Hypothesis

If identity is a logical relation (a=a) and logic is a process independent of a self (i.e., objectively true) then no self can be said to be self-identical from a vantage within that self. That is to say, we can ONLY be objectively self-identical, NOT subjectively self-identical. The "identity" of "cogito ergo sum" dislocates the truth-value of the cogitator and the summary into the ergo-sphere... leaving us to ponder the function of identity (and its validity) between two separable phenomena, rather than the phenomena itself.

Dimensionality

Let us declare a one-dimensional array which we will denote as Arrα(2). This gives us one dimension with two points. Arrα[0] and Arrα[1].
Let us state that Arrα[0]=X₁ and Arrα[1]=X₂.

Now, let us say that every point within this array is possessed of an attribute. This attribute is therefore a property of that point, and the point is likewise a property of its attribute. Thus, if there is an attribute of "redness", we could say that there is some function R(X₁) which represents the redness of X₁, and a function X₁(R) which expresses the X₁-ness of Red. (Before you object, consider whether your hometown is "as boring as Fresno." Fresno is not only a place which has some level of boring-ness... there is some level of boring-ness beyond which we might as well be in Fresno)

We have created a second-dimension. The comparative dimension.

Let us say that X₂ is also possessed of redness (even if that value is equal to 0). There is then a function R(X₂) which expressess the redness of X₂ and also a function X₂(R) which express the X₂-ness of Red. Thus, in considering the redness of X, we find that we need a two-dimensional array as follows:

X1	X2
R(X1)	R(X2)

Let us call this two-dimensional system Arrß(2,2).
Arrß[0,0] = X₁, R(X₁)
Arrß[0,1] = X₁, R(X₂)
Arrß[1,0] = X₂, R(X₁)
Arrß[1,1] = X₂, R(X₂)
We know for a fact that we still have only two points (X₁ and X₂) which are both elements of Arrα(2). But the mere consideration of the attributes of points X₁ and X₂ necessitated the creation of a new array with a new dimension Arrß(2,2) which contains twice as many points as we know to exist. That is to say, half the points contained with Arrß(2,2) are potentialities. Each point represents either a "fact" or a "conjecture".

"What if X₂ were the color of R(X₁)?"

"Why, then, R(X₂) = R(X₁)"!

And thus, X₁=X₂ with respect to redness by means of their identity within the dimension of R.

So, there must be another one-dimensional array, ArrR(2) containing the values of R(X₁) and R(X₂), and thus we could more accurately express the dimensionality of Arrß(2,2) as Arrß(Arrα(2), ArrR(2)).

If we were then to stipulate that R(X₁)=R(X₂), we could say that the size of ArrR is 1, which we denote ArrR(1). That would then alter Arrß(Arrα(2), ArrR(1)) as follows:
Arrß[0,0] = X₁, R(X₁)
Arrß[1,0] = X₂, R(X₁)

Though the number of points within the array shrink, the number of dimensions remains constant. It merely becomes an array wherein all ArrR() of Arrß(Arrα(), ArrR()) are identical.

How can this be?
IF
R(X₁)=R(X₁) for any point in ArrR() of Arrß(Arrα(), ArrR()),
BUT
R(X₁) is a property of any point of Arrß(Arrα(), ArrR())
AND
Arrß[0,0] != Arrß[1,0],
THEN
how can it be possible that R(X₁) is both a component of Arrß[0,0] AND Arrß[1,0] yet also self-identical? For there are at least two properties of R(X₁):
X₁(R(X₁)) and X₂(R(X₁)) that are not identical.

Within the scope of ArrR(0) there is no paradox. But within the scope of Arrß(Arrα(), ArrR()) there is an obvious paradox. The self-identity of ArrR[0] within Arrß(Arrα(), ArrR()) must itself be reliant upon a dimension... a dimension of self-identity?

If you took logic in college, feel free to groan and depart... I'm not here to impress you.

In order for the values of R(X₁) to be identical within the Array of Arrß(Arrα(), ArrR()) there must be some function X() whereby X(Arrß(0,0)) = X(Arrß(1,0)).

This function would take as an argument the points within the array of Arrß(Arrα(), ArrR()) and yield a one-dimensional array of it's own - ArrX() - where each value within the array of X corresponds to the derivation of a point within the larger array of Arrß(Arrα(), ArrR()).

This of course means that we now need a new, three-dimensional array to express the relations between our points:
ArrΓ(Arrα(), ArrR(), ArrX()).

And on and on it goes...

Why the FUCK Does This Matter?

So, allow me to return to my pet theory of the moment - consciousness is a function of space/time.

Stipulate that the universe is a sequence of property-bearing points contained wtihin a four-dimensional array of ArrU(x,y,z,t). Now, let us speculate that consciousness is a PROPERTY of any given point A with coordinates ArrU[x,y,z,t]. Thus, there would be some function X(A) which expresses the consciousness of point A. In order to argue that the consciousness of another point, B, is continuously joined with the function of X(A) we need to at a minimum establish that X(A) = X(B). Thus, we need a new dimension, ArrΩ() which will allow us to say that for any given point [x,y,z,t,Ω] that Ω is self-identical. Any given life, then, becomes a finite five-dimensional array within space time of ordered points - Arr∞(x,y,z,t,Ω). (Pun intended)

But, what's more... in order for that ordered set of consciousness to apprehend itself as continously self-identical it would need to resort to ANOTHER dimension (call it the logical dimension) to draw a cohesive relationship of self-identity between the separate points within the larger matrix of the consciousness. And the points of THAT dimension would require yet another dimension (call it the justified dimension) which would draw the various instances of logicality into some relation of self-identity.

And on and on it goes...