[JDHill]

...I must be missing your point, but the 'issue at heart', imo, is the infinite divisibility of a linear continuum...and how Zeno's puzzle misses the point in that regard, by posing it as a profound paradox, rather than a simple fact.

[Thomas An.]

...I must be missing your point, but the 'issue at heart', imo, is the infinite divisibility of a linear continuum...and how Zeno's puzzle misses the point in that regard, by posing it as a profound paradox, rather than a simple fact.

Well, Zeno is not really missing a point (he knew exactly what he was doing) and mathematics is actually doing the same things. There is a reason why we still feel compleled to include the term 'lim' in calculations instead of just dropping it altogether.

In the example I could have worked out the numbers so that the movement is bounded by a whole number. So for example if the turtle speed was 5m/s (half of the Achiles speed) then their movement is limited by the value 2000m (which is a whole number) .... but it wasn't the point.

[dutch_designer]

Sorry, didn't read all the previous posts, but doesn't this come down to slowing down time asymptotically to a moment where time no longer exists?
Same question would be if you jump off a building, you will never hit the ground, because you constantly half the distance that you travel. Factor in the time distance and you'll see that you cross those distances in shorter and shorter times. At one point the distance between the ground and yourself will be too small to fit a single atom between and then the whole premise doesn't count any more and you make contact. The time in which you traveled these distances will determine the force of impact..
This whole mind experiment is really cool, but it relies on fooling the brain by disregarding some factors that come into play when you move.

[Thomas An.]

...At one point the distance between the ground and yourself will be too small to fit a single atom between and then the whole premise doesn't count any more and you make contact ...

Yeah, but you can't really argue it that way, because in that you are no longer operating on pure theoretical logic. The thing is that atom is not the smallest portion of matter ... there is also electrons, neutron, quarks .... and even smaller particles ... and then we go down to pure energy .... and then what is that (pure energy) made of ?

Suppose you repeat your experiment but instead of dropping yourself out of a building, you drop a neutrino ....

Sorry, didn't read all the previous posts, but doesn't this come down to slowing down time asymptotically to a moment where time no longer exists?

Yes, that is the point.
"it requires infinite observer speed which would entail consumption of the universe in an attempt to pause time"

[Mihai]

But you agree that there is no such thing as infinitely small? You're saying we cannot observe the exact "moment" those two will meet, only make an approximation?

[CJElven]

The Observer will never see the moment the runner passes the tortoise. There are several possible reasons for this.

1) He accelerates towards infinity, and eventually exists at all points in the universe at the same instant...and uses up all the energy in the cosmos, destroying everything; thus not allowing the runner and tortoise to meet.

2) He accelerates towards infinity, and eventually exists at all points in the universe at the same instant. He can never escape this instant because time no longer has any meaning, since the 4th dimension is the thing that keeps the various forces of the universe separated.

3) [Most likely] He tries to accelerate towards infinity, and would eventually exist at all points in the universe, but missed the moment in which the tortoise and runner met because for him time stopped, therefore not ALLOWING him to accelerate any faster.

Basically, this hypothesis runs smack into the relativity limit. When time stops for you, you can no longer accelerate past that point. I don't know if something can be accelerated past the speed of light. But something could not accelerate ITSELF past the speed of light, since time has stopped for that object in relation to the universe.

This, of course, assumes Einsteinian physics still rule, and have not been supplanted.

[Thomas An.]

The Observer will never see the moment the runner passes the tortoise. There are several possible reasons for this.

1) He accelerates towards infinity, and eventually exists at all points in the universe at the same instant...and uses up all the energy in the cosmos, destroying everything; thus not allowing the runner and tortoise to meet.

2) He accelerates towards infinity, and eventually exists at all points in the universe at the same instant. He can never escape this instant because time no longer has any meaning, since the 4th dimension is the thing that keeps the various forces of the universe separated.

3) [Most likely] He tries to accelerate towards infinity, and would eventually exist at all points in the universe, but missed the moment in which the tortoise and runner met because for him time stopped, therefore not ALLOWING him to accelerate any faster.

Basically, this hypothesis runs smack into the relativity limit. When time stops for you, you can no longer accelerate past that point. I don't know if something can be accelerated past the speed of light. But something could not accelerate ITSELF past the speed of light, since time has stopped for that object in relation to the universe.

This, of course, assumes Einsteinian physics still rule, and have not been supplanted.

Exactly ! This is IMHO the best approach for a theoretical answer to this paradox.

[CJElven]

Of course, accelerating something else past the speed of light means you need to find something that moves faster than the speed of light already and use that as the accelerator...but neutrinos (which may or may not move faster than light...committee is still out) are very small, and difficult to manage, so we'll see.

As far as I know, neutrinos are the only things that they have found that may move faster than c, right? Or am I substituting a different particle for the correct one?

[misterasset]

Okay, I've been debating whether or not to try to argue how stupid this "paradox" is, or to just make fun of it... so I'll do both.

The observer can't even reach infinity speed because he can't even reach the speed of light. To accelerate to the speed of light means your mass becomes infinite which is impossible. So the rest of the paradox is pointless in my opinion. But more importantly, why would he use that approach. Just watch for the second the runner's foot comes crashing down on the tortise and *boom* the runner has past him. End of discussion. I know I know, that isn't the spirit of the "paradox"

But I did find the solution. If you click "recalcualte normals" everything works out. That solution works everytime.

[tom]

But I did find the solution. If you click "recalcualte normals" everything works out.

Isn't this what we do everyday
1-Wake up
2-Recalculate normals
3-Render
4-Sleep

Well, this problem proves the first 3 dimension (XYZ space) is absolutely glued to a 4th dimension, therefore we call it time. But I think it's not the time we perceive in everyday life. The problem starts when we want to measure something. Theoretically it is not possible to make an unbiased measurement. Imagine you want to measure a length or weight of something. First, you need to make sure the absolute limits of it. However we just assume it starts here and ends up here. We measure everything biased, this is a fact. Even if we put unbiased rational values instead of cropped variables, it still doesn't lead us to solution as long as we cannot define absolute limits due to subdivision. Theoretically we divide everything and no matter how small it's, it's still large enough to divide once again and size is truly relative. So the "time" happens. It's annoying, yes. As long as we have "things made of other things forever", it means it's unbiased and this invites the annoying variable "time". This thing (time) occurs when we have something near/in/over another thing. So, as long as we have 3 dimensions and as long as they are not absolute quantities, time will stay forever. I welcome fractals at this point. See klein bottle, it has no volume but it has a limited(?) surface. Limited to what? Is it for example 10 sqcm? or 10.0001? or more?... Hmm, I'm not sure if we can make absolute measurements as long as we keep dividing things theoretically. IMHO, time is misperception of subdivision. But subdivision is a fact and as long as we have this rule in this game, I'm afraid it won't be possible to carry our calculations up to a global level. We will always stuck valid only in a range with blurry margins. Like it happened in gravity,em vs. quantum mechanics. Let's say strings are there...what next? No matter what is next, it will be there and time will trap us forever.

[Thomas An.]

But you agree that there is no such thing as infinitely small? You're saying we cannot observe the exact "moment" those two will meet, only make an approximation?

Well it wasn't the point. You seem to be hovering around the same area with JD ... which is a parallel topic (related) but still tangential to the point made in regards to the paradox.

You are refering to infinitescimal subdivisions (precision) ... and JD brought up the the numerical example of 1/3=0.3333.)

As a matter of fact (strictly theoretically speaking) infinitescimals are imprtant and cannot be neglected. In this case 1/3 * 3 = 0.3333... * 3 which brings as to whether 1 = 0.999....

As a matter of fact if we neglect infinitescimals then we get into new falacies as seen bellow.

[tom]

In this case 1/3 * 3 = 0.3333... * 3 which brings as to whether 1 = 0.999....

From my perspective:
0.333333..... is something more realistic, why? Because now you may all reject that it might at one point go like that 0.33333333.......333333433333...
So, I don't agree it's absolutely equal to 1/3 as long as there is no absolute 1 and no absolute 3. They are just numbers. In real life quantities can be 0.999...... or 1.000000..... but I don't believe in integers. Kill me now. Therefore beside integers, I also can't believe such number 0.333... with pure infinity with the number "3" after comma in perfect stability with no flaw. If it goes to infinity, it would surely has noise after some precision and as long as we assume this perfection, our solutions will always remain biased so we won't be able to hack the trap.

[misterasset]

..... but I don't believe in integers. Kill me now.

1 Water Bottle

[tom]

...another "1 water bottle"


It's not possible to clone something "as is" (with all atoms, strings etc) , so they won't be the same "1 water bottle" forever.
So;

1<>1 ?

Makes sense?

[Thomas An.]

Tom,

The thing is that you are not following the spirit of the paradox.
The paradox is constructed on pure mathematical logic and you need to attack it on pure mathematical logic to uncover a fallacy.

Introducing imperfections is like cheating ... yes, we can go out and do the test ... no problems there. (but it is not the point).

Mathematical logic involves purity ... and you need to use the same purity to unwravel this thing.

Again, the issue is related to relativity ... and the observer that wishes to prove the Zeno assumption would require to pause time (by moving at infinite speed). If infinite speed is a possibility then so is the Zeno assumption ... if infinite speed not a possibility then the Zeno assumption is flawed.