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Zeno's Paradox

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time -- which is to say, it can never be completed. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even be begun. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion


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really something to think about. if you want to find sources i got it from wikipedia.org.

DISCUSS and dont be shy about throwing in some other ones !!!

Haha thats kinda clever.

While he travelling say, the one sixteenth, he is also travelling the one eigth and the one quarter. So he is actually travelling ALL infinite of the finite distances at once, which doesnt make them infinite at all.... ?

hmm I'm pretty sure i just explained absolutely nothing... meh... hooray for the limit to infinity expression in maths I guess?

SOmeone give us some more, I'm bored lol.

naw, i'm typing this message aren't i?

thefillersweetcityjesus wrote:The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion

Only when restricted to those literal parameters...obviously it can't be the case.

Apply the same to the motion of, say...the electron...and The Universe ceases to exist.

Coma Divine wrote:

thefillersweetcityjesus wrote:The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion

Only when restricted to those literal parameters...obviously it can't be the case.

Apply the same to the motion of, say...the electron...and The Universe ceases to exist.

WHHHOOOOO!!!!!!! (Universe being destroyd noises)...(Small amount of time passing)....(Me putting on Devin)...'It's not so bad'.

It's not a finite number of distances. It's not very infinite, but it is infinite. Case closed.

EDIT: Damn I misread it.

OK, here's a couple of points from what I've gathered so far:

1) Zeno's argument essentially revolves around the concept of distance as being infinitely divisible and therefore infinite in its own right.

However, if motion is an illusion then it stands to reason that all ideas/concepts that are connected to/created by it must also be part of the illusion, for they are influenced by a false phenomenon. Zeno presents distance as being intrinsically linked to the concept of motion (as a yardstick by which any possible motion may be measured), using the former to discredit the latter.

However, if motion is illusory how can any perception of it - including those regarding distance - be reliable to prove anything? His logic doesn't convincingly portray distance as any kind of solid source of information - in fact, his stretching of the parameters of distance suggests the very opposite. Is he not therefore essentially undermining the validity of the very material on which he has based his claims? For if he's right, his entire argument is based on false ideas and is, arguably, itself illusory by its very nature.

2) Zeno infinitely divides the distance to the bus, and it is perfectly logical (In a purely semantic sense) that each of these divisions represent infinite distances. However, the fact remains that for these fractions to retain their logical integrity the first distance (i.e. that original distance from Homer to the bus) must remain constant: otherwise, the fractions lose all meaning.

After all, what is a fraction? A part of a clearly defined whole. If there is no whole, there is nothing left for the fraction to be a fraction OF, and it by definition ceases to be a fraction.

Zeno's paradox works on the assumption that becuse the fractions can be extrapolated infinitely, the distance itself extends into infinity. However, this idea doesn't take into account the basic nature of fractions, which require a set and constant amount to be part of. Zeno treats each fraction he divides from the original distance as if it were an individual stretch of distance in and of itself, when by definition they can only be seen as a slice of that very first distance, which has not actually changed - all that has changed is Zeno's perception of each fraction's relation to the whole, i.e. as somehow causing the length of the whole to somehow grow. This, however, clashes with the notion that the fractions are infinitely divisble, for to change the overall distance nayturally changes the nature of each fraction.

This also challenges his second problem, as claiming that being able to halve the first distance negates that first distance completely is putting the cart before the horse. Like I said before, fractions cannot exist without a whole to be part of, so to say that the fraction makes the whole impossible can only make the fraction itself impossible in turn, leaving us with nothing.

Overall, Zeno's argument, like most paradoxes, is an exercise in the most clinical of logic; however, it essentially quashes the very principles on which it builds itself, a paradox of logic eating itself...

...At least, that's what I think (Anyone still reading?) I'm glad Zeno found a job that suited him, 'cause he'd have made a really crappy ambulance driver:

RADIO: Vehicle 42, we have a cardiac arrest at 85 Elm St, please proceed-

ZENO: Sorry Base, I can't do that.

RADIO: Vehicle 42, please repeat your last statement.

ZENO: I said I can't go to Elm St, Base

RADIO: Vehicle 42, are you experiencing some kind of mechanical difficulty?

ZENO: Nope.

RADIO: Well, what's the problem then?

ZENO: Motion, that's the problem. It's impossible.

PAUSE.

RADIO: Sorry?

ZENO: Motion. It's impossible, you know.

RADIO: Vehicle 42, I'm sure I don't have to remind you that this is an emergency situation-

ZENO: Think about it. Elm St from here - it's what, 2 or 3 k's? Well, you can divide that couple of k's into an infinite amount of finite distances, which must take an infinite time to travel and can therefore never be completed...

RADIO: Vehicle 42, scrub that job. Fucker just died on us. Please return to Base for a good kicking.

RADIO CRACKLES OFF

ZENO: Can't do it...


Arf arf

Cav wrote:OK, here's a couple of points from what I've gathered so far:

1) Zeno's argument essentially revolves around the concept of distance as being infinitely divisible and therefore infinite in its own right.

However, if motion is an illusion then it stands to reason that all ideas/concepts that are connected to/created by it must also be part of the illusion, for they are influenced by a false phenomenon. Zeno presents distance as being intrinsically linked to the concept of motion (as a yardstick by which any possible motion may be measured), using the former to discredit the latter.

<SNIP>

Dude, chill. I think you're overanalysing this just a bit

The thing goes like this:
1. assume that the distance to the bus is finite.
2. Then it can be divided into an infinite number of smaller distances which must all be covered before you reach the bus.
3. The total distance of an infinite number of finite (and positive) distances is infinite.
4. Hence the total distance to the bus is infinite.

Clearly, 1 contradicts 4, and that's why it's called a paradox. It's just a thought experiment.

Anyway, there are two separate ways to resolve the problem. All mathematicians know that an infinite sum of positive terms can very well converge to a finite number provided the terms "become small enough", so step 3 above would be wrong.

Alternatively, space and hence distance could be quantised in some way, so that there exists a smallest possible unit of length (sometimes called the Planck length). If this is true, then the distance couldn't be divided into an infinite number of bits, but only a very large one. Once you have distances that are as short as the Plank length they can't be divided anymore - there is no half of a Planck length.

(I know that nobody really knows it the Planck length works like this, but it might. )

grrrv wrote:Dude, chill. I think you're overanalysing this just a bit

The thing goes like this:
1. assume that the distance to the bus is finite.
2. Then it can be divided into an infinite number of smaller distances which must all be covered before you reach the bus.
3. The total distance of an infinite number of finite (and positive) distances is infinite.
4. Hence the total distance to the bus is infinite.

Clearly, 1 contradicts 4, and that's why it's called a paradox. It's just a thought experiment.

Hey, I was just having fun with it. I love this stuff I understood that it was a paradox, but I was just joining in on the experiment. After all, the fun isn't that it's contradictory - the fun lies in working out why it's contradictory, right? Wouldn't want you thinking this was something I was getting worked up over.

I wouldn't say I was overanalysing it, in fact what you put down is basically pretty similar to what I did (You put it a bit more succinctly, mind: not knowing much about physics or mathematics, I just have to work out how to put this stuff best myself, hence the rambling!)

I like chocolate.

JuZ wrote:I like chocolate.

Now there's some thinking I can get behind

Cav wrote:Hey, I was just having fun with it. I love this stuff I understood that it was a paradox, but I was just joining in on the experiment. After all, the fun isn't that it's contradictory - the fun lies in working out why it's contradictory, right? Wouldn't want you thinking this was something I was getting worked up over.

I wouldn't say I was overanalysing it, in fact what you put down is basically pretty similar to what I did (You put it a bit more succinctly, mind: not knowing much about physics or mathematics, I just have to work out how to put this stuff best myself, hence the rambling!)

Yeah, no problem. From your first message I got the impression that you were attacking the paradox a bit too aggressively. Too aggressively for your own well-being, I mean I guess I just misunderstood the tone of it.

Cav wrote:

JuZ wrote:I like chocolate.

Now there's some thinking I can get behind

I agree. In fact I'll eat some right now.

grrrv wrote:

Cav wrote:Hey, I was just having fun with it. I love this stuff I understood that it was a paradox, but I was just joining in on the experiment. After all, the fun isn't that it's contradictory - the fun lies in working out why it's contradictory, right? Wouldn't want you thinking this was something I was getting worked up over.

I wouldn't say I was overanalysing it, in fact what you put down is basically pretty similar to what I did (You put it a bit more succinctly, mind: not knowing much about physics or mathematics, I just have to work out how to put this stuff best myself, hence the rambling!)

Yeah, no problem. From your first message I got the impression that you were attacking the paradox a bit too aggressively. Too aggressively for your own well-being, I mean I guess I just misunderstood the tone of it.

Nah, that's cool. It was just enthusiasm

Then, according to Zeno, a gun would never kill anyone, since the bullet has to travel half the distance towards its target, and a quarter of the distance....etc

By the way, he should have realised his distance theory was crap when he walked towards his desk to write it down

Ah, brain exercise!

Take ONE

The impossibility of motion is created if and only if the person about to move counts all the divisions in real time. As if he must get aquainted with the procedure, y'know, like "Hmm, so before I reach that point I must reach that one and before that one this one and before this one ..." Since there is no indivisible distance, he actually never moves, he is still counting, thus Zeno is right and there is NO PARADOX. If you see the man move, you must be imagining things because he's still there looking for the point to move first to.

*********

Take TWO

Zeno and I are standing two metres apart.

Zeno: You can't come closer.
Me: How come?
Zeno: *tells his story*
Me: Hmm, interesting. Let's see ...
Zeno: Wait! What are you doing?
Me: I'm coming over.
Zeno: But you can't!
Me: OK, we'll see ...
Zeno: No no no, I told you, before you come here, you must come half way and th...
Me: I'm coming half way too.
Zeno: But wait! Before that you must come half of that.
Me: Yes, no problem.
Zeno: Way problem. You can't even start, man!
Me: How's that again?
Zeno: Ok, tell me which is the first point you are going to reach?
Me: What? I don't know ... the one nearest to me I guess ...
Zeno: I can always prove to you that the point you think you'll reach first is actually not the first you'll reach.
Me: And?
Zeno: So you'll never reach ANY point, silly.
Me: Eeee, wait, I'll reach that one you say it's the ... first ... the one before my first ...
Zeno: That won't be the first one either.
Me: Wait, there's gotta be a point to start!
Zeno: Sorry, no point.
Me: *stares at Zeno*
Zeno: *stares at me*
Time: *passes*
Tension: *increases*
Is our hero beaten? He still does not move! Zeno proved his point by making him THINK instead of ACT! Zeno successfully stopped every attempt of our hero to move and now ...
Me: Aha! Gotcha! You say that I can't move because there is no firts point that could be reached because between me and the point is always the distance which never becomes the minimal constant, for it decreases into infinity?
Zeno: You can put it that way, yes.
Me: Ok, check this! I won't be reaching for any point, I won't be travelling any distance because the contact between me and everything that surrounds me HAS NO DISTANCE. The point where I end and the environment begins has no spacial value. I can't be measured, it is only a place where two points touch each other.
Zeno: Yeees?
Me: So my motion, this motion you see here *moves* ...
Zeno: AAAAAAAAAAHH!!! *falls to the ground*
Me: ... is created by pushing this contact point - which is no point at all - through space in the direction I want. So I actually don't reach any first point, because this first point is ALWAYS AHEAD OF ME as I'm pushing ME through space.
Zeno: Fuck!
Me: Not only that motion is possible, you don't even need to pass the distance to move. Everything is in motion all the time, there is no way of telling the difference between stillness and motion.

**********

Take THREE

The illusion of motion can be perceived as an illusion only by those who are not subject to that particular illusion. We, Zeno included, obviously ARE subject to this illusion of motion, which makes this illusion a NECESSARY illusion. There is no paradox! Zeno is right, all motion is illusion, but without this illusion ...

**********

Take FOUR

There is a precise moment which enables Zeno to fuck with our heads: the link between impossibility and illusion (with a touch of people's narcissism). Zeno's point is: what we perceive as motion is an illusion. His reasoning goes: "If motion is impossible and there de facto IS motion, this motion can only be an illusionary motion." OK! This holds water, people! There is no logical paradox! YET! It comes when WE make the unjustified link BACK, FROM the illusion TO impossibility. Only when we think "If I can move NOW *waves*, but Zeno proved that this very motion of mine is IMPOSSIBLE, something's not right!", we have a paradox. But the funny thing is, Zeno never said there can be no motion at all! His underlying premise is: the impossibility of motion DOESN'T mean there is NO motion. There IS motion, an illusionary one, but motion nonetheless.

So our poor narcissistic self is having a hard time accepting that his praised existence can be easily reduced to a mere illusion, and rather transposes that into a paradox, a mindplay useful for keeping the truely depressing thoughts of "Am I deceived?" away.