The Paradox of Motion

The universe is a machine that is not documented.

At any one instant, an object must be at rest, an idea captured by this illustration of a moving dancer. Since this is ture for all instants, surely the object will always be at rest, so how can motion arise? The Greek philosopher Zeno posed this paradox as a challenge to the belief that time consists of a succession of discrete instants.

Calculus applies to continuous, rather than discrete, motion. But on 1st analysis, the very idea of continuous motion seems to be paradoxical.

The differential calculus is a collection of techniques for the manipulation of patterns. (The word calculus is a Latin word that means ‘pebble’ — recall that early counting systems involved the physical manipulation of pebbles.)

The basic operation of the differential calculus is a process known as differentiation. The aim of differentiation is to obtain the rate of change of some changing quantity. In order to do this, the value, or position, or path of that quantity has to be given by means of an appropriate formula. Differentiation then acts upon that formula to produce another formula that gives the rate of change. Thus, differentiation is a process for turning formulas into other formulas.

Consider that, at a particular instant in time, any object must be at a particular…