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Zeno's Paradoxes

Zeno's Dichotomy Paradox

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 ¹/₄ of the way, but before we get ¹/₄ of the way we must reach ¹/₈ of the way etc. This argument makes us realize 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's Arrow Paradox

If 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).

Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution.

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.

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