though it has no last term, does in general converge to a definite limit. Accordingly there is a class of limits l{s) w^hich is the class of the limits of those members of g'(e„) v^hich have homologues throughout the series q{s) as n indefinitely increases. We can represent this state- ment diagrammatically by using an arrow (-*) to mean 'converges to.' Then
ei, e^, ^3, ..., Bn, e^+i, ... ■* nothing, and
?(«i). 9(^2)'? (^3). •■■,q(en),qien+i), ...^lis).
The mutual relations betv^^een the limits in the set
l{s), and also betvsreen these limits and the limits in
other sets l{s'), lis"), ..., v^hich arise from other
abstractive sets s', s", etc., have a pecuHar simplicity.
Thus the set 5 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.
82 THE CONCEPT OF NATURE [ch.
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 ex- trinsic character of an abstractive set determines a definite intrinsic character is the reason of the import- ance 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 bafiles 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 con- verge 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
Iv] Method of Extensive Abstraction 83
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 con- vergence which lead to the approximation to different types of intrinsic characters as limits.
We now pass to the investigation of possible con- nexions 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 ex- tension every succeeding member of the small end of q is part oie. 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/).
Two abstractive sets may each cover the other. When
this is the case I shall call the two sets 'equal in ab-
stractive force.' When there is no danger of misunder-
standing I shall shorten this phrase by simply saying
that the two abstractive sets are ' equal.' The possibility
84 THE CONCEPT OF NATURE L^h.
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 ^, 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 ^ we can find an event z which is part of jy, 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 ab-
stractive 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^ covers an abstractive set q, then any abstractive set belonging to the abstractive element of which ^ is a member will cover any abstractive set belonging to the element of which ^ 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 ab- stractive 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
Iv] Method of Extensive Abstraction 85
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 oi A in a. sense
includes the intrinsic character of 5. 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 ex- plains 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. EucHd
has expressed for all time the general idea of a point,
86 THE CONCEPT OF NATURE [CH.
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 repre- sent 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 time- less 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 corre-
sponding 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 hiving 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
Iv] Method of Extensive Abstraction 87
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^
There is however a difficulty involved in this definition
if adopted without some further addition, and I am now
not satisfied vdth the way in which I attempted to get
over that difficulty in the paper referred to.
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 a be the name of any condition which some abstractive sets fulfil. I say that an abstractive set is
1 Cf. 'La Thdorie Relationniste de I'Espace,' Rev. de MSta- physique et de Morale, vol. xxiii, 1916.
88 THE CONCEPT OF NATURE [ch.
* a -prime' when it has the two properties, (i) that it
satisfies the condition o and (ii) that it is covered by
every abstractive set which both is covered by it and
satisfies the condition a .
In other words you cannot get any abstractive set
satisfying the condition a which exhibits intrinsic
character more simple than that of a a -prime.
There are also the correlative abstractive sets which
I call the sets of a-antiprimes. An abstractive set is a
cr -antiprime when it has the two properties, (i) that it
satisfies the condition a and (ii) that it covers every
abstractive set which both covers it and satisfies the
condition a. In other words you cannot get any ab-
stractive set satisfying the condition a which exhibits
an intrinsic character more complex than that of a
CT -antiprime.
The intrinsic character of a o-prime has a certain
minimum of fullness among those abstractive sets which
are subject to the condition of satisfying a\ whereas
the intrinsic character of a a -antiprime has a corre-
sponding maximum of fullness, and includes all it can
in the circumstances.
Let us first consider what help the notion of anti-
primes could give us in the definition of moments
which we gave in the last lecture. Let the condition
«T 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
<T-antiprime, where the condition a has this special
meaning. It will be found on consideration (i) that
each abstractive set forming a moment is a cr-antiprime,