I 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 dura- tions 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 itAis 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 indi- vidualities of the entities of an allied group of slightly differing durations. This is one instance of the in- determinateness 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 com-
pletely 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
6o THE CONCEPT OF NATURE [ch.
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 pro-
perties : (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
Iii] Time 6 1
duration of any temporal extension, but in descending the series the temporal extension progressively con- tracts 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 con- sidered. 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
62 THE CONCEPT OF NATUKt i^^n.
difficulty is to express our meaning in terms of the imme-
diate 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^.
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 defini- tion (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.'
1 Cf . An Enquiry concerning the Principles of Natural Knowledge, Cambridge University Press, 1919.
Ill] TIME 6;
The durations which enter into the composition o
a moment all belong to one family. Thus there is on*
family of moments corresponding to one family o
durations. Also if we take two moments of the samt
family, among the durations which enter into the com
position of one moment the smaller durations ar(
completely separated from the smaller durations whicl
enter into the composition of the other moment. Thui
the two moments in their intrinsic properties mus
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 tw(
moments of the associated family of moments whicl
are the boundary moments of that duration, t
'boundary moment' of a duration can be defined ii
this way. There are durations of the same family as th(
given duration which overlap it but are not containe(
in it. Consider an abstractive set of such durations
Such a set defines a moment which is just as mucl
without the duration as within it. Such a moment is ;
boundary moment of the duration. Also we call upoi
our sense-awareness of the passage of nature to inforn
us that there are two such boundary moments, namel;
the earlier one and the later one. We will call them th(
initial and the final boundaries.
There are also moments of the same family such tha
the shorter durations in their composition are entirely
separated from the given duration. Such moments wil
be said to lie ' outside' the given duration. Again othe
moments of the family are such that the shorter dura
tions in their composition are parts of the given dura
tion. Such moments are said to lie 'within' the givei
64 THE CONCEPT OF NATURE lch.
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^ 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 corre-
sponds to retrogression towards the past.
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
^ Cf . Enquiry,
Ill] TIME 65
abstraction. What I have done is to give precise defini- tions 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
66 THE CONCEPT Ul- IN A lURr- l—
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.