27. The Relation of Extension,Fundamental Properties. 27.1 The fact that event extends over event will be expressed by the abbreviation . Thus '' is to be read 'extends over' and is the symbol for the fundamental relation of extension.
27.2 Some properties of essential for the method of extensive abstraction are,
(i) implies that is distinct from , namely, 'part' here means 'proper part':
(ii) Every event extends over other events and is itself part of other events: the set of events which an event e extends over is called the set of parts of :
(iii) If the parts of are also parts of and and are distinct, then :
(iv) The relation is transitive, i.e. if and , then :
(v) If , there are events such as where and :
(vi) If and are any two events, there are events such as where and .
It follows from (i) and (iv) that and are inconsistent. Properties (ii) and (v) and (vi) together postulate something like the existence of an ether; but it is not necessary here to pursue the analogy.
28. Intersection,Separation and Dissection. 28.1 Two events 'intersect' when they have parts in common. Intersection, as thus defined, includes the case when one event extends over the other, since is transitive. If every intersector of also intersects , then either or and are identical.
Events which do not intersect are said to be 'separated.' A 'separated set' of events is a set of events of which any two are separated from each other.
28.2 A 'dissection' of an event is a separated set such that the set of intersectors of its members is identical with the set of intersectors of the event. Thus a dissection is a non-overlapping exhaustive analysis of an event into a set of parts, and conversely the dissected event is the one and only event of which that set is a dissection. There will always be an indefinite number of dissections of any given event.
If , there are dissections of of which is a member. It follows that if is part of , there are always events separated from which are also parts of .
29. The Junction of Events. 29.1 Two events and are 'joined' when there is a third event such that (i) intersects both and , and (ii) there is a dissection of of which each member is a part of , or of , or of both.
The concept of the continuity of nature arises entirely from this relation of the junction between two events. Two joined events are continuous one with the other. Intersecting events are necessarily joined; but the notion of junction is wider than that of intersection, for it is possible for two separated events to be joined. Two events which are joined have that relation to each other necessary for the existence of one event which extends over them and over no extraneous events. Two events which are both separated and joined are said to be 'adjoined.'
29.2 An event is said to 'injoin' an event when (i) extends over , and (ii) there is some third event which is separated from and adjoined to .
Fig. 4.
In this definition a property of the boundary of an event first makes its appearance. The assumption that examples of the relation of injunction hold is a long step towards a theory of such boundaries, as the annexed diagram illustrates. It is important to note that injunction has been defined purely in terms of extension.
If and is separated from and adjoins , then adjoins .
29.3 Injunction and adjunction are the closest types of boundary union possible respectively for an event with its part and for a pair of separated events. The geometry for events is four-dimensional, but in the three-dimensional analogue such a surface union for a pair of volumes would be the existence of a finite area of surface in common.
[Note that spatial diagrams, such as the one above, are to some extent misleading in that they emphasise the spatial character of events at the expense of their temporal character. The temporal character is very far from being represented by an extra dimension producing an ordinary four-dimensional euclidean geometry.]
30. Abstractive Classes. 30.1 A set of events is called an 'abstractive class' when (i) of any two of its members one extends over the other, and (ii) there is no event which is extended over by every event of the set.
The properties of an abstractive class secure that its members form a series in which the predecessors extend over their successors, and that the extension of the members of the series (as we pass towards the 'converging end' comprising the smaller members) diminishes without limit; so that there is no end to the series in this direction along it and the diminution of the extension finally excludes any assignable event. Thus any property of the individual events which survives throughout members of the series as we pass towards the converging end is a property belonging to an ideal simplicity which is beyond that of any one assignable event. There is no one event which the series marks out, but the series itself is a route of approximation towards an ideal simplicity of 'content.' The systematic use of these abstractive classes is the 'method of extensive abstraction.' All the spatial and temporal concepts can be defined by means of them.
30.2 One class of events—, say—is said to 'cover' another class of events—, say—when every member of a extends over some member of .
If a be an abstractive class and covers , then must have an infinite number of members and there can be no event which is extended over by every member of . For any member of , however small, extends over some member of . The usual case of covering is when both classes, and , are abstractive classes; then each member of , the covering class, extends over the whole converging end of subsequent to the first member of which it extends over.
30.3 Two classes of events are called '-equal' when each covers the other. Evidently such classes cannot have a finite number of members. Inequality is a relation in which two abstractive classes can stand to each other. The relation is symmetrical and transitive, and every abstractive class is -equal to itself.
[Note. Abstractive classes and the relation of 'covering' can be illustrated by spatial diagrams, with the same caution as to their possibly misleading character.
fig5Fig. 5.
Consider a series of squares, concentric and similarly situated. Let the lengths of the sides of the successive squares, stated in order of diminishing size, be Then each square extends over all the subsequent squares of the set. Also let namely, let tend to zero as increases indefinitely. Then the set forms an abstractive class.
Again, consider a series of rectangles, concentric and similarly situated. Let the lengths of the sides of the successive rectangles, stated in order of diminishing size, be (), (),...(),....
Thus one pair of opposite sides is of the same length throughout the whole series. Then each rectangle extends over all the subsequent rectangles. Let , tend to zero as increases indefinitely. Then the set forms an abstractive class.
Evidently the set of squares converges to a point, and the set of rectangles to a straight line. Similarly, using three dimensions and volumes, we can thus diagrammatically find abstractive classes which converge to areas. If we suppose the centre of the set of squares to be the same as that of the set of rectangles, and place the squares so that their sides are parallel to the sides of the rectangles, then the set of rectangles covers the set of squares, but the set of squares does not cover the set of rectangles.
Again, consider a set of concentric circles with their common centre at the centre of the squares, and let each circle be inscribed in one of the squares, and let each square have one of the circles inscribed in it. Then the circles form an abstractive class converging to their common centre. The set of squares covers the set of circles and the set of circles covers the set of squares. Accordingly the two sets are -equal.]
31. Primes and Antiprimes. 31.1 An abstractive class is called 'prime in respect to the formative condition ' [whatever condition '' may be] when (i) it satisfies the condition , and (ii) it is covered by every other abstractive class satisfying the same condition .
For brevity an abstractive class which is prime in respect to a formative condition is called a -prime. Evidently two -primes, with the same formative condition in the two cases, are -equal.
31.2 An abstractive class is called 'antiprime in respect to the formative condition ' [whatever condition '' may be] when (i) it satisfies the condition , and (ii) it covers every other abstractive class satisfying the same condition . For brevity an abstractive class which is antiprime in respect to a formative condition is called a -antiprime. Evidently two -antiprimes, with the same formative condition in the two cases, are -equal.
31.3 Let be any assigned formative condition, let be the condition of 'being a -prime,' and let be the condition of 'being a -antiprime.' Thus an abstractive class, which satisfies the condition , (i) satisfies the condition , and (ii) is covered by every other abstractive class satisfying the same condition .
Hence any two abstractive classes which satisfy the condition cover each other. Hence every class which satisfies the condition is covered by every other class which satisfies the same condition . That is to say, every such class is a -prime. Analogously, it is a -antiprime.
Similarly the -antiprimes are the -primes and -antiprimes.
A formative condition will be called 'regular for primes' when (i) there are -primes and (ii) the set of abstractive classes -equal to any one assigned -prime is identical with the complete set of -primes; and will be called 'regular for antiprimes' when (i) there are -antiprimes and (ii) the set of abstractive classes -equal to any one assigned -antiprime is identical with the complete set of -antiprimes. Thus if be a formative condition regular for primes, the set of -primes is the same as the set of abstractive classes -equal to -primes; and if be a formative condition regular for antiprimes, the set of -antiprimes is the same as the set of abstractive classes -equal to -antiprimes.
31.4 Errors arise unless we remember the existence of some exceptional abstractive classes. Since we assume that each event has a definite demarcation we know that the laws of nature ordinarily assumed in science will issue in ascribing to each event a definite boundary which will be a spatial surface prolonged into three dimensions by reason of its time-extension. Thus the possibilities of the spatial contact of surfaces are reproduced in the three-dimensional boundaries of events. Abstractive classes exist whose converging ends converge to elements [instantaneous points, or routes, or etc.] on the surface of one of the members of the class. In such a case, as we pass down the abstractive class towards its converging end, after some definite member of the class the remaining members, all extended over by , have some form of internal contact with the boundary of . The closest form of such contact is to be injoined in . But there will also be more abstract types of point-contact or of line-contact which we have not defined here, but know about from their occurrence in geometry. If we merely exclude such cases without explicit definition, we are really appealing to fundamental relations and properties which have not been explicitly recognised. We must use definitions based solely upon those properties of the relation which have been made explicit. We cannot explicitly take account of point-contact till points have been defined.
32. Abstractive Elements. 32.1 A 'finite abstractive element deduced from the formative condition ' is the set of events which are members of -primes, where is a formative condition regular for primes. The element is said to be 'deduced' from its formative condition .
An 'infinite abstractive element deduced from the formative condition ' is the set of events which are members of -antiprimes, where is a formative condition regular for antiprimes. The element is said to be 'deduced' from its formative condition .
The abstractive elements are the set of finite and infinite abstractive elements.
32.2 An abstractive element deduced from a regular formative condition is such that every abstractive class formed out of its members either covers all -primes [element finite] or is covered by all -antiprimes [element infinite]. Thus it represents a set of equivalent routes of approximation guided by the condition that each route is to satisfy the condition .
32.3 An abstractive element will be said to 'inhere' in any event which is a member of it. Two elements such that there are abstractive classes covered by both are said to 'intersect' in those abstractive classes.
One abstractive element may cover another abstractive element. The elements of the utmost simplicity will be those which cover no other abstractive elements. These are elements which in euclidean phrase may be said to be 'without parts and without magnitude.' It will be our business to classify some of the more important types of elements. The elements of the greatest complexity will be those which can cover elements of all types. These will be 'moments.'
A point of nomenclature is important. We shall name individual abstractive elements by capital latin letters, classes of elements by capital or small latin letters, and also, as heretofore, events by small latin letters. will continue to denote the fundamental relation of extension from which all the relations here considered are derived.