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 con- ceptual. I do not understand the virtue of these phrases. I suppose that it is meant that the space is the concep- tion 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 pro- duction of the relevant entities in nature applies whether space be relative or absolute. On the theory of relative
Iv] Method of Extensive Abstraction 97
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 vs^alked 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 Cam-
bridge 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 under-
stand 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 assump- tion 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 dis- covered interconnexions between red as a factor in nature and other factors in nature, for example waves of light which are waves of electromagnetic disturbances.
W.N. 7
98 THE CONCEPT OF NATURE [ch.iv
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 con-
nexions constitutes the scientific explanation of our
vision of colour. In like manner the dependence of
the character of space on the character of time con-
stitutes 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 ab-
stracts 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 con-
struction 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 ex-
position of these definitions and the preliminary ex-
planations 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 a. There will be one moment of
the temporal series of a which covers the given event-
particle. Thus the position of the event-particle in the
temporal series a is defined by this moment, which we
loo THE CONCEPT OF NATURE [CH.
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-dimen-
V] SPACE AND MOTION loi
sional 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, pro-
vided 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 intersect-
ing 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 illus- trating 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 a stand for the fact that each of the events of any abstractive set satisfying it has all the event-particles of some particular soUd lying
102 THE CONCEPT OF NATURE [ch.
in it. Then the group of all the a -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 pre-
vents 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
v.] SPACE AND MOTION 103
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 por-
tions of surfaces are 'momental areas.'
It is unnecessary at this stage to enter into the com-
plexity 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 abstrac-
tive 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 investiga-
tion 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 re-
ference to the definitions of volumes and surfaces.