# Zeno’s Paradox Deconstructed – Zen’s Paradox, Ze’s Paradox, and Z’s Paradox

by Antone Roundy | 8 Comments | Vision

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.

December 31st, 2005 at 12:15 am

"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."Then a line (segment) being a real or imaginary mark positioned in relation to (non-existent) fixed

pointsof reference.....cannot exist!July 21st, 2007 at 1:49 pm

I think I agree with most if not all of this blog. This was interesting:

"Iâ€™ll start there: a line isnâ€™t made up of a bunch of pointsâ€“points are just descriptions of positions on a line."

As was this:

"space-time, or the four-dimensional space in which motion occurs, isnâ€™t made up of a bunch of instantsâ€“an instant is just a description of a â€œpositionâ€ in space-time."

In both of those quotes, Antone Roundy tells us what those things are NOT made up of but he does say what they ARE made up of. Here are my thoughts on Zenoâ€™s paradox: It is true that IF you could infinitely divide the amount distance something can move, then you could not move because if there are an infinite number of halfway points, and each of them takes a certain amount of time to get there, then it would take you an infinite amount of time to get there. Therefore, I conclude that there must be a smallest amount of movement possible for anything. At some point, it either moved or it didn't move. One could call this 'binary movement' because at some point something either moved or it didn't move, with no possible half way point. A bit is the smallest amount of information imaginable because it specifies that something is either true or false, 1 or 0, on or off, yes or no. I suppose you could refer this smallest amount of movement possible as one unit of space. But if there is "binary movement", then there has to be "binary time" as well because otherwise, there would be a point in time, at which something moved halfway between the 2 points which make up a length of the smallest amount of movement possible, which would mean that it was not the smallest amount of movement possible. Of course if you are moving from point A to point B, and you keep only moving by halfway, you would never get there, because you are moving by a smaller amount each time (not just smaller but half the distance of your previous movement), so there has to be a point at with you or anything else cannot move half a that distance. This makes it seem like time is something more than just a concept because you cannot move the smallest amount of movement possible in less than the smallest amount of time possible, which gives it a real world property of the smallest amount of itself. You could refer to this as one unit of time. So now I have this concept of single units of space and single units of time and the slowest anything can ever move is with a 1:1 ratio between space and time units. I am not completely sure about any of this, but sounds very logical to me. Let me know what you think.

January 6th, 2008 at 4:44 am

In reponse to INTPnerd's arguement:

If, like you say, there is a single smallest unit of movement, such as the binary movement, then it would undoubtably be movement over an amount of space. We also know that space is able to be represented by a number. Whether that number be one inch or one centemeter etc. However, if it is movement over space then theoretically that space can get smaller, because numbers can always become smaller. Therefore Space, time and anything represented by numbers can become smaller and smaller into infinity.

Please respond with any well thought out arguements.

June 9th, 2008 at 7:26 pm

I certainly do not have all of this fully fleshed out, however, the "single smallest unit of movement" is a conflation of one of the three dimensions of space: depth/length with time, the fourth dimension.

Time is a relative measurement and not a physical constant like length/depth: It consists of the distance between any two points and the time/measuremnt it takes for an object to move between those two points, i.e. the second is 1/60th within a minute and is a specific measurement of space/distance/time and is not an entirely arbitary (Sun/earth rotations) but still completely relative measurement as expressed through the classic instrument of time measurement: the 60 second/60 minute/24 hour western time-piece/watch/clock.

This must, in turn, be opposed/juxtaposed to other equally measured, but relativeistically determined distances, which include miles/kilometers, longitude/latitudes. Time cannot be, but is exluded from the Zeno's (Un)Paradox as his object/subject only takes " if it is movement over space then theoretically that space can get smaller, because numbers can always become smaller" which is the length/depth of the movement and NOT the time (fourth dimension) that is the actual transitive productive force that initiates the movement from the two points.

Zeno displaces "time" and thereby creates an infinite (albeit tantilizing) ultimately "absurd" regression:

"For me to act it takes 1 second to... 1 millisecond to... ," this, also, is an inf. regre. but instead of time being removed, the spatial element of length/depth has been removed.

"I," Zeno and possibly INTPnerd make the cognitive error of removing one of the 4 dimensions of space/time--which function interdependently and without regard to cognition. They combine mutually to create: space--length (or depth), width, height [which cumulatively create matter, "me" and you] and time [as an intrinsic combinatorial element of space, cannot be removed except cognitively]).

Zeno commutes the fourth element of time into length/depth, when he says, "one must take the half step of a half step of a... ?!?!" Time, which is the distance between two points, becomes depth or its length (THAT IS THE POINT A TO B and verily, as Roundy states, are just "a bunch of pointsâ€“points [which] are just descriptions of positions on a line."). That is, the depth or length of the step and is erroneously conflated by Zeno (without malicious intent, I'm sure) as the "time" it takes to make the first step:

â€œspace-time, or the four-dimensional space in which motion occurs, isnâ€™t made up of a bunch of instantsâ€“an instant is just a description of a â€œpositionâ€ in space-time.â€

There are no discrete moments in time [1] such as there are discrete steps or movements relative to two points in time-space because if there were we would then never move as Zeno hypothesized, but wrongly conflated: depth/length of the step as the relative (and fourth dimenstion of) movement it takes for the step, i.e. the time itself it takes to make the movement relative to point A and B.

1. See also Jeff Lynds at: http://www.peterlynds.net.nz/papers.html. Particularly, "Zeno's Paradoxes: A Timely Solution."

June 9th, 2008 at 8:41 pm

This blog post was actually written in reponse to Lynds' paper. The gist of the paper is that if you think of an instant as an very small slice of time, then there is motion within every instant -- just less and less of it as you measure out smaller and smaller instants. You can never slice time so thin that there's no room for motion in it.

I think that Lynds errs (or fails to give the best answer to the riddle) in not recognizing that, for example, a point is a ZERO dimensional abstraction rather than an infinitely small thing.

In other words, a point has no size, not because it's an object too infinitely small to contain any size, but because it's not an object at all -- just an abstraction -- a precise description of a position (and the position isn't an infinitely small area of space either).

An apple is not a small apple pie. It's what you're left with when you remove the sauce, the crust, etc. from an apple pie -- it's an object lacking certain parts of the nature of an apple pie. "Size" is not part of the nature of a point any more than crust is part of the nature of an apple.

Time is not part of the nature of an "instant" (an "instant" is what you're left with after you remove time from reality).

Once you've grasped the notion of completely discarding time when thinking about instants, it's simply unnecessary to go through all the complexity of slicing time really small and figuring out whether there's motion in the slice.

Motion is impossible in an instant because instants are abstractions from reality which contain no time and time is a necessary component of motion.

July 10th, 2008 at 6:29 pm

Now all 3-dimensional space is made up of lines between points, but those lines and points don't exist in reality. If by definition no point actually exists, then you really can't have a starting point or an ending point. So saying that you are six feet tall doesn't work because any point on you measured to any other point is simply an abstract concept measured along a line that doesn't exist. The difficulty about saying things don't exist at some smallest part is that you can take it back to the point that really all time and space is equally abstract as a single point or an instant.

The very fact that we CAN measure time and space might very well indicate that there really is some measure to a point or an instant - we're just not capable of perceiving them.

Remember that relativity tells us that time slows to a stop in the universe as you approach the speed of light. So the fastest thing in the universe causes everything else to be the slowest. So perhaps as you get larger, the smallest space gets smaller until you occupy all space, essentially becoming the smallest thing in the universe by virtue of being the largest.

If you are the largest(smallest) thing in the universe, you can never get to your destination because your destination lies the farthest possible distance from you - which is also the smallest possible distance from you - zero.

So the farther our goal, the faster we go, the relative distances and times change until we simply become our goal - having never gone anywhere because where we started has become where we are.

July 10th, 2008 at 7:22 pm

Chris, I'm not sure whether to think that you're joking, or whether you missed the ... ahem ... point.

A point doesn't exist PHYSICALLY, it has no PHYSICAL dimensions, but that doesn't mean we can't use it as a tool of thought to describe a location, either within another tool of thought like a line, a plane, or a 3-D space, or within the tangible space-time continuum (assuming the space-time continuum is the top level of reality).

Likewise, we can use a line segment as a tool of thought to talk about, for example, the relative positions of other abstract or physical things (like the top of your head and the bottoms of your feet).

Space, time, objects, etc. do of course exist, and obviously we can measure them. But when we describe the LENGTH of an object, for example, there's no need to say "it's three feet long, 1 inch thick, cylindrical in shape, and lasts for 25 years." Most of that information is immaterial to what we're interested in -- its length.

So we use a term, "length", to describe it. And the "length" that we're talking about isn't a trillionth of an inch thick or any other thickness. It just plain doesn't have any thickness to it.

A line segment equal in length to the object likewise doesn't have any thickness. The essence of a line segment is all about its length, position, and orientation -- not its thickness.

April 23rd, 2009 at 4:13 am

The Zeno paradox is only a paradox by specifically limiting the parameters so far from reality so as to create the paradox inherently.

In a universe consisting solely of 2 points and an attempt to travel between them such travel IS inherently impossible. It is only by the addition of a 3rd or more points can any motion be experienced.

The simple answer is that to travel to any given point you must only travel "half the distance" to a point 2 times as far away. You movement to the destination point from the starting point must inherently be referenced in relation to a 3rd point. It is absolutely correct that in a system consisting solely of 2 points there cannot be movement.... and reality does not remotely map to that criteria. Really one of the stupidest "paradoxes" ever. Like saying, If you had no legs, you couldn't walk, because you have to have legs to walk.... duh...