and

q(e1), q(e2), q(e3), ..., 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 s.p.a.ce 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 pa.s.s 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 a.s.sumption 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 s.p.a.ce and time, and we now turn to the consideration of the properties involved in the formation of special cla.s.ses of such elements. In my last lecture I have already investigated one cla.s.s 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 s.p.a.ce. The first thing to do is to get hold of the cla.s.s of abstractive elements which are in some sense the points of s.p.a.ce.

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 ent.i.ties as these ideal events. These points will not be the points of an external timeless s.p.a.ce but of instantaneous s.p.a.ces. We ultimately want to arrive at the timeless s.p.a.ce 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 s.p.a.ces 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 s.p.a.ce--will be a cla.s.s of event-particles.

Furthermore there is a separate timeless s.p.a.ce corresponding to each separate temporal series, that is to each separate family of durations.

We will come back to points in timeless s.p.a.ces 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 s.p.a.ce being each a cla.s.s 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'Es.p.a.ce,' _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 s be the name of any condition which some abstractive sets fulfil. I say that an abstractive set is 's-prime' when it has the two properties, (i) that it satisfies the condition s and (ii) that it is covered by every abstractive set which both is covered by it and satisfies the condition s.

In other words you cannot get any abstractive set satisfying the condition s which exhibits intrinsic character more simple than that of a s-prime.

There are also the correlative abstractive sets which I call the sets of s-antiprimes. An abstractive set is a s-antiprime when it has the two properties, (i) that it satisfies the condition s and (ii) that it covers every abstractive set which both covers it and satisfies the condition s. In other words you cannot get any abstractive set satisfying the condition s which exhibits an intrinsic character more complex than that of a s-antiprime.

The intrinsic character of a s-prime has a certain minimum of fullness among those abstractive sets which are subject to the condition of satisfying s; whereas the intrinsic character of a s-antiprime has a corresponding maximum of fullness, and includes all it can in the circ.u.mstances.

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 s be the property of being a cla.s.s 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 s-antiprime, where the condition s has this special meaning. It will be found on consideration (i) that each abstractive set forming a moment is a s-antiprime, where s 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 's' 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 's' 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 s.p.a.ce 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 s.p.a.ce. In another aspect it is got at by ignoring the circ.u.mambient s.p.a.ce, 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 s.p.a.ce, its extrinsic character, and its intrinsic character. The same is true of any other spatial element. For example an instantaneous volume in instantaneous s.p.a.ce 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 s.p.a.ce, we must evidently be quite clear as to what we mean by instantaneous s.p.a.ce in itself. Instantaneous s.p.a.ce 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 s.p.a.ce at that instant. Instantaneous s.p.a.ce must be an a.s.semblage of abstractive elements considered in their mutual relations. Thus an instantaneous s.p.a.ce is the a.s.semblage of abstractive elements covered by some one moment, and it is the instantaneous s.p.a.ce 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 s.p.a.ce 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 a.s.semblage 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 s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce M. Also the common intersection of A, B, and M is the intersection of the two planes in the s.p.a.ce M, namely it is a straight line in the s.p.a.ce 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 a.s.semblage of abstractive elements which are each covered (or 'lie in') all four moments. The three-dimensional property of instantaneous s.p.a.ce 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 s.p.a.ce of a time-system. Accordingly an instantaneous plane in the instantaneous s.p.a.ce 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 a.s.semblage 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 s.p.a.ce 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 const.i.tutes 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 ent.i.ties 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 ent.i.ties posited in sense-awareness. A definite event-particle is defined in reference to a definite punct in the following manner: Let the condition s mean the property of covering all the abstractive elements which are members of that punct; so that an abstractive set which satisfies the condition s is an abstractive set which covers every abstractive element belonging to the punct. Then the definition of the event-particle a.s.sociated with the punct is that it is the group of all the s-primes, where s has this particular meaning.

It is evident that--with this meaning of s--every abstractive set equal to a s-prime is itself a s-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 a.s.sociated with some given punct, which we will call p, it is as follows: The event-particle a.s.sociated with p is the group of abstractive cla.s.ses each of which has the two properties (i) that it covers every abstractive set in p and (ii) that all the abstractive sets which also satisfy the former condition as to p and which it covers, also cover it.

An event-particle has position by reason of its a.s.sociation with a punct, and conversely the punct gains its derived character as a route of approximation from its a.s.sociation 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 cla.s.ses 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 s.p.a.ce 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 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 s.p.a.ce of moment A. Thus the parallelism of moments in a temporal series begets the parallelism of levels in an instantaneous s.p.a.ce, and thence--as it is easy to see--the parallelism of rects. Accordingly the Euclidean property of s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ce is immediately derived from time-order. For consider the s.p.a.ce of a moment M. Let a be the name of a time-system to which M does not belong. Let A1, A2, A3 etc. be moments of a in the order of their occurrences. Then A1, A2, A3, etc.

intersect M in parallel levels l1, l2, l3, etc. Then the relative order of the parallel levels in the s.p.a.ce of M is the same as the relative order of the corresponding 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 instantaneous s.p.a.ce. Accordingly the various modes of deriving spatial order from diverse time-systems must harmonise with one spatial order in each instantaneous s.p.a.ce. In this way also diverse time-orders are comparable.

We have two great questions still on hand to be settled before our theory of s.p.a.ce is fully adjusted. One of these is the question of the determination of the methods of measurement within the s.p.a.ce, in other words, the congruence-theory of the s.p.a.ce. The measurement of s.p.a.ce 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 s.p.a.ce and for time. Secondly there is the determination of the timeless s.p.a.ce which corresponds to any particular time-system with its infinite set of instantaneous s.p.a.ces in its successive moments. This is the s.p.a.ce--or rather, these are the s.p.a.ces--of physical science. It is very usual to dismiss this s.p.a.ce by saying that this is conceptual. I do not understand the virtue of these phrases. I suppose that it is meant that the s.p.a.ce is the conception of something in nature. Accordingly if the s.p.a.ce 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 s.p.a.ce 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 ent.i.ties in nature applies whether s.p.a.ce be relative or absolute. On the theory of relative s.p.a.ce, it may perhaps be argued that there is no timeless s.p.a.ce for physical science, and that there is only the momentary series of instantaneous s.p.a.ces.

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 s.p.a.ce into another s.p.a.ce? 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 s.p.a.ce for that instant is 52 miles from London at 11 o'clock this morning in the appropriate instantaneous s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ces are thus correlated into one s.p.a.ce by any method which is available on the current theories of s.p.a.ce.

You will have noticed that by the aid of the a.s.sumption of alternative time-systems, we are arriving at an explanation of the character of s.p.a.ce. 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 ent.i.ty 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.