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[Craniumonempty]

The notion that the idea of something implies existence has been rejected by most philosophers from Kant to Kierkegaard, and this rejection gives rise to the Boolean interpretation of classical logic.

No, that's not what I meant. Did I say that? I might have to re-read my post. I meant that when something does exist, then idea comes to exist. That idea as a concept still exists when the thing is gone which is another concept, but because the idea still exists, the object can be said not to be there (as far as the observer can tell). The negation as far as existence goes, doesn't mean that the object stopped existing, because as far as we can tell currently, objects are composed of smaller objects and so on. Those smaller objects may still be there, but them still fitting the original idea is no longer there, so in concept it's not there, even though in reality everything that was there before still is. Did that make any sense? I hope so.

[...] I cut some since I wanted to add more of the same thing

EDIT: I think I may have a better way to explain:

I'm hoping this will be clearer.

Let's start with the dimensional x and y axis lines on the Cartesian coordinate system, except only using the positive sides for each (let's say for this explanation the negative numbers don't exist). All numbers in x will be designated with "~" and all numbers in y will be designated with "`". All operations are as we see them currently, so lets try to make some rules for translation from one number line to the other.

We know:

`1 + `2 = `3
`4 - `2 = `1
~8 + ~3 = ~11
~6 - ~1 = ~5

So, what about:

`2 - `8 = (`2 - `2) - `6= 0 - `6

Which we can just write "-`6". Let's say the magnitudes are the same so |`1| = |~1| and that all operations can translate through the 0 to the other line.

So:
-`6 = ~6

We can also notice:
`8 - ~2 = `8 – (-`2) = `8 + `2 = `10
`8 + ~2 = `8 + (-`2) = `8 - `2 = `6
~2 - `8 = ~2 - (-~8) = ~2 + ~8 = ~10
~2 + `8 = ~2 + (-~8) = ~2 - ~8 = `6

So for the y we could designate "-" instead of "`" and leave x without a sign to clean it up:

-8 - 2 = 8 – (- (-2)) = -8 + (-2) = -8 – 2 = -10
-8 + 2 = 8 + (-(-2)) = -8 - (-2) = -8 + 2 = -6
2 - -8 = 2 + 8 = 10
2 + -8 = 2 – 8 = -6

Now multiplication:

`1 * `2 = `2
`4 * `2 = `8
~8 * ~3 = ~24
~6 * ~1 = ~6

`2 * ~8 = `2 * (0-`8) = `2 * -`8

Well, multiplication alone can't translate through the 0, so we are going to have to make a rule. We could do like addition and multiply it out in the opposite direction from the first number, but that doesn't give easy to use results:

`2 * -`8 = ~14
`8 * -`2 = ~8

You know the answer of course is to multiply it out and switch the entire side of the result, because it's almost the way it's done currently:

`2 * -`8 = ~16
`8 * -`2 = ~16

~2 * -~8 = `16
~8 * -~2 = `16

How does it look cleaned up (`becomes -):

-2 * -(-8) = -2 * 8 = 16
-8 * -(-2) = -8 * 2 = 16

2 * -(8) = -16
8 * -(2) = -16

This doesn't fit the original rule, the rule needs to change again. First, let's look at cleaning up the original multiplication examples:

-1 * -2 = -2
-4 * -2 = -8
8 * 3 = 24
6 * 1 = 6

That looks a little odd, so we could make two opposite signs negative and two the same positive:

-1 * -2 = 2
-4 * -2 = 8
8 * 3 = 24
6 * 1 = 6

Then for the other examples:

-2 * 8 = -16
-8 * 2 = -16

2 * -(8) = -16
8 * -(2) = -16

Yeah, that works. That's how we do it currently. There is another way to approach this that gives the same result from the same line of thinking, but that will wait for later.

- It might be worth noting, that the original formulas give an X formation through 0 if we go to a 2D plane, but when the new rules are applied it pushes the result to show only one side of that X. Addition/subtraction forms a straight line in the 2D plane, and multiplication/division makes an X if we assume that the translations can fit for negatives.

Now lets go to exponentiation. In my opinion, the way we interpret the exponent is correct, but like the multiplication, missing something. However, that's another discussion. To focus on the base, I'm going to drop designation from the exponent and for now assume that it can be from either line just using the magnitude:

`8^2 = `64
`2^3 = `8
~8^2 = ~64
~2^3 = ~8

Cleaned up:

(-8)^2 = -64
(-2)^3 = -8
8^2 = 64
2^3 = 8

Well, that doesn't look like how we have it currently, so maybe just the magnitude?

|-8|^2 = 64
|-2|^3 = 8
8^2 = 64
2^3 = 8

One of those should have remained negative, so we are going to have to call a negation on each multiplication that is below the exponentiation (basically make it fit to the rule that we created to make multiplication easier) which will be designated "+" because it's slightly different than the standard negation:

(+|-8|)^2 = 64
(+|-2|)^3 = -8
8^2 = 64
2^3 = 8

Hmm, so the only problem would be going back from even powers.

sqrt(-64) wouldn't exist in this system, so we are going to have to pull it out.

8*sqrt(-1) or 8i

So are the negative numbers anything like the x and y lines that I was using here? Both of those lines were equal and able to translate, so how different is that then negative numbers?

I'm not even saying that imaginary numbers should go away, but this is how things look to me. I simply think that there is more to the interpretation of negative numbers than simply through predefined operations. How did they get to those operations in the first place? That is where my problem lies.

EDIT2:

You can look at operators as having a dimensionality, but not quite the same as spacial dimensions:

2 = 2 (0 dimension because nothing is done)
S(1) = 2 (1 dimension operation, you can think of the negative of this operation as another single dimension operator also)
1+2 = 3 as 1 dimension and 2 + (-1) = 1 or (-2) + 1 = -1 as the other (2 dimensional operation, with subtraction as the other)
I'm not going to list more, because of the way we break them down currently, but multiplication/division are 4 dimensional and exponentiation is 8 dimensional and up.

How it appears to me is that we try to fit all the operations into as few dimensions as possible for ease of use, which is fine unless it changes the outcome. Basically, we are changing the outcome of exponentiation which is fine, but we lose all the other aspects of the operation especially when we try to go higher like with tetration.