"Motion is impossible" claims modern Zeno

[thinkandmull]

Yes but when does it reach zero? There is no final step of the series? A line seems to have the qualities of the finite and the infinite at the same time. Its seems almost like a round square to me

 

[A.T.]

A line seems to have the qualities of the finite and the infinite at the same time.

It has nothing to do with the qualities of a line, just with numbers which we use to quantify a line's length, a square's area or whatever. Positive numbers can be divided into infinitely many positive non-zero parts.

 

[Dale]

I struggle to understand how a limit can exist for something in space like a line.

Do you understand how a limit can exist for an infinite series of real numbers?

 

[thinkandmull]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

 

[votingmachine]

Think of the series:
0.3 + 0.03 + 0.003 +0.0003 + 0.00003 + ...

The series goes on for an infinite amount of terms, but the limit is 1/3.

 

[jbriggs444]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

A "limit" is, roughly speaking, the single, exact real point that is approached more and more closely by an infinite sequence -- if that sequence converges at all. The point is that the sequence does not need to have a last step in order to have a limit that it approaches.

In mathematics, the real number line are constructed in such a way that convergent sequences always have limits. If you have a sequence of numbers that get closer and closer to one another in a particular way (https://en.wikipedia.org/wiki/Cauchy_sequence) then there will always be a real number that is the limit of the sequence.

A course in Real Analysis would often take you through the definitions and theorems that make clear exactly how this works.

 

[thinkandmull]

If someone is always taking half-steps, he will never get to the destination. This is so counter-intuitive. If you are ever getting closer to something, how could you never reach it? The eternity of future's time seems to be bigger than any infinity of points

 

[jbriggs444]

If someone is always taking half-steps, he will never get to the destination. This is so counter-intuitive. If you are ever getting closer to something, how could you never reach it? The eternity of future's time seems to be bigger than any infinity of points

That's Zeno's paradox. If someone is always taking steps in half times, that sequence of steps will not cover all times out to eternity.

 

[votingmachine]

That's Zeno's paradox. If someone is always taking steps in half times, that sequence of steps will not cover all times out to eternity.

If you run a marathon in 2 hours, you also run two half-marathons in 1 hour each. Or 4 quarter-marathons in 1/2 hour each. Or 100 one-hundredth-marathons in 0.02 hours each. You can apply any division you want and you will always apply the same division to the time and to the distance.

It seems confusing because Zeno phrases it in terms of a sequence that is ahead of you, and therefore there are an infinite amount of steps. But you can ALWAYS divide the distance and the time by any arbitrary number, including the infinite halving sequence (Run a half-marathon, then a quarter-marathon, then an eighth-marathon, then ...).

The answer is that there is always a corresponding division of the time. Pick ANY number of divisions for that marathon. Divide the distance by that number. Divide the time by that number. Then multiply them together and get the original answer. In my example you will always get 1-marathon-per-2-hours.

 

[thinkandmull]

So we must say that continuous motion has a special quality that is a "sum greater than its parts", so to speak. Because the infinite half steps need to be passed over before one gets from A to B, but I think the continuous motion is greater than the motion that is taking half steps. But greater how?

 

[A.T.]

So we must say that continuous motion has a special quality that is a "sum greater than its parts", so to speak.

No, it has nothing to with motion being special. You can divide any number like that. An it's not greater than the sum of its parts.

 

[thinkandmull]

Than we are faced with the same paradox as Zeno's Arrow paradox. I think motion is "one step ahead" of matter so to speak. I don't know how else to say it. When you cross a finite segment, you pass over infinite points, and there is no final step. There are no discrete "one two three" steps to take and then wow! you are at your destination. I'm thinking that motion must be more mysterious, something about it that we are not getting straight from math, but can understand better with physics

 

[PeroK]

Than we are faced with the same paradox as Zeno's Arrow paradox. I think motion is "one step ahead" of matter so to speak. I don't know how else to say it. When you cross a finite segment, you pass over infinite points, and there is no final step. There are no discrete "one two three" steps to take and then wow! you are at your destination. I'm thinking that motion must be more mysterious, something about it that we are not getting straight from math, but can understand better with physics

My steps are about 1m, so if I have to travel 3m, then it is "one, two, three" and I'm at my destination.

I never thought much of Zeno's paradox, I must admit. Here's my version:

The hare starts 1m behind the tortoise.
The hare starts 1m behind the tortoise.
The hare starts 1m behind the tortoise ...

Ergo, the hare never even moves let alone catches the tortoise.

 

[Dale]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

This is the thing that you need to learn then. Once you have understood the limit of an infinite sum of real numbers then the mapping from the real numbers to a line is trivial.

As far as the infinite sum goes, this is actually straightforward. If you have $\sum _{n=1}^k \frac{1}{2^n}$ then you can fairly easily convince your self (simply by writing it out for a few small values of $k$) that the answer is $1-2^{-k}$. As $k$ becomes larger and larger the sum gets closer and closer to 1. In fact, you can specify any arbitrary $0<\epsilon<<0.5$ and calculate the $k$ necessary to make the sum closer to 1 than $\epsilon$. This is what it means to have a limit. The limit of any finite series is less than 1, but as the finite series gets arbitrarily large the sum gets arbitrarily close to 1. So the infinite sum is equal to 1.

Please start with this website and think and mull it over a bit:
http://www.math.utah.edu/~carlson/teaching/calculus/series.html

 

[A.T.]

...so to speak. I don't know how else to say it.

Use math.