A Brief Consideration of Continuity and it's Implications on Computing

Consider a computer, stripped of its layers of abstraction which expose to users access to its facilities.  At the core of a computer is a binary processor acting on logical operations.  Each such operation is executed at a set increment, controlled by the computers clock speed.  These small operations are combined together through layers of abstraction to produce the robust functionality we have come to expect from a machine such as this.  

On such a system one with proper knowledge could produce an animation of, let's say, a ball rolling across a table.  During the animation the ball traverses the table, rolling from a point we shall call a to a second point which we shall call b.  Once produced the user can watch a playback of the animation and, assuming it was well produced, would not be surprised to see what they would naturally expect to see if a ball were indeed rolling across a table.

Further, consider a true ball in the physical world rolling across a true table from the same point a to the same point b.  Along the path between these points, as many points as you please may be observed  as well, and never can we observe so many points in between that there are not more which can be observed.  This continuity of movement, and of being, is innate to objects in the natural world.  The ball from our example moves continuously along the path from a to b, through the infinity of measurable (measurable either arithmetically or geometrically) points over an amount of time also exhibiting continuity.

In comparison, the animation which is produced consists of a series of frames displayed at a set rate.  Each frame can be said to be a measurement of the position of our virtual ball at a specific point in time.  From this animation select two adjacent frames.  In between these frames could be inserted a third showing the position of the ball at a point in time half way between the point represented in the first frame and the point represented in the  second frame.  Again, selecting this new frame and it's adjacent frame, another new frame could be made to show the point in time representative of the mid point of these two frames.  Such a process could be repeated as many times as we please without reaching a final result.

This is representative of a large limitation of a binary representation of continuity.  While binary operations are able to perform logical processes with accuracy, the representation of continuity is limited to an approximation.  The producer of our ball and table animation is forced to select points in time which will be shown on the frames of the animation and then calculate where the ball would be at that point in time.  While such an approximation is more than suitable for the domain of animation (as the eye can be tricked into seeing continuity) it is less desirable for a true simulation where the interaction of events across a continuous flow of time is to be observed as opposed to the state of events at a discreet point in time.

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-PCM