You use the equation 1+(-1)=0 as a proof of "the space of zero", but I am not following that proof. I thought zero has space because you defined it to. What about the equation 1+(-1)=0 would bring me to the conclusion that 0 has space? What is the space of $2+(-3)$?I think your question is very valid.
If I am comparing these two expressions...
n/(_) : n/(_)
Yes, I agree, there is no way to say which is zero and which is the number 1.
However the op shows that it can be expressed as...
n/(0_) and n/(1_)
n/(0) and n/1
n/0 = n/0_ = n/(_)
n/1 = n/1_ = n/(_)
This then shows exactly when (_) is the dimension of zero, as opposed to the dimension of 1.
So technically when we divide in this system by zero, or any number...it is not a numerical quantity it is a dimensional quantity, that is "placed into".
So if I divide a cookie or zero cookie into the dimensional space of zero, or the dimensional space of any number....that cookie is still there.
So I may only say the cookie divided into the "space" of zero, or the cookie divided into the "space" of 1. "Both of which are the same"
So in any and all semantic statements, I must clarify which "thing" is the cookie, or numerical value, and which "thing" is the space, or dimensional unit.
I claim that zero has space...because of many reasons...
I also claim zero is the absences of a given numerical quantity...(as is currently accepted)...but it is also a quantity of dimension.
"Nothingness" does not exist...so if zero is not "nothingness" and it is also the absence of a numerical quantity...what then is it...the only remaining option is a dimensional unit quantity.
Also the equation 1 + (-1) = 0....proves the space of zero...and that it is equivalent to the space of 1.
We never have a "number" unless a numerical quantity and dimensional unit quantity are "together". This is so in the math and so in the semantics.
I hope I have understood your questions. I hope I have answered them.