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Salient: An Organ of Student Opinion At Victoria University College, Wellington, N. Z. Vol. 24, No. 4. 1961

Can Zeno's Paradoxes be Resolved ?

page 8

Can Zeno's Paradoxes be Resolved ?

On March 22, Mr K. K. Campbell presented a paper to the Philosophical Society entitled "Can Zeno's Paradoxes Be Resolved?"

Zeno, the Greek philosopher presented four arguments against the possibility of motion. The central argument is that the concept of motion is self contradictory. Reality is to be described in propositions not merely self-consistent but necessarily true. Hence motion cannot be attributed to any part of the real world. Zeno said mortal men live in the world of appearance where motion is possible and the apparent world is contradictory. Philosophers have striven to escape Zeno's conclusions for if they cannot physics, mathematics and commonsense must be abandoned as sources of knowledge.

Mr Campbell explained Zeno's arguments. The arguments are in two groups of two each. The first group is the "Dichotomy" and "Achilles and the Tortoise."

The Dichotomy says that you cannot cross the arena. For before you reach the far side you must reach the middle and before you reach the middle you must reach the quarter mark and so on. To assert that a body moves is to assert that it moves through and successively touches an infinite number of points one after another. This is impossible. Therefore motion is impossible.

"Achilles and the Tortoise." Achilles gives the tortoise a start and then sets out after him running 10 times as fast as the tortoise. Before overtaking the tortoise he must reach the point from which the tortoise departed, and after that Achilles must reach the point which the tortoise had reached when he had reached the tortoise's starting point and so on. A world in which the fast cannot catch the slow is a world where motion is impossible. Thus motion is impossible.

The second pair of arguments consists of the "Flying Arrow" and the "Stadium." The "Flying Arrow" says: Whatever occupies a space equal to itself is at rest. But at any given instant in its flight a flying arrow occupies a space equal to itself. Hence, throughout its flight it occupies a space equal to itself. Therefore, it is at rest throughout its flight. Therefore, motion is impossible. "The Stadium" was not discussed by Mr Campbell owing to its complexity and also it has not had the importance of the other three arguments.

Philosophers from Aristotle to Bertrand Russell have tried to locate and account for fallacies in Zeno's arguments. Now these arguments are worded elliptically. In each case at least one argument is suppressed. The dichotomy expressed fully reads:
1.To move from A to B involves the serial completion of an infinite series of acts.
2.An infinite series of acts cannot be completed.
3.So every movement from A to B cannot be completed.
4.If every movement cannot he completed motion is impossible.
5.Therefore motion is impossible.

Successive Philosophers have Charged Zeno with Deductive Blunders, Because of the Omitted Steps. However, it may be shown that the arguments are not fallacious. The other way out is to deny the premises on which the arguments rest. The first pair of arguments depends on the infinite divisibility of space for otherwise there is not always a half-way point to reach or further ground to make up. The second pair of arguments depends on the atomic nature of space for without it the transposition from "arrow stationary at each point" to "arrow always stationary" cannot be made.

Most philosophers have held to the idea of the infinite divisibility of space since they believed the world to be Euclidian. Then the second pair of arguments have no cogency but the first pair become very difficult to deal with. Attempts have been made by philosophers from Aristotle onwards to deal with Zeno's paradoxes. However, to do so involves either conceptual schemes which are odd, paradoxical, or strained; or leads to further paradoxes.

Mr Campbell discussed the arguments used by several philosophers and the difficulties connected with them. Among the more interesting was an attempt by Bertrand Russell who claimed that space and time had properties of the Cantor continuum of real numbers. No paradoxes arise because the infinite classes can be defined so that the terms can be collected at one blow and not member by member. But it has been pointed out that moving from A to B in a cantor continuum is equivalent to beginning at 0 and counting the real numbers up to 1.

The trouble is that:
(i)There is no next number after 0.
(ii)The real numbers are demonstratably non denumerable.

The importance of Zeno's paradoxes lies in the fact that they have shown that premises are matters of choice and not observation. Zeno has pointed out unpalatable consequences of several more tempting ways of ordering our thinking. There is no scheme of concepts which accurately mirrors all and only all of the features of the real world. Nor is there a single clear consistent relationship between objects and concepts.

It can be said that Zeno's paradoxes have been resolved if by that is meant that the concept of motion is restored to a useful role. But it cannot be said that the essential condition of reality is mathematical in the sense that Euclidian or Cantonian mathematics applies to the real world. Zeno's arguments show that the possibility of applied mathematics is a contingent one. But nevertheless mathematics is workably applicable.