Motion is impossible. - Page 6

To return to Zeno’s paradoxes, the solution to all of the mentioned paradoxes then,9 is that there isn’t
an instant in time underlying the body’s motion (if there were, it couldn’t be in motion), and as its
position is constantly changing no matter how small the time interval, and as such, is at no time
determined, it simply doesn’t have a determined position. In the case of the Arrow paradox, there isn’t
an instant in time underlying the arrows motion at which it’s volume would occupy just “one block of
space”, and as its position is constantly changing in respect to time as a result, the arrow is never static
and motionless. The paradoxes of Achilles and the Tortoise and the Dichotomy are also resolved
through this realisation: when the apparently moving body’s associated position and time values are
fractionally dissected in the paradoxes, an infinite regression can then be mathematically induced, and
resultantly, the idea of motion and physical continuity shown to yield contradiction, as such values are
not representative of times at which a body is in that specific precise position, but rather, at which it is
passing through them. The body’s relative position is constantly changing in respect to time, so it is
never in that position at any time. Indeed, and again, it is the very fact that there isn’t a static instant in
time underlying the motion of a body, and that is doesn’t have a determined position at any time while
in motion, that allows it to be in motion in the first instance.

The paradoxes of Achilles and the Tortoise and the Dichotomy are often thought to be solved
through calculus and the summation of an infinite series of progressively small time intervals and
distances, so that the time taken for Achilles to reach his goal (overtake the Tortoise), or to traverse the
said distance in the Dichotomy, is in fact, finite. The faulty logic in Zeno’s argument is often seen to be
the assumption that the sum of an infinite number of numbers is always infinite, when in fact, an
infinite sum, for instance, 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +...., can be mathematically shown to be
equal to a finite number, or in this case, equal to 2.
This type of series is known as a geometric series. A geometric series is a series that begins with one
term and then each successive term is found by multiplying the previous term by some fixed amount,
say x. For the above series, x is equal to 1/2. Infinite geometric series are known to converge (sum to a
finite number) when the multiplicative factor x is less than one. Both the distance to be traversed and
the time taken to do so can be expressed as an infinite geometric series with x less than one. So, the
body in apparent motion traverses an infinite number of "distance intervals" before reaching the said
goal, but because the "distance intervals" are decreasing geometrically, the total distance that it
traverses before reaching that point is not infinite. Similarly, it takes an infinite number of time
intervals for the body to reach its said goal, but the sum of these time intervals is a finite amount of
time.
So, for the above example, with an initial distance of say 10 m, we have,
t = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + .… + 1 / 2 n Difference = 10 / 2 n m
Now we want to take the limit as n goes to infinity to find out when the distance between the body in
apparent motion and its said goal is zero. If we define
S n = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + .… + 1 / 2 n
then, divide by 2 and subtract the two expressions:
S n - 1/2 S n = 1 - 1 / 2 n+1
or equivalently, solve for S n:
S n = 2 ( 1 - 1 / 2 n+1)
So that now S n is a simple sequence, for which we know how to take limits. From the last expression it
is clear that:
lim S n = 2
as n approaches infinity.
Therefore, Zeno’s infinitely many subdivisions of any distance to be traversed can be mathematically
reassembled to give the desired finite answer.

On July 26 2005 17:22 Tontow wrote:






Could you do a proof of that?
I’m not shore 100 is correct Because the way you wrote it:
-The limit of N is infinity. (N = infinity)
- 100N = Infinity
-N/N = Infinity




And remember that you stated that:











Here is how I arrived at the answer of (0 < (X / infinity) <_ 1) / infinity :


I take into account the theorem (It is important to keep this in mind):


X/infinity
X can’t equal 100 because it is impossible to get a repeating decimal that is greater than 1.
X/infinity will eventually have a repeating decimal. And thanks to the theorem, ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”, we will eventually run into a paradox.
-It is impossible for X/infinity to equal 0 and so I use “ 0 < “.
-It is impossible for X to equal anything greater than 1 since we are constantly dividing. However, X can equal 1 thanks to the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1” and so I use “<_ 1”

And so I end up with:



To summarize and simplify for everyone:
(1). Start with any given number.
(2). Continue to divide that number until you end up with a repeating decimal; I can guarantee that the repeating decimal will not be greater than 1.
(3). Given the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”. That repeating decimal is = 1
(4). The given number we have now is 1. And your back at step (1).

Thus: (0 < (X / infinity) <_ 1) / infinity

On July 26 2005 17:33 BigBalls wrote:
what are you even arguing against me???

his problem was lim(n -> infinity) (100n/n), which is 100

To summarize and simplify for everyone:
(1). Start with any given number.
(2). Continue to divide that number until you end up with a repeating decimal; I can guarantee that the repeating decimal will not be greater than 1.
(3). Given the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”. That repeating decimal is = 1
(4). The given number we have now is 1. And your back at step (1).

Thus: (0 < (X / infinity) <_ 1) / infinity

On July 26 2005 19:20 Triton wrote:
Can mathematics provide a number for 1/3?

On July 26 2005 10:21 BigBalls wrote:
sickoflife, they both have the same amount of money.

here is a bit of a trickier example.

Define Z+ = the positive integers. So Z+ = {1,2,3,.....}
2*Z+ = {2,4,6,.....}, the even positive integers.

Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Thus, there exists a 1 to 1 and onto map between these two sets, which is known as a bijection. Thus, there is an infinite set which is a subset of another infinite set, and can be mapped 1-1 AND onto it.

To move into analysis, take the interval (0,1). (0,1) is NOT countable, there are an infinite number of elements in there. furthermore, any interval is not countable, as long as it has positive measure then it is infinite. However, the example before, 2Z+ IS countable. Although it has an infinite number of elements, it has measure 0.

On July 26 2005 19:28 Echo wrote:

1/3 is approximately equal to .3 repeating (infinitely) and is considered an inaccurate calculation. Math has a number for 1/3 already. That would be 1/3

On July 24 2005 22:54 Hydrolisko wrote:
Zeno is a greek philospher who thought motion is impossible. I have to write an essay to refute his argument intelligently (rationally, can't just say stuff like "wow how dumb"). I have an argument in mind but I'd like to see some more insights before I start writing. Here is his argument:

1. Zeno starts by assuming what his opponent says is possible: motion. In particular, the motion of a single body across a finite distance in a finite time.
2. To make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a sprinter, who starts running at one end of a 100 metre straight track, and runs to the other end. Zeno’s opponents (probably including yourself) think that it really is possible that runners can do this. (i.e. They think that it is not just an illusion.) Zeno begins by assuming that his opponents are correct, and then the fun starts.
3. Zeno points out, given that we are assuming that space is continuous, that before the runner can cover the whole 100 metres she has to cover half the distance, i.e. run 50 metres.
4. Zeno then repeats the move just made (point 4) and points out that the move can be repeated an infinite number of times: before the runner can cover 50 metres, she has to run 25, before that she must run 12.5, before that 6.25, etc. Recall that we are assuming that space is continuous, which means that any finite piece of it, such as a sprint track, can be divided into infinitely many parts, which are infinitely small.
5. Zeno then argues that to cover each of these infinitely small parts will take a certain amount of time.
6. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres. But we were assuming that the runner could cover the distance in a finite amount of time, not that she would take forever.
7. We have run into a problem, and Zeno’s conclusion is that motion is impossible, because any finite motion would take forever.

Thoughts?

On July 26 2005 23:19 MPXMX wrote:
is .9 repeating exact same thing as 1?

On July 26 2005 23:19 MPXMX wrote:
is .9 repeating exact same thing as 1?

On July 27 2005 01:20 geod wrote:

No, it isn't.
lim n->oo 0,9...9 = 1, but 0,9...9 < 1.

On July 26 2005 22:18 baal wrote:
Yes and No, i cant remember who proved that infinites in infinites theory

On July 27 2005 05:46 aseq wrote:
stop this rabbit and tortoise thing already.
these stupid philosophers probably skipped math lessons all the time. It's about limits, and we all know that decreasing the time intervals makes no sense whatsoever.