Page:Our knowledge of the external world.djvu/136

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of the given event. It will be found that this class of events is the first instant at which the given event exists, provided every event wholly after some contemporary of the given event is wholly after some initial contemporary of it.

Finally, the series of instants will be compact if, given any two events of which one wholly precedes the other, there are events wholly after the one and simultaneous with something wholly before the other. Whether this is the case or not, is an empirical question; but if it is not, there is no reason to expect the time-series to be compact.^[1]

Thus our definition of instants secures all that mathematics requires, without having to assume the existence of any disputable metaphysical entities.

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The assumptions made concerning time-relations in the above are as follows:—
- I. In order to secure that instants form a series, we assume:
  - (a) No event wholly precedes itself. (An “event” is defined as whatever is simultaneous with something or other.)
  - (b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third.
  - (c) If one event wholly precedes another, it is not simultaneous with it.
  - (d) Of two events which are not simultaneous, one must wholly precede the other.
- II. In order to secure that the initial contemporaries of a given event should form an instant, we assume:
  - (e) An event wholly after some contemporary of a given event is wholly after some initial contemporary of the given event.
- III. In order to secure that the series of instants shall be compact, we assume:
  - (f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.
This assumption entails the consequence that if one event covers the whole of a stretch of time immediately preceding another event, then it must have at least one instant in common with the other event; i.e. it is impossible for one event to cease just before another begins. I do not know whether this should be regarded as inadmissible. For a mathematico-logical treatment of the above topics, cf. N. Wilner, “A Contribution to the Theory of Relative Position,” Proc. Camb. Phil. Soc., xvii. 5, pp. 441-449.

[1] The assumptions made concerning time-relations in the above are as follows:—

I. In order to secure that instants form a series, we assume:
(a) No event wholly precedes itself. (An “event” is defined as whatever is simultaneous with something or other.)

(b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third.

(c) If one event wholly precedes another, it is not simultaneous with it.

(d) Of two events which are not simultaneous, one must wholly precede the other.

II. In order to secure that the initial contemporaries of a given event should form an instant, we assume:
(e) An event wholly after some contemporary of a given event is wholly after some initial contemporary of the given event.

III. In order to secure that the series of instants shall be compact, we assume:
(f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.
This assumption entails the consequence that if one event covers the whole of a stretch of time immediately preceding another event, then it must have at least one instant in common with the other event; i.e. it is impossible for one event to cease just before another begins. I do not know whether this should be regarded as inadmissible. For a mathematico-logical treatment of the above topics, cf. N. Wilner, “A Contribution to the Theory of Relative Position,” Proc. Camb. Phil. Soc., xvii. 5, pp. 441-449.

[2] I. In order to secure that instants form a series, we assume:
(a) No event wholly precedes itself. (An “event” is defined as whatever is simultaneous with something or other.)

(b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third.

(c) If one event wholly precedes another, it is not simultaneous with it.

(d) Of two events which are not simultaneous, one must wholly precede the other.

[3] (a) No event wholly precedes itself. (An “event” is defined as whatever is simultaneous with something or other.)

[4] (b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third.

[5] (c) If one event wholly precedes another, it is not simultaneous with it.

[6] (d) Of two events which are not simultaneous, one must wholly precede the other.

[7] II. In order to secure that the initial contemporaries of a given event should form an instant, we assume:
(e) An event wholly after some contemporary of a given event is wholly after some initial contemporary of the given event.

[8] (e) An event wholly after some contemporary of a given event is wholly after some initial contemporary of the given event.

[9] III. In order to secure that the series of instants shall be compact, we assume:
(f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.

[10] (f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.

[1]