CoffmanPhDIs space-time discrete or continuous?

[FrogPad]

Let me first start by saying I kinda skimmed this thread. I wanted to just add some physical reasoning.

Lets say you have two electrons spaced apart at 1mm. From coulombs law \frac{q^2}{4 \pi \epsilon_0 r^2}, the force would be ~2.3 x 10^-22 N

If we keep dividing the distance by two we have:

1 mm | 2 x 10^-22 N
1/2 mm | 9 x 10^-22 N
1/4 mm | 4 x 10^-21 N

1/2^3 mm | 1.5 x 10^-20 N

1/2^100 mm | 4 x 10^38 NFrom wikipedia,
"The force of Earth's gravity on a human being weighing 70 kg is approximately 700 N."

10^38 N is not going to happen.

I understand you are not arguing on physical grounds, but I remember justifying this to myself in such a way when I read godel, escher, and bach some time ago.

 

[arunbg]

What is the contradiction? What statement is both proven and disproven?

And (IMHO less importantly), what are the hypotheses, proof, and disproof[/size]?

I agree with you, these things should have been laid out at the beginning of the thread.
The statement that is being proven and disproven both, is "the hero catches up with the turtle".

Taking finite time to catch up with the turtle is the "proof". Taking an infinite no. of steps to get there(the natural no. counting example) would be the "disproof".

 

[onemind]

Here we go with pedantic semantics again.

Call it Zenos enigma if it makes you happy.

 

[Cane_Toad]

Here we go with pedantic semantics again.

Well, "pedantic" is a simple disparagement, so let's look at "semantics":

se·man·tics (s-mntks)
n. (used with a sing. or pl. verb)
1. Linguistics The study or science of meaning in language.
2. Linguistics The study of relationships between signs and symbols and what they represent. Also called semasiology.
3. The meaning or the interpretation of a word, sentence, or other language form: We're basically agreed; let's not quibble over semantics.

You're using only the third definition. If you *apply* semantics to Zeno's paradox, you can make progress. "Semantics" is a tool, not just an insult.

If you want to insult somebody properly, try something like:

soph·is·try (sf-str)
n. pl. soph·is·tries
1. Plausible but fallacious argumentation.
2. A plausible but misleading or fallacious argument.

Sophist Any of a group of professional fifth-century b.c. Greek philosophers and teachers who speculated on theology, metaphysics, and the sciences, and who were later characterized by Plato as superficial manipulators of rhetoric and dialectic.

which, of course, is the description given to Zeno by his contemporaries.

Call it Zenos enigma if it makes you happy.

It's not even an enigma. It's a trick question. Either you look wide-eyed at a word problem, or you deconstruct it, and apply rigor.

As Hurkyl said:

Well, (actual) paradoxes aren't matters of opinion: either you have, or you have not exhibited an argument that derives a contradiction from a specified set of hypotheses. What hypotheses do you think lead to a paradox? What contradiction is derived? What is the proof?

I gave my attempt at this in my earlier post.

If you don't do a rigorous elucidation, then you fall into Zeno's word trap. Zeno sets a false arena.

Onemind wants to have cool paradox that bends the mind, so he's following Zeno's lead into believing that Zeno's conclusion is the only one. Onemind has also carried Zeno's false lead when constructing a mathematical version for himself.

Here is a page with a *GOOD* paradox. It is clear, well stated, and teaches a principle, instead of dazzling. Try to wrap your mind around this, and you'll end up with an understanding, not stuck in a trap leading nowhere:

http://en.wikipedia.org/wiki/Ladder_paradox"

Here is a page with a big list:

http://en.wikipedia.org/wiki/List_of_paradoxes"

Zeno's paradox is listed under *philosophical*, not mathematical, not physical, not logic.

There is even a category which is very similar to Zeno's paradoxes:

Vagueness

* Ship of Theseus (a.k.a. George Washington's or Grandfather's old axe): It seems like you can replace any component of a ship, and it will still be the same ship. So you can replace them all, or one at a time, and it will still be the same ship. But then you can take all the original pieces, and assemble them into a ship. That, too, is the same ship you started with.
* 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. Similarly, one hair can't make the difference between being bald and not being bald. But then if you remove one hair at a time, you will never become bald.

Ignoring the context in which Zeno's paradoxes were given is a mistake, because there are clues as to why it is worded so.

 

[onemind]

lol, your whole post is a good example of pedantic semantics.

Zeno's paradox is listed under *philosophical*

No sht. From the first post:

probably a naive philosophical question that either has an obvious answer you all know about or has no answer and is right up there with the meaning of life.

I don't care about how mathematicians deal with the paradox of finite infinity hence this post being moved to the philosophy section.

I get and accept your math concepts but it doesn't solve the deeper philisophical question.


Its like talking to a wall.

 

[Hurkyl]

I agree with you, these things should have been laid out at the beginning of the thread.
The statement that is being proven and disproven both, is "the hero catches up with the turtle".

Taking finite time to catch up with the turtle is the "proof". Taking an infinite no. of steps to get there(the natural no. counting example) would be the "disproof".

How precisely do you conclude that Achilles doesn't catch up with the turtle?

I virtually never see anyone carry out this essential step in making it a paradox -- they simply take a giant logical leap from "there are infinitely many intermediate steps" to "Achilles does not catch up". (Actually, they usually leap straight towards "Paradox", which is even worse)


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.

 

[Hurkyl]

I don't care about how mathematicians deal with the paradox of finite infinity hence this post being moved to the philosophy section.

You want to know how philosophers deal with infinity? See this passage from Wikipedia:

Modern discussion of the infinite is now regarded as part of set theory and mathematics. This discussion is generally avoided by philosophers.
​

You keep saying "finite infinity" -- if you want to have any sort of discussion about that, you need to actually say what you mean. Allow me to direct you to the philosophy guidelines.

 

[neurocomp2003]

"What is the contradiction? What statement is both proven and disproven?":

taking both math & science into consideration(or perhaps i can't)...
PROVEN: The mathematical standpoint
There are an infinite amount of R in [0,1] (or |[0,1]| = inf. ).

?PROVEN?: The scientific standpoint
experimentally we know that we can translate from [0,1] given a defined coordinate system. And we will need to pass between all points in [0,1]

?CONTRADICTION?: Combining the two standpoints, to tranverse the finite boundary conditions(in time [0,1] and space [0,1]) we need to take an infinite amount of actions.

What is this kinda of problem called if not a paradox? just a problem?

 

[Hurkyl]

What is this kinda of problem called if not a paradox? just a problem?

A conclusion that is not a logical contradiction, but simply defies one's intuition, is a pseudoparadox. (alas, the word is often shortened to simply paradox when no confusion can arise. When confusion can arise, you should use the proper word)

 

[neurocomp2003]

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.

 

[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 &lt; x_n &lt; 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).