Mathematicians and their Contributions

Zeno of Elea

For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.

Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.

The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum

1/2 + 1/4 + 1/8 + ... = 1

On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + ... backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that 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 the distance). Aristotle argues against the paradox by claiming:-

... for time is not composed of indivisible 'nows', no more than is any other magnitude.

However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.