Zeno’s paradox | Hacker News Reader

Zeno’s paradox

joeldavidhamkins.substack.com

47 pointsswarmy63 comments

A little while back I did some reading on Zeno's paradox (really Zeno's paradoxes --- he had several, not all of which seem to have survived). I had always understood them as an interesting early attempt to grapple with the concept of infinity, but now more or less solved by modern calculus. And from a purely mathematical standpoint this basically holds.

But what surprised me was that Zeno was not fundamentally interested in the mathematical implications of his paradoxes. He was more interested in their philosophical implications. To give a bit of context, Zeno was a student of Parmenides, who was perhaps the purest of the monists. Parmenides held that all things are a single unity and that any change is an illusion. Most other Greek philosophers found this idea absurd since it seems pretty self-evident that things are changing around us --- arrows fly through the air, runners race around a track.

Zeno's purpose in devising these paradoxes was to show that it wasn't so self-evident that things were actually moving at all. And it was a counterpoint to the Pythagoreans, who believed that all things began in the number one, but then proceeded in multiples of the number one (the number two, three, four, etc., and through the numbers, all things in the universe since the Pythagoreans believed that all matter was fundamentally composed of integers). In the arrow paradox, Zeno was essentially trying to show that it was logically absurd to start with a unity and go to a multiple as the Pythagoreans held. And in the Achilles and stadium paradoxes Zeno was trying to show that it was absurd to start with a multiple and go to a unity.

We are smarter than that. Now we know the world is composed of IEEE floats.

Calculus doesn’t solve this. Creating a way to solve sums of infinite series does not itself prove anything about the mechanism of the physical world. Yes, obviously motion is possible. And 1/2 + 1/4 + 1/8 … = 1. But that doesn’t tell you how the motion can actually go through all those steps in finite time. It might very well be that the only reason the universe works is because it’s discrete, not continuous.

The total time taken you can show is bounded by 2 which is finite.

Zeno also knew that arrows arrive at their target in finite time. That's why it is a paradox.

Thanks for the background! I hadn’t known about its place as a philosophical battle ground.

A much more intuitive explanation is this:

We are concerned really with how long it takes to travel this distance. We also know, from real life, that it is, in fact, possible to travel it.

So Zeno is telling us "where am I wrong, show me", this is what a "paradox it".

Assume that we are running at a constant velocity for the entire distance.

As the distance shortens so does the TIME needed to travel it.

As the distance becomes infintessimaly small so does the TIME NEEDED TO TRAVEL IT.

So when you reach "the limit" you travel "through it" in an "instant".

So your velocity doesn't change as the time decreases alongside the distance traveled.

So the point is "don't look at the distance only", since in order to TRAVEL a given distance, one needs to MOVE, and movement implies TIME, which means VELOCITY comes into play.

So while there are many of those smaller 1/2's, you travel through them faster and faster.

This is true. But Zeno would still counter:

Infinitely many time intervals, however short, cannot have passed after finite time.

True.

Then I would counter with:

We traveled 1/2 of the way, so we still have 1/2 to travel.

Since we travel at constant speed the 2nd half of the entire way cannot take longer to travel than the 1st half of the entire way.

So each subdivision can NOT ADD more time than the previous 1/2 interval.

So we have an upper bound on time, since time can never be larger.

So while we do have an infinite amount of time intervals, the sum will never grow, it's upwards limited.

But since each "parent interval" is upwards limited, it's irrelevant to how many "child" intervals you subdivide it as the parent time interval is always upwards limited.

So no matter how many times you subdivide, the TOTAL TIME never grows, thus time is not infinite.

I like this explanation. Every other argument I've seen against Zeno's paradox either takes as a given that an infinite series can converge to a finite sum (which I don't think is a self evident truth) or relies on theorems about infinite sums that weren't rigorously proven until well over 1000 years later. This is the only counterargument I've seen that seems like it would have held up in Zeno's time.

I like this point, but I'm not sure exactly how well concepts such as velocity and time, etc. were understood. They were certainly thinking about it, but even the idea of no action meaning constant motion (without other forces) was about 1500+ years away.

I think Zeno might have invented calculus if it weren't for the fact that math wasn't even nearly sophisticated[1] enough to admit any sensible formalization of the ideas.

[1] Perhaps it was even just because the power of symbolic algebra hadn't been fully realized. Geometry and logic only takes you so far.

Infinitesimals exist precisely to create an reciprocal quantity to infinities.

Infinitely many finitely short time intervals cannot have passed after finite time, but infinitely many infinitesimally short time intervals can. In the same way that one way of conceiving infinity is to imagine it as an arbitrarily large value at any given expansion, but infinity in the limit, an infinitesimal is an arbitrarily small value at any given expansion, and zero in the limit.

That's the whole point. The atomic distance of a subdivided step goes to infinitesimal at the same rate that the time associated with the distance does, so we're fine!

Infinitesimals are cool but I think this is not really about infinitesimals - it is about ordinary real numbers.

(so I think this is not true: "Infinitely many finitely short time intervals cannot have passed after finite time")

(I agree here: "infinitely many infinitesimally" is finite) (an infinitesimal is smaller than any real number, especially smaller than 1/n for every natural number)

https://en.wikipedia.org/wiki/Infinitesimal

"Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632."

Yes they can. Zeno is wrong. There are an infinite amount of points between 0 and 1.

> Infinitely many time intervals, however short, cannot have passed after finite time.

This isn't true, because we know from math that infinite series can converge to finite sums. In particular, 1/2 + 1/4 + 1/8 + 1/2^n does, in fact, converge to 1.

No one seems to be bold enough to actually go this far, so I’ll make the claim:

Zeno’s paradox is wrong because its key premise is wrong: there aren’t infinitely many time intervals or distances, or arbitrarily small time intervals or distances.

That just raises a new paradox: how can you jump across those discrete distances without passing through the space between?

With my feet, usually.

Solvitur ambulando, as St. Augustine and/or Lewis Carroll would say.

If space is discrete, then there is no space in between. If you shift a bit to the right or left, it does not pass through some in between space, it disappears in one position and appears in one next to it.

Similar to Richard Feynman’s famous answer to the question “why do I feel the force of a magnet from several inches away?” What counts as a satisfying answer depends on the person, but the general answer is “that’s how the physical world works.” This one doesn’t particularly feel like a “paradox.”

I would say it is the other way around, it is totally feasible to traverse an infinite number of intervals in a fixed time. As I also responded to a different comment, just assume space is E³. Would you want to argue that you can not move or reach the goal under that assumption?

But if it's an infinite number of spatial intervals, then it's also an infinite number of temporal intervals, so you can't argue they cancel without quantifying how fast they grow.

Sure, but we know from the setup and I think it is fair to just assume a constant velocity which gives us this countably infinite sequence of distances and times, both getting shorter geometrically. And their partial sums have a finite limit. In essence you seem to be stuck because you have to complete an infinite number of tasks, but that is just the wrong measure, i.e. you must measure the time or distance, not the number of tasks in order to determine if you can complete all the tasks. Just walk from start to goal, there is no task in Xeno's infinite task list where he could say that you did not complete it. The problem is that Xeno can not give you the complete list of tasks if he tries to enumerate them one by one, not that you can not complete them all. Instead of enumerating them, he should just tell you to visit all places 2^-n for all natural n, you walk one meter and are done.

The way I had the 0.999…=1 concept explained to me (evidently late, I was out of high school when I encountered it) is that there are infinities which are larger or smaller than other ones.

I now know this more familiarly as sets: the infinite set of half distances in Zeno’s paradox is a subset of another infinite set in the same paradox, where the distance to travel is greater. If, for instance, your destination is the chemist down the road, that’s a smaller[1] infinite set of half distances than traversing all of space.

Because we are living beings who move beyond our initial destinations, with compounding goals we reach the smaller infinite set because it’s a subset of a larger one. And because we're living beings who are mortal, we eventually cease movement presumably at some increment less than 1/1 of some particular destination or goal.

There, not a paradox! We are simultaneously able to move because there is no singular infinite subset of distances to traverse, and unable to move at a distinct point in time because we die before we exhaust the infinite superset of other distances to traverse, but we’ve already traversed some subset of it by that point.

1: peanuts/1, and this is the first time I’ve got to reference the same joke on HN twice in totally different contexts just a few days apart.

My eight year old son (who watches a lot of youtube videos about physics stuff) countered with the comment that there can’t be infinitely many time intervals because eventually they get down to Planck time and time doesn’t really make sense past that point.

>> Infinitely many time intervals, however short, cannot have passed after finite time.

If they are finite time intervals that's true. But not true for infinitessimally short time intervals. If Zeno doesn't like this, we can say the time intervals are just as large as his distances. He can't argue infinitessimally short distances and not accept the same for time intervals. Playing dumb here backfires.

Actually, Zeno's argument actually was that it was impossible to move. He was a disciple of the earlier philosopher Parmenides who had argued the change if any kind was impossible and therefore an illusion. This was a hot debate in pre-socratic philosophy.

i'm guessing it was difficult to get them to change their position on this topic?

> Assume that we are running at a constant velocity for the entire distance.

Doesn’t this assumption rely on a resolution of Zeno’s paradox, to go from zero to a constant velocity?

Some Dilbert Zeno's linked below...

The first comic can actually be a pragmatic answer (used slightly tongue in cheek) at status report meetings. I have used this line twice and it sounded plausible enough that no more questions were asked of me that week :)

[1] https://dilbert.com/strip/2016-01-31

[2] https://dilbert.com/strip/2014-02-02

When we did Zeno's Paradox at school, I came up with a "harder" variation: We have a switch that turns a light bulb off and on; the light bulb is off; we switch it on after a second; we switch it off again after another half second; we switch it on again after another quarter of a second, etc., etc. After a total of two seconds: is the light bulb on or off?

The answer is off, because as the rate of switching the light on and off gets to a certain point, a massive amounts of heat is generated which melts the circuit.

The fun thing about infinities is that they don't exist, and you can't prove they exist because you don't have time or accurate enough measuring devices to do it.

This analogous to the "grim reaper paradox"

Spacetime is not a Euclidean nor Cartesian space. Assuming you can subdivide it as much as you want is a big assumption.

> Before you complete the move from A to B , however, you must of course have gotten half way there.

This is where the paradox falls apart for me. There's no reason to assume that there is always a halfway there.

Just assume space is E³. Would you want to argue that it is impossible to move in E³? If not, then the resolution can not [only] be that you can not indefinitly divide space intervals in half.

> Just assume space is E³

To me, that's just begging the question. You can't assume that.

> Would you want to argue that it is impossible to move in E³?

I don't know what gave you that impression. Yes, it's possible to move in Euclidean space. That has no bearing on Zeno's paradox which is not about Euclidean space.

You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space, otherwise it would not matter whether or not our universe is Euclidean, correct? So finally, if space being Euclidean is what separates being able to move in our universe but not in the one portrait by the paradox, would this then not imply that either you can not move or reach the goal in Euclidean space or that there must be another reason that prevents the consequences of the paradox from applying to Euclidean space?

> You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space

What I should have said is that you cannot assume that spacetime can be infinitely divided. Instead I started that spacetime is not what people usually assume it to be.

The paradox (as stated) doesn't assume Euclidean space, although Euclidean space satisfies the assumptions of the paradox.

Finally, if spacetime is not Euclidean then applying the paradox to a Euclidean space is just a thought experiment and everything will depend on your definitions and how you define movement in your fictional space.

What Zeno was trying to get at with his paradox was a takedown of the idea of infinity applying to the real world.

To move from 0 to 1 in R starting at time 0 and arriving at time 1 means x(t) = max(0, min(t, 1)) ignoring the problem of infinite acceleration at 0 and 1 which we could smooth out. I would find it paradoxical if that motion was not possible. All our laws of physics assume und describe motion in space build on top of the reals and tell you how it works and they can perfectly describe the motion I sketched at the top. If anything, then should expect to not be able to move in the real world if space can not be indefinitely divided because the laws showing that this motion is possible do not apply.

I think the rather straight forward solution is to simply say that you can do infinitely many things in some fixed time. Where is the problem in saying after walking one meter that by doing so you managed to traverse an infinite number of intervals? This is only problematic if you think that doing an infinite number of things requires an infinite amount of time, but why would it?

Firstly, there is a little conflation between the Euclidean space and a Euclidean space with a time element going on here. It's not important to your argument but I wanted to call attention to it.

Yes, one way to resolve the paradox is to allow time to be infinitely divisible like (Euclidean) space. Then you can describe where something is at any point in time.

But that discussion is not interesting to me. It is NOT true that "all of our laws of physics assume and describe motion in space build on top of the reals." Both relativity and QM tell us that spacetime is NOT Euclidean.

To be precise, relativity - special and general - do not use Euclidean space, but quantum mechanics does. I guess you where thinking of relativistic quantum mechanics or relativistic quantum field theories. But again, irrelevant. I just picked Euclidean space and an implied real time axis because I am not sure how to label the idea properly - do we need a real closed field, would rational numbers do? So do not take Euclidean too literally, understand it as three dimensions of space and one of time that all can be divided indefinitely. If we reach a point where this is not good enough, we can refine it.

And I think you still did not get my point. If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time. So I argue you would then have to accept the consequences of the paradox in this hypothetical universe, would you not? And one such hypothetical universe is the universe of classical mechanics, so how would you reconcile that? Was classical mechanics wrong all the time when it described motions in a space where motions are not possible?

> If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time.

This does not follow logically. We have a sequence of assumptions and an argument. Just because there is an example of a system where the assumptions do not hold, that does not mean the argument is sound when the assumptions do hold.

And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.)

Okay, now we are converging. I spelled this out in my first comment but was sometimes lazy later on and did not mention it every time.

[...] then the resolution can not [only] be that you can not indefinitly divide space intervals in half.

So your position is essentially the paradox is resolved in our universe because the unstated assumption about the divisibility of space is not satisfied, but even if it was satisfied, Xeno's reasoning would still be wrong. So there is some more fundamental flaw in the reasoning but it doesn't really matter to you because you have a resolution for our universe by denying one of the assumption.

Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental.

But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible. So let me offer you a different view that does not hinge on the divisibility of space and time.

Xeno says, to reach the goal you first have to move half the way. Fine, I can do that. Then he says but in order to do that, you first have to move quarter of the way. Fine, I can do that. Then he says but in order to do that, you first have to move one eight of the way. Fine, I can do that. Then he says in order...and I get annoyed. Just tell me the first thing I have to do so we can start actually doing something. And now Xeno is in trouble, it is not that you can not do what he asks you to do, the problem is that Xeno can not make up his mind what you should do first.

Xeno can pick any arbitrarily large number, subdivide that often, and give you a list of things to do and you can easily do them, no matter how large a number he picked. You could even say, hey Xeno, let me just move to the goal. Does doing so include the task of going half of the way? Yes. Does this include the task of going quarter of the way? Yes. An eight? Yes? Is there any task in your infinite list that I did not complete? No.

The reasoning in the other version is more or less the same, you go half the way, then half the way of the rest, half the way of the rest, ... This time you seem stuck because Xeno can not make up his mind what the last task is you have to do, but again you can do all of them. This version has the nice property that you can actually start doing something as Xeno can at least tell you what the first task is, so you move half the way. And then another quarter. Another eight.

And while it might seem as if you are stuck as Xeno keeps throwing new tasks at you, you are not. We just have this tendency to think of tasks as requiring some fixed time and doing infinitely many of them will take forever. But if you keep track of time as Xeno throws more and more tasks at you, you will notice that time grinds to a halt. You complete one task after another but as you move with constant velocity each one takes less and less time and the total time approaches some limit slower and slower. So it takes you forever when measured in number of tasks but only something finite when measured in time. And again, just walking to the goal will include any task that Xeno would ever have asked you to do, so he can do something better with his time than trying to write a complete list of tasks for you.

And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.)

Wrong is the wrong term, it is an approximation. Take special relativity, in the limit of all velocities going to zero, you recover Newtonian mechanics exactly. Take general relativity, in the limit of slow motions, small masses, and low energies, you recover Newtonian mechanics exactly. The same for quantum mechanics. So if you approximately satisfy the assumptions, you will get a result with a very small error.

EDIT: I just thought of a variation of the problem, assuming indefinite divisibility of time. Xeno asks you to be in a certain place at 12:00. And 11:30. And 11:15. And 11:07:30. And so on. Admittedly being in a place is somewhat different than moving to somewhere, but would we call this a paradox? If we just stay at this place from 11:00 to 12:00 we can satisfy all requests Xeno makes, we can be there at all the times from any arbitrary finite or infinite list of times between 11:00 and 12:00. The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.

> Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental.

Yes, I agree, our knowledge is imperfect and the truth may be different.

> But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible.

Here we disagree. I think the whole point of the exercise from Zeno's point-of-view is to make you question your assumptions.

> So let me offer you a different view that does not hinge on the divisibility of space and time.

I didn't find your resolution very satisfying - it seemed to boil down to "motion is clearly possible by demonstration" and "you can't make an infinite list". Both of these things are part of Zeno's argument (the second in a different form).

> And again, just walking to the goal will include any task that Xeno would ever have asked you to do

When you get down to the quantum scale, is this still true? Does it even make sense to talk about a human moving a very small fraction of the width of an atom?

> The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.

Would it surprise you to learn that I think the universe is at most countable, if not finite?

Here we disagree. I think the whole point of the exercise from Zeno's point-of-view is to make you question your assumptions.

Sure, but we also agree that it would be weird if it was impossible to move in Euclidean space, so I would say further questioning is warranted, what other assumptions are wrong that also resolve the paradox for Euclidean space.

I would say the core of my argument is that Xeno essentially argues that you can not do an infinite number of things in finite time at that this is a wrong assumption. Look, I can not even finish telling you all the things you have to do, how could you possibly do them all?

And as I said, just being in one place from 11:00 to 12:00 makes you do being at that place at all times between 11:00 to 12:00, independent of whether there are finitely many, countably many, or uncountably many instants in that interval. It is also noteworthy that doing something here is a pretty weak form of doing something, being at a place at some instant does not take any time at all or the smallest possible amount of time if time is not continuous. But similarly is moving infinitesimal distances in the original paradox a pretty weak form of doing something.

But now that I think about it again, my argument can not replace your argument as I rely on the indefinit divisibility. So what I would really want to say is probably either space and time or not indefinitely divisibel, then Xeno can not make an infinite list of tasks to do, or space and time are indefinitely divisibel, but then it is also possible to do infinitely many tasks in finite time notwithstanding Xeno's inability to enumerate them.

Would it surprise you to learn that I think the universe is at most countable, if not finite?

Not really, I am also sympathetic towards ideas like loop quantum gravity. I also always liked - for no good reason - the idea of a closed universe but the observed flatness seems to point into a different direction. On the other hand someone pointed out to me some time ago that there are closed flat topologies, for example a torus, which I knew but never considered when thinking about the universe.

Think of it this way - if we allow ourselves to infinitely divide space, eventually we are talking about distances much, much smaller than the Planck distance (QM concept in case you are not familiar). At those scales, we can't even pinpoint the position of an electron. How can you talk about the movement of an arrow over than distance? You do really think that the equations of motion are accurately describing the physically reality at that point?

You are missing my point. If being unable to divide space and time over and over again is the true and only resolution to the paradox, then in a hypothetical world where you can divide space and time indefinitely, the paradox would not be resolved. So I ask, are you willing to argue that if our spacetime was Euclidean, then we could not move? I would find that claim absurd, after all classical mechanics describes motion in Euclidean space. Is classical mechanics nonsense because it describes motion in a space where motion is not possible?

I said:

>> Yes, one way to resolve the paradox is to allow time to be infinitely divisible like (Euclidean) space. Then you can describe where something is at any point in time.

Then you said:

> So I ask, are you willing to argue that if our spacetime was Euclidean, then we could not move?

I'm sure how to move forward from there and I am not interested in discussing Zeno's paradox relative to Euclidean space and Newtonian mechanics about which we seem to be in violent agreement.

On the one hand you say, the reason we can move is that our space and time are not indefinitely divisible, on the other hand you say even in a hypothetical universe with indefinitely divisible space and time motion can exist.

I'm saying my problem with Zeno's argument in setting up the paradox is assuming that spacetime is infinitely divisible.

That doesn't mean that's the "reason we can move", it means that's why the argument is flawed.

Now Euclidean space with Newtonian motion is a mathematical construction. It has infinitely divisible space and time. That allows for the resolution of Zeno's paradox. It was constructed to describe motion so this is not a surprise and not really interesting.

The difference is that spacetime, aka reality[1], does not allow for space to be infinitely divided. So you can't apply Zeno's argument (which was his point).

In summary, Zeno's paradox is resolved by infinitely divisible time in Newtonian mechanics. You can't apply Zeno's argument to spacetime, because his assumptions are not satisfied.

(Hopefully no autocorrect issues, moved to my phone.)

[1] Of course I am actually taking about our best models of reality, relatively and QM, which are works in progress and not even consistent. So I could be all wet in one hundred years.

1. Zeno makes some assumptions and proves from them that it is impossible to move.

2. A quick experiment proves that you can move around anyway.

3. Therefore Zeno's reasoning must be wrong, either he assumed something false or his reasoning itself is flawed.

4. You say, Xeno's reasoning is fine, in the setting he describes it is indeed impossible to move, but Xeno has an unstated assumption - that space and time are indefinitely divisible - but in our universe, because our spacetime is not indefinitely divisible, Xeno's reasoning - which is fine if all assumptions are satisfied - just does not apply.

This is where the paradox falls apart for me. There's no reason to assume that there is always a halfway there.

5. All assumptions of Xeno's reasoning are satisfied in a hypothetical universe with Euclidean space - and classical mechanics is some sense represents such universe - and because Xeno's reasoning is fine in principle if all the assumptions are satisfied and the hypothetical universe satisfies by assumption, Xeno's reasoning applies in the hypothetical universe and therefore you can not move in the hypothetical universe.