstraction. It is not the substratum of the factor, but the very
factor itself as bared in thought. Thus what is a mere procedure of mind
in the translation of sense-awareness into discursive knowledge has been
transmuted into a fundamental character of nature. In this way matter
has emerged as being the metaphysical substratum of its properties, and
the course of nature is interpreted as the history of matter.

Plato and Aristotle found Greek thought preoccupied with the quest for
the simple substances in terms of which the course of events could be
expressed. We may formulate thise of mind in the question, What is
nature made of? The answers which their genius gave to this question,
and more particularly the concepts which underlay the terms in which
they framed their answers, have determined the unquestioned
presuppositions as to time, space and matter which have reigned in
science.

In Plato the forms of thought are more fluid than in Aristotle, and
therefore, as I venture to think, the more valuable. Their importance
consists in the evidence they yield of cultivated thought about nature
before it had been forced into a uniform mould by the long tradition of
scientific philosophy. For example in the _Timaeus_ there is a
presupposition, somewhat vaguely expressed, of a distinction between the
general becoming of nature and the measurable time of nature. In a later
lecture I have to distinguish between what I call the passage of nature
and particular time-systems which exhibit certain characteristics of
that passage. I will not go so far as to claim Plato in direct support
of this doctrine, but I do think that the sections of the _Timaeus_
which deal with time become clearer if my distinction is admitted.

This is however a digression. I am now concerned with the origin of the
scientific doctrine of matter in Greek thought. In the _Timaeus_ Plato
asserts that nature is made of fire and earth with air and water as
intermediate between them, so that 'as fire is to air so is air to
water, and as air is to water so is water to earth.' He also suggests a
molecular hypothesis for these four elements. In this hypothesis
everything depends on the shape of the atoms; for earth it is cubical
and for fire it is pyramidal. To-day physicists are again discussing
the structure of the atom, and its shape is no slight factor in that
structure. Plato's guesses read much more fantastically than does
Aristotle's systematic analysis; but in some ways they are more
valuable. The main outline of his ideas is comparable with that of
modern science. It embodies concepts which any theory of natural
philosophy must retain and in some sense must explain. Aristotle asked
the fundamental question, What do we mean by 'substance'? Here the
reaction between his philosophy and his logic worked very unfortunately.
In his logic, the fundamental type of affirmative proposition is the
attribution of a predicate to a subject. Accordingly, amid the many
current uses of the term 'substance' which he analyses, he emphasises
its meaning as 'the ultimate substratum which is no longer predicated of
anything else.'

The unquestioned acceptance of the Aristotelian logic has led to an
ingrained tendency to postulate a substratum for whatever is disclosed
in sense-awareness, namely, to look below what we are aware of for the
substance in the sense of the 'concrete thing.' This is the origin of
the modern scientific concept of matter and of ether, namely they are
the outcome of this insistent habit of postulation.

Accordingly ether has been invented by modern science as the substratum
of the events which are spread through space and time beyond the reach
of ordinary ponderable matter. Personally, I think that predication is a
muddled notion confusing many different relations under a convenient
common form of speech. For example, I hold that the relation of green to
a blade of grass is entirely different from the relation of green to
the event which is the life history of that blade for some short period,
and is different from the relation of the blade to that event. In a
sense I call the event the situation of the green, and in another sense
it is the situation of the blade. Thus in one sense the blade is a
character or property which can be predicated of the situation, and in
another sense the green is a character or property of the same event
which is also its situation. In this way the predication of properties
veils radically different relations between entities.

Accordingly 'substance,' which is a correlative term to 'predication,'
shares in the ambiguity. If we are to look for substance anywhere, I
should find it in events which are in some sense the ultimate substance
of nature.

Matter, in its modern scientific sense, is a return to the Ionian effort
to find in space and time some stuff which composes nature. It has a
more refined signification than the early guesses at earth and water by
reason of a certain vague association with the Aristotelian idea of
substance.

Earth, water, air, fire, and matter, and finally ether are related in
direct succession so far as concerns their postulated characters of
ultimate substrata of nature. They bear witness to the undying vitality
of Greek philosophy in its search for the ultimate entities which are
the factors of the fact disclosed in sense-awareness. This search is the
origin of science.

The succession of ideas starting from the crude guesses of the early
Ionian thinkers and ending in the nineteenth century ether reminds us
that the scientific doctrine of matter is really a hybrid through which
philosophy passed on its way to the refined Aristotelian concept of
substance and to which science returned as it reacted against
philosophic abstractions. Earth, fire, and water in the Ionic philosophy
and the shaped elements in the _Timaeus_ are comparable to the matter
and ether of modern scientific doctrine. But substance represents the
final philosophic concept of the substratum which underlies any
attribute. Matter (in the scientific sense) is already in space and
time. Thus matter represents the refusal to think away spatial and
temporal characteristics and to arrive at the bare concept of an
individual entity. It is this refusal which has caused the muddle of
importing the mere procedure of thought into the fact of nature. The
entity, bared of all characteristics except those of space and time, has
acquired a physical status as the ultimate texture of nature; so that
the course of nature is conceived as being merely the fortunes of matter
in its adventure through space.

Thus the origin of the doctrine of matter is the outcome of uncritical
acceptance of space and time as external conditions for natural
existence. By this I do not mean that any doubt should be thrown on
facts of space and time as ingredients in nature. What I do mean is 'the
unconscious presupposition of space and time as being that within which
nature is set.' This is exactly the sort of presupposition which tinges
thought in any reaction against the subtlety of philosophical criticism.
My theory of the formation of the scientific doctrine of matter is that
first philosophy illegitimately transformed the bare entity, which is
simply an abstraction necessary for the method of thought, into the
metaphysical substratum of these factors in nature which in various
senses are assigned to entities as their attributes; and that, as a
second step, scientists (including philosophers who were scientists) in
conscious or unconscious ignoration of philosophy presupposed this
substratum, _qua_ substratum for attributes, as nevertheless in time and
space.

This is surely a muddle. The whole being of substance is as a substratum
for attributes. Thus time and space should be attributes of the
substance. This they palpably are not, if the matter be the substance of
nature, since it is impossible to express spatio-temporal truths without
having recourse to relations involving relata other than bits of matter.
I waive this point however, and come to another. It is not the substance
which is in space, but the attributes. What we find in space are the red
of the rose and the smell of the jasmine and the noise of cannon. We
have all told our dentists where our toothache is. Thus space is not a
relation between substances, but between attributes.

Thus even if you admit that the adherents of substance can be allowed to
conceive substance as matter, it is a fraud to slip substance into space
on the plea that space expresses relations between substances. On the
face of it space has nothing to do with substances, but only with their
attributes. What I mean is, that if you choose--as I think wrongly--to
construe our experience of nature as an awareness of the attributes of
substances, we are by this theory precluded from finding any analogous
direct relations between substances as disclosed in our experience. What
we do find are relations between the attributes of substances. Thus if
matter is looked on as substance in space, the space in which it finds
itself has very little to do with the space of our experience.

The above argument has been expressed in terms of the relational theory
of space. But if space be absolute--namely, if it have a being
independent of things in it--the course of the argument is hardly
changed. For things in space must have a certain fundamental relation to
space which we will call occupation. Thus the objection that it is the
attributes which are observed as related to space, still holds.

The scientific doctrine of matter is held in conjunction with an
absolute theory of time. The same arguments apply to the relations
between matter and time as apply to the relations between space and
matter. There is however (in the current philosophy) a difference in the
connexions of space with matter from those of time with matter, which I
will proceed to explain.

Space is not merely an ordering of material entities so that any one
entity bears certain relations to other material entities. The
occupation of space impresses a certain character on each material
entity in itself. By reason of its occupation of space matter has
extension. By reason of its extension each bit of matter is divisible
into parts, and each part is a numerically distinct entity from every
other such part. Accordingly it would seem that every material entity is
not really one entity. It is an essential multiplicity of entities.
There seems to be no stopping this dissociation of matter into
multiplicities short of finding each ultimate entity occupying one
individual point. This essential multiplicity of material entities is
certainly not what is meant by science, nor does it correspond to
anything disclosed in sense-awareness. It is absolutely necessary that
at a certain stage in this dissociation of matter a halt should be
called, and that the material entities thus obtained should be treated
as units. The stage of arrest may be arbitrary or may be set by the
characteristics of nature; but all reasoning in science ultimately drops
its space-analysis and poses to itself the problem, 'Here is one
material entity, what is happening to it as a unit entity?' Yet this
material entity is still retaining its extension, and as thus extended
is a mere multiplicity. Thus there is an essential atomic property in
nature which is independent of the dissociation of extension. There is
something which in itself is one, and which is more than the logical
aggregate of entities occupying points within the volume which the unit
occupies. Indeed we may well be sceptical as to these ultimate entities
at points, and doubt whether there are any such entities at all. They
have the suspicious character that we are driven to accept them by
abstract logic and not by observed fact.

Time (in the current philosophy) does not exert the same disintegrating
effect on matter which occupies it. If matter occupies a duration of
time, the whole matter occupies every part of that duration. Thus the
connexion between matter and time differs from the connexion between
matter and space as expressed in current scientific philosophy. There is
obviously a greater difficulty in conceiving time as the outcome of
relations between different bits of matter than there is in the
analogous conception of space. At an instant distinct volumes of space
are occupied by distinct bits of matter. Accordingly there is so far no
intrinsic difficulty in conceiving that space is merely the resultant of
relations between the bits of matter. But in the one-dimensional time
the same bit of matter occupies different portions of time. Accordingly
time would have to be expressible in terms of the relations of a bit of
matter with itself. My own view is a belief in the relational theory
both of space and of time, and of disbelief in the current form of the
relational theory of space which exhibits bits of matter as the relata
for spatial relations. The true relata are events. The distinction which
I have just pointed out between time and space in their connexion with
matter makes it evident that any assimilation of time and space cannot
proceed along the traditional line of taking matter as a fundamental
element in space-formation.

The philosophy of nature took a wrong turn during its development by
Greek thought. This erroneous presupposition is vague and fluid in
Plato's _Timaeus_. The general groundwork of the thought is still
uncommitted and can be construed as merely lacking due explanation and
the guarding emphasis. But in Aristotle's exposition the current
conceptions were hardened and made definite so as to produce a faulty
analysis of the relation between the matter and the form of nature as
disclosed in sense-awareness. In this phrase the term 'matter' is not
used in its scientific sense.

I will conclude by guarding myself against a misapprehension. It is
evident that the current doctrine of matter enshrines some fundamental
law of nature. Any simple illustration will exemplify what I mean. For
example, in a museum some specimen is locked securely in a glass case.
It stays there for years: it loses its colour, and perhaps falls to
pieces. But it is the same specimen; and the same chemical elements and
the same quantities of those elements are present within the case at the
end as were present at the beginning. Again the engineer and the
astronomer deal with the motions of real permanences in nature. Any
theory of nature which for one moment loses sight of these great basic
facts of experience is simply silly. But it is permissible to point out
that the scientific expression of these facts has become entangled in a
maze of doubtful metaphysics; and that, when we remove the metaphysics
and start afresh on an unprejudiced survey of nature, a new light is
thrown on many fundamental concepts which dominate science and guide the
progress of research.




CHAPTER II

THEORIES OF THE BIFURCATION OF NATURE


In my previous lecture I criticised the concept of matter as the
substance whose attributes we perceive. This way of thinking of matter
is, I think, the historical reason for its introduction into science,
and is still the vague view of it at the background of our thoughts
which makes the current scientific doctrine appear so obvious. Namely we
conceive ourselves as perceiving attributes of things, and bits of
matter are the things whose attributes we perceive.

In the seventeenth century the sweet simplicity of this aspect of matter
received a rude shock. The transmission doctrines of science were then
in process of elaboration and by the end of the century were
unquestioned, though their particular forms have since been modified.
The establishment of these transmission theories marks a turning point
in the relation between science and philosophy. The doctrines to which I
am especially alluding are the theories of light and sound. I have no
doubt that the theories had been vaguely floating about before as
obvious suggestions of common sense; for nothing in thought is ever
completely new. But at that epoch they were systematised and made exact,
and their complete consequences were ruthlessly deduced. It is the
establishment of this procedure of taking the consequences seriously
which marks the real discovery of a theory. Systematic doctrines of
light and sound as being something proceeding from the emitting bodies
were definitely established, and in particular the connexion of light
with colour was laid bare by Newton.

The result completely destroyed the simplicity of the 'substance and
attribute' theory of perception. What we see depends on the light
entering the eye. Furthermore we do not even perceive what enters the
eye. The things transmitted are waves or--as Newton thought--minute
particles, and the things seen are colours. Locke met this difficulty by
a theory of primary and secondary qualities. Namely, there are some
attributes of the matter which we do perceive. These are the primary
qualities, and there are other things which we perceive, such as
colours, which are not attributes of matter, but are perceived by us as
if they were such attributes. These are the secondary qualities of
matter.

Why should we perceive secondary qualities? It seems an extremely
unfortunate arrangement that we should perceive a lot of things that are
not there. Yet this is what the theory of secondary qualities in fact
comes to. There is now reigning in philosophy and in science an
apathetic acquiescence in the conclusion that no coherent account can be
given of nature as it is disclosed to us in sense-awareness, without
dragging in its relations to mind. The modern account of nature is not,
as it should be, merely an account of what the mind knows of nature; but
it is also confused with an account of what nature does to the mind. The
result has been disastrous both to science and to philosophy, but
chiefly to philosophy. It has transformed the grand question of the
relations between nature and mind into the petty form of the interaction
between the human body and mind.

Berkeley's polemic against matter was based on this confusion introduced
by the transmission theory of light. He advocated, rightly as I think,
the abandonment of the doctrine of matter in its present form. He had
however nothing to put in its place except a theory of the relation of
finite minds to the divine mind.

But we are endeavouring in these lectures to limit ourselves to nature
itself and not to travel beyond entities which are disclosed in
sense-awareness.

Percipience in itself is taken for granted. We consider indeed
conditions for percipience, but only so far as those conditions are
among the disclosures of perception. We leave to metaphysics the
synthesis of the knower and the known. Some further explanation and
defence of this position is necessary, if the line of argument of these
lectures is to be comprehensible.

The immediate thesis for discussion is that any metaphysical
interpretation is an illegitimate importation into the philosophy of
natural science. By a metaphysical interpretation I mean any discussion
of the how (beyond nature) and of the why (beyond nature) of thought and
sense-awareness. In the philosophy of science we seek the general
notions which apply to nature, namely, to what we are aware of in
perception. It is the philosophy of the thing perceived, and it should
not be confused with the metaphysics of reality of which the scope
embraces both perceiver and perceived. No perplexity concerning the
object of knowledge can be solved by saying that there is a mind knowing
it[2].

[2] Cf. _Enquiry_, preface.

In other words, the ground taken is this: sense-awareness is an
awareness of something. What then is the general character of that
something of which we are aware? We do not ask about the percipient or
about the process, but about the perceived. I emphasise this point
because discussions on the philosophy of science are usually extremely
metaphysical--in my opinion, to the great detriment of the subject.

The recourse to metaphysics is like throwing a match into the powder
magazine. It blows up the whole arena. This is exactly what scientific
philosophers do when they are driven into a corner and convicted of
incoherence. They at once drag in the mind and talk of entities in the
mind or out of the mind as the case may be. For natural philosophy
everything perceived is in nature. We may not pick and choose. For us
the red glow of the sunset should be as much part of nature as are the
molecules and electric waves by which men of science would explain the
phenomenon. It is for natural philosophy to analyse how these various
elements of nature are connected.

In making this demand I conceive myself as adopting our immediate
instinctive attitude towards perceptual knowledge which is only
abandoned under the influence of theory. We are instinctively willing to
believe that by due attention, more can be found in nature than that
which is observed at first sight. But we will not be content with less.
What we ask from the philosophy of science is some account of the
coherence of things perceptively known.

This means a refusal to countenance any theory of psychic additions to
the object known in perception. For example, what is given in perception
is the green grass. This is an object which we know as an ingredient in
nature. The theory of psychic additions would treat the greenness as a
psychic addition furnished by the perceiving mind, and would leave to
nature merely the molecules and the radiant energy which influence the
mind towards that perception. My argument is that this dragging in of
the mind as making additions of its own to the thing posited for
knowledge by sense-awareness is merely a way of shirking the problem of
natural philosophy. That problem is to discuss the relations _inter se_
of things known, abstracted from the bare fact that they are known.
Natural philosophy should never ask, what is in the mind and what is in
nature. To do so is a confession that it has failed to express relations
between things perceptively known, namely to express those natural
relations whose expression is natural philosophy. It may be that the
task is too hard for us, that the relations are too complex and too
various for our apprehension, or are too trivial to be worth the trouble
of exposition. It is indeed true that we have gone but a very small way
in the adequate formulation of such relations. But at least do not let
us endeavour to conceal failure under a theory of the byplay of the
perceiving mind.

What I am essentially protesting against is the bifurcation of nature
into two systems of reality, which, in so far as they are real, are real
in different senses. One reality would be the entities such as electrons
which are the study of speculative physics. This would be the reality
which is there for knowledge; although on this theory it is never known.
For what is known is the other sort of reality, which is the byplay of
the mind. Thus there would be two natures, one is the conjecture and the
other is the dream.

Another way of phrasing this theory which I am arguing against is to
bifurcate nature into two divisions, namely into the nature apprehended
in awareness and the nature which is the cause of awareness. The nature
which is the fact apprehended in awareness holds within it the greenness
of the trees, the song of the birds, the warmth of the sun, the hardness
of the chairs, and the feel of the velvet. The nature which is the cause
of awareness is the conjectured system of molecules and electrons which
so affects the mind as to produce the awareness of apparent nature. The
meeting point of these two natures is the mind, the causal nature being
influent and the apparent nature being effluent.

There are four questions which at once suggest themselves for discussion
in connexion with this bifurcation theory of nature. They concern (i)
causality, (ii) time, (iii) space, and (iv) delusions. These questions
are not really separable. They merely constitute four distinct starting
points from which to enter upon the discussion of the theory.

Causal nature is the influence on the mind which is the cause of the
effluence of apparent nature from the mind. This conception of causal
nature is not to be confused with the distinct conception of one part of
nature as being the cause of another part. For example, the burning of
the fire and the passage of heat from it through intervening space is
the cause of the body, its nerves and its brain, functioning in certain
ways. But this is not an action of nature on the mind. It is an
interaction within nature. The causation involved in this interaction is
causation in a different sense from the influence of this system of
bodily interactions within nature on the alien mind which thereupon
perceives redness and warmth.

The bifurcation theory is an attempt to exhibit natural science as an
investigation of the cause of the fact of knowledge. Namely, it is an
attempt to exhibit apparent nature as an effluent from the mind because
of causal nature. The whole notion is partly based on the implicit
assumption that the mind can only know that which it has itself produced
and retains in some sense within itself, though it requires an exterior
reason both as originating and as determining the character of its
activity. But in considering knowledge we should wipe out all these
spatial metaphors, such as 'within the mind' and 'without the mind.'
Knowledge is ultimate. There can be no explanation of the 'why' of
knowledge; we can only describe the 'what' of knowledge. Namely we can
analyse the content and its internal relations, but we cannot explain
why there is knowledge. Thus causal nature is a metaphysical chimera;
though there is need of a metaphysics whose scope transcends the
limitation to nature. The object of such a metaphysical science is not
to explain knowledge, but exhibit in its utmost completeness our concept
of reality.

However, we must admit that the causality theory of nature has its
strong suit. The reason why the bifurcation of nature is always creeping
back into scientific philosophy is the extreme difficulty of exhibiting
the perceived redness and warmth of the fire in one system of relations
with the agitated molecules of carbon and oxygen, with the radiant
energy from them, and with the various functionings of the material
body. Unless we produce the all-embracing relations, we are faced with a
bifurcated nature; namely, warmth and redness on one side, and
molecules, electrons and ether on the other side. Then the two factors
are explained as being respectively the cause and the mind's reaction to
the cause.

Time and space would appear to provide these all-embracing relations
which the advocates of the philosophy of the unity of nature require.
The perceived redness of the fire and the warmth are definitely related
in time and in space to the molecules of the fire and the molecules of
the body.

It is hardly more than a pardonable exaggeration to say that the
determination of the meaning of nature reduces itself principally to the
discussion of the character of time and the character of space. In
succeeding lectures I shall explain my own view of time and space. I
shall endeavour to show that they are abstractions from more concrete
elements of nature, namely, from events. The discussion of the details
of the process of abstraction will exhibit time and space as
interconnected, and will finally lead us to the sort of connexions
between their measurements which occur in the modern theory of
electromagnetic relativity. But this is anticipating our subsequent line
of development. At present I wish to consider how the ordinary views of
time and space help, or fail to help, in unifying our conception of
nature.

First, consider the absolute theories of time and space. We are to
consider each, namely both time and space, to be a separate and
independent system of entities, each system known to us in itself and
for itself concurrently with our knowledge of the events of nature. Time
is the ordered succession of durationless instants; and these instants
are known to us merely as the relata in the serial relation which is the
time-ordering relation, and the time-ordering relation is merely known
to us as relating the instants. Namely, the relation and the instants
are jointly known to us in our apprehension of time, each implying the
other.

This is the absolute theory of time. Frankly, I confess that it seems to
me to be very unplausible. I cannot in my own knowledge find anything
corresponding to the bare time of the absolute theory. Time is known to
me as an abstraction from the passage of events. The fundamental fact
which renders this abstraction possible is the passing of nature, its
development, its creative advance, and combined with this fact is
another characteristic of nature, namely the extensive relation between
events. These two facts, namely the passage of events and the extension
of events over each other, are in my opinion the qualities from which
time and space originate as abstractions. But this is anticipating my
own later speculations.

Meanwhile, returning to the absolute theory, we are to suppose that time
is known to us independently of any events in time. What happens in time
occupies time. This relation of events to the time occupied, namely this
relation of occupation, is a fundamental relation of nature to time.
Thus the theory requires that we are aware of two fundamental relations,
the time-ordering relation between instants, and the time-occupation
relation between instants of time and states of nature which happen at
those instants.

There are two considerations which lend powerful support to the reigning
theory of absolute time. In the first place time extends beyond nature.
Our thoughts are in time. Accordingly it seems impossible to derive time
merely from relations between elements of nature. For in that case
temporal relations could not relate thoughts. Thus, to use a metaphor,
time would apparently have deeper roots in reality than has nature. For
we can imagine thoughts related in time without any perception of
nature. For example we can imagine one of Milton's angels with thoughts
succeeding each other in time, who does not happen to have noticed that
the Almighty has created space and set therein a material universe. As a
matter of fact I think that Milton set space on the same absolute level
as time. But that need not disturb the illustration. In the second place
it is difficult to derive the true serial character of time from the
relative theory. Each instant is irrevocable. It can never recur by the
very character of time. But if on the relative theory an instant of time
is simply the state of nature at that time, and the time-ordering
relation is simply the relation between such states, then the
irrevocableness of time would seem to mean that an actual state of all
nature can never return. I admit it seems unlikely that there should
ever be such a recurrence down to the smallest particular. But extreme
unlikeliness is not the point. Our ignorance is so abysmal that our
judgments of likeliness and unlikeliness of future events hardly count.
The real point is that the exact recurrence of a state of nature seems
merely unlikely, while the recurrence of an instant of time violates our
whole concept of time-order. The instants of time which have passed, are
passed, and can never be again.

Any alternative theory of time must reckon with these two considerations
which are buttresses of the absolute theory. But I will not now continue
their discussion.

The absolute theory of space is analogous to the corresponding theory of
time, but the reasons for its maintenance are weaker. Space, on this
theory, is a system of extensionless points which are the relata in
space-ordering relations which can technically be combined into one
relation. This relation does not arrange the points in one linear series
analogously to the simple method of the time-ordering relation for
instants. The essential logical characteristics of this relation from
which all the properties of space spring are expressed by mathematicians
in the axioms of geometry. From these axioms[3] as framed by modern
mathematicians the whole science of geometry can be deduced by the
strictest logical reasoning. The details of these axioms do not now
concern us. The points and the relations are jointly known to us in our
apprehension of space, each implying the other. What happens in space,
occupies space. This relation of occupation is not usually stated for
events but for objects. For example, Pompey's statue would be said to
occupy space, but not the event which was the assassination of Julius
Caesar. In this I think that ordinary usage is unfortunate, and I hold
that the relations of events to space and to time are in all respects
analogous. But here I am intruding my own opinions which are to be
discussed in subsequent lectures. Thus the theory of absolute space
requires that we are aware of two fundamental relations, the
space-ordering relation, which holds between points, and the
space-occupation relation between points of space and material objects.

[3] Cf. (for example) _Projective Geometry_ by Veblen and Young, vol. i.
1910, vol. ii. 1917, Ginn and Company, Boston, U.S.A.

This theory lacks the two main supports of the corresponding theory of
absolute time. In the first place space does not extend beyond nature in
the sense that time seems to do. Our thoughts do not seem to occupy
space in quite the same intimate way in which they occupy time. For
example, I have been thinking in a room, and to that extent my thoughts
are in space. But it seems nonsense to ask how much volume of the room
they occupied, whether it was a cubic foot or a cubic inch; whereas the
same thoughts occupy a determinate duration of time, say, from eleven to
twelve on a certain date.

Thus whereas the relations of a relative theory of time are required to
relate thoughts, it does not seem so obvious that the relations of a
relative theory of space are required to relate them. The connexion of
thought with space seems to have a certain character of indirectness
which appears to be lacking in the connexion of thought with time.

Again the irrevocableness of time does not seem to have any parallel for
space. Space, on the relative theory, is the outcome of certain
relations between objects commonly said to be in space; and whenever
there are the objects, so related, there is the space. No difficulty
seems to arise like that of the inconvenient instants of time which
might conceivably turn up again when we thought that we had done with
them.

The absolute theory of space is not now generally popular. The knowledge
of bare space, as a system of entities known to us in itself and for
itself independently of our knowledge of the events in nature, does not
seem to correspond to anything in our experience. Space, like time,
would appear to be an abstraction from events. According to my own
theory it only differentiates itself from time at a somewhat developed
stage of the abstractive process. The more usual way of expressing the
relational theory of space would be to consider space as an abstraction
from the relations between material objects.

Suppose now we assume absolute time and absolute space. What bearing
has this assumption on the concept of nature as bifurcated into causal
nature and apparent nature? Undoubtedly the separation between the two
natures is now greatly mitigated. We can provide them with two systems
of relations in common; for both natures can be presumed to occupy the
same space and the same time. The theory now is this: Causal events
occupy certain periods of the absolute time and occupy certain positions
of the absolute space. These events influence a mind which thereupon
perceives certain apparent events which occupy certain periods in the
absolute time and occupy certain positions of the absolute space; and
the periods and positions occupied by the apparent events bear a
determinate relation to the periods and positions occupied by the causal
events.

Furthermore definite causal events produce for the mind definite
apparent events. Delusions are apparent events which appear in temporal
periods and spatial positions without the intervention of these causal
events which are proper for influencing of the mind to their perception.

The whole theory is perfectly logical. In these discussions we cannot
hope to drive an unsound theory to a logical contradiction. A reasoner,
apart from mere slips, only involves himself in a contradiction when he
is shying at a _reductio ad absurdum_. The substantial reason for
rejecting a philosophical theory is the 'absurdum' to which it reduces
us. In the case of the philosophy of natural science the 'absurdum' can
only be that our perceptual knowledge has not the character assigned to
it by the theory. If our opponent affirms that his knowledge has that
character, we can only--after making doubly sure that we understand
each other--agree to differ. Accordingly the first duty of an expositor
in stating a theory in which he disbelieves is to exhibit it as logical.
It is not there where his trouble lies.

Let me summarise the previously stated objections to this theory of
nature. In the first place it seeks for the cause of the knowledge of
the thing known instead of seeking for the character of the thing known:
secondly it assumes a knowledge of time in itself apart from events
related in time: thirdly it assumes a knowledge of space in itself apart
from events related in space. There are in addition to these objections
other flaws in the theory.

Some light is thrown on the artificial status of causal nature in this
theory by asking, why causal nature is presumed to occupy time and
space. This really raises the fundamental question as to what
characteristics causal nature should have in common with apparent
nature. Why--on this theory--should the cause which influences the mind
to perception have any characteristics in common with the effluent
apparent nature? In particular, why should it be in space? Why should it
be in time? And more generally, What do we know about mind which would
allow us to infer any particular characteristics of a cause which should
influence mind to particular effects?

The transcendence of time beyond nature gives some slight reason for
presuming that causal nature should occupy time. For if the mind
occupies periods of time, there would seem to be some vague reason for
assuming that influencing causes occupy the same periods of time, or at
least, occupy periods which are strictly related to the mental periods.
But if the mind does not occupy volumes of space, there seems to be no
reason why causal nature should occupy any volumes of space. Thus space
would seem to be merely apparent in the same sense as apparent nature is
merely apparent. Accordingly if science is really investigating causes
which operate on the mind, it would seem to be entirely on the wrong
tack in presuming that the causes which it is seeking for have spatial
relations. Furthermore there is nothing else in our knowledge analogous
to these causes which influence the mind to perception. Accordingly,
beyond the rashly presumed fact that they occupy time, there is really
no ground by which we can determine any point of their character. They
must remain for ever unknown.

Now I assume as an axiom that science is not a fairy tale. It is not
engaged in decking out unknowable entities with arbitrary and fantastic
properties. What then is it that science is doing, granting that it is
effecting something of importance? My answer is that it is determining
the character of things known, namely the character of apparent nature.
But we may drop the term 'apparent'; for there is but one nature, namely
the nature which is before us in perceptual knowledge. The characters
which science discerns in nature are subtle characters, not obvious at
first sight. They are relations of relations and characters of
characters. But for all their subtlety they are stamped with a certain
simplicity which makes their consideration essential in unravelling the
complex relations between characters of more perceptive insistence.

The fact that the bifurcation of nature into causal and apparent
components does not express what we mean by our knowledge is brought
before us when we realise our thoughts in any discussion of the causes
of our perceptions. For example, the fire is burning and we see a red
coal. This is explained in science by radiant energy from the coal
entering our eyes. But in seeking for such an explanation we are not
asking what are the sort of occurrences which are fitted to cause a mind
to see red. The chain of causation is entirely different. The mind is
cut out altogether. The real question is, When red is found in nature,
what else is found there also? Namely we are asking for an analysis of
the accompaniments in nature of the discovery of red in nature. In a
subsequent lecture I shall expand this line of thought. I simply draw
attention to it here in order to point out that the wave-theory of light
has not been adopted because waves are just the sort of things which
ought to make a mind perceive colours. This is no part of the evidence
which has ever been adduced for the wave-theory, yet on the causal
theory of perception, it is really the only relevant part. In other
words, science is not discussing the causes of knowledge, but the
coherence of knowledge. The understanding which is sought by science is
an understanding of relations within nature.

So far I have discussed the bifurcation of nature in connexion with the
theories of absolute time and of absolute space. My reason has been that
the introduction of the relational theories only weakens the case for
bifurcation, and I wished to discuss this case on its strongest grounds.

For instance, suppose we adopt the relational theory of space. Then the
space in which apparent nature is set is the expression of certain
relations between the apparent objects. It is a set of apparent
relations between apparent relata. Apparent nature is the dream, and
the apparent relations of space are dream relations, and the space is
the dream space. Similarly the space in which causal nature is set is
the expression of certain relations between the causal objects. It is
the expression of certain facts about the causal activity which is going
on behind the scenes. Accordingly causal space belongs to a different
order of reality to apparent space. Hence there is no pointwise
connexion between the two and it is meaningless to say that the
molecules of the grass are in any place which has a determinate spatial
relation to the place occupied by the grass which we see. This
conclusion is very paradoxical and makes nonsense of all scientific
phraseology. The case is even worse if we admit the relativity of time.
For the same arguments apply, and break up time into the dream time and
causal time which belong to different orders of reality.

I have however been discussing an extreme form of the bifurcation
theory. It is, as I think, the most defensible form. But its very
definiteness makes it the more evidently obnoxious to criticism. The
intermediate form allows that the nature we are discussing is always the
nature directly known, and so far it rejects the bifurcation theory. But
it holds that there are psychic additions to nature as thus known, and
that these additions are in no proper sense part of nature. For example,
we perceive the red billiard ball at its proper time, in its proper
place, with its proper motion, with its proper hardness, and with its
proper inertia. But its redness and its warmth, and the sound of the
click as a cannon is made off it are psychic additions, namely,
secondary qualities which are only the mind's way of perceiving nature.
This is not only the vaguely prevalent theory, but is, I believe, the
historical form of the bifurcation theory in so far as it is derived
from philosophy. I shall call it the theory of psychic additions.

This theory of psychic additions is a sound common-sense theory which
lays immense stress on the obvious reality of time, space, solidity and
inertia, but distrusts the minor artistic additions of colour, warmth
and sound.

The theory is the outcome of common-sense in retreat. It arose in an
epoch when the transmission theories of science were being elaborated.
For example, colour is the result of a transmission from the material
object to the perceiver's eye; and what is thus transmitted is not
colour. Thus colour is not part of the reality of the material object.
Similarly for the same reason sounds evaporate from nature. Also warmth
is due to the transfer of something which is not temperature. Thus we
are left with spatio-temporal positions, and what I may term the
'pushiness' of the body. This lands us to eighteenth and nineteenth
century materialism, namely, the belief that what is real in nature is
matter, in time and in space and with inertia.

Evidently a distinction in quality has been presupposed separating off
some perceptions due to touch from other perceptions. These
touch-perceptions are perceptions of the real inertia, whereas the other
perceptions are psychic additions which must be explained on the causal
theory. This distinction is the product of an epoch in which physical
science has got ahead of medical pathology and of physiology.
Perceptions of push are just as much the outcome of transmission as are
perceptions of colour. When colour is perceived the nerves of the body
are excited in one way and transmit their message towards the brain, and
when push is perceived other nerves of the body are excited in another
way and transmit their message towards the brain. The message of the one
set is not the conveyance of colour, and the message of the other set is
not the conveyance of push. But in one case colour is perceived and in
the other case the push due to the object. If you snip certain nerves,
there is an end to the perception of colour; and if you snip certain
other nerves, there is an end to the perception of push. It would appear
therefore that any reasons which should remove colour from the reality
of nature should also operate to remove inertia.

Thus the attempted bifurcation of apparent nature into two parts of
which one part is both causal for its own appearance and for the
appearance of the other part, which is purely apparent, fails owing to
the failure to establish any fundamental distinction between our ways of
knowing about the two parts of nature as thus partitioned. I am not
denying that the feeling of muscular effort historically led to the
formulation of the concept of force. But this historical fact does not
warrant us in assigning a superior reality in nature to material inertia
over colour or sound. So far as reality is concerned all our
sense-perceptions are in the same boat, and must be treated on the same
principle. The evenness of treatment is exactly what this compromise
theory fails to achieve.

The bifurcation theory however dies hard. The reason is that there
really is a difficulty to be faced in relating within the same system of
entities the redness of the fire with the agitation of the molecules. In
another lecture I will give my own explanation of the origin of the
difficulty and of its solution.

Another favourite solution, the most attenuated form which the
bifurcation theory assumes, is to maintain that the molecules and ether
of science are purely conceptual. Thus there is but one nature, namely
apparent nature, and atoms and ether are merely names for logical terms
in conceptual formulae of calculation.

But what is a formula of calculation? It is presumably a statement that
something or other is true for natural occurrences. Take the simplest of
all formulae, Two and two make four. This--so far as it applies to
nature--asserts that if you take two natural entities, and then again
two other natural entities, the combined class contains four natural
entities. Such formulae which are true for any entities cannot result in
the production of the concepts of atoms. Then again there are formulae
which assert that there are entities in nature with such and such
special properties, say, for example, with the properties of the atoms
of hydrogen. Now if there are no such entities, I fail to see how any
statements about them can apply to nature. For example, the assertion
that there is green cheese in the moon cannot be a premiss in any
deduction of scientific importance, unless indeed the presence of green
cheese in the moon has been verified by experiment. The current answer
to these objections is that, though atoms are merely conceptual, yet
they are an interesting and picturesque way of saying something else
which is true of nature. But surely if it is something else that you
mean, for heaven's sake say it. Do away with this elaborate machinery of
a conceptual nature which consists of assertions about things which
don't exist in order to convey truths about things which do exist. I am
maintaining the obvious position that scientific laws, if they are true,
are statements about entities which we obtain knowledge of as being in
nature; and that, if the entities to which the statements refer are not
to be found in nature, the statements about them have no relevance to
any purely natural occurrence. Thus the molecules and electrons of
scientific theory are, so far as science has correctly formulated its
laws, each of them factors to be found in nature. The electrons are only
hypothetical in so far as we are not quite certain that the electron
theory is true. But their hypothetical character does not arise from the
essential nature of the theory in itself after its truth has been
granted.

Thus at the end of this somewhat complex discussion, we return to the
position which was affirmed at its beginning. The primary task of a
philosophy of natural science is to elucidate the concept of nature,
considered as one complex fact for knowledge, to exhibit the fundamental
entities and the fundamental relations between entities in terms of
which all laws of nature have to be stated, and to secure that the
entities and relations thus exhibited are adequate for the expression of
all the relations between entities which occur in nature.

The third requisite, namely that of adequacy, is the one over which all
the difficulty occurs. The ultimate data of science are commonly assumed
to be time, space, material, qualities of material, and relations
between material objects. But data as they occur in the scientific laws
do not relate all the entities which present themselves in our
perception of nature. For example, the wave-theory of light is an
excellent well-established theory; but unfortunately it leaves out
colour as perceived. Thus the perceived redness--or, other colour--has
to be cut out of nature and made into the reaction of the mind under the
impulse of the actual events of nature. In other words this concept of
the fundamental relations within nature is inadequate. Thus we have to
bend our energies to the enunciation of adequate concepts.

But in so doing, are we not in fact endeavouring to solve a metaphysical
problem? I do not think so. We are merely endeavouring to exhibit the
type of relations which hold between the entities which we in fact
perceive as in nature. We are not called on to make any pronouncement as
to the psychological relation of subjects to objects or as to the status
of either in the realm of reality. It is true that the issue of our
endeavour may provide material which is relevant evidence for a
discussion on that question. It can hardly fail to do so. But it is only
evidence, and is not itself the metaphysical discussion. In order to
make clear the character of this further discussion which is out of our
ken, I will set before you two quotations. One is from Schelling and I
extract the quotation from the work of the Russian philosopher Lossky
which has recently been so excellently translated into English[4]--'In
the "Philosophy of Nature" I considered the subject-object called nature
in its activity of self-constructing. In order to understand it, we must
rise to an intellectual intuition of nature. The empiricist does not
rise thereto, and for this reason in all his explanations it is always
_he himself_ that proves to be constructing nature. It is no wonder,
then, that his construction and that which was to be constructed so
seldom coincide. A _Natur-philosoph_ raises nature to independence, and
makes it construct itself, and he never feels, therefore, the necessity
of opposing nature as constructed (_i.e._ as experience) to real
nature, or of correcting the one by means of the other.'

[4] _The Intuitive Basis of Knowledge_, by N. O. Lossky, transl. by Mrs
Duddington, Macmillan and Co., 1919.

The other quotation is from a paper read by the Dean of St Paul's before
the Aristotelian Society in May of 1919. Dr Inge's paper is entitled
'Platonism and Human Immortality,' and in it there occurs the following
statement: 'To sum up. The Platonic doctrine of immortality rests on the
_independence_ of the spiritual world. The spiritual world is not a
world of unrealised ideals, over against a real world of unspiritual
fact. It is, on the contrary, the real world, of which we have a true
though very incomplete knowledge, over against a world of common
experience which, as a complete whole, is not real, since it is
compacted out of miscellaneous data, not all on the same level, by the
help of the imagination. There is no world corresponding to the world of
our common experience. Nature makes abstractions for us, deciding what
range of vibrations we are to see and hear, what things we are to notice
and remember.'

I have cited these statements because both of them deal with topics
which, though they lie outside the range of our discussion, are always
being confused with it. The reason is that they lie proximate to our
field of thought, and are topics which are of burning interest to the
metaphysically minded. It is difficult for a philosopher to realise that
anyone really is confining his discussion within the limits that I have
set before you. The boundary is set up just where he is beginning to get
excited. But I submit to you that among the necessary prolegomena for
philosophy and for natural science is a thorough understanding ofng of the
types of entities, and types of relations among those entities, which
are disclosed to us in our perceptions of nature.




CHAPTER III

TIME


The two previous lectures of this course have been mainly critical. In
the present lecture I propose to enter upon a survey of the kinds of
entities which are posited for knowledge in sense-awareness. My purpose
is to investigate the sorts of relations which these entities of various
kinds can bear to each other. A classification of natural entities is
the beginning of natural philosophy. To-day we commence with the
consideration of Time.

In the first place there is posited for us a general fact: namely,
something is going on; there is an occurrence for definition.

This general fact at once yields for our apprehension two factors, which
I will name, the 'discerned' and the 'discernible.' The discerned is
comprised of those elements of the general fact which are discriminated
with their own individual peculiarities. It is the field directly
perceived. But the entities of this field have relations to other
entities which are not particularly discriminated in this individual
way. These other entities are known merely as the relata in relation to
the entities of the discerned field. Such an entity is merely a
'something' which has such-and-such definite relations to some definite
entity or entities in the discerned field. As being thus related, they
are--owing to the particular character of these relations--known as
elements of the general fact which is going on. But we are not aware of
them except as entities fulfilling the functions of relata in these
relations.

Thus the complete general fact, posited as occurring, comprises both
sets of entities, namely the entities perceived in their own
individuality and other entities merely apprehended as relata without
further definition. This complete general fact is the discernible and it
comprises the discerned. The discernible is all nature as disclosed in
that sense-awareness, and extends beyond and comprises all of nature as
actually discriminated or discerned in that sense-awareness. The
discerning or discrimination of nature is a peculiar awareness of
special factors in nature in respect to their peculiar characters. But
the factors in nature of which we have this peculiar sense-awareness are
known as not comprising all the factors which together form the whole
complex of related entities within the general fact there for
discernment. This peculiarity of knowledge is what I call its
unexhaustive character. This character may be metaphorically described
by the statement that nature as perceived always has a ragged edge. For
example, there is a world beyond the room to which our sight is confined
known to us as completing the space-relations of the entities discerned
within the room. The junction of the interior world of the room with the
exterior world beyond is never sharp. Sounds and subtler factors
disclosed in sense-awareness float in from the outside. Every type of
sense has its own set of discriminated entities which are known to be
relata in relation with entities not discriminated by that sense. For
example we see something which we do not touch and we touch something
which we do not see, and we have a general sense of the space-relations
between the entity disclosed in sight and the entity disclosed in touch.
Thus in the first place each of these two entities is known as a relatum
in a general system of space-relations and in the second place the
particular mutual relatiothese two entities as related to each
other in this general system is determined. But the general system of
space-relations relating the entity discriminated by sight with that
discriminated by sight is not dependent on the peculiar character of the
other entity as reported by the alternative sense. For example, the
space-relations of the thing seen would have necessitated an entity as a
relatum in the place of the thing touched even although certain elements
of its character had not been disclosed by touch. Thus apart from the
touch an entity with a certain specific relation to the thing seen would
have been disclosed by sense-awareness but not otherwise discriminated
in respect to its individual character. An entity merely known as
spatially related to some discerned entity is what we mean by the bare
idea of 'place.' The concept of place marks the disclosure in
sense-awareness of entities in nature known merely by their spatial
relations to discerned entities. It is the disclosure of the discernible
by means of its relations to the discerned.

This disclosure of an entity as a relatum without further specific
discrimination of quality is the basis of our concept of significance.
In the above example the thing seen was significant, in that it
disclosed its spatial relations to other entities not necessarily
otherwise entering into consciousness. Thus significance is relatedness,
but it is relatedness with the emphasis on one end only of the relation.

For the sake of simplicity I have confined the argument to spatial
relations; but the same considerations apply to temporal relations. The
concept of 'period of time' marks the disclosure in sense-awareness of
entities in nature known merely by their temporal relations to
discerned entities. Still further, this separation of the ideas of space
and time has merely been adopted for the sake of gaining simplicity of
exposition by conformity to current language. What we discern is the
specific character of a place through a period of time. This is what I
mean by an 'event.' We discern some specific character of an event. But
in discerning an event we are also aware of its significance as a
relatum in the structure of events. This structure of events is the
complex of events as related by the two relations of extension and
cogredience. The most simple expression of the properties of this
structure are to be found in our spatial and temporal relations. A
discerned event is known as related in this structure to other events
whose specific characters are otherwise not disclosed in that immediate
awareness except so far as that they are relata within the structure.

The disclosure in sense-awareness of the structure of events classifies
events into those which are discerned in respect to some further
individual character and those which are not otherwise disclosed except
as elements of the structure. These signified events must include events
in the remote past as well as events in the future. We are aware of
these as the far off periods of unbounded time. But there is another
classification of events which is also inherent in sense-awareness.
These are the events which share the immediacy of the immediately
present discerned events. These are the events whose characters together
with those of the discerned events comprise all nature present for
discernment. They form the complete general fact which is all nature now
present as disclosed in that sense-awareness. It is in this second
classification of events that the differentiation of space from time
takes its origin. The germ of space is to be found in the mutual
relations of events within the immediate general fact which is all
nature now discernible, namely within the one event which is the
totality of present nature. The relations of other events to this
totality of nature form the texture of time.

The unity of this general present fact is expressed by the concept of
simultaneity. The general fact is the whole simultaneous occurrence of
nature which is now for sense-awareness. This general fact is what I
have called the discernible. But in future I will call it a 'duration,'
meaning thereby a certain whole of nature which is limited only by the
property of being a simultaneity. Further in obedience to the principle
of comprising within nature the whole terminus of sense-awareness,
simultaneity must not be conceived as an irrelevant mental concept
imposed upon nature. Our sense-awareness posits for immediate
discernment a certain whole, here called a 'duration'; thus a duration
is a definite natural entity. A duration is discriminated as a complex
of partial events, and the natural entities which are components of this
complex are thereby said to be 'simultaneous with this duration.' Also
in a derivative sense they are simultaneous with each other in respect
to this duration. Thus simultaneity is a definite natural relation. The
word 'duration' is perhaps unfortunate in so far as it suggests a mere
abstract stretch of time. This is not what I mean. A duration is a
concrete slab of nature limited by simultaneity which is an essential
factor disclosed in sense-awareness.

Nature is a process. As in the case of everything directly exhibited in
sense-awareness, there can be no explanation of this characteristic of
nature. All that can be done is to use language which may speculatively
demonstrate it, and also to express the relation of this factor in
nature to other factors.

It is an exhibition of the process of nature that each duration happens
and passes. The process of nature can also be termed the passage of
nature. I definitely refrain at this stage from using the word 'time,'
since the measurable time of science and of civilised life generally
merely exhibits some aspects of the more fundamental fact of the passage
of nature. I believe that in this doctrine I am in full accord with
Bergson, though he uses 'time' for the fundamental fact which I call the
'passage of nature.' Also the passage of nature is exhibited equally in
spatial transition as well as in temporal transition. It is in virtue of
its passage that nature is always moving on. It is involved in the
meaning of this property of 'moving on' that not only is any act of
sense-awareness just that act and no other, but the terminus of each act
is also unique and is the terminus of no other act. Sense-awareness
seizes its only chance and presents for knowledge something which is for
it alone.

There are two senses in which the terminus of sense-awareness is unique.
It is unique for the sense-awareness of an individual mind and it is
unique for the sense-awareness of all minds which are operating under
natural conditions. There is an important distinction between the two
cases. (i) For one mind not only is the discerned component of the
general fact exhibited in any act of sense-awareness distinct from the
discerned component of the general fact exhibited in any other act of
sense-awareness of that mind, but the two corresponding durations which
are respectively related by simultaneity to the two discerned components
are necessarily distinct. This is an exhibition of the temporal passage
of nature; namely, one duration has passed into the other. Thus not only
is the passage of nature an essential character of nature in its _role_
of the terminus of sense-awareness, but it is also essential for
sense-awareness in itself. It is this truth which makes time appear to
extend beyond nature. But what extends beyond nature to mind is not the
serial and measurable time, which exhibits merely the character of
passage in nature, but the quality of passage itself which is in no way
measurable except so far as it obtains in nature. That is to say,
'passage' is not measurable except as it occurs in nature in connexion
with extension. In passage we reach a connexion of nature with the
ultimate metaphysical reality. The quality of passage in durations is a
particular exhibition in nature of a quality which extends beyond
nature. For example passage is a quality not only of nature, which is
the thing known, but also of sense-awareness which is the procedure of
knowing. Durations have all the reality that nature has, though what
that may be we need not now determine. The measurableness of time is
derivative from the properties of durations. So also is the serial
character of time. We shall find that there are in nature competing
serial time-systems derived from different families of durations. These
are a peculiarity of the character of passage as it is found in nature.
This character has the reality of nature, but we must not necessarily
transfer natural time to extra-natural entities. (ii) For two minds, the
discerned components of the general facts exhibited in their respective
acts of sense-awareness must be different. For each mind, in its
awareness of nature is aware of a certain complex of related natural
entities in their relations to the living body as a focus. But the
associated durations may be identical. Here we are touching on that
character of the passage nature which issues in the spatial relations of
simultaneous bodies. This possible identity of the durations in the case
of the sense-awareness of distinct minds is what binds into one nature
the private experiences of sentient beings. We are here considering the
spatial side of the passage of nature. Passage in this aspect of it also
seems to extend beyond nature to mind.

It is important to distinguish simultaneity from instantaneousness. I
lay no stress on the mere current usage of the two terms. There are two
concepts which I want to distinguish, and one I call simultaneity and
the other instantaneousness. I hope that the words are judiciously
chosen; but it really does not matter so long as I succeed in explaining
my meaning. Simultaneity is the property of a group of natural elements
which in some sense are components of a duration. A duration can be all
nature present as the immediate fact posited by sense-awareness. A
duration retains within itself the passage of nature. There are within
it antecedents and consequents which are also durations which may be the
complete specious presents of quicker consciousnesses. In other words a
duration retains temporal thickness. Any concept of all nature as
immediately known is always a concept of some duration though it may be
enlarged in its temporal thickness beyond the possible specious present
of any being known to us as existing within nature. Thus simultaneity is
an ultimate factor in nature, immediate for sense-awareness.

Instantaneousness is a complex logical concept of a procedure in thought
by which constructed logical entities are produced for the sake of the
simple expression in thought of properties of nature. Instantaneousness
is the concept of all nature at an instant, where an instant is
conceived as deprived of all temporal extension. For example we conceive
of the distribution of matter in space at an instant. This is a very
useful concept in science especially in applied mathematics; but it is a
very complex idea so far as concerns its connexions with the immediate
facts of sense-awareness. There is no such thing as nature at an instant
posited by sense-awareness. What sense-awareness delivers over for
knowledge is nature through a period. Accordingly nature at an instant,
since it is not itself a natural entity, must be defined in terms of
genuine natural entities. Unless we do so, our science, which employs
the concept of instantaneous nature, must abandon all claim to be
founded upon observation.

I will use the term 'moment' to mean 'all nature at an instant.' A
moment, in the sense in which the term is here used, has no temporal
extension, and is in this respect to be contrasted with a duration which
has such extension. What is directly yielded to our knowledge by
sense-awareness is a duration. Accordingly we have now to explain how
moments are derived from durations, and also to explain the purpose
served by their introduction.

A moment is a limit to which we approach as we confine attention to
durations of minimum extension. Natural relations among the ingredients
of a duration gain in complexity as we consider durations of increasing
temporal extension. Accordingly there is an approach to ideal simplicity
as we approach an ideal diminution of extension.

The word 'limit' has a precise signification in the logic of number and
even in the logic of non-numerical one-dimensional series. As used here
it is so far a mere metaphor, and it is necessary to explain directly
the concept which it is meant to indicate.

Durations can have the two-termed relational property of extending one
over the other. Thus the duration which is all nature during a certain
minute extends over the duration which is all nature during the
30th second of that minute. This relation of 'extending
over'--'extension' as I shall call it--is a fundamental natural relation
whose field comprises more than durations. It is a relation which two
limited events can have to each other. Furthermore as holding between
durations the relation appears to refer to the purely temporal
extension. I shall however maintain that the same relation of extension
lies at the base both of temporal and spatial extension. This discussion
can be postponed; and for the present we are simply concerned with the
relation of extension as it occurs in its temporal aspect for the
limited field of durations.

The concept of extension exhibits in thought one side of the ultimate
passage of nature. This relation holds because of the special character
which passage assumes in nature; it is the relation which in the case of
durations expresses the properties of 'passing over.' Thus the duration
which was one definite minute passed over the duration which was its
30th second. The duration of the 30th second was part of the duration of
the minute. I shall use the terms 'whole' and 'part' exclusively in this
sense, that the 'part' is an event which is extended over by the other
event which is the 'whole.' Thus in my nomenclature 'whole' and 'part'
refer exclusively to this fundamental relation of extension; and
accordingly in this technical usage only events can be either wholes or
parts.

The continuity of nature arises from extension. Every event extends over
other events, and every event is extended over by other events. Thus in
the special case of durations which are now the only events directly
under consideration, every duration is part of other durations; and
every duration has other durations which are parts of it. Accordingly
there are no maximum durations and no minimum durations. Thus there is
no atomic structure of durations, and the perfect definition of a
duration, so as to mark out its individuality and distinguish it from
highly analogous durations over which it is passing, or which are
passing over it, is an arbitrary postulate of thought. Sense-awareness
posits durations as factors in nature but does not clearly enable
thought to use it as distinguishing the separate individualities of the
entities of an allied group of slightly differing durations. This is one
instance of the indeterminateness of sense-awareness. Exactness is an
ideal of thought, and is only realised in experience by the selection of
a route of approximation.

The absence of maximum and minimum durations does not exhaust the
properties of nature which make up its continuity. The passage of nature
involves the existence of a family of durations. When two durations
belong to the same family either one contains the other, or they overlap
each other in a subordinate duration without either containing the
other; or they are completely separate. The excluded case is that of
durations overlapping in finite events but not containing a third
duration as a common part.

It is evident that the relation of extension is transitive; namely as
applied to durations, if duration A is part of duration B, and
duration B is part of duration C, then A is part of C. Thus the
first two cases may be combined into one and we can say that two
durations which belong to the same family _either_ are such that there
are durations which are parts of both _or_ are completely separate.

Furthermore the converse of this proposition holds; namely, if two
durations have other durations which are parts of both _or_ if the two
durations are completely separate, then they belong to the same family.

The further characteristics of the continuity of nature--so far as
durations are concerned--which has not yet been formulated arises in
connexion with a family of durations. It can be stated in this way:
There are durations which contain as parts any two durations of the same
family. For example a week contains as parts any two of its days. It is
evident that a containing duration satisfies the conditions for
belonging to the same family as the two contained durations.

We are now prepared to proceed to the definition of a moment of time.
Consider a set of durations all taken from the same family. Let it have
the following properties: (i) of any two members of the set one contains
the other as a part, and (ii) there is no duration which is a common
part of every member of the set.

Now the relation of whole and part is asymmetrical; and by this I mean
that if A is part of B, then B is not part of A. Also we have
already noted that the relation is transitive. Accordingly we can easily
see that the durations of any set with the properties just enumerated
must be arranged in a one-dimensional serial order in which as we
descend the series we progressively reach durations of smaller and
smaller temporal extension. The series may start with any arbitrarily
assumed duration of any temporal extension, but in descending the
series the temporal extension progressively contracts and the successive
durations are packed one within the other like the nest of boxes of a
Chinese toy. But the set differs from the toy in this particular: the
toy has a smallest box which forms the end box of its series; but the
set of durations can have no smallest duration nor can it converge
towards a duration as its limit. For the parts either of the end
duration or of the limit would be parts of all the durations of the set
and thus the second condition for the set would be violated.

I will call such a set of durations an 'abstractive set' of durations.
It is evident that an abstractive set as we pass along it converges to
the ideal of all nature with no temporal extension, namely, to the ideal
of all nature at an instant. But this ideal is in fact the ideal of a
nonentity. What the abstractive set is in fact doing is to guide thought
to the consideration of the progressive simplicity of natural relations
as we progressively diminish the temporal extension of the duration
considered. Now the whole point of the procedure is that the
quantitative expressions of these natural properties do converge to
limits though the abstractive set does not converge to any limiting
duration. The laws relating these quantitative limits are the laws of
nature 'at an instant,' although in truth there is no nature at an
instant and there is only the abstractive set. Thus an abstractive set
is effectively the entity meant when we consider an instant of time
without temporal extension. It subserves all the necessary purposes of
giving a definite meaning to the concept of the properties of nature at
an instant. I fully agree that this concept is fundamental in the
expression of physical science. The difficulty is to express our
meaning in terms of the immediate deliverances of sense-awareness, and I
offer the above explanation as a complete solution of the problem.

In this explanation a moment is the set of natural properties reached by
a route of approximation. An abstractive series is a route of
approximation. There are different routes of approximation to the same
limiting set of the properties of nature. In other words there are
different abstractive sets which are to be regarded as routes of
approximation to the same moment. Accordingly there is a certain amount
of technical detail necessary in explaining the relations of such
abstractive sets with the same convergence and in guarding against
possible exceptional cases. Such details are not suitable for exposition
in these lectures, and I have dealt with them fully elsewhere[5].

[5] Cf. _An Enquiry concerning the Principles of Natural Knowledge_,
Cambridge University Press, 1919.

It is more convenient for technical purposes to look on a moment as
being the class of all abstractive sets of durations with the same
convergence. With this definition (provided that we can successfully
explain what we mean by the 'same convergence' apart from a detailed
knowledge of the set of natural properties arrived at by approximation)
a moment is merely a class of sets of durations whose relations of
extension in respect to each other have certain definite peculiarities.
We may term these connexions of the component durations the 'extrinsic'
properties of a moment; the 'intrinsic' properties of the moment are the
properties of nature arrived at as a limit as we proceed along any one
of its abstractive sets. These are the properties of nature 'at that
moment,' or 'at that instant.'

The durations which enter into the composition of a moment all belong to
one family. Thus there is one family of moments corresponding to one
family of durations. Also if we take two moments of the same family,
among the durations which enter into the composition of one moment the
smaller durations are completely separated from the smaller durations
which enter into the composition of the other moment. Thus the two
moments in their intrinsic properties must exhibit the limits of
completely different states of nature. In this sense the two moments are
completely separated. I will call two moments of the same family
'parallel.'

Corresponding to each duration there are two moments of the associated
family of moments which are the boundary moments of that duration. A
'boundary moment' of a duration can be defined in this way. There are
durations of the same family as the given duration which overlap it but
are not contained in it. Consider an abstractive set of such durations.
Such a set defines a moment which is just as much without the duration
as within it. Such a moment is a boundary moment of the duration. Also
we call upon our sense-awareness of the passage of nature to inform us
that there are two such boundary moments, namely the earlier one and the
later one. We will call them the initial and the final boundaries.

There are also moments of the same family such that the shorter
durations in their composition are entirely separated from the given
duration. Such moments will be said to lie 'outside' the given duration.
Again other moments of the family are such that the shorter durations in
their composition are parts of the given duration. Such moments are said
to lie 'within' the given duration or to 'inhere' in it. The whole
family of parallel moments is accounted for in this way by reference to
any given duration of the associated family of durations. Namely, there
are moments of the family which lie without the given duration, there
are the two moments which are the boundary moments of the given
duration, and the moments which lie within the given duration.
Furthermore any two moments of the same family are the boundary moments
of some one duration of the associated family of durations.

It is now possible to define the serial relation of temporal order among
the moments of a family. For let A and C be any two moments of the
family, these moments are the boundary moments of one duration d of
the associated family, and any moment B which lies within the duration
d will be said to lie between the moments A and C. Thus the
three-termed relation of 'lying-between' as relating three moments A,
B, and C is completely defined. Also our knowledge of the passage of
nature assures us that this relation distributes the moments of the
family into a serial order. I abstain from enumerating the definite
properties which secure this result, I have enumerated them in my
recently published book[6] to which I have already referred. Furthermore
the passage of nature enables us to know that one direction along the
series corresponds to passage into the future and the other direction
corresponds to retrogression towards the past.

[6] Cf. _Enquiry_.

Such an ordered series of moments is what we mean by time defined as a
series. Each element of the series exhibits an instantaneous state of
nature. Evidently this serial time is the result of an intellectual
process of abstraction. What I have done is to give precise definitions
of the procedure by which the abstraction is effected. This procedure is
merely a particular case of the general method which in my book I name
the 'method of extensive abstraction.' This serial time is evidently not
the very passage of nature itself. It exhibits some of the natural
properties which flow from it. The state of nature 'at a moment' has
evidently lost this ultimate quality of passage. Also the temporal
series of moments only retains it as an extrinsic relation of entities
and not as the outcome of the essential being of the terms of the
series.

Nothing has yet been said as to the measurement of time. Such
measurement does not follow from the mere serial property of time; it
requires a theory of congruence which will be considered in a later
lecture.

In estimating the adequacy of this definition of the temporal series as
a formulation of experience it is necessary to discriminate between the
crude deliverance of sense-awareness and our intellectual theories. The
lapse of time is a measurable serial quantity. The whole of scientific
theory depends on this assumption and any theory of time which fails to
provide such a measurable series stands self-condemned as unable to
account for the most salient fact in experience. Our difficulties only
begin when we ask what it is that is measured. It is evidently something
so fundamental in experience that we can hardly stand back from it and
hold it apart so as to view it in its own proportions.

We have first to make up our minds whether time is to be found in nature
or nature is to be found in time. The difficulty of the latter
alternative--namely of making time prior to nature--is that time then
becomes a metaphysical enigma. What sort of entities are its instants
or its periods? The dissociation of time from events discloses to our
immediate inspection that the attempt to set up time as an independent
terminus for knowledge is like the effort to find substance in a shadow.
There is time because there are happenings, and apart from happenings
there is nothing.

It is necessary however to make a distinction. In some sense time
extends beyond nature. It is not true that a timeless sense-awareness
and a timeless thought combine to contemplate a timeful nature.
Sense-awareness and thought are themselves processes as well as their
termini in nature. In other words there is a passage of sense-awareness
and a passage of thought. Thus the reign of the quality of passage
extends beyond nature. But now the distinction arises between passage
which is fundamental and the temporal series which is a logical
abstraction representing some of the properties of nature. A temporal
series, as we have defined it, represents merely certain properties of a
family of durations--properties indeed which durations only possess
because of their partaking of the character of passage, but on the other
hand properties which only durations do possess. Accordingly time in the
sense of a measurable temporal series is a character of nature only, and
does not extend to the processes of thought and of sense-awareness
except by a correlation of these processes with the temporal series
implicated in their procedures.

So far the passage of nature has been considered in connexion with the
passage of durations; and in this connexion it is peculiarly associated
with temporal series. We must remember however that the character of
passage is peculiarly associated with the extension of events, and that
from this extension spatial transition arises just as much as temporal
transition. The discussion of this point is reserved for a later lecture
but it is necessary to remember it now that we are proceeding to discuss
the application of the concept of passage beyond nature, otherwise we
shall have too narrow an idea of the essence of passage.

It is necessary to dwell on the subject of sense-awareness in this
connexion as an example of the way in which time concerns mind, although
measurable time is a mere abstract from nature and nature is closed to
mind.

Consider sense-awareness--not its terminus which is nature, but
sense-awareness in itself as a procedure of mind. Sense-awareness is a
relation of mind to nature. Accordingly we are now considering mind as a
relatum in sense-awareness. For mind there is the immediate
sense-awareness and there is memory. The distinction between memory and
the present immediacy has a double bearing. On the one hand it discloses
that mind is not impartially aware of all those natural durations to
which it is related by awareness. Its awareness shares in the passage of
nature. We can imagine a being whose awareness, conceived as his private
possession, suffers no transition, although the terminus of his
awareness is our own transient nature. There is no essential reason why
memory should not be raised to the vividness of the present fact; and
then from the side of mind, What is the difference between the present
and the past? Yet with this hypothesis we can also suppose that the
vivid remembrance and the present fact are posited in awareness as in
their temporal serial order. Accordingly we must admit that though we
can imagine that mind in the operation of sense-awareness might be free
from any character of passage, yet in point of fact our experience of
sense-awareness exhibits our minds as partaking in this character.

On the other hand the mere fact of memory is an escape from transience.
In memory the past is present. It is not present as overleaping the
temporal succession of nature, but it is present as an immediate fact
for the mind. Accordingly memory is a disengagement of the mind from the
mere passage of nature; for what has passed for nature has not passed
for mind.

Furthermore the distinction between memory and the immediate present is
not so clear as it is conventional to suppose. There is an intellectual
theory of time as a moving knife-edge, exhibiting a present fact without
temporal extension. This theory arises from the concept of an ideal
exactitude of observation. Astronomical observations are successively
refined to be exact to tenths, to hundredths, and to thousandths of
seconds. But the final refinements are arrived at by a system of
averaging, and even then present us with a stretch of time as a margin
of error. Here error is merely a conventional term to express the fact
that the character of experience does not accord with the ideal of
thought. I have already explained how the concept of a moment
conciliates the observed fact with this ideal; namely, there is a
limiting simplicity in the quantitative expression of the properties of
durations, which is arrived at by considering any one of the abstractive
sets included in the moment. In other words the extrinsic character of
the moment as an aggregate of durations has associated with it the
intrinsic character of the moment which is the limiting expression of
natural properties.

Thus the character of a moment and the ideal of exactness which it
enshrines do not in any way weaken the position that the ultimate
terminus of awareness is a duration with temporal thickness. This
immediate duration is not clearly marked out for our apprehension. Its
earlier boundary is blurred by a fading into memory, and its later
boundary is blurred by an emergence from anticipation. There is no sharp
distinction either between memory and the present immediacy or between
the present immediacy and anticipation. The present is a wavering
breadth of boundary between the two extremes. Thus our own
sense-awareness with its extended present has some of the character of
the sense-awareness of the imaginary being whose mind was free from
passage and who contemplated all nature as an immediate fact. Our own
present has its antecedents and its consequents, and for the imaginary
being all nature has its antecedent and its consequent durations. Thus
the only difference in this respect between us and the imaginary being
is that for him all nature shares in the immediacy of our present
duration.

The conclusion of this discussion is that so far as sense-awareness is
concerned there is a passage of mind which is distinguishable from the
passage of nature though closely allied with it. We may speculate, if we
like, that this alliance of the passage of mind with the passage of
nature arises from their both sharing in some ultimate character of
passage which dominates all being. But this is a speculation in which we
have no concern. The immediate deduction which is sufficient for us is
that--so far as sense-awareness is concerned--mind is not in time or in
space in the same sense in which the events of nature are in time, but
that it is derivatively in time and in space by reason of the peculiar
alliance of its passage with the passage of nature. Thus mind is in time
and in space in a sense peculiar to itself. This has been a long
discussion to arrive at a very simple and obvious conclusion. We all
feel that in some sense our minds are here in this room and at this
time. But it is not quite in the same sense as that in which the events
of nature which are the existences of our brains have their spatial and
temporal positions. The fundamental distinction to remember is that
immediacy for sense-awareness is not the same as instantaneousness for
nature. This last conclusion bears on the next discussion with which I
will terminate this lecture. This question can be formulated thus, Can
alternative temporal series be found in nature?

A few years ago such a suggestion would have been put aside as being
fantastically impossible. It would have had no bearing on the science
then current, and was akin to no ideas which had ever entered into the
dreams of philosophy. The eighteenth and nineteenth centuries accepted
as their natural philosophy a certain circle of concepts which were as
rigid and definite as those of the philosophy of the middle ages, and
were accepted with as little critical research. I will call this natural
philosophy 'materialism.' Not only were men of science materialists, but
also adherents of all schools of philosophy. The idealists only differed
from the philosophic materialists on question of the alignment of nature
in reference to mind. But no one had any doubt that the philosophy of
nature considered in itself was of the type which I have called
materialism. It is the philosophy which I have already examined in my
two lectures of this course preceding the present one. It can be
summarised as the belief that nature is an aggregate of material and
that this material exists in some sense _at_ each successive member of a
one-dimensional series of extensionless instants of time. Furthermore
the mutual relations of the material entities at each instant formed
these entities into a spatial configuration in an unbounded space. It
would seem that space--on this theory--would be as instantaneous as the
instants, and that some explanation is required of the relations between
the successive instantaneous spaces. The materialistic theory is however
silent on this point; and the succession of instantaneous spaces is
tacitly combined into one persistent space. This theory is a purely
intellectual rendering of experience which has had the luck to get
itself formulated at the dawn of scientific thought. It has dominated
the language and the imagination of science since science flourished in
Alexandria, with the result that it is now hardly possible to speak
without appearing to assume its immediate obviousness.

But when it is distinctly formulated in the abstract terms in which I
have just stated it, the theory is very far from obvious. The passing
complex of factors which compose the fact which is the terminus of
sense-awareness places before us nothing corresponding to the trinity of
this natural materialism. This trinity is composed (i) of the temporal
series of extensionless instants, (ii) of the aggregate of material
entities, and (iii) of space which is the outcome of relations of
matter.

There is a wide gap between these presuppositions of the intellectual
theory of materialism and the immediate deliverances of sense-awareness.
I do not question that this materialistic trinity embodies important
characters of nature. But it is necessary to express these characters in
terms of the facts of experience. This is exactly what in this lecture I
have been endeavouring to do so far as time is concerned; and we have
now come up against the question, Is there only one temporal series? The
uniqueness of the temporal series is presupposed in the materialist
philosophy of nature. But that philosophy is merely a theory, like the
Aristotelian scientific theories so firmly believed in the middle ages.
If in this lecture I have in any way succeeded in getting behind the
theory to the immediate facts, the answer is not nearly so certain. The
question can be transformed into this alternative form, Is there only
one family of durations? In this question the meaning of a 'family of
durations' has been defined earlier in this lecture. The answer is now
not at all obvious. On the materialistic theory the instantaneous
present is the only field for the creative activity of nature. The past
is gone and the future is not yet. Thus (on this theory) the immediacy
of perception is of an instantaneous present, and this unique present is
the outcome of the past and the promise of the future. But we deny this
immediately given instantaneous present. There is no such thing to be
found in nature. As an ultimate fact it is a nonentity. What is
immediate for sense-awareness is a duration. Now a duration has within
itself a past and a future; and the temporal breadths of the immediate
durations of sense-awareness are very indeterminate and dependent on the
individual percipient. Accordingly there is no unique factor in nature
which for every percipient is pre-eminently and necessarily the present.
The passage of nature leaves nothing between the past and the future.
What we perceive as present is the vivid fringe of memory tinged with
anticipation. This vividness lights up the discriminated field within a
duration. But no assurance can thereby be given that the happenings of
nature cannot be assorted into other durations of alternative families.
We cannot even know that the series of immediate durations posited by
the sense-awareness of one individual mind all necessarily belong to the
same family of durations. There is not the slightest reason to believe
that this is so. Indeed if my theory of nature be correct, it will not
be the case.

The materialistic theory has all the completeness of the thought of the
middle ages, which had a complete answer to everything, be it in heaven
or in hell or in nature. There is a trimness about it, with its
instantaneous present, its vanished past, its non-existent future, and
its inert matter. This trimness is very medieval and ill accords with
brute fact.

The theory which I am urging admits a greater ultimate mystery and a
deeper ignorance. The past and the future meet and mingle in the
ill-defined present. The passage of nature which is only another name
for the creative force of existence has no narrow ledge of definite
instantaneous present within which to operate. Its operative presence
which is now urging nature forward must be sought for throughout the
whole, in the remotest past as well as in the narrowest breadth of any
present duration. Perhaps also in the unrealised future. Perhaps also in
the future which might be as well as the actual future which will be. It
is impossible to meditate on time and the mystery of the creative
passage of nature without an overwhelming emotion at the limitations of
human intelligence.




CHAPTER IV

THE METHOD OF EXTENSIVE ABSTRACTION


To-day's lecture must commence with the consideration of limited events.
We shall then be in a position to enter upon an investigation of the
factors in nature which are represented by our conception of space.

The duration which is the immediate disclosure of our sense-awareness is
discriminated into parts. There is the part which is the life of all
nature within a room, and there is the part which is the life of all
nature within a table in the room. These parts are limited events. They
have the endurance of the present duration, and they are parts of it.
But whereas a duration is an unlimited whole and in a certain limited
sense is all that there is, a limited event possesses a completely
defined limitation of extent which is expressed for us in
spatio-temporal terms.

We are accustomed to associate an event with a certain melodramatic
quality. If a man is run over, that is an event comprised within certain
spatio-temporal limits. We are not accustomed to consider the endurance
of the Great Pyramid throughout any definite day as an event. But the
natural fact which is the Great Pyramid throughout a day, meaning
thereby all nature within it, is an event of the same character as the
man's accident, meaning thereby all nature with spatio-temporal
limitations so as to include the man and the motor during the period
when they were in contact.

We are accustomed to analyse these events into three factors, time,
space, and material. In fact, we at once apply to them the concepts of
the materialistic theory of nature. I do not deny the utility of this
analysis for the purpose of expressing important laws of nature. What I
am denying is that anyone of these factors is posited for us in
sense-awareness in concrete independence. We perceive one unit factor in
nature; and this factor is that something is going on then--there. For
example, we perceive the going-on of the Great Pyramid in its relations
to the goings-on of the surrounding Egyptian events. We are so trained,
both by language and by formal teaching and by the resulting
convenience, to express our thoughts in terms of this materialistic
analysis that intellectually we tend to ignore the true unity of the
factor really exhibited in sense-awareness. It is this unit factor,
retaining in itself the passage of nature, which is the primary concrete
element discriminated in nature. These primary factors are what I mean
by events.

Events are the field of a two-termed relation, namely the relation of
extension which was considered in the last lecture. Events are the
things related by the relation of extension. If an event A extends
over an event B, then B is 'part of' A, and A is a 'whole' of
which B is a part. Whole and part are invariably used in these
lectures in this definite sense. It follows that in reference to this
relation any two events A and B may have any one of four relations
to each other, namely (i) A may extend over B, or (ii) B may
extend over A, or (iii) A and B may both extend over some third
event C, but neither over the other, or (iv) A and B may be
entirely separate. These alternatives can obviously be illustrated by
Euler's diagrams as they appear in logical textbooks.

The continuity of nature is the continuity of events. This continuity is
merely the name for the aggregate of a variety of properties of events
in connexion with the relation of extension.

In the first place, this relation is transitive; secondly, every event
contains other events as parts of itself; thirdly every event is a part
of other events; fourthly given any two finite events there are events
each of which contains both of them as parts; and fifthly there is a
special relation between events which I term 'junction.'

Two events have junction when there is a third event of which both
events are parts, and which is such that no part of it is separated from
both of the two given events. Thus two events with junction make up
exactly one event which is in a sense their sum.

Only certain pairs of events have this property. In general any event
containing two events also contains parts which are separated from both
events.

There is an alternative definition of the junction of two events which I
have adopted in my recent book[7]. Two events have junction when there
is a third event such that (i) it overlaps both events and (ii) it has
no part which is separated from both the given events. If either of
these alternative definitions is adopted as the definition of junction,
the other definition appears as an axiom respecting the character of
junction as we know it in nature. But we are not thinking of logical
definition so much as the formulation of the results of direct
observation. There is a certain continuity inherent in the observed
unity of an event, and these two definitions of junction are really
axioms based on observation respecting the character of this continuity.

[7] Cf. _Enquiry_.

The relations of whole and part and of overlapping are particular cases
of the junction of events. But it is possible for events to have
junction when they are separate from each other; for example, the upper
and the lower part of the Great Pyramid are divided by some imaginary
horizontal plane.

The continuity which nature derives from events has been obscured by the
illustrations which I have been obliged to give. For example I have
taken the existence of the Great Pyramid as a fairly well-known fact to
which I could safely appeal as an illustration. This is a type of event
which exhibits itself to us as the situation of a recognisable object;
and in the example chosen the object is so widely recognised that it has
received a name. An object is an entity of a different type from an
event. For example, the event which is the life of nature within the
Great Pyramid yesterday and to-day is divisible into two parts, namely
the Great Pyramid yesterday and the Great Pyramid to-day. But the
recognisable object which is also called the Great Pyramid is the same
object to-day as it was yesterday. I shall have to consider the theory
of objects in another lecture.

The whole subject is invested with an unmerited air of subtlety by the
fact that when the event is the situation of a well-marked object, we
have no language to distinguish the event from the object. In the case
of the Great Pyramid, the object is the perceived unit entity which as
perceived remains self-identical throughout the ages; while the whole
dance of molecules and the shifting play of the electromagnetic field
are ingredients of the event. An object is in a sense out of time. It is
only derivatively in time by reason of its having the relation to events
which I term 'situation.' This relation of situation will require
discussion in a subsequent lecture.

The point which I want to make now is that being the situation of a
well-marked object is not an inherent necessity for an event. Wherever
and whenever something is going on, there is an event. Furthermore
'wherever and whenever' in themselves presuppose an event, for space and
time in themselves are abstractions from events. It is therefore a
consequence of this doctrine that something is always going on
everywhere, even in so-called empty space. This conclusion is in accord
with modern physical science which presupposes the play of an
electromagnetic field throughout space and time. This doctrine of
science has been thrown into the materialistic form of an all-pervading
ether. But the ether is evidently a mere idle concept--in the
phraseology which Bacon applied to the doctrine of final causes, it is a
barren virgin. Nothing is deduced from it; and the ether merely
subserves the purpose of satisfying the demands of the materialistic
theory. The important concept is that of the shifting facts of the
fields of force. This is the concept of an ether of events which should
be substituted for that of a material ether.

It requires no illustration to assure you that an event is a complex
fact, and the relations between two events form an almost impenetrable
maze. The clue discovered by the common sense of mankind and
systematically utilised in science is what I have elsewhere[8] called
the law of convergence to simplicity by diminution of extent.

[8] Cf. _Organisation of Thought_, pp. 146 et seq. Williams and Norgate,
1917.

If A and B are two events, and A' is part of A and B' is part
of B, then in many respects the relations between the parts A' and
B' will be simpler than the relations between A and B. This is the
principle which presides over all attempts at exact observation.

The first outcome of the systematic use of this law has been the
formulation of the abstract concepts of Time and Space. In the previous
lecture I sketched how the principle was applied to obtain the
time-series. I now proceed to consider how the spatial entities are
obtained by the same method. The systematic procedure is identical in
principle in both cases, and I have called the general type of procedure
the 'method of extensive abstraction.'

You will remember that in my last lecture I defined the concept of an
abstractive set of durations. This definition can be extended so as to
apply to any events, limited events as well as durations. The only
change that is required is the substitution of the word 'event' for the
word 'duration.' Accordingly an abstractive set of events is any set of
events which possesses the two properties, (i) of any two members of the
set one contains the other as a part, and (ii) there is no event which
is a common part of every member of the set. Such a set, as you will
remember, has the properties of the Chinese toy which is a nest of
boxes, one within the other, with the difference that the toy has a
smallest box, while the abstractive class has neither a smallest event
nor does it converge to a limiting event which is not a member of the
set.

Thus, so far as the abstractive sets of events are concerned, an
abstractive set converges to nothing. There is the set with its members
growing indefinitely smaller and smaller as we proceed in thought
towards the smaller end of the series; but there is no absolute minimum
of any sort which is finally reached. In fact the set is just itself and
indicates nothing else in the way of events, except itself. But each
event has an intrinsic character in the way of being a situation of
objects and of having parts which are situations of objects and--to
state the matter more generally--in the way of being a field of the life
of nature. This character can be defined by quantitative expressions
expressing relations between various quantities intrinsic to the event
or between such quantities and other quantities intrinsic to other
events. In the case of events of considerable spatio-temporal extension
this set of quantitative expressions is of bewildering complexity. If
e be an event, let us denote by q(e) the set of quantitative expressions
defining its character including its connexions with the rest of nature.
Let e_1, e_2, e_3, etc. be an abstractive set, the members being so
arranged that each member such as e_{n} extends over all the succeeding
members such as e_{n+1}, e_{n+2} and so on. Then corresponding to the
series

    e_1, e_2, e_3, ..., e_{n}, e_{n+1}, ...,

there is the series

    q(e_1), q(e_2), q(e_3), ..., q(e_{n}), q(e_{n+1}), ....

Call the series of events s and the series of quantitative expressions
q(s). The series s has no last term and no events which are contained
in every member of the series. Accordingly the series of events
converges to nothing. It is just itself. Also the series q(s) has no
last term. But the sets of homologous quantities running through the
various terms of the series do converge to definite limits. For example
if Q_1 be a quantitative measurement found in q(e_1), and Q_2 the homologue
to Q_1 to be found in q(e_2), and Q_3 the homologue to Q_1 and Q_2 to be
found in q(e_3), and so on, then the series

    Q_1, Q_2, Q_3, ..., Q_{n}, Q_{n+1}, ...,

though it has no last term, does in general converge to a definite
limit. Accordingly there is a class of limits l(s) which is the
class of the limits of those members of q(e_{n}) which have
homologues throughout the series q(s) as n indefinitely increases.
We can represent this statement diagrammatically by using an arrow (-->)
to mean 'converges to.' Then

    e_1, e_2, e_3, ..., e_{n}, e_{n+1}, ... --> nothing,

and

    q(e_1), q(e_2), q(e_3), ..., q(e_{n}), q(e_{n+1}), ... --> l(s).

The mutual relations between the limits in the set l(s), and also
between these limits and the limits in other sets l(s'), l(s"), ...,
which arise from other abstractive sets s', s", etc., have a peculiar
simplicity.

Thus the set s does indicate an ideal simplicity of natural relations,
though this simplicity is not the character of any actual event in s.
We can make an approximation to such a simplicity which, as estimated
numerically, is as close as we like by considering an event which is far
enough down the series towards the small end. It will be noted that it
is the infinite series, as it stretches away in unending succession
towards the small end, which is of importance. The arbitrarily large
event with which the series starts has no importance at all. We can
arbitrarily exclude any set of events at the big end of an abstractive
set without the loss of any important property to the set as thus
modified.

I call the limiting character of natural relations which is indicated by
an abstractive set, the 'intrinsic character' of the set; also the
properties, connected with the relation of whole and part as concerning
its members, by which an abstractive set is defined together form what I
call its 'extrinsic character.' The fact that the extrinsic character of
an abstractive set determines a definite intrinsic character is the
reason of the importance of the precise concepts of space and time. This
emergence of a definite intrinsic character from an abstractive set is
the precise meaning of the law of convergence.

For example, we see a train approaching during a minute. The event which
is the life of nature within that train during the minute is of great
complexity and the expression of its relations and of the ingredients of
its character baffles us. If we take one second of that minute, the more
limited event which is thus obtained is simpler in respect to its
ingredients, and shorter and shorter times such as a tenth of that
second, or a hundredth, or a thousandth--so long as we have a definite
rule giving a definite succession of diminishing events--give events
whose ingredient characters converge to the ideal simplicity of the
character of the train at a definite instant. Furthermore there are
different types of such convergence to simplicity. For example, we can
converge as above to the limiting character expressing nature at an
instant within the whole volume of the train at that instant, or to
nature at an instant within some portion of that volume--for example
within the boiler of the engine--or to nature at an instant on some area
of surface, or to nature at an instant on some line within the train, or
to nature at an instant at some point of the train. In the last case the
simple limiting characters arrived at will be expressed as densities,
specific gravities, and types of material. Furthermore we need not
necessarily converge to an abstraction which involves nature at an
instant. We may converge to the physical ingredients of a certain point
track throughout the whole minute. Accordingly there are different types
of extrinsic character of convergence which lead to the approximation to
different types of intrinsic characters as limits.

We now pass to the investigation of possible connexions between
abstractive sets. One set may 'cover' another. I define 'covering' as
follows: An abstractive set p covers an abstractive set q when every
member of p contains as its parts some members of q. It is evident
that if any event e contains as a part any member of the set q, then
owing to the transitive property of extension every succeeding member of
the small end of q is part of e. In such a case I will say that the
abstractive set q 'inheres in' the event e. Thus when an abstractive
set p covers an abstractive set q, the abstractive set q inheres
in every member of p.

Two abstractive sets may each cover the other. When this is the case I
shall call the two sets 'equal in abstractive force.' When there is no
danger of misunderstanding I shall shorten this phrase by simply saying
that the two abstractive sets are 'equal.' The possibility of this
equality of abstractive sets arises from the fact that both sets, p
and q, are infinite series towards their small ends. Thus the equality
means, that given any event x belonging to p, we can always by
proceeding far enough towards the small end of q find an event y
which is part of x, and that then by proceeding far enough towards the
small end of p we can find an event z which is part of y, and so
on indefinitely.

The importance of the equality of abstractive sets arises from the
assumption that the intrinsic characters of the two sets are identical.
If this were not the case exact observation would be at an end.

It is evident that any two abstractive sets which are equal to a third
abstractive set are equal to each other. An 'abstractive element' is the
whole group of abstractive sets which are equal to any one of
themselves. Thus all abstractive sets belonging to the same element are
equal and converge to the same intrinsic character. Thus an abstractive
element is the group of routes of approximation to a definite intrinsic
character of ideal simplicity to be found as a limit among natural
facts.

If an abstractive set p covers an abstractive set q, then any
abstractive set belonging to the abstractive element of which p is a
member will cover any abstractive set belonging to the element of which
q is a member. Accordingly it is useful to stretch the meaning of the
term 'covering,' and to speak of one abstractive element 'covering'
another abstractive element. If we attempt in like manner to stretch the
term 'equal' in the sense of 'equal in abstractive force,' it is obvious
that an abstractive element can only be equal to itself. Thus an
abstractive element has a unique abstractive force and is the construct
from events which represents one definite intrinsic character which is
arrived at as a limit by the use of the principle of convergence to
simplicity by diminution of extent.

When an abstractive element A covers an abstractive element B, the
intrinsic character of A in a sense includes the intrinsic character
of B. It results that statements about the intrinsic character of B
are in a sense statements about the intrinsic character of A; but the
intrinsic character of A is more complex than that of B.

The abstractive elements form the fundamental elements of space and
time, and we now turn to the consideration of the properties involved in
the formation of special classes of such elements. In my last lecture I
have already investigated one class of abstractive elements, namely
moments. Each moment is a group of abstractive sets, and the events
which are members of these sets are all members of one family of
durations. The moments of one family form a temporal series; and,
allowing the existence of different families of moments, there will be
alternative temporal series in nature. Thus the method of extensive
abstraction explains the origin of temporal series in terms of the
immediate facts of experience and at the same time allows for the
existence of the alternative temporal series which are demanded by the
modern theory of electromagnetic relativity.

We now turn to space. The first thing to do is to get hold of the class
of abstractive elements which are in some sense the points of space.
Such an abstractive element must in some sense exhibit a convergence to
an absolute minimum of intrinsic character. Euclid has expressed for all
time the general idea of a point, as being without parts and without
magnitude. It is this character of being an absolute minimum which we
want to get at and to express in terms of the extrinsic characters of
the abstractive sets which make up a point. Furthermore, points which
are thus arrived at represent the ideal of events without any extension,
though there are in fact no such entities as these ideal events. These
points will not be the points of an external timeless space but of
instantaneous spaces. We ultimately want to arrive at the timeless space
of physical science, and also of common thought which is now tinged with
the concepts of science. It will be convenient to reserve the term
'point' for these spaces when we get to them. I will therefore use the
name 'event-particles' for the ideal minimum limits to events. Thus an
event-particle is an abstractive element and as such is a group of
abstractive sets; and a point--namely a point of timeless space--will be
a class of event-particles.

Furthermore there is a separate timeless space corresponding to each
separate temporal series, that is to each separate family of durations.
We will come back to points in timeless spaces later. I merely mention
them now that we may understand the stages of our investigation. The
totality of event-particles will form a four-dimensional manifold, the
extra dimension arising from time--in other words--arising from the
points of a timeless space being each a class of event-particles.

The required character of the abstractive sets which form
event-particles would be secured if we could define them as having the
property of being covered by any abstractive set which they cover. For
then any other abstractive set which an abstractive set of an
event-particle covered, would be equal to it, and would therefore be a
member of the same event-particle. Accordingly an event-particle could
cover no other abstractive element. This is the definition which I
originally proposed at a congress in Paris in 1914[9]. There is however
a difficulty involved in this definition if adopted without some further
addition, and I am now not satisfied with the way in which I attempted
to get over that difficulty in the paper referred to.

[9] Cf. 'La Theorie Relationniste de l'Espace,' _Rev. de Metaphysique et
de Morale_, vol. XXIII, 1916.

The difficulty is this: When event-particles have once been defined it
is easy to define the aggregate of event-particles forming the boundary
of an event; and thence to define the point-contact at their boundaries
possible for a pair of events of which one is part of the other. We can
then conceive all the intricacies of tangency. In particular we can
conceive an abstractive set of which all the members have point-contact
at the same event-particle. It is then easy to prove that there will be
no abstractive set with the property of being covered by every
abstractive set which it covers. I state this difficulty at some length
because its existence guides the development of our line of argument. We
have got to annex some condition to the root property of being covered
by any abstractive set which it covers. When we look into this question
of suitable conditions we find that in addition to event-particles all
the other relevant spatial and spatio-temporal abstractive elements can
be defined in the same way by suitably varying the conditions.
Accordingly we proceed in a general way suitable for employment beyond
event-particles.

Let {sigma} be the name of any condition which some abstractive sets
fulfil. I say that an abstractive set is '{sigma}-prime' when it has the
two properties, (i) that it satisfies the condition {sigma} and (ii)
that it is covered by every abstractive set which both is covered by it
and satisfies the condition {sigma}.

In other words you cannot get any abstractive set satisfying the
condition {sigma} which exhibits intrinsic character more simple than
that of a {sigma}-prime.

There are also the correlative abstractive sets which I call the sets of
{sigma}-antiprimes. An abstractive set is a {sigma}-antiprime when it
has the two properties, (i) that it satisfies the condition {sigma} and
(ii) that it covers every abstractive set which both covers it and
satisfies the condition {sigma}. In other words you cannot get any
abstractive set satisfying the condition {sigma} which exhibits an
intrinsic character more complex than that of a {sigma}-antiprime.

The intrinsic character of a {sigma}-prime has a certain minimum of
fullness among those abstractive sets which are subject to the condition
of satisfying {sigma}; whereas the intrinsic character of a
{sigma}-antiprime has a corresponding maximum of fullness, and includes
all it can in the circumstances.

Let us first consider what help the notion of antiprimes could give us
in the definition of moments which we gave in the last lecture. Let the
condition {sigma} be the property of being a class whose members are all
durations. An abstractive set which satisfies this condition is thus an
abstractive set composed wholly of durations. It is convenient then to
define a moment as the group of abstractive sets which are equal to some
{sigma}-antiprime, where the condition {sigma} has this special meaning.
It will be found on consideration (i) that each abstractive set forming
a moment is a {sigma}-antiprime, where {sigma} has this special meaning,
and (ii) that we have excluded from membership of moments abstractive
sets of durations which all have one common boundary, either the initial
boundary or the final boundary. We thus exclude special cases which are
apt to confuse general reasoning. The new definition of a moment, which
supersedes our previous definition, is (by the aid of the notion of
antiprimes) the more precisely drawn of the two, and the more useful.

The particular condition which '{sigma}' stood for in the definition of
moments included something additional to anything which can be derived
from the bare notion of extension. A duration exhibits for thought a
totality. The notion of totality is something beyond that of extension,
though the two are interwoven in the notion of a duration.

In the same way the particular condition '{sigma}' required for the
definition of an event-particle must be looked for beyond the mere
notion of extension. The same remark is also true of the particular
conditions requisite for the other spatial elements. This additional
notion is obtained by distinguishing between the notion of 'position'
and the notion of convergence to an ideal zero of extension as exhibited
by an abstractive set of events.

In order to understand this distinction consider a point of the
instantaneous space which we conceive as apparent to us in an almost
instantaneous glance. This point is an event-particle. It has two
aspects. In one aspect it is there, where it is. This is its position in
the space. In another aspect it is got at by ignoring the circumambient
space, and by concentrating attention on the smaller and smaller set of
events which approximate to it. This is its extrinsic character. Thus a
point has three characters, namely, its position in the whole
instantaneous space, its extrinsic character, and its intrinsic
character. The same is true of any other spatial element. For example an
instantaneous volume in instantaneous space has three characters,
namely, its position, its extrinsic character as a group of abstractive
sets, and its intrinsic character which is the limit of natural
properties which is indicated by any one of these abstractive sets.

Before we can talk about position in instantaneous space, we must
evidently be quite clear as to what we mean by instantaneous space in
itself. Instantaneous space must be looked for as a character of a
moment. For a moment is all nature at an instant. It cannot be the
intrinsic character of the moment. For the intrinsic character tells us
the limiting character of nature in space at that instant. Instantaneous
space must be an assemblage of abstractive elements considered in their
mutual relations. Thus an instantaneous space is the assemblage of
abstractive elements covered by some one moment, and it is the
instantaneous space of that moment.

We have now to ask what character we have found in nature which is
capable of according to the elements of an instantaneous space different
qualities of position. This question at once brings us to the
intersection of moments, which is a topic not as yet considered in these
lectures.

The locus of intersection of two moments is the assemblage of
abstractive elements covered by both of them. Now two moments of the
same temporal series cannot intersect. Two moments respectively of
different families necessarily intersect. Accordingly in the
instantaneous space of a moment we should expect the fundamental
properties to be marked by the intersections with moments of other
families. If M be a given moment, the intersection of M with another
moment A is an instantaneous plane in the instantaneous space of M; and
if B be a third moment intersecting both M and A, the intersection of M
and B is another plane in the space M. Also the common intersection of
A, B, and M is the intersection of the two planes in the space M, namely
it is a straight line in the space M. An exceptional case arises if B
and M intersect in the same plane as A and M. Furthermore if C be a
fourth moment, then apart from special cases which we need not consider,
it intersects M in a plane which the straight line (A, B, M) meets. Thus
there is in general a common intersection of four moments of different
families. This common intersection is an assemblage of abstractive
elements which are each covered (or 'lie in') all four moments. The
three-dimensional property of instantaneous space comes to this, that
(apart from special relations between the four moments) any fifth moment
either contains the whole of their common intersection or none of it. No
further subdivision of the common intersection is possible by means of
moments. The 'all or none' principle holds. This is not an _a priori_
truth but an empirical fact of nature.

It will be convenient to reserve the ordinary spatial terms 'plane,'
'straight line,' 'point' for the elements of the timeless space of a
time-system. Accordingly an instantaneous plane in the instantaneous
space of a moment will be called a 'level,' an instantaneous straight
line will be called a 'rect,' and an instantaneous point will be called
a 'punct.' Thus a punct is the assemblage of abstractive elements which
lie in each of four moments whose families have no special relations to
each other. Also if P be any moment, either every abstractive element
belonging to a given punct lies in P, or no abstractive element of
that punct lies in P.

Position is the quality which an abstractive element possesses in virtue
of the moments in which it lies. The abstractive elements which lie in
the instantaneous space of a given moment M are differentiated from
each other by the various other moments which intersect M so as to
contain various selections of these abstractive elements. It is this
differentiation of the elements which constitutes their differentiation
of position. An abstractive element which belongs to a punct has the
simplest type of position in M, an abstractive element which belongs
to a rect but not to a punct has a more complex quality of position, an
abstractive element which belongs to a level and not to a rect has a
still more complex quality of position, and finally the most complex
quality of position belongs to an abstractive element which belongs to a
volume and not to a level. A volume however has not yet been defined.
This definition will be given in the next lecture.

Evidently levels, rects, and puncts in their capacity as infinite
aggregates cannot be the termini of sense-awareness, nor can they be
limits which are approximated to in sense-awareness. Any one member of a
level has a certain quality arising from its character as also belonging
to a certain set of moments, but the level as a whole is a mere logical
notion without any route of approximation along entities posited in
sense-awareness.

On the other hand an event-particle is defined so as to exhibit this
character of being a route of approximation marked out by entities
posited in sense-awareness. A definite event-particle is defined in
reference to a definite punct in the following manner: Let the condition
{sigma} mean the property of covering all the abstractive elements which
are members of that punct; so that an abstractive set which satisfies
the condition {sigma} is an abstractive set which covers every
abstractive element belonging to the punct. Then the definition of the
event-particle associated with the punct is that it is the group of all
the {sigma}-primes, where {sigma} has this particular meaning.

It is evident that--with this meaning of {sigma}--every abstractive set
equal to a {sigma}-prime is itself a {sigma}-prime. Accordingly an
event-particle as thus defined is an abstractive element, namely it is
the group of those abstractive sets which are each equal to some given
abstractive set. If we write out the definition of the event-particle
associated with some given punct, which we will call {pi}, it is as
follows: The event-particle associated with {pi} is the group of
abstractive classes each of which has the two properties (i) that it
covers every abstractive set in {pi} and (ii) that all the abstractive
sets which also satisfy the former condition as to {pi} and which it
covers, also cover it.

An event-particle has position by reason of its association with a
punct, and conversely the punct gains its derived character as a route
of approximation from its association with the event-particle. These two
characters of a point are always recurring in any treatment of the
derivation of a point from the observed facts of nature, but in general
there is no clear recognition of their distinction.

The peculiar simplicity of an instantaneous point has a twofold origin,
one connected with position, that is to say with its character as a
punct, and the other connected with its character as an event-particle.
The simplicity of the punct arises from its indivisibility by a moment.

The simplicity of an event-particle arises from the indivisibility of
its intrinsic character. The intrinsic character of an event-particle is
indivisible in the sense that every abstractive set covered by it
exhibits the same intrinsic character. It follows that, though there are
diverse abstractive elements covered by event-particles, there is no
advantage to be gained by considering them since we gain no additional
simplicity in the expression of natural properties.

These two characters of simplicity enjoyed respectively by
event-particles and puncts define a meaning for Euclid's phrase,
'without parts and without magnitude.'

It is obviously convenient to sweep away out of our thoughts all these
stray abstractive sets which are covered by event-particles without
themselves being members of them. They give us nothing new in the way of
intrinsic character. Accordingly we can think of rects and levels as
merely loci of event-particles. In so doing we are also cutting out
those abstractive elements which cover sets of event-particles, without
these elements being event-particles themselves. There are classes of
these abstractive elements which are of great importance. I will
consider them later on in this and in other lectures. Meanwhile we will
ignore them. Also I will always speak of 'event-particles' in preference
to 'puncts,' the latter being an artificial word for which I have no
great affection.

Parallelism among rects and levels is now explicable.

Consider the instantaneous space belonging to a moment A, and let A
belong to the temporal series of moments which I will call {alpha}.
Consider any other temporal series of moments which I will call {beta}.
The moments of {beta} do not intersect each other and they intersect the
moment A in a family of levels. None of these levels can intersect, and
they form a family of parallel instantaneous planes in the instantaneous
space of moment A. Thus the parallelism of moments in a temporal series
begets the parallelism of levels in an instantaneous space, and
thence--as it is easy to see--the parallelism of rects. Accordingly the
Euclidean property of space arises from the parabolic property of time.
It may be that there is reason to adopt a hyperbolic theory of time and
a corresponding hyperbolic theory of space. Such a theory has not been
worked out, so it is not possible to judge as to the character of the
evidence which could be brought forward in its favour.

The theory of order in an instantaneous space is immediately derived
from time-order. For consider the space of a moment M. Let {alpha} be
the name of a time-system to which M does not belong. Let A_1, A_2, A_3
etc. be moments of {alpha} in the order of their occurrences. Then A_1,
A_2, A_3, etc. intersect M in parallel levels l_1, l_2, l_3, etc. Then
the relative order of the parallel levels in the space of M is the same
as the relative order of the corresponding moments in the time-system
{alpha}. Any rect in M which intersects all these levels in its set of
puncts, thereby receives for its puncts an order of position on it. So
spatial order is derivative from temporal order. Furthermore there are
alternative time-systems, but there is only one definite spatial order
in each instantaneous space. Accordingly the various modes of deriving
spatial order from diverse time-systems must harmonise with one spatial
order in each instantaneous space. In this way also diverse time-orders
are comparable.

We have two great questions still on hand to be settled before our
theory of space is fully adjusted. One of these is the question of the
determination of the methods of measurement within the space, in other
words, the congruence-theory of the space. The measurement of space will
be found to be closely connected with the measurement of time, with
respect to which no principles have as yet been determined. Thus our
congruence-theory will be a theory both for space and for time. Secondly
there is the determination of the timeless space which corresponds to
any particular time-system with its infinite set of instantaneous spaces
in its successive moments. This is the space--or rather, these are the
spaces--of physical science. It is very usual to dismiss this space by
saying that this is conceptual. I do not understand the virtue of these
phrases. I suppose that it is meant that the space is the conception of
something in nature. Accordingly if the space of physical science is to
be called conceptual, I ask, What in nature is it the conception of? For
example, when we speak of a point in the timeless space of physical
science, I suppose that we are speaking of something in nature. If we
are not so speaking, our scientists are exercising their wits in the
realms of pure fantasy, and this is palpably not the case. This demand
for a definite Habeas Corpus Act for the production of the relevant
entities in nature applies whether space be relative or absolute. On the
theory of relative space, it may perhaps be argued that there is no
timeless space for physical science, and that there is only the
momentary series of instantaneous spaces.

An explanation must then be asked for the meaning of the very common
statement that such and such a man walked four miles in some definite
hour. How can you measure distance from one space into another space? I
understand walking out of the sheet of an ordnance map. But the meaning
of saying that Cambridge at 10 o'clock this morning in the appropriate
instantaneous space for that instant is 52 miles from London at
11 o'clock this morning in the appropriate instantaneous space for that
instant beats me entirely. I think that, by the time a meaning has been
produced for this statement, you will find that you have constructed
what is in fact a timeless space. What I cannot understand is how to
produce an explanation of meaning without in effect making some such
construction. Also I may add that I do not know how the instantaneous
spaces are thus correlated into one space by any method which is
available on the current theories of space.

You will have noticed that by the aid of the assumption of alternative
time-systems, we are arriving at an explanation of the character of
space. In natural science 'to explain' means merely to discover
'interconnexions.' For example, in one sense there is no explanation of
the red which you see. It is red, and there is nothing else to be said
about it. Either it is posited before you in sense-awareness or you are
ignorant of the entity red. But science has explained red. Namely it has
discovered interconnexions between red as a factor in nature and other
factors in nature, for example waves of light which are waves of
electromagnetic disturbances. There are also various pathological
states of the body which lead to the seeing of red without the
occurrence of light waves. Thus connexions have been discovered between
red as posited in sense-awareness and various other factors in nature.
The discovery of these connexions constitutes the scientific explanation
of our vision of colour. In like manner the dependence of the character
of space on the character of time constitutes an explanation in the
sense in which science seeks to explain. The systematising intellect
abhors bare facts. The character of space has hitherto been presented as
a collection of bare facts, ultimate and disconnected. The theory which
I am expounding sweeps away this disconnexion of the facts of space.




CHAPTER V

SPACE AND MOTION


The topic for this lecture is the continuation of the task of explaining
the construction of spaces as abstracts from the facts of nature. It was
noted at the close of the previous lecture that the question of
congruence had not been considered, nor had the construction of a
timeless space which should correlate the successive momentary spaces of
a given time-system. Furthermore it was also noted that there were many
spatial abstractive elements which we had not yet defined. We will first
consider the definition of some of these abstractive elements, namely
the definitions of solids, of areas, and of routes. By a 'route' I mean
a linear segment, whether straight or curved. The exposition of these
definitions and the preliminary explanations necessary will, I hope,
serve as a general explanation of the function of event-particles in the
analysis of nature.

We note that event-particles have 'position' in respect to each other.
In the last lecture I explained that 'position' was quality gained by a
spatial element in virtue of the intersecting moments which covered it.
Thus each event-particle has position in this sense. The simplest mode
of expressing the position in nature of an event-particle is by first
fixing on any definite time-system. Call it {alpha}. There will be one
moment of the temporal series of {alpha} which covers the given
event-particle. Thus the position of the event-particle in the temporal
series {alpha} is defined by this moment, which we will call M. The
position of the particle in the space of M is then fixed in the ordinary
way by three levels which intersect in it and in it only. This procedure
of fixing the position of an event-particle shows that the aggregate of
event-particles forms a four-dimensional manifold. A finite event
occupies a limited chunk of this manifold in a sense which I now proceed
to explain.

Let e be any given event. The manifold of event-particles falls into
three sets in reference to e. Each event-particle is a group of equal
abstractive sets and each abstractive set towards its small-end is
composed of smaller and smaller finite events. When we select from these
finite events which enter into the make-up of a given event-particle
those which are small enough, one of three cases must occur. Either (i)
all of these small events are entirely separate from the given event
e, or (ii) all of these small events are parts of the event e, or
(iii) all of these small events overlap the event e but are not parts
of it. In the first case the event-particle will be said to 'lie
outside' the event e, in the second case the event-particle will be
said to 'lie inside' the event e, and in the third case the
event-particle will be said to be a 'boundary-particle' of the event
e. Thus there are three sets of particles, namely the set of those
which lie outside the event e, the set of those which lie inside the
event e, and the boundary of the event e which is the set of
boundary-particles of e. Since an event is four-dimensional, the
boundary of an event is a three-dimensional manifold. For a finite event
there is a continuity of boundary; for a duration the boundary consists
of those event-particles which are covered by either of the two bounding
moments. Thus the boundary of a duration consists of two momentary
three-dimensional spaces. An event will be said to 'occupy' the
aggregate of event-particles which lie within it.

Two events which have 'junction' in the sense in which junction was
described in my last lecture, and yet are separated so that neither
event either overlaps or is part of the other event, are said to be
'adjoined.'

This relation of adjunction issues in a peculiar relation between the
boundaries of the two events. The two boundaries must have a common
portion which is in fact a continuous three-dimensional locus of
event-particles in the four-dimensional manifold.

A three-dimensional locus of event-particles which is the common portion
of the boundary of two adjoined events will be called a 'solid.' A solid
may or may not lie completely in one moment. A solid which does not lie
in one moment will be called 'vagrant.' A solid which does lie in one
moment will be called a volume. A volume may be defined as the locus of
the event-particles in which a moment intersects an event, provided that
the two do intersect. The intersection of a moment and an event will
evidently consist of those event-particles which are covered by the
moment and lie in the event. The identity of the two definitions of a
volume is evident when we remember that an intersecting moment divides
the event into two adjoined events.

A solid as thus defined, whether it be vagrant or be a volume, is a mere
aggregate of event-particles illustrating a certain quality of position.
We can also define a solid as an abstractive element. In order to do so
we recur to the theory of primes explained in the preceding lecture. Let
the condition named {sigma} stand for the fact that each of the events
of any abstractive set satisfying it has all the event-particles of some
particular solid lying in it. Then the group of all the {sigma}-primes
is the abstractive element which is associated with the given solid. I
will call this abstractive element the solid as an abstractive element,
and I will call the aggregate of event-particles the solid as a locus.
The instantaneous volumes in instantaneous space which are the ideals of
our sense-perception are volumes as abstractive elements. What we really
perceive with all our efforts after exactness are small events far
enough down some abstractive set belonging to the volume as an
abstractive element.

It is difficult to know how far we approximate to any perception of
vagrant solids. We certainly do not think that we make any such
approximation. But then our thoughts--in the case of people who do think
about such topics--are so much under the control of the materialistic
theory of nature that they hardly count for evidence. If Einstein's
theory of gravitation has any truth in it, vagrant solids are of great
importance in science. The whole boundary of a finite event may be
looked on as a particular example of a vagrant solid as a locus. Its
particular property of being closed prevents it from being definable as
an abstractive element.

When a moment intersects an event, it also intersects the boundary of
that event. This locus, which is the portion of the boundary contained
in the moment, is the bounding surface of the corresponding volume of
that event contained in the moment. It is a two-dimensional locus.

The fact that every volume has a bounding surface is the origin of the
Dedekindian continuity of space.

Another event may be cut by the same moment in another volume and this
volume will also have its boundary. These two volumes in the
instantaneous space of one moment may mutually overlap in the familiar
way which I need not describe in detail and thus cut off portions from
each other's surfaces. These portions of surfaces are 'momental areas.'

It is unnecessary at this stage to enter into the complexity of a
definition of vagrant areas. Their definition is simple enough when the
four-dimensional manifold of event-particles has been more fully
explored as to its properties.

Momental areas can evidently be defined as abstractive elements by
exactly the same method as applied to solids. We have merely to
substitute 'area' for a 'solid' in the words of the definition already
given. Also, exactly as in the analogous case of a solid, what we
perceive as an approximation to our ideal of an area is a small event
far enough down towards the small end of one of the equal abstractive
sets which belongs to the area as an abstractive element.

Two momental areas lying in the same moment can cut each other in a
momental segment which is not necessarily rectilinear. Such a segment
can also be defined as an abstractive element. It is then called a
'momental route.' We will not delay over any general consideration of
these momental routes, nor is it important for us to proceed to the
still wider investigation of vagrant routes in general. There are
however two simple sets of routes which are of vital importance. One is
a set of momental routes and the other of vagrant routes. Both sets can
be classed together as straight routes. We proceed to define them
without any reference to the definitions of volumes and surfaces.

The two types of straight routes will be called rectilinear routes and
stations. Rectilinear routes are momental routes and stations are
vagrant routes. Rectilinear routes are routes which in a sense lie in
rects. Any two event-particles on a rect define the set of
event-particles which lie between them on that rect. Let the
satisfaction of the condition {sigma} by an abstractive set mean that
the two given event-particles and the event-particles lying between them
on the rect all lie in every event belonging to the abstractive set. The
group of {sigma}-primes, where {sigma} has this meaning, form an
abstractive element. Such abstractive elements are rectilinear routes.
They are the segments of instantaneous straight lines which are the
ideals of exact perception. Our actual perception, however exact, will
be the perception of a small event sufficiently far down one of the
abstractive sets of the abstractive element.

A station is a vagrant route and no moment can intersect any station in
more than one event-particle. Thus a station carries with it a
comparison of the positions in their respective moments of the
event-particles covered by it. Rects arise from the intersection of
moments. But as yet no properties of events have been mentioned by which
any analogous vagrant loci can be found out.

The general problem for our investigation is to determine a method of
comparison of position in one instantaneous space with positions in
other instantaneous spaces. We may limit ourselves to the spaces of the
parallel moments of one time-system. How are positions in these various
spaces to be compared? In other words, What do we mean by motion? It is
the fundamental question to be asked of any theory of relative space,
and like many other fundamental questions it is apt to be left
unanswered. It is not an answer to reply, that we all know what we mean
by motion. Of course we do, so far as sense-awareness is concerned. I am
asking that your theory of space should provide nature with something to
be observed. You have not settled the question by bringing forward a
theory according to which there is nothing to be observed, and by then
reiterating that nevertheless we do observe this non-existent fact.
Unless motion is something as a fact in nature, kinetic energy and
momentum and all that depends on these physical concepts evaporate from
our list of physical realities. Even in this revolutionary age my
conservatism resolutely opposes the identification of momentum and
moonshine.

Accordingly I assume it as an axiom, that motion is a physical fact. It
is something that we perceive as in nature. Motion presupposes rest.
Until theory arose to vitiate immediate intuition, that is to say to
vitiate the uncriticised judgments which immediately arise from
sense-awareness, no one doubted that in motion you leave behind that
which is at rest. Abraham in his wanderings left his birthplace where it
had ever been. A theory of motion and a theory of rest are the same
thing viewed from different aspects with altered emphasis.

Now you cannot have a theory of rest without in some sense admitting a
theory of absolute position. It is usually assumed that relative space
implies that there is no absolute position. This is, according to my
creed, a mistake. The assumption arises from the failure to make another
distinction; namely, that there may be alternative definitions of
absolute position. This possibility enters with the admission of
alternative time-systems. Thus the series of spaces in the parallel
moments of one temporal series may have their own definition of absolute
position correlating sets of event-particles in these successive spaces,
so that each set consists of event-particles, one from each space, all
with the property of possessing the same absolute position in that
series of spaces. Such a set of event-particles will form a point in the
timeless space of that time-system. Thus a point is really an absolute
position in the timeless space of a given time-system.

But there are alternative time-systems, and each time-system has its own
peculiar group of points--that is to say, its own peculiar definition of
absolute position. This is exactly the theory which I will elaborate.

In looking to nature for evidence of absolute position it is of no use
to recur to the four-dimensional manifold of event-particles. This
manifold has been obtained by the extension of thought beyond the
immediacy of observation. We shall find nothing in it except what we
have put there to represent the ideas in thought which arise from our
direct sense-awareness of nature. To find evidence of the properties
which are to be found in the manifold of event-particles we must always
recur to the observation of relations between events. Our problem is to
determine those relations between events which issue in the property of
absolute position in a timeless space. This is in fact the problem of
the determination of the very meaning of the timeless spaces of physical
science.

In reviewing the factors of nature as immediately disclosed in
sense-awareness, we should note the fundamental character of the percept
of 'being here.' We discern an event merely as a factor in a determinate
complex in which each factor has its own peculiar share.

There are two factors which are always ingredient in this complex, one
is the duration which is represented in thought by the concept of all
nature that is present now, and the other is the peculiar _locus standi_
for mind involved in the sense-awareness. This _locus standi_ in nature
is what is represented in thought by the concept of 'here,' namely of an
'event here.'

This is the concept of a definite factor in nature. This factor is an
event in nature which is the focus in nature for that act of awareness,
and the other events are perceived as referred to it. This event is part
of the associated duration. I call it the 'percipient event.' This event
is not the mind, that is to say, not the percipient. It is that in
nature from which the mind perceives. The complete foothold of the mind
in nature is represented by the pair of events, namely, the present
duration which marks the 'when' of awareness and the percipient event
which marks the 'where' of awareness and the 'how' of awareness. This
percipient event is roughly speaking the bodily life of the incarnate
mind. But this identification is only a rough one. For the functions of
the body shade off into those of other events in nature; so that for
some purposes the percipient event is to be reckoned as merely part of
the bodily life and for other purposes it may even be reckoned as more
than the bodily life. In many respects the demarcation is purely
arbitrary, depending upon where in a sliding scale you choose to draw
the line.

I have already in my previous lecture on Time discussed the association
of mind with nature. The difficulty of the discussion lies in the
liability of constant factors to be overlooked. We never note them by
contrast with their absences. The purpose of a discussion of such
factors may be described as being to make obvious things look odd. We
cannot envisage them unless we manage to invest them with some of the
freshness which is due to strangeness.

It is because of this habit of letting constant factors slip from
consciousness that we constantly fall into the error of thinking of the
sense-awareness of a particular factor in nature as being a two-termed
relation between the mind and the factor. For example, I perceive a
green leaf. Language in this statement suppresses all reference to any
factors other than the percipient mind and the green leaf and the
relation of sense-awareness. It discards the obvious inevitable factors
which are essential elements in the perception. I am here, the leaf is
there; and the event here and the event which is the life of the leaf
there are both embedded in a totality of nature which is now, and within
this totality there are other discriminated factors which it is
irrelevant to mention. Thus language habitually sets before the mind a
misleading abstract of the indefinite complexity of the fact of
sense-awareness.

What I now want to discuss is the special relation of the percipient
event which is 'here' to the duration which is 'now.' This relation is a
fact in nature, namely the mind is aware of nature as being with these
two factors in this relation.

Within the short present duration the 'here' of the percipient event has
a definite meaning of some sort. This meaning of 'here' is the content
of the special relation of the percipient event to its associated
duration. I will call this relation 'cogredience.' Accordingly I ask for
a description of the character of the relation of cogredience. The
present snaps into a past and a present when the 'here' of cogredience
loses its single determinate meaning. There has been a passage of nature
from the 'here' of perception within the past duration to the different
'here' of perception within the present duration. But the two 'heres' of
sense-awareness within neighbouring durations may be indistinguishable.
In this case there has been a passage from the past to the present, but
a more retentive perceptive force might have retained the passing nature
as one complete present instead of letting the earlier duration slip
into the past. Namely, the sense of rest helps the integration of
durations into a prolonged present, and the sense of motion
differentiates nature into a succession of shortened durations. As we
look out of a railway carriage in an express train, the present is past
before reflexion can seize it. We live in snippits too quick for
thought. On the other hand the immediate present is prolonged according
as nature presents itself to us in an aspect of unbroken rest. Any
change in nature provides ground for a differentiation among durations
so as to shorten the present. But there is a great distinction between
self-change in nature and change in external nature. Self-change in
nature is change in the quality of the standpoint of the percipient
event. It is the break up of the 'here' which necessitates the break up
of the present duration. Change in external nature is compatible with a
prolongation of the present of contemplation rooted in a given
standpoint. What I want to bring out is that the preservation of a
peculiar relation to a duration is a necessary condition for the
function of that duration as a present duration for sense-awareness.
This peculiar relation is the relation of cogredience between the
percipient event and the duration. Cogredience is the preservation of
unbroken quality of standpoint within the duration. It is the
continuance of identity of station within the whole of nature which is
the terminus of sense-awareness. The duration may comprise change within
itself, but cannot--so far as it is one present duration--comprise
change in the quality of its peculiar relation to the contained
percipient event.

In other words, perception is always 'here,' and a duration can only be
posited as present for sense-awareness on condition that it affords one
unbroken meaning of 'here' in its relation to the percipient event. It
is only in the past that you can have been 'there' with a standpoint
distinct from your present 'here.'

Events there and events here are facts of nature, and the qualities of
being 'there' and 'here' are not merely qualities of awareness as a
relation between nature and mind. The quality of determinate station in
the duration which belongs to an event which is 'here' in one
determinate sense of 'here' is the same kind of quality of station which
belongs to an event which is 'there' in one determinate sense of
'there.' Thus cogredience has nothing to do with any biological
character of the event which is related by it to the associated
duration. This biological character is apparently a further condition
for the peculiar connexion of a percipient event with the percipience of
mind; but it has nothing to do with the relation of the percipient event
to the duration which is the present whole of nature posited as the
disclosure of the percipience.

Given the requisite biological character, the event in its character of
a percipient event selects that duration with which the operative past
of the event is practically cogredient within the limits of the
exactitude of observation. Namely, amid the alternative time-systems
which nature offers there will be one with a duration giving the best
average of cogredience for all the subordinate parts of the percipient
event. This duration will be the whole of nature which is the terminus
posited by sense-awareness. Thus the character of the percipient event
determines the time-system immediately evident in nature. As the
character of the percipient event changes with the passage of
nature--or, in other words, as the percipient mind in its passage
correlates itself with the passage of the percipient event into another
percipient event--the time-system correlated with the percipience of
that mind may change. When the bulk of the events perceived are
cogredient in a duration other than that of the percipient event, the
percipience may include a double consciousness of cogredience, namely
the consciousness of the whole within which the observer in the train is
'here,' and the consciousness of the whole within which the trees and
bridges and telegraph posts are definitely 'there.' Thus in perceptions
under certain circumstances the events discriminated assert their own
relations of cogredience. This assertion of cogredience is peculiarly
evident when the duration to which the perceived event is cogredient is
the same as the duration which is the present whole of nature--in other
words, when the event and the percipient event are both cogredient to
the same duration.

We are now prepared to consider the meaning of stations in a duration,
where stations are a peculiar kind of routes, which define absolute
position in the associated timeless space.

There are however some preliminary explanations. A finite event will be
said to extend throughout a duration when it is part of the duration
and is intersected by any moment which lies in the duration. Such an
event begins with the duration and ends with it. Furthermore every event
which begins with the duration and ends with it, extends throughout the
duration. This is an axiom based on the continuity of events. By
beginning with a duration and ending with it, I mean (i) that the event
is part of the duration, and (ii) that both the initial and final
boundary moments of the duration cover some event-particles on the
boundary of the event.

Every event which is cogredient with a duration extends throughout that
duration.

It is not true that all the parts of an event cogredient with a duration
are also cogredient with the duration. The relation of cogredience may
fail in either of two ways. One reason for failure may be that the part
does not extend throughout the duration. In this case the part may be
cogredient with another duration which is part of the given duration,
though it is not cogredient with the given duration itself. Such a part
would be cogredient if its existence were sufficiently prolonged in that
time-system. The other reason for failure arises from the
four-dimensional extension of events so that there is no determinate
route of transition of events in linear series. For example, the tunnel
of a tube railway is an event at rest in a certain time-system, that is
to say, it is cogredient with a certain duration. A train travelling in
it is part of that tunnel, but is not itself at rest.

If an event e be cogredient with a duration d, and d' be any
duration which is part of d. Then d' belongs to the same time-system
as d. Also d' intersects e in an event e' which is part of e
and is cogredient with d'.

Let P be any event-particle lying in a given duration d. Consider
the aggregate of events in which P lies and which are also cogredient
with d. Each of these events occupies its own aggregate of
event-particles. These aggregates will have a common portion, namely the
class of event-particle lying in all of them. This class of
event-particles is what I call the 'station' of the event-particle P
in the duration d. This is the station in the character of a locus. A
station can also be defined in the character of an abstractive element.
Let the property {sigma} be the name of the property which an
abstractive set possesses when (i) each of its events is cogredient with
the duration d and (ii) the event-particle P lies in each of its events.
Then the group of {sigma}-primes, where {sigma} has this meaning, is an
abstractive element and is the station of P in d as an abstractive
element. The locus of event-particles covered by the station of P in d
as an abstractive element is the station of P in d as a locus. A station
has accordingly the usual three characters, namely, its character of
position, its extrinsic character as an abstractive element, and its
intrinsic character.

It follows from the peculiar properties of rest that two stations
belonging to the same duration cannot intersect. Accordingly every
event-particle on a station of a duration has that station as its
station in the duration. Also every duration which is part of a given
duration intersects the stations of the given duration in loci which are
its own stations. By means of these properties we can utilise the
overlappings of the durations of one family--that is, of one
time-system--to prolong stations indefinitely backwards and forwards.
Such a prolonged station will be called a point-track. A point-track is
a locus of event-particles. It is defined by reference to one particular
time-system, {alpha} say. Corresponding to any other time-system these
will be a different group of point-tracks. Every event-particle will lie
on one and only one point-track of the group belonging to any one
time-system. The group of point-tracks of the time-system {alpha} is the
group of points of the timeless space of {alpha}. Each such point
indicates a certain quality of absolute position in reference to the
durations of the family associated with {alpha}, and thence in reference
to the successive instantaneous spaces lying in the successive moments
of {alpha}. Each moment of {alpha} will intersect a point-track in one
and only one event-particle.

This property of the unique intersection of a moment and a point-track
is not confined to the case when the moment and the point-track belong
to the same time-system. Any two event-particles on a point-track are
sequential, so that they cannot lie in the same moment. Accordingly no
moment can intersect a point-track more than once, and every moment
intersects a point-track in one event-particle.

Anyone who at the successive moments of {alpha} should be at the
event-particles where those moments intersect a given point of {alpha}
will be at rest in the timeless space of time-system {alpha}. But in any
other timeless space belonging to another time-system he will be at a
different point at each succeeding moment of that time-system. In other
words he will be moving. He will be moving in a straight line with
uniform velocity. We might take this as the definition of a straight
line. Namely, a straight line in the space of time-system {beta} is the
locus of those points of {beta} which all intersect some one point-track
which is a point in the space of some other time-system. Thus each point
in the space of a time-system {alpha} is associated with one and only
one straight line of the space of any other time-system {beta}.
Furthermore the set of straight lines in space {beta} which are thus
associated with points in space {alpha} form a complete family of
parallel straight lines in space {beta}. Thus there is a one-to-one
correlation of points in space {alpha} with the straight lines of a
certain definite family of parallel straight lines in space {beta}.
Conversely there is an analogous one-to-one correlation of the points in
space {beta} with the straight lines of a certain family of parallel
straight lines in space {alpha}. These families will be called
respectively the family of parallels in {beta} associated with {alpha},
and the family of parallels in {alpha} associated with {beta}. The
direction in the space of {beta} indicated by the family of parallels in
{beta} will be called the direction of {alpha} in space {beta}, and the
family of parallels in {alpha} is the direction of {beta} in space
{alpha}. Thus a being at rest at a point of space {alpha} will be moving
uniformly along a line in space {beta} which is in the direction of
{alpha} in space {beta}, and a being at rest at a point of space {beta}
will be moving uniformly along a line in space {alpha} which is in the
direction of {beta} in space {alpha}.

I have been speaking of the timeless spaces which are associated with
time-systems. These are the spaces of physical science and of any
concept of space as eternal and unchanging. But what we actually
perceive is an approximation to the instantaneous space indicated by
event-particles which lie within some moment of the time-system
associated with our awareness. The points of such an instantaneous space
are event-particles and the straight lines are rects. Let the
time-system be named {alpha}, and let the moment of time-system {alpha}
to which our quick perception of nature approximates be called M. Any
straight line r in space {alpha} is a locus of points and each point is
a point-track which is a locus of event-particles. Thus in the
four-dimensional geometry of all event-particles there is a
two-dimensional locus which is the locus of all event-particles on
points lying on the straight line r. I will call this locus of
event-particles the matrix of the straight line r. A matrix intersects
any moment in a rect. Thus the matrix of r intersects the moment M in a
rect {rho}. Thus {rho} is the instantaneous rect in M which occupies at
the moment M the straight line r in the space of {alpha}. Accordingly
when one sees instantaneously a moving being and its path ahead of it,
what one really sees is the being at some event-particle A lying in the
rect {rho} which is the apparent path on the assumption of uniform
motion. But the actual rect {rho} which is a locus of event-particles is
never traversed by the being. These event-particles are the
instantaneous facts which pass with the instantaneous moment. What is
really traversed are other event-particles which at succeeding instants
occupy the same points of space {alpha} as those occupied by the
event-particles of the rect {rho}. For example, we see a stretch of road
and a lorry moving along it. The instantaneously seen road is a portion
of the rect {rho}--of course only an approximation to it. The lorry is
the moving object. But the road as seen is never traversed. It is
thought of as being traversed because the intrinsic characters of the
later events are in general so similar to those of the instantaneous
road that we do not trouble to discriminate. But suppose a land mine
under the road has been exploded before the lorry gets there. Then it is
fairly obvious that the lorry does not traverse what we saw at first.
Suppose the lorry is at rest in space {beta}. Then the straight line r
of space {alpha} is in the direction of {beta} in space {alpha}, and the
rect {rho} is the representative in the moment M of the line r of space
{alpha}. The direction of {rho} in the instantaneous space of the moment
M is the direction of {beta} in M, where M is a moment of time-system
{alpha}. Again the matrix of the line r of space {alpha} will also be
the matrix of some line s of space {beta} which will be in the direction
of {alpha} in space {beta}. Thus if the lorry halts at some point P of
space {alpha} which lies on the line r, it is now moving along the line
s of space {beta}. This is the theory of relative motion; the common
matrix is the bond which connects the motion of {beta} in space {alpha}
with the motions of {alpha} in space {beta}.

Motion is essentially a relation between some object of nature and the
one timeless space of a time-system. An instantaneous space is static,
being related to the static nature at an instant. In perception when we
see things moving in an approximation to an instantaneous space, the
future lines of motion as immediately perceived are rects which are
never traversed. These approximate rects are composed of small events,
namely approximate routes and event-particles, which are passed away
before the moving objects reach them. Assuming that our forecasts of
rectilinear motion are correct, these rects occupy the straight lines in
timeless space which are traversed. Thus the rects are symbols in
immediate sense-awareness of a future which can only be expressed in
terms of timeless space.

We are now in a position to explore the fundamental character of
perpendicularity. Consider the two time-systems {alpha} and {beta}, each
with its own timeless space and its own family of instantaneous moments
with their instantaneous spaces. Let M and N be respectively a moment of
{alpha} and a moment of {beta}. In M there is the direction of {beta}
and in N there is the direction of {alpha}. But M and N, being moments
of different time-systems, intersect in a level. Call this level
{lambda}. Then {lambda} is an instantaneous plane in the instantaneous
space of M and also in the instantaneous space of N. It is the locus of
all the event-particles which lie both in M and in N.

In the instantaneous space of M the level {lambda} is perpendicular to
the direction of {beta} in M, and in the instantaneous space of N the
level {lambda} is perpendicular to the direction of {alpha} in N. This
is the fundamental property which forms the definition of
perpendicularity. The symmetry of perpendicularity is a particular
instance of the symmetry of the mutual relations between two
time-systems. We shall find in the next lecture that it is from this
symmetry that the theory of congruence is deduced.

The theory of perpendicularity in the timeless space of any time-system
{alpha} follows immediately from this theory of perpendicularity in each
of its instantaneous spaces. Let {rho} be any rect in the moment M of
{alpha} and let {lambda} be a level in M which is perpendicular to
{rho}. The locus of those points of the space of {alpha} which intersect
M in event-particles on {rho} is the straight line r of space {alpha},
and the locus of those points of the space of {alpha} which intersect M
in event-particles on {lambda} is the plane l of space {alpha}. Then the
plane l is perpendicular to the line r.

In this way we have pointed out unique and definite properties in nature
which correspond to perpendicularity. We shall find that this discovery
of definite unique properties defining perpendicularity is of critical
importance in the theory of congruence which is the topic for the next
lecture.

I regret that it has been necessary for me in this lecture to administer
such a large dose of four-dimensional geometry. I do not apologise,
because I am really not responsible for the fact that nature in its most
fundamental aspect is four-dimensional. Things are what they are; and it
is useless to disguise the fact that 'what things are' is often very
difficult for our intellects to follow. It is a mere evasion of the
ultimate problems to shirk such obstacles.




CHAPTER VI

CONGRUENCE


The aim of this lecture is to establish a theory of congruence. You must
understand at once that congruence is a controversial question. It is
the theory of measurement in space and in time. The question seems
simple. In fact it is simple enough for a standard procedure to have
been settled by act of parliament; and devotion to metaphysical
subtleties is almost the only crime which has never been imputed to any
English parliament. But the procedure is one thing and its meaning is
another.

First let us fix attention on the purely mathematical question. When the
segment between two points A and B is congruent to that between the
two points C and D, the quantitative measurements of the two
segments are equal. The equality of the numerical measures and the
congruence of the two segments are not always clearly discriminated, and
are lumped together under the term equality. But the procedure of
measurement presupposes congruence. For example, a yard measure is
applied successively to measure two distances between two pairs of
points on the floor of a room. It is of the essence of the procedure of
measurement that the yard measure remains unaltered as it is transferred
from one position to another. Some objects can palpably alter as they
move--for example, an elastic thread; but a yard measure does not alter
if made of the proper material. What is this but a judgment of
congruence applied to the train of successive positions of the yard
measure? We know that it does not alter because we judge it to be
congruent to itself in various positions. In the case of the thread we
can observe the loss of self-congruence. Thus immediate judgments of
congruence are presupposed in measurement, and the process of
measurement is merely a procedure to extend the recognition of
congruence to cases where these immediate judgments are not available.
Thus we cannot define congruence by measurement.

In modern expositions of the axioms of geometry certain conditions are
laid down which the relation of congruence between segments is to
satisfy. It is supposed that we have a complete theory of points,
straight lines, planes, and the order of points on planes--in fact, a
complete theory of non-metrical geometry. We then enquire about
congruence and lay down the set of conditions--or axioms as they are
called--which this relation satisfies. It has then been proved that
there are alternative relations which satisfy these conditions equally
well and that there is nothing intrinsic in the theory of space to lead
us to adopt any one of these relations in preference to any other as the
relation of congruence which we adopt. In other words there are
alternative metrical geometries which all exist by an equal right so far
as the intrinsic theory of space is concerned.

Poincare, the great French mathematician, held that our actual choice
among these geometries is guided purely by convention, and that the
effect of a change of choice would be simply to alter our expression of
the physical laws of nature. By 'convention' I understand Poincare to
mean that there is nothing inherent in nature itself giving any peculiar
_role_ to one of these congruence relations, and that the choice of one
particular relation is guided by the volitions of the mind at the other
end of the sense-awareness. The principle of guidance is intellectual
convenience and not natural fact.

This position has been misunderstood by many of Poincare's expositors.
They have muddled it up with another question, namely that owing to the
inexactitude of observation it is impossible to make an exact statement
in the comparison of measures. It follows that a certain subset of
closely allied congruence relations can be assigned of which each member
equally well agrees with that statement of observed congruence when the
statement is properly qualified with its limits of error.

This is an entirely different question and it presupposes a rejection of
Poincare's position. The absolute indetermination of nature in respect
of all the relations of congruence is replaced by the indetermination of
observation with respect to a small subgroup of these relations.

Poincare's position is a strong one. He in effect challenges anyone to
point out any factor in nature which gives a preeminent status to the
congruence relation which mankind has actually adopted. But undeniably
the position is very paradoxical. Bertrand Russell had a controversy
with him on this question, and pointed out that on Poincare's principles
there was nothing in nature to determine whether the earth is larger or
smaller than some assigned billiard ball. Poincare replied that the
attempt to find reasons in nature for the selection of a definite
congruence relation in space is like trying to determine the position of
a ship in the ocean by counting the crew and observing the colour of
the captain's eyes.

In my opinion both disputants were right, assuming the grounds on which
the discussion was based. Russell in effect pointed out that apart from
minor inexactitudes a determinate congruence relation is among the
factors in nature which our sense-awareness posits for us. Poincare asks
for information as to the factor in nature which might lead any
particular congruence relation to play a preeminent _role_ among the
factors posited in sense-awareness. I cannot see the answer to either of
these contentions provided that you admit the materialistic theory of
nature. With this theory nature at an instant in space is an independent
fact. Thus we have to look for our preeminent congruence relation amid
nature in instantaneous space; and Poincare is undoubtedly right in
saying that nature on this hypothesis gives us no help in finding it.

On the other hand Russell is in an equally strong position when he
asserts that, as a fact of observation, we do find it, and what is more
agree in finding the same congruence relation. On this basis it is one
of the most extraordinary facts of human experience that all mankind
without any assignable reason should agree in fixing attention on just
one congruence relation amid the indefinite number of indistinguishable
competitors for notice. One would have expected disagreement on this
fundamental choice to have divided nations and to have rent families.
But the difficulty was not even discovered till the close of the
nineteenth century by a few mathematical philosophers and philosophic
mathematicians. The case is not like that of our agreement on some
fundamental fact of nature such as the three dimensions of space. If
space has only three dimensions we should expect all mankind to be aware
of the fact, as they are aware of it. But in the case of congruence,
mankind agree in an arbitrary interpretation of sense-awareness when
there is nothing in nature to guide it.

I look on it as no slight recommendation of the theory of nature which I
am expounding to you that it gives a solution of this difficulty by
pointing out the factor in nature which issues in the preeminence of one
congruence relation over the indefinite herd of other such relations.

The reason for this result is that nature is no longer confined within
space at an instant. Space and time are now interconnected; and this
peculiar factor of time which is so immediately distinguished among the
deliverances of our sense-awareness, relates itself to one particular
congruence relation in space.

Congruence is a particular example of the fundamental fact of
recognition. In perception we recognise. This recognition does not
merely concern the comparison of a factor of nature posited by memory
with a factor posited by immediate sense-awareness. Recognition takes
place within the present without any intervention of pure memory. For
the present fact is a duration with its antecedent and consequent
durations which are parts of itself. The discrimination in
sense-awareness of a finite event with its quality of passage is also
accompanied by the discrimination of other factors of nature which do
not share in the passage of events. Whatever passes is an event. But we
find entities in nature which do not pass; namely we recognise
samenesses in nature. Recognition is not primarily an intellectual act
of comparison; it is in its essence merely sense-awareness in its
capacity of positing before us factors in nature which do not pass. For
example, green is perceived as situated in a certain finite event within
the present duration. This green preserves its self-identity throughout,
whereas the event passes and thereby obtains the property of breaking
into parts. The green patch has parts. But in talking of the green patch
we are speaking of the event in its sole capacity of being for us the
situation of green. The green itself is numerically one self-identical
entity, without parts because it is without passage.

Factors in nature which are without passage will be called objects.
There are radically different kinds of objects which will be considered
in the succeeding lecture.

Recognition is reflected into the intellect as comparison. The
recognised objects of one event are compared with the recognised objects
of another event. The comparison may be between two events in the
present, or it may be between two events of which one is posited by
memory-awareness and the other by immediate sense-awareness. But it is
not the events which are compared. For each event is essentially unique
and incomparable. What are compared are the objects and relations of
objects situated in events. The event considered as a relation between
objects has lost its passage and in this aspect is itself an object.
This object is not the event but only an intellectual abstraction. The
same object can be situated in many events; and in this sense even the
whole event, viewed as an object, can recur, though not the very event
itself with its passage and its relations to other events.

Objects which are not posited by sense-awareness may be known to the
intellect. For example, relations between objects and relations between
relations may be factors in nature not disclosed in sense-awareness but
known by logical inference as necessarily in being. Thus objects for our
knowledge may be merely logical abstractions. For example, a complete
event is never disclosed in sense-awareness, and thus the object which
is the sum total of objects situated in an event as thus inter-related
is a mere abstract concept. Again a right-angle is a perceived object
which can be situated in many events; but, though rectangularity is
posited by sense-awareness, the majority of geometrical relations are
not so posited. Also rectangularity is in fact often not perceived when
it can be proved to have been there for perception. Thus an object is
often known merely as an abstract relation not directly posited in
sense-awareness although it is there in nature.

The identity of quality between congruent segments is generally of this
character. In certain special cases this identity of quality can be
directly perceived. But in general it is inferred by a process of
measurement depending on our direct sense-awareness of selected cases
and a logical inference from the transitive character of congruence.

Congruence depends on motion, and thereby is generated the connexion
between spatial congruence and temporal congruence. Motion along a
straight line has a symmetry round that line. This symmetry is expressed
by the symmetrical geometrical relations of the line to the family of
planes normal to it.

Also another symmetry in the theory of motion arises from the fact that
rest in the points of {beta} corresponds to uniform motion along a
definite family of parallel straight lines in the space of {alpha}. We
must note the three characteristics, (i) of the uniformity of the motion
corresponding to any point of {beta} along its correlated straight line
in {alpha}, and (ii) of the equality in magnitude of the velocities
along the various lines of {alpha} correlated to rest in the various
points of {beta}, and (iii) of the parallelism of the lines of this
family.

We are now in possession of a theory of parallels and a theory of
perpendiculars and a theory of motion, and from these theories the
theory of congruence can be constructed. It will be remembered that a
family of parallel levels in any moment is the family of levels in which
that moment is intersected by the family of moments of some other
time-system. Also a family of parallel moments is the family of moments
of some one time-system. Thus we can enlarge our concept of a family of
parallel levels so as to include levels in different moments of one
time-system. With this enlarged concept we say that a complete family of
parallel levels in a time-system {alpha} is the complete family of
levels in which the moments of {alpha} intersect the moments of {beta}.
This complete family of parallel levels is also evidently a family lying
in the moments of the time-system {beta}. By introducing a third
time-system {gamma}, parallel rects are obtained. Also all the points of
any one time-system form a family of parallel point-tracks. Thus there
are three types of parallelograms in the four-dimensional manifold of
event-particles.

In parallelograms of the first type the two pairs of parallel sides are
both of them pairs of rects. In parallelograms of the second type one
pair of parallel sides is a pair of rects and the other pair is a pair
of point-tracks. In parallelograms of the third type the two pairs of
parallel sides are both of them pairs of point-tracks.

The first axiom of congruence is that the opposite sides of any
parallelogram are congruent. This axiom enables us to compare the
lengths of any two segments either respectively on parallel rects or on
the same rect. Also it enables us to compare the lengths of any two
segments either respectively on parallel point-tracks or on the same
point-track. It follows from this axiom that two objects at rest in any
two points of a time-system {beta} are moving with equal velocities in any
other time-system {alpha} along parallel lines. Thus we can speak of the
velocity in {alpha} due to the time-system {beta} without specifying any
particular point in {beta}. The axiom also enables us to measure time in
any time-system; but does not enable us to compare times in different
time-systems.

The second axiom of congruence concerns parallelograms on congruent
bases and between the same parallels, which have also their other pairs
of sides parallel. The axiom asserts that the rect joining the two
event-particles of intersection of the diagonals is parallel to the rect
on which the bases lie. By the aid of this axiom it easily follows that
the diagonals of a parallelogram bisect each other.

Congruence is extended in any space beyond parallel rects to all rects
by two axioms depending on perpendicularity. The first of these axioms,
which is the third axiom of congruence, is that if ABC is a triangle
of rects in any moment and D is the middle event-particle of the base
BC, then the level through D perpendicular to BC contains A when
and only when AB is congruent to AC. This axiom evidently expresses
the symmetry of perpendicularity, and is the essence of the famous pons
asinorum expressed as an axiom.

The second axiom depending on perpendicularity, and the fourth axiom of
congruence, is that if r and A be a rect and an event-particle in the
same moment and AB and AC be a pair of rectangular rects intersecting r
in B and C, and AD and AE be another pair of rectangular rects
intersecting r in D and E, then either D or E lies in the segment BC and
the other one of the two does not lie in this segment. Also as a
particular case of this axiom, if AB be perpendicular to r and in
consequence AC be parallel to r, then D and E lie on opposite sides of B
respectively. By the aid of these two axioms the theory of congruence
can be extended so as to compare lengths of segments on any two rects.
Accordingly Euclidean metrical geometry in space is completely
established and lengths in the spaces of different time-systems are
comparable as the result of definite properties of nature which indicate
just that particular method of comparison.

The comparison of time-measurements in diverse time-systems requires two
other axioms. The first of these axioms, forming the fifth axiom of
congruence, will be called the axiom of 'kinetic symmetry.' It expresses
the symmetry of the quantitative relations between two time-systems when
the times and lengths in the two systems are measured in congruent
units.

The axiom can be explained as follows: Let {alpha} and {beta} be the names
of two time-systems. The directions of motion in the space of {alpha} due
to rest in a point of {beta} is called the '{beta}-direction in {alpha}'
and the direction of motion in the space of {beta} due to rest in a
point of {alpha} is called the '{alpha}-direction in {beta}.' Consider a
motion in the space of {alpha} consisting of a certain velocity in the
{beta}-direction of {alpha} and a certain velocity at right-angles to
it. This motion represents rest in the space of another time-system--call
it {pi}. Rest in {pi} will also be represented in the space of {beta} by
a certain velocity in the {alpha}-direction in {beta} and a certain
velocity at right-angles to this {alpha}-direction. Thus a certain
motion in the space of {alpha} is correlated to a certain motion in the
space of {beta}, as both representing the same fact which can also be
represented by rest in {pi}. Now another time-system, which I will name
{sigma}, can be found which is such that rest in its space is
represented by the same magnitudes of velocities along and perpendicular
to the {alpha}-direction in {beta} as those velocities in {alpha}, along
and perpendicular to the {beta}-direction, which represent rest in {pi}.
The required axiom of kinetic symmetry is that rest in {sigma} will be
represented in {alpha} by the same velocities along and perpendicular to
the {beta}-direction in {alpha} as those velocities in {beta} along and
perpendicular to the {alpha}-direction which represent rest in {pi}.

A particular case of this axiom is that relative velocities are equal
and opposite. Namely rest in {alpha} is represented in {beta} by a
velocity along the {alpha}-direction which is equal to the velocity
along the {beta}-direction in {alpha} which represents rest in {beta}.

Finally the sixth axiom of congruence is that the relation of congruence
is transitive. So far as this axiom applies to space, it is superfluous.
For the property follows from our previous axioms. It is however
necessary for time as a supplement to the axiom of kinetic symmetry. The
meaning of the axiom is that if the time-unit of system {alpha} is
congruent to the time-unit of system {beta}, and the time-unit of system
{beta} is congruent to the time-unit of system {gamma}, then the
time-units of {alpha} and {gamma} are also congruent.

By means of these axioms formulae for the transformation of
measurements made in one time-system to measurements of the same facts
of nature made in another time-system can be deduced. These formulae
will be found to involve one arbitrary constant which I will call k.

It is of the dimensions of the square of a velocity. Accordingly four
cases arise. In the first case k is zero. This case produces
nonsensical results in opposition to the elementary deliverances of
experience. We put this case aside.

In the second case k is infinite. This case yields the ordinary
formulae for transformation in relative motion, namely those formulae
which are to be found in every elementary book on dynamics.

In the third case, k is negative. Let us call it -c squared, where c
will be of the dimensions of a velocity. This case yields the formulae
of transformation which Larmor discovered for the transformation of
Maxwell's equations of the electromagnetic field. These formulae were
extended by H. A. Lorentz, and used by Einstein and Minkowski as the
basis of their novel theory of relativity. I am not now speaking of
Einstein's more recent theory of general relativity by which he deduces
his modification of the law of gravitation. If this be the case which
applies to nature, then c must be a close approximation to the
velocity of light _in vacuo_. Perhaps it is this actual velocity. In
this connexion '_in vacuo_' must not mean an absence of events, namely
the absence of the all-pervading ether of events. It must mean the
absence of certain types of objects.

In the fourth case, k is positive. Let us call it h squared, where h
will be of the dimensions of a velocity. This gives a perfectly possible
type of transformation formulae, but not one which explains any facts
of experience. It has also another disadvantage. With the assumption of
this fourth case the distinction between space and time becomes unduly
blurred. The whole object of these lectures has been to enforce the
doctrine that space and time spring from a common root, and that the
ultimate fact of experience is a space-time fact. But after all mankind
does distinguish very sharply between space and time, and it is owing to
this sharpness of distinction that the doctrine of these lectures is
somewhat of a paradox. Now in the third assumption this sharpness of
distinction is adequately preserved. There is a fundamental distinction
between the metrical properties of point-tracks and rects. But in the
fourth assumption this fundamental distinction vanishes.

Neither the third nor the fourth assumption can agree with experience
unless we assume that the velocity c of the third assumption, and the
velocity h of the fourth assumption, are extremely large compared to
the velocities of ordinary experience. If this be the case the formulae
of both assumptions will obviously reduce to a close approximation to
the formulae of the second assumption which are the ordinary formulae of
dynamical textbooks. For the sake of a name, I will call these textbook
formulae the 'orthodox' formulae.

There can be no question as to the general approximate correctness of
the orthodox formulae. It would be merely silly to raise doubts on this
point. But the determination of the status of these formulae is by no
means settled by this admission. The independence of time and space is
an unquestioned presupposition of the orthodox thought which has
produced the orthodox formulae. With this presupposition and given the
absolute points of one absolute space, the orthodox formulae are
immediate deductions. Accordingly, these formulae are presented to our
imaginations as facts which cannot be otherwise, time and space being
what they are. The orthodox formulae have therefore attained to the
status of necessities which cannot be questioned in science. Any attempt
to replace these formulae by others was to abandon the _role_ of
physical explanation and to have recourse to mere mathematical formulae.

But even in physical science difficulties have accumulated round the
orthodox formulae. In the first place Maxwell's equations of the
electromagnetic field are not invariant for the transformations of the
orthodox formulae; whereas they are invariant for the transformations of
the formulae arising from the third of the four cases mentioned above,
provided that the velocity c is identified with a famous
electromagnetic constant quantity.

Again the null results of the delicate experiments to detect the earth's
variations of motion through the ether in its orbital path are explained
immediately by the formulae of the third case. But if we assume the
orthodox formulae we have to make a special and arbitrary assumption as
to the contraction of matter during motion. I mean the Fitzgerald-Lorentz
assumption.

Lastly Fresnel's coefficient of drag which represents the variation of
the velocity of light in a moving medium is explained by the formulae of
the third case, and requires another arbitrary assumption if we use the
orthodox formulae.

It appears therefore that on the mere basis of physical explanation
there are advantages in the formulae of the third case as compared with
the orthodox formulae. But the way is blocked by the ingrained belief
that these latter formulae possess a character of necessity. It is
therefore an urgent requisite for physical science and for philosophy to
examine critically the grounds for this supposed necessity. The only
satisfactory method of scrutiny is to recur to the first principles of
our knowledge of nature. This is exactly what I am endeavouring to do in
these lectures. I ask what it is that we are aware of in our
sense-perception of nature. I then proceed to examine those factors in
nature which lead us to conceive nature as occupying space and
persisting through time. This procedure has led us to an investigation
of the characters of space and time. It results from these
investigations that the formulae of the third case and the orthodox
formulae are on a level as possible formulae resulting from the basic
character of our knowledge of nature. The orthodox formulae have thus
lost any advantage as to necessity which they enjoyed over the serial
group. The way is thus open to adopt whichever of the two groups best
accords with observation.

I take this opportunity of pausing for a moment from the course of my
argument, and of reflecting on the general character which my doctrine
ascribes to some familiar concepts of science. I have no doubt that some
of you have felt that in certain aspects this character is very
paradoxical.

This vein of paradox is partly due to the fact that educated language
has been made to conform to the prevalent orthodox theory. We are thus,
in expounding an alternative doctrine, driven to the use of either
strange terms or of familiar words with unusual meanings. This victory
of the orthodox theory over language is very natural. Events are named
after the prominent objects situated in them, and thus both in language
and in thought the event sinks behind the object, and becomes the mere
play of its relations. The theory of space is then converted into a
theory of the relations of objects instead of a theory of the relations
of events. But objects have not the passage of events. Accordingly space
as a relation between objects is devoid of any connexion with time. It
is space at an instant without any determinate relations between the
spaces at successive instants. It cannot be one timeless space because
the relations between objects change.

A few minutes ago in speaking of the deduction of the orthodox formulae
for relative motion I said that they followed as an immediate deduction
from the assumption of absolute points in absolute space. This reference
to absolute space was not an oversight. I know that the doctrine of the
relativity of space at present holds the field both in science and
philosophy. But I do not think that its inevitable consequences are
understood. When we really face them the paradox of the presentation of
the character of space which I have elaborated is greatly mitigated. If
there is no absolute position, a point must cease to be a simple entity.
What is a point to one man in a balloon with his eyes fixed on an
instrument is a track of points to an observer on the earth who is
watching the balloon through a telescope, and is another track of points
to an observer in the sun who is watching the balloon through some
instrument suited to such a being. Accordingly if I am reproached with
the paradox of my theory of points as classes of event-particles, and of
my theory of event-particles as groups of abstractive sets, I ask my
critic to explain exactly what he means by a point. While you explain
your meaning about anything, however simple, it is always apt to look
subtle and fine spun. I have at least explained exactly what I do mean
by a point, what relations it involves and what entities are the relata.
If you admit the relativity of space, you also must admit that points
are complex entities, logical constructs involving other entities and
their relations. Produce your theory, not in a few vague phrases of
indefinite meaning, but explain it step by step in definite terms
referring to assigned relations and assigned relata. Also show that your
theory of points issues in a theory of space. Furthermore note that the
example of the man in the balloon, the observer on earth, and the
observer in the sun, shows that every assumption of relative rest
requires a timeless space with radically different points from those
which issue from every other such assumption. The theory of the
relativity of space is inconsistent with any doctrine of one unique set
of points of one timeless space.

The fact is that there is no paradox in my doctrine of the nature of
space which is not in essence inherent in the theory of the relativity
of space. But this doctrine has never really been accepted in science,
whatever people say. What appears in our dynamical treatises is Newton's
doctrine of relative motion based on the doctrine of differential motion
in absolute space. When you once admit that the points are radically
different entities for differing assumptions of rest, then the orthodox
formulae lose all their obviousness. They were only obvious because you
were really thinking of something else. When discussing this topic you
can only avoid paradox by taking refuge from the flood of criticism in
the comfortable ark of no meaning.

The new theory provides a definition of the congruence of periods of
time. The prevalent view provides no such definition. Its position is
that if we take such time-measurements so that certain familiar
velocities which seem to us to be uniform are uniform, then the laws of
motion are true. Now in the first place no change could appear either as
uniform or non-uniform without involving a definite determination of the
congruence for time-periods. So in appealing to familiar phenomena it
allows that there is some factor in nature which we can intellectually
construct as a congruence theory. It does not however say anything about
it except that the laws of motion are then true. Suppose that with some
expositors we cut out the reference to familiar velocities such as the
rate of rotation of the earth. We are then driven to admit that there is
no meaning in temporal congruence except that certain assumptions make
the laws of motion true. Such a statement is historically false. King
Alfred the Great was ignorant of the laws of motion, but knew very well
what he meant by the measurement of time, and achieved his purpose by
means of burning candles. Also no one in past ages justified the use of
sand in hour-glasses by saying that some centuries later interesting
laws of motion would be discovered which would give a meaning to the
statement that the sand was emptied from the bulbs in equal times.
Uniformity in change is directly perceived, and it follows that mankind
perceives in nature factors from which a theory of temporal congruence
can be formed. The prevalent theory entirely fails to produce such
factors.

The mention of the laws of motion raises another point where the
prevalent theory has nothing to say and the new theory gives a complete
explanation. It is well known that the laws of motion are not valid for
any axes of reference which you may choose to take fixed in any rigid
body. You must choose a body which is not rotating and has no
acceleration. For example they do not really apply to axes fixed in the
earth because of the diurnal rotation of that body. The law which fails
when you assume the wrong axes as at rest is the third law, that action
and reaction are equal and opposite. With the wrong axes uncompensated
centrifugal forces and uncompensated composite centrifugal forces
appear, due to rotation. The influence of these forces can be
demonstrated by many facts on the earth's surface, Foucault's pendulum,
the shape of the earth, the fixed directions of the rotations of
cyclones and anticyclones. It is difficult to take seriously the
suggestion that these domestic phenomena on the earth are due to the
influence of the fixed stars. I cannot persuade myself to believe that a
little star in its twinkling turned round Foucault's pendulum in the
Paris Exhibition of 1861. Of course anything is believable when a
definite physical connexion has been demonstrated, for example the
influence of sunspots. Here all demonstration is lacking in the form of
any coherent theory. According to the theory of these lectures the axes
to which motion is to be referred are axes at rest in the space of some
time-system. For example, consider the space of a time-system {alpha}.
There are sets of axes at rest in the space of {alpha}. These are
suitable dynamical axes. Also a set of axes in this space which is
moving with uniform velocity without rotation is another suitable set.
All the moving points fixed in these moving axes are really tracing out
parallel lines with one uniform velocity. In other words they are the
reflections in the space of {alpha} of a set of fixed axes in the space
of some other time-system {beta}. Accordingly the group of dynamical
axes required for Newton's Laws of Motion is the outcome of the
necessity of referring motion to a body at rest in the space of some one
time-system in order to obtain a coherent account of physical
properties. If we do not do so the meaning of the motion of one portion
of our physical configuration is different from the meaning of the
motion of another portion of the same configuration. Thus the meaning of
motion being what it is, in order to describe the motion of any system
of objects without changing the meaning of your terms as you proceed
with your description, you are bound to take one of these sets of axes
as axes of reference; though you may choose their reflections into the
space of any time-system which you wish to adopt. A definite physical
reason is thereby assigned for the peculiar property of the dynamical
group of axes.

On the orthodox theory the position of the equations of motion is most
ambiguous. The space to which they refer is completely undetermined and
so is the measurement of the lapse of time. Science is simply setting
out on a fishing expedition to see whether it cannot find some procedure
which it can call the measurement of space and some procedure which it
can call the measurement of time, and something which it can call a
system of forces, and something which it can call masses, so that these
formulae may be satisfied. The only reason--on this theory--why anyone
should want to satisfy these formulae is a sentimental regard for
Galileo, Newton, Euler and Lagrange. The theory, so far from founding
science on a sound observational basis, forces everything to conform to
a mere mathematical preference for certain simple formulae.

I do not for a moment believe that this is a true account of the real
status of the Laws of Motion. These equations want some slight
adjustment for the new formulae of relativity. But with these
adjustments, imperceptible in ordinary use, the laws deal with
fundamental physical quantities which we know very well and wish to
correlate.

The measurement of time was known to all civilised nations long before
the laws were thought of. It is this time as thus measured that the laws
are concerned with. Also they deal with the space of our daily life.
When we approach to an accuracy of measurement beyond that of
observation, adjustment is allowable. But within the limits of
observation we know what we mean when we speak of measurements of space
and measurements of time and uniformity of change. It is for science to
give an intellectual account of what is so evident in sense-awareness.
It is to me thoroughly incredible that the ultimate fact beyond which
there is no deeper explanation is that mankind has really been swayed by
an unconscious desire to satisfy the mathematical formulae which we call
the Laws of Motion, formulae completely unknown till the seventeenth
century of our epoch.

The correlation of the facts of sense-experience effected by the
alternative account of nature extends beyond the physical properties of
motion and the properties of congruence. It gives an account of the
meaning of the geometrical entities such as points, straight lines, and
volumes, and connects the kindred ideas of extension in time and
extension in space. The theory satisfies the true purpose of an
intellectual explanation in the sphere of natural philosophy. This
purpose is to exhibit the interconnexions of nature, and to show that
one set of ingredients in nature requires for the exhibition of its
character the presence of the other sets of ingredients.

The false idea which we have to get rid of is that of nature as a mere
aggregate of independent entities, each capable of isolation. According
to this conception these entities, whose characters are capable of
isolated definition, come together and by their accidental relations
form the system of nature. This system is thus thoroughly accidental;
and, even if it be subject to a mechanical fate, it is only accidentally
so subject.

With this theory space might be without time, and time might be without
space. The theory admittedly breaks down when we come to the relations
of matter and space. The relational theory of space is an admission that
we cannot know space without matter or matter without space. But the
seclusion of both from time is still jealously guarded. The relations
between portions of matter in space are accidental facts owing to the
absence of any coherent account of how space springs from matter or how
matter springs from space. Also what we really observe in nature, its
colours and its sounds and its touches are secondary qualities; in other
words, they are not in nature at all but are accidental products of the
relations between nature and mind.

The explanation of nature which I urge as an alternative ideal to this
accidental view of nature, is that nothing in nature could be what it is
except as an ingredient in nature as it is. The whole which is present
for discrimination is posited in sense-awareness as necessary for the
discriminated parts. An isolated event is not an event, because every
event is a factor in a larger whole and is significant of that whole.
There can be no time apart from space; and no space apart from time; and
no space and no time apart from the passage of the events of nature. The
isolation of an entity in thought, when we think of it as a bare 'it,'
has no counterpart in any corresponding isolation in nature. Such
isolation is merely part of the procedure of intellectual knowledge.

The laws of nature are the outcome of the characters of the entities
which we find in nature. The entities being what they are, the laws must
be what they are; and conversely the entities follow from the laws. We
are a long way from the attainment of such an ideal; but it remains as
the abiding goal of theoretical science.




CHAPTER VII

OBJECTS


The ensuing lecture is concerned with the theory of objects. Objects are
elements in nature which do not pass. The awareness of an object as some
factor not sharing in the passage of nature is what I call
'recognition.' It is impossible to recognise an event, because an event
is essentially distinct from every other event. Recognition is an
awareness of sameness. But to call recognition an awareness of sameness
implies an intellectual act of comparison accompanied with judgment. I
use recognition for the non-intellectual relation of sense-awareness
which connects the mind with a factor of nature without passage. On the
intellectual side of the mind's experience there are comparisons of
things recognised and consequent judgments of sameness or diversity.
Probably 'sense-recognition' would be a better term for what I mean by
'recognition.' I have chosen the simpler term because I think that I
shall be able to avoid the use of 'recognition' in any other meaning
than that of 'sense-recognition.' I am quite willing to believe that
recognition, in my sense of the term, is merely an ideal limit, and that
there is in fact no recognition without intellectual accompaniments of
comparison and judgment. But recognition is that relation of the mind to
nature which provides the material for the intellectual activity.

An object is an ingredient in the character of some event. In fact the
character of an event is nothing but the objects which are ingredient in
it and the ways in which those objects make their ingression into the
event. Thus the theory of objects is the theory of the comparison of
events. Events are only comparable because they body forth permanences.
We are comparing objects in events whenever we can say, 'There it is
again.' Objects are the elements in nature which can 'be again.'

Sometimes permanences can be proved to exist which evade recognition in
the sense in which I am using that term. The permanences which evade
recognition appear to us as abstract properties either of events or of
objects. All the same they are there for recognition although
undiscriminated in our sense-awareness. The demarcation of events, the
splitting of nature up into parts is effected by the objects which we
recognise as their ingredients. The discrimination of nature is the
recognition of objects amid passing events. It is a compound of the
awareness of the passage of nature, of the consequent partition of
nature, and of the definition of certain parts of nature by the modes of
the ingression of objects into them.

You may have noticed that I am using the term 'ingression' to denote the
general relation of objects to events. The ingression of an object into
an event is the way the character of the event shapes itself in virtue
of the being of the object. Namely the event is what it is, because the
object is what it is; and when I am thinking of this modification of the
event by the object, I call the relation between the two 'the ingression
of the object into the event.' It is equally true to say that objects
are what they are because events are what they are. Nature is such that
there can be no events and no objects without the ingression of objects
into events. Although there are events such that the ingredient objects
evade our recognition. These are the events in empty space. Such events
are only analysed for us by the intellectual probing of science.

Ingression is a relation which has various modes. There are obviously
very various kinds of objects; and no one kind of object can have the
same sort of relations to events as objects of another kind can have. We
shall have to analyse out some of the different modes of ingression
which different kinds of objects have into events.

But even if we stick to one and the same kind of objects, an object of
that kind has different modes of ingression into different events.
Science and philosophy have been apt to entangle themselves in a
simple-minded theory that an object is at one place at any definite
time, and is in no sense anywhere else. This is in fact the attitude of
common sense thought, though it is not the attitude of language which is
naively expressing the facts of experience. Every other sentence in a
work of literature which is endeavouring truly to interpret the facts of
experience expresses differences in surrounding events due to the
presence of some object. An object is ingredient throughout its
neighbourhood, and its neighbourhood is indefinite. Also the
modification of events by ingression is susceptible of quantitative
differences. Finally therefore we are driven to admit that each object
is in some sense ingredient throughout nature; though its ingression may
be quantitatively irrelevant in the expression of our individual
experiences.

This admission is not new either in philosophy or science. It is
obviously a necessary axiom for those philosophers who insist that
reality is a system. In these lectures we are keeping off the profound
and vexed question as to what we mean by 'reality.' I am maintaining the
humbler thesis that nature is a system. But I suppose that in this case
the less follows from the greater, and that I may claim the support of
these philosophers. The same doctrine is essentially interwoven in all
modern physical speculation. As long ago as 1847 Faraday in a paper in
the _Philosophical Magazine_ remarked that his theory of tubes of force
implies that in a sense an electric charge is everywhere. The
modification of the electromagnetic field at every point of space at
each instant owing to the past history of each electron is another way
of stating the same fact. We can however illustrate the doctrine by the
more familiar facts of life without recourse to the abstruse
speculations of theoretical physics.

The waves as they roll on to the Cornish coast tell of a gale in
mid-Atlantic; and our dinner witnesses to the ingression of the cook
into the dining room. It is evident that the ingression of objects into
events includes the theory of causation. I prefer to neglect this aspect
of ingression, because causation raises the memory of discussions based
upon theories of nature which are alien to my own. Also I think that
some new light may be thrown on the subject by viewing it in this fresh
aspect.

The examples which I have given of the ingression of objects into events
remind us that ingression takes a peculiar form in the case of some
events; in a sense, it is a more concentrated form. For example, the
electron has a certain position in space and a certain shape. Perhaps it
is an extremely small sphere in a certain test-tube. The storm is a
gale situated in mid-Atlantic with a certain latitude and longitude, and
the cook is in the kitchen. I will call this special form of ingression
the 'relation of situation'; also, by a double use of the word
'situation,' I will call the event in which an object is situated 'the
situation of the object.' Thus a situation is an event which is a
relatum in the relation of situation. Now our first impression is that
at last we have come to the simple plain fact of where the object really
is; and that the vaguer relation which I call ingression should not be
muddled up with the relation of situation, as if including it as a
particular case. It seems so obvious that any object is in such and such
a position, and that it is influencing other events in a totally
different sense. Namely, in a sense an object is the character of the
event which is its situation, but it only influences the character of
other events. Accordingly the relations of situation and influencing are
not generally the same sort of relation, and should not be subsumed
under the same term 'ingression.' I believe that this notion is a
mistake, and that it is impossible to draw a clear distinction between
the two relations.

For example, Where was your toothache? You went to a dentist and pointed
out the tooth to him. He pronounced it perfectly sound, and cured you by
stopping another tooth. Which tooth was the situation of the toothache?
Again, a man has an arm amputated, and experiences sensations in the
hand which he has lost. The situation of the imaginary hand is in fact
merely thin air. You look into a mirror and see a fire. The flames that
you see are situated behind the mirror. Again at night you watch the
sky; if some of the stars had vanished from existence hours ago, you
would not be any the wiser. Even the situations of the planets differ
from those which science would assign to them.

Anyhow you are tempted to exclaim, the cook is in the kitchen. If you
mean her mind, I will not agree with you on the point; for I am only
talking of nature. Let us think only of her bodily presence. What do you
mean by this notion? We confine ourselves to typical manifestations of
it. You can see her, touch her, and hear her. But the examples which I
have given you show that the notions of the situations of what you see,
what you touch, and what you hear are not so sharply separated out as to
defy further questioning. You cannot cling to the idea that we have two
sets of experiences of nature, one of primary qualities which belong to
the objects perceived, and one of secondary qualities which are the
products of our mental excitements. All we know of nature is in the same
boat, to sink or swim together. The constructions of science are merely
expositions of the characters of things perceived. Accordingly to affirm
that the cook is a certain dance of molecules and electrons is merely to
affirm that the things about her which are perceivable have certain
characters. The situations of the perceived manifestations of her bodily
presence have only a very general relation to the situations of the
molecules, to be determined by discussion of the circumstances of
perception.

In discussing the relations of situation in particular and of ingression
in general, the first requisite is to note that objects are of radically
different types. For each type 'situation' and 'ingression' have their
own special meanings which are different from their meanings for other
types, though connexions can be pointed out. It is necessary therefore
in discussing them to determine what type of objects are under
consideration. There are, I think, an indefinite number of types of
objects. Happily we need not think of them all. The idea of situation
has its peculiar importance in reference to three types of objects which
I call sense-objects, perceptual objects and scientific objects. The
suitability of these names for the three types is of minor importance,
so long as I can succeed in explaining what I mean by them.

These three types form an ascending hierarchy, of which each member
presupposes the type below. The base of the hierarchy is formed by the
sense-objects. These objects do not presuppose any other type of
objects. A sense-object is a factor of nature posited by sense-awareness
which (i), in that it is an object, does not share in the passage of
nature and (ii) is not a relation between other factors of nature. It
will of course be a relatum in relations which also implicate other
factors of nature. But it is always a relatum and never the relation
itself. Examples of sense-objects are a particular sort of colour, say
Cambridge blue, or a particular sort of sound, or a particular sort of
smell, or a particular sort of feeling. I am not talking of a particular
patch of blue as seen during a particular second of time at a definite
date. Such a patch is an event where Cambridge blue is situated.
Similarly I am not talking of any particular concert-room as filled with
the note. I mean the note itself and not the patch of volume filled by
the sound for a tenth of a second. It is natural for us to think of the
note in itself, but in the case of colour we are apt to think of it
merely as a property of the patch. No one thinks of the note as a
property of the concert-room. We see the blue and we hear the note. Both
the blue and the note are immediately posited by the discrimination of
sense-awareness which relates the mind to nature. The blue is posited as
in nature related to other factors in nature. In particular it is
posited as in the relation of being situated in the event which is its
situation.

The difficulties which cluster around the relation of situation arise
from the obstinate refusal of philosophers to take seriously the
ultimate fact of multiple relations. By a multiple relation I mean a
relation which in any concrete instance of its occurrence necessarily
involves more than two relata. For example, when John likes Thomas there
are only two relata, John and Thomas. But when John gives that book to
Thomas there are three relata, John, that book, and Thomas.

Some schools of philosophy, under the influence of the Aristotelian
logic and the Aristotelian philosophy, endeavour to get on without
admitting any relations at all except that of substance and attribute.
Namely all apparent relations are to be resolvable into the concurrent
existence of substances with contrasted attributes. It is fairly obvious
that the Leibnizian monadology is the necessary outcome of any such
philosophy. If you dislike pluralism, there will be only one monad.

Other schools of philosophy admit relations but obstinately refuse to
contemplate relations with more than two relata. I do not think that
this limitation is based on any set purpose or theory. It merely arises
from the fact that more complicated relations are a bother to people
without adequate mathematical training, when they are admitted into the
reasoning.

I must repeat that we have nothing to do in these lectures with the
ultimate character of reality. It is quite possible that in the true
philosophy of reality there are only individual substances with
attributes, or that there are only relations with pairs of relata. I do
not believe that such is the case; but I am not concerned to argue about
it now. Our theme is Nature. So long as we confine ourselves to the
factors posited in the sense-awareness of nature, it seems to me that
there certainly are instances of multiple relations between these
factors, and that the relation of situation for sense-objects is one
example of such multiple relations.

Consider a blue coat, a flannel coat of Cambridge blue belonging to some
athlete. The coat itself is a perceptual object and its situation is not
what I am talking about. We are talking of someone's definite
sense-awareness of Cambridge blue as situated in some event of nature.
He may be looking at the coat directly. He then sees Cambridge blue as
situated practically in the same event as the coat at that instant. It
is true that the blue which he sees is due to light which left the coat
some inconceivably small fraction of a second before. This difference
would be important if he were looking at a star whose colour was
Cambridge blue. The star might have ceased to exist days ago, or even
years ago. The situation of the blue will not then be very intimately
connected with the situation (in another sense of 'situation') of any
perceptual object. This disconnexion of the situation of the blue and
the situation of some associated perceptual object does not require a
star for its exemplification. Any looking glass will suffice. Look at
the coat through a looking glass. Then blue is seen as situated behind
the mirror. The event which is its situation depends upon the position
of the observer.

The sense-awareness of the blue as situated in a certain event which I
call the situation, is thus exhibited as the sense-awareness of a
relation between the blue, the percipient event of the observer, the
situation, and intervening events. All nature is in fact required,
though only certain intervening events require their characters to be of
certain definite sorts. The ingression of blue into the events of nature
is thus exhibited as systematically correlated. The awareness of the
observer depends on the position of the percipient event in this
systematic correlation. I will use the term 'ingression into nature' for
this systematic correlation of the blue with nature. Thus the ingression
of blue into any definite event is a part statement of the fact of the
ingression of blue into nature.

In respect to the ingression of blue into nature events may be roughly
put into four classes which overlap and are not very clearly separated.
These classes are (i) the percipient events, (ii) the situations, (iii)
the active conditioning events, (iv) the passive conditioning events. To
understand this classification of events in the general fact of the
ingression of blue into nature, let us confine attention to one
situation for one percipient event and to the consequent _roles_ of the
conditioning events for the ingression as thus limited. The percipient
event is the relevant bodily state of the observer. The situation is
where he sees the blue, say, behind the mirror. The active conditioning
events are the events whose characters are particularly relevant for the
event (which is the situation) to be the situation for that percipient
event, namely the coat, the mirror, and the state of the room as to
light and atmosphere. The passive conditioning events are the events of
the rest of nature.

In general the situation is an active conditioning event; namely the
coat itself, when there is no mirror or other such contrivance to
produce abnormal effects. But the example of the mirror shows us that
the situation may be one of the passive conditioning events. We are then
apt to say that our senses have been cheated, because we demand as a
right that the situation should be an active condition in the
ingression.

This demand is not so baseless as it may seem when presented as I have
put it. All we know of the characters of the events of nature is based
on the analysis of the relations of situations to percipient events. If
situations were not in general active conditions, this analysis would
tell us nothing. Nature would be an unfathomable enigma to us and there
could be no science. Accordingly the incipient discontent when a
situation is found to be a passive condition is in a sense justifiable;
because if that sort of thing went on too often, the _role_ of the
intellect would be ended.

Furthermore the mirror is itself the situation of other sense-objects
either for the same observer with the same percipient event, or for
other observers with other percipient events. Thus the fact that an
event is a situation in the ingression of one set of sense-objects into
nature is presumptive evidence that that event is an active condition in
the ingression of other sense-objects into nature which may have other
situations.

This is a fundamental principle of science which it has derived from
common sense.

I now turn to perceptual objects. When we look at the coat, we do not in
general say, There is a patch of Cambridge blue; what naturally occurs
to us is, There is a coat. Also the judgment that what we have seen is
a garment of man's attire is a detail. What we perceive is an object
other than a mere sense-object. It is not a mere patch of colour, but
something more; and it is that something more which we judge to be a
coat. I will use the word 'coat' as the name for that crude object which
is more than a patch of colour, and without any allusion to the
judgments as to its usefulness as an article of attire either in the
past or the future. The coat which is perceived--in this sense of the
word 'coat'--is what I call a perceptual object. We have to investigate
the general character of these perceptual objects.

It is a law of nature that in general the situation of a sense-object is
not only the situation of that sense-object for one definite percipient
event, but is the situation of a variety of sense-objects for a variety
of percipient events. For example, for any one percipient event, the
situation of a sense-object of sight is apt also to be the situations of
sense-objects of sight, of touch, of smell, and of sound. Furthermore
this concurrence in the situations of sense-objects has led to the
body--_i.e._ the percipient event--so adapting itself that the
perception of one sense-object in a certain situation leads to a
subconscious sense-awareness of other sense-objects in the same
situation. This interplay is especially the case between touch and
sight. There is a certain correlation between the ingressions of
sense-objects of touch and sense-objects of sight into nature, and in a
slighter degree between the ingressions of other pairs of sense-objects.
I call this sort of correlation the 'conveyance' of one sense-object by
another. When you see the blue flannel coat you subconsciously feel
yourself wearing it or otherwise touching it. If you are a smoker, you
may also subconsciously be aware of the faint aroma of tobacco. The
peculiar fact, posited by this sense-awareness of the concurrence of
subconscious sense-objects along with one or more dominating
sense-objects in the same situation, is the sense-awareness of the
perceptual object. The perceptual object is not primarily the issue of a
judgment. It is a factor of nature directly posited in sense-awareness.
The element of judgment comes in when we proceed to classify the
particular perceptual object. For example, we say, That is flannel, and
we think of the properties of flannel and the uses of athletes' coats.
But that all takes place after we have got hold of the perceptual
object. Anticipatory judgments affect the perceptual object perceived by
focussing and diverting attention.

The perceptual object is the outcome of the habit of experience.
Anything which conflicts with this habit hinders the sense-awareness of
such an object. A sense-object is not the product of the association of
intellectual ideas; it is the product of the association of
sense-objects in the same situation. This outcome is not intellectual;
it is an object of peculiar type with its own particular ingression into
nature.

There are two kinds of perceptual objects, namely, 'delusive perceptual
objects' and 'physical objects.' The situation of a delusive perceptual
object is a passive condition in the ingression of that object into
nature. Also the event which is the situation will have the relation of
situation to the object only for one particular percipient event. For
example, an observer sees the image of the blue coat in a mirror. It is
a blue coat that he sees and not a mere patch of colour. This shows that
the active conditions for the conveyance of a group of subconscious
sense-objects by a dominating sense-object are to be found in the
percipient event. Namely we are to look for them in the investigations
of medical psychologists. The ingression into nature of the delusive
sense-object is conditioned by the adaptation of bodily events to the
more normal occurrence, which is the ingression of the physical object.

A perceptual object is a physical object when (i) its situation is an
active conditioning event for the ingression of any of its component
sense-objects, and (ii) the same event can be the situation of the
perceptual object for an indefinite number of possible percipient
events. Physical objects are the ordinary objects which we perceive when
our senses are not cheated, such as chairs, tables and trees. In a way
physical objects have more insistent perceptive power than
sense-objects. Attention to the fact of their occurrence in nature is
the first condition for the survival of complex living organisms. The
result of this high perceptive power of physical objects is the
scholastic philosophy of nature which looks on the sense-objects as mere
attributes of the physical objects. This scholastic point of view is
directly contradicted by the wealth of sense-objects which enter into
our experience as situated in events without any connexion with physical
objects. For example, stray smells, sounds, colours and more subtle
nameless sense-objects. There is no perception of physical objects
without perception of sense-objects. But the converse does not hold:
namely, there is abundant perception of sense-objects unaccompanied by
any perception of physical objects. This lack of reciprocity in the
relations between sense-objects and physical objects is fatal to the
scholastic natural philosophy.

There is a great difference in the _roles_ of the situations of
sense-objects and physical objects. The situations of a physical object
are conditioned by uniqueness and continuity. The uniqueness is an ideal
limit to which we approximate as we proceed in thought along an
abstractive set of durations, considering smaller and smaller durations
in the approach to the ideal limit of the moment of time. In other
words, when the duration is small enough, the situation of the physical
object within that duration is practically unique.

The identification of the same physical object as being situated in
distinct events in distinct durations is effected by the condition of
continuity. This condition of continuity is the condition that a
continuity of passage of events, each event being a situation of the
object in its corresponding duration, can be found from the earlier to
the later of the two given events. So far as the two events are
practically adjacent in one specious present, this continuity of passage
may be directly perceived. Otherwise it is a matter of judgment and
inference.

The situations of a sense-object are not conditioned by any such
conditions either of uniqueness or of continuity. In any durations
however small a sense-object may have any number of situations separated
from each other. Thus two situations of a sense-object, either in the
same duration or in different durations, are not necessarily connected
by any continuous passage of events which are also situations of that
sense-object.

The characters of the conditioning events involved in the ingression of
a sense-object into nature can be largely expressed in terms of the
physical objects which are situated in those events. In one respect this
is also a tautology. For the physical object is nothing else than the
habitual concurrence of a certain set of sense-objects in one situation.
Accordingly when we know all about the physical object, we thereby know
its component sense-objects. But a physical object is a condition for
the occurrence of sense-objects other than those which are its
components. For example, the atmosphere causes the events which are its
situations to be active conditioning events in the transmission of
sound. A mirror which is itself a physical object is an active condition
for the situation of a patch of colour behind it, due to the reflection
of light in it.

Thus the origin of scientific knowledge is the endeavour to express in
terms of physical objects the various _roles_ of events as active
conditions in the ingression of sense-objects into nature. It is in the
progress of this investigation that scientific objects emerge. They
embody those aspects of the character of the situations of the physical
objects which are most permanent and are expressible without reference
to a multiple relation including a percipient event. Their relations to
each other are also characterised by a certain simplicity and
uniformity. Finally the characters of the observed physical objects and
sense-objects can be expressed in terms of these scientific objects. In
fact the whole point of the search for scientific objects is the
endeavour to obtain this simple expression of the characters of events.
These scientific objects are not themselves merely formulae for
calculation; because formulae must refer to things in nature, and the
scientific objects are the things in nature to which the formulae refer.

A scientific object such as a definite electron is a systematic
correlation of the characters of all events throughout all nature. It is
an aspect of the systematic character of nature. The electron is not
merely where its charge is. The charge is the quantitative character of
certain events due to the ingression of the electron into nature. The
electron is its whole field of force. Namely the electron is the
systematic way in which all events are modified as the expression of its
ingression. The situation of an electron in any small duration may be
defined as that event which has the quantitative character which is the
charge of the electron. We may if we please term the mere charge the
electron. But then another name is required for the scientific object
which is the full entity which concerns science, and which I have called
the electron.

According to this conception of scientific objects, the rival theories
of action at a distance and action by transmission through a medium are
both incomplete expressions of the true process of nature. The stream of
events which form the continuous series of situations of the electron is
entirely self-determined, both as regards having the intrinsic character
of being the series of situations of that electron and as regards the
time-systems with which its various members are cogredient, and the flux
of their positions in their corresponding durations. This is the
foundation of the denial of action at a distance; namely the progress of
the stream of the situations of a scientific object can be determined by
an analysis of the stream itself.

On the other hand the ingression of every electron into nature modifies
to some extent the character of every event. Thus the character of the
stream of events which we are considering bears marks of the existence
of every other electron throughout the universe. If we like to think of
the electrons as being merely what I call their charges, then the
charges act at a distance. But this action consists in the modification
of the situation of the other electron under consideration. This
conception of a charge acting at a distance is a wholly artificial one.
The conception which most fully expresses the character of nature is
that of each event as modified by the ingression of each electron into
nature. The ether is the expression of this systematic modification of
events throughout space and throughout time. The best expression of the
character of this modification is for physicists to find out. My theory
has nothing to do with that and is ready to accept any outcome of
physical research.

The connexion of objects with space requires elucidation. Objects are
situated in events. The relation of situation is a different relation
for each type of object, and in the case of sense-objects it cannot be
expressed as a two-termed relation. It would perhaps be better to use a
different word for these different types of the relation of situation.
It has not however been necessary to do so for our purposes in these
lectures. It must be understood however that, when situation is spoken
of, some one definite type is under discussion, and it may happen that
the argument may not apply to situation of another type. In all cases
however I use situation to express a relation between objects and events
and not between objects and abstractive elements. There is a derivative
relation between objects and spatial elements which I call the relation
of location; and when this relation holds, I say that the object is
located in the abstractive element. In this sense, an object may be
located in a moment of time, in a volume of space, an area, a line, or a
point. There will be a peculiar type of location corresponding to each
type of situation; and location is in each case derivative from the
corresponding relation of situation in a way which I will proceed to
explain.

Also location in the timeless space of some time-system is a relation
derivative from location in instantaneous spaces of the same
time-system. Accordingly location in an instantaneous space is the
primary idea which we have to explain. Great confusion has been
occasioned in natural philosophy by the neglect to distinguish between
the different types of objects, the different types of situation, the
different types of location, and the difference between location and
situation. It is impossible to reason accurately in the vague concerning
objects and their positions without keeping these distinctions in view.
An object is located in an abstractive element, when an abstractive set
belonging to that element can be found such that each event belonging to
that set is a situation of the object. It will be remembered that an
abstractive element is a certain group of abstractive sets, and that
each abstractive set is a set of events. This definition defines the
location of an element in any type of abstractive element. In this sense
we can talk of the existence of an object at an instant, meaning thereby
its location in some definite moment. It may also be located in some
spatial element of the instantaneous space of that moment.

A quantity can be said to be located in an abstractive element when an
abstractive set belonging to the element can be found such that the
quantitative expressions of the corresponding characters of its events
converge to the measure of the given quantity as a limit when we pass
along the abstractive set towards its converging end.

By these definitions location in elements of instantaneous spaces is
defined. These elements occupy corresponding elements of timeless
spaces. An object located in an element of an instantaneous space will
also be said to be located at that moment in the timeless element of the
timeless space which is occupied by that instantaneous element.

It is not every object which can be located in a moment. An object which
can be located in every moment of some duration will be called a
'uniform' object throughout that duration. Ordinary physical objects
appear to us to be uniform objects, and we habitually assume that
scientific objects such as electrons are uniform. But some sense-objects
certainly are not uniform. A tune is an example of a non-uniform object.
We have perceived it as a whole in a certain duration; but the tune as a
tune is not at any moment of that duration though one of the individual
notes may be located there.

It is possible therefore that for the existence of certain sorts of
objects, _e.g._ electrons, minimum quanta of time are requisite. Some
such postulate is apparently indicated by the modern quantum theory and
it is perfectly consistent with the doctrine of objects maintained in
these lectures.

Also the instance of the distinction between the electron as the mere
quantitative electric charge of its situation and the electron as
standing for the ingression of an object throughout nature illustrates
the indefinite number of types of objects which exist in nature. We can
intellectually distinguish even subtler and subtler types of objects.
Here I reckon subtlety as meaning seclusion from the immediate
apprehension of sense-awareness. Evolution in the complexity of life
means an increase in the types of objects directly sensed. Delicacy of
sense-apprehension means perceptions of objects as distinct entities
which are mere subtle ideas to cruder sensibilities. The phrasing of
music is a mere abstract subtlety to the unmusical; it is a direct
sense-apprehension to the initiated. For example, if we could imagine
some lowly type of organic being thinking and aware of our thoughts, it
would wonder at the abstract subtleties in which we indulge as we think
of stones and bricks and drops of water and plants. It only knows of
vague undifferentiated feelings in nature. It would consider us as given
over to the play of excessively abstract intellects. But then if it
could think, it would anticipate; and if it anticipated, it would soon
perceive for itself.

In these lectures we have been scrutinising the foundations of natural
philosophy. We are stopping at the very point where a boundless ocean of
enquiries opens out for our questioning.

I agree that the view of Nature which I have maintained in these
lectures is not a simple one. Nature appears as a complex system whose
factors are dimly discerned by us. But, as I ask you, Is not this the
very truth? Should we not distrust the jaunty assurance with which every
age prides itself that it at last has hit upon the ultimate concepts in
which all that happens can be formulated? The aim of science is to seek
the simplest explanations of complex facts. We are apt to fall into the
error of thinking that the facts are simple because simplicity is the
goal of our quest. The guiding motto in the life of every natural
philosopher should be, Seek simplicity and distrust it.




CHAPTER VIII

SUMMARY


There is a general agreement that Einstein's investigations have one
fundamental merit irrespective of any criticisms which we may feel
inclined to pass on them. They have made us think. But when we have
admitted so far, we are most of us faced with a distressing perplexity.
What is it that we ought to think about? The purport of my lecture this
afternoon will be to meet this difficulty and, so far as I am able, to
set in a clear light the changes in the background of our scientific
thought which are necessitated by any acceptance, however qualified, of
Einstein's main positions. I remember that I am lecturing to the members
of a chemical society who are not for the most part versed in advanced
mathematics. The first point that I would urge upon you is that what
immediately concerns you is not so much the detailed deductions of the
new theory as this general change in the background of scientific
conceptions which will follow from its acceptance. Of course, the
detailed deductions are important, because unless our colleagues the
astronomers and the physicists find these predictions to be verified we
can neglect the theory altogether. But we may now take it as granted
that in many striking particulars these deductions have been found to be
in agreement with observation. Accordingly the theory has to be taken
seriously and we are anxious to know what will be the consequences of
its final acceptance. Furthermore during the last few weeks the
scientific journals and the lay press have been filled with articles as
to the nature of the crucial experiments which have been made and as to
some of the more striking expressions of the outcome of the new theory.
'Space caught bending' appeared on the news-sheet of a well-known
evening paper. This rendering is a terse but not inapt translation of
Einstein's own way of interpreting his results. I should say at once
that I am a heretic as to this explanation and that I shall expound to
you another explanation based upon some work of my own, an explanation
which seems to me to be more in accordance with our scientific ideas and
with the whole body of facts which have to be explained. We have to
remember that a new theory must take account of the old well-attested
facts of science just as much as of the very latest experimental results
which have led to its production.

To put ourselves in the position to assimilate and to criticise any
change in ultimate scientific conceptions we must begin at the
beginning. So you must bear with me if I commence by making some simple
and obvious reflections. Let us consider three statements, (i)
'Yesterday a man was run over on the Chelsea Embankment,' (ii)
'Cleopatra's Needle is on the Charing Cross Embankment,' and (iii)
'There are dark lines in the Solar Spectrum.' The first statement about
the accident to the man is about what we may term an 'occurrence,' a
'happening,' or an 'event.' I will use the term 'event' because it is
the shortest. In order to specify an observed event, the place, the
time, and character of the event are necessary. In specifying the place
and the time you are really stating the relation of the assigned event
to the general structure of other observed events. For example, the man
was run over between your tea and your dinner and adjacently to a
passing barge in the river and the traffic in the Strand. The point
which I want to make is this: Nature is known to us in our experience as
a complex of passing events. In this complex we discern definite mutual
relations between component events, which we may call their relative
positions, and these positions we express partly in terms of space and
partly in terms of time. Also in addition to its mere relative position
to other events, each particular event has its own peculiar character.
In other words, nature is a structure of events and each event has its
position in this structure and its own peculiar character or quality.

Let us now examine the other two statements in the light of this general
principle as to the meaning of nature. Take the second statement,
'Cleopatra's Needle is on the Charing Cross Embankment.' At first sight
we should hardly call this an event. It seems to lack the element of
time or transitoriness. But does it? If an angel had made the remark
some hundreds of millions of years ago, the earth was not in existence,
twenty millions of years ago there was no Thames, eighty years ago there
was no Thames Embankment, and when I was a small boy Cleopatra's Needle
was not there. And now that it is there, we none of us expect it to be
eternal. The static timeless element in the relation of Cleopatra's
Needle to the Embankment is a pure illusion generated by the fact that
for purposes of daily intercourse its emphasis is needless. What it
comes to is this: Amidst the structure of events which form the medium
within which the daily life of Londoners is passed we know how to
identify a certain stream of events which maintain permanence of
character, namely the character of being the situations of Cleopatra's
Needle. Day by day and hour by hour we can find a certain chunk in the
transitory life of nature and of that chunk we say, 'There is
Cleopatra's Needle.' If we define the Needle in a sufficiently abstract
manner we can say that it never changes. But a physicist who looks on
that part of the life of nature as a dance of electrons, will tell you
that daily it has lost some molecules and gained others, and even the
plain man can see that it gets dirtier and is occasionally washed. Thus
the question of change in the Needle is a mere matter of definition. The
more abstract your definition, the more permanent the Needle. But
whether your Needle change or be permanent, all you mean by stating that
it is situated on the Charing Cross Embankment, is that amid the
structure of events you know of a certain continuous limited stream of
events, such that any chunk of that stream, during any hour, or any day,
or any second, has the character of being the situation of Cleopatra's
Needle.

Finally, we come to the third statement, 'There are dark lines in the
Solar Spectrum.' This is a law of nature. But what does that mean? It
means merely this. If any event has the character of being an exhibition
of the solar spectrum under certain assigned circumstances, it will also
have the character of exhibiting dark lines in that spectrum.

This long discussion brings us to the final conclusion that the concrete
facts of nature are events exhibiting a certain structure in their
mutual relations and certain characters of their own. The aim of science
is to express the relations between their characters in terms of the
mutual structural relations between the events thus characterised. The
mutual structural relations between events are both spatial and
temporal. If you think of them as merely spatial you are omitting the
temporal element, and if you think of them as merely temporal you are
omitting the spatial element. Thus when you think of space alone, or of
time alone, you are dealing in abstractions, namely, you are leaving out
an essential element in the life of nature as known to you in the
experience of your senses. Furthermore there are different ways of
making these abstractions which we think of as space and as time; and
under some circumstances we adopt one way and under other circumstances
we adopt another way. Thus there is no paradox in holding that what we
mean by space under one set of circumstances is not what we mean by
space under another set of circumstances. And equally what we mean by
time under one set of circumstances is not what we mean by time under
another set of circumstances. By saying that space and time are
abstractions, I do not mean that they do not express for us real facts
about nature. What I mean is that there are no spatial facts or temporal
facts apart from physical nature, namely that space and time are merely
ways of expressing certain truths about the relations between events.
Also that under different circumstances there are different sets of
truths about the universe which are naturally presented to us as
statements about space. In such a case what a being under the one set of
circumstances means by space will be different from that meant by a
being under the other set of circumstances. Accordingly when we are
comparing two observations made under different circumstances we have to
ask 'Do the two observers mean the same thing by space and the same
thing by time?' The modern theory of relativity has arisen because
certain perplexities as to the concordance of certain delicate
observations such as the motion of the earth through the ether, the
perihelion of mercury, and the positions of the stars in the
neighbourhood of the sun, have been solved by reference to this purely
relative significance of space and time.

I want now to recall your attention to Cleopatra's Needle, which I have
not yet done with. As you are walking along the Embankment you suddenly
look up and say, 'Hullo, there's the Needle.' In other words, you
recognise it. You cannot recognise an event; because when it is gone, it
is gone. You may observe another event of analogous character, but the
actual chunk of the life of nature is inseparable from its unique
occurrence. But a character of an event can be recognised. We all know
that if we go to the Embankment near Charing Cross we shall observe an
event having the character which we recognise as Cleopatra's Needle.
Things which we thus recognise I call objects. An object is situated in
those events or in that stream of events of which it expresses the
character. There are many sorts of objects. For example, the colour
green is an object according to the above definition. It is the purpose
of science to trace the laws which govern the appearance of objects in
the various events in which they are found to be situated. For this
purpose we can mainly concentrate on two types of objects, which I will
call material physical objects and scientific objects. A material
physical object is an ordinary bit of matter, Cleopatra's Needle for
example. This is a much more complicated type of object than a mere
colour, such as the colour of the Needle. I call these simple objects,
such as colours or sounds, sense-objects. An artist will train himself
to attend more particularly to sense-objects where the ordinary person
attends normally to material objects. Thus if you were walking with an
artist, when you said 'There's Cleopatra's Needle,' perhaps he
simultaneously exclaimed 'There's a nice bit of colour.' Yet you were
both expressing your recognition of different component characters of
the same event. But in science we have found out that when we know all
about the adventures amid events of material physical objects and of
scientific objects we have most of the relevant information which will
enable us to predict the conditions under which we shall perceive
sense-objects in specific situations. For example, when we know that
there is a blazing fire (_i.e._ material and scientific objects
undergoing various exciting adventures amid events) and opposite to it a
mirror (which is another material object) and the positions of a man's
face and eyes gazing into the mirror, we know that he can perceive the
redness of the flame situated in an event behind the mirror--thus, to a
large extent, the appearance of sense-objects is conditioned by the
adventures of material objects. The analysis of these adventures makes
us aware of another character of events, namely their characters as
fields of activity which determine the subsequent events to which they
will pass on the objects situated in them. We express these fields of
activity in terms of gravitational, electromagnetic, or chemical forces
and attractions. But the exact expression of the nature of these fields
of activity forces us intellectually to acknowledge a less obvious type
of objects as situated in events. I mean molecules and electrons. These
objects are not recognised in isolation. We cannot well miss Cleopatra's
Needle, if we are in its neighbourhood; but no one has seen a single
molecule or a single electron, yet the characters of events are only
explicable to us by expressing them in terms of these scientific
objects. Undoubtedly molecules and electrons are abstractions. But then
so is Cleopatra's Needle. The concrete facts are the events
themselves--I have already explained to you that to be an abstraction
does not mean that an entity is nothing. It merely means that its
existence is only one factor of a more concrete element of nature. So an
electron is abstract because you cannot wipe out the whole structure of
events and yet retain the electron in existence. In the same way the
grin on the cat is abstract; and the molecule is really in the event in
the same sense as the grin is really on the cat's face. Now the more
ultimate sciences such as Chemistry or Physics cannot express their
ultimate laws in terms of such vague objects as the sun, the earth,
Cleopatra's Needle, or a human body. Such objects more properly belong
to Astronomy, to Geology, to Engineering, to Archaeology, or to Biology.
Chemistry and Physics only deal with them as exhibiting statistical
complexes of the effects of their more intimate laws. In a certain
sense, they only enter into Physics and Chemistry as technological
applications. The reason is that they are too vague. Where does
Cleopatra's Needle begin and where does it end? Is the soot part of it?
Is it a different object when it sheds a molecule or when its surface
enters into chemical combination with the acid of a London fog? The
definiteness and permanence of the Needle is nothing to the possible
permanent definiteness of a molecule as conceived by science, and the
permanent definiteness of a molecule in its turn yields to that of an
electron. Thus science in its most ultimate formulation of law seeks
objects with the most permanent definite simplicity of character and
expresses its final laws in terms of them.

Again when we seek definitely to express the relations of events which
arise from their spatio-temporal structure, we approximate to simplicity
by progressively diminishing the extent (both temporal and spatial) of
the events considered. For example, the event which is the life of the
chunk of nature which is the Needle during one minute has to the life of
nature within a passing barge during the same minute a very complex
spatio-temporal relation. But suppose we progressively diminish the time
considered to a second, to a hundredth of a second, to a thousandth of a
second, and so on. As we pass along such a series we approximate to an
ideal simplicity of structural relations of the pairs of events
successively considered, which ideal we call the spatial relations of
the Needle to the barge at some instant. Even these relations are too
complicated for us, and we consider smaller and smaller bits of the
Needle and of the barge. Thus we finally reach the ideal of an event so
restricted in its extension as to be without extension in space or
extension in time. Such an event is a mere spatial point-flash of
instantaneous duration. I call such an ideal event an 'event-particle.'
You must not think of the world as ultimately built up of
event-particles. That is to put the cart before the horse. The world we
know is a continuous stream of occurrence which we can discriminate into
finite events forming by their overlappings and containings of each
other and separations a spatio-temporal structure. We can express the
properties of this structure in terms of the ideal limits to routes of
approximation, which I have termed event-particles. Accordingly
event-particles are abstractions in their relations to the more concrete
events. But then by this time you will have comprehended that you cannot
analyse concrete nature without abstracting. Also I repeat, the
abstractions of science are entities which are truly in nature, though
they have no meaning in isolation from nature.

The character of the spatio-temporal structure of events can be fully
expressed in terms of relations between these more abstract
event-particles. The advantage of dealing with event-particles is that
though they are abstract and complex in respect to the finite events
which we directly observe, they are simpler than finite events in
respect to their mutual relations. Accordingly they express for us the
demands of an ideal accuracy, and of an ideal simplicity in the
exposition of relations. These event-particles are the ultimate elements
of the four-dimensional space-time manifold which the theory of
relativity presupposes. You will have observed that each event-particle
is as much an instant of time as it is a point of space. I have called
it an instantaneous point-flash. Thus in the structure of this
space-time manifold space is not finally discriminated from time, and
the possibility remains open for diverse modes of discrimination
according to the diverse circumstances of observers. It is this
possibility which makes the fundamental distinction between the new way
of conceiving the universe and the old way. The secret of understanding
relativity is to understand this. It is of no use rushing in with
picturesque paradoxes, such as 'Space caught bending,' if you have not
mastered this fundamental conception which underlies the whole theory.
When I say that it underlies the whole theory, I mean that in my opinion
it ought to underlie it, though I may confess some doubts as to how far
all expositions of the theory have really understood its implications
and its premises.

Our measurements when they are expressed in terms of an ideal accuracy
are measurements which express properties of the space-time manifold.
Now there are measurements of different sorts. You can measure lengths,
or angles, or areas, or volumes, or times. There are also other sorts of
measures such as measurements of intensity of illumination, but I will
disregard these for the moment and will confine attention to those
measurements which particularly interest us as being measurements of
space or of time. It is easy to see that four such measurements of the
proper characters are necessary to determine the position of an
event-particle in the space-time manifold in its relation to the rest of
the manifold. For example, in a rectangular field you start from one
corner at a given time, you measure a definite distance along one side,
you then strike out into the field at right angles, and then measure a
definite distance parallel to the other pair of sides, you then rise
vertically a definite height and take the time. At the point and at the
time which you thus reach there is occurring a definite instantaneous
point-flash of nature. In other words, your four measurements have
determined a definite event-particle belonging to the four-dimension
space-time manifold. These measurements have appeared to be very simple
to the land-surveyor and raise in his mind no philosophic difficulties.
But suppose there are beings on Mars sufficiently advanced in
scientific invention to be able to watch in detail the operations of
this survey on earth. Suppose that they construe the operations of the
English land-surveyors in reference to the space natural to a being on
Mars, namely a Martio-centric space in which that planet is fixed. The
earth is moving relatively to Mars and is rotating. To the beings on
Mars the operations, construed in this fashion, effect measurements of
the greatest complication. Furthermore, according to the relativistic
doctrine, the operation of time-measurement on earth will not correspond
quite exactly to any time-measurement on Mars.

I have discussed this example in order to make you realise that in
thinking of the possibilities of measurement in the space-time manifold,
we must not confine ourselves merely to those minor variations which
might seem natural to human beings on the earth. Let us make therefore
the general statement that four measurements, respectively of
independent types (such as measurements of lengths in three directions
and a time), can be found such that a definite event-particle is
determined by them in its relations to other parts of the manifold.

If (p_1, p_2, p_3, p_4) be a set of measurements of this system, then
the event-particle which is thus determined will be said to have p_1,
p_2, p_3, p_4 as its co-ordinates in this system of measurement. Suppose
that we name it the p-system of measurement. Then in the same p-system
by properly varying (p_1, p_2, p_3, p_4) every event-particle that has
been, or will be, or instantaneously is now, can be indicated.
Furthermore, according to any system of measurement that is natural to
us, three of the co-ordinates will be measurements of space and one will
be a measurement of time. Let us always take the last co-ordinate to
represent the time-measurement. Then we should naturally say that (p_1,
p_2, p_3) determined a point in space and that the event-particle
happened at that point at the time p_4. But we must not make the mistake
of thinking that there is a space in addition to the space-time
manifold. That manifold is all that there is for the determination of
the meaning of space and time. We have got to determine the meaning of a
space-point in terms of the event-particles of the four-dimensional
manifold. There is only one way to do this. Note that if we vary the
time and take times with the same three space co-ordinates, then the
event-particles, thus indicated, are all at the same point. But seeing
that there is nothing else except the event-particles, this can only
mean that the point (p_1, p_2, p_3) of the space in the p-system is
merely the collection of event-particles (p_1, p_2, p_3, [p_4]), where
p_4 is varied and (p_1, p_2, p_3) is kept fixed. It is rather
disconcerting to find that a point in space is not a simple entity; but
it is a conclusion which follows immediately from the relative theory of
space.

Furthermore the inhabitant of Mars determines event-particles by another
system of measurements. Call his system the q-system. According to him
(q_1, q_2, q_3, q_4) determines an event-particle, and (q_1, q_2, q_3)
determines a point and q_4 a time. But the collection of event-particles
which he thinks of as a point is entirely different from any such
collection which the man on earth thinks of as a point. Thus the q-space
for the man on Mars is quite different from the p-space for the
land-surveyor on earth.

So far in speaking of space we have been talking of the timeless space
of physical science, namely, of our concept of eternal space in which
the world adventures. But the space which we see as we look about is
instantaneous space. Thus if our natural perceptions are adjustable to
the p-system of measurements we see instantaneously all the
event-particles at some definite time p_4, and observe a succession
of such spaces as time moves on. The timeless space is achieved by
stringing together all these instantaneous spaces. The points of an
instantaneous space are event-particles, and the points of an eternal
space are strings of event-particles occurring in succession. But the
man on Mars will never perceive the same instantaneous spaces as the man
on the earth. This system of instantaneous spaces will cut across the
earth-man's system. For the earth-man there is one instantaneous space
which is the instantaneous present, there are the past spaces and the
future spaces. But the present space of the man on Mars cuts across the
present space of the man on the earth. So that of the event-particles
which the earth-man thinks of as happening now in the present, the man
on Mars thinks that some are already past and are ancient history, that
others are in the future, and others are in the immediate present. This
break-down in the neat conception of a past, a present, and a future is
a serious paradox. I call two event-particles which on some or other
system of measurement are in the same instantaneous space 'co-present'
event-particles. Then it is possible that A and B may be co-present,
and that A and C may be co-present, but that B and C may not be
co-present. For example, at some inconceivable distance from us there
are events co-present with us now and also co-present with the birth of
Queen Victoria. If A and B are co-present there will be some systems
in which A precedes B and some in which B precedes A. Also there
can be no velocity quick enough to carry a material particle from A to
B or from B to A. These different measure-systems with their
divergences of time-reckoning are puzzling, and to some extent affront
our common sense. It is not the usual way in which we think of the
Universe. We think of one necessary time-system and one necessary space.
According to the new theory, there are an indefinite number of
discordant time-series and an indefinite number of distinct spaces. Any
correlated pair, a time-system and a space-system, will do in which to
fit our description of the Universe. We find that under given conditions
our measurements are necessarily made in some one pair which together
form our natural measure-system. The difficulty as to discordant
time-systems is partly solved by distinguishing between what I call the
creative advance of nature, which is not properly serial at all, and any
one time series. We habitually muddle together this creative advance,
which we experience and know as the perpetual transition of nature into
novelty, with the single-time series which we naturally employ for
measurement. The various time-series each measure some aspect of the
creative advance, and the whole bundle of them express all the
properties of this advance which are measurable. The reason why we have
not previously noted this difference of time-series is the very small
difference of properties between any two such series. Any observable
phenomena due to this cause depend on the square of the ratio of any
velocity entering into the observation to the velocity of light. Now
light takes about fifty minutes to get round the earth's orbit; and the
earth takes rather more than 17,531 half-hours to do the same. Hence all
the effects due to this motion are of the order of the ratio of one to
the square of 10,000. Accordingly an earth-man and a sun-man have only
neglected effects whose quantitative magnitudes all contain the factor
1/10^8. Evidently such effects can only be noted by means of the most
refined observations. They have been observed however. Suppose we
compare two observations on the velocity of light made with the same
apparatus as we turn it through a right angle. The velocity of the earth
relatively to the sun is in one direction, the velocity of light
relatively to the ether should be the same in all directions. Hence if
space when we take the ether as at rest means the same thing as space
when we take the earth as at rest, we ought to find that the velocity of
light relatively to the earth varies according to the direction from
which it comes.

These observations on earth constitute the basic principle of the famous
experiments designed to detect the motion of the earth through the
ether. You all know that, quite unexpectedly, they gave a null result.
This is completely explained by the fact that, the space-system and the
time-system which we are using are in certain minute ways different from
the space and the time relatively to the sun or relatively to any other
body with respect to which it is moving.

All this discussion as to the nature of time and space has lifted above
our horizon a great difficulty which affects the formulation of all the
ultimate laws of physics--for example, the laws of the electromagnetic
field, and the law of gravitation. Let us take the law of gravitation
as an example. Its formulation is as follows: Two material bodies
attract each other with a force proportional to the product of their
masses and inversely proportional to the square of their distances.
In this statement the bodies are supposed to be small enough to be
treated as material particles in relation to their distances; and we
need not bother further about that minor point. The difficulty to which
I want to draw your attention is this: In the formulation of the law one
definite time and one definite space are presupposed. The two masses are
assumed to be in simultaneous positions.

But what is simultaneous in one time-system may not be simultaneous in
another time-system. So according to our new views the law is in this
respect not formulated so as to have any exact meaning. Furthermore an
analogous difficulty arises over the question of distance. The distance
between two instantaneous positions, _i.e._ between two event-particles,
is different in different space-systems. What space is to be chosen?
Thus again the law lacks precise formulation, if relativity is accepted.
Our problem is to seek a fresh interpretation of the law of gravity in
which these difficulties are evaded. In the first place we must avoid
the abstractions of space and time in the formulation of our fundamental
ideas and must recur to the ultimate facts of nature, namely to events.
Also in order to find the ideal simplicity of expressions of the
relations between events, we restrict ourselves to event-particles. Thus
the life of a material particle is its adventure amid a track of
event-particles strung out as a continuous series or path in the
four-dimensional space-time manifold. These event-particles are the
various situations of the material particle. We usually express this
fact by adopting our natural space-time system and by talking of the
path in space of the material particle as it exists at successive
instants of time.

We have to ask ourselves what are the laws of nature which lead the
material particle to adopt just this path among event-particles and no
other. Think of the path as a whole. What characteristic has that path
got which would not be shared by any other slightly varied path? We are
asking for more than a law of gravity. We want laws of motion and a
general idea of the way to formulate the effects of physical forces.

In order to answer our question we put the idea of the attracting masses
in the background and concentrate attention on the field of activity of
the events in the neighbourhood of the path. In so doing we are acting
in conformity with the whole trend of scientific thought during the last
hundred years, which has more and more concentrated attention on the
field of force as the immediate agent in directing motion, to the
exclusion of the consideration of the immediate mutual influence between
two distant bodies. We have got to find the way of expressing the field
of activity of events in the neighbourhood of some definite
event-particle E of the four-dimensional manifold. I bring in a
fundamental physical idea which I call the 'impetus' to express this
physical field. The event-particle E is related to any neighbouring
event-particle P by an element of impetus. The assemblage of all the
elements of impetus relating E to the assemblage of event-particles in
the neighbourhood of E expresses the character of the field of
activity in the neighbourhood of E. Where I differ from Einstein is
that he conceives this quantity which I call the impetus as merely
expressing the characters of the space and time to be adopted and thus
ends by talking of the gravitational field expressing a curvature in the
space-time manifold. I cannot attach any clear conception to his
interpretation of space and time. My formulae differ slightly from his,
though they agree in those instances where his results have been
verified. I need hardly say that in this particular of the formulation
of the law of gravitation I have drawn on the general method of
procedure which constitutes his great discovery.

Einstein showed how to express the characters of the assemblage of
elements of impetus of the field surrounding an event-particle E in
terms of ten quantities which I will call J_{11}, J_{12}
(=J_{21}), J_{22}, J_{23}(=J_{32}), etc. It will be noted that
there are four spatio-temporal measurements relating E to its
neighbour P, and that there are ten pairs of such measurements if we
are allowed to take any one measurement twice over to make one such
pair. The ten J's depend merely on the position of E in the
four-dimensional manifold, and the element of impetus between E and
P can be expressed in terms of the ten J's and the ten pairs of the
four spatio-temporal measurements relating E and P. The numerical
values of the J's will depend on the system of measurement adopted,
but are so adjusted to each particular system that the same value is
obtained for the element of impetus between E and P, whatever be the
system of measurement adopted. This fact is expressed by saying that the
ten J's form a 'tensor.' It is not going too far to say that the
announcement that physicists would have in future to study the theory of
tensors created a veritable panic among them when the verification of
Einstein's predictions was first announced.

The ten J's at any event-particle E can be expressed in terms of two
functions which I call the potential and the 'associate-potential' at
E. The potential is practically what is meant by the ordinary
gravitation potential, when we express ourselves in terms of the
Euclidean space in reference to which the attracting mass is at rest.
The associate-potential is defined by the modification of substituting
the direct distance for the inverse distance in the definition of the
potential, and its calculation can easily be made to depend on that of
the old-fashioned potential. Thus the calculation of the J's--the
coefficients of impetus, as I will call them--does not involve anything
very revolutionary in the mathematical knowledge of physicists. We now
return to the path of the attracted particle. We add up all the elements
of impetus in the whole path, and obtain thereby what I call the
'integral impetus.' The characteristic of the actual path as compared
with neighbouring alternative paths is that in the actual paths the
integral impetus would neither gain nor lose, if the particle wobbled
out of it into a small extremely near alternative path. Mathematicians
would express this by saying, that the integral impetus is stationary
for an infinitesimal displacement. In this statement of the law of
motion I have neglected the existence of other forces. But that would
lead me too far afield.

The electromagnetic theory has to be modified to allow for the presence
of a gravitational field. Thus Einstein's investigations lead to the
first discovery of any relation between gravity and other physical
phenomena. In the form in which I have put this modification, we deduce
Einstein's fundamental principle, as to the motion of light along its
rays, as a first approximation which is absolutely true for infinitely
short waves. Einstein's principle, thus partially verified, stated in my
language is that a ray of light always follows a path such that the
integral impetus along it is zero. This involves that every element of
impetus along it is zero.

In conclusion, I must apologise. In the first place I have considerably
toned down the various exciting peculiarities of the original theory and
have reduced it to a greater conformity with the older physics. I do not
allow that physical phenomena are due to oddities of space. Also I have
added to the dullness of the lecture by my respect for the audience. You
would have enjoyed a more popular lecture with illustrations of
delightful paradoxes. But I know also that you are serious students who
are here because you really want to know how the new theories may affect
your scientific researches.




CHAPTER IX

THE ULTIMATE PHYSICAL CONCEPTS


The second chapter of this book lays down the first principle to be
guarded in framing our physical concept. We must avoid vicious
bifurcation. Nature is nothing else than the deliverance of
sense-awareness. We have no principles whatever to tell us what could
stimulate mind towards sense-awareness. Our sole task is to exhibit in
one system the characters and inter-relations of all that is observed.
Our attitude towards nature is purely 'behaviouristic,' so far as
concerns the formulation of physical concepts.

Our knowledge of nature is an experience of activity (or passage). The
things previously observed are active entities, the 'events.' They are
chunks in the life of nature. These events have to each other relations
which in our knowledge differentiate themselves into space-relations and
time-relations. But this differentiation between space and time, though
inherent in nature, is comparatively superficial; and space and time are
each partial expressions of one fundamental relation between events
which is neither spatial nor temporal. This relation I call 'extension.'
The relation of 'extending over' is the relation of 'including,' either
in a spatial or in a temporal sense, or in both. But the mere
'inclusion' is more fundamental than either alternative and does not
require any spatio-temporal differentiation. In respect to extension two
events are mutually related so that either (i) one includes the other,
or (ii) one overlaps the other without complete inclusion, or (iii)
they are entirely separate. But great care is required in the
definition of spatial and temporal elements from this basis in order to
avoid tacit limitations really depending on undefined relations and
properties.

Such fallacies can be avoided by taking account of two elements in our
experience, namely, (i) our observational 'present,' and (ii) our
'percipient event.'

Our observational 'present' is what I call a 'duration.' It is the whole
of nature apprehended in our immediate observation. It has therefore the
nature of an event, but possesses a peculiar completeness which marks
out such durations as a special type of events inherent in nature. A
duration is not instantaneous. It is all that there is of nature with
certain temporal limitations. In contradistinction to other events a
duration will be called infinite and the other events are finite[10]. In
our knowledge of a duration we distinguish (i) certain included events
which are particularly discriminated as to their peculiar
individualities, and (ii) the remaining included events which are only
known as necessarily in being by reason of their relations to the
discriminated events and to the whole duration. The duration as a whole
is signified[11] by that quality of relatedness (in respect to
extension) possessed by the part which is immediately under observation;
namely, by the fact that there is essentially a beyond to whatever is
observed. I mean by this that every event is known as being related to
other events which it does not include. This fact, that every event is
known as possessing the quality of exclusion, shows that exclusion is as
positive a relation as inclusion. There are of course no merely
negative relations in nature, and exclusion is not the mere negative of
inclusion, though the two relations are contraries. Both relations are
concerned solely with events, and exclusion is capable of logical
definition in terms of inclusion.

[10] Cf. note on 'significance,' pp. 197, 198.

[11] Cf. Ch. III, pp. 51 et seq.

Perhaps the most obvious exhibition of significance is to be found in
our knowledge of the geometrical character of events inside an opaque
material object. For example we know that an opaque sphere has a centre.
This knowledge has nothing to do with the material; the sphere may be a
solid uniform billiard ball or a hollow lawn-tennis ball. Such knowledge
is essentially the product of significance, since the general character
of the external discriminated events has informed us that there are
events within the sphere and has also informed us of their geometrical
structure.

Some criticisms on 'The Principles of Natural Knowledge' show that
difficulty has been found in apprehending durations as real
stratifications of nature. I think that this hesitation arises from the
unconscious influence of the vicious principle of bifurcation, so deeply
embedded in modern philosophical thought. We observe nature as extended
in an immediate present which is simultaneous but not instantaneous, and
therefore the whole which is immediately discerned or signified as an
inter-related system forms a stratification of nature which is a
physical fact. This conclusion immediately follows unless we admit
bifurcation in the form of the principle of psychic additions, here
rejected.

Our 'percipient event' is that event included in our observational
present which we distinguish as being in some peculiar way our
standpoint for perception. It is roughly speaking that event which is
our bodily life within the present duration. The theory of perception
as evolved by medical psychology is based on significance. The distant
situation of a perceived object is merely known to us as signified by
our bodily state, _i.e._ by our percipient event. In fact perception
requires sense-awareness of the significations of our percipient event
together with sense-awareness of a peculiar relation (situation) between
certain objects and the events thus signified. Our percipient event is
saved by being the whole of nature by this fact of its significations.
This is the meaning of calling the percipient event our standpoint for
perception. The course of a ray of light is only derivatively connected
with perception. What we do perceive are objects as related to events
signified by the bodily states excited by the ray. These signified
events (as is the case of images seen behind a mirror) may have very
little to do with the actual course of the ray. In the course of
evolution those animals have survived whose sense-awareness is
concentrated on those significations of their bodily states which are on
the average important for their welfare. The whole world of events is
signified, but there are some which exact the death penalty for
inattention.

The percipient event is always here and now in the associated present
duration. It has, what may be called, an absolute position in that
duration. Thus one definite duration is associated with a definite
percipient event, and we are thus aware of a peculiar relation which
finite events can bear to durations. I call this relation 'cogredience.'
The notion of rest is derivative from that of cogredience, and the
notion of motion is derivative from that of inclusion within a duration
without cogredience with it. In fact motion is a relation (of varying
character) between an observed event and an observed duration, and
cogredience is the most simple character or subspecies of motion. To sum
up, a duration and a percipient event are essentially involved in the
general character of each observation of nature, and the percipient
event is cogredient with the duration.

Our knowledge of the peculiar characters of different events depends
upon our power of comparison. I call the exercise of this factor in our
knowledge 'recognition,' and the requisite sense-awareness of the
comparable characters I call 'sense-recognition.' Recognition and
abstraction essentially involve each other. Each of them exhibits an
entity for knowledge which is less than the concrete fact, but is a real
factor in that fact. The most concrete fact capable of separate
discrimination is the event. We cannot abstract without recognition, and
we cannot recognise without abstraction. Perception involves
apprehension of the event and recognition of the factors of its
character.

The things recognised are what I call 'objects.' In this general sense
of the term the relation of extension is itself an object. In practice
however I restrict the term to those objects which can in some sense or
other be said to have a situation in an event; namely, in the phrase
'There it is again' I restrict the 'there' to be the indication of a
special event which is the situation of the object. Even so, there are
different types of objects, and statements which are true of objects of
one type are not in general true of objects of other types. The objects
with which we are here concerned in the formulation of physical laws are
material objects, such as bits of matter, molecules and electrons. An
object of one of these types has relations to events other than those
belonging to the stream of its situations. The fact of its situations
within this stream has impressed on all other events certain
modifications of their characters. In truth the object in its
completeness may be conceived as a specific set of correlated
modifications of the characters of all events, with the property that
these modifications attain to a certain focal property for those events
which belong to the stream of its situations. The total assemblage of
the modifications of the characters of events due to the existence of an
object in a stream of situations is what I call the 'physical field' due
to the object. But the object cannot really be separated from its field.
The object is in fact nothing else than the systematically adjusted set
of modifications of the field. The conventional limitation of the object
to the focal stream of events in which it is said to be 'situated' is
convenient for some purposes, but it obscures the ultimate fact of
nature. From this point of view the antithesis between action at a
distance and action by transmission is meaningless. The doctrine of this
paragraph is nothing else than another way of expressing the
unresolvable multiple relation of an object to events.

A complete time-system is formed by any one family of parallel
durations. Two durations are parallel if either (i) one includes the
other, or (ii) they overlap so as to include a third duration common to
both, or (iii) are entirely separate. The excluded case is that of two
durations overlapping so as to include in common an aggregate of finite
events but including in common no other complete duration. The
recognition of the fact of an indefinite number of families of parallel
durations is what differentiates the concept of nature here put forward
from the older orthodox concept of the essentially unique time-systems.
Its divergence from Einstein's concept of nature will be briefly
indicated later.

The instantaneous spaces of a given time-system are the ideal
(non-existent) durations of zero temporal thickness indicated by routes
of approximation along series formed by durations of the associated
family. Each such instantaneous space represents the ideal of nature at
an instant and is also a moment of time. Each time-system thus possesses
an aggregate of moments belonging to it alone. Each event-particle lies
in one and only one moment of a given time-system. An event-particle has
three characters[12]: (i) its extrinsic character which is its character
as a definite route of convergence among events, (ii) its intrinsic
character which is the peculiar quality of nature in its neighbourhood,
namely, the character of the physical field in the neighbourhood, and
(iii) its position.

[12] Cf. pp. 82 et seq.

The position of an event-particle arises from the aggregate of moments
(no two of the same family) in which it lies. We fix our attention on
one of these moments which is approximated to by the short duration of
our immediate experience, and we express position as the position in
this moment. But the event-particle receives its position in moment M
in virtue of the whole aggregate of other moments M{'}, M{''}, etc.,
in which it also lies. The differentiation of M into a geometry of
event-particles (instantaneous points) expresses the differentiation of
M by its intersections with moments of alien time-systems. In this way
planes and straight lines and event-particles themselves find their
being. Also the parallelism of planes and straight lines arises from the
parallelism of the moments of one and the same time-system intersecting
M. Similarly the order of parallel planes and of event-particles on
straight lines arises from the time-order of these intersecting moments.
The explanation is not given here[13]. It is sufficient now merely to
mention the sources from which the whole of geometry receives its
physical explanation.

[13] Cf. _Principles of Natural Knowledge_, and previous chapters of the
present work.

The correlation of the various momentary spaces of one time-system is
achieved by the relation of cogredience. Evidently motion in an
instantaneous space is unmeaning. Motion expresses a comparison between
position in one instantaneous space with positions in other
instantaneous spaces of the same time-system. Cogredience yields the
simplest outcome of such comparison, namely, rest.

Motion and rest are immediately observed facts. They are relative in the
sense that they depend on the time-system which is fundamental for the
observation. A string of event-particles whose successive occupation
means rest in the given time-system forms a timeless point in the
timeless space of that time-system. In this way each time-system
possesses its own permanent timeless space peculiar to it alone, and
each such space is composed of timeless points which belong to that
time-system and to no other. The paradoxes of relativity arise from
neglecting the fact that different assumptions as to rest involve the
expression of the facts of physical science in terms of radically
different spaces and times, in which points and moments have different
meanings.

The source of order has already been indicated and that of congruence is
now found. It depends on motion. From cogredience, perpendicularity
arises; and from perpendicularity in conjunction with the reciprocal
symmetry between the relations of any two time-systems congruence both
in time and space is completely defined (cf. _loc. cit._).

The resulting formulae are those for the electromagnetic theory of
relativity, or, as it is now termed, the restricted theory. But there is
this vital difference: the critical velocity c which occurs in these
formulae has now no connexion whatever with light or with any other fact
of the physical field (in distinction from the extensional structure of
events). It simply marks the fact that our congruence determination
embraces both times and spaces in one universal system, and therefore if
two arbitrary units are chosen, one for all spaces and one for all
times, their ratio will be a velocity which is a fundamental property of
nature expressing the fact that times and spaces are really comparable.

The physical properties of nature are expressed in terms of material
objects (electrons, etc.). The physical character of an event arises
from the fact that it belongs to the field of the whole complex of such
objects. From another point of view we can say that these objects are
nothing else than our way of expressing the mutual correlation of the
physical characters of events.

The spatio-temporal measurableness of nature arises from (i) the
relation of extension between events, and (ii) the stratified character
of nature arising from each of the alternative time-systems, and (iii)
rest and motion, as exhibited in the relations of finite events to
time-systems. None of these sources of measurement depend on the
physical characters of finite events as exhibited by the situated
objects. They are completely signified for events whose physical
characters are unknown. Thus the spatio-temporal measurements are
independent of the objectival physical characters. Furthermore the
character of our knowledge of a whole duration, which is essentially
derived from the significance of the part within the immediate field of
discrimination, constructs it for us as a uniform whole independent, so
far as its extension is concerned, of the unobserved characters of
remote events. Namely, there is a definite whole of nature,
simultaneously now present, whatever may be the character of its remote
events. This consideration reinforces the previous conclusion. This
conclusion leads to the assertion of the essential uniformity of the
momentary spaces of the various time-systems, and thence to the
uniformity of the timeless spaces of which there is one to each
time-system.

The analysis of the general character of observed nature set forth above
affords explanations of various fundamental observational facts:
({alpha}) It explains the differentiation of the one quality of
extension into time and space. ({beta}) It gives a meaning to the
observed facts of geometrical and temporal position, of geometrical and
temporal order, and of geometrical straightness and planeness. ({gamma})
It selects one definite system of congruence embracing both space and
time, and thus explains the concordance as to measurement which is in
practice attained. ({delta}) It explains (consistently with the theory
of relativity) the observed phenomena of rotation, _e.g._ Foucault's
pendulum, the equatorial bulge of the earth, the fixed senses of
rotation of cyclones and anticyclones, and the gyro-compass. It does
this by its admission of definite stratifications of nature which are
disclosed by the very character of our knowledge of it. ({epsilon}) Its
explanations of motion are more fundamental than those expressed in
({delta}); for it explains what is meant by motion itself. The observed
motion of an extended object is the relation of its various situations
to the stratification of nature expressed by the time-system fundamental
to the observation. This motion expresses a real relation of the object
to the rest of nature. The quantitative expression of this relation will
vary according to the time-system selected for its expression.

This theory accords no peculiar character to light beyond that accorded
to other physical phenomena such as sound. There is no ground for such a
differentiation. Some objects we know by sight only, and other objects
we know by sound only, and other objects we observe neither by light nor
by sound but by touch or smell or otherwise. The velocity of light
varies according to its medium and so does that of sound. Light moves in
curved paths under certain conditions and so does sound. Both light and
sound are waves of disturbance in the physical characters of events; and
(as has been stated above, p. 188) the actual course of the light is of
no more importance for perception than is the actual course of the
sound. To base the whole philosophy of nature upon light is a baseless
assumption. The Michelson-Morley and analogous experiments show that
within the limits of our inexactitude of observation the velocity of
light is an approximation to the critical velocity 'c' which expresses
the relation between our space and time units. It is provable that the
assumption as to light by which these experiments and the influence of
the gravitational field on the light-rays are explained is deducible _as
an approximation_ from the equations of the electromagnetic field. This
completely disposes of any necessity for differentiating light from
other physical phenomena as possessing any peculiar fundamental
character.

It is to be observed that the measurement of extended nature by means of
extended objects is meaningless apart from some observed fact of
simultaneity inherent in nature and not merely a play of thought.
Otherwise there is no meaning to the concept of one presentation of your
extended measuring rod AB. Why not AB' where B' is the end B
five minutes later? Measurement presupposes for its possibility nature
as a simultaneity, and an observed object present then and present now.
In other words, measurement of extended nature requires some inherent
character in nature affording a rule of presentation of events.
Furthermore congruence cannot be defined by the permanence of the
measuring rod. The permanence is itself meaningless apart from some
immediate judgment of self-congruence. Otherwise how is an elastic
string differentiated from a rigid measuring rod? Each remains the same
self-identical object. Why is one a possible measuring rod and the other
not so? The meaning of congruence lies beyond the self-identity of the
object. In other words measurement presupposes the measurable, and the
theory of the measurable is the theory of congruence.

Furthermore the admission of stratifications of nature bears on the
formulation of the laws of nature. It has been laid down that these laws
are to be expressed in differential equations which, as expressed in any
general system of measurement, should bear no reference to any other
particular measure-system. This requirement is purely arbitrary. For a
measure-system measures something inherent in nature; otherwise it has
no connexion with nature at all. And that something which is measured
by a particular measure-system may have a special relation to the
phenomenon whose law is being formulated. For example the gravitational
field due to a material object at rest in a certain time-system may be
expected to exhibit in its formulation particular reference to spatial
and temporal quantities of that time-system. The field can of course be
expressed in any measure-systems, but the particular reference will
remain as the simple physical explanation.


NOTE: ON THE GREEK CONCEPT OF A POINT

The preceding pages had been passed for press before I had the pleasure
of seeing Sir T. L. Heath's _Euclid in Greek_[14]. In the original
Euclid's first definition is

    semeion estin, ou meros outhen.

I have quoted it on p. 86 in the expanded form taught to me in
childhood, 'without parts and without magnitude.' I should have
consulted Heath's English edition--a classic from the moment of its
issue--before committing myself to a statement about Euclid. This is
however a trivial correction not affecting sense and not worth a note. I
wish here to draw attention to Heath's own note to this definition in
his _Euclid in Greek_. He summarises Greek thought on the nature of a
point, from the Pythagoreans, through Plato and Aristotle, to Euclid. My
analysis of the requisite character of a point on pp. 89 and 90 is in
complete agreement with the outcome of the Greek discussion.

[14] Camb. Univ. Press, 1920.


NOTE: ON SIGNIFICANCE AND INFINITE EVENTS

The theory of significance has been expanded and made more definite in
the present volume. It had already been introduced in the _Principles of
Natural Knowledge_ (cf. subarticles 3.3 to 3.8 and 16.1, 16.2, 19.4, and
articles 20, 21). In reading over the proofs of the present volume, I
come to the conclusion that in the light of this development my
limitation of infinite events to durations is untenable. This limitation
is stated in article 33 of the _Principles_ and at the beginning of
Chapter IV (p. 74) of this book. There is not only a significance of the
discerned events embracing the whole present duration, but there is a
significance of a cogredient event involving its extension through a
whole time-system backwards and forwards. In other words the essential
'beyond' in nature is a definite beyond in time as well as in space [cf.
pp. 53, 194]. This follows from my whole thesis as to the assimilation
of time and space and their origin in extension. It also has the same
basis in the analysis of the character of our knowledge of nature. It
follows from this admission that it is possible to define point-tracks
[_i.e._ the points of timeless spaces] as abstractive elements. This is
a great improvement as restoring the balance between moments and points.
I still hold however to the statement in subarticle 35.4 of the
_Principles_ that the intersection of a pair of non-parallel durations
does not present itself to us as one event. This correction does not
affect any of the subsequent reasoning in the two books.

I may take this opportunity of pointing out that the 'stationary events'
of article 57 of the _Principles_ are merely cogredient events got at
from an abstract mathematical point of view.




INDEX

_In the case of terms of frequent occurrence, only those occurrences are
indexed which are of peculiar importance for the elucidation of
meaning._

  A [_or_ an], 11

  Abraham, 105

  Absolute position, 105, 106, 114, 188

  Abstraction, 33, 37, 168, 171, 173;
    extensive, 65, 79, 85

  Abstractive element, 84;
    set, 61, 79

  Action at a distance, 159, 190

  Action by transmission, 159, 190

  Active conditions, 158

  Activity, field of, 170, 181

  Adjunction, 101

  Aggregate, 23

  Alexander, Prof., viii

  Alexandria, 71

  Alfred the Great, 137

  Anticipation, 69

  Anti-prime, 88

  Apparent nature, 31, 39

  Area, 99;
    momental, 103;
    vagrant, 103

  Aristotelian logic, 150

  Aristotle, 16, 17, 18, 24, 197

  Associate-potential, 183

  Atom, 17

  Attribute, 21, 26, 150

  Awareness, 3

  Axiom, 36, 121

  Axioms of congruence, 128 et seqq.


  Bacon, Francis, 78

  Behaviouristic, 185

  Bergson, 54

  Berkeley, 28

  Between, 64

  Beyond, 186, 198

  Bifurcation, vi, 30, 185, 187

  Boundary, 100;
    moment, 63;
    particle, 100

  Broad, C. D., viii


  Calculation, formula of, 45, 158

  Cambridge, 97

  Causal nature, 31, 39

  Causation, 31, 146

  Centrifugal force, 138

  Change, uniformity of, 140

  Character, extrinsic, 82, 89, 90, 113, 191;
    intrinsic, 80, 82, 90, 113, 191

  Charge, 160

  Closure of nature, 4

  Coefficient of drag, 133

  Coefficients of impetus, 183

  Cogredience, 110, 188

  Coherence, 29

  Comparison, 124, 125, 143, 189

  Complex, 13

  Conceptual nature, 45;
    space, 96

  Concrete facts, 167, 171, 189

  Conditioning events, 152

  Conditions, active, 158

  Congruence, 65, 96, 118, 120, 127, 196

  Continuity, 157;
    Dedekindian, 102;
    of events, 76;
    of nature, 59, 76

  Convention, 121

  Convergence, 62, 79;
    law of, 82

  Conveyance, 154, 155

  Co-present, 177

  Covering, 83

  Creative advance, 178

  Critical velocity, 193, 195

  Curvature of space-time, 182

  Cyclone, 194


  Dedekindian continuity, 102

  Definite, 53, 194, 198

  Delusions, 31, 38

  Delusive perceptual object, 153

  Demarcation of events, 144

  Demonstrative phrase, 6

  Descriptive phrase, 6, 10

  Differential equations, 196

  Discrimination, 14, 50, 144

  Diversification of nature, 15

  Duddington, Mrs, 47

  Duration, 37, 53, 55, 186

  Durations, families of, 59, 73, 190

  Dynamical axes, 138


  Einstein, vii, 102, 131, 164, 165, 181, 182, 183, 184, 191

  Electromagnetic field, 179

  Electron, 30, 146, 158, 171

  Element, 17;
    abstractive, 84

  Elliptical phraseology, 7

  Empty space, 145

  Entity, 5, 13

  Equal in abstractive force, 83

  Error, 68

  Ether, 18, 78, 160;
    material, 78;
    of events, 78

  Euclid, 85, 94, 197

  Euler, 140

  Event, 15, 52, 75, 165;
    percipient, 107, 152, 186

  Event-particle, 86, 93, 94, 172, 191

  Events, conditioning, 152;
    continuity of, 76;
    demarcation of, 144;
    ether of, 78;
    infinite, 197, 198;
    limited, 74;
    passage of, 34;
    signified, 52;
    stationary, 198;
    stream of, 167;
    structure of, 52, 166

  Exclusion, 186

  Explanation, 97, 141

  Extended nature, 196

  Extension, 22, 58, 75, 185

  Extensive abstraction, 65, 79, 85

  Extrinsic character, 82, 89, 90, 113, 191;
    properties, 62


  Fact, 12, 13

  Factors, 12, 13, 15

  Facts, concrete, 167, 171

  Family of durations, 59, 63, 73;
    of moments, 63

  Faraday, 146

  Field, gravitational, 197;
    of activity, 170, 181;
    physical, 190

  Finite truths, 12

  Fitzgerald, 133

  Formula of calculation, 45, 158

  Foucault, 138, 194

  Four-dimensional manifold, 86

  Fresnel, 133

  Future, the, 72, 177


  Galileo, 139

  Geometrical order, 194

  Geometry, 36;
    metrical, 129

  Gravitation, 179 et seqq.

  Gravitational field, 197

  Greek philosophy, 16;
    thought, 197

  Gyro-compass, 194


  Heath, Sir T. L., 197

  Here, 107


  Idealists, 70

  Immediacy, 52;
    of perception, 72

  Impetus, 181, 182;
    coefficients of, 183;
    integral, 183

  Inclusion, 186

  Individuality, 13

  Infinite events, 197, 198

  Inge, Dr, 48

  Ingredient, 14

  Ingression, 144, 145, 148, 152

  Inherence, 83

  Inside, 106

  Instant, 33, 35, 57

  Instantaneous plane, 91;
    present, 72;
    spaces, 86, 90, 177

  Instantaneousness, 56, 57

  Intersection, locus of, 90

  Intrinsic character, 80, 82, 90, 113, 191;
    properties, 62

  Ionian thinkers, 19

  Irrelevance, infinitude of, 12

  Irrevocableness, 35, 37

  It, 8


  Julius Caesar, 36

  Junction, 76, 101


  Kinetic energy, 105;
    symmetry, 129

  Knowledge, 28, 32


  Lagrange, 140

  Larmor, 131

  Law of convergence, 82

  Laws of motion, 137, 139;
    of nature, 196

  Leibnizian monadology, 150

  Level, 91, 92

  Light, 195;
    ray of, 188;
    velocity of, 131

  Limit, 57

  Limited events, 74

  Location, 160, 161

  Locke, 27

  Locus, 102;
    of intersection, 90

  London, 97

  Lorentz, H. A., 131, 133

  Lossky, 47


  Manifold, four-dimensional, 86;
    space-time, 173

  Material ether, 78;
    object, 169

  Materialism, 43, 70

  Matrix, 116

  Matter, 16, 17, 19, 20, 26

  Maxwell, 131, 133

  Measurableness, 196;
    of nature, 193

  Measurement, 96, 120, 174, 196;
    of time, 65, 140

  Measure-system, 196

  Memory, 68

  Metaphysics, 28, 32

  Metrical geometry, 129

  Michelson-Morley, 195

  Milton, 35

  Mind, 27, 28

  Minkowski, viii, 131

  Molecule, 32, 171

  Moment, 57, 60, 88

  Momental area, 103;
    route, 103

  Momentum, 105

  Motion, 105, 114, 117, 127, 188, 192

  Multiplicity, 22


  Natural philosophy, 29, 30

  Natural science, philosophy of, 46

  Nature, 3;
    apparent, 31, 39;
    causal, 31, 39;
    conceptual, 45;
    continuity of, 59, 76;
    discrimination of, 144;
    extended, 196;
    laws of, 196;
    passage of, 54;
    stratification of, 194, 196;
    system of, 146

  Newton, 27, 136, 139, 140


  Object, 77, 125, 143, 169, 189;
    delusive perceptual, 155;
    material, 169;
    perceptual, 153;
    physical, 155, 157;
    scientific, 158, 169;
    uniform, 162

  Occupation, 22, 34, 36, 100, 101

  Order, source of, 192;
    spatial, 95, 194;
    temporal, 64, 95, 194

  Organisation of thought, 79

  Outside, 63, 100


  Paradox, 192

  Parallel, 63, 127;
    durations, 190

  Parallelism, 95, 191

  Parallelogram, 127

  Paris, 87, 138

  Parliament, 120

  Part, 14, 15, 58

  Passage of events, 34;
    of nature, 54

  Past, the, 72, 177

  Perception, 3

  Perceptual objects, 149, 153

  Percipience, 28

  Percipient event, 107, 152, 186, 187

  Period of time, 51

  Permanence, 144

  Perpendicularity, 117, 127, 193

  Philosophy, 1;
    natural, 29, 30;
    of natural science, 46;
    of the sciences, 2

  Physical field, 190;
    object, 155, 156, 157

  Physics, speculative, 30

  Place, 51

  Plane, 191;
    instantaneous, 91

  Plato, 16, 17, 18, 24, 197

  Poincare, 121, 122, 123

  Point, 35, 89, 91, 114, 173, 176

  Point-flash, 172, 173

  Point of space, 85

  Point, timeless, 192

  Point-track, 113, 198

  Pompey, 36

  Position, 89, 90, 92, 93, 99, 113, 191;
    absolute, 105, 106, 114, 188

  Potential, 183;
    associate-, 183

  Predicate, 18

  Predication, 18

  Present, the, 69, 72, 177;
    instantaneous, 72;
    observational, 186

  Primary qualities, 27

  Prime, 88

  Process, 53, 54;
    of nature, 54

  Psychic additions, 29, 187

  Punct, 92, 93, 94

  Pythagoreans, 197


  Quality, 27

  Quantum of time, 162

  Quantum theory, 162


  Ray of light, 188

  Reality, 30;
    of durations, 55, 187

  Recognition, 124, 143, 189

  Rect, 91, 92

  Recurrence, 35

  Relative motion, 117;
    velocity, 130

  Relativity, 169;
    restricted theory of, 193

  Rest, 105, 114, 188, 192

  Rotation, 138, 194

  Route, 99;
    momental, 103;
    straight, 103

  Russell, Bertrand, 11, 122, 123


  Schelling, 47

  Science, 2;
    metaphysical, 32

  Scientific objects, 149, 158, 169

  Secondary qualities, 27

  Self-congruence, 196

  Self-containedness of nature, 4

  Sense-awareness, 3, 67

  Sense-object, 149, 170

  Sense-perception, 3, 14

  Sense-recognition, 143, 189

  Series, temporal, 66, 70, 85, 178

  Set, abstractive, 61, 79

  Significance, 51, 186, 187, 188, 194, 197, 198

  Signified events, 52

  Simplicity, 163, 173

  Simultaneity, 53, 56, 196

  Situation, 15, 78, 147, 148, 152, 160, 189

  Solid, 99, 101, 102;
    vagrant, 101

  Sound, 195

  Space, 16, 17, 31, 33, 79;
    empty, 145;
    timeless, 86, 106, 114;
    uniformity of, 194

  Spaces, instantaneous, 86, 90

  Space-system, 179

  Space-time manifold, 173

  Spatial-order, 95

  Spatio-temporal structure, 173

  Speculative demonstration, 6

  Speculative physics, 30

  Standpoint for perception, 107, 188

  Station, 103, 104, 113

  Stationary events, 198

  Straight line, 91, 114, 191;
    route, 103

  Stratification of nature, 187, 194, 196

  Stream of events, 167

  Structure of events, 52, 166

  Structure, spatio-temporal, 173

  Subject, 18

  Substance, 16, 18, 19, 150

  Substratum, 16, 18, 21

  Symmetry, 118, 126;
    kinetic, 129

  System of nature, 146

  System, time-, 192


  Tarner, Edward, v, 1

  Temporal order, 64, 95, 194

  Temporal series, 66, 70, 85

  Tensor, 182

  Terminus, 4

  The, 11

  Theory, quantum, 162

  There, 110, 189

  This, 11

  Thought, 3, 14

  Timaeus, the, 17, 20, 24

  Time, 16, 17, 31, 33, 49, 79;
    measurement of, 140;
    quantum of, 162;
    transcendence of, 39

  Time-series, 178, also cf. Temporal series

  Time-system, see Time-series, also 91, 97, 104, 179, 192

  Timeless point, 192;
    space, 86, 106, 114, 177

  Totality, 89

  Transcendence of time, 39

  Transmission, 26, 28;
    action by, 159, 190

  Tubes of force, 146


  Unexhaustiveness, 50

  Uniform object, 162

  Uniformity of change, 140;
    of space, 194


  Vagrant area, 103;
    solid, 101

  Veblen and Young, 36

  Velocity, critical, 193, 195;
    of light, 131, 195;
    relative, 130

  Volume, 92, 101


  When, 107

  Where, 107

  Whole, 58

  Within, 63


  Young, Veblen and, 36





n, 63


  Young, Veblen and, 36






  When, 107

  Where, 107

  Whole, 58

  Within, 63


  Young, Veblen and, 36