CoffmanPhDIs space-time discrete or continuous?

[Hurkyl]

ah...

but how does a "logical contradiction" arise then? Because it seems straight forward that one should never be able to tranverse time or space if we employ the definition of the REAL number system and limits.

I gave an example of how one might derive a contradiction in my post #66.

 

[neurocomp2003]

"for example, we rejected the hypothesis that any task is divisible into finitely many atomic tasks." can you give an example?

 

[Hurkyl]

"for example, we rejected the hypothesis that any task is divisible into finitely many atomic tasks." can you give an example?

Er, what do you mean?

I gave an argument, with four of its hypotheses listed. If I tried to make the same argument without making assumption #1, then the first step in the proof I sketched is a non-sequitor, and thus the derivation is flawed -- it would no longer be a paradox, but simply an incorrect argument.

 

[arunbg]

A sample way to carry out this step, (which bears similarity, I think, to something I once read) is to make the assumptions:

(1) Any task can be decomposed into finitely many atomic tasks.
(2) Any region of space larger than a single point can be divided into two subregions.
(3) Movement is a task.
(4) A movement task across two consecutive regions of space can be decomposed into two movement tasks, one per region of space.

and from this, a contradiction can be formally derived. (Decompose movement into atomic tasks, apply divisibility of space, apply the divisibility of movement tasks, and this contradicts the atomicity of the original decomposition)

And when it's presented clearly like this, we see that this particular argument would not survive if, for example, we rejected the hypothesis that any task is divisible into finitely many atomic tasks.

From what I understand from your hypotheses and subsequent contradiction is that, a task in the way you have defined it, cannot be divided into finite atomic tasks, due to the infinite divisibility of space that you have included.
Remove the first argument, and there is no contradiction.

But where does that lead us in the context of Zeno's whatever-you-like-to-call-it?
How about something like,

1) Atomic tasks are by definition indivisible.
2)Movement is an atomic task.
3)Any region of space larger than a single point can be divided into two subregions.
4)To undergo motion from regions A to B, one has to carry out a "succession" of movement tasks (which are atomic).

Note that in the fourth hypothesis, movement is not divided into further atomic entities. However to go from A to B, you clearly need to go through an infinite no of atomic tasks(movements) due to divisibility of space. I don't think there is a contradiction here.
What do you think?

 

[Cane_Toad]

I think Hurkyl's post was to construct a contradiction, thus the deliberate requirement that a task can be divided into finite subtasks.

 

[Cane_Toad]

Onemind, at this point YOU are the wall. You have quit saying anything helpful, and are spouting complaints. If you want somebody to understand you, you have to do a better job of explaining your point. Aren't you wondering why "nobody is getting your point"? Is everybody stupid but you?

As far as I can tell, you're just baiting us now. I don't know whether you hope somebody will return your attacks, but you've stopped being polite.

 

[arunbg]

I think Hurkyl's post was to construct a contradiction, thus the deliberate requirement that a task can be divided into finite subtasks.

Of course, I understood that very well. I think in my earlier post, I might have appeared trying to oppose him. But when I said

Remove the first argument, and there is no contradiction.

I was only trying to clarify if that was what he really meant, not suggesting anything new

The real point of my post, was whether his hypotheses led us to a solution to Zeno's problem; I tend to think otherwise. I think my group of hypotheses can also used, so that there is no contradiction, but still Zeno's problem remains.
But I might be missing something ...

Onemind, please understand that on these forums being rude or making personal atacks are just not going to get you anywhere near your purpose.
I know it can be quite frustrating not able to understand or agree with what others say at times, but loosen up a bit okay.

Cheers

 

[Cane_Toad]

The real point of my post, was whether his hypotheses led us to a solution to Zeno's problem; I tend to think otherwise. I think my group of hypotheses can also used, so that there is no contradiction, but still Zeno's problem remains.
But I might be missing something ...

The problem with Zeno's version is that it is too poorly defined, which I think is also Hurkyl's point. Zeno's version would have to be rewritten in order to apply a decent logical process to it, for example as Hurkyl did, albeit with an omission of turtles.

 

[arunbg]

But the problem with Hurkyl's post is that there is no paradox, only a contradiction. IMHO he wasn't trying to give a logical alternative version of the "paradox".

 

[Cane_Toad]

But the problem with Hurkyl's post is that there is no paradox, only a contradiction. IMHO he wasn't trying to give a logical alternative version of the "paradox".

A paradox is [often] based on contradiction. Hurkyl's version is very close to the heart of what Zeno was doing. Maybe what you want is something that makes you think that it isn't a contradiction even though it is, or visa versa. Most paradoxes exhibit this quality, and it is often in clever wording that leads the reader via misdirection.

There are other paradoxes that are contradictory based on differing perspectives, like the ladder paradox.

Some are just juxtapositions of seeming contradictory concepts, like "standing is more tiring than walking".

P.S. I and others have tried to show that the solution to Zeno's paradox only comes when you actually understand what Zeno is *not* saying. So, in one way of looking at it, there is no solution because it is a trick question. You have to rewrite it in clear terms, in which case it stops looking like Zeno's. Look back at the previous posts and see if it makes sense to you.

 

[neurocomp2003]

Hurkyl: Ignore the Achilles/Turtle terminology...

And restate zeno's example as simply
- A Traveller moving along a standard unit axis.
- can the traveller ever reach 1 or 1.1. in the interval [0,1.1]
(taking into consideration the R number system) in a finite
number of actions.
- if not then what can we take of the phrase "an infinite number of actions to reach finite bounds". Doesn't this imply the person should never reach these bounds or in fact that they shouldn't be able to move at all? Because for every distance from the traveller to the destination the space in between, by R, divides into another infinite # of points to be acted upon.

Now your hypothesis(1) that you rejected...cna you explain how it will fit into this picture to clear things up for my understanding.(both for rejecting and accepting.

 

[arunbg]

I was mentioning that in Hurkyl's post he put forth his own set of hypotheses and came out with a contradiction, thereby proving that the set of conjectures were wrong "as a whole". This is not a paradox, but rather a proof or disproof, whatever you want to call it.
The contradictions involved with paradoxes are of a different kind, the hypotheses have both a proof as well as a disproof.

As I was going through your post no 56, I note you said

If Achilles does stop at each point, he is in essence forever chosing to allow the turtle to stay ahead, which is no paradox either.

Now in the actual version, Achilles is not made to stop at the previous position of the tortoise. Consider you are taking snapshots of the race each time Achilles reaches the turtles prior position. As you said if he is allowed to stop, the turtle would definitely get forward and hence no paradox clearly.

To make it clearer, the turtle has a headstart of some distance, and Achilles sees the turtle some distance ahead, runs to the position he saw the turtle(not stopping after he gets there). Obviously this has taken some time during which the turtle has moved foreward a bit, and the process continues. If you consider the series of events in this way, how can Achilles ever get past the turtle?

If you want a more formal wordplay-less approach, consider Achilles and the turtles as subatomic particles with high and low K.Es respectively heading for a target atom

 

[Cane_Toad]

To make it clearer, the turtle has a headstart of some distance, and Achilles sees the turtle some distance ahead, runs to the position he saw the turtle(not stopping after he gets there). Obviously this has taken some time during which the turtle has moved foreward a bit, and the process continues. If you consider the series of events in this way, how can Achilles ever get past the turtle?

It's all in the details. If Achilles is not required to stop, and is running at 5m/s and the turtle at 0.05m/s, it can be easily seen that once Achilles is close, i.e. less then 5m, in any time around 1s it takes the turtle to move to his new position, Achilles will have blown past him.

I don't see how you can attempt to make it a paradox unless you imply that Achilles has to stop, or at least decrease his speed toward 0, at each half way marker and allow the turtle to advance. If he is doing this he is either deliberately keeping the turtle ahead of him, or his destination is always behind the turtle, meaning he isn't trying to catch the turtle, so that doesn't seem like a paradox to me.

 

[arunbg]

That's why I asked you to think that you are taking snapshots of the race; or you are watching a video of the race, and pause at the required moments.
Does that work?

 

[Cane_Toad]

Zeno's is really no different in its principle of misdirection than:

* Sorites paradox: One grain of sand is not a heap. If you don't have a heap, then adding only one grain of sand won't give you a heap. Then no number of grains of sand will make a heap.

In this case the reason for the "paradox" is easy to see. It is omission, and choice of details. There is no definition of when a heap becomes a heap.

These paradoxes seem puzzling because they are using some underlying logical or mathematical principle and wrapping a real world situation around it to confuse the issue.

If you say (sorry, I don't do logic syntax much) :

1) A heap > 1 (assuming this is what was meant, not "heap != 1")
2) If total sand <= 1 then total sand + 1 != heap
3a) ( ( for any total sand ) + 1 ) < heap
or
3b) total sand + n < heap

then it shows it obvious flaws. Step 2 is incorrect, and step 3 is unsupported.

Zeno's paradox is similar, only more complex.

 

[Cane_Toad]

That's why I asked you to think that you are taking snapshots of the race; or you are watching a video of the race, and pause at the required moments.
Does that work?

If you decide to take snapshots only at each of the half way points, then you have decided to only look behind the turtle by definition, so it has become your choice never to see when Achilles reaches the turtle.

 

[Hurkyl]

Hurkyl: Ignore the Achilles/Turtle terminology...

Let f be a continuous, increasing, real-valued function, with f(0) = 0, and f(1) = 1.

Then there exists an infinite sequence of points $\{ x_n \}_{n = 1}^{+\infty}$ such that, for every n, $0 < x_n < 1$ and $f(x_n) = 1 - 2^{-n}$.

The union of the intervals $[x_n, x_{n + 1}]$ is the interval [x_1, 1) which does not include 1. Similarly, the union of the intervals $[f(x_n), f(x_{n+1})]$ is $[1/2, 1)$ which also does not include 1.

 

[Cane_Toad]

Dammit, neurocomp, see what you did!

 

[neurocomp2003]

thats what i wanted to see...now I'm a bit confused how that relates to the rest of his post #66.

 

[arunbg]

If you decide to take snapshots only at each of the half way points, then you have decided to only look behind the turtle by definition, so it has become your choice never to see when Achilles reaches the turtle.

No that was not my point. All I'm saying that if we take snapshots of the race at those particular points, the process seems endless, because there is always going to be a point when the turtle is ahead of Achilles. The idea is that we never get a chance to see Achilles crossing the turtle, not that we are biasing observation with Achilles always behind the turtle(which in fact happens but leads to no contradictions).

 

[Cane_Toad]

No that was not my point. All I'm saying that if we take snapshots of the race at those particular points, the process seems endless, because there is always going to be a point when the turtle is ahead of Achilles. The idea is that we never get a chance to see Achilles crossing the turtle, not that we are biasing observation with Achilles always behind the turtle(which in fact happens but leads to no contradictions).

That it seems endless is irrelevant. Are you choosing a set of points which has the possibility of eventually seeing Achilles reach the turtle, or not.

The snapshot doesn't bias the observation, you have already biased it by your choice of how Achilles is allowed to move, and therefore the set of possible points.

 

[arunbg]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

 

[Cane_Toad]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points.

 

[neurocomp2003]

"By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points...:"
And that is the point =]

 

[Hurkyl]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

Physical intuition and simple demonstration.

I can simply consider the set of points {start of race, point where Achilles catches the turtle}. :tongue:

Or, to be fancy, I can consider the set of points:
{your infinite set of points} union {the point where Achilles catches the turtle}

Or, really, I can just consider
{all of the points from the starting point to the point where Achilles catches the turtle, inclusive}

And that's only if I limit myself to points that are involved in the race; at my discretion, I can consider points before Achilles enters the race, and points after Achilles passed the turtle.


More importantly, (and the point I've been trying to drive home this entire time), why would you ever think that

the set of "points" when watching Achilles behind the turtle is never-ending​

suggests

you cannot expect to have a set of "points" where Achilles catches up?​

If it helps, the following statement is false:

Every ordered set X can be "counted" -- that is, there exists a function f from X into some interval of integers such that x < y if and only if f(x) < f(y). (For any x, y in X)​

 

[Cane_Toad]

"By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points...:"
And that is the point =]

No, it was a statement left unfinished as an exercise for why that approach will lead nowhere. (Other approaches have been posted.)

The idea is, as Hurkyl illustrates above, that you have to pick a useful set of points, and it doesn't matter if that set is big, just that its extent does you some good. But now I'm just repeated myself...

 

[neurocomp2003]

but by your logic don't you create a finite set of points? and thus a finite stepsize?

Because if u consider the path to be Continuous, then at what point does the person go from Ai to Ai+1. or in terms of numbers [0,1]
at what point do they go from 0 to 1 through stepping?

 

[Saladsamurai]

Awesome...

 

[Pythagorean]

but by your logic don't you create a finite set of points? and thus a finite stepsize?

Because if u consider the path to be Continuous, then at what point does the person go from Ai to Ai+1. or in terms of numbers [0,1]
at what point do they go from 0 to 1 through stepping?

You might be confusing the length of a number as it is expressed:

.7529293829829380980329480928344... (and so on)

with it's value.

That 'irrational' number above is not more than .8 just because it has more numbers than .8

it doesn't take infinite time to pass through that point because that's not quite how the math works.

Furthermore, the atoms that make me up (and even the elctrons) can be defined within a finite amount of decimals (i.e. we can round of that irrational number to have a finite number of decimal places without losing any information about a person passing a point).

There's also the convention of units so that it doesn't even have to be expressed as a decimal (i.e. nanometers in place of 10^-9 meters).

 

[arunbg]

{all of the points from the starting point to the point where Achilles catches the turtle, inclusive}

Sure, can you clarify what you mean by considering the set of all points from start to finish?
Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.
There is nothing physically wrong with watching things from the turtle's perspective.

Physical intuition and simple demonstration.

Of course, we do know that Achilles will cross the turtle(physical knowledge), the "paradox" is that we can't counter Zeno's arguments, by simply saying that the whole scenario was not taken into account.

No, it was a statement left unfinished as an exercise for why that approach will lead nowhere. (Other approaches have been posted.)

The idea is, as Hurkyl illustrates above, that you have to pick a useful set of points, and it doesn't matter if that set is big, just that its extent does you some good. But now I'm just repeated myself...

Note that we are not trying to get a better or the "right" approach here, but rather to find flaws in an argument set.

 

[Cane_Toad]

Sure, can you clarify what you mean by considering the set of all points from start to finish?
Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.
There is nothing physically wrong with watching things from the turtle's perspective.

Zeno makes it ambiguous as to which set of points we are supposed to be using in order to view his paradox.

His paradox is based on this ambiguity, IMHO. He sets up the story focusing on the points behind the turtle (infinite bifurcation), but he then states the problem in a context that supposed includes all the points up to and beyond the turtle. He leaves it ambiguous, and thus creates a confusion, and not a true paradox. This is why I say it is a trick question.

Note that we are not trying to get a better or the "right" approach here, but rather to find flaws in an argument set.

Finding a "right" approach is just a possible part of the deconstruction to understand the problem.

 

[Hurkyl]

Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.

HOW? By what means do you arrive at that conclusion?

If you were simply trying to argue that Achilles doesn't pass the turtle during the time when he's catching up, but I do not see how you are arriving at the stronger conclusion that Achilles doesn't pass the turtle at any other time. How are you ruling that out?

 

[-Job-]

There are an infinite number of (infinitely small) steps between A and B. But each of those steps require infinitely small time to cross.

The time required to go from A to B is the addition of an infinity of infinitely small time units.

Zeno's paradox is proportional to saying, since there are an infinite number of steps between A and B, and because it takes us an infinitely small amount of time (i.e. 0) to cross an infinitely small step (moving at some speed c), then the time required to go from A to B is 0.

 

[neurocomp2003]

Pythagorean: no one said anything about infinite time. The conversation is about the concept of infinite actions in a finite space [0,1] & time bounds [0,1]. Or as hurkyl put it perhaps we cannot assume infinite actions(or to my understanding of his points).

Put it another way how does motion/time evolution occur when we assumse space to be continuous?

 

[arunbg]

If you were simply trying to argue that Achilles doesn't pass the turtle during the time when he's catching up, but I do not see how you are arriving at the stronger conclusion that Achilles doesn't pass the turtle at any other time.

Why not, if Achilles doesn't get to the turtle ever when he's catching up, how can he pass the turtle at any other time? Of course all this is in context of Zeno's arguments, not actual physical experience.
Infinitesimally small distances and times are approached, however absolute 0 is never reached.
Again I state the problem for clarity(sorry I couldn't exclude the terminologies)

The turtle has a head start on Achilles, who is actually faster. At a certain time during the race, Achilles reaches the initial position of the turtle. However this process has taken some time, during which the turtle has moved a little foreward. The process continues(like a series of mini-races).

How can Achilles catch up with the turtle, if it always takes him some finite time to get to the turtle's last position(however small), during which the turtle would have moved a small distance foreward ?

Finding a "right" approach is just a possible part of the deconstruction to understand the problem.

I partly agree, the "right" approach is essential, but it should also be able to explain why the other approach was wrong. I don't think anyone so far has done that.

Zeno's paradox is proportional to saying, since there are an infinite number of steps between A and B, and because it takes us an infinitely small amount of time (i.e. 0) to cross an infinitely small step (moving at some speed c), then the time required to go from A to B is 0.

Can you clarify how you arrived at this conclusion from Zeno's arguments?