In many ways this represents a return to yesterday's blog entry.
However my intention here is to highlight the enormous gap that I would see as between a truly comprehensive mathematical approach (that coherently combines both quantitative and qualitative aspects of understanding) and the highly reduced conventional model that spuriously attempts to maintain its monopoly with respect to mathematical truth.
If we take the - apparently - simple expression 1^4, this seems somewhat trivial from a conventional (Type 1) mathematical perspective.
Here the numbers are viewed in a merely cardinal (quantitative) sense.
This again means that each number unit is treated in a totally homogeneous fashion (devoid of any qualitative distinction). So from the perspective 4 can be broken down as 1 + 1 + 1 + 1 (where each cardinal unit is considered as identical).
Likewise also with respect to the multiplication 1^4 = 1 * 1 * 1 * 1 (with each unit again treated as identical in quantitative terms).
And then again because it ignores any qualitative distinction attaching to each unit the 4 (as dimension with respect to 1 as base number) itself can have no significance from a merely quantitative perspective.
So form a Type 1 perspective 1^4 is thereby quickly reduced as indistinguishable in quantitative terms from 1^1.
So, as I said from such a perspective, 1^4 seems a somewhat trivial expression.
Contrast this now with Type 3 interpretation.
The starting point here is that each number can be given both a quantitative and qualitative meaning, Yet such meanings (which - are - relatively distinct) yet can then be also combined in a dynamically interactive manner.
Thus 4 as dimensional number now also can be given an ordinal (qualitative) interpretation which is inversely related with its corresponding root.
Therefore 4 as ordinal dimension i.e. 1^4 (where it serves as a holistic means of interpretation with respect to the 4th dimension) has its quantitative complementary counterpart as 1^(1/4).
Now this quantitative root = i is a number that simultaneously has both a linear and circular significance (in quantitative terms). In other words from a geometrical perspective, it lies as the extreme point on the imaginary line drawn from the centre of the circle of unit radius (in the complex plane) where it meets the circular circumference. So here at this point the very notion of line and circle coincide.
Then in corresponding qualitative terms, it relates to the qualitative interpretation of the imaginary number i, which equally combines an intersection of both linear and circular type understanding.
So from the ordinal (qualitative) perspective each of the "units" comprising 4 are distinct.
So in this context of 4, the 1st, 2nd, 3rd, and 4th units are of a distinctive qualitative nature, which are "seen" through conversion from the linear to the corresponding circular aspect of the number system.
This implies that we cannot properly understand the qualitative meaning associated with the 4th dimension without - in this context - providing meaning to the 1st, 2nd, and 3rd dimensions also.
Once again, this has its quantitative counterpart equivalent in the fact that we have - again in this context - 4 roots of units i, - 1, - i and + 1 respectively.
Thterefore, the deeper significance of multiple roots of a number is that they inevitably refer to underlying ordinal qualitative type distinctions (which cannot be reduced in a quantitative manner).
And because Type 1 Mathematics makes no provision for qualitative distinctions, it can give multiple solutions to polynomial equations no proper interpretation!
So in understanding 4 properly in qualitative terms we also include the understanding 1, 2, and 3 in a distinctive manner.
However Type 3 enables the combination of both cardinal and ordinal type interpretation.
So from this context 1^4 also requires 1^3, 1^2 and 1^1 which inversely are related to 1^1, 1^(3/4), 1^(2/4) and ^(1/4) as the counterpart quantitative roots.
And here is where the cardinal notion of 4 as dimension comes in, for when we raise each of the root expressions to the power of 4 we obtain 1^4, 1^3, 1^2 and 1^1 thereby directly switching - relatively - from a quantitative to a qualitative context.
So we have now combined the cardinal notion of dimension with its corresponding ordinal equivalent.
Put another way, we have combined linear with circular understanding (which is implied by the very geometrical notion of roots) which then must be given both linear and circular interpretation in qualitative terms.
Finally the true meaning of 1^4 strictly implies a direct coincidence of linear and circular type notions (from both the quantitative and qualitative perspectives).
So the true meaning of the expression is one of ineffable mystery (as here both quantitative and qualitative aspects are identical).
However in the very attempt to unravel such mystery we must necessarily attempt to separate to a degree both aspects (quantitative and qualitative).
So in the actual realm of phenomenal type understanding, a degree of uncertainty necessarily attaches to any mathematical expression (including the "simple" one here i.e. 1^4).
So one can perhaps see how the rich mystery inherent in very simplest of mathematical expressions evaporates completely from a Type 1 perspective (where it is treated as utterly trivial).
And such extreme reductionism of truth is what we still misleadingly refer to as Mathematics.
Type 3 can thus indeed provide the appropriate perspective to appreciate why Type 1 Mathematics can reach such a conclusion.
So it provides the context to show that while Type 1 certainly has the appearance of being "absolute", that this in fact represents but an extreme distortion of the much more authentic nature of what properly constitues mathematical truth.