IV Motion and Time The Classical Period It is arguably fair to say that much of Greek metaphysical thought was an attempt to answer the question, What is constant in a world of change? To an extent the question was culturally conditioned. Economic and political instabilities encouraged such questions if they did not obviously trigger them. However, the question is natural. We all want to know what will change and cannot be counted on, and what can be trusted to remain the same. Since motion is one type of change, motion became one of the phenomena of which the Greeks sought explanations. Most of the Ionians responded by positing some basic element as the unchanging stuff of the world. In effect, the invention of elements answered the question with the reply, The elements never change; change is an interchange or motion of the changeless elements. Water for Thales, air for Anaximenes, and the boundless for Anaximander; the basic stuff changed through processes such as condensation or melting. The processes were the causes of motion. Thus began a centuries-long dialogue among Greek thinkers, a series of responses and counter-responses. An Ionian with a different, disturbing thought was Heraclitus of Ephesus. The riddler, as he was called, insisted that change is all there is. Change is caused by conflict, a war of opposites. The material embodiment of this strife is fire. Elea, a Greek colony in southern Italy, produced a number of important thinkers. One of them, Parmenides (b.515 BC), may have visited Athens and met Socrates. He placed stress on the law of contradiction. If, as Heraclitus claimed, opposites are at war, what is the opposite of being? Nonbeing is the only possible answer but to say nonbeing exists is a contradiction and must be nonsense. Thus, no opposite of being exists and no war can occur. Nothing really changes, then, because all things already have being. Change can only be an illusion. Wisely, Parmenides died and left to his student, Zeno (b.495 BC), the task of elaborating and defending these ideas. This Zeno did with a series of paradoxes aimed at showing the contradictions inherent in any other view of motion and change. The most famous is that of the race between Achilles and the tortoise. Retelling it in modern terms we might assume Achilles is ten times as fast as the tortoise and the tortoise has a 100 m head start. When Achilles has covered 100 m, the tortoise is 10 m ahead. When Achilles has run another 10 m, the tortoise is 1 m ahead and so on. At the end of every such time interval, the tortoise will still be ahead! The conclusion is that Achilles can never catch the tortoise, though, of course, we know he can. The error in Zenos discussion was not obvious to ancient hearers anymore than it is to a general audience today. He erroneously assumed that the sum of an infinite series (of time intervals) is always infinite regardless of the size of the terms. His fundamental conclusion is that Achilles will need an infinite amount of time to catch the tortoise. The sum of an infinite number of steps (an infinite series) is not necessarily equal to infinity. Some series do sum to infinity but many do not. The series Zeno had in mind is not infinite and clearly has a finite sum as we can see if we add terms (see Appendix D). The total time taken by Achilles to travel 100 m, then 10 m, then 1 m and so on is not infinite but actually not long at all. He will catch the tortoise after running 111.1 m. The time required will not be infinite, as Zeno assumed, but will simply be 111.1 m divided by Achilles speed. Our intuition that he will quickly catch the slow tortoise is correct. But Zenos contemporaries did not understand the root of the paradox and it troubled them inordinately. In particular, it made Greek thinkers suspicious of anything having to do with infinity or an infinite numer of steps. Anaxagorus, followed by Leucippus and Democritus, gave an atomistic response to Zeno. The error in Zenos paradox was, they thought, the assumption that distance can be indefinitely subdivided. Eventually, the irreducible atoms will prevent the endless dividing Zeno had in mind. So the analysis was erroneous. Motion, for the atomists, was the normal state of matter, needing no explanation. Platos response to Zeno was to let change reside in the world of appearances while the world of forms was changeless and timeless. Thus, Plato agreed that change was an illusion in the sense that all appearance is illusion. Time was the movement of the heavens, a moving image of the form or principle of soul. Since the heavenly motions were mensurable, Plato had some sense of quantifying motion. However, there is nothing in Plato that qualifies as a law of motion unless we count the view that heavenly motio