CHAPTER IXDURATIONS, MOMENTS AND TIME-SYSTEMS33. Antiprimes, Durations and Moments.33.1Among the constants of externality discussed inPart IIwas the reference of events to durations which are, in a sense, complete wholes of nature. A duration has thus in some sense an unlimited extension, though it is bounded in its temporal extent. Although we have not yet in our investigation ofdistinguished between spatial and temporal extension, durations can nevertheless be defined in terms ofby this unlimited aspect of their extents. Namely, we assume that there are no other events with the same unlimited property. Accordingly, any abstractive class which is composed purely of durations can only be covered by abstractive classes which also are composed purely of durations.33.2An abstractive classis called an 'absolute antiprime' whenis itself one of the antiprimes which satisfy the formative condition of covering. In other words, an absolute antiprime is an abstractive class which covers every abstractive class which covers it.If an abstractive class be an absolute antiprime, it is evident that the formative condition of 'covering it' is regular for antiprimes. Thus the set of events which are members of the absolute antiprimes which cover some one assigned absolute antiprime constitutes an abstractive element. Such an element will be called a 'moment.' Thus a moment is an abstractive element deduced from the condition of covering an absolute antiprime.Only events of a certain type can be members of an absolute antiprime, namely events which inPart IIhave been called 'durations.' Only durations can extend over durations, and accordingly all the members of a moment are durations.33.3 We may conceive of a duration as a sort of temporal thickness (or, slab) of nature[6]. In an absolute antiprime we have a series of temporal thicknesses successively packed one inside the other and converging towards the ideal of no thickness. An absolute antiprime indicates the ideal of an extensionless moment of time.[6]The slab of nature forming a duration is limited in its temporal dimension and unlimited in its spatial dimensions. Thus it represents a finite time and infinite space.fig07Fig. 7.For example let the horizontal line represent the time; and assume nature to be spatially one-dimensional, so that an unlimited vertical line in the diagram represents space at an instant.fig08Fig. 8.Then the area between the unlimited parallel linesandrepresents a duration. Also the area betweenandrepresents another duration which is extended over by the duration bounded byand. But infig. 7we have assumed only one time-system, which is the Newtonian hypothesis. Suppose there are many time-systems and consider two such systemsand. These are represented by two lines inclined to each other. A duration of time-systemis represented by the area betweenand, and a duration of time-systemis represented by the area betweenand. Two such durations necessarily intersect and also can neither completely extend over the other.These diagrams are crude illustrations of some properties of durations and are in many respects misleading as the sequel will show.The set of moments which inhere in a duration are completely characteristic of that duration, and vice versa. A moment is to be conceived as an abstract of all nature at an instant. No abstractive element can cover a moment except that moment itself. A moment is a route of approximation to all nature which has lost its (essential) temporal extension; thus it is nature under the aspect of a three-dimensional instantaneous space. This is the ideal to which we endeavour to approximate in our exact observations.34. Parallelism and Time-Systems.34.1If the Newtonian theory of relativity were true, no pair of durations would lack durations extending over both of them, namely larger durations including both the given durations. But on the electromagnetic theory of relativity this is not necessarily the case, namely some pairs of durations are extended over by a family of durations and some are not. We shall adopt the electromagnetic theory of relativity.A pair of durations both of which are parts of the same duration are called 'parallel'; and also a pair of moments such that there are durations in which both inhere are called 'parallel.'Parallelism has the usual properties of transitiveness, symmetry and reflexiveness. Also two durations which do not intersect are parallel; and parallel moments which are not identical never intersect. If two parallel durations intersect there is a duration which is their complete intersection, but there are no durations among the common parts of two durations which are not parallel. Two moments which are not parallel necessarily intersect.34.2Two durations which are parallel to the same duration are parallel to each other; thus it is evident that each absolute antiprime and each moment must be composed of parallel durations.A 'family of parallel durations' is formed by all the durations parallel to a given duration, including that duration itself. Evidently any two members of such a family are parallel, and no duration out of the family is parallel to any duration of the family.Analogously to such families of parallel durations, there are families of parallel moments, with the property that no two moments of the same family intersect and that any moment out of a given family intersects every moment belonging to the family.The durations which are the members of the various moments of a given family of moments themselves form a family of parallel durations. Thus corresponding to a family of parallel durations there is one and only one family of parallel moments; and corresponding to a family of parallel moments there is one and only one family of parallel durations. A pair of such corresponding families, one of durations and the other of moments, form the 'time-system' associated with either of the two families.Evidently each duration belongs to one and only one family of parallel durations; and thus each duration belongs to one and only one time-system. Also each moment belongs to one and only one family of parallel moments; and thus each moment belongs to one and only one time-system. Thus two distinct time-systems have no durations in common and no moments in common. But every event not a duration is contained in some durations of any given time-system. Furthermore there will be a minimum duration in a given time-system which is the duration 'when' the event happened in that time-system; namely, the minimum duration has the properties (i) that it extends over the event and (ii) that every duration which is part of it intersects the event.34.3The moments of a time-system are arranged in serial order in this way:(i) A duration belonging to a time-system is 'bounded' by a moment of the same time-system when each duration in which that moment inheres intersects the given duration and also intersects events separated from the given duration:(ii) Every duration has two such bounding moments, and every pair of parallel moments bound one duration of that time-system:(iii) A momentof a time-system 'lies between' two momentsandof the same time-system wheninheres in the duration whichandbound:(iv) This relation of 'lying between' has the following properties which generate continuous serial order in each time-system, namely,() Of any three moments of the same time-system, one of them lies between the other two:() If the momentlies between the momentsand, and the momentlies between the momentsand, thenlies betweenand:() There are not four moments in the same time-system such that one of them lies between each pair of the remaining three:() The serial-order among moments of the same time-system has the Cantor-Dedekind type of continuity.Nothing has yet been said about the measurement of the lapse of time. This topic will be considered as part of the general theory of congruence.35. Levels, Rects, and Puncts.35.1The electromagnetic theory of relativity is obviously the more general of the two. It has also the merit of providing definitions of flatness, of straightness, of punctual position, of parallelism, of time-order and spatial order as interconnected phenomena, and (with the help of cogredience) of perpendicularity and of congruence. The theory of extension has also provided the definition of a duration. It is a remarkable fact that the characteristic concepts of time and of geometry should thus be exhibited as arising out of the nature of things as expressed by the two fundamental relations of extension and cogredience. It has already been explained that a moment is the route of approximation towards an instantaneous three-dimensional whole of nature. The set of abstractive elements and abstractive classes covered by both of two non-parallel moments is the locus which is their common intersection. Such a locus will be called a 'level' in either moment. A level is in fact an instantaneous plane in the instantaneous space of any moment in which it lies. But we reserve the conventional spatial terms, such as 'plane,' for the time-less spaces to be defined later. Accordingly the word 'level' is used here.35.2An indefinite number of non-parallel moments will intersect each other in the same level, forming their complete intersection; and one level will never be merely a (logical) part of another level. Let three mutually intersecting moments (, say) intersect in the levels. Then three cases can arise:either(i) the levels are all identical [this will happen if any two are identical],or(ii) no pair of the levels intersect,or(iii) a pair of the levels, sayand, intersect. In case (i) the three moments are called 'co-level.' In case (ii) there are special relations of parallelism of levels, to be considered later. In case (iii) the locus of abstractive elements and abstractive classes which forms the intersection ofandwill be called a 'rect'; let this rect be named. Thenis also the complete intersection ofand, and of and, and of the three moments. When three moments have a rect as their complete intersection they are called 'co-rect.' A rect is an instantaneous straight line in the instantaneous three-dimensional space of any moment in which it lies. But, as before, the conventional space-nomenclature is avoided in connection with instantaneous spaces.35.3For four distinct moments there are four possible cases in respect to their intersection. In case (i) there is no common intersection: in case (ii) there is a common intersection which is a level: in case (iii) there is a common intersection which is a rect: in case (iv) there is a common intersection which is neither a rect nor a level; in this case the common intersection will be called a 'punct.'Consider four momentswhich constitute an instance of case (iv). Letbe the level which is the intersection ofand, and let be the rect which is the intersection of. Then the rectdoes not lie in the level. The rectintersects the levelin the common intersection of the four moments. This common intersection is an instantaneous point in the instantaneous spaces of the moments. In accordance with our practice of avoiding the conventional spatial terms when speaking of an instantaneous space, we have called this intersection a 'punct.' Since space is three-dimensional, any moment either covers every member of a given punct or covers none of its members. A punct represents the ideal of the maximum simplicity of absolute position in the instantaneous space of a moment in which it lies.35.4It is tempting, on the mathematical analogy of four-dimensional space, to assert the existence of unlimited events which may be called the complete intersections of pairs of non-parallel durations. It is dangerous however blindly to follow spatial analogies; and I can find no evidence for such unlimited events, forming the complete intersections of pairs of intersecting durations, except in the excluded case of parallelism when the complete intersection (if it exist) is itself a duration. Accordingly, apart from parallelism, it may be assumed that the events extended over by a pair of intersecting durations are all finite events. No change in the sequel is required if the existence of such infinite events be asserted.36. Parallelism and Order.36.1Two levels which are the intersections of one moment with two parallel moments are called 'parallel.' Two parallel levels do not intersect, and conversely two levels in the same moment which do not intersect are parallel.In any moment there will be a complete system of levels parallel to a given level in that moment, and such levels will be parallel to each other.Similarly 'parallel' rects are defined by the intersection of parallel levels with a given level, all in one moment. Thus within any moment the whole theory of euclidean parallelism (so far as it is non-metrical) follows, and need not be further elaborated except to note the existence of parallelograms.36.2The definitions of parallel levels and of parallel rects can be extended to include levels and rects which are not co-momental:(i) Two levels,and′ are parallel ifis the intersection of momentsand, and′ of moments′ and′, whereis parallel to′ andto′:(ii) Two rects,and′, are parallel ifis the intersection of co-momental levelsand, and′ of comomental levels′ and′, whereis parallel to′ andto′.A moment and a rect which do not intersect are parallel. A rect either intersects a moment in one punct, or is parallel to it, or is contained in it.36.3The essential characteristic of space is bound up in what may be termed the 'repetition property' of parallelism. This repetition property is an essential element in congruence as will be seen later; also the homogeneity of space depends on it. Examples of the repetition property are as follows: if a rect intersects any moment in one and only one punct, then it intersects each moment of that time-system in one and only one punct: if a level intersects any moment in one and only one rect, then it intersects any moment of that time-system in one and only one rect. But we must not apply the theory of repetition in parallelism mechanically without attention to the nature of the property concerned. For example, if a rect is incident in a moment, it does not intersect any other moment of the same time-system, and thereforeà fortioriis not incident in any of them; and analogously for a level incident in a moment.36.4Puncts on a rect have an order which is derivative from the order of moments in a time-system and which connects the orders of various time-systems. The puncts on any given rectwill respectively be incident in the moments of any time-systemto which the rect is not parallel. Any moment ofwill contain one punct of, and any punct ofwill lie in one moment of. Thus the puncts ofhave derivatively the order of the moments of. Again letbe another such time-system. Then the puncts ofhave derivatively the order of the moments of. But it is found that these two orders for puncts onare identical, namely there is only one order for the puncts onto be obtained in this way. By means of these puncts on rects the orders of moments of different time-systems are correlated. Thus the existence of order in the instantaneous spaces of moments is explained; but the theory of congruence has not yet been entered upon.36.5The set of puncts, rects and levels in any one moment thus form a complete three-dimensional euclidean geometry, of which the meaning of the metrical properties has not yet been investigated. It is no necessary here to enunciate the fundamental propositions [such as two puncts defining a rect, and so on] from which the whole theory can be deduced so far as metrical relations are not concerned.
33. Antiprimes, Durations and Moments.33.1Among the constants of externality discussed inPart IIwas the reference of events to durations which are, in a sense, complete wholes of nature. A duration has thus in some sense an unlimited extension, though it is bounded in its temporal extent. Although we have not yet in our investigation ofdistinguished between spatial and temporal extension, durations can nevertheless be defined in terms ofby this unlimited aspect of their extents. Namely, we assume that there are no other events with the same unlimited property. Accordingly, any abstractive class which is composed purely of durations can only be covered by abstractive classes which also are composed purely of durations.
33.2An abstractive classis called an 'absolute antiprime' whenis itself one of the antiprimes which satisfy the formative condition of covering. In other words, an absolute antiprime is an abstractive class which covers every abstractive class which covers it.
If an abstractive class be an absolute antiprime, it is evident that the formative condition of 'covering it' is regular for antiprimes. Thus the set of events which are members of the absolute antiprimes which cover some one assigned absolute antiprime constitutes an abstractive element. Such an element will be called a 'moment.' Thus a moment is an abstractive element deduced from the condition of covering an absolute antiprime.
Only events of a certain type can be members of an absolute antiprime, namely events which inPart IIhave been called 'durations.' Only durations can extend over durations, and accordingly all the members of a moment are durations.
33.3 We may conceive of a duration as a sort of temporal thickness (or, slab) of nature[6]. In an absolute antiprime we have a series of temporal thicknesses successively packed one inside the other and converging towards the ideal of no thickness. An absolute antiprime indicates the ideal of an extensionless moment of time.
[6]The slab of nature forming a duration is limited in its temporal dimension and unlimited in its spatial dimensions. Thus it represents a finite time and infinite space.fig07Fig. 7.For example let the horizontal line represent the time; and assume nature to be spatially one-dimensional, so that an unlimited vertical line in the diagram represents space at an instant.fig08Fig. 8.Then the area between the unlimited parallel linesandrepresents a duration. Also the area betweenandrepresents another duration which is extended over by the duration bounded byand. But infig. 7we have assumed only one time-system, which is the Newtonian hypothesis. Suppose there are many time-systems and consider two such systemsand. These are represented by two lines inclined to each other. A duration of time-systemis represented by the area betweenand, and a duration of time-systemis represented by the area betweenand. Two such durations necessarily intersect and also can neither completely extend over the other.These diagrams are crude illustrations of some properties of durations and are in many respects misleading as the sequel will show.
[6]The slab of nature forming a duration is limited in its temporal dimension and unlimited in its spatial dimensions. Thus it represents a finite time and infinite space.
fig07Fig. 7.
Fig. 7.
Fig. 7.
For example let the horizontal line represent the time; and assume nature to be spatially one-dimensional, so that an unlimited vertical line in the diagram represents space at an instant.
fig08Fig. 8.
Fig. 8.
Fig. 8.
Then the area between the unlimited parallel linesandrepresents a duration. Also the area betweenandrepresents another duration which is extended over by the duration bounded byand. But infig. 7we have assumed only one time-system, which is the Newtonian hypothesis. Suppose there are many time-systems and consider two such systemsand. These are represented by two lines inclined to each other. A duration of time-systemis represented by the area betweenand, and a duration of time-systemis represented by the area betweenand. Two such durations necessarily intersect and also can neither completely extend over the other.
These diagrams are crude illustrations of some properties of durations and are in many respects misleading as the sequel will show.
The set of moments which inhere in a duration are completely characteristic of that duration, and vice versa. A moment is to be conceived as an abstract of all nature at an instant. No abstractive element can cover a moment except that moment itself. A moment is a route of approximation to all nature which has lost its (essential) temporal extension; thus it is nature under the aspect of a three-dimensional instantaneous space. This is the ideal to which we endeavour to approximate in our exact observations.
34. Parallelism and Time-Systems.34.1If the Newtonian theory of relativity were true, no pair of durations would lack durations extending over both of them, namely larger durations including both the given durations. But on the electromagnetic theory of relativity this is not necessarily the case, namely some pairs of durations are extended over by a family of durations and some are not. We shall adopt the electromagnetic theory of relativity.
A pair of durations both of which are parts of the same duration are called 'parallel'; and also a pair of moments such that there are durations in which both inhere are called 'parallel.'
Parallelism has the usual properties of transitiveness, symmetry and reflexiveness. Also two durations which do not intersect are parallel; and parallel moments which are not identical never intersect. If two parallel durations intersect there is a duration which is their complete intersection, but there are no durations among the common parts of two durations which are not parallel. Two moments which are not parallel necessarily intersect.
34.2Two durations which are parallel to the same duration are parallel to each other; thus it is evident that each absolute antiprime and each moment must be composed of parallel durations.
A 'family of parallel durations' is formed by all the durations parallel to a given duration, including that duration itself. Evidently any two members of such a family are parallel, and no duration out of the family is parallel to any duration of the family.
Analogously to such families of parallel durations, there are families of parallel moments, with the property that no two moments of the same family intersect and that any moment out of a given family intersects every moment belonging to the family.
The durations which are the members of the various moments of a given family of moments themselves form a family of parallel durations. Thus corresponding to a family of parallel durations there is one and only one family of parallel moments; and corresponding to a family of parallel moments there is one and only one family of parallel durations. A pair of such corresponding families, one of durations and the other of moments, form the 'time-system' associated with either of the two families.
Evidently each duration belongs to one and only one family of parallel durations; and thus each duration belongs to one and only one time-system. Also each moment belongs to one and only one family of parallel moments; and thus each moment belongs to one and only one time-system. Thus two distinct time-systems have no durations in common and no moments in common. But every event not a duration is contained in some durations of any given time-system. Furthermore there will be a minimum duration in a given time-system which is the duration 'when' the event happened in that time-system; namely, the minimum duration has the properties (i) that it extends over the event and (ii) that every duration which is part of it intersects the event.
34.3The moments of a time-system are arranged in serial order in this way:
(i) A duration belonging to a time-system is 'bounded' by a moment of the same time-system when each duration in which that moment inheres intersects the given duration and also intersects events separated from the given duration:
(ii) Every duration has two such bounding moments, and every pair of parallel moments bound one duration of that time-system:
(iii) A momentof a time-system 'lies between' two momentsandof the same time-system wheninheres in the duration whichandbound:
(iv) This relation of 'lying between' has the following properties which generate continuous serial order in each time-system, namely,
() Of any three moments of the same time-system, one of them lies between the other two:
() If the momentlies between the momentsand, and the momentlies between the momentsand, thenlies betweenand:
() There are not four moments in the same time-system such that one of them lies between each pair of the remaining three:
() The serial-order among moments of the same time-system has the Cantor-Dedekind type of continuity.
Nothing has yet been said about the measurement of the lapse of time. This topic will be considered as part of the general theory of congruence.
35. Levels, Rects, and Puncts.35.1The electromagnetic theory of relativity is obviously the more general of the two. It has also the merit of providing definitions of flatness, of straightness, of punctual position, of parallelism, of time-order and spatial order as interconnected phenomena, and (with the help of cogredience) of perpendicularity and of congruence. The theory of extension has also provided the definition of a duration. It is a remarkable fact that the characteristic concepts of time and of geometry should thus be exhibited as arising out of the nature of things as expressed by the two fundamental relations of extension and cogredience. It has already been explained that a moment is the route of approximation towards an instantaneous three-dimensional whole of nature. The set of abstractive elements and abstractive classes covered by both of two non-parallel moments is the locus which is their common intersection. Such a locus will be called a 'level' in either moment. A level is in fact an instantaneous plane in the instantaneous space of any moment in which it lies. But we reserve the conventional spatial terms, such as 'plane,' for the time-less spaces to be defined later. Accordingly the word 'level' is used here.
35.2An indefinite number of non-parallel moments will intersect each other in the same level, forming their complete intersection; and one level will never be merely a (logical) part of another level. Let three mutually intersecting moments (, say) intersect in the levels. Then three cases can arise:either(i) the levels are all identical [this will happen if any two are identical],or(ii) no pair of the levels intersect,or(iii) a pair of the levels, sayand, intersect. In case (i) the three moments are called 'co-level.' In case (ii) there are special relations of parallelism of levels, to be considered later. In case (iii) the locus of abstractive elements and abstractive classes which forms the intersection ofandwill be called a 'rect'; let this rect be named. Thenis also the complete intersection ofand, and of and, and of the three moments. When three moments have a rect as their complete intersection they are called 'co-rect.' A rect is an instantaneous straight line in the instantaneous three-dimensional space of any moment in which it lies. But, as before, the conventional space-nomenclature is avoided in connection with instantaneous spaces.
35.3For four distinct moments there are four possible cases in respect to their intersection. In case (i) there is no common intersection: in case (ii) there is a common intersection which is a level: in case (iii) there is a common intersection which is a rect: in case (iv) there is a common intersection which is neither a rect nor a level; in this case the common intersection will be called a 'punct.'
Consider four momentswhich constitute an instance of case (iv). Letbe the level which is the intersection ofand, and let be the rect which is the intersection of. Then the rectdoes not lie in the level. The rectintersects the levelin the common intersection of the four moments. This common intersection is an instantaneous point in the instantaneous spaces of the moments. In accordance with our practice of avoiding the conventional spatial terms when speaking of an instantaneous space, we have called this intersection a 'punct.' Since space is three-dimensional, any moment either covers every member of a given punct or covers none of its members. A punct represents the ideal of the maximum simplicity of absolute position in the instantaneous space of a moment in which it lies.
35.4It is tempting, on the mathematical analogy of four-dimensional space, to assert the existence of unlimited events which may be called the complete intersections of pairs of non-parallel durations. It is dangerous however blindly to follow spatial analogies; and I can find no evidence for such unlimited events, forming the complete intersections of pairs of intersecting durations, except in the excluded case of parallelism when the complete intersection (if it exist) is itself a duration. Accordingly, apart from parallelism, it may be assumed that the events extended over by a pair of intersecting durations are all finite events. No change in the sequel is required if the existence of such infinite events be asserted.
36. Parallelism and Order.36.1Two levels which are the intersections of one moment with two parallel moments are called 'parallel.' Two parallel levels do not intersect, and conversely two levels in the same moment which do not intersect are parallel.
In any moment there will be a complete system of levels parallel to a given level in that moment, and such levels will be parallel to each other.
Similarly 'parallel' rects are defined by the intersection of parallel levels with a given level, all in one moment. Thus within any moment the whole theory of euclidean parallelism (so far as it is non-metrical) follows, and need not be further elaborated except to note the existence of parallelograms.
36.2The definitions of parallel levels and of parallel rects can be extended to include levels and rects which are not co-momental:
(i) Two levels,and′ are parallel ifis the intersection of momentsand, and′ of moments′ and′, whereis parallel to′ andto′:
(ii) Two rects,and′, are parallel ifis the intersection of co-momental levelsand, and′ of comomental levels′ and′, whereis parallel to′ andto′.
A moment and a rect which do not intersect are parallel. A rect either intersects a moment in one punct, or is parallel to it, or is contained in it.
36.3The essential characteristic of space is bound up in what may be termed the 'repetition property' of parallelism. This repetition property is an essential element in congruence as will be seen later; also the homogeneity of space depends on it. Examples of the repetition property are as follows: if a rect intersects any moment in one and only one punct, then it intersects each moment of that time-system in one and only one punct: if a level intersects any moment in one and only one rect, then it intersects any moment of that time-system in one and only one rect. But we must not apply the theory of repetition in parallelism mechanically without attention to the nature of the property concerned. For example, if a rect is incident in a moment, it does not intersect any other moment of the same time-system, and thereforeà fortioriis not incident in any of them; and analogously for a level incident in a moment.
36.4Puncts on a rect have an order which is derivative from the order of moments in a time-system and which connects the orders of various time-systems. The puncts on any given rectwill respectively be incident in the moments of any time-systemto which the rect is not parallel. Any moment ofwill contain one punct of, and any punct ofwill lie in one moment of. Thus the puncts ofhave derivatively the order of the moments of. Again letbe another such time-system. Then the puncts ofhave derivatively the order of the moments of. But it is found that these two orders for puncts onare identical, namely there is only one order for the puncts onto be obtained in this way. By means of these puncts on rects the orders of moments of different time-systems are correlated. Thus the existence of order in the instantaneous spaces of moments is explained; but the theory of congruence has not yet been entered upon.
36.5The set of puncts, rects and levels in any one moment thus form a complete three-dimensional euclidean geometry, of which the meaning of the metrical properties has not yet been investigated. It is no necessary here to enunciate the fundamental propositions [such as two puncts defining a rect, and so on] from which the whole theory can be deduced so far as metrical relations are not concerned.