How Does Calculus Resolve Zeno's Paradox of Motion?

[rajeshmarndi]

It is said the sum of infinite series 1/2+1/4+1/8...=1, i.e it say there are infinite points.

It also means if it goes on like this it will approach the limit which is 1 in this case but never ever will it be able to reach 1.

Or in other words it says 1 is the point that would never be reached, infact 1 is the point to which it will be getting closer with every addition of the series.

So how does the solution to this paradox say ,one cover all these infinite points between A and B.

 

[Dickfore]

There is an equal number of points on the first half of the trip as on the whole trip. Ancient Greeks didn't know about this. Also, a sum of an infinite number of terms may be finite. Ancient Greeks didn't know this either (except Archimedes).

 

[rajeshmarndi]

There is an equal number of points on the first half of the trip as on the whole trip.

that is a good information, that's how infinity is.

Also, a sum of an infinite number of terms may be finite.

Does it mean some do and some not, i thought all sum up to finite.

But how does one reach a finite point, how does calculus or limits say, it covered all the infinite point.

 

[pbuk]

The amount of time taken to cover each of these distances decreases at the same rate, so it takes 1/2 of the time to cover 1/2 of the distance... 1/65536 of the time to cover 1/65536 of the distance... and in the limit, in 100% of the time you have covered 100% of the distance.

 

[sophiecentaur]

that is a good information, that's how infinity is.


Does it mean some do and some not, i thought all sum up to finite.

But how does one reach a finite point, how does calculus or limits say, it covered all the infinite point.

This introduces the subject of 'Sums of Series'. the nature of Series is a massive topic and you really have to start at the beginning, with the appropriate Maths skills. It very soon gets very 'Mathsy' but some Mathematicians really take off on it. You mention the word 'Infinity'. There are many different 'infinities', some of which are bigger than others. There is no end to this!

 

[pbuk]

The nature of Series is a massive topic and you really have to start at the beginning

When I was taught about limits, Xeno's paradox was the beginning - it is an excellent place to start IMHO.

 

[rajeshmarndi]

All these is possible only if at all motion is possible OR if it only move to the next adjoining points.

Therefore it doesn't make sense it move from one point to another, since the next adjoining point doesn't seem to exist.

 

[rajeshmarndi]

Suppose we say its speed is 10 km/h, if it is true then the rest is also true, since it is found or observed, it cover at a rate of 10 km per hour.

But it doesn't explain how the motion of 10 km/h is made possible.

 

[DaveC426913]

You're intuiting that each leg of a trip takes the same length of time, so it seems intuitively that an infinite number of legs would take forever to complete.

But note that, as the leg shrinks, so does the time.

So, what is the total duration of 1/2 second + 1/4 second + 1/8 second + ... to infinity?

It is one second.

No matter how many intervals you take, after one second, all infinite number of them will be complete.

 

[rajeshmarndi]

You're intuiting that each leg of a trip takes the same length of time, so it seems intuitively that an infinite number of legs would take forever to complete.

But note that, as the leg shrinks, so does the time.

Ok, I didn't thought of that.

No matter how many intervals you take, after one second, all infinite number of them will be complete.

Does the infinite number get complete and what do you mean by that.

 

[Dickfore]

You mentally divided the segment into an infinite sequence of halved segments. There is no real need for the walker to decrease his pace. Therefore, there is actually a finite number of steps.

Even if you had mentally divided it into infinite number of steps, you would have to be pedantic and keep all of them into account. When you sum the duration of all of them, you get a finite answer, namely, 1 second.

 

[Neandethal00]

It is said the sum of infinite series 1/2+1/4+1/8...=1, i.e it say there are infinite points.

It also means if it goes on like this it will approach the limit which is 1 in this case but never ever will it be able to reach 1.

Or in other words it says 1 is the point that would never be reached, infact 1 is the point to which it will be getting closer with every addition of the series.

So how does the solution to this paradox say ,one cover all these infinite points between A and B.

I personally feel Zero's paradox has never been solved for practical purposes. Most explanations I have read are analytical. The reasons can be any of

1) We do not understand what time actually is
2) We do not know whether time makes changes or changes make time
3) Space and time are discrete. After certain number of divisions, they can not be divided further. After this limit of divisions is reached next division will always point to the point in the grids of time and space not halfway between 2 grid points.

I'll vote for the 3rd case.

 

[rajeshmarndi]

You mentally divided the segment into an infinite sequence of halved segments. There is no real need for the walker to decrease his pace. Therefore, there is actually a finite number of steps.

How does one find it to be finite?
that is what I want to know.

 

[rajeshmarndi]

I personally feel Zero's paradox has never been solved for practical purposes. Most explanations I have read are analytical. The reasons can be any of

I do not know it yet.

1) We do not understand what time actually is

There has to be an observer who observe the change of time flow.

2) We do not know whether time makes changes or changes make time

It is the changes that makes time.

3) Space and time are discrete. After certain number of divisions, they can not be divided further. After this limit of divisions is reached next division will always point to the point in the grids of time and space not halfway between 2 grid points.

I'll vote for the 3rd case.

As I said if one says it is moving at say 10 km/h, there is no problem, since it is been observed covering some distance. That is, there is motion.

But is it right to ask, how motion is happening.

Let say there are two points A and B. Is there a point x that is said to be next to A.

I think there couldn't be any such point. Because then it could also be said there would be infinite points between A and x, as there will be infinite points between x and B, again as there are infinite points between A and B.

Also let say it take t sec from A to x. There is no existence between A and x, as there are no points in between them, only time passed, that is to say it just appeared after t sec at x. It seems it is teleporting from one point to another, else it would be instantaneous i.e A to B also will be instantaneously.

 

[Dickfore]

How does one find it to be finite?
that is what I want to know.

The distance is finite. The length of the step is finite. If you divide a finite number by another finite number, you get a finite number.

 

[rajeshmarndi]

The distance is finite. The length of the step is finite. If you divide a finite number by another finite number, you get a finite number.

I still didn't get you.

So how much you will be dividing.

For example, how much a 5m(finite) distance can be divided into.

 

[Jobrag]

You know how far your journey is, you aim for twice the distance, and arrive on your first step.

 

[Dickfore]

I still didn't get you.

So how much you will be dividing.

For example, how much a 5m(finite) distance can be divided into.

How big is your step?

 

[rajeshmarndi]

How big is your step?

Sorry I misunderstood step as step of a procedure.

Yes the distances as well as the length of the step are finite. Even the length of the step say 1m long is also to be covered, since motion is occurring.

If you say a step cover 1m long at t sec, then there is no problem. Since we are agreeing with motion as it happen. But if we question motion itself, I do not understand how to answer.

So what I replied to Neandethal still apply here.