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 limita-
tions so as to include the man and the motor during the
period when they were in contact.
CH.iv] METHOD OF EXTENSIVE ABSTRACTION 75
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 inde-
pendence. JWe 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 sur-
rounding 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 ' ^, and ^ is a 'whole of
which 5 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
76 THE CONCEPT OF NATURE [ch.
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^.
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
^ Of. Enquiry.
Iv] Method of Extensive Abstraction 77
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 con-
tinuity.
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 through-
78 THE CUNUErr~u¥~rnrTTTRc. l^^n.
out 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 some- thing 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 phraseo- logy 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
Iv] Method of Extensive Abstraction 79
utilised in science is what I have elsew^here ^ called the
law^ of convergence to simplicity by diminution of
extent.
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 pro-
cedure is identical in principle in both cases, and I
have called the general t}'pe 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 con- tains 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
^ Cf. Organisation of Thought, pp. 146 et seq. Williams and Norgate, 1917.
8o THE CONCEPT Ut JNAi uku l^«.
event nor does it converge to a limiting event which is ftot a member of the set.
Thus, so far as the abstractive sets of events are con- cerned, an abstractive set converges to nothing. There is the set with its members grovidng 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 con- siderable spatio-temporal extension this set of quanti- tative expressions is of bewildering complexity. If e be an event, let us denote by q (e) the set of quanti- tative expressions defining its character including its connexions with the rest of nature. Let e^, €2, e^, etc. be an abstractive set, the members being so arranged that each member such as e^ extends over all the suc- ceeding members such as e^+i, e^+^y and so on. Then corresponding to the series
^l> ^2' ^3' •••■> ^ni ^n + lf •••>
there is the series
Q (^i)> Q («2)> q (^3). •••> ? (en), q (e^+i), ....
Call the series of events 5 and the series of quanti-
tative expressions q (s). The series 5 has no last term and
IV] METHOD OF EXTENSIVE ABSTRACTION 8i
no events w^hich are contained in every member of the
series. Accordingly the series of events converges to
nothing. It is just itself. Also the series q {$) 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^^ be a
quantitative measurement found in q {e^, and Q^ the
homologue to Q^ to be found in q {e^, and Q^ the
homologue to Q^ and Q^ to be found in q {e^, and so on,
then the series
yi> ^2' >CZy '"■> icny >^n+li •••>