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
limitations so as to include the man and the motor during the period
when they were in contact.
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 independence. We 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 surrounding 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' A, and A is a 'whole' of
which B 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 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[7]. 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 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 continuity.
[7] Cf. _Enquiry_.
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 throughout 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 something 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
phraseology 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 utilised in science is what I have elsewhere[8] called
the law of convergence to simplicity by diminution of extent.
[8] Cf. _Organisation of Thought_, pp. 146 et seq. Williams and Norgate,
1917.
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 procedure is identical in
principle in both cases, and I have called the general type 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 contains 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 event
nor does it converge to a limiting event which is not a member of the
set.
Thus, so far as the abstractive sets of events are concerned, an
abstractive set converges to nothing. There is the set with its members
growing 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 considerable spatio-temporal extension
this set of quantitative expressions is of bewildering complexity. If
e be an event, let us denote by q(e) the set of quantitative expressions
defining its character including its connexions with the rest of nature.
Let e_1, e_2, e_3, etc. be an abstractive set, the members being so
arranged that each member such as e_{n} extends over all the succeeding
members such as e_{n+1}, e_{n+2} and so on. Then corresponding to the
series
e_1, e_2, e_3, ..., e_{n}, e_{n+1}, ...,
there is the series
q(e_1), q(e_2), q(e_3), ..., q(e_{n}), q(e_{n+1}), ....
Call the series of events s and the series of quantitative expressions
q(s). The series s has no last term and no events which are contained
in every member of the series. Accordingly the series of events
converges to nothing. It is just itself. Also the series q(s) 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_1 be a quantitative measurement found in q(e_1), and Q_2 the homologue
to Q_1 to be found in q(e_2), and Q_3 the homologue to Q_1 and Q_2 to be
found in q(e_3), and so on, then the series
Q_1, Q_2, Q_3, ..., Q_{n}, Q_{n+1}, ...,
though it has no last term, does in general converge to a definite
limit. Accordingly there is a class of limits l(s) which is the
class of the limits of those members of q(e_{n}) which have
homologues throughout the series q(s) as n indefinitely increases.
We can represent this statement diagrammatically by using an arrow (-->)
to mean 'converges to.' Then
e_1, e_2, e_3, ..., e_{n}, e_{n+1}, ... --> nothing,
and
q(e_1), q(e_2), q(e_3), ..., q(e_{n}), q(e_{n+1}), ... --> l(s).
The mutual relations between the limits in the set l(s), and also
between these limits and the limits in other sets l(s'), l(s"), ...,
which arise from other abstractive sets s', s", etc., have a peculiar
simplicity.
Thus the set s 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, 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 extrinsic character of
an abstractive set determines a definite intrinsic character is the
reason of the importance 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 baffles 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 converge 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 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 convergence which lead to the approximation to
different types of intrinsic characters as limits.
We now pass to the investigation of possible connexions 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 extension every succeeding member of
the small end of q is part of e. 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 p.
Two abstractive sets may each cover the other. When this is the case I
shall call the two sets 'equal in abstractive force.' When there is no
danger of misunderstanding I shall shorten this phrase by simply saying
that the two abstractive sets are 'equal.' The possibility 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 p, 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 p we can find an event z which is part of y, 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 abstractive 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 p covers an abstractive set q, then any
abstractive set belonging to the abstractive element of which p is a
member will cover any abstractive set belonging to the element of which
q 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 abstractive 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 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 of A in a sense includes the intrinsic character
of B. 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 explains 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. Euclid has expressed for all
time the general idea of a point, 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 represent 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 timeless 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 corresponding 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 having 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 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[9]. There is however
a difficulty involved in this definition if adopted without some further
addition, and I am now not satisfied with the way in which I attempted
to get over that difficulty in the paper referred to.
[9] Cf. 'La Theorie Relationniste de l'Espace,' _Rev. de Metaphysique et
de Morale_, vol. XXIII, 1916.
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 {sigma} be the name of any condition which some abstractive sets
fulfil. I say that an abstractive set is '{sigma}-prime' when it has the
two properties, (i) that it satisfies the condition {sigma} and (ii)
that it is covered by every abstractive set which both is covered by it
and satisfies the condition {sigma}.
In other words you cannot get any abstractive set satisfying the
condition {sigma} which exhibits intrinsic character more simple than
that of a {sigma}-prime.
There are also the correlative abstractive sets which I call the sets of
{sigma}-antiprimes. An abstractive set is a {sigma}-antiprime when it
has the two properties, (i) that it satisfies the condition {sigma} and
(ii) that it covers every abstractive set which both covers it and
satisfies the condition {sigma}. In other words you cannot get any
abstractive set satisfying the condition {sigma} which exhibits an
intrinsic character more complex than that of a {sigma}-antiprime.
The intrinsic character of a {sigma}-prime has a certain minimum of
fullness among those abstractive sets which are subject to the condition
of satisfying {sigma}; whereas the intrinsic character of a
{sigma}-antiprime has a corresponding maximum of fullness, and includes
all it can in the circumstances.
Let us first consider what help the notion of antiprimes could give us
in the definition of moments which we gave in the last lecture. Let the
condition {sigma} 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
{sigma}-antiprime, where the condition {sigma} has this special meaning.
It will be found on consideration (i) that each abstractive set forming
a moment is a {sigma}-antiprime, where {sigma} 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 '{sigma}' 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 '{sigma}' 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 obtained by distinguishing between the notion of 'position'
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 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
different families necessarily intersect. Accordingly in the
instantaneous space of a moment we should expect the fundamental
properties to be marked by the intersections 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 instantaneous space of M; and
if B 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
A, 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 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 abstractive 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 approximated 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 to exhibit this
character of being a route of approximation marked out by entities
posited in sense-awareness. A definite event-particle is defined in
reference to a definite punct in the following manner: Let the condition
{sigma} mean the property of covering all the abstractive elements which
are members of that punct; so that an abstractive set which satisfies
the condition {sigma} 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 {sigma}-primes, where {sigma} has this particular meaning.
It is evident that--with this meaning of {sigma}--every abstractive set
equal to a {sigma}-prime is itself a {sigma}-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 {pi}, it is as
follows: The event-particle associated with {pi} is the group of
abstractive classes each of which has the two properties (i) that it
covers every abstractive set in {pi} and (ii) that all the abstractive
sets which also satisfy the former condition as to {pi} 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 with the event-particle. These two
characters of a point are always recurring in any treatment of the
derivation of a point from the observed facts of nature, but in general
there is no clear recognition of their distinction.
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 connected 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 considering them since we gain no additional
simplicity in the expression of natural properties.
These two characters of simplicity enjoyed respectively 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 preference
to 'puncts,' the latter being an artificial word for which I have no
great affection.
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 {alpha}.
Consider any other temporal series of moments which I will call {beta}.
The moments of {beta} do not intersect each other and they intersect the
moment A 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 hyperbolic 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 {alpha} be
the name of a time-system to which M does not belong. Let A_1, A_2, A_3
etc. be moments of {alpha} in the order of their occurrences. Then A_1,
A_2, A_3, etc. intersect M in parallel levels l_1, l_2, l_3, etc. Then
the relative order of the parallel levels in the space of M is the same
as the relative order of the corresponding moments in the time-system
{alpha}. 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 instantaneous 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 time-orders
are comparable.
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 conceptual. I do not understand the virtue of these
phrases. I suppose that it is meant that the space is the conception 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 production of the relevant
entities in nature applies whether space be relative or absolute. On the
theory of relative 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 walked 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 Cambridge 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 understand 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 assumption 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
discovered interconnexions between red as a factor in nature and other
factors in nature, for example waves of light which are waves of
electromagnetic disturbances. 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 connexions constitutes the scientific explanation
of our vision of colour. In like manner the dependence of the character
of space on the character of time constitutes 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.