Zeno’s Paradox

Zeno’s Paradox calls into question the possibility of motion. It has a few variants, each made by Zeno of Elea, a Greek philosopher of the 5th century BC.

Paradoxes are arguments with premises that appear obviously true or reasonable and yet lead to a conclusion that is blatantly absurd or false. For example, many logical paradoxes start with assumptions that look completely reasonable (e.g. “This sentence is false” is either true or false), and yet lead to contradictions (e.g. “This sentence is false” is both true and false).

The key to solving paradoxes involves either showing that one or more of the premises are false, and so do not lead to the absurd conclusion, or else showing that the absurd conclusion does not have to follow from the premises (i.e. the conclusion can rightly be false even when the premises are true), or else showing that the supposed absurd result is not actually absurd.

So we already know the conclusion with Zeno’s Paradox—the denial of the possibility of motion. There’s no need to get into how absurd this sounds. Indeed, I will take it for granted that it is absurd. So how does Zeno get us there?

The first version of Zeno’s Paradox I’ll cover is called the Arrow Paradox. Consider an object that is typically considered to be moving, such as that of a flying arrow. The arrow will occupy a specific position in space at any given instance. However, in occupying an exact position in space, as opposed to some other position, it is not in motion. Since the arrow is merely a series of occupied places, the arrow is therefore never in motion.

My presentation of the argument is slightly more drawn out in order to give, I hope, a clearer picture of what is going on. For reference, here is Aristotle’s more concise version of this argument: “[Zeno] says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.” (Physics Book VI, section 9) (1)

There are two responses to the arrow paradox that I now tend to take. One says that while motion may not occur in an instant, this does not prevent motion from being, because motion is composed of more than one instant; it is in fact a change in the position of the object from one instance to the next. Such a view seems plausible and intuitive. This is, more or less, Bertrand Russell’s response, called the at-at theory of motion, in which, as Wesely C. Salmon outlines:

 “. . . motion is a function that pairs points of space with instants of time. To move from A to B is simply to occupy the intervening points at the intervening moments. It consists in being at particular points at the corresponding instants (hence the name of the theory). There is no additional question as to how the arrow gets from point A to point B; the answer has already been given: by being at the intervening points at the intervening moments.” (2)

The second response I now take says that there is no such thing as an instance of time that is without duration. Even logically speaking, or in an ideal sense, we should have a problem with such a notion. For we might ask how long an instance lasts. If it lasts but an instant—again how long is that? If it lasts no time at all, then we should think we are dealing with nothing at all. And if it lasts some amount of time, on the other hand, then we seem to be moving away from Zeno’s premise of an instance of time where the arrow is necessarily stationary. For all we know, objects in motion are always in motion, even in their respective instances of time, since they admit of duration. The result is that an object in motion will be in motion for any duration of time we care to deal with.

Peter Lynds draws a very similar conclusion in denying static instances of time. It is clear that what he is against is the notion of some interval of time that is static, and which would make anything in motion be likewise static under that time. (3)

Lynds’ further contention is that a thing in motion is never fully determinate with regard to its position (If you are interested, please read his article here: http://www.quantum-gravitation.de/media/f483977d7e58f426ffff8253ffffffef.pdf). Although I won’t go into great detail on this point, I will note how I think it should be resisted. For one point of possible contention, we should be careful not to deny that objects can occupy one discrete position in space as opposed to some other. Perhaps the way that it occupies that space is what should be qualified, as opposed to its occupying space full stop. The place of the arrow that is on the side of the direction it is headed in is perhaps occupying a particular place for entering it, while the tail end is coming out of the place it is leaving. Without going into further detail about this, such a picture would never deny that this occurs relative to certain, specific and exact places and not others. Such places are presupposed in order for the picture of the movement of the arrow to be intelligible at all. And so, the location of a moving object would still be capable of being determinate. Also, there should be an answer to exactly how much space is being traversed in any amount of time. That is, it should be determinate exactly which spaces are being exited and entered; perhaps taken together this makes such spaces the traversal space of a moving object. It is not very clear what else is required that would make such moving objects determinate with regard to their position, unless ‘position’ itself just meant that what is in it is not moving, which Lynds and I reject. In short, I see no reason to term such moving objects ‘indeterminate’ with regard to their position. If ‘determinate’ means non-moving, then even so, exactly where something is moving is non-moving. The area of traversal does not itself move, and thus objects can be determinately placed as traversing in it. Other than this minor point, however, I find Lynds’ approach to be very helpful in coming to deny static instances of time, or of durationless instances. Without a durationless instant, it is hard to see how dividing instances gets the desired result of undermining motion, since motion is unproblematic granted there is a duration.

Another version of Zeno’s Paradox is called the Dichotomy Paradox.

Consider crossing a room. How can you get across it? For in order to cross it, you must first go halfway across. Yet in order to go halfway, you must first go half of halfway. Yet in order to go half of halfway, you must first go half of half of halfway. And so on it will go. So how could you even begin, given that any starting point that you entertain traversing will necessitate first traversing its half? Also, there are an infinite number of steps, if you could begin at all. How could you complete an infinite number of steps in a finite time?

One answer goes that space is not actually infinitely divisible (perhaps it is potentially infinitely divisible). There is a smallest unit of space, which today is hypothesized to be the Planck length (from the physicist Max Planck). If there is such a smallest unit of space, then such units may be traversed in successive moments according to the at-at theory of motion or according to the fact that there are no moments without duration, as previously explained.

However, what if there is no smallest unit of space? In the context of Zeno’s Paradox, is it granted that objects, moving or non-moving, take up space? A skeptical argument might go: how could that be, for in order to take up some amount of space, it would have to take up an infinite amount of spaces that compose it? However, does this successfully grant us skepticism concerning that things take up space? It is unclear at best: after all, it does not seem like an object’s taking up some space has to complete an infinite number of steps, or have to complete some step it cannot begin, as in the Dichotomy Paradox. The object’s being in some space does not seem logically hindered from being there. Yet if this is right, then what is to stop an object from proceeding to any other space whatsoever, granted the at-at theory of motion? That is, why does a moving object have to first be in some particular space before it gets to another? Why not just show up two spaces ahead in the next moment just because it is in the nature of the thing to move in such a way? That is, is it illegal for an object to skip ahead some spaces in its motion? It does not seem illegal from a logical point of view. The at-at theory presents a ready picture for how this can happen.

With this thought in hand, it does not seem like a stretch to just make motion into something more discrete, such as in frames on a computer screen. Consider a game like pong. So what if the ball moves by shutting off certain squares behind and lighting up others ahead? How is the notion of motion damaged for this? It does not seem to be in the slightest. Obviously, the squares that are lit and turned off can be composed of lengths less than that of the squares themselves. Yet, the ball in pong does not need to traverse these lengths in any continuous fashion—one moment it is not there, and in another it is. If we ask how it can be there at some moment, for having to take up an infinite number of increasingly smaller spaces, then the right answer seems to be that it takes up parts of its space for taking up the whole of its space, just as any static or supposedly static object does and which we find unproblematic.

There is something in the Dichotomy Paradox like a reversed order of explanation. We should explain that something moves halfway across a room for moving all the way across. At least moving halfway is necessitated by moving all the way. On the other hand, we should never explain that something moved all the way across a room for moving halfway across it, unless there is some further explanation for how moving halfway necessitated its moving all the way. Zeno’s Paradox seems to demand an explanation for moving partway while denying that moving all the way is in any way reasonable. But an object’s moving some amount seems to be a basic fact, one that can perhaps be explained by appealing to external causal factors that act on it, or internal ones having to do with its nature, but beyond which cannot be further explained. Its moving granted a cause is just a brute fact. The why question seems to be at an end here. An object takes up some space because without that it would not be an object, and it moves into other spaces for its intrinsic nature or causal forces acting upon it.

The reason for skepticism has to do with the notion that an infinite number of steps needs to be performed in a finite time. After all, we tend to think of the steps in a task as serial—that is, beginning with step one and ending with the final step, with each step taking some amount of time. That’s how much of our work proceeds; and of course if we had a task that had an infinite number of steps we could not complete it without taking forever. However, not all tasks are serial. Just like being in a place happens all at once, so does moving to a next place. There is just no problem with thinking that movement to a next place in some amounts of time need to involve anything more than a step, composed of non-serial, perhaps infinite, steps (e.g. into a region of space where that space is thought to be infinitely divisible) that are completed in the time. All we need to do is resist the assumption that such steps are serial. Being there is not a series of steps, despite being spatially composed, and neither is moving. A discrete one-step motion is what can get an object from one position to another. Note too that this need not say anything about whether space is discrete or continuous. It merely says that the objects and their movement through space can be in discrete blocks that are all at once, on or off.

On the other hand, continuous motion also need not be jeopardized. As with the Arrow Paradox, there is never an instant of time where the thing moving across the room is static. It is always already on its way through some given amount of space, given any amount of time. Perhaps it is wondered how a static object begins to move. This would be a relevant and excellent question. For now, I see this as merely another occurrence completed in one concrete, non-serially composed, step. And I don’t note anything that might undermine such a notion.

The final version I will cover is the Achilles and the Tortoise Paradox.

By now, you will likely see how to resolve it (or how I would resolve it) as I present it. Consider Achilles who races a tortoise. Since Achilles is very fast, he will be handicapped a little by giving the tortoise a head start. Yet, now the tortoise must win, for Achilles will never overtake him. For, when Achilles starts running, he will need to reach the tortoise where he is, but since the tortoise still moves, once Achilles reaches that spot, he will need to reach where the tortoise have moved to, yet once at that spot, the tortoise will have moved still further, and will need to get there. But once there—you guessed it—the tortoise will have moved ever so slightly more. Since this would be similarly divided forever, Achilles will never reach the tortoise.

To give a brief response using the tools we’ve used with the other versions, Achilles will indeed reach and surpass the tortoise, because given that any moment of Achilles’ and the tortoises’ movement will have a duration, the movement in that duration will be of a greater distance for Achilles than it is for the tortoise. Consider Achilles gaining on the tortoise (this is an illustration that commonly accompanies the paradox, but doesn’t the image contradict the other versions of Zeno’s Paradox!?—No, it’s always only apparent motion!). This is only so if Achilles’ movement is greater than that of the tortoise. Given the greater movement relative to the tortoise at any given duration, it is therefore only a matter of time before Achilles surpasses the tortoise.

This has been an overview of three of Zeno’s Paradoxes against the possibility of motion, with responses to each one. When we understand moments as always having a duration, or if we make use of an at-at theory of motion, it becomes clear how to resolve the versions of the paradox I’ve presented.

I will hopefully write on at least one other version of Zeno’s paradox in the future.



(1) Aristotle. Physics Book VI, section 9.  http://classics.mit.edu/Aristotle/physics.6.vi.html

(2) Salmon, Wesely C. Causality and Explanationhttps://books.google.com/books?id=uPRbOOv1YxUC&pg=PA198&lpg=PA198&dq=at+at+theory+of+motion+russell&hl=en#v=onepage&q=at%20at%20theory%20of%20motion%20russell&f=false

(3) Lynds, Peter. Zeno’s Paradox: A Timely Solutionhttp://www.quantum-gravitation.de/media/f483977d7e58f426ffff8253ffffffef.pdf

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