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Achilles and the Tortoise

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Zeno's second paradox of motion, of Achilles and the tortoise, is probably the best known of his four paradoxes of motion. In this problem, the fleet Greek warrior runs a race against a slow-moving tortoise. Assume Achilles runs at ten times the speed of the tortoise (1 meter per second to 0.1 meter per second). The tortoise is given a 100-meter handicap in a race that is 1,000 meters. By the time Achilles reaches the tortoise's starting point T0, the tortoise will have moved on to point T1. Soon, Achilles will reach point T1, but by then the tortoise would have moved on to T2, and so on, ad infinitum. Every time Achilles reaches a point where the tortoise has just been, the tortoise has moved on a bit. Although the distances between the two runners will diminish rapidly, Achilles can never catch up with the tortoise, or so it would seem.

Weird...

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• 6 months later...

Motion itself is a truthful resolution of an underlying paradox to our perspective. Motion is made up of time and distance. Truth may be communicated instantly between objects but to a particular perspective it must communicate the order and structure of the objects it is communicating and this communication has a revealed order which we call time. Distance is simply the degree of truthful separation. Distance or space has a time element to it as further objects must be communicated after near objects but time is much more than distance or space because it involves "other" events where A has to happen before B which has to happen before C in a linear vector sort of manner. This being said truth itself is discrete and Achilles smallest halfway point will reach it and cross it in a flash.

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• 1 month later...

Zeno simply describes Achilles' approach to the tortoise in vanishingly small increments of distance.

It does not describe what happens after that. In fact, the tortoise does not even have to be present.

We could examine his separation from a fixed wall at geometrically decreasing intervals of time.

The fact that an infinite number of snapshots can be taken before the time of intercept does not prove

the wall will not be reached or the tortoise overtaken. It's a pseudo paradox.

Any supposed difficulty here belies lack of familiarity with the nature of real numbers.

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• 1 month later...

This paradox can obviously be proven false by Achilles actually racing the tortoise, but its explanation is quite complicated. A simplified version of the explanation is that while Achilles must reach the spot where the tortoise was, he will most likely be moving so swiftly as to pass that point and get ahead of the tortoise mid-stride. This paradox only works if both Achilles and the tortoise go a certain length, stop, and start again.

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