The Edge Is A Ghost?

[cabfrank]

I find it interesting too. No need to stop.

[JSumm]

*Sorry if me and Synov are going off topic* I find it interesting, but I'll leave it be, and get back to the knives. You get the last word here Synov. Thank you for the discussion.

Heck no you better not! The topic is about the ghost. Seems rather on topic to me and very interesting. Even if my simple mind cannot keep up. Carry on! May just insert some humor here and there to keep us simple minded people (speaking for myself) engaged. Kinda like a picture book, but super fascinating none the less.

[Synov]

Well in this case we were talking about the "process" of halving space, so that process has no limit? Or in other words you could half space infinitely without end?

That's the same as asking if there's a final number of the form 1/2^n. There isn't. There are infinite integers n and infinite numbers of the form 1/2^n.

Or we could say (hypothetically) drive from point A to beyond the boundaries of the known Universe and keep driving, forever. How many miles is forever? It's undefined as it's a number without end.

Sure, if the universe is infinitely large and we ignore practical limitations like running out of gas. What stops a photon from traveling forever? The amount of miles would be infinite. If infinite miles is "undefined" then so is "forever," which just means infinite time. Neither are undefined. You may not personally agree with their use, but mathematics certainly rigorously defines them, and infinity is even used in physics.

I get that you can look at a sequence of fractions and say, it sums to 1. But if you can divide it infinitely, .01, .001, .0001, forever, there's no limit to how small you can make the division.

Correct, there is no limit to how small they can be, just as there is no limit to how high the integers go. You accept the latter without accepting the former, when really they are the same thing.

Do you not find it philosophically interesting that between any unit is an infinite amount of space. It's like asking, what's the biggest number in existence. We say infinity as a place holder. Because there is no such number, you can always count to a higher number.

There are an infinite amount of numbers between any two integers, but not space. The space between two adjacent integers is always 1. Infinity is not "the biggest number" it is simply a number larger than any integer. There are multiple infinities of different sizes just like there are multiple integers, but that's a whole other topic.

Maybe I can explain the idea of the paradox like this, if you were to watch an object move half the distance closer to another object, and half, and half... as it approached the object, at a certain point, as the half distances became smaller and smaller, the object would appear to stop moving.

No, because the process you described happens all around us. Any time any distance is crossed, you are watching the distance being halved infinitely. Again, you are forgetting that the speed of crossing remains constant, so the shorter the distance the shorter the time and there is no observable slow down.

You can round up and call it close enough to one. But it isn't.

Again, you're talking about some finite step in the process, not the entire process. The infinite sum is exactly 1. Of course if you only look at some partial step in the process, you don't reach 1. Try to remember that how you cut up a distance is completely arbitrary and changes nothing about the total distance or how long it takes to cross.

[ZrowsN1s]

@Synov the crowd says keep going

So I'll go a little more. You are describing the paradox.

"That's the same as asking if there's a final number of the form 1/2^n. There isn't. There are infinite integers n and infinite numbers of the form 1/2^n."

"There are an infinite amount of numbers between any two integers, but not space. The space between two adjacent integers is always 1."

And here is where the pardox lies. Every number represents or describes a point in space. Like notches on a ruler. There are infinite numbers between two points. Do you see the pardox? It is both correct that a mile is a mile, and that you can divide it infinitely.

Calculus comes in and says, you can divide the area of a square into infinitely smaller parts, but when you add it all up, the area is the same. You can divide a mile into infinitely smaller parts, but it's still a mile. And it declares problem solved, it's a square, or it's a mile. But calculus is skipping over the pardox. Skipping to the end. You can't write out the terms of an infinite sequence, because the sequence never ends. It's 1,2,3....infinity. Or 1/2, 1/4, 1/8,.... 0. The pardox is, if you don't skip to the end, if you don't throw in the "..." and say well it all sums to 1 in the end, you can go on forever.


"Of course if you only look at some partial step in the process, you don't reach 1."

Exactly. The nature of the pardox is the thought that you can stay in the partial steps of dividing the distance infinitely.

If you could make yourself smaller, or zoom in, so you were relative to the next smaller step, you could do so infinitely. Which is why I said before an object would appear to stop moving (to someome without a zoomed in view), it hasn't stopped, it's still moving at the same speed. But it is moving a distance so small as to be imperceptible to an observer. If you zoomed in small enough, relatively it would look like there was still a mile between the objects. Go half that distance, zoom in again, still looks like a mile relatively. You could do this forever going smaller and smaller. Logically this works. I think even mathematically it works. And yet we touch things all the time, we travel miles, the area of a square is the area of a square, a mile is a mile, we can move. That's the pardox.

[Synov]

And here is where the pardox lies. Every number represents or describes a point in space. Like notches on a ruler. There are infinite numbers between two points. Do you see the pardox? It is both correct that a mile is a mile, and that you can divide it infinitely.

I don't see the paradox. How does one contradict the other? Again, I think you're incorrectly assuming that infinite parts means infinite distance or time.

You can't write out the terms of an infinite sequence, because the sequence never ends. It's 1,2,3....infinity. Or 1/2, 1/4, 1/8,.... 0. The pardox is, if you don't skip to the end, if you don't throw in the "..." and say well it all sums to 1 in the end, you can go on forever.

But there is no need to write out the terms in order to sum them, so again I don't see the paradox. If you can find the limit of the sequence of finite sums then you can do the infinite sum. Calculus is great because it allows us to do infinite operations in finite time, but that's not a paradox, it's a useful tool.

The nature of the pardox is the thought that you can stay in the partial steps of dividing the distance infinitely.

I think you're trying to say that the paradox is that you can complete an endless number of steps. Well you can, if the time to complete each step is getting smaller and smaller. Again, you keep on incorrectly assuming that an infinite amount of steps takes infinite time without actually calculating the amount of time it would take.

If you could make yourself smaller, or zoom in, so you were relative to the next smaller step, you could do so infinitely. Which is why I said before an object would appear to stop moving (to someome without a zoomed in view), it hasn't stopped, it's still moving at the same speed. But it is moving a distance so small as to be imperceptible to an observer.

That movement would be over a time equally as small, so it would impossible to observe a "stop." All you would see is the object moving at a constant speed across the entire distance. Dividing the distance up into small parts changes nothing about what you observe.

If you zoomed in small enough, relatively it would look like there was still a mile between the objects. Go half that distance, zoom in again, still looks like a mile relatively. You could do this forever going smaller and smaller. Logically this works. I think even mathematically it works. And yet we touch things all the time, we travel miles, the area of a square is the area of a square, a mile is a mile, we can move. That's the pardox.

Again, there's no contradiction there, so no paradox. The only contradiction is in the hidden assumptions you add to the problem.

[ZrowsN1s]

You don't see the contradiction? It doesn't matter if the time it takes to cross the next half distance gets faster with each step, there's always another distance to cross. You can't reach the end of an infinite set of steps no matter how fast you go through them. There are endless 'infinite' steps.

[benja-man]

What calculus does is to give you means to determine the value of 1/2 + 1/4 + 1/8 + ... in finitely many steps. Those steps are not summing up a finite amount of fractions but instead a number of logical steps (what we call a mathematical proof) to arrive at the conclusion that the value of this infinite sum is without question equal to 1.

To question this means to question the logical steps involved - which is totally a valid thing we do in mathematical logic / axiomatic set theory - but then you are not talking about individual arithmetic operations but about the logical foundations involved.

[Synov]

You don't see the contradiction? It doesn't matter if the time it takes to cross the next half distance gets faster with each step, there's always another distance to cross. You can't reach the end of an infinite set of steps no matter how fast you go through them. There are endless 'infinite' steps.

But it does matter, because you're saying something can't happen in finite time without actually doing the math to determine that it can't. Again, an infinite amount of steps does not imply an infinite amount of time to complete them all. But that's the fundamental assumption your argument is making without justification.

Also, you're not reaching the end of an infinite set in the sense that there is a final step, you're reaching the end in the sense that you have completed every step in the set. The former is true when completing every step in a finite set, but not for infinite sets. It's a subtle difference, but one that becomes very important when dealing with infinite sets. You're trying to apply the logic of finite sets to infinite sets where it doesn't apply.

[Jesla]

At this rate we might as well talk politics…. I think an awesome thread just got trashed.

[ZrowsN1s]

You don't see the contradiction? It doesn't matter if the time it takes to cross the next half distance gets faster with each step, there's always another distance to cross. You can't reach the end of an infinite set of steps no matter how fast you go through them. There are endless 'infinite' steps.

But it does matter, because you're saying something can't happen in finite time without actually doing the math to determine that it can't. Again, an infinite amount of steps does not imply an infinite amount of time to complete them all. But that's the fundamental assumption your argument is making without justification.
.....

Let me try again. We agree that you can divide a unit into infinite steps, yes? I think we agree. If we don't agree on this, we don't agree on the premise of the pardox.

Imagine you are trying to write out every one of those steps, or take every one of those infinite steps. Can you explain mathematically or logically how you can write out infinite steps, or take infinite steps, at any speed?

At this rate we might as well talk politics…. I think an awesome thread just got trashed.

Apologies, some wanted to hear it. We will wrap it up.

[Mushroom]

I, for one, have thoroughly enjoyed following along with this thread!

[JSumm]

Me too

[Synov]

Let me try again. We agree that you can divide a unit into infinite steps, yes? I think we agree. If we don't agree on this, we don't agree on the premise of the pardox.

Of course.

Imagine you are trying to write out every one of those steps, or take every one of those infinite steps. Can you explain mathematically or logically how you can write out infinite steps, or take infinite steps, at any speed?

Yes, let's say the first step takes t amount of time, and each subsequent step you reduce the amount of time it takes by a fraction f. Then the total amount of time to complete every step is the infinite sum

t+t(1-f)+t(1-f)^2+t(1-f)^3+...

This is called a geometric series and its sum is t/f. If you want a proof of this, I can write it out for you. Since t is finite and f is not zero, this means infinite steps will be completed in finite time. QED.

[ZrowsN1s]

Let me try again. We agree that you can divide a unit into infinite steps, yes? I think we agree. If we don't agree on this, we don't agree on the premise of the pardox.

Of course.

Imagine you are trying to write out every one of those steps, or take every one of those infinite steps. Can you explain mathematically or logically how you can write out infinite steps, or take infinite steps, at any speed?

Yes, let's say the first step takes t amount of time, and each subsequent step you reduce the amount of time it takes by a fraction f. Then the total amount of time to complete every step is the infinite sum

t+t(1-f)+t(1-f)^2+t(1-f)^3+...

This is called a geometric series and its sum is t/f. If you want a proof of this, I can write it out for you. Since t is finite and f is not zero, this means infinite steps will be completed in finite time. QED.

This (highlighted) makes no logical sense what so ever. The tell for this exists in the '...'

Where does this end ? '...'
t+t(1-f)+t(1-f)^2+t(1-f)^3+...

And if you reduce the time to complete the next step by a fraction, don't you then run into the same logical problem of the fraction of time being able to be infinitely smaller? 1/2 the time, 1/4 the time, 1/8, 1/16, 1/32, '...' infinity. Just like the mile can be broken up into infinite steps, you can break an hour into infinitely smaller fractions of time. The process of dividing a mile infinitely smaller, by an infinitely smaller fraction of time, can continue infinitely.

All the proof's do is show you that a mile is a mile, and an hour is an hour, and no matter how you slice them, it's an hour and a mile. That's well and and good, and matches the reality we observe. But the proofs do nothing to address the pardox and logic of the endless process of infinity. And the endless process of infinitely dividing something. They merely show you what happens if you skip the paradox with a '...' .

The proofs don't address the logical contradiction of an 'end' to 'endless' steps.
The lack of logic of 'complete every step' in an 'infinite' series of steps. You can't complete them by definition of 'infinite'. What's the last number that comes before infinity? (Infinity -1 ) seriously though.

[Synov]

Where does this end ? '...'

What do you mean by "end?" It doesn't have a final term, if that's what you mean. It's an infinite sum.

And if you reduce the time to complete the next step by a fraction (or another way to say it would be to increase the speed to complete the next step), don't you then run into the same logical problem of the fraction of time being able to be infinitely small?

I don't know what you mean by "infinitely small" or what the logical problem is. The amount of time for the nth step is t(1-f)^n which is always a real, non-zero number. So not infinitely small by any definition I can think of. If you mean there is no limit to how small this number gets, then yes. But that's not infinitely small. That would be like saying there is an infinitely large integer because there is no limit to how high the integers can go. No, every integer is finite, and that's fine.

Just like the mile can be broken up into infinite steps, you can break an hour into infinitely smaller fractions of time.

Sure.

All the proof's do is show you that a mile is a mile, and an hour is an hour, and no matter how you slice them, it's an hour and a mile. That's well and and good, and matches the reality we observe. But the proofs do nothing to address the pardox and logic of the endless process of infinity.

So far you haven't presented a paradox to address in the first place. You asked me to show the mathematics of how infinite steps could be completed at any speed. I showed that if the speed is getting faster by the same ratio at each step, then infinite steps can be completed in finite time.

The proofs do nothing to address the logical contradiction of an 'end' to 'endless' steps.

There is no contradiction there because endless in amount doesn't mean endless in time, as I've already shown. You're just conflating two different "ends."

The lack of logic of 'complete every step' in an infinite series of steps. You can't complete them by definition of 'infinite'. What's the last number that comes before infinity? (Infinity -1 ) seriously though.

You can complete all of them in finite time, as I already showed. Again, I think you're confusing the lack of a final step with being unable to complete infinite steps. By definition, infinite steps don't have a final step, but that has nothing to do with how long it takes to complete every one.

[VooDooChild]

Maybe just start a metaphysics discussion in the off topic section. I have a degree in mathematics and minored in philosophy. While I do appreciate these discussions, this one seems to have a lot of arguing semantics.

[ZrowsN1s]

Where does this end ? '...'

What do you mean by "end?" It doesn't have a final term, if that's what you mean. It's an infinite sum.

And if you reduce the time to complete the next step by a fraction (or another way to say it would be to increase the speed to complete the next step), don't you then run into the same logical problem of the fraction of time being able to be infinitely small?

I don't know what you mean by "infinitely small" or what the logical problem is. The amount of time for the nth step is t(1-f)^n which is always a real, non-zero number. So not infinitely small by any definition I can think of. If you mean there is no limit to how small this number gets, then yes. But that's not infinitely small. That would be like saying there is an infinitely large integer because there is no limit to how high the integers can go. No, every integer is finite, and that's fine.

Just like the mile can be broken up into infinite steps, you can break an hour into infinitely smaller fractions of time.

Sure.

All the proof's do is show you that a mile is a mile, and an hour is an hour, and no matter how you slice them, it's an hour and a mile. That's well and and good, and matches the reality we observe. But the proofs do nothing to address the pardox and logic of the endless process of infinity.

So far you haven't presented a paradox to address in the first place. You asked me to show the mathematics of how infinite steps could be completed at any speed. I showed that if the speed is getting faster by the same ratio at each step, then infinite steps can be completed in finite time.

The proofs do nothing to address the logical contradiction of an 'end' to 'endless' steps.

There is no contradiction there because endless in amount doesn't mean endless in time, as I've already shown. You're just conflating two different "ends."

The lack of logic of 'complete every step' in an infinite series of steps. You can't complete them by definition of 'infinite'. What's the last number that comes before infinity? (Infinity -1 ) seriously though.

You can complete all of them in finite time, as I already showed. Again, I think you're confusing the lack of a final step with being unable to complete infinite steps. By definition, infinite steps don't have a final step, but that has nothing to do with how long it takes to complete every one.

I don't think you've shown finite time. As I showed in my example just like distance, you can divide time infinitely. Just like the mile. You're taking the thousand foot view of it and seeing the whole as finite. As I keep saying the paradox lies in the process of dividing. Which can be done infinitely. Imagine taking a math test where the instructions said you had to write out every term of the infinite sequence (show your work, every term) before you can solve. You'd never solve it with all the time in the world.

The ability to use the phrase 'lack of a final step' and 'able to complete the steps' in the same sentence and not see the contradiction is a head scratcher for me. Ok, I guess we will agree to disagree. But thank you very much for the interesting discussion. If nothing else you made me look at the problem more thoroughly than before, and I appreciate the conversation and time you took. Observed reality is definitely on your side of the argument.

Maybe just start a metaphysics discussion in the off topic section. I have a degree in mathematics and minored in philosophy. While I do appreciate these discussions, this one seems to have a lot of arguing semantics.

You are not wrong, I will cease and desist.

[Synov]

I don't think you've shown finite time.

I gave you a mathematical proof. Can you explain what part of it is wrong?

You're taking the thousand foot view of it and seeing the whole as finite.

No, I calculated it from the bottom up.

As I keep saying the paradox lies in the process of dividing. Which can be done infinitely. Imagine taking a math test where the instructions said you had to write out every term of the infinite sequence (show your work, every term) before you can solve. You'd never solve it with all the time in the world.

I could do it in finite time and finite space if I was able to write each term faster and smaller than the one before by the same fraction, as I've already shown and you've failed to refute. Saying that I would not be able to do this if it took the same amount of time to write each term or some other practical limitation neither shows a paradox nor does it refute that it can be done with a different method.

The ability to use the phrase 'lack of a final step' and 'able to complete the steps' in the same sentence and not see the contradiction is a head scratcher for me.

I don't understand how it can be a head scratcher for you when I already explained that completing the steps means doing every step in finite time, which doesn't require a final step. Only finite amounts of steps have a final step that is completed when every step is completed. We're just going in circles at this point because you won't respond to what I've already explained.

[ZrowsN1s]

I explained it pretty clearly. And you didn't satisfactorily refute it in my opinion. Your answers just dance around it. We both agree we've reached the end of a productive discussion on this topic. Thank you again for the effort, I hope no hard feelings regardless of disagreement.

-Matt

*I hope at least some people enjoyed the thought excercise and discussion.

[horzuff]

I'll give it a shot, without the equations and formulas:
Let's say You have a 100 m distance to cross with 1m/s speed. And let's say You take the journey by going in stints of half the remaining distance, so first 50m in 50s, then 25m in 25s and so on. Without stopping, constant speed. Will You have reached the end by 75s? no, You're 75% of the way there. Will You get there after 99.99999999999s? No, You're 99.99999999999% there. But will You get there after 100s? Yup, welcome to the other side .

Remember, You're never stopping, so it doesn't matter how small You've divided the distance into and how many parts there are as You always cover them at the same speed, so the sum of time to travel a certain distance at constant speed is always the same.

You're trying to argue, that by the sole fact that You can still imagine dividing the remaining part into smaller bits disproves the fact of how long it takes to get to the other side.