NationStates Jolt Archive

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

Although there are an infinite number of intervals these intervals become infinitely small over time so there is no paradox. This is a simplified explanation as I can't remember the formula for infinite series which tend to a finite limit.

I bet my Dad that I could disprove the Zeno's paradox

what, just the one?

For some reason, profs never accept Xeno's paradox as a legitimate reason for not handing an assignment in on time...

Although there are an infinite number of intervals these intervals become infinitely small over time so there is no paradox.

so you can do an infinite number of things, provided they are very small?

Although there are an infinite number of intervals these intervals become infinitely small over time so there is no paradox. This is a simplified explanation as I can't remember the formula for infinite series which tend to a finite limit.

But going on a purely mathematical basis, each distance can always be divided, and so it will always take some time to get from the midpoint to the next midpoint, no matter how small. So it seems impossible mathematically.

Zeno posed many paradoxes, this is known as Achilles and the Tortoise.

It is very much possible to travel from A to B. Anyone doing so will complete the infinitely many intervals (an interval being the space between each ‘midpoint’) in a finite time because each successive interval is smaller than the one before it, and thus can be traveled across more quickly than the one that precedes it.

Why? Well, you have to get a bit mathematical here, and define the sum of an infinite series as the limit to which the sequence of its successive partial sums converges.

So, if A to B is 1 mile, then the intervals you traverse are, in fractions of a mile, 1/2. 1/4, 1/8, 1/16, and so on. The partial sums are:

1/2
1/2 + 1/4 = 3/4
1/2 + 1/4 + 1/8 = 7/8
1/2 + 1/4 + 1/8 + 1/16 = 15/16 and so on.

The sequence of partial sums goes:

1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, 511/512,

and so on, for ever, getting closer an closer to 1. 1 is the limit, and the sum, of the series.

Nevertheless, you can still point out that you never quite to B (1 mile), since there is no last traversal of an interval; no last bit to get to 1 mile. However, as you must be somewhere, you must be at B. You cannot be short of 1 mile because you would have only traveled over a finite series of intervals, and would have more intervals (infinitely more, in fact) to make. Moreover you cannot be beyond 1 mile because there is no interval from B to any point beyond B which is covered by a traversal.

So you must have traveled the 1 mile between A and B.

For some reason, profs never accept Xeno's paradox as a legitimate reason for not handing an assignment in on time...
The reason being, of course, that the time horizon in question has been successfully passed. >_>;

Here's a question though: is the disproof of this paradox evidence of a finite and discrete packet of time?

simple don't use math:

getting to point B is always an objective and as we know having aims and goals is the human condition thus as a end of all equation, point B is invalid as is the question itself

Alternatively…

Point B is subject to the laws of odds and as such if you are infinitely travelling to point B the point will in fact travel to you!

yeah I gave up :(

But going on a purely mathematical basis, each distance can always be divided, and so it will always take some time to get from the midpoint to the next midpoint, no matter how small. So it seems impossible mathematically.

Practically, this means:
Imagine "0" and "1". These two are absolute numbers, which do definitely exist. Now, although the difference between them is "1", it contains an infinite number of numbers: 0.1, 0.2, 0.21, 0.213, 0.2134, 0.912354325... and so on. Yet, the difference is clearly not infinte, although containing infinite many numbers, since an infinite number of these infinite numbers is infinitely small, since they are infinite divisions of "1".
I think that is what this is supposed to work like.

so you can do an infinite number of things, provided they are very small?
That is true, except for "very small" - they must be infinitely small - as small as those infinite numbers i mentioned above. Imagine any number divided by "0". "Nothing" fits into anything an infinte amount of times. In theory, this is practical, yet, in rl, of course, such does not exist. And it is not possible to do that with "0" amount of time, when the things you want to do take longer than "0". Infinitely small is not possible here, it must be "0".

Practically, this means:
Imagine "0" and "1". These two are absolute numbers, which do definitely exist. Now, although the difference between them is "1", it contains an infinite number of numbers: 0.1, 0.2, 0.21, 0.213, 0.2134, 0.912354325... and so on. Yet, the difference is clearly not indefinte, since an infinte number of these infinite numbers is infinitely small, since they are infinite divisions of "1".
I think that is what this is supposed to work like.

It also kind of links to the idea that the distance between 0.99999.... and 1 is infinately small, = 0

Now I know there is a mathematical solution for this involving using sigma notation or something.

Just add time to the equation. The flaw in this paradox is that it talks about distances and time, but only takes distance into account.

EDIT: like this:

It is very much possible to travel from A to B. Anyone doing so will complete the infinitely many intervals (an interval being the space between each ‘midpoint’) in a finite time because each successive interval is smaller than the one before it, and thus can be traveled across more quickly than the one that precedes it.

But going on a purely mathematical basis, each distance can always be divided, and so it will always take some time to get from the midpoint to the next midpoint, no matter how small. So it seems impossible mathematically.

Nope. You wil get a 0/0 or an inf/inf.

Here's a question though: is the disproof of this paradox evidence of a finite and discrete packet of time?

and if so, how does that hold up against the arrow paradox?

Couldn't you just write a big A and a big B on a couple of bits of paper, put one at each end of the garden and video yourself walking from A to B? :p

Couldn’t you just write a big A and a big B on a couple of bits of paper, put one at each end of the garden and video yourself walking from A to B? :p
Well, yes you could, but that wouldn’t solve the problem. Just as letting go of a ball and watching it fall to the ground doesn’t explain how gravity works.

Couldn't you just write a big A and a big B on a couple of bits of paper, put one at each end of the garden and video yourself walking from A to B? :p

zeno's the point is that that would just be an illusion.

in fact, motion itself is impossible, as at any particular instant 'now' an allegedly moving object would be at rest. at an instantaneous slice of time, now, an object allegedly in motion occupies the exact same amount of space as it would when at rest, and is just sitting there exactly as if it were at rest (though possibly hanging in the air or whatever) - motion does not happen in an instant. and if time is just a series of nows, then there cannot be any motion, as all objects are at rest in each of those instantaneous nows individually.

It reminds me somewhat of the part of the Hitchhiker's Guide to the Galaxy...

The Universe (Some information to help you live in it.)
Size: Infinite.
Imports: None. (You cannot import things into an infinite space, there being no outside to import things from.)
Exports: None. (See Imports.)
Rainfall: None. (Rain cannot fall because in an infinite space there is no up for it to fall down from.)
Population: None. (It is known that there are an infinite number of worlds, however there is only a finite number of inhabited worlds. Since any number divided by infinity is as near to nothing as to make no difference, you can safely say that anyone you might meet on your travels are just a figment of a deranged imagination.)
Currency: None. (Actually there are three exchangeable currencies, but the Altarian Dollar has recently collapsed, the Flanian Pobble Bead is only exchangeable for other Flanian Pobble Beads and the Triganic Pu doesn't really count as money. It's exchange rate of six Ningis to one Pu is simple enough, but as a Ningi is a triangular rubber coin six thousand eight hundred miles along each side, no one has ever collected enough to own one Pu. Ningis are not negotiable currency because the Galactic banks refuse to deal in fiddling small change. From this is it possible to surmise that the Galactic banks are also a figment of a deranged imagination.)

I'm scared by how much of that I remembered from memory... I'm going to go for a lay down now. XD

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't.

*points and laughs*
Hahahahahahahahahahahahahahahahahahahahhahahahahahaha.


Ha.

It also kind of links to the idea that the distance between 0.99999.... and 1 is infinately small, = 0

How'd you get to .999...?

Just add time to the equation. The flaw in this paradox is that it talks about distances and time, but only takes distance into account.

It doesn't? Presumably you're moving at a certain rate, distance per time. You go an infinite number of non-zero distances. Each distance, being non-zero, takes some amount of time to traverse...

The nth term for the series is (2**n)-1/2**n where ** indicates to the power of.

The nth term for the series is (2**n)-1/2**n where ** indicates to the power of.

Personally, I like ^.

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

There's a simple way to disprove this. Set up a point A and a point B, and walk from the first to the second. Presto.

Personally, I like ^.

I considered ut, I realy did. But I've been doing some crypto stuff lately and ^ is the operator for bitwise exclusive or (xor).

Zeno's paradox requires that you can only travel half the distance from your position to point B, which, of course, is easily disproved.

Besides, wasn't Zeno a Roman emperor that got his armies whipped by the barbarian hordes in their destructive, pillaging sweep across Europe?

Besides, wasn’t Zeno a Roman emperor that got his armies whipped by the barbarian hordes in their destructive, pillaging sweep across Europe?
Nope, he was a pre-socratic Greek philosopher, living from about 490-420 BCE.

Zeno's paradox requires that you can only travel half the distance from your position to point B, which, of course, is easily disproved.

Besides, wasn't Zeno a Roman emperor that got his armies whipped by the barbarian hordes in their destructive, pillaging sweep across Europe?

Nero?

There's a simple way to disprove this. Set up a point A and a point B, and walk from the first to the second. Presto.

except that according to zeno, "all is one" and our senses deceive us, and you would therefore have proven nothing.

Nero?

Oh well. They rhyme, at least.

Zeno's paradox is solvable.

An odd way to think about is if you consider the first term you have 0.5 and are half way to 1. Now add a half of a half (1/4) and you will have 3/4 (1 minus 1/4).

Take this to the extreme. If you get to the nth term then what you are adding will be infintely small hence the difference between the sum of your series and 2 is infintely small. Hence the sum of your series is equal to 2.

Sorry, I can't remember the actual maths, but that's the reasoning behind it as I remember.

Zeno's paradox is solvable.

An odd way to think about is if you consider the first term you have 0.5 and are half way to 1. Now add a half of a half (1/4) and you will have 3/4 (1 minus 1/4).

Take this to the extreme. If you get to the nth term then what you are adding will be infintely small hence the difference between the sum of your series and 2 is infintely small. Hence the sum of your series is equal to 2.

So, 1 over 2 to the infinite power is zero, you're saying?

So, 1 over 2 to the infinite power is zero, you're saying?

No i think he is saying the distance between 1 and 1/2^infinity is 0.

For some reason, profs never accept Xeno's paradox as a legitimate reason for not handing an assignment in on time...

I called into work once that I couldn't come in because of Xeno's paradox.

Actually got away with it, too...

No i think he is saying the distance between 1 and 1/2^infinity is 0.

Nah, the distance remaining is always 1/2^n. He'd said at the nth term (I assume he meant infinity, not n) the distance remaining was infinitely small, and thus zero.

i wud say that

the very fact you said it takes X amount of time to travel half the distance implies that it takes X amount of time to travel the other half so it takes 2X to travel the whole, u then just kept dividing X by 2 and produced infinite numbers, but when the infinite numbers are added together they produce a finite number

its like gravitational potential and measuring the energy taken to throw an object to infinity and it works out that a finite amount of energy can push an object infinite distance away from anuva object

i wud say that

the very fact you said it takes X amount of time to travel half the distance implies that it takes X amount of time to travel the other half so it takes 2X to travel the whole

who said anything of the sort?

Zeno's paradox is solvable.

An odd way to think about is if you consider the first term you have 0.5 and are half way to 1. Now add a half of a half (1/4) and you will have 3/4 (1 minus 1/4).

Take this to the extreme. If you get to the nth term then what you are adding will be infintely small hence the difference between the sum of your series and 2 is infintely small. Hence the sum of your series is equal to 2.

this slightly misses the point. that "last" infinitely small step isn't merely the last step you have to take to get there (not that there is a last one anyways...). there are infinitely many points you must first move to before you can even have gone 1/100000000000 of the distance. in order to get from point A to point B, our traveler must make an infinite number of journeys. but one clearly cannot do an infinite number of things in a finite amount of time. therefore they don't move at all.

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

I know the answer. Trust me on this, I'm in college.

There is definitely a way to express it in sigma notation. Let the finite distance equal 1. For simplicity's sake, your starting point (point A) will be 0 and the destination (point B) will be 1.

Therefore, the first distance you travel will be 1/2. The second distance 1/4, the third 1/8, and so forth. The nth distance is 1/(2^n), or (1/2)^n.

In a summation, you might express it as the summation of 1 to infinity of (1/2)^n.

Here is the formula for a geometric series summation, courtesy of wikipedia:

http://upload.wikimedia.org/math/e/d/b/edb842841d4c9135e28f583706204f3d.png

Now, the upper and lower bounds must be integers; so it is possible to work backwards by manipulating this formula to match our conditions. Specifically, if we let m equal one and n equal infinity, then the left-hand side of the equation will match what we want to solve.

Substitute 1/2 for x, and througth algebra, the right-hand side can be reduced to (infinity is represented by 'oo'):

= [(1/2)^(oo + 1) - (1/2)^1] / (1/2 - 1)

= [(1/2)*(1/2)^(oo) - 1/2] / (-1/2)

= -(1/2)^(oo) + 1

Which is simply: 1 - (1/2)^(oo)

Now, all that is left to do is take the limit of the this expression as an arbitrary variable (call it z) approaches infinity.

limit as z->oo of: 1 - (1/2)^z

The second term approaches zero, leaving you with a total distance traveled = 1, which is also the distance between point A and point B ( 1-0 = 1 ).

This assumes that in the specific problem you described, your velocity, as you traverse the distance, remains constant. You stress that the amount of time will be infinite; and this would be the case, if you were slowing down at a rate proportional to the intervals. But if you are moving with a constant speed, for example, 1 unit per second, the total time taken will simply be one second.

Another solution to this paradox is the reality that space is not infinitely divisible, i.e. there is a point far short of infinity that it is no longer possible to travel an even shorter distance.

QED, thank you very much.

*bows*

So, 1 over 2 to the infinite power is zero, you're saying?Damn, I knew I wasn't right...

this slightly misses the point. that "last" infinitely small step isn't merely the last step you have to take to get there (not that there is a last one anyways...). there are infinitely many points you must first move to before you can even have gone 1/100000000000 of the distance. in order to get from point A to point B, our traveler must make an infinite number of journeys. but one clearly cannot do an infinite number of things in a finite amount of time. therefore they don't move at all.Yes, but if you are considered to be moving at a constant speed then then each step will take smaller amount of time.

Damn, I knew I wasn't right...

No, you are correct. It is indeed zero. Strictly speaking, something to the infinity power is not a number, but the limit of (1/2)^n as n approaches infinity is zero.

actually yes I am. As n -> oo 1/2n -> 0. Hence at the infinite step 1/2n = 0.

edit: wait.. Ginnoria's proof seems to back up what I said. However it is a rather limited proof. As in it takes a formula which implies the answer. bleh... (if we set the answer to be 1 then the answer will be 1...)

I got... I remember the proof. at least I think I do.

If the series is 1, 1/2, 1/4, 1/8 ... (approaches 2)
Then the summation of the series is 1 + 1/2 +1/4 + 1/8 ... + 1/[2^(n-1)] + 1/(2^n)
Which can be stated as (1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^(n-1) + (1/2)^n
If we times this summation by (1-1/2) then we get

(1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^(n-1) + (1/2)^n
----[(1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^n + (1/2)^(n+1)]
Which is equal to - (1/2)^0 - (1/2)^(n+1)

Now we divide by (1-1/2) to get back to the original summation
(1/2)^0 - (1/2)^(n+1)
= -------------------------
(1 - 1/2)

1 - (1/2)^(n+1)
= -------------------------
1/2

= 2 - (1/2)^n

= 2 - 0

= 2

There. Proof from first principles. Unless you want me to go into why 1/(x^n) tends to 0 where n=oo and x is > |1|

Zeno's paradox requires that you can only travel half the distance from your position to point B, which, of course, is easily disproved.

Besides, wasn't Zeno a Roman emperor that got his armies whipped by the barbarian hordes in their destructive, pillaging sweep across Europe?

Flavius Zeno (c. 425 -491), original name Tarasicodissa or Trascalissaeus, Eastern Roman Emperor (February 9, 474 - April 9, 491) was one of the more prominent of the early Byzantine emperors. Domestic revolts and religious dissension plagued his reign which nevertheless succeeded to some extent in foreign issues. He presided over the official end of the Roman Empire in the west under Julius Nepos and Romulus Augustus, while at the same time contributing much to stabilizing the empire in the east. He was thus the first emperor of the eastern Roman empire to rule without a western counterpart, making him the first emperor of a fully independent Byzantine Empire :p

Here's the thing. Zeno assumes that you travel exactly half of the distance to B at once, then stop, then move half of the remaining distance to B, then stop...Nobody travels like this, because to do so means that you can never reach your destination. Most people travel at a set pace. Even if I travel 1/5 of the distance between A and B and stop, then travel 1/5 of the distance between A and B, etc, I will reach B in 5 stops.

In Zeno's paradox, you never reach your destination because you are, every time you stop, decreasing the distance that you travel, unto an infinitely small distance.

Again, nobody travels in this manner. One would, in fact, find it incredibly difficult to do so once the distances involved became smaller than the diameter of, say, a hydrogen atom. At some point, your vessel/leg has a minimum distance that it is possible to intentionally travel, which will be greater than half the remaining distance to B, and you will reach B.

Then, your head will explode.

My favourite one of Zeno's paradoxes is the one about the arrow in mid flight.

At any point in time the arrow is filling a space exactly the same size as itself, and is therefore absolutely still. Thus at any point in its flight the arrow is not moving.

actually yes I am. As n -> oo 1/2n -> 0. Hence at the infinite step 1/2n = 0.

edit: wait.. Ginnoria's proof seems to back up what I said. However it is a rather limited proof. As in it takes a formula which implies the answer. bleh... (if we set the answer to be 1 then the answer will be 1...)

I got... I remember the proof. at least I think I do.

If the series is 1, 1/2, 1/4, 1/8 ... (approaches 2)
Then the summation of the series is 1 + 1/2 +1/4 + 1/8 ... + 1/[2^(n-1)] + 1/(2^n)
Which can be stated as (1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^(n-1) + (1/2)^n
If we times this summation by (1-1/2) then we get

(1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^(n-1) + (1/2)^n
----[(1/2)^1 + (1/2)^2 + (1/2)^3 ... (1/2)^n + (1/2)^(n+1)]
Which is equal to - (1/2)^0 - (1/2)^(n+1)

Now we divide by (1-1/2) to get back to the original summation
(1/2)^0 - (1/2)^(n+1)
= -------------------------
(1 - 1/2)

1 - (1/2)^(n+1)
= -------------------------
1/2

= 2 - (1/2)^n

= 2 - 0

= 2

There. Proof from first principles. Unless you want me to go into why 1/(x^n) tends to 0 where n=oo and x is > |1|

Psh. Mine looks prettier. You use the same formula, just in a different way.

Our answers appear to differ by 1, but that's just because you are beginning your summation at 0, which is fine, as long as it's clear that B = 2.

But anyway, good work. Take that, Zeno. :cool:

Psh. Mine looks prettier. You use the same formula, just in a different way.

Our answers appear to differ by 1, but that's just because you are beginning your summation at 0, which is fine, as long as it's clear that B = 2.

But anyway, good work. Take that, Zeno. :cool:Yes, but I prove that the formula works, whereas you just assume that it works. :p

Yes, I wrote in the post that the series approaches 2.

Yes, I think we annilated Zeno, but to be fair we had 1500 years of knowledge to work with. Whereas Zeno was just an idiot.

Maths is so like a language. If you don't use it you forget it, and fast. Once you start using it again it slowly comes back. At the beginning of this thread I was 'duh... what's the name for that weird 'E' that means you add all the numbers up?'

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

Unless I'm mistaken in my understanding of what you wrote, I can prove it wrong right now.

I'm currently about to stand up.

I just stood. Now I'm going to walk across the room. My destination was the wall approximately five feet away from me, and as I left point A (My computer chair), I walked to point B (The Wall). I touched the wall, and returned to my seat. Therefore, I did go to my waypoint, and ultimately reached point "B".

Yes, but I prove that the formula works, whereas you just assume that it works. :p

Yes, I wrote in the post that the series approaches 2.

Yes, I think we annilated Zeno, but to be fair we had 1500 years of knowledge to work with. Whereas Zeno was just an idiot.

Maths is so like a language. If you don't use it you forget it, and fast. Once you start using it again it slowly comes back. At the beginning of this thread I was 'duh... what's the name for that weird 'E' that means you add all the numbers up?'

That's why I hate proofs. Why reinvent the wheel, someone else proved that formula works so I don't have to.

My favourite one of Zeno's paradoxes is the one about the arrow in mid flight.

mine too

In Zeno's paradox, you never reach your destination because you are, every time you stop, decreasing the distance that you travel, unto an infinitely small distance.

Again, nobody travels in this manner. One would, in fact, find it incredibly difficult to do so once the distances involved became smaller than the diameter of, say, a hydrogen atom. At some point, your vessel/leg has a minimum distance that it is possible to intentionally travel, which will be greater than half the remaining distance to B, and you will reach B.

So in order to solve Zeno's poser you suggest a quantum structure of space and instantaneous travel between neighbouring quantum locations? I believe a quantum theory of time is also required for your cunning ploy to work. Sledgehammer to crack a nut there, pal.

Another solution to this paradox is the reality that space is not infinitely divisible

this one actually more closely addresses what is being got at with the paradox. of course, if true then you might have some interesting metaphysical and mathematical implications forced on you. it certainly hasn't been the historically intuitive notion.

additionally, if you hold the same for time (which you have to in order to get out this way from other variations) that falls right in to his arrow paradox. what zeno is up to is showing that whether you think that space and time are either infinitely divisible or made up of discrete segments, motion/change is impossible.

That's why I hate proofs. Why reinvent the wheel, someone else proved that formula works so I don't have to.It demonstrates understanding. Anyone can find the right formula for the situation and then plug numbers in. It takes intelligence to understand the mechanics behind the formula. Could you prove sin^2x + cos^2x = 1?

This is the kind of stuff you'd have to take hallucinigenics to understand. But then, you'd start to understand the Wondertwins. Can't ever understand Aquaman no matter how many funny pills you take. He's that messed up. Isn't that like the "Y=1/X" graph? No matter how hard you work at it, the graph never touches "Y=0". Complex and mind boggling, like finding out how Wonderwoman flies her invisible jet or how Batman can type things into his Batcomputer when the Batbuttons aren't Batmarked at Batall. I wonder how many times Aquaman has accidentally nuked Canada just by trying to balance his checkbook on the Batcomputer.

Anyway, the concept of infiniti is mindboggling. Not used to the boundless and limitless, we can not understand what we can't imagine. Since imagination is restricted by brain power, we can never percieve infiniti. It's like how time is theoretically infinite. No matter when it is, there was always a past and there will always be a future. Something had to always exist because, according to scientific law, something can never come from nothing. In the grand scheme of things, beginning and end are impossible outside the specific. My brain is hurting now. Me go away and not think until brain go not hurt no anymore.

There is a similar problem that involves two cars traveling down the road. The car in back starts to pass but will never be able to because you can't find the exact point where the cars will be even. If they are never even then obviously the passing car will never get around the slower car.

There is a similar problem that involves two cars traveling down the road. The car in back starts to pass but will never be able to because you can't find the exact point where the cars will be even. If they are never even then obviously the passing car will never get around the slower car.

achilles and the tortoise, with the names changed

It demonstrates understanding. Anyone can find the right formula for the situation and then plug numbers in. It takes intelligence to understand the mechanics behind the formula. Could you prove sin^2x + cos^2x = 1?

Of course.

sin^2x = 1/2 - (1/2)*cos 2x
sin^2x = 1/2 + (1/2)*cos 2x

1/2 - (1/2)*cos 2x + 1/2 + (1/2)*cos 2x = 1
[(1/2)*cos 2x - (1/2)*cos 2x] + 1/2 + 1/2 = 1
0 + 1/2 + 1/2 = 1

I shouldn't have to prove theorems that I use to prove something else. Saying "I used these 40 theorems to prove this mathematical principle, but I ALSO proved each of the 40 theorems I used" doesn't make your proof any more complete than if you didn't prove those theorems.

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

Simply. Your destination is not Point B. It is a point beyond Point B. You pass through Point B as you are getting to this third point, and stop there.

Mwa. I win.

Didn't the mods tell you not to ask help on your homework here?


Because that's what this sounds like.

Of course.

sin^2x = 1/2 - (1/2)*cos 2x
sin^2x = 1/2 + (1/2)*cos 2x etc.

And can you also prove it in a way that shows you understand why it is one ? Not just the formulas, but the meaning behind them ?

And can you also prove it in a way that shows you understand why it is one ? Not just the formulas, but the meaning behind them ?Yeah... I thought the proof had something to do with integration and how cosx = sin(x + π/2)

Nope, I was wrong. It has something to do with the unit circle
http://library.thinkquest.org/C0121962/identi1.gif

Pythagoras' theorm states that a^2 + b^2 ≡ c^2

Definitions
Sine = opposite/hypotenuse
Cosine = adjacent/hypotenuse

Using the triangle OBP
edit: x is the angle POB

OP = 1 (the definition of a unit circle r=1)

Sin(x) = BP/hypotenuse
Sin(x) = OA/hypotenuse
Sin(x) = OA

Cos(x) = OB/hypotenuse
Cos(x) = OB

OA^2 + OB^2 ≡ OP^2
Thus
Sin^2(x) + Cos^2(x) ≡ 1

And can you also prove it in a way that shows you understand why it is one ? Not just the formulas, but the meaning behind them ?

Well, the only way I know how to do that explicitly is by showing that it is derived from the Pythagorean Theorem.

cos^2(x) + sin^2(x) = 1

For any right triangle, a^2 + b^2 = c^2. Also, cos(x)*c (where x is the angle opposite b) = a, and sin(x)*c = a. The hypontenuse is c, and a and b are the sides.

Put them into the Pythagorean Theorem, and you have (cos(x)*c)^2 + (sin(x)*c)^2 = c^2. Then divide by c^2 and you have that expression.

I suppose next that you will ask me to prove the Pythagorean Theorem, and next explain each basic tenet of trigonometry, and why 1 + 1 = 2. Who cares? It has nothing to do with showing that Zeno's paradox is false. The distance traveled is expressed by a summation; the summation is solved with a formula. Who cares where the formula comes from? I'm not interested in making new theorems, if I was I'd be taking Theoretical Mathematics 306, which I'm not, and don't care to. This was a problem, I solved it, I shouldn't have to explain why mathematics works in the process.

Well, the only way I know how to do that explicitly is by showing that it is derived from the Pythagorean Theorem.

cos^2(x) + sin^2(x) = 1

For any right triangle, a^2 + b^2 = c^2. Also, cos(x)*c (where x is the angle opposite b) = a, and sin(x)*c = a. The hypontenuse is c, and a and b are the sides.

Put them into the Pythagorean Theorem, and you have (cos(x)*c)^2 + (sin(x)*c)^2 = c^2. Then divide by c^2 and you have that expression.

I suppose next that you will ask me to prove the Pythagorean Theorem, and next explain each basic tenet of trigonometry, and why 1 + 1 = 2. Who cares? It has nothing to do with showing that Zeno's paradox is false. The distance traveled is expressed by a summation; the summation is solved with a formula. Who cares where the formula comes from? I'm not interested in making new theorems, if I was I'd be taking Theoretical Mathematics 306, which I'm not, and don't care to. This was a problem, I solved it, I shouldn't have to explain why mathematics works in the process.
But the proof for Pythagoras' Theorm is cool. Especially how it can be done so many ways.

But the proof for Pythagoras' Theorm is cool. Especially how it can be done so many ways.

I do admit, the thing with the folding piece of paper is pretty cool. ;)

Didn't the mods tell you not to ask help on your homework here?

Because that's what this sounds like.

that'd be fucking ridiculous homework. on par with "solve the problem of free will"

Well, the only way I know how to do that explicitly is by showing that it is derived from the Pythagorean Theorem.

Nope - the unit circle explanation was quite nice :)

You can however also prove it mathematically by using the exp (i x) notation. But that is not understanding; that is using the definition of a sine.

Here's the thing. Zeno assumes that you travel exactly half of the distance to B at once, then stop, then move half of the remaining distance to B, then stop...Nobody travels like this, because to do so means that you can never reach your destination. Most people travel at a set pace. Even if I travel 1/5 of the distance between A and B and stop, then travel 1/5 of the distance between A and B, etc, I will reach B in 5 stops.

In Zeno's paradox, you never reach your destination because you are, every time you stop, decreasing the distance that you travel, unto an infinitely small distance.

Again, nobody travels in this manner. One would, in fact, find it incredibly difficult to do so once the distances involved became smaller than the diameter of, say, a hydrogen atom. At some point, your vessel/leg has a minimum distance that it is possible to intentionally travel, which will be greater than half the remaining distance to B, and you will reach B.

Then, your head will explode.

...That's, not quite what it means, I imagine...

...That's, not quite what it means, I imagine...

but strawmen are so much easier to deal with...

Hmm, how did my thread get to Pythagorus?

(and no i can't be bothered to read through the thread myself :p )

Bleh, damned Zeno.

Besides, what does it matter if you never get to your destination? You get infinitely close, so it's all good.

The Hotel Infinite is more fun anyway.

Bleh, damned Zeno.

Besides, what does it matter if you never get to your destination? You get infinitely close, so it's all good.

The Hotel Infinite is more fun anyway.I wish Zeno's theorm worked for time. Then I'd have an excuse for never having to hand in my essays. :D

I wish Zeno's theorm worked for time. Then I'd have an excuse for never having to hand in my essays. :D

I think the problem with Zeno's paradox is that he never finished it. He got infinitely close though.

I think the problem with Zeno's paradox is that he never finished it. He got infinitely close though.I think this is almost as bad as the time when I laughed at the chemistry joke 'If you're not part of the solution you're part of the precipitate'. I shall remove myself from the genepool shortly.

Besides, what does it matter if you never get to your destination? You get infinitely close, so it's all good.

no you don't. you don't even get to start. you don't get halfway and halfway again. the paradox is that in order to go halfway in the first place, you must go halfway to there first. and in order to get to that quarter mark, you must again first go halfway. and so on to infinity. but if it goes to infinity, then there is never a first move that is possible to make. and therefore motion is logically impossible.

no you don't. you don't even get to start. you don't get halfway and halfway again. the paradox is that in order to go halfway in the first place, you must go halfway to there first. and in order to get to that quarter mark, you must again first go halfway. and so on to infinity. but if it goes to infinity, then there is never a first move that is possible to make. and therefore motion is logically impossible.

You know what, I've never actually thought about it that way.

I wish Zeno's theorm worked for time.

they do. or at least, they address similar issues with time too. and even if they didn't, we could easily think up some variations that would.

You know what, I've never actually thought about it that way.

well, people express it the other way because his paradox of achilles and the tortoise does go that direction, and it makes it look as if the mathematical 'solution' actually addresses the problem. which is convenient, as then the paradox looks resolved. but it misses what zeno was getting at.

no you don't. you don't even get to start. you don't get halfway and halfway again. the paradox is that in order to go halfway in the first place, you must go halfway to there first. and in order to get to that quarter mark, you must again first go halfway. and so on to infinity. but if it goes to infinity, then there is never a first move that is possible to make. and therefore motion is logically impossible.

There are no words for how much I wish I could punch Zeno for the confusion he's caused me.

There are no words for how much I wish I could punch Zeno for the confusion he's caused me.

Luckily, your fist can never go to his face.

Luckily, your fist can never go to his face.

any sensation of pain to the contrary must just be in his imagination

any sensation of pain to the contrary must just be in his imagination

And if he falls over he'll be content in the fact that he'll never hit the ground.

Nope - the unit circle explanation was quite nice :)

You can however also prove it mathematically by using the exp (i x) notation. But that is not understanding; that is using the definition of a sine.

The unit circle explanation was precisely the same as mine. He just used a picture. If you look closely the math is the same :)

the only inveral that matters is the ONE between point a and b, which never changes. It always remains exactly what it was when you started. Your distance to it only gets smaller as yo travel. It doesn't halve repeatedly. Therefore you can indeed travel from point a to b, as there is no point at which that infinitely small half actually occurs.

the only inveral that matters is the ONE between point a and b, which never changes. It always remains exactly what it was when you started. Your distance to it only gets smaller as yo travel. It doesn't halve repeatedly. Therefore you can indeed travel from point a to b, as there is no point at which that infinitely small half actually occurs.

true or false - to move in a line from point A to point B, you must go to point C which is located between A and B.

true or false - to move in a line from point A to point C, you must go to point D which is located between A and C.

true or false - there are infinitely many such points that can be made by halving the distance.

true or false - infinity has an end.

true or false - to move in a line from point A to point B, you must go to point C which is located between A and B.

true or false - to move in a line from point A to point C, you must go to point D which is located between A and C.

true or false - there are infinitely many such points that can be made by halving the distance.

true or false - infinity has an end.

false. You must only move from point a to b. moving through the location of point C is possible but not part of the equation of the trip. The only actual math involved is a to b. that's the trick of this theory. If you refuse to look at it, it disappears like Schroedingers cat.

true in the sense of theory, false in the sense of actual practice.

false. infinity is inherantly endless, however the distance from a to b is not infinite, only the number of halfway points you can create theoretically point which again you don't need to consider when moving from point a to b.


In other words, theory and reality don't necessarily converge. When there is a discrepency between theory and reality, then reality must take precedence.

The unit circle explanation was precisely the same as mine. He just used a picture. If you look closely the math is the same :)Indeed, but I think I gave the clearer explanation. ;)

Indeed, but I think I gave the clearer explanation. ;)

A picture is worth a thousand words. Making me more concise.

(In reality,
<---- Too lazy to look up a picture)

I don't understand why Zeno's Paradox is even a thing. All Zeno shows is that you can't measure how long it takes to get there by using his method - suggesting you can't get there is a non sequitur.

Telling me that I can't measure smething by a particular method that you made up isn't informative. I can't measure your speed by stuffing a giant wooden alpaca with potato salad, either, but I'm not about to declare that "Llewdor's Paradox".

I don't understand why Zeno's Paradox is even a thing. All Zeno shows is that you can't measure how long it takes to get there by using his method - suggesting you can't get there is a non sequitur.

Telling me that I can't measure smething by a particular method that you made up isn't informative. I can't measure your speed by stuffing a giant wooden alpaca with potato salad, either, but I'm not about to declare that "Llewdor's Paradox".

I might declare that Llewdor's Paradox on your behalf.....

This is exactly why I never go anywhere. Damn infinite points....

I might declare that Llewdor's Paradox on your behalf.....
I was kind of hoping someone would.

A mathematician, a physicist and an engineer were asked to answer the following question:

A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance apart at time 0, they are at , at , at , and so on.)

When do they meet at the center of the dance hall?

The mathematician said they would never actually meet because the series is infinite.
The physicist said they would meet when time equals infinity.
The engineer said that within one minute they would be close enough for all practical purposes.

I bet my Dad that I could disprove the Zeno's paradox, but i've just realised that I can't. It seems impossible so I was wondering if anyone on NSG can do it.

Incase you don't know about it, it's basicly this:

Say you wanted to get from point A to point B
to travel to point B, you must travel half the distance first, then once you have arrived at the mid point you must travel half of the remaining distance. But once you have arrived at the midpoint of the remaining distance, you still need to travel the remaining half of that distance. This goes on ad infinitum, because it will always take some time, no matter how small, to travel half the remaining distance. And since the remaining distance can be divided into half, it will take an infinate amount of time to reach point B.
So basically you can never reach point B.

Now I know there is a mathematical solution for this involving using sigma notation or something.

After you devide and devide, the distance becomes smaller than your foot. Hence you reach the destination. Voila...

false. You must only move from point a to b.

ridiculous

A----------C----------B

The only actual math involved is a to b.

what math? the question is about the nature of space.

true in the sense of theory, false in the sense of actual practice.

explain to me how to avoid this in 'actual practice':

A----D----C----------B

false. infinity is inherantly endless, however the distance from a to b is not infinite

who is talking about infinite distance?

When there is a discrepency between theory and reality, then reality must take precedence.

this is exactly what zeno thinks he is doing

I don't understand why Zeno's Paradox is even a thing. All Zeno shows is that you can't measure how long it takes to get there by using his method

that isn't what he showed

I don't understand why Zeno's Paradox is even a thing. All Zeno shows is that you can't measure how long it takes to get there by using his method - suggesting you can't get there is a non sequitur.

Telling me that I can't measure smething by a particular method that you made up isn't informative. I can't measure your speed by stuffing a giant wooden alpaca with potato salad, either, but I'm not about to declare that "Llewdor's Paradox".

Aye. I think the interest with Zeno's Paradox has to do with the fact that it boggles folk's minds when they think about it, not that it's inherently a problem.

ridiculous

A----------C----------B



what math? the question is about the nature of space.



explain to me how to avoid this in 'actual practice':

A----D----C----------B



who is talking about infinite distance?



this is exactly what zeno thinks he is doing


I don't see no C.

A-----B


So where is the problem? You're just insisting that I look at it from yourpoint of veiw, which I could give a damn about. I don't believe in C, therefore it doesnt exist.
And no, the question is a matter of philosophy and veiwpoint. WHich is why you will forever run around in little circles and starve while I walk from my house to the corner groceries and buy my dinner.

I don't believe in C, therefore it doesnt exist.

does this ever work?

does this ever work?

The problem is not really that you won't arrive ... its that we don't have an adequete relationship between space and time.

5*5*5*5...... is bigger than 2*2*2*2..... even though both are infinite.

The problem with zenos paradox is that there is no way to tell whether the distance remaining decreases faster than the time taken to travel it.

does this ever work?

seems to for you.:rolleyes:


*you walked into that one*

seems to for you.:rolleyes:


*you walked into that one*

what the fuck are you talking about?

Here's a question though: is the disproof of this paradox evidence of a finite and discrete packet of time?

Or just the opposite?

"Lean closely, and I'll tell you the answer," said the Zen master. <Whack!> :D

But going on a purely mathematical basis, each distance can always be divided, and so it will always take some time to get from the midpoint to the next midpoint, no matter how small. So it seems impossible mathematically.

An infinite sum can converge to a finite number. For example,
1/2 + 1/4 + 1/8 + ... = 1
or more precisely: (summation from i = 1 to infinity of 1/2^i) = 1
Another example of the same sum: Suppose you cut a piece of paper in half an infinite number of times, if you put all of these pieces together, you cannot have more than the 1 piece you started with.

that'd be fucking ridiculous homework. on par with "solve the problem of free will"
I recall having a discussion about this. We cam to the conclusion that for free will, the supernatural is required. This is because without supernatural, we must explain everything using natural laws. All thinking is done in your brain, in which everything happens on a small scale where the effects of quantum physics are noticeable. Thus, even though we cannot say for sure what will happen in your neural network because its activities are based on probabilities.

Do not think that this means your actions are completely random. Neural networks for machine learning can also use some sorts of randomization. Either that or I didn't follow that lecture.

This thread is good stuff, very philosophical.

But going on a purely mathematical basis, each distance can always be divided,
Nope. The minimum possible distance is the Planck length. It is physically impossible for there to be any distance shorter.

Nope. The minimum possible distance is the Planck length. It is physically impossible for there to be any distance shorter.

In this case, would spatial/motion modeling thus be better addressed by discrete rather than continuous mathematics?

How would this affect the use of calculus (which, at least at my current low level of study, uses limits at infinity) for analysis?

Also, is the concept of an angle also limited by quantum steps? Can an angle be infinitely divided, or is it discretely quantatized such that its associated rays must "tick" by intervals associated with the Planck distance?

Can spatial constructs with presence at planck segments have aspects whose distance might have irrational values, and if so, how does this reconcile with a spatial Planck grid?

I'm a beginner, so I apologize if my questions are stupid or meaningless. I honestly would like to understand these things better. Thanks.

Nope. The minimum possible distance is the Planck length. It is physically impossible for there to be any distance shorter.

isn't it more that quantum effects completely dominate at that scale? 'cause we know how long the planck length is: 1.6 x 10^-35 m. clearly we can conceptually make a smaller length, just make it 10^-40 or some such.

isn't it more that quantum effects completely dominate at that scale? 'cause we know how long the planck length is: 1.6 x 10^-35 m. clearly we can conceptually make a smaller length, just make it 10^-40 or some such.

We can conceptually make a smaller length, but said length does not actually exist. Nothing can be smaller than the Planck length. There's also a minimum length of time as well, but I forget what it is.

isn't it more that quantum effects completely dominate at that scale? 'cause we know how long the planck length is: 1.6 x 10^-35 m. clearly we can conceptually make a smaller length, just make it 10^-40 or some such.

In the case of a continuous spatial model with dimensions that occupy something like the real number line, would the convering series theorems offer reasonable answer to Zeno's Paradox?

We can conceptually make a smaller length, but said length does not actually exist. Nothing can be smaller than the Planck length. There's also a minimum length of time as well, but I forget what it is.

also planck - dude's got a lock on the fundamentally small. but my understanding is that its all to do with measurement, not existence. as in, we cannot know anything about what the universe was like before the planck time (when it was smaller than the planck length), but there actually was a t=0 regardless.

also planck - dude's got a lock on the fundamentally small. but my understanding is that its all to do with measurement, not existence. as in, we cannot know anything about what the universe was like before the planck time (when it was smaller than the planck length), but there actually was a t=0 regardless.

So, is any speed only truly measured in units of length that are integer multiples of the Planck Length divided by units of time that are similar increments of the Planck time??

In the case of a continuous spatial model with dimensions that occupy something like the real number line, would the convering series theorems offer reasonable answer to Zeno's Paradox?

not exactly - the real problem is not in figuring out how to add up the distance, but in getting there. if space is infinitely divisible, then there is no first move that is possible to make, as there would always be more things you'd have to do before.

the planck time and length thing may actually address this somewhat, now that i think about it. space and time may still be infinitely divisible, but at divisions smaller than those units, motion doesn't happen the way we think of it happening up here in bigscaleland. down there, you actually can occupying an infinite number of points all at once due to quantum effects.

not exactly - the real problem is not in figuring out how to add up the distance, but in getting there. if space is infinitely divisible, then there is no first move that is possible to make, as there would always be more things you'd have to do before.

the planck time and length thing may actually address this somewhat, now that i think about it. space and time may still be infinitely divisible, but at divisions smaller than those units, motion doesn't happen the way we think of it happening up here in bigscaleland. down there, you actually can occupying an infinite number of points all at once due to quantum effects.

So, instead of the object occupying a continuous and changing series of infinite points, are we talking about the object itself being a probability distribution wave that is only observable as a discrete object at waypoints governed by planck units?

I'm sorry, I clearly need to up my math kung fu before I can pose any kind of real question here.