Zeno's Paradoxes

One enduring feature of pre-Socratic Philosophy is the group of arguments known as 'Zeno's Paradoxes'. The essence of them runs as follows--an arrow shot towards a target will first cover half the distance to the latter, and then half of the remaining distance, and then half of that remainder, etc. But (1/2)+(1/4)+(1/8), etc. will never add up to 1, meaning that the arrow can never reach the target, a conclusion at odds with the plain fact that it does. Zeno devised this kind of example in order to demonstrate a main doctrine of his colleague Parmenides, that Motion is illusory. It took two millennia for a compelling solution to arrive, as the invention of Calculus, and its Integral function, demonstrates how the arrow finally reaches its destination, thus disproving the thesis that Motion is unreal. In contrast, Bergson drew an entirely different lesson from the Paradox, and challenged its very premise. He inverts the Parmenidean principle, and argues that the example shows only that the intellect can only inadequately cognize Motion, i. e. that all that is demonstrated is that the sequence of static representations will never reach the end, the fallacy being that no static representation is identical to any Motion of any duration. Hence, it is the Intellect, not Motion, that is of an inferior reality. In the spirit of Bergson, another analysis is that 'half the distance to' presupposes something that does not at that point exist--the traversing of the full distance, which is meaningful only retrospectively upon completion--and, hence, is meaningless. Thus, the problem is not so much that the arrow never arrives, but that it can never get started under any description given as a fraction of the completed motion. This confusion is a based on the presupposition of the existence of the Future, a problem discussed in previous postings, and is of psychological and ethical significance beyond that of a mere intellectual puzzle.