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-   -   2=1 (http://forums.netphoria.org/showthread.php?t=14791)

bonsor 03-07-2002 09:23 PM
Quote:
Originally posted by Irrelevant:
have you heard of Zeno's Paradox? here's an example i got off the web:

A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.

Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

and calculus deals with these limits. i don't know if it solves this problem, but it does negate its effect.
The only reason space is allegedly infinitely divisible is because we apply the abstract concept of mathematics to it.

For instance, the quantam leap, the phenomenon on electrons traveling from different electron shells of an atom. When they do that, they don't (appear to) exist anywhere between the two electron shells, which in a way helps prove that space is definitely divisible.

twice 03-07-2002 09:43 PM
*head implodes*

jenny4ever 03-07-2002 11:35 PM
holy shit lawson was just telling me about this like a week ago.

Nate the Grate 03-07-2002 11:45 PM
Let's talk about PI!

*awaits discussion*

raindrops + sunshowers 03-07-2002 11:45 PM
Dorks.

Jaggie 03-08-2002 12:16 AM
I agree with both of you, Irrelevent and ******. I don't know if there is an answer really. Numbers are arbitrary.

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i wasted all my years, been chasing all my fears

Mayfuck 03-08-2002 12:18 AM
There's a lengthy answer to Zeno's Paradox, but it has to do with the collapse of waves and quantum mechanics, and other stuff that goes over my head.

SPFreak37 03-08-2002 12:44 AM
Quote:
Originally posted by pale_princess:
u + me = us.
that's what i'm talking about!!


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dying of boredom...i'll try it all

Ammy 03-08-2002 12:45 AM
this thread makes me all warm inside.

scouse_dave 03-08-2002 02:31 PM
Quote:
Originally posted by Irrelevant:
have you heard of Zeno's Paradox? here's an example i got off the web:

A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.

Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal.
well, this would certainly be true if 'infinity', a much misunderstood word, was a definitive number. in reality, it is agreed that there are various types of infinities, some 'larger' than others, or diverging more rapidly than others.

in this paradox, you're essentially summing an infinite series of increasingly small distances which sum to 100m (50+25***2.5+etc). corresponding to this series is the time taken to traverse these distances. each of these specific times is finite, but there is not a hard and fast rule with regards to summing an infinite series of finite numbers. in some instances, the total diverges to infinity and in others it converges to a finite answer, as in this example.

Assuming that the runner maintains a constant pace throughout the race, and it takes him X seconds to reach the 50m mark, then the total time for the race is:

T = X + X/2 + X/4 + X/8 + X/16 + ... (TO INFINITY)
= sum(X/(2^(n-1)) ; where n corresponds to the nth number in the series

This can be proven to sum to 2X (i'm sure as hell not gonna go into that now). in essence, the subsequent terms added on to the total each time are not 'strong' enough to break free from a finite asymptotic constraint, and hence this may be considered a 'weak' infinity. (otherwise it would sum to infiity)

hence the time it takes the runner to reach the 50m mark is exactly half the total race time. this is a result which you'd expect.

yeah, it's weird, but then so is infinity. deal with it. technically, it's not a paradox either...it's just phrased in a manner to deliberately confuse people. a paradox holds up to scrutiny. and yes, i'm a bit bored.

Jaggie 03-08-2002 02:59 PM
Quote:
Originally posted by scouse_dave:
well, this would certainly be true if 'infinity', a much misunderstood word, was a definitive number. in reality, it is agreed that there are various types of infinities, some 'larger' than others, or diverging more rapidly than others.

in this paradox, you're essentially summing an infinite series of increasingly small distances which sum to 100m (50+25***2.5+etc). corresponding to this series is the time taken to traverse these distances. each of these specific times is finite, but there is not a hard and fast rule with regards to summing an infinite series of finite numbers. in some instances, the total diverges to infinity and in others it converges to a finite answer, as in this example.

Assuming that the runner maintains a constant pace throughout the race, and it takes him X seconds to reach the 50m mark, then the total time for the race is:

T = X + X/2 + X/4 + X/8 + X/16 + ... (TO INFINITY)
= sum(X/(2^(n-1)) ; where n corresponds to the nth number in the series

This can be proven to sum to 2X (i'm sure as hell not gonna go into that now). in essence, the subsequent terms added on to the total each time are not 'strong' enough to break free from a finite asymptotic constraint, and hence this may be considered a 'weak' infinity. (otherwise it would sum to infiity)

hence the time it takes the runner to reach the 50m mark is exactly half the total race time. this is a result which you'd expect.

yeah, it's weird, but then so is infinity. deal with it. technically, it's not a paradox either...it's just phrased in a manner to deliberately confuse people. a paradox holds up to scrutiny. and yes, i'm a bit bored.
OK, so what's your take on my earlier question?

scouse_dave 03-08-2002 03:42 PM
Quote:
Originally posted by Jaggie:
I've always pondered something...How can any number truly be defined? How can we say something is 2. There are infinite amount of numbers in between 1 and 2 (1.5, 1.55, 1.555, etc), thus one can never actually make it to 2. Does that make sense?
numbers are totally man-made. before mathematicians, there wasn't a '2' - it didn't exist. in fact there's a case to say that it doesn't exist now. http://www.netphoria.org/wwwboard/tongue.gif
Even if you accept that numbers exist, it's hard to argue against them being so fundamentally basic that even giving a common-sense definition to a friend is impossible. How do you explain to someone 'two' without holding up two bananas and saying "look!!! TWO bananas!!!" or whatever?? we're all so comfortable with numbers cos we use them all the time, not cos we understand what they are.

It's true that by starting from 1 and slowly increasing in increments, it may be impossible to reach 2, but i don't see why that's a problem. again, we've invented an increasingly complex number system because various modern-day practices require more accuracy; engineering etc. greeks knew that pi was roughly 22/7, yet they were quite happy approximating it to 3 most of the time.

so i don't even know what you're trying to say. if we were being picky we couldn't count up to 2, that's true; but we invented 2 first http://www.netphoria.org/wwwboard/smile.gif it's just not a problem.

Quote:
Originally posted by Jaggie:
It's kinda like putting a definite number as to the distance to the end of the universe. Assuming the universe is infinite, we can place an arbitrary value (for example, EU can stand for the distance to the end of the universe) but this value will never be defined because there is an infinite amount of numbers in between.
well as far as i know, the universe is still expanding, so this number EU couldn't ever be defined anyway. my maths is better than my physics, so i'm probably not the best person to ask on this. i don't see why the infinite amount of numbers inbetween is a problem tho. you could divide your shoelaces into infinitely small segments; but that doesn't stop you saying that they're 12 inches long or whatever.

maths is natural. the way in which we define maths is completely artificial

Jaggie 03-08-2002 05:44 PM
But that's my whole point, numbers are not only artificial, but completely undefined. If something is undefined, how can it be used with any validity. The EU example was the same idea. You said that the universe is expanding and that the true distance to the end is undefined, thus how can we use that number in a formula?

So very sad about me 03-08-2002 08:18 PM
Quote:
Originally posted by Jaggie:
The EU example was the same idea. You said that the universe is expanding and that the true distance to the end is undefined, thus how can we use that number in a formula?
cos even tho it's changing, you're still able to calculate it?

i dunno, i'm not sure about any of this.

i think the universe discussion should be entirely separate from the numbers one...

pink_ribbon_scars 03-08-2002 10:31 PM
Quote:
Originally posted by Jaggie:
But that's my whole point, numbers are not only artificial, but completely undefined. If something is undefined, how can it be used with any validity. The EU example was the same idea. You said that the universe is expanding and that the true distance to the end is undefined, thus how can we use that number in a formula?

numbers aren't completely undefined, are they? they ARE defined, and that's what makes them what they are. (?)

Jaggie 03-08-2002 11:53 PM
Quote:
Originally posted by pink_ribbon_scars:

numbers aren't completely undefined, are they? they ARE defined, and that's what makes them what they are. (?)
But what we're saying is that because there are infinite amounts of numbers in between a set of numbers, nothing except for 0 can be defined.

bonsor 03-09-2002 12:17 AM
Quote:
Originally posted by Jaggie:
But what we're saying is that because there are infinite amounts of numbers in between a set of numbers, nothing except for 0 can be defined.
Yes it can.

1 is 1. 2 is 2. Numbers are defined. That is why mathematics exists.

Jaggie 03-09-2002 01:49 AM
Quote:
Originally posted by ******:
Quote:
Originally posted by Jaggie:
But what we're saying is that because there are infinite amounts of numbers in between a set of numbers, nothing except for 0 can be defined.
Yes it can.

1 is 1. 2 is 2. Numbers are defined. That is why mathematics exists.
What is 2 defined as? It cannot be defined because all it is is 1 + 1. But there are infinite amount of numbers between 1 and 2 (also between 0 and 1). Thus we can substitute infinity for any number other than 0. So then the formula becomes infinity plus infinity = 2? No, can't be, you cannot add an undefinable (infinity) number to anything. Infinity plus 1 still equals infinity.

Mustard 03-09-2002 04:02 AM
i like inner toobs! http://www.netphoria.org/wwwboard/smile.gif

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If God is inside of me like everybody says he is, then I hope he likes enchiladas, cause that's what he's getting.

the_boy 03-09-2002 04:17 AM
Quote:
Originally posted by scouse_dave:

Even if you accept that numbers exist, it's hard to argue against them being so fundamentally basic that even giving a common-sense definition to a friend is impossible. How do you explain to someone 'two' without holding up two bananas and saying "look!!! TWO bananas!!!" or whatever?? we're all so comfortable with numbers cos we use them all the time, not cos we understand what they are.
Ya, i think you hit the nail on the head with this here. This is exactly like an discussion we had in one of my Philosophy Of Language classes. Basically I think that understanding exactly what numbers are and what they mean is much less important than understanding what they represent . So long as we have this knowledge, we can proceed with mathematics and our daily lives without having to worry about all this "Zeno's Paradox" bullshit which will probably never really have an answer. And thats good enough for me. http://www.netphoria.org/wwwboard/smile.gif


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