Zenos' paradox
 

Zenos is a greek philosipher and he came with one of the weirdest paradoxes ive ever seen, cause theoreticaly it should work. This is the paradox as quoted from the book "A Brief History Of Infinity" by Brian Clegg.

That paradox just plain scares me, I mean I know i can get up an move away, but being me, sometimes I fear I won't, but then again I am insane.
I just wanna see what you think on it.

I've heard this before, and I'm sorry to say I've always thought of it as crap. It's inherently easy disprove it by getting up and walking across any plain old room. You can set any number of arbitrary points along a line segment, but that does nothing to change the length of the segment nor the rate at which a particular person could travel along that length. As quirky food for thought on the nature of time, reality, and some other abstract things, it's useful and fun, but reality is so obviously contrary to the idea.

So it's crap. But good crap. Plus random philosophical tangents always intrigue me.

Zenos was legitimately insane. Not kidding.

The theory doesn't take into account the fact that people will walk 1/2, and then 1/2. It assumes you can not continue until you reach an earlier point. But by continuing you reach all of the necessary points.

This was used in a chick flick once to explain how the girl scientist was falling for the boy scientist despite all rationale. Ah, love...

Eh. That paradox actually said that Achilles - the invincible guy from Greek mythology - couldn't beat a turtle in a race if it had a 1/2 of the distance of advantage. I think that's stupid.

I don't think you're quite understanding it...

Ok, I'm going to use a bunch of periods to denote a line, with a comma making the half way point an a line marking me.

|.....,.....

Now, I have to reach that comma to get to the end, right? So I do.

.....|.....

However, I now have a shorter line ahead of me with another halfway point.

|..,..

So I reach that halfway point.

..|..

But now there's another one...

|.,.

So on and so forth.

And as that I can't teleport I can't just 'skip' the halfway point. Even if I step over it I'm still reaching that point.

So I never have to reach an earlier point. It's always a point beyond me that I have to travel to.

And, if you take VERY small distances into consideration, it's not actually fully incorrect. Even at physical touching there is a certain distance between particles.

But if you bring in quatum mechanices (I am not positive this is correct, I only read it somewhere.) There is only a certain amount of minimun distance one could travel. At one point you would hit that marker to where you would always break the halfway point no matter how much you held back.

I believe Inbred is referring to the Planck Length, or if you need to add dimensions, the Planck square or cube. After that point space and time as we know it (with our current understanding) breaks down. I guess that's where the Zenos Paradox goes haywire. ;)

Krylo gave a much better explanation of it, but again, its horribly flawed. The paradox seems to require that you reach these segmented points before you can move on. Boiled down, its throwing infinity in your face and going 'Haha, puny finite creature.' Luckily, in our finite room, there is no paradox. Why? Because of the flaw in the paradox itself. It assumes right off you can never take a step because that step is infinitely dividing itself by two, OR your steps are divided by two every time you take a step, meaning you never reach your destination.

Stupid people and their over analysis! You are creating these halfway points that don't exist, or fit in with physics! It's not as if you stop upon reaching this halfway point, nor is that the differing of distance that you travel has any effect of your speed or ability to travel. No matter what distance you go over, you are moving at a constant speed until friction stops you.

|.....,..... You're going this distance at 3 mph.

.....|..... Now you've reached the halfway point, still going 3mph.

|..,.. Still going 3mph, but due to the fact you still have some way to go, you make a new half space.

..|.. Now you've reached the center (3mph still) but in the bigger sense ........|..

|.,. But halfway it's a smaller distance, and you'll cross it quickly, in bigger ........|.

|,. Now you're halfway is so tiny that you'll cross it twice in a mere few milliseconds

and reach the end .........|

In other words, the more fractioned the distance, the less time it takes to go over it, till it takes the smallest millesecond to go over it. So you'll always reach the other side of a room because despite all these halfway marks you're making, you are crossing them in very very small amount of time.

Besides, the very definition of a paradox is a that something that works shouldnt logically, yet still works. Like most of Star war's early ships and vehicles.

But in order to cross a point you have to meet it first. And then you have to reach the next halfway point, etc. etc. no matter how many halfway points you cross.

But it does break down at plank lengths.

AND being there by our perception isn't being there on an atomic level.

You know, I think it was me that brought this up once before, when I was playing devil's advocate in some thread.

The only important thing I have to say is that the philosopher in question was named Zeno, not Zenos, and he lived in Elea, and is most commonly referred to as Zeno of Elea.

I understand it, I was just saying what people do realistically, rather than theoretically, and I didn't explain it well.

Walking any distance, you walk an infinite amount of fractions of that distance theoretically, and if there is a definite planck length as some people are claiming, than in real life you'd cover a finite amount of fractions.

Uh, zeno's paradox isn't a paradox at all. You can solve it using calculus in like... 2 seconds.

The time it takes for me to get from whereever to halfway to the door is X seconds. As i continually repeat this, not only does distance go to a tiny amount, but SO DOES THE TIME IT TAKES ME. Thus the sum of all these distances dx over time dt is not = to retarded, its just = x, which is my original door to me distance.

Basically, you can use l'hopital's rule, and eliminate the factors which cause the infinity over infinity.

i just wrecked a hole in paradox.

Or to put it succinctly, Zenos paradox in essence states that if your chasing someone and are decelerating at the right rate you will never catch them but will always cover distance between you and him.

If states otherwise it seems to put our view of the world into question, but stated like that it seems simplistic and stupid. There are more interesting paradoxes.


• A = B
• A+A = A+B (adding A to both sides)
• 2A = A+B (simplifying)
• 2A-2B = A+B-2B (subtracting 2B from both sides)
• 2A-2B = A-B (simplifying)
• 2(A-B) = 1(A-B) (factoring)
• 2 = 1 (cancelling)
its interesting to see how long it takes people to find the flaw.

you're dividing by 0

Yes I am. And it took my accountant dad a good five minutes to see that. And there are some co workers who still believe all of mathematics is a lie.

Oh ya of course YOU were the first person to calculate it and all, and as soon as Zenos made this paradox no one moved ever again, till now.

The deal with this paradox is that its just showing that the whole idea of motion as a continuos process that could be divided up as much as you like was untenable.

Zenos didnt stop moving after he figured this out because he believed he couldnt. He wanted to prove infinity, he used it as a hypothetical situation.

And to whomever said this was the achilles tortoise theory, your wrong, their both theories of Zenos which are extremely similar but are worded differently.

but it can, even in the way you explain it. i even gave you the happy math to do it. in fact, calculus relies pretty much on dividing things up into infinity tiny pieces... so by using it and disproving the paradox, i've done what you said i can't. If you want, you can get a nice little equation, and for every slap happy x you put in (each sliver of time you want to check) there'll be an answer for you.

Thank you for the accolades, i'm sure the world can rest easy now that i've liberated them. Donations and support can be sent to my house.. PM me for an address.

On a more important note, apperantly there's a smallest possible length and a smallest possible unit of time, but since i'm lazy, i won't link any physics journals.

This pardox flaw is that it thinks you have to stop at that point before contiuning. Most us us would cross over it. Well I think.

In the proof showen with the equation where does he divide by zero?

For the love of Pete! I think I already said it was the Planck Length (Or Square in 2D, Cube in 3D). As for the exact number, 10^-33 centimeters if memory serves me. The shortest unit of time is, and how unoriginally, the Planck Time, the time it takes light to cross one Planck Length (something like 10^-43 seconds).

But really, those units are so utterly useless in this discussion. I mean, atomic nuclei are on the, what, 10^-15 scale?

Edit: Right, and uh...Lucas is right. Some quick magic with L'Hospital's Rule and the paradox is worked out mathematically. As per usual its tough to convert it into practical thinking...though, I do like the logic behind it.

*smacks self in the head* Forget the fancy math there is a very easy way around this. Simply put you cover a set amount of distance everytime you step. Your motion is infact discontinous. Meaning that you cover a set amount of distance stop and cover that, or nearly so, amount of distance again. So as time goes on and you travel greater and greater precentages of the remaining distance.

Actually the very heart of this paradox is the lack of the consideration of time. Since even today we don't have a very clear understanding of time it still seems to work logically. That is until you do the math and force time into consideration. About the only thing the paradox proves is the inability to consider motion without considering time for without time we wouldn't be moving. (At least not in the same sense of the word.)

If A=B then A-B=0

hence why dx is proportional to dt... i already said everything you did. =)

O ya. I knew that. I was just testing you, to show my interior knowlege.
It's superior not interior.
Uh, ya.

Yes but you used mainly math and did something no one during his time period could. You disproved the paradox by showing why it doesn't work. I went a little deeper and explained the flaw in the logic the seemed to work. That being he was considering motion without taking into account time.

hence dt. t in dt is time, unless i'm measuring... uh.. teslas.

I never said you didn't mention time, you saw and exploited that weakness in the argument. What you did not do was say anything at all about that weakness. You simply stated that the math doesn't work out, and did that by introducing a variable that wasn't in the original problem without explaining why. Basically, you used mathmatical know how to see a logical hole without actually having to do the logic.

I on the other hand spotted where he went wrong. As in Zeno never consider the effect of speed and thereby time. You jumped straight to equations of motion sidestepping the problem of having to add in the abstract nature of time by making it abstract. Then used unrelated math rules to show why it works, which is actually how a lot of things are down. This took into account the base underlying flaw of Zeno's logic but never really stated outwardly.

I took the less traveled route and found why logic the seemingly worked in words didn't work out mathmatically. That being he Zeno wasn't thinking of motion in the same way we understand it, and he was negelecting time entirely. The merit of your way is that we get the answer and disprove the paradox but you miss a bit of the reasoning behind it. Like the fundementally different ways you and he treated motion. Indeed, after I said something you became quite aware of his missing consideration of time. However, I highly doubt when you were constructing that mathmatical proof you were thinking, or at least you weren't aware you were thinking, that since he had not considered the time it would take to cover the distance he had made an error.

It could be that you did notice the lack of time and from there luanched into the math proof with abstract rules. That suggests that you found the logic no quite enough to completely prove the point, which it most certainly does. Math is a powerfull tool for getting answers but the greatest minds have always reasoned their way through a problem. Einstein hated math and because of that we have general relativity, which absurdly enough is unbearably math intensive. Special relativity and relativity in general grew from thought expierments and the math fell into place later. Einstein was simply able to intuitively handle abstract concepts that today we have to simplify into math.

In short trust in the force Luke(Sorry really bad joke). If the logic works then trust it. If your unsure check it with the math but don't use the math as a reason for the logic. If the math works there has to be a nonmathmatical logical reason for it. Finding the reason is the real challenge, and is the path to truly understanding something.

I'd take that apart piece by piece, but its rather obvious that math (especially in this instance) is logic. Math is just the logical extension of patterns. So basically you're just wasting time rehashing what i put down with a grand total of 5 characters ( dx/dt )

more importantly, you seem to have forgotten that i said
Ta-Dah! you've been repeating what i've been saying all along =D
next time, if my post has like 5 lines, be sure not to skip 2 thru 4.

Logic yes but weak logic that uses rules actually exterior to the problem. Notice how you had to use an intergration then invoked L'Hopital's rule as a final proof to your math proof. If you go back in time and try to use that on Zeno's he'd look at you like you were crazy.

That's just rude as you seem to have skipped all this from my post.

Let's take a look at what you posted.

Implicit in that is everything I said but no where do you actually come out and state it. That is the power of math and it's weakness. Anyone can do the math with a little training but understanding the logic buried there is another story. So as I said before you introduced time but never said why it had to be there or anything about it not being in the original problem. Basically, you showed how the paradox doesn't work; while I dug in a bit and found the why. You used propieties of numbers, equations, and mathmatical operators basically going outside the problem for the tools that answer the question. My way stays completely with the intrinsic properties of the problem and gives a solid explination of why the math works out and what was missing.

If you really can't see the difference between the logic used in math and good old reasoning then no argument I make will sway you. Indeed, the only argument you'd really trust would contain math and I'm sorry but I can't show you the difference mathmatically. (Which another reason why its better to use reasoning type logic as much as possible. It just has more uses.)

Oh and to clarify the point a bit. With math you can end up with correct answers that make no sense at all. Like the wave particle duality of light and all other particles. When this was proven with math everyone was at a loss as to how a particle could be a wave. That is until someone took the step and said it's not the particle that's waving but the probility of finding it at a location. That insight was no where in the math and was a quantitatively proven later with math, but not before the quantitative leap using reasoning.

Incorrect, i said exactly what the weakness was, and that was....
The only difference is that you didn't like the fact that i used math. Seriously, this isn't a big deal, you're just pretty much beating on a drum that doesn't exist. Saying that using math to describe a problem gives an incomplete view is totally correct, but not when i caps lock-ed the most important part in text.

If i had set up the infinite sum, changed it into an integral from 0 to infinity, and evaluated whether or not the time taken was finite or infinite, then fine, you'd be perfectly justified in saying that my math didn't talk about the weakness inherant in the paradox. Did i do that? No.

Basically, you're making this a bigger issue than it is; you repeated what i said... no biggie. If you have any real beef with that, i'm sad to say i'm going to ignore you, because creating a rivalry here is totally useless, especially when the paradox has already been nicely debunked. To me, this thread is pretty much closed.

You've managed to completely misinterpert everything I said. The dispute is not over the content so much as how you pharsed in. In that regard you were correct in saying I didn't like the math. I personally love to do math but I know that it is inherently difficult to form a clear mental picture straight from math. I don't care that you said everything I said. The entire point of me saying it again was to make more apperant want went wrong. Saying: Says aboslutely nothing at all about how or why time is centeral to the paradox, unless you are really good with the math and can imediately see the physical reasoning behind it. (Sorry to say most people can't.) It says what is happening with respect to time but it doesn't describe why you can't consider motion without time. Nor does it say anything about Zeno not considering time intially. Our difference of opinion is purley ontological and semantical but that doesn't make it any less important. So really you mentioned the central issue of the paradox in passing but you never said anything about why it was central or even that it was central.

Oh and yes you did pass over it slightly be using the math you did and saying that but not nearly deep enough to really get at the core reason. That reason being Zeno was trying to describe motion without using time not just that he never said anything about time.

As for simply ignoring me that is plain inconsiderate and rude, more so than flaming in my opinion. When someone flames the due it through lack of inteligence but for an otherwise inteligent person to close out another person trying to have a reasonable desicussion because of a difference of opinion, no matter how insignificant either side thinks that difference is, smacks of arogance. I've been perfectly civil in trying to explain the very subtle difference between what you said and what I said and being dismissive is not flattering. As my argumentative writing teacher would say, you must treat all oponents fairly and evenly, not dismissing any out of hand; doing so will not change the minds of those dismissed and will weaken your case in the eyes of even those you treated fairly. Not to mention it makes it seem like you think your opinion and interpertation is more valid than anyone elses while providing absolutely no proof either way.

I know that. I like my phrasing.

Speed is measured in meters per second. distance over time. wow. that was totally difficult.

there is no discussion here: you like your phrasing, i like mine, that's all there is. i know exactly what you've been saying the entire time, and your entire argument rests on preference, as does mine. we can't take this any further, nor do i want to.

Do you honestly not see it? you're getting worked up over simple semantics: let it go.

There is only not a discussion here if you simply refuse to discuss anything you believe/prefer. Preference are based on facts and reasoning even if the one holding them is not aware of them. Thus preferences can be discussed as much as one wants. I've been kind enough to give my reasoning behind why I think my pharasing does a slightly better job. You thus far have not bothered to do that. Your reluctance to discuss further means either you're lazy, sorry about that, or you're so set in your ways you fear what might come of changing them. Oh and I'm not worked up at all. If I was worked up I'd be knocking on your door with a bat and not typing, bad temper and all.

It's always good to know that Nuklear Power forums's intellectuals consider to knock before solving an argument with a bat.

As for phrasing, I couldn't care less if you did it all in spanish, with egyptian runes for numerical digits.

do you honestly think that i don't know where preference comes from? i know there are facts and reasoning behind it, but quite frankly, since this isn't a thread about whether or not you prefer to use calculus or if you prefer to use half a page to describe the same thing that dx/dt does, i don't feel like i really have to go that indepth about why i prefer using a tiny amount of space to describe the so-called paradoxical motion. If you want to make a thread entitled:

"THE MERITS OF DX/DT VERSUS LONG REAMS OF TEXT SAYING THE SAME THING"

then by all means, do so. The paradox, i.e. the topic of this thread, is done with. Its gone, we killed it.

I find it funny, however, that when i DO refute your statements of logic leading to preference you don't take up the slack and counter refute. In light of that, i'm done (again) with this thread. I have interest in flogging dead horses.

Amen.

I forget what movie it was in, but this paradox was used as a threat, from someone stupid against someone smart. The stupid guy summed it up as "So, if I shoot you, this bullet will never reach you, is that right? Still think you're smarter than me?" Cracked me up.

Back to your regularly scheduled paradox.

There is no holy rule that says we can't take a thread whose orignal topic has been exhausted and change it a bit. It's how most normal threads stay alive. It also keeps the boards somewhat cleaner by cutting down on the number of half used threads. If there were still another discussion going on I'd say we should move on but as this is the only one going I see no need. We are discussing solving paradoxs in general with a concentration on the disproof of the one Zeno came up with. That is a perfectly good topic to segway into to keep a thread going and not have to start a new one. I am, however, now of topic.

Because you didn't. You repeatedly stated we said the same thing. Or said that your math proved the same thing so any difference was irrelevent. That's not a logical refutation. Those are statments of opinion and inherently undebatable unless you give some reasoning behind them. Which you didn't do. Aside from saying they say the same thing so who cares.

If you knew that then why did you say this:
Because clearly while opinions and preferences can't be used to persuade and debate they are often the subject of debates. They can be the subject of debates because their foundation lies in fact in one way or another. Your two statments are self contridictory. (That by the way is a refutation.)

So it's NOT just me that SithDarth latches on to and tries to beat into the ground! I feel so much better about myself.

perhaps the shoddiest refutation ever. Your argument only makes sense if preference has an absolute base in fact, which it most certainly doesn't. Your position is ENTIRELY based on your preference; you think that your explanation was clearer and cut right to the heart of the issue.

Awesome.

My explanation does the EXACT SAME THING, only you didn't like the fact that i used math. Am i going to debate the semantics of this again and again? are you seriously not catching on to my reasoning, which i've let slip out in little nice phrases for you to look at? i think every single post i've made has refered to your explanation as long, as compared with my concise answer:

I like math because it takes very little room to show something in this case; you like text because you feel its not "abstract" like math is.

Wow, i just argued the entire discussion in 2 lines.

oh, but i did.

Yes it is. I set out to prove theorem X, i find 2 ways to do it, both say the same thing about the theorem in question (i.e. that its true).

happy?

I'm sorry if it offends you put I don't like being misunderstood or misunderstanding someone else. So I always make the effort to try and clear up any percieved misunderstandings.

Doesn't offend me in the least: it just wastes my time when the issue is so pedantic and trivial. If we were talking about some complex geopolitical issue, fine, but this just doesn't fit that category.

Did you completely and totally miss all this:
This is all support. Sure you made a trite summary but it lost all my support. For all your conciseness you left out so much of the meat and support.

That would be an opinion without any support what so ever and is another example of my centeral point being that being concise usually means cutting out some support or making it harder to dig out.

Edit: and if your time is in such great demand that you can't bear to clearly state your reasoning then you shouldn't even be in a discussion.

I suggest that once again, you go and review my original post with its nice capslock enhanced point about time. I read everything you wrote, and I still think none of it applies, since its obvious that i didn't just randomly apply math. Math was used to support that capslocked statement, not the other way around. Quite frankly, i think you should go back and read my original statement. Using math wasn't a coverup in the slightest, as you seem to think it is, but its used as an explanation as to why the very simple proportionality that distance traveled and time turns out to be so powerful in destroying to so-called paradoxical statement.

In the end... is just about as frankly as i can say it, only that i didn't and someone else did.

Oh and one last thing Teachers don't re-answer the same question in class 8 times in a row, so i don't think i'm obliged to do the same.

That is one of my major points. Had you actually said that in the beging in addition to doing the math instead of just saying what you did then we wouldn't be discussing this. The burden is on you to show me exactly how and what you are thinking. It's not my job to get inside your head and deduce what you know or don't know. You can look at what you said and see it all in there because you wrote it. I can't because I didn't which is why I favour using more words and less math, which I've said.

The good ones do or they drag the kid aside later and hash it out or the student drags the teacher aside later and asks. No sane person would ask the same question over and over if they understood and were satisified the question. You are not obligated if you don't want but that just means I'll be left wondering what the hell you were talking about.

actually, the burden is on you to show that my statement isn't the same as yours at all. i don't have to prove anything, since according to me both of our statements say the exact same thing, which is what you don't agree with.

means you shouldn't fall asleep in class.

Do you even read half of what I type. I'm not disputing we said basically the same thing. (The fact you seem so hung up on it suggests a need for validation that you got the answer first and therefor must be better and more correct.) I wasn't even talking about our actual argument I was talking about your first statement. At that point it's your burden to fully explain everything your thinking and reasoning in a way anyone can make out. Which has been my entire point.

That's just plain mean. There are legit reasons why a student who is paying attention might not understand. I see it every time I attend a physics lecture. Sure I hate hearing the same thing seven ways because someone can't get their mind around it but it helps them and in the long run everyone else in class.

and i clearly was. What's your point?
and there are plenty of reasons why a student who isn't paying attention would try to ask lots of questions to get back to par with the class.