Zeno’s paradox

[antognini]

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.

[analog31]

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

[bmitc]

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

[noyoudumbdolt]

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.

[oifjsidjf]

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.

[hackandthink]

This is true. But Zeno would still counter:

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

[astine]

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.

[bmitc]

> 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?

[mmerlin]

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

[Archelaos]

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?

[ludston]

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.

[thunkle]

This analogous to the "grim reaper paradox"

[drdec]

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.

[danbruc]

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.

[askiiart]

See also: Supertasks, Vsauce

https://youtu.be/ffUnNaQTfZE

[bmitc]

The article never really seems to circle back to Zeno’s paradox.

I think the key thing in analyzing Zeno’s paradox, that doesn’t seem to be mentioned in the article, is that it rests upon the assumption of a continuum. It isn’t clear that continuums actually exist in reality. For example, consider an analog of the paradox where you are filling a glass with water. If we take as fact that the smallest thing we can actually call water is an H2O molecule, then by filling the glass, all we are doing is filling it with a finite amount of H2O molecules. And the discussion of when is the glass full revolves around some discussion of packing density.

[meroes]

https://youtu.be/zEdQbMziYfE

Here's JDH speaking about these paradoxes recently, that his substack is how he's publishing a book, and more stuff

[hackandthink]

There is so much good stuff from Mr. Joel David Hamkins Mathoverflow.

https://mathoverflow.net/users/1946/joel-david-hamkins

https://golem.ph.utexas.edu/category/2011/08/the_settheoreti...

[NKosmatos]

Very nice and well illustrated explanation, I really liked the key phrase “…an infinite sum with a finite value” that explains this paradox.

[karmakaze]

Another way to resolve these is to note that in addition to subdividing distances, times are also being subdivided. Simply choose constant velocities and a larger time than the time of surpassing the passing point. In effect you were only choosing to look before the point of interest with greater magnifications rather than look past it.

[scoofy]

I see Zeno's Paradox is just trying to apply ℵ1 numbers to an ℵ2 space. When we are talking about starting position, we can use 0, and terminus as 1, but what is the first unit of movement? It's an infinitely small number, and isn't describable because it isn't discrete.

[ShamelessC]

I don’t think this explanation is as intuitive as you suggest.

[danbruc]

This is avoided be the other version, first you have to go half the way, then half the way of the rest, ad infinitum.

[wprl]

It’s easy. Just try to go twice as far, then you’ll be there after the first iteration.

Or keep halving until you reach the Planck length. This length cannot be divided into smaller lengths, so you can just walk the remaining finite distance.

[Incipient]

In a simple (simplistic?) approach, even if you divide 1 second into infinite small timesteps, is it still not just 1 second?