Here as elsewhere what concerns us is the possibility of the continuity of objects as they move through space. Spatial change can be understood without continuous objects as a succession of states of matter and void, where the locations of matter differ. This difference of location, like all difference, involves relation. This relation may be conceived of as either a relation to previous arrangements of matter (previous sets of relations) or to absolute locations defined by points.
The definition of absolute location, for a finite observer, must be in terms of matter, for no other method could serve to communicate the reference point. Even if it were not necessary to communicate to another, one must nonetheless name the location for oneself by reference to matter that one takes to be stable. Otherwise change will sweep locations out of the mind and leave us incapable of measurement.
Thus, experiential verification of spatial change requires a constant point around which the world may turn. The natural choice for this is the body, as its endurance over time is ensured, at least to the extent that knowledge can be relevant.
Choosing the body as the point of pivoting, we seem to grant to it continuous endurance through time. Is this necessarily the case, or is it possible that at each moment the body is made anew and ceases to be what it was? No, it must endure, lest its reforging make it irrelevant as a point of reference.
All change is in time. Spatial change is the change of spatial relations over time. Spatial relations are measured by distances between extended objects. The body being one such, and here employed as the pivot, we measure distance as distance from the body. Objects cannot draw nearer or further without enduring. Otherwise, a new object, though closer, would occur; it would be nearer than the old one, but there would be no object drawing closer.
If the essence of an object is that which endures through change, the endurance of external objects through spatial change amounts to a specification of the object’s essence, such that location ceases to define the essence and the object becomes capable of spatial motion. By contrast, a spatial point could be understood as something who’s essence is defined by its spatial location.
But in addition to this we must establish the continuity of the motion, not simply of the object that moves. Continuity consists in the absence of gaps, so that in any motion there is no space that was uninhabited, from the origin point to the last. So we ask, why must motion be continuous? Or indeed, must motion be continuous?
It seems possible to imagine a world wherein objects change their location without continuity, and it has been suggested by some that this is indeed the case. So why does it seem to run so counter to common sense? Our theory makes no allowance for common sense, and cannot derive it except as useful belief. It seems possible then that objects could simply move in a spastic and spontaneous manner, so it needs to be explained why their motion is continuous.
Perhaps causality might hold the key to their continuity. But if causality is understood as the essential relation between one state and its successor, then this seems unlikely. It could as easily cause spastic motion as continuous.
The point of reference, we have seen, is, like the object, continuous in time. Furthermore it is continuous in space, given that it does not move and remains fixed while the world moves around it. This seems to be a kind of forced continuity, granted by immobility.
This suggests the following. Motion is spastic, discontinuous. But the change in location, taken as motion, ie, as a difference in space over time, gives us a distance. This distance permits, like any mathematical quantity, of infinite subdivision, if not in nature then in representation. This possibility of subdivision would imply the object’s existence at each point. If the object’s change can be divided into smaller distances, it can be divided into smaller changes. These smaller changes would posit the object at both the origin and the termination point. Thus the possibility of infinite subdivision would lead us to posit a continuity of motion.
In other words, motion is change of space over time. If space is infintiely divisible, ie, continuous, then the motion too must continuous.