Reaching the Speed of Light: A Thought Experiment on Halving Distance Traveled

[jbriggs444]

@ShaunM I would strongly recommend going over this post multiple times until you understand it completely. This post completely resolves Zeno’s paradox, and it does so on Zeno’s own terms without avoiding the issue raised by Zeno.

If you look for it, there is a quibble that can be raised. Although we talk about the "sum" of an infinite series as if it were a sum and treat it [correctly, as long as it is absolutely convergent] as a sum, the "sum" of an infinite series is technically a limit rather than a sum. It is the limit of the sequence of partial sums.

The naive picture of computing a sum by adding to a running total and reporting the result after the last term has been added in does not work for infinite series because there is no last term. So instead, one reports the value that is approached, if any.

 

[russ_watters]

I can't however see how I can 'decide' what's going on (or can I?) and that's precisely the problem I have. What is the object actually doing? Speeding up, slowing down, is its speed constant and why doesn't it seem to reach point B?

It's *your* scenario. You get to decide how to make it. But hopefully if you try applying math/numbers to it you will recognize it is self-contradictory. Specifically if your time steps are equal in length your speed is dropping, not increasing. 1m/s, 1/2m/s, 1/4m/s, etc.

 

[Dale]

If you look for it, there is a quibble that can be raised. Although we talk about the "sum" of an infinite series as if it were a sum and treat it [correctly, as long as it is absolutely convergent] as a sum, the "sum" of an infinite series is technically a limit rather than a sum. It is the limit of the sequence of partial sums.

The naive picture of computing a sum by adding to a running total and reporting the result after the last term has been added in does not work for infinite series because there is no last term. So instead, one reports the value that is approached, if any.

I wouldn't call that a quibble. That is part of Zeno's problem setup. He computes this limit (incorrectly) and claims that it is not finite. You cannot answer the problem on Zeno's terms without computing the same limit (correctly) and showing that it is finite.

 

[ShaunM]

It might help you to learn about infinite series. Mainly that the sum of infinitely many non-zero positive numbers can still be finite. For example:

1/2 + 1/4 + 1/8 + ... = 1

If you move 1m at 1m/s you have these steps:

1/2m + 1/4m + 1/8m + ... = 1m

Which take these times:

1/2s + 1/4s + 1/8s + ... = 1s

So the total time is still 1s.

Many thanks. That is cooincidentally exactly what is contained in this video I have just found:



I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

Im enjoying this very much however and I am glad to have stumbled across such an old idea and you lot.

 

[PeroK]

Many thanks. That is cooincidentally exactly what is contained in this video I have just found:



I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

Im enjoying this very much however and I am glad to have stumbled across such an old idea and you lot.

The mathematics of infinite sums can be put on a rigorous basis, but that is not necessary.

Zeno adds $1/2 + 1/4 +1/8 \dots$. Now,

1) This sum might eventually reach any number. In fact the series $1/2 + 1/3 +1/4 \dots$ does just that.

In which case, time passes and things move.

2) this sum might never reach $1$. In which case it is a poor model for the passage of time, which experimentally would appear to tick along.

3) this sum might have no meaning, in which case again it is a poor model.

Fundamentally, case 2 is the critical one. What Zeno is really saying is: here is a model for the passage of time. Time moves in increasingly small steps, in some sense, and can never reach $1s$.

The refutation is simply to let $1s$ pass by whatever measure of time you adopt.

It's a simple mathematical fact that the finite sums above never reaches the value $1$. But, that makes it a poor model for the passage of time. And, it certainly doesn't compel time to behave according to Zeno's model.

 

[A.T.]

I still don't understand why that makes 1m

You start with 1m. You subdivide it into more and more, smaller and smaller pieces. But you never add to or abstract from it, so it remains 1m:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

The sum doesn't change in those steps, so no matter how often you repeat it, it is still 1.
This way to write it also avoids the issue mentioned by @jbriggs444.

 

[russ_watters]

I think this point is critical:

It's a simple mathematical fact that the finite sums above never reaches the value $1$. But, that makes it a poor model for the passage of time. And, it certainly doesn't compel time to behave according to Zeno's model.

Zeno did the math wrong, but even if you do the math right, it is still a poor way to describe motion. It's the model that's wrong, not reality.

Specifically:
If you graph time over displacement of a constant walking speed with a zero in the middle instead of at the end, you get a simple sloped line.

if you graph the number of samples over distance you get a hyperbole with a discontinuity where the model breaks.

 

[jbriggs444]

I still don't understand why that makes 1m as that would mean, as stated in the video, this infinite sequence has an end which seems to my untrained self to be a contradiction:)

The sequence is bounded. [There is a single fixed value such that every sequence element is less than or equal to that value]

The sequence has no last term. [For every element in the sequence, there is another element with a larger value]

The two are not contradictory.

 

[Herman Trivilino]

This post completely resolves Zeno’s paradox, and it does so on Zeno’s own terms without avoiding the issue raised by Zeno

Not this particular one. You still have to add an infinite number of terms to get the final finite sum. Thus requiring an infinite number of steps to get there.

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

 

[russ_watters]

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

The problem gets worse if you don't intend to stop walking when you cross the goal line. The function is discontinuous at 0.

 

[russ_watters]

Zeno's model vs the normal model, in graphical form:

I think the problem gets clearer if you arbitrarily move the "0" displacement.

 

[PeroK]

Not this particular one. You still have to add an infinite number of terms to get the final finite sum. Thus requiring an infinite number of steps to get there.

This assumes that steps can be made as small as you like while still maintaining a known velocity. That assumption is false because it requires a violation of the HUP.

I don't remember saying that!

It was actually @Dale who said that.

You need to take a course in real analysis.

The HUP is not relevant to Zeno's argument. QM shows that the naive analysis of motion, ultimately for small enough particles, doesn't apply. It's actually the expected values of measurements that behave classically.

 

[ShaunM]

You start with 1m. You subdivide it into more and more, smaller and smaller pieces. But you never add to or abstract from it, so it remains 1m:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

The sum doesn't change in those steps, so no matter how often you repeat it, it is still 1.
This way to write it also avoids the issue mentioned by @jbriggs444.

Thanks. To my (admittedly untrained) eye that's hard to understand. Working backwards from 1m to 0 in this way seems to to me to lead to the exact same problem. there's always one discrete part of 1 missing.

Im actually quite comfortable with the thought that 1 is never reached as it seems to object isn't actually trying to get there anyway.

 

[A.T.]

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

there's always one discrete part of 1 missing.

How much is missing in which line?

Im actually quite comfortable with the thought that 1 is never reached

You shouldn't be, because it's nonsense.

 

[PeroK]

Im actually quite comfortable with the thought that 1 is never reached as it seems to object isn't actually trying to get there anyway.

Given this is a physics forum and not a new age philosophy forum, I'm going to quote Richard Feynman:

If your theory doesn't agree with experiment, then it doesn't matter what your name is or how clever you are, it's wrong.

 

[ShaunM]

Given this is a physics forum and not a new age philosophy forum, I'm going to quote Richard Feynman:

If your theory doesn't agree with experiment, then it doesn't matter what your name is or how clever you are, it's wrong.

Quite right. I am having problems however understanding how 1 is reached. I appreciate you all trying to explain.

 

[ShaunM]

How much is missing in which line?

You shouldn't be, because it's nonsense.

The lines don't seem to me to be what we are talking about, sorry if I am not getting something.

Ive never studied math or physics outside of school but I do think about these things and some things I stumble across bother and interest me. To quote Manuel however, 'I know nothing'.

I know its nonsense simply from my own experience but 1/2 + 1/4 + 1/8... doesn't seem to me to ever reach 1 either.

 

[ShaunM]

It seems my understanding of math is not sufficient to allow me to understand how 1 is ever reached but thanks anyway to all of you, I really appreciate your time. Its enough for me to know that people with more knowledge can assure me that 1 is reached. The fact that this conclusion is also visible in my real experiences is a bonus:)

 

[A.T.]

The lines don't seem to me to be what we are talking about, sorry if I am not getting something.

I'm talking about this:

1 =
1/2 + 1/2 =
1/2 + 1/4 + 1/4 =
1/2 + 1/4 + 1/8 + 1/8 =
1/2 + 1/4 + 1/8 + 1/16 + 1/16 =
...

Do you see that each line sums up to 1? Do you see that repeating the subdivision infinitely many times won't change that?

 

[Ibix]

Thanks. To my (admittedly untrained) eye that's hard to understand.

He's generating each line by copying the line above but with the last term replaced by two halves of it. The sum is one, and it never stops being one no matter how many times you split the last term.

Think of actually doing it - start with a 1m ruler and cut it in half. The total length is still 1m. Cut one half in half (two quarter-meters). The total length is still 1m. Cut one of the quarter meters into two eighth meters. The total length is still 1m. That's all @A.T. is doing, just stated algebraically. The point is that no step in this process ever changes the length of the ruler - the total length is always 1m. That's a (rather boring) pattern that won't change no matter how many times you keep subdividing the last piece.

In practice, with a real ruler, it will of course end when you get to the atomic scale. But you only know this because we've done experiments and we know that matter is made of atoms - extra knowledge you bring to the problem. If you don't know about atoms, you can propose just keeping dividing the ruler into smaller and smaller segments forever, and the total length stays 1m.

So the only problem with dividing a ruler into infinitely many pieces is that we know a ruler is actually made of a finite number of pieces. We don't know that about space itself, though, so there might be absolutely no problem about thinking about subdividing it infinitely many times.

The other problem you have, I think, is that you are thinking that you have divided the task of moving from A to B into infinitely many subtasks, so it can't be completed. But you've forgotten to think about how much time each subtask takes. Assuming that the object is moving at constant speed (which is why I asked a couple of pages ago), the amount of time each step takes is half the previous one. Just like with the ruler, simply dividing the amount of time up infinitely many times isn't a problem in principle. It may turn out to be one in practice, but we have no evidence of that, and your argument doesn't provide one.

The fundamental problem is that you are trying to impose discrete thinking onto an example that can be well explained by continuous variables. If time and space are continuous (which is a reasonable assumption as far as we are aware) then you can continue subdividing them infinitely, and the total distance and time remain unchanged. If your objection to that is "there must be a first step in moving" then the only answer we can give you is "sorry, you're wrong" and ask you exactly when you think the "keep on subdividing" will suddenly stop adding to one, or when a further subdivision is impossible - and you cannot answer that. It feels strange, I agree, but it's perfectly self-consistent.

 

[ShaunM]

I'm talking about this:Do you see that each line sums up to 1? Do you see that repeating the subdivision infinitely many times won't change that?

I obviously see that each line sums up to 1. I don't see where you are getting the second 'half' from each time you reduce. This is the missing half we are talking about and there it is each time. It seems to me this is being magicked out of thin air but I understand I don't know enough to comprehend why this isn't so.

 

[Ibix]

I obviously see that each line sums up to 1. I don't see where you are getting the second 'half' from each time you reduce. This is the missing half we are talking about and there it is each time. It seems to me this is being magicked out of thin air but I understand I don't know enough to comprehend why this isn't so.

See the first two paragraphs of my last post.

 

[ShaunM]

He's generating each...

Thanks for this reply.

I do know about atoms and I understand fully what you are saying about the ruler, no matter how small its cut up, is still the length it always was. What I don't think it will ever be however is no ruler. No matter how many times its cup up you would never reach 0.

"sorry, you're wrong"

I really like this answer and I mean that sincerely:) I assume I am wrong I am just trying to understand why.

 

[A.T.]

I understand fully what you are saying about the ruler, no matter how small its cut up, is still the length it always was.

Then the time to travel along will also be always be the same.

What I don't think it will ever be however is no ruler. No matter how many times its cup up you would never reach 0.

The point was to reach the end in finite time, not in zero time.

 

[jbriggs444]

In practice, with a real ruler, it will of course end when you get to the atomic scale.

And before. As a carpenter you learn not to ignore the width of the kerf.

 

[ShaunM]

Then the time to travel along will also be always be the same.The point was to reach the end in finite time, not in zero time.

I think the way I am seeing it is that trying to reach 1 on the way 'up' is the same as trying to reach '0' on the way down. Is this not so?

 

[jbriggs444]

I think the way I am seeing it is that trying to reach 1 on the way 'up' is the same as trying to reach '0' on the way down. Is this not so?

Either way you have to pass by an uncountable infinity of points to get there. Passing a countable subset is a trivial matter.

 

[russ_watters]

Im actually quite comfortable with the thought that 1 is never reached as it seems to object isn't actually trying to get there anyway.

See my graphs in post #41 (the first two in particular): there is zero difference in the motion when described using Zeno's logic, it just presents slightly differently as an uneven spacing of points on the graph. The object gets to the goal just as surely as you have no trouble walking across a room.

What would cause the object never to reach the goal is if each step took equal time. Then it would be obvious that the total time would be infinite.

 

[Herman Trivilino]

What would cause the object never to reach the goal is if each step took equal time. Then it would be obvious that the total time would be infinite.

Right, because the number of steps is infinite.

 

[sophiecentaur]

Right, because the number of steps is infinite.

There is no actual need to bring Infinity (which is yet another of those concepts that make people uneasy) into it explicitly if you just do what the introduction to differential calculus does when it describes velocity as the limit of δx/δt as δt approaches zero. The concept of Limits is not too intuitive but it is a good way into a lot of these sort of problems.
That Zeno has done us no favours over the years,