For communication purposes, attention will be initially confined here to the natural number system. The purpose of the exercise is to demonstrate that - appropriately understood - there are in fact two distinct natural number systems relating to the quantitative and qualitative aspects of number respectively.
In Type 1 Mathematics the natural number system would be expressed as
1, 2, 3, 4, 5, ..... and in conventional terms diverges to infinity.
However implicit within each of these natural numbers is a default dimensional number of 1.
Properly understood - relative to each of the natural numbers (as quantitatively understood) - the default dimensional number is of a qualitative nature.
Therefore, because the very nature of Type 1 Mathematics is ultimately defined in 1-dimensional (linear) terms whereby the qualitative aspect of understanding is reduced to the quantitative, the dimensional aspect is conveniently ignored.
Thus expressed in a more comprehensive manner the natural number system for Type 1 Mathematics would be written
1^1, 2^1, 3^1, 4^1, 5^1, .....
However there is an alternative natural number system - with a direct qualitative significance - that equally can be constructed. In this context we have a fixed default base quantity = 1, with each successive number involving a change in the dimensional number involved.
Therefore expressed in a more comprehensive manner the number system for Type 2 Mathematics would be written:
1^1, 1^2, 1^3, 1^4, 1^5, .....
Now when we try to view this latter number system from a Type 1 perspective it seems of little use as the quantitative value in each case remains unchanged as 1.
Thus to see the significance of this from a Type 2 perspective we switch to a circular - rather than linear - appreciation.
And the secret to this circular dimensional appreciation (of a qualitative nature) is the recognition that such dimensions have a close inverse relationship with their corresponding roots in quantitative terms.
We dealt for example with the dimension 2 in the last blog.
Thus to appreciate the qualitative nature of 2-dimensional interpretation we obtain the two roots of unity (which as we have seen are + 1 and - 1 respectively).
Now whereas in quantitative terms these two roots are interpreted with respect to linear either/or logic so that the square root can be either + 1 or - 1 respectively, in corresponding qualitative dimensional terms terms this same relationship is interpreted with respect to circular both/and logic which is often expressed as the complementarity of opposites. This is what I refer to in my writings as Integral 1 understanding which is the minimum required for true integral appreciation.
So from a 2-dimensional logical perspective, all interpretation is seen to combine positive and negative poles of understanding (which are considered complementary).
What this means in effect for Mathematics is that - when appreciated in a more refined cognitive manner - any mathematical object such as a number that is given an objective existence (as external to the enquiring must equally be given a corresponding mental interpretation (through the perception of that number).
So for example one cannot form knowledge of the number 2 (as an external mathematical object) without the mental perception corresponding to the number 2, which is - relatively - subjective and internal in direction.
So in the inevitable interaction of all mathematical experience from this context two poles are inevitably always involved that are positive and negative with respect to each other. And such is the essence of 2-dimensional interpretation.
So what I have been at pains to demonstrate here is how the number 2 - when appreciated in its true qualitative dimensional sense, has an intimate holistic relevance for the overall manner in which we logically interpret mathematical relationships.
So there is not just one such manner corresponding to the standard either/or logical system of 1-dimensional interpretation but potentially an infinite number!
Expressed in an alternative manner what this entails is that rather than being based on merely (conscious) rational means of interpretation that Mathematics properly combines both (conscious) rational and (unconscious) intuitive means of recognition.
And clearly from a more comprehensive perspective both intuition and reason should be combined in formal interpretation!
So in qualitative terms the holistic significance of any number (as representing a dimension) is that it can be directly associated with a unique means of configuring the relationship between conscious and unconscious modes of understanding. And each of such configurations thereby defines a unique logical manner of interpreting mathematical relationships.
Fro example the important number 4 (as dimension) is inversely associated with the 4 corresponding roots of 1 which are + 1, - 1, + i and - i respectively.
Therefore 4-dimensional interpretation entails a more refined type of understanding that is based on the complementarity of both real and imaginary opposites in experience (which I have developed at length in my writings and refer to as Integral 2 understanding).
However, once again what is truly remarkable is that in principle a unique logical interpretation (reflecting a special mix of refined rational and intuitive type appreciation) can be associated with every number as dimension.
So just as Type 1 Mathematics can be shown to have a remarkable resonance with reality (as quantitatively understood), Type 2 Mathematics will eventually be shown to have similar remarkable resonance with reality (as qualitatively understood).