Tackling a paradox: Do we or don't we arrive at the destination?

Essay by qalaghCollege, Undergraduate February 2004

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Suppose I'm to walk on a straight road from A to B: I wear a pedometer, (as well as my sneakers) and set off. The distance (AB) is 100 meter. I come to a place where my pedometer shows 50. "Half of the way", I think and keep going, roughly estimating the remained distance: I'm going to walk half of the remained 50, then half of the remained 25, then half of the remained 12.5, then half of the remained 6.25, then half of the remained 3.125, then... Will I really arrive at B? Mathematically speaking, NO! I may get so so close to B, but never arrive at B itself. Then, why I really do arrive at B?

It's perhaps not that easy to answer. However, I just try to afford some rough conjecture: Focusing on the different nature of Bs (arriving points) may shed some light upon the dilemma: Realistically speaking, B is a place, not a single point.

It's a tree, or stone, or a cottage at the end of the road, where is 100 meter away from the starting station, A. Why we arrive at B? Well, because we take B as some visible, concrete spatial thing. B occupies some space; it has a three-dimensional nature NOT comparable to the mathematical B which is just a point, consisting of no dimension. I can claim that I actually never arrive at point B (some invisible theoretical thing) at the end of my journey, but I do arrive at that B (a stone, or a tree, or even an extremely narrow upright beam) which is placed right 100 meter away from the starting point A. Before launching into any further inferences, let's discuss the very nature of point itself: Mathematically speaking, what is a point? Is it some...