PART IVTHE THEORY OF OBJECTSCHAPTER XIVTHE LOCATION OF OBJECTS53. Location.53.1We conceive objects as located in space. This conception of location in space is distinct from that of being situated in an event, though the two concepts are closely allied by a determinate connection. The notion of the situation of an object is logically indefinable being one of the ultimate data of science; the notion of the location of an object is definable in terms of the notion of its situation.An object is said to be 'located' in an abstractive element if there is a simple abstractive class 'converging' to the element and such that each of its members is a situation of the object.In general when an object is located in an abstractive element there will be many simple abstractive classes converging to the element and such that each of their members is a situation of the object. In any specific case of location usually all abstractive classes of a certain type will possess the required property.It follows from this definition that, in the primary signification of location, an object is located in an element of instantaneous space. The notion of location in an element of time-less space follows derivatively by correlating the elements of instantaneous space to the elements of time-less space in the way already described. In our immediate thoughts which follow perception we make a jump from the situation of an object within the short specious present to its location in instantaneous space, and thence by further reflexion to its location in time-less space. Thus location in space is always an ideal of thought and never a fact of perception. An object may be located in a volume, an area, a route, or an event-particle of instantaneous space, and thence derivatively it will be located in a volume, or an area, or a segment, or a point of timeless space.53.2In considering the scientific object it is the occupied event which corresponds to the situation of the physical object. The occupied event is the situation of the charge, in so far as the single scientific object is conceived as an (ideal) physical object.53.3There are evidently many different kinds of location which satisfy the general definition of location in an abstractive element, even when the kind of abstractive element is assigned. These differences mainly arise from differences in the relations of objects to parts of their situations. An object is an atomic entity and as such is related to its situations. But a situation is an event with parts of various kinds, and we have to consider the various kinds of relationships which objects may have to various kinds of parts of their situations.For example, if the sense-object 'redness, of a definite shade' be located in an area, it will be located in any portion of that area; and this arises from the fact that if it be situated in an event, it is also situated in any portion of that event. But it is not true that if a chair be situated in an event, that the chair—as one atomic object—is situated in any part of the event though it is so situated in some parts. Again a tune cannot be situated in any event comprised in a duration too short for the successive notes to be sounded. Thus for a tune a minimum quantum of time is necessary.54. Uniform Objects.54.1It will be convenient to classify objects according as they do or do not satisfy certain important conditions respecting their relations to their situations.'Uniform' objects are objects with a certain smoothness in their temporal relations, so that they require no minimum quantum of time-lapse in the events which are their situations. These are objects which can be said to exist 'at a given moment.' For example, a tune is not an uniform object; but a chair, as ordinarily recognised, is such an object. The example of the chair, and the dissolution of its continuous materials with specific physical constants into assemblages of electrons, warn us that a problem remains over for discussion after we shall have defined the meaning to be assigned to 'uniformity.'54.2In order to explain more precisely the theory of uniform objects, it is convenient to make a few definitions:A 'slice' of an eventin a time-systemis that part oflying between two moments of, where both moments intersect. The two moments are called the terminal moments of the slice, and the volumes in which the terminal moments intersectare called the terminal volumes. For brevity a slice ofin the time-systemis called an '-slice of.'It follows from the continuity of events that any-moment lying between the terminal moments of an-slice ofintersectsin a volume. Such a volume is called an-section of the slice. A slice is itself an event which stretches throughout the duration bounded by its terminal moments. Thus if the duration be the specious present for some percipient, the slice ofis the part of the event e which falls within that specious present.54.3The properties of uniform objects will be enunciated as a set of laws regulating their character.Law I. Ifbe any time-system andbe a situation of an uniform object, then an-slice ofexists which is a situation of.Law II. Ifbe any time-system andbe a situation of an uniform objectand′ be an-slice ofwhich is a situation of, then every-slice of′ is a situation of.Law I can roughly be construed as meaning that if an uniform objecthas been situated in any event, then there is some period of time (in any time-system) during which it has existed; and in the same way Law II means that if an uniform object has existed during any period of time, then it has existed during any shorter period within that period. These laws are obvious as applied to uniform objects, but not so obvious for objects in general, as 'object' is here defined. For example a musical note cannot exist in a period of time shorter than its period of vibration, and a percipient whose specious present was too short could not hear it. It follows from law II that if an uniform object O is situated in an eventand′ be an-slice ofwhich is a situation of, then an abstractive class of-slices converging to any-section of′ can be found such thatis situated in each member of the class. Hence evidentlyis located in every-section of′. This is the conception of an uniform object being located in a spatial volume at a durationless moment of time.With certain explanations and limitations laws I and II apply to many types of objects. In fact it requires an effort to realise that there are cases to which they do not apply. They have been stated above in the most formal manner to exhibit the fact that, when they do apply, they are empirical laws of nature and notà priorilogical truths.55. Components of Objects.55.1The concept of a 'component' of a main object is difficult to make precise. A component of an objectis another distinct object′ such that (i) wheneveris situated in an event, there is an event′, which is eitheritself or a part of, in which′ is situated, and (ii)′ may also be situated in an event″ which is not a situation ofor any part of a situation of.Thus a component is necessary to the main object, but the main object is not necessary to the component. For example, a certain note may be necessary for a certain tune, but the note can be sounded without the tune. The main object requires its component, but the component does not require the main object.But this general idea of a component is not of great importance apart from further specialisation. There are many such specialisations; but in science there are three which are of peculiar importance, namely, 'concurrent components,' 'extensive components' and 'causal components.'55.2An object′ is a 'concurrent' component of an objectwhen it is a component of, and ifbe any situation of, there is an event′ which is part ofand is such that (i) it is a situation of′ and (ii) it is cut in a slice which is a situation of′ by any duration which cutsin a slice which is a situation of.Thus a concurrent component lasts concurrently with the main object in any time-system.
53. Location.53.1We conceive objects as located in space. This conception of location in space is distinct from that of being situated in an event, though the two concepts are closely allied by a determinate connection. The notion of the situation of an object is logically indefinable being one of the ultimate data of science; the notion of the location of an object is definable in terms of the notion of its situation.
An object is said to be 'located' in an abstractive element if there is a simple abstractive class 'converging' to the element and such that each of its members is a situation of the object.
In general when an object is located in an abstractive element there will be many simple abstractive classes converging to the element and such that each of their members is a situation of the object. In any specific case of location usually all abstractive classes of a certain type will possess the required property.
It follows from this definition that, in the primary signification of location, an object is located in an element of instantaneous space. The notion of location in an element of time-less space follows derivatively by correlating the elements of instantaneous space to the elements of time-less space in the way already described. In our immediate thoughts which follow perception we make a jump from the situation of an object within the short specious present to its location in instantaneous space, and thence by further reflexion to its location in time-less space. Thus location in space is always an ideal of thought and never a fact of perception. An object may be located in a volume, an area, a route, or an event-particle of instantaneous space, and thence derivatively it will be located in a volume, or an area, or a segment, or a point of timeless space.
53.2In considering the scientific object it is the occupied event which corresponds to the situation of the physical object. The occupied event is the situation of the charge, in so far as the single scientific object is conceived as an (ideal) physical object.
53.3There are evidently many different kinds of location which satisfy the general definition of location in an abstractive element, even when the kind of abstractive element is assigned. These differences mainly arise from differences in the relations of objects to parts of their situations. An object is an atomic entity and as such is related to its situations. But a situation is an event with parts of various kinds, and we have to consider the various kinds of relationships which objects may have to various kinds of parts of their situations.
For example, if the sense-object 'redness, of a definite shade' be located in an area, it will be located in any portion of that area; and this arises from the fact that if it be situated in an event, it is also situated in any portion of that event. But it is not true that if a chair be situated in an event, that the chair—as one atomic object—is situated in any part of the event though it is so situated in some parts. Again a tune cannot be situated in any event comprised in a duration too short for the successive notes to be sounded. Thus for a tune a minimum quantum of time is necessary.
54. Uniform Objects.54.1It will be convenient to classify objects according as they do or do not satisfy certain important conditions respecting their relations to their situations.
'Uniform' objects are objects with a certain smoothness in their temporal relations, so that they require no minimum quantum of time-lapse in the events which are their situations. These are objects which can be said to exist 'at a given moment.' For example, a tune is not an uniform object; but a chair, as ordinarily recognised, is such an object. The example of the chair, and the dissolution of its continuous materials with specific physical constants into assemblages of electrons, warn us that a problem remains over for discussion after we shall have defined the meaning to be assigned to 'uniformity.'
54.2In order to explain more precisely the theory of uniform objects, it is convenient to make a few definitions:
A 'slice' of an eventin a time-systemis that part oflying between two moments of, where both moments intersect. The two moments are called the terminal moments of the slice, and the volumes in which the terminal moments intersectare called the terminal volumes. For brevity a slice ofin the time-systemis called an '-slice of.'
It follows from the continuity of events that any-moment lying between the terminal moments of an-slice ofintersectsin a volume. Such a volume is called an-section of the slice. A slice is itself an event which stretches throughout the duration bounded by its terminal moments. Thus if the duration be the specious present for some percipient, the slice ofis the part of the event e which falls within that specious present.
54.3The properties of uniform objects will be enunciated as a set of laws regulating their character.
Law I. Ifbe any time-system andbe a situation of an uniform object, then an-slice ofexists which is a situation of.
Law II. Ifbe any time-system andbe a situation of an uniform objectand′ be an-slice ofwhich is a situation of, then every-slice of′ is a situation of.
Law I can roughly be construed as meaning that if an uniform objecthas been situated in any event, then there is some period of time (in any time-system) during which it has existed; and in the same way Law II means that if an uniform object has existed during any period of time, then it has existed during any shorter period within that period. These laws are obvious as applied to uniform objects, but not so obvious for objects in general, as 'object' is here defined. For example a musical note cannot exist in a period of time shorter than its period of vibration, and a percipient whose specious present was too short could not hear it. It follows from law II that if an uniform object O is situated in an eventand′ be an-slice ofwhich is a situation of, then an abstractive class of-slices converging to any-section of′ can be found such thatis situated in each member of the class. Hence evidentlyis located in every-section of′. This is the conception of an uniform object being located in a spatial volume at a durationless moment of time.
With certain explanations and limitations laws I and II apply to many types of objects. In fact it requires an effort to realise that there are cases to which they do not apply. They have been stated above in the most formal manner to exhibit the fact that, when they do apply, they are empirical laws of nature and notà priorilogical truths.
55. Components of Objects.55.1The concept of a 'component' of a main object is difficult to make precise. A component of an objectis another distinct object′ such that (i) wheneveris situated in an event, there is an event′, which is eitheritself or a part of, in which′ is situated, and (ii)′ may also be situated in an event″ which is not a situation ofor any part of a situation of.
Thus a component is necessary to the main object, but the main object is not necessary to the component. For example, a certain note may be necessary for a certain tune, but the note can be sounded without the tune. The main object requires its component, but the component does not require the main object.
But this general idea of a component is not of great importance apart from further specialisation. There are many such specialisations; but in science there are three which are of peculiar importance, namely, 'concurrent components,' 'extensive components' and 'causal components.'
55.2An object′ is a 'concurrent' component of an objectwhen it is a component of, and ifbe any situation of, there is an event′ which is part ofand is such that (i) it is a situation of′ and (ii) it is cut in a slice which is a situation of′ by any duration which cutsin a slice which is a situation of.
Thus a concurrent component lasts concurrently with the main object in any time-system.