Sunday, December 17, 2006


"You cannot even start."

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." (Aristotle Physics VI:9, 239b10)

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be represented as:

This description requires one to complete an infinite number of steps, which Zeno maintains is an impossibility. A more simple way of looking at it is to appreciate that any number divided by 2 can never be 0, and any number that is not 0 can be divided by 2. As such, if one considers that arriving at some destination point means 0 segments of distance left to travel, we would thus be faced with contending with a mathematically impossible x/2=0, where x is that remaining distance to the point of destination.

As long as there is some measurable distance left to travel, and dividing that remaining distance results in a number that is not 0, and given that any measure of distance divided by two cannot result in a 0 distance remaining, Xeno has argued that one must thus never reach 0, never reach the destination.

This sequence also presents a second problem in that it contains no first distance to run (no "x"), for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. Again, back to the mathematics of it, x/0="undefined". The answer to x/0 is that there is no answer. As such, part of the semantic problem is to discuss travel over a finite ("defined") distance using language devoid of a finite ("defined") definition. x/0 is undefined, and thus not useable when discussing travel of a finite (defined) distance.


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