CoffmanPhDIs space-time discrete or continuous?

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The discussion centers around the philosophical implications of Zeno's paradox and whether space-time is discrete or continuous. Participants explore the nature of movement, questioning if mathematical concepts like limits and infinitesimals accurately represent physical reality. There is a debate on whether movement can be understood as continuous or if it is fundamentally discrete, with some arguing that human perception shapes our understanding of these concepts. The conversation also touches on the limitations of human perception and the potential existence of realities beyond our sensory experiences. Ultimately, the discussion highlights the complex relationship between mathematical models and the true nature of existence.
  • #51
yet you used the terminology "continuous/continuity"...would you care to elaborate on your understanding of this terminology, and how you would describe "physical/reality" concept "motion"?

I can't explain it.

All i know is that in reality, i can move an object from A to B but i don't know how. I understand how to get around this problem with the use of calculus to create useful models of reality and realize that the models are not reality.

Hence my statement a few pages back that i will die ignorant like everyone else :)

And for the record, i am not trying to be a smart ass and accept the possibility that you guys know something i don't and am unable to understand because of my own lack of mathematics education but i doubt it.
 
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  • #52
The paradox is childish, but the reflections it sparked are not

Well i guess i am a child then because zenos paradox still baffles me.
 
  • #53
It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead.

Only the parts of the story that support the paradox are adequately stated. It doesn't say that Achilles has to stop at each point, which is left as an implication(?). If he doesn't than there is no paradox.

The reaction time of Achilles each time he stops isn't specified, so:
Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.

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.

The other implication is that Achilles *must* go only 1/2 the distance, and no further, but we'll let that slide, since that's the good part.

Zeno could simply have said, if you halve something and halve the result, and repeat, would you ever have nothing left? This is the clear idea behind it all, but instead Zeno chooses a clever wording that creates an apparent paradox out of a straightforward concept, the paradox being "Achilles can never reach the turtle". There is no paradox in the previous statement of infinite reduction, only the question of whether you think there is an indivisible limit or not.

So, Zeno's "proof" only served to cloud and confuse, which was what he was attempting, since he was in a charged debate, not actually trying to find the answer.

On the other hand,
In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...
— Bertrand Russell, The Principles of Mathematics (1903)1

So, if you're still in awe of the paradoxes, you can count him on your side.

Personally, I think he's just being contrarian.

Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite.

So much for being the "foundation of a mathematical renaissance" after two thousand years. Not only that, but it was the "failure" of his proofs which stood the test of time, so giving him the credit for the success in the area of infinities is accidental at best. Remember he was trying to show the falacy of infinitely divisible space in reality.

The paradoxes lasted this long because they were cute, and had real world objects in them doing odd things. Zeno was just an early spin doctor, a master of sound bites. He didn't invent the concept, just stuck words around it. People thought they were fun, so they were retold, and of course they mystified the layman. Aesop's fables lasted for the same reason, but they were a lot better.

Blah, the more I think about it, the more I dislike him. The world is full of disingenuous people doing things like that, using their brilliance to warp ideas to suit their purposes, and is very much the worse for it.
 
  • #54
Only the parts of the story that support the paradox are adequately stated. It doesn't say that Achilles has to stop at each point, which is left as an implication(?). If he doesn't than there is no paradox.

The reaction time of Achilles each time he stops isn't specified, so:

What in the world are you talking about mate? Reaction time of achilles? I think your reading too much into the story.

Forget the story and look at it the way in terms of point A to B as i outlined in the first post.

Talk about disingenuous people using ideas to suit them. :rolleyes:
 
  • #55
Onemind, if you look at Hurkyl's posts, you would see that nowhere is he stating that there is no paradox at all. He is simply trying to clarify how mathematics has resolved the paradox to some extent.
For eg, it is believed that Zeno was ignorant that the sum of the infinite geometrically decreasing time intervals taken by Achilles to reach the previous positions of the tortoise, was in fact finite.

But although the total time taken to catch up is finite(which is real worldish), it takes him an infinite no. of steps(which is non-real worldish), this is the heart of the paradox today. I believe this still remains a paradox.
 
  • #56
onemind said:
What in the world are you talking about mate? Reaction time of achilles? I think your reading too much into the story.

Forget the story and look at it the way in terms of point A to B as i outlined in the first post.

You clearly ignored my point: either you pay attention to the words, as if they were intended as a serious question, or you ignore them, and treat it as a mathematical problem.

As I understand it, Zeno's intent wasn't a pure mathematical problem. Taken as a real-world problem it has many flaws. Taken as a mathematical problem, it isn't a paradox, it just introduces a few concepts about infinity and limits.

You're making it a paradox by introducing poorly defined time elements in your A to B post. There were many replies about this.

Talk about disingenuous people using ideas to suit them. :rolleyes:

Honestly, what did I write all that for? Show that you were at least paying attention by posting something displaying effort on your part. Wise-ass remarks are unwelcome if you haven't earned the right.
 
  • #57
Thanks Arnbg.

If that is the case then this wouldn't have went on for 4 pages of the same explanations.
 
  • #58
arunbg said:
But although the total time taken to catch up is finite(which is real worldish), it takes him an infinite no. of steps(which is non-real worldish), this is the heart of the paradox today. I believe this still remains a paradox.
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?
 
  • #59
the paradox: there are infinite amount of steps(action) to get to the bounding conditions(time/space bounds)?
 
  • #60
neurocomp2003 said:
the paradox: there are infinite amount of steps(action) to get to the bounding conditions(time/space bounds)?
What is the contradiction? What statement is both proven and disproven?

And (IMHO less importantly), what are the hypotheses, proof, and disproof[/size]?
 
  • #61
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.
 
  • #62
Hurkyl said:
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".
 
  • #63
Here we go with pedantic semantics again.

Call it Zenos enigma if it makes you happy.
 
  • #64
onemind said:
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.
 
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  • #65
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 philosphy section.

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


Its like talking to a wall.
 
  • #66
arunbg said:
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.
 
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  • #67
onemind said:
I don't care about how mathematicians deal with the paradox of finite infinity hence this post being moved to the philosphy 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.
 
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  • #68
"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?
 
  • #69
neurocomp2003 said:
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)
 
  • #70
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.
 
  • #71
neurocomp2003 said:
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.
 
  • #72
"for example, we rejected the hypothesis that any task is divisible into finitely many atomic tasks." can you give an example?
 
  • #73
neurocomp2003 said:
"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.
 
  • #74
Hurkyl said:
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?
 
  • #75
I think Hurkyl's post was to construct a contradiction, thus the deliberate requirement that a task can be divided into finite subtasks.
 
  • #76
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.
 
  • #77
Cane_Toad said:
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 :smile:

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
 
  • #78
arunbg said:
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.
 
  • #79
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".
 
  • #80
arunbg said:
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.
 
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  • #81
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.
 
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  • #82
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
Cane_Toad 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 :biggrin:
 
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  • #83
arunbg said:
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.
 
  • #84
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?
 
  • #85
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.
 
  • #86
arunbg said:
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.
 
  • #87
neurocomp2003 said:
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.
 
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  • #88
Dammit, neurocomp, see what you did! :smile:
 
  • #89
thats what i wanted to see...now I'm a bit confused how that relates to the rest of his post #66.
 
  • #90
Cane_Toad said:
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).
 
  • #91
arunbg said:
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.
 
  • #92
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?
 
  • #93
arunbg said:
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.
 
  • #94
"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 =]
 
  • #95
arunbg said:
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}. :-p

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)​
 
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  • #96
neurocomp2003 said:
"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...
 
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  • #97
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?
 
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  • #98
Awesome...
 
  • #99
neurocomp2003 said:
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).
 
  • #100
hurkyl said:
{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.
Hurkyl said:
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.

Cane_Toad said:
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.
 
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