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Consider the two time-systems α and β, each with its own timeless space and its own family of instantaneous moments with their instantaneous spaces. Let M and N be respectively a moment of α and a moment of β. In M there is the direction of β and in N there is the direction of α. But M and N, being moments of different time-systems, intersect in a level. Call this level λ.
Accordingly the various modes of deriving spatial order from diverse time-systems must harmonise with one spatial order in each instantaneous space. In this way also diverse time-orders are comparable. We have two great questions still on hand to be settled before our theory of space is fully adjusted.
The axiom also enables us to measure time in any time-system; but does not enable us to compare times in different time-systems. The second axiom of congruence concerns parallelograms on congruent bases and between the same parallels, which have also their other pairs of sides parallel.
Thus a being at rest at a point of space α will be moving uniformly along a line in space β which is in the direction of α in space β, and a being at rest at a point of space β will be moving uniformly along a line in space α which is in the direction of β in space α. I have been speaking of the timeless spaces which are associated with time-systems.
Such a set of event-particles will form a point in the timeless space of that time-system. Thus a point is really an absolute position in the timeless space of a given time-system. But there are alternative time-systems, and each time-system has its own peculiar group of points that is to say, its own peculiar definition of absolute position. This is exactly the theory which I will elaborate.
The assumption arises from the failure to make another distinction; namely, that there may be alternative definitions of absolute position. This possibility enters with the admission of alternative time-systems.
Durations have all the reality that nature has, though what that may be we need not now determine. The measurableness of time is derivative from the properties of durations. So also is the serial character of time. We shall find that there are in nature competing serial time-systems derived from different families of durations.
Any rect in M which intersects all these levels in its set of puncts, thereby receives for its puncts an order of position on it. So spatial order is derivative from temporal order. Furthermore there are alternative time-systems, but there is only one definite spatial order in each instantaneous space.
Namely, there is a definite whole of nature, simultaneously now present, whatever may be the character of its remote events. This consideration reinforces the previous conclusion. This conclusion leads to the assertion of the essential uniformity of the momentary spaces of the various time-systems, and thence to the uniformity of the timeless spaces of which there is one to each time-system.
In a later lecture I have to distinguish between what I call the passage of nature and particular time-systems which exhibit certain characteristics of that passage. I will not go so far as to claim Plato in direct support of this doctrine, but I do think that the sections of the Timaeus which deal with time become clearer if my distinction is admitted. This is however a digression.
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