CHAPTER XFINITE ABSTRACTIVE ELEMENTS37. Absolute Primes and Event-Particles.37.1It follows from the principles of convergence to simplicity with diminution of extent that, for exhibiting the relations between events in their utmost simplicity, abstractive elements of minimum complexity are required, that is, elements which converge towards the ideal of an atomic event. This requisite exacts that the formative condition from which the 'atomic' element is deduced should be such as to impose the minimum of restriction on convergence.37.2An abstractive class which is prime in respect to the formative condition of 'covering all the elements and abstractive classes constituting some assigned punct' is called an 'absolute prime.'Evidently the condition satisfied by an absolute prime is regular for primes. The abstractive element deduced from an absolute prime is called an 'event-particle.' An event-particle is the route of approximation to an atomic event, which is an ideal satisfied by no actual event.An abstractive class which is antiprime in respect to the formative condition of 'being a member of some assigned punct' is evidently an absolute prime. In fact this set of antiprimes is identical with the set of absolute primes.An event-particle is an instantaneous point viewed in the guise of an atomic event. The punct which an event-particle covers gives it an absolute position in the instantaneous space of any moment in which it lies. Event-particles on a rect lie in the order derived from the puncts which they cover.37.3The complete set of event-particles inhering in an event will be called the set 'analysing' that event. A set of event-particles can only analyse one event, and an event can be analysed by only one set of event-particles.An event-particle 'bounds' an eventwhen every event in which the event-particle inheres intersects bothand events separated from. The set of event-particles bounding an event is called the 'boundary' of that event. A boundary can only bound one event and every event has a boundary.Event-particles which neither inhere in an event nor bound it are said to lie 'outside' it.The existence of boundaries enables the contact of events to be defined, namely, events are in 'contact' when their boundaries have one or more event-particles in common. The adjunction of events implies contact but not vice versa; since adjunction requires that a solid of the boundaries should be in common. But we define the notion of solid by means of that of adjunction, and not conversely.37.4Ifandare distinct event-particles, there are events separated from each other in whichandrespectively inhere.Two events intersect if there are event-particles each inhering in both events; and conversely, there are event-particles inhering in both events if they intersect.37.5The fact that the instantaneous geometry within a moment is three-dimensional leads to the conclusion that the geometry for all event-particles will be four-dimensional. It is to be noted however that the straight lines for this four-dimensional geometry have so far only been defined for event-particles which are co-momental, namely the rects. Event-particles which are not co-momental will be called 'sequent.' Straight lines of the four-dimensional space joining sequent event-particles will be defined inChapter XI.37.6The theory of contact is based on the four-dimensionality of the geometry of event-particles. Some results of that datum are now to be noted.A 'simple' abstractive class is an abstractive class for which there is no one event-particle on the boundaries of all those members of the converging end, which succeed some given member of the class; namely, for a simple abstractive class there is no one event-particle at which all members of the converging end have contact.Absolute antiprimes and absolute primes are simple abstractive classes. The 'atomic' property of an absolute prime is expressed by the theorem, that an absolute prime is a simple abstractive class which is covered by every simple abstractive class which it covers. The property of 'instantaneous completeness' exhibited by an absolute antiprime is expressed by the theorem, that an absolute antiprime is an abstractive class which covers every abstractive class that covers it.38. Routes.38.1Event-particles are abstractive elements of atomic simplicity. Routes are abstractive elements in which is found the first advance towards increasing complexity.A 'linear' abstractive class is a simple abstractive class () which (i) covers two event-particlesand(called the end-points), and (ii) is such that no selection of the event-particles which it covers can be the complete set of event-particles covered by another simple abstractive class, provided that the selection comprisesandand does not comprise all the event-particles covered by. The condition (i) secures that a linear abstractive class converges to an element of higher complexity than an event-particle; and the condition (ii) secures that it has the linear type of continuity.A 'linear prime' is an abstractive class which is prime in respect to the formative condition of (i) being covered by an assigned linear abstractive class covering two assigned end-points and (ii) being itself a linear abstractive class covering the same assigned end-points. This formative condition is evidently regular for primes.A 'route' is the abstractive element deduced from a linear prime. The two assigned event-particles which occur as end-points in the definition of the linear prime from which a route is deduced are called the 'end-points' of that route. A route is said to lie between its end-points.38.2A route is a linear segment, straight or curved, between two event-particles, co-momental or sequent. There are an indefinite number of routes between a given pair of event-particles as end-points. A route will cover an infinite number of event-particles in addition to its end-points. The continuity of events issues in a theory of the continuity of routes.Ifandbe any two event-particles covered by a route, there is one and only one route withandas end-points which is covered by.If,andbe any three event-particles covered by a route, thenis said to 'lie between'andon the routeifis covered by that route withandas end-points which is covered by.The particles on any route are arranged in a continuous serial order by his relation of 'lying between' holding for triads of points on it. The necessary and sufficient conditions which the relation must satisfy to produce this serial order are detailed in (iv) of34.3.38.3A route may, or may not, be covered by a moment. If it is so covered it is called a 'co-momental route.' A 'rectilinear route' is a route such that all the event-particles which it covers lie on a rect. In a rectilinear route the order of the event-particles on the rect agrees with the order of the event-particles as defined by the relation of 'lying between' as defined for the route.Between any two event-particles on a rect there is one and only one rectilinear route. Ifandbe two event-particles on a rect, the rectilinear route between them can also be defined as the element deduced from the prime with the formative condition of being a simple abstractive class which coversandand all the event-particles betweenandon the rect.38.4Among the routes which are not co-momental, the important type is that here named 'kinematic routes.' A 'kinematic route' is a route (i) whose end-points are sequent and (ii) such that each moment, which in any time-system lies between the two moments covering the end-points, covers one and only one event-particle on the route, and (iii) all the event-particles of the route are so covered.The event-particles covered by a kinematic route represent a possible path for a 'material particle.' But this anticipates later developments of the subject, since the concept of a 'material particle' has not yet been defined.39. Solids.39.1A 'solid prime' is a prime with the formative condition of being a simple abstractive class which covers all the event-particles shared in common by both boundaries of two adjoined events. This formative condition is evidently regular for primes. A 'solid' is the abstractive element deduced from a solid prime.39.2If two event-particles are covered by a solid, there are an indefinite number of routes between them covered by the same solid.A solid may or may not be covered by a moment. If it is so covered, it is called 'co-momental.'A solid which is not co-momental is called 'vagrant.' The properties of vagrant solids are assuming importance in connection with Einstein's theory of gravitation; the consideration of these properties is not undertaken in this enquiry. Co-momental solids are also called 'volumes.' Volumes are capable of a simpler definition which is given in the next article.40. Volumes.40.1A 'volume prime' is a prime with the formative condition of being a simple abstractive class which covers all the event-particles inhering in an assigned event and covered by an assigned moment. If there are no such particles, there will be no corresponding volume prime. This formative condition is evidently regular for primes.A 'volume' is the abstractive element deduced from a volume prime. A volume is thus the section of an event made by a moment.40.2Any volume is covered by the assigned moment of which mention occurs in its definition. Thus every volume (as here defined) is co-momental. Also a volume only covers those event-particles of which mention occurs in its definition. This set of event-particles is completely characteristic of the volume, and may be considered as the volume conceived as a locus of event-particles.In exactly the same way a solid, or a route, is completely defined by the event-particles which it covers and vice versa. Thus solids and routes can be conceived as loci of event-particles.The concrete event itself is also defined by (or, analysed by) the event-particles inhering in it, and such a set of event-particles defines only one event. Thus an event can be looked on as a locus of event-particles. An event′ which is part of an eventis defined in this way by a set of event-particles which are some of the set defining. This fact is the reason for the confusion of the logical 'all' and 'some' with the physical 'whole' and 'part' which apply solely to events. An event is also uniquely defined by the set of event-particles which form its boundary.
37. Absolute Primes and Event-Particles.37.1It follows from the principles of convergence to simplicity with diminution of extent that, for exhibiting the relations between events in their utmost simplicity, abstractive elements of minimum complexity are required, that is, elements which converge towards the ideal of an atomic event. This requisite exacts that the formative condition from which the 'atomic' element is deduced should be such as to impose the minimum of restriction on convergence.
37.2An abstractive class which is prime in respect to the formative condition of 'covering all the elements and abstractive classes constituting some assigned punct' is called an 'absolute prime.'
Evidently the condition satisfied by an absolute prime is regular for primes. The abstractive element deduced from an absolute prime is called an 'event-particle.' An event-particle is the route of approximation to an atomic event, which is an ideal satisfied by no actual event.
An abstractive class which is antiprime in respect to the formative condition of 'being a member of some assigned punct' is evidently an absolute prime. In fact this set of antiprimes is identical with the set of absolute primes.
An event-particle is an instantaneous point viewed in the guise of an atomic event. The punct which an event-particle covers gives it an absolute position in the instantaneous space of any moment in which it lies. Event-particles on a rect lie in the order derived from the puncts which they cover.
37.3The complete set of event-particles inhering in an event will be called the set 'analysing' that event. A set of event-particles can only analyse one event, and an event can be analysed by only one set of event-particles.
An event-particle 'bounds' an eventwhen every event in which the event-particle inheres intersects bothand events separated from. The set of event-particles bounding an event is called the 'boundary' of that event. A boundary can only bound one event and every event has a boundary.
Event-particles which neither inhere in an event nor bound it are said to lie 'outside' it.
The existence of boundaries enables the contact of events to be defined, namely, events are in 'contact' when their boundaries have one or more event-particles in common. The adjunction of events implies contact but not vice versa; since adjunction requires that a solid of the boundaries should be in common. But we define the notion of solid by means of that of adjunction, and not conversely.
37.4Ifandare distinct event-particles, there are events separated from each other in whichandrespectively inhere.
Two events intersect if there are event-particles each inhering in both events; and conversely, there are event-particles inhering in both events if they intersect.
37.5The fact that the instantaneous geometry within a moment is three-dimensional leads to the conclusion that the geometry for all event-particles will be four-dimensional. It is to be noted however that the straight lines for this four-dimensional geometry have so far only been defined for event-particles which are co-momental, namely the rects. Event-particles which are not co-momental will be called 'sequent.' Straight lines of the four-dimensional space joining sequent event-particles will be defined inChapter XI.
37.6The theory of contact is based on the four-dimensionality of the geometry of event-particles. Some results of that datum are now to be noted.
A 'simple' abstractive class is an abstractive class for which there is no one event-particle on the boundaries of all those members of the converging end, which succeed some given member of the class; namely, for a simple abstractive class there is no one event-particle at which all members of the converging end have contact.
Absolute antiprimes and absolute primes are simple abstractive classes. The 'atomic' property of an absolute prime is expressed by the theorem, that an absolute prime is a simple abstractive class which is covered by every simple abstractive class which it covers. The property of 'instantaneous completeness' exhibited by an absolute antiprime is expressed by the theorem, that an absolute antiprime is an abstractive class which covers every abstractive class that covers it.
38. Routes.38.1Event-particles are abstractive elements of atomic simplicity. Routes are abstractive elements in which is found the first advance towards increasing complexity.
A 'linear' abstractive class is a simple abstractive class () which (i) covers two event-particlesand(called the end-points), and (ii) is such that no selection of the event-particles which it covers can be the complete set of event-particles covered by another simple abstractive class, provided that the selection comprisesandand does not comprise all the event-particles covered by. The condition (i) secures that a linear abstractive class converges to an element of higher complexity than an event-particle; and the condition (ii) secures that it has the linear type of continuity.
A 'linear prime' is an abstractive class which is prime in respect to the formative condition of (i) being covered by an assigned linear abstractive class covering two assigned end-points and (ii) being itself a linear abstractive class covering the same assigned end-points. This formative condition is evidently regular for primes.
A 'route' is the abstractive element deduced from a linear prime. The two assigned event-particles which occur as end-points in the definition of the linear prime from which a route is deduced are called the 'end-points' of that route. A route is said to lie between its end-points.
38.2A route is a linear segment, straight or curved, between two event-particles, co-momental or sequent. There are an indefinite number of routes between a given pair of event-particles as end-points. A route will cover an infinite number of event-particles in addition to its end-points. The continuity of events issues in a theory of the continuity of routes.
Ifandbe any two event-particles covered by a route, there is one and only one route withandas end-points which is covered by.
If,andbe any three event-particles covered by a route, thenis said to 'lie between'andon the routeifis covered by that route withandas end-points which is covered by.
The particles on any route are arranged in a continuous serial order by his relation of 'lying between' holding for triads of points on it. The necessary and sufficient conditions which the relation must satisfy to produce this serial order are detailed in (iv) of34.3.
38.3A route may, or may not, be covered by a moment. If it is so covered it is called a 'co-momental route.' A 'rectilinear route' is a route such that all the event-particles which it covers lie on a rect. In a rectilinear route the order of the event-particles on the rect agrees with the order of the event-particles as defined by the relation of 'lying between' as defined for the route.
Between any two event-particles on a rect there is one and only one rectilinear route. Ifandbe two event-particles on a rect, the rectilinear route between them can also be defined as the element deduced from the prime with the formative condition of being a simple abstractive class which coversandand all the event-particles betweenandon the rect.
38.4Among the routes which are not co-momental, the important type is that here named 'kinematic routes.' A 'kinematic route' is a route (i) whose end-points are sequent and (ii) such that each moment, which in any time-system lies between the two moments covering the end-points, covers one and only one event-particle on the route, and (iii) all the event-particles of the route are so covered.
The event-particles covered by a kinematic route represent a possible path for a 'material particle.' But this anticipates later developments of the subject, since the concept of a 'material particle' has not yet been defined.
39. Solids.39.1A 'solid prime' is a prime with the formative condition of being a simple abstractive class which covers all the event-particles shared in common by both boundaries of two adjoined events. This formative condition is evidently regular for primes. A 'solid' is the abstractive element deduced from a solid prime.
39.2If two event-particles are covered by a solid, there are an indefinite number of routes between them covered by the same solid.
A solid may or may not be covered by a moment. If it is so covered, it is called 'co-momental.'
A solid which is not co-momental is called 'vagrant.' The properties of vagrant solids are assuming importance in connection with Einstein's theory of gravitation; the consideration of these properties is not undertaken in this enquiry. Co-momental solids are also called 'volumes.' Volumes are capable of a simpler definition which is given in the next article.
40. Volumes.40.1A 'volume prime' is a prime with the formative condition of being a simple abstractive class which covers all the event-particles inhering in an assigned event and covered by an assigned moment. If there are no such particles, there will be no corresponding volume prime. This formative condition is evidently regular for primes.
A 'volume' is the abstractive element deduced from a volume prime. A volume is thus the section of an event made by a moment.
40.2Any volume is covered by the assigned moment of which mention occurs in its definition. Thus every volume (as here defined) is co-momental. Also a volume only covers those event-particles of which mention occurs in its definition. This set of event-particles is completely characteristic of the volume, and may be considered as the volume conceived as a locus of event-particles.
In exactly the same way a solid, or a route, is completely defined by the event-particles which it covers and vice versa. Thus solids and routes can be conceived as loci of event-particles.
The concrete event itself is also defined by (or, analysed by) the event-particles inhering in it, and such a set of event-particles defines only one event. Thus an event can be looked on as a locus of event-particles. An event′ which is part of an eventis defined in this way by a set of event-particles which are some of the set defining. This fact is the reason for the confusion of the logical 'all' and 'some' with the physical 'whole' and 'part' which apply solely to events. An event is also uniquely defined by the set of event-particles which form its boundary.