Paradox!

[Archimedes]

Well, which interval is it that you feel completes the hour and gets you to noon?

In the real world, if you imagine dividing up the real hour between 11 and noon into 30 minutes, 15 minutes, 7.5 minutes, etc. ad nauseum into ever small and smaller intervals half the size of the previous one, which interval gets you to noon?

Does thinking about a finite length of time this way somehow cause clocks to run slower and slower as the hour progresses and thus prevent clocks from striking noon?

 

[Tom Sawyer]

Well, which interval is it that you feel completes the hour and gets you to noon?

In the real world, if you imagine dividing up the real hour between 11 and noon into 30 minutes, 15 minutes, 7.5 minutes, etc. ad nauseum into ever small and smaller intervals half the size of the previous one, which interval gets you to noon?

Does thinking about a finite length of time this way somehow cause clocks to run slower and slower as the hour progresses and thus prevent clocks from striking noon?

None of the intervals get you to noon. I'm not sure what you're trying to get at with the time slowing down stuff, but you're just taking smaller and smaller steps forward and not ever actually arriving there and the discussion is about what happens in each of those steps. There's not a difference in this scenario when you're talking about what you're doing within a 15 minute interval and talking about what you're doing within a 0.00000000000000000000002 second interval.

That interval never actually gets down to zero.

If you're talking about how to implement the problem in the real world as opposed to just as a pure math solution, then the time it takes to carry out the operation exceeds the interval for the operation quite quickly and you reach noon after just a few iterations. If you decide you're viewing it as a pure math problem and the details of how to implement it in the real world aren't being factored in, then the consecutive intervals never stop and the question of what happens in the real world implementation if you go to a time period after they stop isn't then related to the question because you decided in the premise that the real world implementations aren't part of the problem and you're simply looking at the equation.

 

[Archimedes]

No, you aren't subtracting infinity from infinity. The number of balls would equal (1 -1) + (1 - 1) + (1 - 1) + (1 - 1)... = 0 + 0 + 0 + 0...

Why isn't thinking about it as (1+1+1+1+1+1.....) - (1+1+1+1+1+1.....) just as valid as way as you are doing here? Isn't there a 1:1 correspondence between the balls being added and removed either way? One way of looking at it (yours) shows it as 0 balls being added infinitely many times and another way of looking at it sees it as infinitely many balls being added in an hour and infinitely many balls being removed in the same hour.

And I think this is beero's point. Our everyday fast and loose thinking about infinity leads to contradictions and paradoxes depending upon how you look at the problem.

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That interval never actually gets down to zero.

So you're saying that an hour after 11 O'Clock it won't be noon?

 

[Tom Sawyer]

That interval never actually gets down to zero.

So you're saying that an hour after 11 O'Clock it won't be noon?

How could you possibly think that that's what I'm saying?

I'm saying that 11:59.50 isn't noon.
Then 11:59.75 isn't noon
Then 11:59.875 isn't noon
Then 11:59.935 isn't noon
Then 11:59.96875 isn't noon
Then 11:59.984375 isn't noon
Then 11:59.9921875 isn't noon
Then 11:59.99609375 isn't noon

And so on and so on. If you continue halving the difference between the current time and noon, you'll never reach noon.

 

[Archimedes]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

But you have to halve the current time to half way to noon. And from then to the next halfway point to noon. And from then to the next halfway point. Ad infinitum.

So why does noon come and go every day if what you say is true?

 

[Kharakov]

So you're saying that an hour after 11 O'Clock it won't be noon?

1000 years ago it wasn't.

 

[Archimedes]

So you're saying that an hour after 11 O'Clock it won't be noon?

1000 years ago it wasn't.

I'm presuming for the purposes of the thread that changes in the earth's rotational period, relativistic effects, etc. aren't being taken into consideration if that's the kind of thing you're getting at.

 

[Kharakov]

So you're saying that an hour after 11 O'Clock it won't be noon?

1000 years ago it wasn't.

I'm presuming for the purposes of the thread that changes in the earth's rotational period, relativistic effects, etc. aren't being taken into consideration if that's the kind of thing you're getting at.

Different direction. I was thinking that the etymology of the word noon might have to do with the midpoint of the day being the "reflection" of the sun over a specific axis at some point in time, because the word is a palindrome "no" "on". I looked up the etymology of the word Noon on etymonline, and found that 1000 years ago, noon meant 3pm. So... I was dropping a clever reference that I thought was rather interesting, and if anyone has a feeling I was also doing something else, you're right.

Noon came from non, came from nona, which meant the 9th hour, or 3pm.

 

[Tom Sawyer]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

But you have to halve the current time to half way to noon. And from then to the next halfway point to noon. And from then to the next halfway point. Ad infinitum.

So why does noon come and go every day if what you say is true?

Ya, that's the entire point. It's why the problem needs to be discussed as a pure math problem as opposed to a math problem which can be implemented in the real world.

 

[Deepak]

That interval never actually gets down to zero.

So you're saying that an hour after 11 O'Clock it won't be noon?

How could you possibly think that that's what I'm saying?

I'm saying that 11:59.50 isn't noon.
Then 11:59.75 isn't noon
Then 11:59.875 isn't noon
Then 11:59.935 isn't noon
Then 11:59.96875 isn't noon
Then 11:59.984375 isn't noon
Then 11:59.9921875 isn't noon
Then 11:59.99609375 isn't noon

And so on and so on. If you continue halving the difference between the current time and noon, you'll never reach noon.

If at 6:55 I imagine two points 0, and 1 then imagine a line being slowly drawn between from point 0 which reaches point 1 at 7:00 does the line never reach 1? Does it never reach 7:00?

Or alternately does either my imagination or time become quantized, and what is the limit on the resolution in this case?

 

[Tom Sawyer]

If at 6:55 I imagine two points 0, and 1 then imagine a line being slowly drawn between from point 0 which reaches point 1 at 7:00 does the line never reach 1? Does it never reach 7:00?

Or alternately does either my imagination or time become quantized, and what is the limit on the resolution in this case?

Well, at what point do you feel it reaches point 1? If, for every given interval, you halve the distance between point 0 and point 1, how many intervals do you go through before reaching point 1?

 

[Jimmy Higgins]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

In the question at hand, Noon is an asymptote.

 

[Deepak]

If at 6:55 I imagine two points 0, and 1 then imagine a line being slowly drawn between from point 0 which reaches point 1 at 7:00 does the line never reach 1? Does it never reach 7:00?

Or alternately does either my imagination or time become quantized, and what is the limit on the resolution in this case?

Well, at what point do you feel it reaches point 1? If, for every given interval, you halve the distance between point 0 and point 1, how many intervals do you go through before reaching point 1?

I asked you first

 

[Tom Sawyer]

If at 6:55 I imagine two points 0, and 1 then imagine a line being slowly drawn between from point 0 which reaches point 1 at 7:00 does the line never reach 1? Does it never reach 7:00?

Or alternately does either my imagination or time become quantized, and what is the limit on the resolution in this case?

Well, at what point do you feel it reaches point 1? If, for every given interval, you halve the distance between point 0 and point 1, how many intervals do you go through before reaching point 1?

I asked you first

I say you never reach point 1. When do you say you reach point 1?

 

[Archimedes]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

In the question at hand, Noon is an asymptote.

So does noon arrive after an hour has passed or not? Time passes at a constant rate of 1 hour per hour in the question at hand, which is why I'm struggling to understand the claim that a period of one hour never passes in the scenario being discussed.

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If at 6:55 I imagine two points 0, and 1 then imagine a line being slowly drawn between from point 0 which reaches point 1 at 7:00 does the line never reach 1? Does it never reach 7:00?

Or alternately does either my imagination or time become quantized, and what is the limit on the resolution in this case?

Well, at what point do you feel it reaches point 1? If, for every given interval, you halve the distance between point 0 and point 1, how many intervals do you go through before reaching point 1?

I asked you first

I say you never reach point 1. When do you say you reach point 1?

In Deepak's scenario? At 7 O'Clock, 5 minutes after his scenario starts at 6:55

 

[Jimmy Higgins]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

In the question at hand, Noon is an asymptote.

So does noon arrive after an hour has passed or not?

Time isn't running linearly in the posed hypothetical.

 

[Tom Sawyer]

In Deepak's scenario? At 7 O'Clock, 5 minutes after his scenario starts at 6:55

OK yes, if you spend 5 minutes slowly drawing a line between two points, it will take you 5 minutes to draw that line. That's not a relevant scenario to the rest of the thread, however.

For the scenario my question was asking, you never reach point 1.

 

[Archimedes]

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

In the question at hand, Noon is an asymptote.

So does noon arrive after an hour has passed or not?

Time isn't running linearly in the posed hypothetical.

What makes you say that? Nothing in the proposed scenario suggests anything of the sort.

 

[Archimedes]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

 

[Jimmy Higgins]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

What step of ball manipulation is he performing when it becomes Noon?