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#601
08-10-2012, 05:48 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Francis Vaughan I'm not sure if this is a question or a statement, or a rhetorical question. You still didn't answer my questions above - what does "completed" mean, and why must we perform this completion? Swap the viewpoint about. As I walk I cover an uncountable number of points. Any question about labelling or otherwise selecting from this set of points as I travel isn't part of my travel. Perhaps some Platonic universe keeps track, but it has nothing to do with me or my movement. Given that it is possible to construct specifications for an uncountable number of these subset selections, I guess this Platonic universe simply has an uncountable set of subsets of the points I pass though. Makes no difference to me. It is just way of specifying the selection of points. I am not required to know about these selections in order to move. So I can't see how they stop me from moving.
Whether you are keeping track has no bearing on the issue. It's like saying it doesn't matter how an airplane flies, it just does. The distance you are covering contains and infinite number of intervals. It does not matter if you are aware. The issue pertains to how its possible for you to complete the series when they are infinite or endless. I find it difficult to see how one could argue that one can and regular does progress through an infinite number of points during the movement form point A to point B, but for the the 9s in .999... it's different.
#602
08-10-2012, 06:00 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Saltire From that same article: "...in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes." I can't understand how you can accept a nonzero number so small that its square is zero, but still have trouble with 0.9...=1.
I'll try to explain. Limits were not a mathematical discovery per-say. Like the proof that the Square Root of 2 is irrational or that Pi is irrational. Limits were constructed, devised as a means to avoid issues such as 1/infinity.

For example the Limit of 1/x as x-> infinity = 0? Why don't we just say 1/infinity = 0 then? Because that would lead to all kinds of contradictions. So that tells me there is something missing here. Something is not right about that assumption.
#603
08-10-2012, 06:03 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x I never argue that one does indeed arrive at B. The question... the paradox is HOW?
You're asking "how" by saying "you shouldn't be able to, by the reasoning that you never reach the end as you go through the sequence of halfway marks. So how is it possible?" I'm answering that question by pointing out the reasoning is flawed.

I don't know how else to answer. "How?" By moving forward at a constant nonzero speed would be one way to do it. If that's not a sufficient answer, then explain why.

Last edited by Frylock; 08-10-2012 at 06:05 PM.
#604
08-10-2012, 06:05 PM
 Frylock Guest Join Date: Jun 2001
In your post 590 you very explicitly argue that you cannot reach B. You do not say you actually think B can't be reached, but you offer an argument that B can't be reached, and ask, in light of that argument, how it's possible to arrive at B. I'm pointing out that the argument that B can't be reached is flawed in the first place--and so the motivation to solve the puzzle of how one reaches B is lost. There's no puzzle any longer, once the argument is shown to be flawed.
#605
08-10-2012, 06:22 PM
 Lumpy Charter Member Join Date: Aug 1999 Location: Minneapolis, Minnesota US Posts: 10,922
For what it's worth, I recently read a fascinating article http://io9.com/5932366/how-do-you-co...rses?tag=space which introduced me to the concept of p-adic numbers- an alternative to the entire real number system, and one arguably more useful than the standard reals.
#606
08-10-2012, 06:22 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock You're asking "how" by saying "you shouldn't be able to, by the reasoning that you never reach the end as you go through the sequence of halfway marks. So how is it possible?" I'm answering that question by pointing out the reasoning is flawed. I don't know how else to answer. "How?" By moving forward at a constant nonzero speed would be one way to do it. If that's not a sufficient answer, then explain why.
I would just ask you to read about the apparent paradox of motion. I am sure you can find many good articles. I don't know how to explain it any better than I have, sorry.
#607
08-10-2012, 06:26 PM
 zombywoof Guest Join Date: Jul 2009
Quote:
The operative word here, of course, being "apparent"!
#608
08-10-2012, 06:28 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x I would just ask you to read about the apparent paradox of motion. I am sure you can find many good articles. I don't know how to explain it any better than I have, sorry.
I know all about it. You don't get the paradox without reasoning that leads to the conclusion that you can't reach B. And indeed, you did offer reasoning that you can't reach B. I pointed out the flaw in that reasoning. Hence, there is now no longer reasoning on the table which says you can't reach B. Without that reasoning, there is no longer a paradox to solve.
#609
08-10-2012, 06:28 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock In your post 590 you very explicitly argue that you cannot reach B. You do not say you actually think B can't be reached, but you offer an argument that B can't be reached, and ask, in light of that argument, how it's possible to arrive at B. I'm pointing out that the argument that B can't be reached is flawed in the first place--and so the motivation to solve the puzzle of how one reaches B is lost. There's no puzzle any longer, once the argument is shown to be flawed.
In post 590 I am simply restating the apparent paradox and asking for your resolution. Your solution is you just keep walking. That does not really address it. Except in a most Zen like fashion. But not in any logical way. There are at least a couple solutions which show that there is no paradox. I am asking for yours. Just that you keep walking... that is not really one that makes any sense.
#610
08-10-2012, 06:33 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x In post 590 I am simply restating the apparent paradox and asking for your resolution. Your solution is you just keep walking. That does not really address it. Except in a most Zen like fashion. But not in any logical way. There are at least a couple solutions which show that there is no paradox. I am asking for yours. Just that you keep walking... that is not really one that makes any sense.
My solution is, (as I said,) that the argument that you can't reach B is invalid, because it illegitimately draws the conclusion that you "never" reach B when the premises actually only support a conclusion that you "don't reach B at any of the points in time where you hit a 1/2^n mark on the line from A to B."

Last edited by Frylock; 08-10-2012 at 06:34 PM.
#611
08-10-2012, 06:37 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock I know all about it. You don't get the paradox without reasoning that leads to the conclusion that you can't reach B. And indeed, you did offer reasoning that you can't reach B. I pointed out the flaw in that reasoning. Hence, there is now no longer reasoning on the table which says you can't reach B. Without that reasoning, there is no longer a paradox to solve.
LOL....

The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would "SEEM" as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong.

Premise A) any distance contains an infinite (endless) number of halfway points
Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway...
Conclusion: Since it is endless you can never complete this motion

Yes the argument is flawed, because we can move, but where is the flaw?

Last edited by erik150x; 08-10-2012 at 06:38 PM.
#612
08-10-2012, 06:45 PM
 Trinopus Member Join Date: Dec 2002 Location: San Diego, CA Posts: 4,759
re Zeno's Paradox, I've often wondered why it is granted that we can move half the distance from A to B. Shouldn't the paradox recursively prohibit the first step, which is usually granted?

Not only can Herakles not outrun the tortoise, he can't even take the first step in the race. The starting pistol (sorry, sling) goes off, and mighty Herakles is frozen in amber, unable even to take half a step... or half of a half-step... or half of half of a half-step...

Why do we grant that he can take the first great many steps, but only invoke the paradox on the "last" one?

ETA: anyway, I take a great big stride -- from my desk toward my refrigerator -- and shout, "I refute it thus!

Last edited by Trinopus; 08-10-2012 at 06:46 PM. Reason: quip
#613
08-10-2012, 06:48 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by erik150x LOL.... The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would "SEEM" as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. Premise A) any distance contains an infinite (endless) number of halfway points Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway... Conclusion: Since it is endless you can never complete this motion Yes the argument is flawed, because we can move, but where is the flaw?
The flaw is that your conclusion does not follow from the premises. You are equivocating between different senses of "endless", as Frylock has pointed out. What more is there to say? Premise B) is true in the sense that every event in the sequence of events is followed by another one in that sequence. But this does not mean that the entire sequence cannot be followed by some other event (as indeed it is).

Consider this analogous argument:
Premise: Any ruler contains an infinite (endless) number of halfway points: the 1 inch mark, the 1/2 inch mark, the 1/4 inch mark, etc.
Conclusion: Since it is endless, the stick cannot have an end. There cannot be a 0 inch mark on the stick.

Last edited by Indistinguishable; 08-10-2012 at 06:49 PM.
#614
08-10-2012, 06:48 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Trinopus re Zeno's Paradox, I've often wondered why it is granted that we can move half the distance from A to B. Shouldn't the paradox recursively prohibit the first step, which is usually granted? Not only can Herakles not outrun the tortoise, he can't even take the first step in the race. The starting pistol (sorry, sling) goes off, and mighty Herakles is frozen in amber, unable even to take half a step... or half of a half-step... or half of half of a half-step... Why do we grant that he can take the first great many steps, but only invoke the paradox on the "last" one?
It is usually easier to make the argument about not reaching B, first. But yes the further conclusion is that you can make A to B shorter and shorter as to prohibit any movement at all. It's too confusing to start out with, okay let's say you have an infinitely small distance A-B... people start thinking about what that means... and you never get anywhere... pun intended
#615
08-10-2012, 06:49 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x LOL.... The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would "SEEM" as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. Premise A) any distance contains an infinite (endless) number of halfway points Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway... Conclusion: Since it is endless you can never complete this motion Yes the argument is flawed, because we can move, but where is the flaw?
See my previous post--post 610--where I repeat what the flaw is. No, the flaw is not "We move by stepping forward at a positive speed." That is the reasoning that we can move, but it is not the flaw in the reasoning that we can't move. For the flaw in the reasoning that we can't move, see post 610 (and two previous posts in this conversation but I won't hammer on that point...)

ETA And now, notice, Indistinguishable is making the very same point I make in post 610 and elsewhere. The flaw is that the argument against motion is invalid. We explain why above. That is what you asked for. We have delivered it. Several times.

Last edited by Frylock; 08-10-2012 at 06:51 PM.
#616
08-10-2012, 06:55 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Indistinguishable The flaw is that your conclusion does not follow from the premises. You are equivocating between different senses of "endless", as Frylock has pointed out. What more is there to say? Premise B) is true in the sense that every event in the sequence of events is followed by another one in that sequence. But this does not mean that the entire sequence cannot be followed by some other event (as indeed it is). Consider this analogous argument: Premise: Any ruler contains an infinite (endless) number of halfway points: the 1 inch mark, the 1/2 inch mark, the 1/4 inch mark, etc. Conclusion: Since it is endless, the stick cannot have an end. There cannot be a 0 inch mark on the stick. Who would buy that?
Well, first lets be clear this is not MY conclusion... I merely stating the apparent paradox.

RE"But this does not mean that the entire sequence cannot be followed by some other even" ... indeed it could move over an endless/infinite set of points... but how do you ever get to the end/complete moving over and endless set of points? Getting to the end, implies an end. But that contradicts the idea that they are endless...
#617
08-10-2012, 07:00 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock My solution is, (as I said,) that the argument that you can't reach B is invalid, because it illegitimately draws the conclusion that you "never" reach B when the premises actually only support a conclusion that you "don't reach B at any of the points in time where you hit a 1/2^n mark on the line from A to B."
Who said anything about time? There are endless points... you must reach the end to get to B? But if they are endless how do you reach the end? You have explained nothing.
#618
08-10-2012, 07:02 PM
 Indistinguishable Guest Join Date: Apr 2007
There is no end within the sequence. But there is an item outside the sequence which comes after every item in the sequence. You can get to this item which comes after the entire sequence, because it exists. You can't get to a last item within the sequence, because there's no such thing. But the goal isn't to get to a last item within the sequence. To goal is to get to each particular item within the sequence at some time, which you can do. And you can even, after having done all that, get to another item (outside the sequence) which comes every item in the sequence.

In lexicographic order, there is no end to the strings beginning with the character A; I can construct the sequence A, AA, AAA, AAAA, in which each string would be placed in a dictionary before the next one. This is an infinite, endless sequence. But that doesn't mean there's no string which comes after all of them; AB comes after all of them.

Last edited by Indistinguishable; 08-10-2012 at 07:03 PM.
#619
08-10-2012, 07:04 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x Well, first lets be clear this is not MY conclusion...
We know this.
Quote:
 I merely stating the apparent paradox.
. We know this as well. We've resolved it by showing how the paradox's reasoning is invalid.
Quote:
 RE"But this does not mean that the entire sequence cannot be followed by some other even" ... indeed it could move over an endless/infinite set of points... but how do you ever get to the end/complete moving over and endless set of points? Getting to the end, implies an end. But that contradicts the idea that they are endless...
We don't need to know how, because you've offered no successful argument against the view that we can do it. We showed how the argument you offered turns out to be invalid.

Once we've shown that the argument against motion is invalid, there is no longer a "how" question to answer. "How?" in this context means "How is it possible in the face of its apparent impossibility?" Once we've shown that the apparent impossibility is based on flawed reasoning (as we have done above) we have thereby answered your "how" question.

Last edited by Frylock; 08-10-2012 at 07:04 PM.
#620
08-10-2012, 07:05 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x Who said anything about time?
. I did. What is illegitimate about it?
#621
08-10-2012, 07:07 PM
 Indistinguishable Guest Join Date: Apr 2007
erik150x, I will point to this again because you did not respond to it:
Quote:
 Originally Posted by Indistinguishable Consider this analogous argument: Premise: Any ruler contains an infinite (endless) number of halfway points: the 1 inch mark, the 1/2 inch mark, the 1/4 inch mark, etc. Conclusion: Since it is endless, the stick cannot have an end. There cannot be a 0 inch mark on the stick. Who would buy that?

Last edited by Indistinguishable; 08-10-2012 at 07:07 PM.
#622
08-10-2012, 07:09 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Indistinguishable There is no end within the sequence. But there is an item outside the sequence which comes after every item in the sequence. You can get to this item which comes after the entire sequence, because it exists. You can't get to a last item within the sequence, because there's no such thing. But the goal isn't to get to a last item within the sequence. To goal is to get to each particular item within the sequence at some time, which you can do. And you can even, after having done all that, get to another item (outside the sequence) which comes every item in the sequence. In lexicographic order, there is no end to the strings beginning with the character A; I can construct the sequence A, AA, AAA, AAAA, in which each string would be placed in a dictionary before the next one. This is an infinite, endless sequence. But that doesn't mean there's no string which comes after all of them; AB comes after all of them.
ugh... And dear, Sir, since there is no end to the sequence A, AA, AAA, AAAA, ... how do you propose I finish reading those in order to get to AB?
#623
08-10-2012, 07:14 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Indistinguishable erik150x, I will point to this again because you did not respond to it:
Ugh,,,, you guys are killing me.

The point of the paradox is not to convince any one that we cannot move or rulers don't exists. So why are you asking me if I buy it? It's irrelevant.
#624
08-10-2012, 07:14 PM
 Indistinguishable Guest Join Date: Apr 2007

I am pointing out how an endless sequence can nonetheless be followed by something which comes after each of its items. In the case of movement across the room, there is an endless sequence of events which, nonetheless, is followed by another event (your having reached the other side of the room). And there is no problem with this. That is all I am noting.

What would you say was the problem with the reasoning in the context of rulers? I would say the problem is that it is nonsense; there is no rule of logic which will take us from the premise to the conclusion. That's all I have to say. If the argument does not justify itself, then I don't have to listen to it.

Last edited by Indistinguishable; 08-10-2012 at 07:16 PM.
#625
08-10-2012, 07:23 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock We know this. . We know this as well. We've resolved it by showing how the paradox's reasoning is invalid. We don't need to know how, because you've offered no successful argument against the view that we can do it. We showed how the argument you offered turns out to be invalid. Once we've shown that the argument against motion is invalid, there is no longer a "how" question to answer. "How?" in this context means "How is it possible in the face of its apparent impossibility?" Once we've shown that the apparent impossibility is based on flawed reasoning (as we have done above) we have thereby answered your "how" question.
You don't show how the reasoning is flawed. You simply state that the conclusion is wrong, and granted it is wrong, but it follows valid from the premise. In order to resolve the paradox one must either show how the premises are false or how the conclusion does not follow from the premisses.

Here would be possible resolutions:
1) distance in the real word is not infinitely (endlessly) divisible.
or
2) it is not impossible to complete an endless series of tasks
#626
08-10-2012, 07:37 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Indistinguishable Who said anything about reading them? I am pointing out how an endless sequence can nonetheless be followed by something which comes after each of its items. In the case of movement across the room, there is an endless sequence of events which, nonetheless, is followed by another event (your having reached the other side of the room). And there is no problem with this. That is all I am noting. What would you say was the problem with the reasoning in the context of rulers? I would say the problem is that it is nonsense; there is no rule of logic which will take us from the premise to the conclusion. That's all I have to say. If the argument does not justify itself, then I don't have to listen to it.
Who is claiming there cannot be another infinite set beyond an initial infinite set? why do you bother to point that out, what is the point of you statement?
#627
08-10-2012, 07:41 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Indistinguishable Who said anything about reading them? I am pointing out how an endless sequence can nonetheless be followed by something which comes after each of its items. In the case of movement across the room, there is an endless sequence of events which, nonetheless, is followed by another event (your having reached the other side of the room). And there is no problem with this. That is all I am noting. What would you say was the problem with the reasoning in the context of rulers? I would say the problem is that it is nonsense; there is no rule of logic which will take us from the premise to the conclusion. That's all I have to say. If the argument does not justify itself, then I don't have to listen to it.
I would like to know how you ever completed writing the A, AA, AAA, AAA... section of your dictionary since it endless. I do not doubt the AB section exists, but just wondering how you managed to get to it writing it since there would be no end to writing your A, AA, AAA, ... section?

Last edited by erik150x; 08-10-2012 at 07:42 PM.
#628
08-10-2012, 07:42 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x Ugh,,,, you guys are killing me. The point of the paradox is not to convince any one that we cannot move or rulers don't exists. So why are you asking me if I buy it? It's irrelevant.
No one here thinks that is the point of the paradox, and no one here thinks you buy its conclusion. We are showing that the reasoning you gave (which, again, we know is not your reasoning) is flawed. Since it is flawed, the paradox is thereby resolved.

Indistinguishable is showing that the reasoning is flawed by giving an exactly parallel argument, and pointing out that the parallel argument is obviously invalid. (Hence the "would you buy that?" comment.) (And, in case you don't already know, I'll note that this is a standard move called "giving a counterexample to the form of the argument.) Since the parallel argument is invalid, so also the original argument is invalid. And since the original argument is invalid (and, again, we know it's not your argument), the paradox is resolved.

For a paradox to exist here, there needs to be good reasoning on hand for the conclusion that motion is impossible. The reasoning you've given for that conclusion, however, is not good reasoning. It has prima facie plausibility, but Indistinguishable and I have explained why that plausibility is only apparent.
#629
08-10-2012, 07:45 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x You don't show how the reasoning is flawed. You simply state that the conclusion is wrong, and granted it is wrong, but it follows valid from the premise.
No. This is utterly incorrect. I have not once in this thread attempted to refute the argument by simply disagreeing with the conclusion. I have in fact explicitly repudiated the idea of refuting the argument in that way. Instead, I have stated the flaw in the argument--explained why its conclusion does not follow from its premises--now three times. (Post 610 being the most recent.)

Last edited by Frylock; 08-10-2012 at 07:46 PM.
#630
08-10-2012, 07:45 PM
 Saltire Charter Member Join Date: Nov 1999 Location: Seattle, WA, USA Posts: 3,447
Quote:
 Originally Posted by erik150x I'll try to explain. Limits were not a mathematical discovery per-say. Like the proof that the Square Root of 2 is irrational or that Pi is irrational. Limits were constructed, devised as a means to avoid issues such as 1/infinity. For example the Limit of 1/x as x-> infinity = 0? Why don't we just say 1/infinity = 0 then? Because that would lead to all kinds of contradictions. So that tells me there is something missing here. Something is not right about that assumption.
You should probably state what assumption you're objecting to. I don't see anything missing. If our system of mathematics states that you can't do arithmetic with infinity as if it were just some number, then 1/infinity cannot be equal to anything in that system.

However, try responding to my point: If you think it's possible to have a non-zero value x such that x2=0, why isn't it possible for 0.9... to equal 1? To me, the first is a very abstract concept, while the second is just an artifact of decimal notation. So if you claim to understand the philosophy of infinitesimals (which I can't claim myself), you shouldn't have any trouble with the point of this thread, since it's far less abstruse. Do you not agree that 3/3=1?

(By the way, I currently don't have internet access at home, and I'm about to leave work for the weekend. So, if you reply to this and I don't respond, please forgive me. I will respond sometime Monday.)
#631
08-10-2012, 08:02 PM
 Francis Vaughan Guest Join Date: Sep 2009
Quote:
 So why are you asking me if I buy it? It's irrelevant.
Correct. It is irrelevant.

Why do you persist in keeping bringing it up?
#632
08-10-2012, 08:04 PM
 Frylock Guest Join Date: Jun 2001
Read this whole post, please, though it may seem to repeat some things that were said before. As I find myself repeating points, I believe there is some serious miscommunication happening here. You think I'm saying things I'm not saying, and you don't know that I am saying the things I am actually saying. Here it all is starkly laid out, in paraphrased dialogue form.

You: I don't actually think that motion is impossible, but nevertheless, here is an argument that motion is impossible. I'm curious what flaw you can find in it. Please don't refer to limits in your answer.

Me: Okay.

You: Okay, here's the argument. To move from A to B, you must first move to the point halfway between A and B--call it C. Then, to get to B, you still have to move to a point halfway between C and B. Call that D. And still even then, to get to B, you have to move to a point halfway between D and B. This goes on forever. So, as you can see, you must move through an infinite number of points before you reach B. So, you never reach B.

Me: But in general, just because you must move through an infinite number of points before reaching a point, that does not mean you can never reach that point. It does mean you never reach that point while moving through the prior points. But after you've moved through all those points, you're free to move to that goal point. That is why the conclusion of the argument does not follow from its premises. That is the flaw in the argument.

Last edited by Frylock; 08-10-2012 at 08:05 PM.
#633
08-10-2012, 08:48 PM
 Trinopus Member Join Date: Dec 2002 Location: San Diego, CA Posts: 4,759
Quote:
 Originally Posted by Saltire . . . Do you not agree that 3/3=1?
It seems to depend on the notation! In binary, since 1/3 is an unending repeating numerical expansion, no. In trinary, where 1/3 is just .1, it's easy!

Quote:
 Originally Posted by Frylock . . . I believe there is some serious miscommunication happening here. . . .
Understatement of the millennium! (But it's early; we have lots of time.)

Quote:
 . . . But in general, just because you must move through an infinite number of points before reaching a point, that does not mean you can never reach that point. It does mean you never reach that point while moving through the prior points. But after you've moved through all those points, you're free to move to that goal point. That is why the conclusion of the argument does not follow from its premises. That is the flaw in the argument.
You might say, we move through whole bunches (infinities!) of points all in a "glomp." We are never obliged to identify, name, or construct the points we pass through. The paradox tricks us by constructing one particular sequence of named points -- 1/2, 1/4, 1/8 -- through which we pass. But that simply isn't how real motion works. We pass through points without naming them.

Similarly, in geometry, we accept that a line is a "locus" of points. It isn't a specified, nominated set of points. We say things like y = 7x + 5 for x > 3 and x < 5. Voila, a line segment, excluding its end-points. No one could ever print out a "roster" of every point in it, but this impossibility doesn't mean that it is not possible for a car, or a bullet, or a molecule, to follow the path that the function describes.
#634
08-10-2012, 09:24 PM
 Francis Vaughan Guest Join Date: Sep 2009
Quote:
 Originally Posted by erik150x Whether you are keeping track has no bearing on the issue.
Correct. So why bother to select some particular set of points?

Quote:
 It's like saying it doesn't matter how an airplane flies, it just does.
No it isn't. An airplane is a physical entity subject to the laws of physics. The number line, and the reals are not physical entities, and are not subject to the laws of physics. They are subject to manipulation by the rules we select for them.

Quote:
 The distance you are covering contains and infinite number of intervals. It does not matter if you are aware.
Correct.

Quote:
 The issue pertains to how its possible for you to complete the series when they are infinite or endless.
And there is your fallacy. You don't like having an infinite number of points in a finite interval.

My point was that there are an uncountable number of series in the interval, not just yours. Completing the series is hardly something special. Do we have to "complete" every such series? It would seem that if we have to pick one arbitrary series, we must have to complete them all.

Quote:
 I find it difficult to see how one could argue that one can and regular does progress through an infinite number of points during the movement form point A to point B, but for the the 9s in .999... it's different.
But it isn't different. There are a different number of points, because one is a countable set, and the other uncountable, but that is all. It is the attempt to define infinity in a different manner to that generally accepted (Aleph Null) that causes issues. In either case you pass though an infinitude of points. 0.9999.... or 1/2, 1/4, 1/8... is simply a selection from those you passed though.

In the end, any attempt to appeal to motion or Zeno seems to fail on the same problem: a wish not to allow infinity to behave the way it does. To somehow make it behave like a finite number, and thus subject to the ordinary rules. But it isn't finite, that is by definition. It doesn't adhere to the intuitive rules of numbers, and trying to make it do so doomed from the outset.

Last edited by Francis Vaughan; 08-10-2012 at 09:26 PM.
#635
08-10-2012, 09:31 PM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by erik150x Premise A) any distance contains an infinite (endless) number of halfway points Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway... Conclusion: Since it is endless you can never complete this motion [snip]...where is the flaw?
The argument above relies on an unstated assumption that an endless sequence of events cannot be completed. That is true if the events have a (real) non-zero duration. But presumably, you're allowing for "events" with a duration of zero. In that case, though, I do not know why anyone would think that an endless sequence of such (zero-duration) events could not be completed. What is your argument for that substantive, non-obvious unstated assumption?

Is your argument for this unstated assumption like this?

P3) Completion of a sequence of events implies that the sequence has an end.
P4) An endless sequence of events has no end.
Conclusion) An endless sequence of events can have no end.

If that is your argument, then, assuming "end" means "final member of a sequence," the flaw is that P3 is false. It is not true that completion of a sequence of events implies that the sequence has a final member. If you think otherwise, you need to justify that substantive and non-obvious premise.

But is that not what you mean by "end?"

Or alternatively, is that not your argument for the unstated assumption I identified in the first argument above?
#636
08-11-2012, 02:29 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Frylock The argument above relies on an unstated assumption that an endless sequence of events cannot be completed. That is true if the events have a (real) non-zero duration. But presumably, you're allowing for "events" with a duration of zero. In that case, though, I do not know why anyone would think that an endless sequence of such (zero-duration) events could not be completed. What is your argument for that substantive, non-obvious unstated assumption? Is your argument for this unstated assumption like this? P3) Completion of a sequence of events implies that the sequence has an end. P4) An endless sequence of events has no end. Conclusion) An endless sequence of events can have no end. If that is your argument, then, assuming "end" means "final member of a sequence," the flaw is that P3 is false. It is not true that completion of a sequence of events implies that the sequence has a final member. If you think otherwise, you need to justify that substantive and non-obvious premise. But is that not what you mean by "end?" Or alternatively, is that not your argument for the unstated assumption I identified in the first argument above?
Okay I think we are finally getting to a meeting of minds here Frylock...

RE: "The argument above relies on an unstated assumption that an endless sequence of events cannot be completed." - I agree with you 100% that is the flaw.

Let's go back a few posts to the following post you made as it is also relevant to the statements you are making above I think.

"If "the end" means "the final member," then yes I do suggest that we never reach the end.

But we reach something. Just not "the end" in the sense of "the final member."

There's also a problem in interpreting your term "reach," btw. What does "reach" mean to you in this context? "

Fist let's address what I mean by reach. I don't understand why this needs clarification, but so be it. By "reach" I mean you are there. In Zeno's Paradox of motion "reaching" each half way point means you are AT that point. Will further define being at that point by the tip of your nose being inline with the point that marks the halfway point. The tip of your nose would be further defined by the furthest physical point towards the direction of motion that would be considered part of your nose. I hope that clearly enough defines reach for you

But the heart of our disagreement may come down to this part:

Frylock: But we reach something. Just not "the end" in the sense of "the final member."

What do we "reach" as we complete the sequence of an infinite (endless) number of halfway points if not the end as in final member, (yes I would take final member and end to be the same thing... I see no other way to possibly take it).

Let's take the set of all natural numbers, ok? (Don't worry we are just borrowing them, we'll give them back when we're done)

Your task is to name them off (all natural numbers) 1 by 1 in order, i.e. "1, 2, 3, ...". I am also going to grant you super powers.

You will be able to name them as fast as you like, in fact infinitely fast. So I don't expect it should take you long. The only concern I have is how you will know when you are done naming them?

Last edited by erik150x; 08-11-2012 at 02:32 AM.
#637
08-11-2012, 02:46 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Lumpy For what it's worth, I recently read a fascinating article http://io9.com/5932366/how-do-you-co...rses?tag=space which introduced me to the concept of p-adic numbers- an alternative to the entire real number system, and one arguably more useful than the standard reals.
Thanks! I will definitely check it out. I have been currently reading up on hyperreals.
#638
08-11-2012, 03:26 AM
 nivlac Charter Member Join Date: May 2001 Location: Golden State Posts: 2,340
Quote:
 Originally Posted by erik150x LOL.... The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would "SEEM" as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. Premise A) any distance contains an infinite (endless) number of halfway points Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway... Conclusion: Since it is endless you can never complete this motion Yes the argument is flawed, because we can move, but where is the flaw?
The flaw is the assumption that you cannot add an infinite number of numbers and arrive at a finite number. The "infinite number of numbers" in this case are all the half-way distances. The "finite number" is the total distance traveled. The infinite number of numbers to be added is called an infinite series. An infinite series can sum to a finite number if the terms of the series get progressively smaller fast enough. That's the crux of the solution. You may want to pick up an elementary calculus book and read the section on infinite series.
#639
08-11-2012, 04:23 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by nivlac The flaw is the assumption that you cannot add an infinite number of numbers and arrive at a finite number. The "infinite number of numbers" in this case are all the half-way distances. The "finite number" is the total distance traveled. The infinite number of numbers to be added is called an infinite series. An infinite series can sum to a finite number if the terms of the series get progressively smaller fast enough. That's the crux of the solution. You may want to pick up an elementary calculus book and read the section on infinite series.
First of all no assumption of the sort is being made. The distance is to be traveled is already given earlier on in this thread. Secondly no one is asserting the that the halves do not add up to a finite number (although there is some debate about that being .999... or 1). But at this point we are on a bit of a tangent. If you actually want to be brought up to speed on this I can point you to the relevant issue.

Post #574

Last edited by erik150x; 08-11-2012 at 04:24 AM.
#640
08-11-2012, 07:43 AM
 Frylock Guest Join Date: Jun 2001
Since the scenario is strictly impossible, it shouldn't be surprising if my answer is strange. It is logically impossible for me, at any point during the counting process, to correctly think "okay, that was the last one--no more numbers to count." So at no point during the counting process will I know that I am finished. But at some point after the interval within which I was doing the counting, I will find that I am not counting any more. (Even though there will have been no counting number that I missed.) At that point, I will know that I am done.
#641
08-11-2012, 08:07 AM
 Lumpy Charter Member Join Date: Aug 1999 Location: Minneapolis, Minnesota US Posts: 10,922
I'm not sure if this has already been brought up (it's been a long thread), but I think that the presumption that you're taking each term in a series one at a time (.9, .99, .999 etc.) is implicitly taking it as countable, when we know that the reals are uncountable.
#642
08-11-2012, 08:08 AM
 Frylock Guest Join Date: Jun 2001
Quote:
 Originally Posted by Lumpy I'm not sure if this has already been brought up (it's been a long thread), but I think that the presumption that you're taking each term in a series one at a time (.9, .99, .999 etc.) is implicitly taking it as countable, when we know that the reals are uncountable.
The reals are uncountable, but that series is countable.
#643
08-11-2012, 01:05 PM
 Bosstone Member Join Date: Mar 2001 Location: Phoenix, AZ Posts: 15,368
This thread is moving pretty fast, but I haven't seen this referenced in the recent discussion about Zeno's Paradox.

It seems that whenever someone gets stuck on Zeno's, it's assumed that it takes the same amount of time to go from 0.5 to 0.75 as it did to go from 0 to 0.5. It didn't, it took half the time.

Sure, you have to move through an infinite number of halfway points from A to B, but the time it takes you to travel between those halfway points also decreases. I'm not sure why limits were off-limits in the explanation, because that's where the paradox falls apart. As your distance from point B decreases to an infinitesimal number, so too does the time it take decrease to an infinitesimal number, until finally you've covered .9999... of the distance and have .0000... left to go in .0000... seconds. Lo, you have reached point B.

Quote:
 Originally Posted by erik150x Your task is to name them off (all natural numbers) 1 by 1 in order, i.e. "1, 2, 3, ...". I am also going to grant you super powers. You will be able to name them as fast as you like, in fact infinitely fast. So I don't expect it should take you long. The only concern I have is how you will know when you are done naming them?
I have that superpower, and thus in between typing the D and the O of the word "done", I have counted them all.
#644
08-11-2012, 08:18 PM
 Francis Vaughan Guest Join Date: Sep 2009
Quote:
 Originally Posted by Bosstone .... That also answers this I have that superpower, and thus in between typing the D and the O of the word "done", I have counted them all.
To save Erik the trouble - so what was the value of the last one?

That is the point of his task- not that you can't do it - but that you can (and in his view - must) then be able to identify the "last" value.
#645
08-11-2012, 09:04 PM
 Derleth Guest Join Date: Apr 2000
Quote:
 Originally Posted by Lumpy I'm not sure if this has already been brought up (it's been a long thread), but I think that the presumption that you're taking each term in a series one at a time (.9, .99, .999 etc.) is implicitly taking it as countable, when we know that the reals are uncountable.
You are trying to use mathematical terms of art as if they meant what the dictionary says they do, you poor bastard.

'Countable' means "You can take every element of this collection and label it with an integer without missing any." That sequence is, therefore, countable. So is the sequence of nines in 0.999...

Frankly, 'listable' is a much better term for the concept, because what we really mean when we say the reals are 'uncountable' is that "If you tried to make a list of them, even an infinite one, you're guaranteed to leave some out." That's the heart of Cantor's diagonal argument.
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#646
08-11-2012, 09:14 PM
 Frylock Guest Join Date: Jun 2001
Or, for those who get nervous about infinite lists, "No matter what method you come up with for assigning integers to reals, there will always be an infinite number of reals that no integer ends up assigned to."
#647
08-12-2012, 01:35 AM
 Francis Vaughan Guest Join Date: Sep 2009
It occurs to me that part of Erik's reasoning depends upon an ability to be sure of every value of the infinite expansion, which is a very limited case of converging infinite series.

So, rather than pick 1 as the target value, why not pick a transcendental? There are many infinite series that converge to transcendental. We can pick any one of them, perhaps restricting ourselves to those with all positive terms to avoid any oddities.

This one is useful:

Pi2/6 = 1/12 + 1/22 + 1/32 + 1/42 + ...

So, I walk along and cover the distance to Pi2/6. I have covered all of the terms of the infinite sequence, and reach it. So clearly by Erik's reasoning I have the last value of the infinite sequence. I wonder what it is?

Another sequence that is interesting is one of the sequences for Pi itself.

Pi = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) ...

This rattles around Pi from above and below, so we don't cover all values just walking from 0 to Pi. Which raise another interesting question. If I walk from 0 to Pi, I must have covered half of the values in the sequence when reaching Pi. If I can find the "last" value in the sequence this way, I must surely be able to find the "last" value coming from below, and the "last" value if I approach from above. Having both these values I must surely be able to tell if it the "last" value of the total series is greater or less than Pi.
#648
08-12-2012, 02:44 AM
 TATG Guest Join Date: Aug 2008
Quote:
 Originally Posted by Derleth Frankly, 'listable' is a much better term for the concept, because what we really mean when we say the reals are 'uncountable' is that "If you tried to make a list of them, even an infinite one, you're guaranteed to leave some out."
Only if the list must be countably infinite.
#649
08-12-2012, 03:37 AM
 Derleth Guest Join Date: Apr 2000
Quote:
 Originally Posted by TATG Only if the list must be countably infinite.
In that context, "list" meant "series of entries each numbered by an integer". Which was pretty well implied by context.

My point, though, was twofold: One, you can't try to understand terms of art using dictionary definitions. Two, mathematicians sometimes pick really lousy words to use as terms of art.
#650
08-12-2012, 03:48 AM
 TATG Guest Join Date: Aug 2008
Quote:
 Originally Posted by Derleth In that context, "list" meant "series of entries each numbered by an integer". Which was pretty well implied by context.
In which case "list" looks the same as your definition of "countable", except with some added complexity. (I don't see why "list" is meant to be better or more explanatory than "countable". And I took it the "what we really mean" comment was to imply some contrast between the two.)
Quote:
 Originally Posted by Derleth My point, though, was twofold: One, you can't try to understand terms of art using dictionary definitions. Two, mathematicians sometimes pick really lousy words to use as terms of art.
Or reuse them (e.g. "consistency").

Last edited by TATG; 08-12-2012 at 03:49 AM.

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