The Duality of Time Theory

If we consider the path of an arrow in flight, and assuming that space and time are discrete: at each instant of its path, the Arrow occupies some position in space; this is what it means to say that space is discrete. But to occupy a certain position in space is to be at rest in this position! So throughout all the points of the entire path of the Arrow through space, it is in fact at rest, because at any indivisible instance of time it must be at some discrete position, so its instantaneous velocityis always zero. If in an indivisible instant of time the Arrow moved, then indeed this instant of time would be divisible, for example, in a smaller instant of time the Arrow would have moved half that distance. Yet in the end the arrow reaches the target, moving at an average positive speed, which means that the sum of zeros is not equal to zero:!

Figure 2.2: If space is not infinitely divisible, the distance from start to target is composed ofdiscrete points. At any discrete pointalong the flight of the Arrow to its target, its instantaneous velocityis always zero. Yet the Arrow reaches the target at an average positive speed! The Arrow can not move between the discrete points, because space is supposed to be divisible only down to those discrete points, and so are the instants of time when the Arrow occupies these points.

Russell describes this astounding conclusion by: “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever” Salmon (2001).

The strength of the Single Monad Model comes exactly from its innovative explanation of this “miraculous way” of the “change of position” without actual motion. This offers what is known as the “at-at theory of motion”, which agrees that there can be no motion “during” a durationless instant, and contends that all that is required for motion is that the Arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view, motion is a function of position with respect to time. So in fact there is no gradual motion in the common sense that the object leaves its place to occupy new adjacent places, but it is successively re-created in those new places, i.e. motion occurs as a result of discrete change and not infinitesimal transmutation, so the observed objects are always at rest in the different positions that they appear in. This will be discussed and demonstrated further in chapter V.