Here's a geometric example of the problem with a supposed multiplication in which we substitute 0*x with 0.

Imagine looking at a single dimension, x. It's simply a line that potentially spans between +/-infinity. A solution for x is a single point and referenced within that dimension.

In order for x to describe a dimension, various values of x need to exist and be differentiable. If the entire dimension of x could either 1) only be a single value, the dimension is not a line and it wouldn't even describe a constant within it - you could not vary values of x to deferentiate between any of them or 2) if we varied values of x, but none of them could alter a detectable result elsewhere, then similarly the linear dimension of x does not exist relative to that other space and this dimension would instead be no different than an isolated point detached from that space - you could not prove any specific properties about x because its influence could not be traced anywhere.

So imagine we're mapping from a dimension x to a dimension y and we're scaling the dimension x by various constants (it's a bit of a misnoemer to call these different values, constants, because if they're altered, then they aren't really constant, but that's a separate subject) and then mapping this rescaled dimension to y. We could also compare this with viewing a rotation of a line.

If we have values of x range from -1 to +1 and map this with a scaling of 1 to y, we see this line segment mapped to y and similarly span between -1 and +1.

Now if we compress x by a factor of 1/2, then |x|<=1 maps to |y|<=1/2 and we might assume for |x|<1 then |y|<1/2 (I prefer such an interpretation, but the .9r=1 guys would probably prefer to add conditions to this inequality). We can establish a connection with other values on the y dimension by mapping them x=y/(1/2) or x=2y. Notice a potential problem here in that for whatever length the y dimension is, in order to establish a 1 to 1 correlation between points, we either need to quantize these dimensions into identical number of positions or have the y dimension span twice as many positions as x (normally not a problem if we're working with finite quantities, but the problem is over making a generalization to unbounded quantites and treating both dimensions as if they were identical units, as no matter what some infinite/unbounded value might be, it couldn't be twice as large as itself, so the x dimension would have to be compressed relative to y).

As we rotate this further, we could generalize this to a scaling factor, c, and map y=cx, and for values of |x|<=1, we have |y|<=c or for |x|<1, |y|<c, and we can associate all values of y with x as x=y/c.

Of course c could be made as small as we want and if not equal to 0, then we can always associate points in y with points in x and visa versa and for any issues of varied density in the representation, if we simply perform the inverse mapping by scaling by 1/c, we can return to the original 1 to 1 correlation between the x and y dimensions.

But when c=0, any point in the x dimension maps to a single point in the y dimension. It's not simply that finite numbers map to a point, but that the entire dimension of x exists within a single point on y (if you want something different, then you need to work with calculus or define such infinitesimal scalings and retain a manner to distinguish the dimension of x from a point).

Notice that if we write a linear equation:

c(x+y)=0

For values of c!=0, we have a line at a 45 deg angle that solve for this. For example, if c!=0, then c(-2+2)=c*0=0 whereas c(8+5)=13c != 0.

But when c=0 any values of x and y solve this equation and instead of a line, we have a completely filled plane.

I know you'll want to that if we write y=0x, then we can plot this as y=0 and find a line with respect to an XY plane, but notice that y=0 is simply a constant (if y could vary, then the dimension would exist as a line, but in this case y is nothing different than 0, which is not a line, nor does it possess a single degree of freedom to describe a line). If you select to plot y=0 with respect to some other dimension, such as x, this choice is arbitrary as the x dimension has no influence on the value of x, neither does the z dimension, nor any other dimension and if we can simply append unrelated dimensions to a plot, then we might as well describe y=0 as a cubic relationship because we could plot it with respect to multiple other dimensions until we had enough to construct this unrelated abstraction with cubic properties. Notice that y=x^2 is a second order polynomial relationship, and that y=x^1 is a linear relationship and that y=x^0 is a constant and not linear.

Anyway, the reason why a division by zero is undefined is because it's the inverse of a multiplication by zero which is not actually a multiplication at all because it doesn't perform a linear scaling. The undefined division by zero is just a symptom of a deeper definitional problem.

Oh and again to see it in a more geometric context, if we rotate a line toward us so that it appears as a point, the point no longer is capable of describing the orientation or any other linear properties of the line (such as what segments of it might exist). If you hand the point to someone else, without additional context as to what linear properties "hide behind it", the point itself can't be rotated back to reveal this, and if you're trying to prove that this point specifically representated such a line, then you need to not simply prove the point, but the information regarding the line and the context becomes a pairing of the point as well as supplying the manner in which such a point was derived - an unadorned zero doesn't do this.

Notice also that if we changed our definition of the line segment in x above to be |x|<1 and that |y|<c, then if c becomes 0, we have |x|<1 and |y|<0 and the x dimension doesn't even fit into a point on the y dimension, but vanishes completely (a much more appropriate description of what actually occurs).

Reality = whatever exists.

'0' stands for "Does nnot exist".

Therefore, '0' represents non-reality.



g.