[tom]

What is the difference between numbers and quantities? Numbers are virtual symbols but quantities are real. However, we easily used to name these infinite undefined quantities with the available numbers and we feel satisfied at one point. I disagree. It can be 1 water bottle ok but it's a set of quantities with plastic, water, paper, ink etc etc. which are still not limited enough. It's just a naming trick, not precise enough. Math is a biased game here.

[Mihai]

Thomas, sorry that this is all over my head, but where does the paradox lie exactly?

[tom]

First of all, forgive me for hijacking the thread with neighbour subjects as usual.

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.

If you wonder what I think specifically about this. Well, as long as the infinite divisibility of any substane is a fact, infinite speed and infinite space makes sense to me. If we talk infinity of something, it automatically brings the possibility of other infinite things should exist for keeping the equation correct. Therefore it's not paradox to me. Since we can't master time, we won't be able to see the runner could never pass the turtle but damn that time makes bad jokes to us. I mean I do believe he could never pass the turtle and this will remain as a fact I think.

[Thomas An.]

Math is a biased game here.

You don't know that !

Do you know if one neutrino is different than another ? or if one electron is different than another ?

The concept of 1 (integer) is to signify discreet entities (one rock and another rock, one bottle and one pen and another pen).

You are trying to cheat by counting molecules .... but you are counting nonetheless ...

... in any case I know exactly where you are going (as in previous arguments) .... but let me tell you now... I disagree with this approach you are taking

[tom]

I disagree with this approach you are taking

You are welcome, I just decide to stay out of the trap, not to tear the inner walls of it

[Thomas An.]

Therefore it's not paradox to me. Since we can't master time, we won't be able to see the runner could never pass the turtle but damn that time makes bad jokes to us. I mean I do believe he could never pass the turtle and this will remain as a fact I think.

Is infinite speed possible even in theory ?

The thing is ...

• infinite speed implies (by induction) infinite energy.
• Infinite energy is a content of the universe (if universe is infinite)
• In which case infinite speed implies consumption of the universe (including the observer), which results to NULL

At this point we can say that the intent to track the Achiles next position in a sequence results to NULL, therefore the intent is flawed.

[tom]

[*]In which case infinite speed implies consumption of the universe (including the observer), which results to NULL

Do you say;
infinite-infinite=0 ?

[Thomas An.]

[*]In which case infinite speed implies consumption of the universe (including the observer), which results to NULL

Do you say;
infinite-infinite=0 ?

Would you rather say that:
infinite-infinite = infinite ?

[tom]

Would you rather say that:
infinite-infinite = infinite ?

Yes, otherwise if infinite-infinite=0, then infinite/infinite=1 and it's not.

[Thomas An.]

Would you rather say that:
infinite-infinite = infinite ?

Yes, otherwise if infinite-infinite=0, then infinite/infinite=1 and it's not.

Yes, but not in this case ... because the energy of the universe converts to observer speed 1-for-1 (a pure transformation)

[tom]

a pure transformation

As long as you're sure this is what happening, you're absolutely right.

[Thomas An.]

Thomas, sorry that this is all over my head, but where does the paradox lie exactly?

It lies in the notion of first setting up constraints and then arguing that there are indeed constraints. We are first planting the answer and then we pretent of looking for the answer.
(in other words, yes, there are constraints because Zeno put them there by the way of setting up the assumption parameters).

Here it is in other words.

1. Suppose we turn it around and we keep the observer speed constant (instead of letting it go infinite); in an attempt to keep things more normal.
2. Suppose we tell achiles (and the turtle) to stop and WAIT every time he meets the next flag (so that we have time to go and mark the new spot

(1) and (2) from above is our setup (we made the rules for this experiment), but even though we claim that we want to see Achiles meet the turtle ... in reality we are telling him to only meet the next flag. Our true objective is for Achiles to meet the next flag; not the turtle (we say one thing and we do another). As such, its a falsification of intent ... we say we want them to meet but we do not mean it ...

This time, based on (1) and (2) we will observe the race to progressively slow down (instead of the observer blowing up to infinity). Practicaly the whole experiement will come to a halt (because we set it up that way) and the speed of Achiles will become assymptotic to zero. By providing the requirement for Achiles to only meet the next flag (but not the turtle) and since there will always be a next flag behind the turtle no matter how minute the distance, we ensure the race goes on indefinately before the claimed event. Basically we will be sitting there watching paint dry for an eternity because we made certain of it.

It is a very subtle point. Do you see it

[jdp]

if we consider the run as a whole (achilles, the turtle and the path) and then the observer, it happens to be a case of relativity: we are basically comparing two space-time continui, right?.
what I wonder is what happens if we take in consideration the one where the paradox lies: taking in account the relative dimensions of achilles and the turtle within each other and both with the path.
In the paradox zeno never mentioned, for instance, the dimension of the foot of achilles: he will catch the turtle when the distance will be negligible relatively to achilles foot.
Mathematically speaking I think the problem lies exactly in the notion of integer and infinitesimal: what if 1 is not exactly 1 but an oscilatting point in between 0 and 1 or better in between 1 and its infinitesimal predecessor? how big is 1 relatively to the limit to 1?
perhaps is that the paradox hit the hearth of mathematics pointing out its very limit, which resides in postulates and axioms (conventions)?

[Thomas An.]

if we consider the run as a whole (achilles, the turtle and the path) and then the observer, it happens to be a case of relativity: we are basically comparing two space-time continui, right?.
what I wonder is what happens if we take in consideration the one where the paradox lies: taking in account the relative dimensions of achilles and the turtle within each other and both with the path.
In the paradox zeno never mentioned, for instance, the dimension of the foot of achilles: he will catch the turtle when the distance will be negligible relatively to achilles foot.
Mathematically speaking I think the problem lies exactly in the notion of integer and infinitesimal: what if 1 is not exactly 1 but an oscilatting point in between 0 and 1 or better in between 1 and its infinitesimal predecessor? how big is 1 relatively to the limit to 1?
perhaps is that the paradox hit the hearth of mathematics pointing out its very limit, which resides in postulates and axioms (conventions)?

It doesn't have to be a moving "foot"... we could replace the participants with tiny spherical particles (say one proton and another proton) moving smoothly in space at different speeds.

As for the infinitescimal. It is not really as much an issue. Once the parameters of the experiment are set up so that the fast particle is always "looking" behind the slow moving particle ... it is set (deliberately) to converge assymptotically from the back but never trully reach. As such it is not really a challenge to the number theory ... the mathematics correctly describe the event as it happens.

... but I do agree that relativity is involved. In other words it would be an error to consider only the participants but not the observer. We need to know for example that the only way for the Zeno observer to prove his argument is to actually (eventually) "touch" the participants and bias the experiment (and as such his claim becomes false) either because he is forced to adjust their speed and make them slow down or because he himself needs to speed up to infinity absorbing the universe in the process and making the participants disapear (or pausing time ... which also affects the outcome of the experiment). Either way he needs to somehow control them. If there was a king overseeing the race ... the king's job would be to referee and ensure the integrity of the experiment and that Zeno does not interact with the outcome (Zeno is not allowed to "touch" / polute / interfere with the race itself. Zeno needs to operate behind a glass wall) .... but he has to "touch" ... he is forced to. As such Zeno is found guilty of needing to polute the experiment in order to prove his point; thus invalidating his own point.

[Mihai]

Thanks for the other explanation Thomas, it's easier to see now But I still think that even if you were to take the smallest movement increments, there will be a moment when the runner will move at least one step, and the turtle not move any. Is this correct?