Zeno’s Paradox Deconstructed – Zen’s Paradox, Ze’s Paradox, and Z’s Paradox
Zeno's paradox: motion is impossible, because to go from one place to another, you have to first go half way. Then to go the rest of the way, you have to first go half way, and so on, so you never quite get there. In fact, you can never get started, because before you can get half way, you first have to go half of half way, and so on. Let's deconstruct this paradox, simplifying it by removing one dimension at a time to make it easier to understand it's flaw.
Zen's paradox removes one dimension (time), and thus one letter: volume is impossible because before you can stack the top plane on an object, you have to stack the plane half way from the bottom to the top on, and then the one half way between the middle and the top, and so on, so you can never finish building it up.
Ze's paradox: area is impossible because before you can add the last line segment to the far edge of a two dimensional shape, you have to add the line segment that's half way there, and so on, so you can never get to the edge.
Z's paradox: length is impossible because before you can tack the last point onto a line segment, you have to tack the point half way to the end on, and so on, so you can never finish extending it.
The same error underlies all of these so-called paradoxes. Since the point and line case is easiest to explain, I'll start there: a line isn't made up of a bunch of points--points are just descriptions of positions on a line. We tend to imagine points as infinitesimally small objects, but in fact they are not real objects at all--a point is a zero-dimensional concept, completely devoid of length, height and width--not even a real object.
Jumping straight from Z to Zeno, just as lines have something in their nature (length) that is not derived from the points on the line, space-time, or the four-dimensional space in which motion occurs, isn't made up of a bunch of instants--an instant is just a description of a "position" in space-time. We tend to imagine instants as infinitesimally small slices of time, but in fact they are not real at all--an instant is a three-dimensional concept that falls short of being real, because it contains no time, which reality does. Although we name instants based on their temporal position in space-time, instants themselves do not contain any time whatsoever, and thus cannot contain anything that depends on time, like motion.
We tend to imagine that a three dimensional object is real and substantial because we think that we experience three dimensional objects all the time. But in fact, what we are experiencing is four dimensional objects--objects with length, width, height, and duration (time). Just as point and planes are not real things, but just concepts, because they lack at least one spatial dimension, an instant--the three dimensional state of things at a timeless position in space-time--is not a real thing, but just a concept. Without a complete object, including it's temporal attributes, we have only a simplification--an idea--not a real object.
The gap between instants and motion cannot be bridged by adding more instants--not even an infinite number of instants--because instants do not contain time. An apple pie can not be constructed only using apples, because there is no crust in an apple. Even with an infinite number of apples, there is no crust unless you add a crust. Even with an infinite number of instants, there is no space-time unless you add time. Thus motion is impossible if all we have is a collection of instants. If we understood time as well as we understand the three spatial dimensions, this would be as obvious as the fact that you can't fill a three dimensional object if all you have is planes.
Finally, let's restate each of the paradoxes in a way that makes their flaws more apparent. Z: it's impossible for a line to have length, because none of the points that make it up add any length to it. Ze: it's impossible for a two dimensional shape to have area because none of the line segments that make it up add any area to it. Zen: it's impossible for an object to have volume because none of the planes that make it up add any volume to it. And last but not least, Zeno: motion is impossible because none the instants that make up space-time contain any motion.