Dichotomy Paradox

Imagine the following situation:

A runner is trying to run 1km. Before they can run 1km, then must run 0.5km. Before they can run 0.5km, they must run 0.25km. Before they can run 0.25km, they must run 0.125km and so on. This would require completing an infinite number of tasks, which Zeno claimed to be impossible.

This naturally divides the space into intervals:

0.5-1, 0.25-0.5, 0.125-0.25...

Note that Zeno assumes that an infinite number of non-overlapping space intervals can fit within a finite space. But he seems to doubt that an infinite number of non-overlapping time intervals can fit within a finite time. Why the asymmetry?

Archilles and the Tortoise Transformation Solution:

Suppose Achilles is in a foot race with a tortoise over 100m. By the time Archilles runs 100m, the tortoise has advanced a further 1m. By the time he has run the 1m, the tortoise has moved 1cm. By the time he runs the 1cm, the tortoise has moved 0.01cm, ect. Thus whenever Archilles has arrived where the tortoise was, he still has further distance to go

Direct Solution

(This is a better solution that I edited in)

If Achilles were to reach 200m, then given the speeds, the tortoise would be at 102m. If Archilles never reaches 200m, then we can conclude that a finite amount of space can be infinitely divided into intervals. This begs the question, why isn't time infinitely divisible into intervals as well? If we don't want an assymetry between space and time, then we'll accept this, which would mean there's no problem with an infinite number of intervals or instants occuring in a finite period of time. In fact, this will occur regardless of whether the arrow is at motion or rest.

Transformation Solution

(This was the original solution that I posted, but I now have a simpler one).

Let's simplify this and imagine that the tortoise moves half as fast as Archilles. To find when they intersect absent this paradox, let t be how far the tortoise has run and a be how far Archilles has run. So if they intersect, they would intersect at:

2t=a=t+100; that is t=100m, a=200m

Just to be absolutely clear, we haven't assumed that they actually intersect, just calculated where they will intersect if they do.

We can then transform the argument about Archilles always being behind the tortoise into one about Archilles never quite reaching the 200m. This then becomes almost equivalent to the previous paradox. The difference is that the distances are ...1/8, 1/4, 1/2 instead of 1/2, 3/4, 7/8... However, this is really just the Dichotomy Paradox in reverse.

Arrow Paradox:

If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion

Limit Solution

It may be useful to abstract this out to a discussion of a mathematical function (such as ) to avoid becoming tangled in metaphysics.

Even though the arrow moves zero distance in each instant, it doesn't mean that the speed is zero as 0/0 is undefined. We simply need to define speed using limits instead.

Aggregate Solution

The problem setup assumes that a time interval can be divided into instants, which is equivalent to saying that we can aggregate instants to form a proper time interval. If we don't want to create an assymetry between time and space, we need to allow that points in a space can also aggregate to from a proper space interval.

Suppose we represent spatial movement over particular periods of time as segments of a line. Whenever we choose an instant as the time period, the segment will be a point. Suppose we aggregate the spatial segments corresponding to each instant in the time period. We might naively assume that the aggregate of 0-size points must also have size of 0, but we argued in the previous section that these can aggregate to a proper space interval.

Final Thoughts:

Hopefully you agree with me that these resolutions are more satisfying than just invoking geometric series. I'm still working on a write-up for the arrow paradox. I feel my current write-up is 80% there, but still has a few loose ends.

New Comment
7 comments, sorted by Click to highlight new comments since: Today at 2:04 AM

I do find the asymmetry interesting, but isn't the arrow paradox the opposite end of that asymmetry, saying time is composed of infinite moments but doubting the same of space? I'm not sure how saying both are infinitely divisible helps resolve anything. You're using (already-substituted) the equations for distance and velocity, which is essentially just asserting the validity of calculus and taking as given that motion is possible and both space and time are continuous. Which is fine, but wouldn't have convinced Zeno, I think.

Basically, Zeno's 'paradox' is one of pure math that math answered by saying, 'infinite and infinitesimals are not actually problems, just go with it and it'll work out' just like it has with so many other seemingly-intuitive-and-necessary assumptions of Zeno's time. 

One other option is to say both space and time are discrete (Planck intervals?), but move in increments that are fundamentally linked in terms of natural units, but then we run into all the problems of combining relativity with quantum mechanics.

I am curious what Zeno would have thought of timeless physics. That is like turning around and saying, "Time is just another direction, so yes, change is impossible, but the unchanging thing already includes time and you have nowhere to stand to view it from outside."

My current write-up of the arrow paradox is here. In short, I start with a time interval being divisible into point-intervals, then I reverse that and say that an aggregate of point-intervals can cover a proper time-interval. I then suggest that this means that point-intervals can cover a space interval. I think I'm onto something here, but it still doesn't feel completely persuasive to me.

You're using (already-substituted) the equations for distance and velocity, which is essentially just asserting the validity of calculus and taking as given that motion is possible and both space and time are continuous. Which is fine, but wouldn't have convinced Zeno, I think.

I'm confused b/c nominally this claims to be addressing the arrow paradox, but it sounds like you are discussing the equations in Archilles and the Tortoise. The equations are being used to determine where they will cross IF they cross. To be absolutely clear, I'm not assuming they cross. I just need to know where the point will be in order to transform it to the dichotomy paradox (actually it's not quite the dichotomy paradox, but this paradox in reverse). And there's absolutely no need for calculus for a constant velocity.

I'm confused b/c nominally this claims to be addressing the arrow paradox, but it sounds like you are discussing the equations in Archilles and the Tortoise.

Ok, that's fair. That does apply more to the arrow paradox (and the dichotomy paradox, which is the same but with time reversed; if you're allowing the rate equation later on you seem to accept the validity of algebra, which means I can reverse time by substituting a new variable t'=-t (I expect negative numbers would give Zeno a heart attack, of course)). In your arrow writeup you talk about asking what an instant is. But in order for points to form a continuous line or line segment, you need uncountably many of them, one for each real number. Or, you need space and time to only be finitely divisible, and movement to happen in discrete intervals. The former requires allowing the existence of an  actually infinite set, which modern math is fine with but the ancient Greeks mostly were not. 

You don't need to know calculus to calculate distance for fixed speed (or fixed acceleration), but where do the rate equations come from in that case? How did you know to use that equation? I don't know any mathematical (not simply empirical, which we can't really on because if we do, we already empirically know things move and reach destinations, and Achilles catches the tortoise) basis for that other than integrating v=constant to get d=rt.

And the arrow and dichotomy paradoxes are prior to the Achilles paradox. If motion is impossible, Achilles can't catch the tortoise. If it is possible and moving things reach destinations, maybe he can. But by the time you have the possibility of motion and the function d=rt, you have a constructive proof.

So yeah - I would say drawing those lines without claiming they cross reduces the Achilles/tortoise paradox to the dichotomy and arrow paradoxes, which in turn are the same under time reversal. But without knowing how to get d=rt without calculus (which would be admitting the existence of limits and allowing taking sums of infinite series) I can't say if there is still a paradox left.

Well, if you assume that a line is made up of points, but that you can't have an actual infinity, then there's your contradiction right there - "The former requires allowing the existence of an  actually infinite set, which modern math is fine with but the ancient Greeks mostly were not. "

Good point. But if you instead take the line itself as primitive like Euclid did, you still need two points on it to define the line you mean, and while you have the starting points, I'm not sure where your second point is, since you're not assuming there is a point where the two lines cross. How, exactly, do you draw the lines in that case?

Side note: identifying the algebraic equation d=rt with a line requires a coordinate system. What coordinate system are you using? What set of numbers are you drawing the allowed coordinate values from? Do those numbers form a field in which all the basic arithmetic operations are well defined? 

The point I'm trying to get at is: Zeno's paradox is only a paradox because 1) he didn't accept the idea of a completed infinite set, and 2) many of his ideas about math and logic were ungrounded and underspecified. It's very hard to still think of these as paradoxical once you've seen how to build up the foundations of math from set theory (for which the assumptions are very simple) including geometry, arithmetic, algebra, and so on. 

I was just reading on another lesswrong thread about how T-reversal implies or not taking a complex conjugate. There was a metaphor about a stack of pancakes. If you reorder them have you flipped the stack or do you need to flip individual pancakes as well? It seemed that there were implicit time directions even in the static slices. That is if you take a static snapshot of particle and tell "now enter the next moment" it has inscribed in itself which way to go. This seems to be incontrast with the "if it is at rest, it can't be moving" intuition. That is a particle moving to rigth if complex conjugated will move to the left and that bit of information exists even in a perfectly thin view of the particle. So a particle is not just a collection of it's positions in different times.

The argument isn't about the sum but that the task is shown to be a hypertask. Zeno doesn't claim that the time intervals don't fit.

If I have code like:



while (distance<goal):


return distance

it doesn't terminate until it runs into floating point imprecision. In the limit of actually accurate floats it doesn't terminate. This doesnt have that much to do what I am doing with the numbers. Either the decomposition of the moving task is unfair, time doesn't perfectly divide or moving is impossible. Refractoring the loop out of it would be cheating. The general case of having recursion which doesn't bottom out will leave your machine in a livelock. Thus if you have a highly recursive function that doesn't livelock, you know it bottoms out.


In using the alternative analysis you are implying logic about running and keeping inertia. The way the movement is specified ight not be compatible. If I have kids in the backseat of a car and they keep asking "are we there yet?" it is not the case that they keep asking it and I answering it after we have arrived.

The argument can also be understood as commenting on the claim that if you have a process which on every step moves forward in time and never backward in time if you take sufficiently many/infinite steps you will neccessary cover all of time. This is false as we can provide a process that constantly moves forward but doesn't cover all of time. There is some slight of concept in that "never" is ambigious in respect to coordinate time vs step time.

You can take the tortoise location graph and the runner location graph and make a line that bounces between them when the runner is checking their position (draw horizontal line where he check where the tortoise is, draw a vertical line to wait until the runner reaches that point). This line doesn't terminate and it doesn't cross the crossing point of the runners. And this line doesn't cover all of coordinate time despite moving in time and never backing up. Letting you run the process longer and longer won't help you.

I don't think you addressed the actual points but dedistracted yourself fromj some surrounding technicalities.