Because, while the logical discipline that he had developed was sufficient to identify the flaw in Zeno's reasoning, Aristotle did not spot how that could be done. So he instead opted for a much more elaborate and philosophically controversial approach.Why do you think Aristotle invented potential infinite to get out of the paradox? — Walter Pound
Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series such as sum_(i=1)^(infty)(1/2)^i=1 can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances.
2. Assert that one cannot do an infinite number of 'things' in a finite time. — andrewk
Suppose Homer wishes to walk to the end of a path. 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.
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This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.
This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) 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
The current status of the (veridical) paradoxes is what it always has been: they eloquently demonstrate that, when one reasons to a conclusion that contradicts what one confidently observes to be the case, there must be a flaw in the reasoning, and one has to carefully examine it in order to locate it. — andrewk
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