War in a 2d world

[LuthiX]

In your piece about war you said:
"In 2d, the single soldier at the front of the army is the only person that can fight. The battle consists of the two front line soldiers battling it out until one is taken out, then the next person steps up and goes at it. The battle proceeds until one of the armies has been annihilated."

I don't see why the soldiers can not fight next to each other. If 2d has a dimension of infinitely small, width, then the lifeforms of the 2d world would also have a width of infinitely small. Infinitely small width next to infinitely small width is still infinitely small? Please explain why they can not be next to each other.

[Keiji]

The key thing here is that a 2d world is infinitly small in width, not zero in width. 0 + 0 = 0, but if x is an infinitely small number, x + x > x. Which is why you cannot have two 2d figures next to one another.

[LuthiX]

Oh okay thanks, I didn't realise that.

[jinydu]

When I'm only thinking about what happens in a 2D universe, I find it's usually not helpful to think about it as "infinitely thin". Doing so implies comparing it to our 3D universe, which I think is unneccessary in cases like this.

[PWrong]

It's an interesting idea though. There could be a 2D universe in which objects can pass through each other as if they were going around. It wouldn't be consistent with our universe, but it would work on a computer program.

[RQ]

The key thing here is that a 2d world is infinitly small in width, not zero in width. 0 + 0 = 0, but if x is an infinitely small number, x + x > x. Which is why you cannot have two 2d figures next to one another.

This is untrue, and although the facts and mathematical reasoning show that they cannot be to their sides, this explanation is insufficient.

1/infinity=0.
Proof:
1/9=0.111...
1/9(9)=0.111...(9)
9/9=0.111...(9)
1=0.999...

thus 0.000...1=0 and in your case of x+x>x, it would be 0+0>0 which is not a true statement where x=x.

The reason why the infinitely thin soldiers cannot stand next to each other is because they have to bend in the third dimension to do so, and thus make a 2D universe in a 3D universe exist, which is forbidden with standard laws and properties.

Furthermore, the reason why our 3D existence is not fundamentally incorrect is a) because of the anthropic principle, but most importantly, b)

Because with respect to a lower dimensional space, the existence of the higher dimensional space is its "space" divided by 0.
For example a square with an area of 4 has 4 sets of infinitely stacked lines, so with respect to the line, the square has an are of 4/0. 4/0 is undefined, but when you extend it in the next dimension it infinitely covers for each unit length of the next dimension.

The length of the line with respect to the square is simply 2.

[pat]

0.99999... is a very convenient shorthand for the limit as n goes to infinity of the summation as k goes from one to n of the quantity 9 * 10^-k. Now, show me where this 0.0000...1 comes in? are you only taking the limit as n goes to one less than infinity? what is that number? and why are you approximating 9 * (1/9) when you're not approximating (9/9)?

Also, the integral from zero to one of 2 dx is greater than the integral from zero to one of 1 dx. So, I think there are grounds for thinking that x + x > x for infinitesimal x.

Now, surreal numbers... that's where it's at....

[RQ]

Limits?
Simple arithmetic for Christ's sake. Why do you have to complicate something when you just get jumbled results that not even you yourself understand. The 0.0..1 comes from 1-0.99~ which is 0.

[Keiji]

0.0r1 is NOT equal to 0.

0.0r1 is the definition of "an infinitely small number".

Suppose there is a FINITE line somewhere. Now, it has infinite points on it, because a point is infinately smaller than a line. These points are x units apart. If x is finite, then the line would be infinite. If x was zero, the line wouldn't exist. So, x must be 0.0r1, which makes the line finite.

You have already been proven wrong many times RQ, and you are way too persistant.

[RQ]

Ah, you are extending a point to the dimension of a line.
No a point is a point with respect to a point which could only be itself, and has 0 distance with respect to a line, but the length of a line with respect to a point is the length of the line,x, over 0 or x/0, which is undefined. With respect to the line, a point is x/x/0 which is who knows what.

[jinydu]

This is basically my point. The "size" of a 2D world is 0 with respect to a 3D world, but I don't see the point of comparing it to a 3D world. Doesn't it make more sense to measure a 2D world in square meters?

[PWrong]

RQ, just wait a year or two until you start learning calculus. It all makes perfect sense after that.

The thing is, there are conventions for dealing with infinity, such as limits, which you haven't learnt in school yet. Your reasoning is correct for arithmetic, but it doesn't work in the long run. Simple arithmetic only applies perfectly to finite and non-zero numbers.

For instance, if you draw a graph of y=x/x, you should get a straight line with a hole when x=0. It looks odd, but it makes sense in a way.

Pat, what's a surreal number? Is that like .9999...?

[pat]

Surreal numbers are usually represented as two sets, a left and a right. They have the property that there is no element on the right side which is less than or equal to any element of the left side. The value of the number is defined as the "simplest" number 'a' such that no member of the right set is less than or equal to 'a' and such that a is not less than or equal to anything in the left set.

There is a formal definition of "simplest". There is a way to add them, there is a way to subtract them, there is a way to multiply them, there is a way to divide them. When you restrict them to only those Surreal Numbers which have real values, then you get exactly the real numbers. But, there a whole slew of other Surreal Numbers.

Simple examples: { | } = 0, { 0 | } = 1, { 1 | } = { 0, 1 | } = 2....
{ 0 | 1 } = 1/2... { | 0 } = -1....

There are Surreal Numbers for countable infinity, surreal numbers for one over countable infinity, and the whole works from there... countable infinity squared... countable infinity plus pi.... etc....

Of particular interest to this discussion though... if you relax some of the restrictions (with the less-than-or-equal), you can get some funky things like * = { 0 | 0 } where * is not equal to zero, but it is not strictly greater-than or strictly less-than zero. And, * + * = * (which is why I mentioned Surreal Numbers above with the x + x > x thing).

[RQ]

RQ, just wait a year or two until you start learning calculus. It all makes perfect sense after that.

The thing is, there are conventions for dealing with infinity, such as limits, which you haven't learnt in school yet. Your reasoning is correct for arithmetic, but it doesn't work in the long run. Simple arithmetic only applies perfectly to finite and non-zero numbers.

For instance, if you draw a graph of y=x/x, you should get a straight line with a hole when x=0. It looks odd, but it makes sense in a way.

Pat, what's a surreal number? Is that like .9999...?

Are you serious? Man, where have I been.

[RQ]

Surreal numbers are usually represented as two sets, a left and a right. They have the property that there is no element on the right side which is less than or equal to any element of the left side. The value of the number is defined as the "simplest" number 'a' such that no member of the right set is less than or equal to 'a' and such that a is not less than or equal to anything in the left set.

There is a formal definition of "simplest". There is a way to add them, there is a way to subtract them, there is a way to multiply them, there is a way to divide them. When you restrict them to only those Surreal Numbers which have real values, then you get exactly the real numbers. But, there a whole slew of other Surreal Numbers.

Simple examples: { | } = 0, { 0 | } = 1, { 1 | } = { 0, 1 | } = 2....
{ 0 | 1 } = 1/2... { | 0 } = -1....

There are Surreal Numbers for countable infinity, surreal numbers for one over countable infinity, and the whole works from there... countable infinity squared... countable infinity plus pi.... etc....

Of particular interest to this discussion though... if you relax some of the restrictions (with the less-than-or-equal), you can get some funky things like * = { 0 | 0 } where * is not equal to zero, but it is not strictly greater-than or strictly less-than zero. And, * + * = * (which is why I mentioned Surreal Numbers above with the x + x > x thing).

I see.