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 Post subject: A Logical Paradox
Unread postPosted: 12th December, 2011, 1:54 pm 
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So, today we finished the maths class sooner than expected and our teacher told us a logical paradox wich i found pretty interesting and wanted to share with you guys.

Zenon the Eleatis brought confusion among the scolars of his time with the following paradox:
He claimed that Achilies the Eleusinian, who had won the golden medal at the Olympic races two times, could never outrun a turtle which is running 100 times slower that him and he has given it due to justice reasons a precedence of 1 stadium (which is about 150 meters). He supported the claim by saying that when achillies starts, the turtle will already be at a point A. When he reaches point A the turtle would have moved a bit further to point B. When he reaches B the turtle would have moved to C etc. So even though the distance between them will be getting smaller and smaller he will never be able to reach the turtle. :wtf:

So how can you explain this paradox?

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 12th December, 2011, 2:16 pm 
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The paradox is silly, as most paradoxes are, because it depends on faulty reasoning. The paradox can be summarized as follows - in order to walk 1 unit, one must first walk a half of a unit. In order to walk half a unit, one must first walk a fourth of a unit. In order to walk a fourth of a unit, one must first walk an eighth of a unit, and so on. Thus, since this process descends infinitely, one can never complete the initial task as it requires an infinite number of subtasks. This is clearly illogical, since we quite clearly walk a unit every day.

One way to think about the issue is this: I intend to walk an entire unit, and I'm capable of making a step. After I make my first step, which, for convenience, will go a distance L, I am 1 - L units away from my destination. However, there will be a point where these subtasks - being required to walk a distance of (1/2)^n - will amount to a distance smaller than 1 - L. That is, we can find an N so that (1/2)^N + (1/2)^(N+1) + ... < 1 - L. So, the only subtasks left for me to complete are walking 1/2, 1/4, 1/8, ... , (1/2)^(N-1). But this is a finite number of tasks, and there's no paradox involved in completing finitely many tasks.

What's going on here is Zeno is breaking down the distance into infinitely small pieces and then asserting that one most first complete each distance in succession before moving on to the next distance. If this is how we walked, he's right, we'd never move a full unit. In fact, we'd never even be able to begin because it would require that before walking any (1/2)^n units, we first attempt to walk (1/2)^m units for every m > n. But we don't walk like this. We walk a fixed length, and so eventually the sum total of his divisions amounts to less than a full step, and so we traverse all of these distances simultaneously, successfully averting the paradox of infinite descent.

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 12th December, 2011, 4:38 pm 
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e^ipi wrote:
The paradox is silly, as most paradoxes are, because it depends on faulty reasoning. The paradox can be summarized as follows - in order to walk 1 unit, one must first walk a half of a unit. In order to walk half a unit, one must first walk a fourth of a unit. In order to walk a fourth of a unit, one must first walk an eighth of a unit, and so on. Thus, since this process descends infinitely, one can never complete the initial task as it requires an infinite number of subtasks. This is clearly illogical, since we quite clearly walk a unit every day.

One way to think about the issue is this: I intend to walk an entire unit, and I'm capable of making a step. After I make my first step, which, for convenience, will go a distance L, I am 1 - L units away from my destination. However, there will be a point where these subtasks - being required to walk a distance of (1/2)^n - will amount to a distance smaller than 1 - L. That is, we can find an N so that (1/2)^N + (1/2)^(N+1) + ... < 1 - L. So, the only subtasks left for me to complete are walking 1/2, 1/4, 1/8, ... , (1/2)^(N-1). But this is a finite number of tasks, and there's no paradox involved in completing finitely many tasks.

What's going on here is Zeno is breaking down the distance into infinitely small pieces and then asserting that one most first complete each distance in succession before moving on to the next distance. If this is how we walked, he's right, we'd never move a full unit. In fact, we'd never even be able to begin because it would require that before walking any (1/2)^n units, we first attempt to walk (1/2)^m units for every m > n. But we don't walk like this. We walk a fixed length, and so eventually the sum total of his divisions amounts to less than a full step, and so we traverse all of these distances simultaneously, successfully averting the paradox of infinite descent.

Ignoring the fact that our strides cover a fixed length in space, how can we reconcile motion in general with Zeno's paradox? If spacetime is a continuum, then shouldn't the number line analogy hold (the real numbers also form a continuum)? If not, then doesn't that indicate a flaw in our understanding of dense sets?

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 12th December, 2011, 5:15 pm 
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e^ipi wrote:
What's going on here is Zeno is breaking down the distance into infinitely small pieces and then asserting that one most first complete each distance in succession before moving on to the next distance. If this is how we walked, he's right, we'd never move a full unit. In fact, we'd never even be able to begin because it would require that before walking any (1/2)^n units, we first attempt to walk (1/2)^m units for every m > n. But we don't walk like this. We walk a fixed length, and so eventually the sum total of his divisions amounts to less than a full step, and so we traverse all of these distances simultaneously, successfully averting the paradox of infinite descent.

This is the only way this "paradox" ever got to me. Why is it (and why was it ever) logically acceptable to assert that movement happens as it's described in the original statement of the problem? Maybe it's the seemingly rational explanation of the conditions of it that's just taken to be true. Or is there supposed to be some underlying implication/lesson about how logic becomes invalid if basic understanding is overlooked while trying to solve a bigger problem? I remember first hearing this "paradox" explained in a movie (I.Q.) back in the mid-90s during elementary and early middle school, so I didn't really have much of an understanding of math or disciplined reasoning beyond the simple stuff they make kids do at that age. But I remember thinking about this "paradox" intently back then, knowing it's not true but not being able to explain why. Maybe its purpose is to make people question what they know to be true, but I guess the same can be said for most/all logic problems. It's just a little hard to imagine this "paradox" ever actually stumped scholars.

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 12th December, 2011, 5:17 pm 
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acablue wrote:
e^ipi wrote:
The paradox is silly, as most paradoxes are, because it depends on faulty reasoning. The paradox can be summarized as follows - in order to walk 1 unit, one must first walk a half of a unit. In order to walk half a unit, one must first walk a fourth of a unit. In order to walk a fourth of a unit, one must first walk an eighth of a unit, and so on. Thus, since this process descends infinitely, one can never complete the initial task as it requires an infinite number of subtasks. This is clearly illogical, since we quite clearly walk a unit every day.

One way to think about the issue is this: I intend to walk an entire unit, and I'm capable of making a step. After I make my first step, which, for convenience, will go a distance L, I am 1 - L units away from my destination. However, there will be a point where these subtasks - being required to walk a distance of (1/2)^n - will amount to a distance smaller than 1 - L. That is, we can find an N so that (1/2)^N + (1/2)^(N+1) + ... < 1 - L. So, the only subtasks left for me to complete are walking 1/2, 1/4, 1/8, ... , (1/2)^(N-1). But this is a finite number of tasks, and there's no paradox involved in completing finitely many tasks.

What's going on here is Zeno is breaking down the distance into infinitely small pieces and then asserting that one most first complete each distance in succession before moving on to the next distance. If this is how we walked, he's right, we'd never move a full unit. In fact, we'd never even be able to begin because it would require that before walking any (1/2)^n units, we first attempt to walk (1/2)^m units for every m > n. But we don't walk like this. We walk a fixed length, and so eventually the sum total of his divisions amounts to less than a full step, and so we traverse all of these distances simultaneously, successfully averting the paradox of infinite descent.

Ignoring the fact that our strides cover a fixed length in space, how can we reconcile motion in general with Zeno's paradox? If spacetime is a continuum, then shouldn't the number line analogy hold (the real numbers also form a continuum)? If not, then doesn't that indicate a flaw in our understanding of dense sets?


Could you elaborate?

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 12th December, 2011, 5:47 pm 
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e^ipi wrote:
most paradoxes are, because it depends on faulty reasoning


I agree 100%. It's like my favourite: the Butter-Cat Paradox. Assuming cats always land their feet and toast always lands butter side up then if you strap a piece of buttered toast butter side up to a cat what would happen when you dropped it? The reality is the cat would land on its feet but if you use flawed logic it is possible to defy gravity and they will spin around infinitely.

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 14th December, 2011, 2:59 pm 
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acablue wrote:
e^ipi wrote:
The paradox is silly, as most paradoxes are, because it depends on faulty reasoning. The paradox can be summarized as follows - in order to walk 1 unit, one must first walk a half of a unit. In order to walk half a unit, one must first walk a fourth of a unit. In order to walk a fourth of a unit, one must first walk an eighth of a unit, and so on. Thus, since this process descends infinitely, one can never complete the initial task as it requires an infinite number of subtasks. This is clearly illogical, since we quite clearly walk a unit every day.

One way to think about the issue is this: I intend to walk an entire unit, and I'm capable of making a step. After I make my first step, which, for convenience, will go a distance L, I am 1 - L units away from my destination. However, there will be a point where these subtasks - being required to walk a distance of (1/2)^n - will amount to a distance smaller than 1 - L. That is, we can find an N so that (1/2)^N + (1/2)^(N+1) + ... < 1 - L. So, the only subtasks left for me to complete are walking 1/2, 1/4, 1/8, ... , (1/2)^(N-1). But this is a finite number of tasks, and there's no paradox involved in completing finitely many tasks.

What's going on here is Zeno is breaking down the distance into infinitely small pieces and then asserting that one most first complete each distance in succession before moving on to the next distance. If this is how we walked, he's right, we'd never move a full unit. In fact, we'd never even be able to begin because it would require that before walking any (1/2)^n units, we first attempt to walk (1/2)^m units for every m > n. But we don't walk like this. We walk a fixed length, and so eventually the sum total of his divisions amounts to less than a full step, and so we traverse all of these distances simultaneously, successfully averting the paradox of infinite descent.

Ignoring the fact that our strides cover a fixed length in space, how can we reconcile motion in general with Zeno's paradox? If spacetime is a continuum, then shouldn't the number line analogy hold (the real numbers also form a continuum)? If not, then doesn't that indicate a flaw in our understanding of dense sets?



So I've thought more about your question, and I think I know what you're getting at. The way I'm interpreting your question is - if motion is continuous, despite the fact that we may perceive our steps as moving a discrete unit, we still in principle are required to move through arbitrarily many subdivisions of any unit of distance, and so Zeno's paradox hasn't really been addressed. If I'm wrong, let me know, because I don't mean to put words in your mouth. Or on your fingers.

Anyway, I have no idea. I don't know much about physics. In principle, it's possible that motion is not continuous and that we have a minimum, discrete distance we have to move. I feel like I'm still dodging your question, so it's even possible that everything, even down to point-particles, has a minimum, discrete distance it can move. That would effectively side-step Zeno's paradox. But I have no idea, and I'm completely talking out of my ass.

Whether or not that poses a problem for our studying the real numbers is a different issue though. The real numbers are a useful fiction. In its early days, mathematics essentially worked like this: physical phenomena were observed, this physical system was translated into a mathematical problem. Analysis is performed on the problem until a solution is found. The solution is translated from mathematics back to reality. It's why most early mathematicians were also physicists. There just wasn't a strong distinction. Now pure mathematics, where issues like what you're raising here, is just an investigation of this logical structure and rules of inference. We'd like to think that it relates to reality because it's a language often used to describe reality, but it's quite possible that it doesn't. Forming a continuum like the reals (or even a dense set like the rationals), may or may not correspond to something in reality. Even if it doesn't, that's not an issue for most mathematicians.

As an example, take a statement like the continuum hypothesis; one statement of it is that there does not exist a set with cardinality strictly between the integers and the reals. Is it true? It depends on what you mean. It's been shown by (I want to say Godel and Cohen) that it's independent of the usual axioms of set theory, meaning that you could take it to be true and never run into a contradiction, or you could take it to be false and never run into a contradiction. So one answer might be it's neither. It's an old question, so I imagine it's already been heavily explored, but there are statements which depend on it, so it's interesting to examine consequences of accepting or rejecting it.

Another question we could ask is if accepting it (or rejecting it) gets us any closer to reality, either because it's "true" in our universe, or because the consequences of accepting/rejecting it are more firmly rooted in our universe. Yet another question we could ask is if it seems like something that ought to be true. That is, if it feels like a reasonable statement to take as true. Even if it feels like something that's reasonable and jives well with our understanding of our universe, that doesn't make studying its negation any less valuable or interesting.

So, back to your question on studying the reals. Does constructing and working with a continuum like the real numbers seem like something that's true? So far, they've aided in developing mathematics that's helped us understand the natural world better. Many results from calculus, for instance, are not true if you're working over a set like the rationals. For instance, the intermediate value theorem and the mean value theorem are quite false for functions of a rational variable. Yet, both of these results find themselves used in engineering and physics quite often. That suggests that there's something to the fiction we're using, so even if our world isn't well modeled by a continuum, we don't have to give up our real numbers.

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 14th December, 2011, 3:10 pm 
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e^ipi wrote:
So I've thought more about your question, and I think I know what you're getting at. The way I'm interpreting your question is - if motion is continuous, despite the fact that we may perceive our steps as moving a discrete unit, we still in principle are required to move through arbitrarily many subdivisions of any unit of distance, and so Zeno's paradox hasn't really been addressed. If I'm wrong, let me know, because I don't mean to put words in your mouth. Or on your fingers.

That's precisely what I was getting at.

e^ipi wrote:
As an example, take a statement like the continuum hypothesis; one statement of it is that there does not exist a set with cardinality strictly between the integers and the reals. Is it true? It depends on what you mean. It's been shown by (I want to say Godel and Cohen) that it's independent of the usual axioms of set theory, meaning that you could take it to be true and never run into a contradiction, or you could take it to be false and never run into a contradiction. So one answer might be it's neither. It's an old question, so I imagine it's already been heavily explored, but there are statements which depend on it, so it's interesting to examine consequences of accepting or rejecting it.

You're correct. The incompleteness theorem was developed by Godel. Pretty wild how conjectures can have truth values, but we can never know what they are, huh?

  
 
 Post subject: Re: A Logical Paradox
Unread postPosted: 14th December, 2011, 8:01 pm 
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acablue wrote:
e^ipi wrote:
As an example, take a statement like the continuum hypothesis; one statement of it is that there does not exist a set with cardinality strictly between the integers and the reals. Is it true? It depends on what you mean. It's been shown by (I want to say Godel and Cohen) that it's independent of the usual axioms of set theory, meaning that you could take it to be true and never run into a contradiction, or you could take it to be false and never run into a contradiction. So one answer might be it's neither. It's an old question, so I imagine it's already been heavily explored, but there are statements which depend on it, so it's interesting to examine consequences of accepting or rejecting it.

You're correct. The incompleteness theorem was developed by Godel. Pretty wild how conjectures can have truth values, but we can never know what they are, huh?


Oh yeah, the incompleteness theorem is a whole other animal. Although, on the whole, it doesn't bother me so much. Things like Gentzen's proof for the consistency of first order arithmetic helped me reconcile the limitations of Godel's theorem.

  
 
PostThis post was deleted by Denuto on 21st December, 2011, 6:16 am.
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