1=0.999... infinities and box of chocolates..Phliosophy of Math...

[Motor Daddy]

You're still not getting it. There's no end to an infinite line of zeros. You can't put a 1 at the end, because there's no end. That's what infinite means.

I said that was a given. I gave you that when I wrote 1/infinity=0 for you. That is your equation for your theory on the division of paper. The equation for my way is 1/infinity=infinitesimal.

One more time, James:

You say you start with 1 piece of paper and end up with zero paper.

I say I start with 1 piece of paper and end up with 100% of the paper, albeit divided infinitely.

 

[James R]

I said that was a given. I gave you that when I wrote 1/infinity=0 for you. That is your equation for your theory on the division of paper. The equation for my way is 1/infinity=infinitesimal.

That's wrong.

1/infinity = 0.

1/(very large number) is infinitesimal.

You're mixing up the idea of infinity with the idea of an inconceivably large number. They aren't the same thing.

You say you start with 1 piece of paper and end up with zero paper.

I say I start with 1 piece of paper and end up with 100% of the paper, albeit divided infinitely.

No.

I keep all the paper. I have one piece that's 1/2 the original sheet, 1 piece that's 1/4 of the original sheet, 1 piece that's 1/8 of the original sheet, etc.

All up, I have an infinite number of pieces of paper that together make the original sheet.

 

[Motor Daddy]

That's wrong.

1/infinity = 0.

1/(very large number) is infinitesimal.

You're mixing up the idea of infinity with the idea of an inconceivably large number. They aren't the same thing.



No.

I keep all the paper. I have one piece that's 1/2 the original sheet, 1 piece that's 1/4 of the original sheet, 1 piece that's 1/8 of the original sheet, etc.

All up, I have an infinite number of pieces of paper that together make the original sheet.

James, when you divide a piece of paper into 2 pieces (1/2) you have two pieces that are 50% of the original 100%. If you divide the paper into 4 pieces (1/4) then each piece is .25.

Do you think that when you divide a piece of paper into 2 pieces (1/2, or 1 piece of 2, or 50%) that the other piece just vanishes into thin air due to the fact that you are only concerned with 1 of the 2 pieces?

 

[James R]

Motor Daddy:

I use all the pieces

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

No part of the paper vanishes.

First I tear the paper in half. One half I put to one side. I take the other half and tear it in half, making two pieces that are each 1/4 of the original sheet of paper. I put one of those aside, take the other, tear it in half. And so on.

The end result is the sum you see above. No paper is discarded in this process.

 

[Motor Daddy]

Maybe we should talk about cutting a pizza into pieces. There you have it, a nice hot out of the oven pizza. You take your handy dandy pizza cutter and your first cut you cut the pizza into 2 pieces, each .5 of the original pizza. The very next cut you cut the pizza into 4 pieces of .25 of the original.

My statements:

1. The total 100% of the original pizza is always on the plate.
2. No matter how many cuts you make, you can not make the pieces zero percent. Not a chance in 7 Eleven.

 

[James R]

2. No matter how many cuts you make, you can not make the pieces zero percent. Not a chance in 7 Eleven.

And so...?

 

[Motor Daddy]

And so...?

So it means that your pieces are not 0%. If you say you made so many cuts on the pizza that the pieces are 0% then you say there is no pizza.

 

[James R]

Motor Daddy:

Here's what you're describing with your pizza slicing:

1 x 1 = 1 (we start off with 1 piece of pizza)
2 x 1/2 = 1 (now we have 2 slices, each is 1/2 the pizza)
4 x 1/4 = 1 (now we have 4 slices, each is 1/4 of the pizza).
8 x 1/8 = 1
16 x 1/16 = 1
slice
slice
...
1 billion x 1/billion = 1
slice
slice
...
(very large number of slices) x (infinitessimally small size slices) = 1 pizza
slice
slice
...
an infinite number of slices x zero size for each slice =1 pizza.

It seems you're willing to go to the second-last step, but not to the last step.

 

[Motor Daddy]

Motor Daddy:

Here's what you're describing with your pizza slicing:

1 x 1 = 1 (we start off with 1 piece of pizza)
2 x 1/2 = 1 (now we have 2 slices, each is 1/2 the pizza)
4 x 1/4 = 1 (now we have 4 slices, each is 1/4 of the pizza).
8 x 1/8 = 1
16 x 1/16 = 1
slice
slice
...
1 billion x 1/billion = 1
slice
slice
...
(very large number of slices) x (infinitessimally small size slices) = 1 pizza
slice
slice
...

Perfect to this point.

an infinite number of slices x zero size for each slice =1 pizza.

Dead wrong!

 

[James R]

Is it too much to ask why it is dead wrong?

This is just another "because Motor Daddy says so" thing, isn't it?

Just like your fantasy theory of relativity.

 

[Motor Daddy]

Is it too much to ask why it is dead wrong?

This is just another "because Motor Daddy says so" thing, isn't it?

Just like your fantasy theory of relativity, it is disproved by reality.

No problem, James.

Anything times zero is zero. If the pieces are zero size, it doesn't matter how many pieces there are, the percentage of the total pizza is 0%, not 100%.

 

[James R]

No problem, James.

Anything times zero is zero.

Not when it comes to infinity.

If the pieces are zero size, it doesn't matter how many pieces there are, the percentage of the total pizza is 0%, not 100%

No. I cut the pizza into an infinite number of pieces. Not just a very large number of pieces, but an infinite number.

How big do you think the slices are when there are an infinite number of them? If not zero, then what?

 

[Motor Daddy]

Not when it comes to infinity.

So you say you can make things disappear into thin air. I get that. I'd like to see the act sometime. Must be quite the illusion. Einstein would be proud of you, James.

No. I cut the pizza into an infinite number of pieces. Not just a very large number of pieces, but an infinite number.

So are you still cutting or did you stop?

How big do you think the slices are when there are an infinite number of them? If not zero, then what?

Infinitesimal. They have a size greater than zero. It doesn't matter how big they are, they are, and they are greater than zero. They exist! Your pieces don't exist. I will eat and you will be hungry.

 

[James R]

So you say you can make things disappear into thin air. I get that. I'd like to see the act sometime. Must be quite the illusion. Einstein would be proud of you, James.

...

So are you still cutting or did you stop?

If I have cut an infinite number of pieces, it's safe to assume that I've been cutting for longer than the universe has been here. If you like, you can say that I never stopped cutting. If I already have an infinite number of pieces and I cut it a few more times, I'll still have an infinite number of pieces, so it's hard to say when I stop cutting.

Infinitesimal. They have a size greater than zero. It doesn't matter how big they are, they are, and they are greater than zero.

No. The only way I can have an infinite number of pieces is if they each have zero size. If they still have a size, no matter how small, then I've only made a finite number of cuts and I'll need to cut it some more to reach an infinite number of cuts.

 

[Motor Daddy]

If I have cut an infinite number of pieces, it's safe to assume that I've been cutting for longer than the universe has been here. If you like, you can say that I never stopped cutting. If I already have an infinite number of pieces and I cut it a few more times, I'll still have an infinite number of pieces, so it's hard to say when I stop cutting.

It's not hard to say when you'll stop cutting. You will NEVER stop cutting. If you don't eat until you've finished cutting, you never eat!

No. The only way I can have an infinite number of pieces is if they each have zero size. If they still have a size, no matter how small, then I've only made a finite number of cuts and I'll need to cut it some more to reach an infinite number of cuts.

This isn't subtraction, James. We are not subtracting anything to get from 100% to 0%. We are DIVIDING! The total is always 100%, you are just making lines. That's it. Just more lines. How many lines does it take to make the pizza pieces 0%????

 

[James R]

It's not hard to say when you'll stop cutting. You will NEVER stop cutting. If you don't eat until you've finished cutting, you never eat!

That depends how fast I cut. If I can cut infinitely quickly, then I can finish cutting and infinite number of pieces.

This isn't subtraction, James. We are not subtracting anything to get from 100% to 0%. We are DIVIDING! The total is always 100%, you are just making lines. That's it. Just more lines. How many lines does it take to make the pizza pieces 0%????

An infinite number of lines.

And here's a funny thing: once I have an infinite number of lines, if I make even more cuts, so that I have double the number of slices as before, I actually have the same number of slices still!

See, your problem is that you assume that the usual rules of multiplication and division (and addition and subtraction) apply to infinite numbers. They don't.

 

[Motor Daddy]

That depends how fast I cut. If I can cut infinitely quickly, then I can finish cutting and infinite number of pieces.

What do you mean "infinitely quick?"

An infinite number of lines.

And here's a funny thing: once I have an infinite number of lines, if I make even more cuts, so that I have double the number of slices as before, I actually have the same number of slices still!

See, your problem is that you assume that the usual rules of multiplication and division apply to infinite numbers. They don't.

That's not the problem. The problem is you assuming that you can count how many slices you have and produce a finite number. If you do, you lose. So just keep cutting, unless you want to forfeit because you don't want to keep cutting forever??

 

[James R]

What do you mean "infinitely quick?"

I mean if you count the number of cuts I make per minute, say, it's infinite. If I can do that, I can cut the pizza into an infinite number of zero-size pieces in one minute!

That's not the problem. The problem is you assuming that you can count how many slices you have and produce a finite number. If you do, you lose. So just keep cutting, unless you want to forfeit because you don't want to keep cutting forever??

No need to cut forever if I can cut infinitely quickly. I only need a minute!

 

[Motor Daddy]

I mean if you count the number of cuts I make per minute, say, it's infinite. If I can do that, I can cut the pizza into an infinite number of zero-size pieces in one minute!

Every cut you make you are counting, right? So how many cuts did you already make in the last 1 minute?

No need to cut forever if I can cut infinitely quickly. I only need a minute!

If you cut faster doesn't your clock mess up?

 

[James R]

Every cut you make you are counting, right? So how many cuts did you already make in the last 1 minute?

An infinite number.

If you cut faster doesn't your clock mess up?

Not at this rate, no.
---

Look, remember I asked you earlier whether you had an in-principle objection to the maths or an objection on the grounds of mere practicality?

You said you had an in-principle objection, but all I'm seeing from you here is arguments about practicalities.