Iv] Method of Extensive Abstraction 89
where a 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 V 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 ' a ' 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 ob- tained by distinguishing between the notion of 'posi- tion ' 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
90 THE CONCEPT OF NATURE [ch.
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 diiferent
families necessarily intersect. Accordingly in the in-
Iv] Method of Extensive Abstraction 91
stantaneous space of a moment we should expect the
fundamental properties to be marked by the inter-
sections 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 instan-
taneous space of M; and if 5 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 ^, 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
92 THE CONCEPT OF NATURE [ch.
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 ab-
stractive 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 approxi- mated 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
Iv] Method of Extensive Abstraction 93
to exhibit this character of being a route of approxi-
mation marked out by entities posited in sense-aware-
ness. A definite event-particle is defined in reference
to a definite punct in the foUovsdng manner: Let the
condition a mean the property of covering all the
abstractive elements which are members of that punct ;
so that an abstractive set which satisfies the condition a
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 or -primes, where a has this
particular meaning.
It is evident that— with this meaning of a — every abstractive set equal to a a -prime is itself a o- -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 tt, it is as follows: The event-particle as- sociated vdth 77 is the group of abstractive classes each of which has the two properties (i) that it covers every abstractive set in -n and (ii) that all the abstractive sets which also satisfy the former condition as to 77 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 wdth the event-particle. These two
characters of a point are always recurring in any treat-
ment of the derivation of a point from the observed
facts of nature, but in general there is no clear recogni-
tion of their distinction.
THE CONCEPT OF NATURE [ch.
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 con-
nected 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 con-
sidering them since we gain no additional simplicity
in the expression of natural properties.
These two characters of simplicity enjoyed respec-
tively 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 pre-
ference to 'puncts,' the latter being an artificial word
for which I have no great affection.
Iv] Method of Extensive Abstraction 95
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 a. Consider any other
temporal series of moments which I will call ^. The
moments of /3 do not intersect each other and they
intersect the moment ^ 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 hyper-
bolic 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 a be the name of a time- system to which M does not belong. Let A^^, A^, A^, etc. be moments of a in the order of their occurrences. Then Af^, A2, A^, etc. intersect M in parallel levels /j, 4, 4, etc. Then the relative order of the parallel levels in the space of M is the same as the relative order of the corre- sponding moments in the time-system a. 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 instan-
96 THE CONCEPT OF NATUKt l^w.
taneous 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 timie-orders are com-
parable.