I will elaborate further on the significance of the three key polarities (corresponding to the holistic mathematical interpretation of 2, 4 and 8 dimensions respectively).
The first set of horizontal polarities (corresponding to the relationship between internal and external) as we have seen are positive and negative with respect to each other in integral terms.
The significance of this is that all scientific understanding of reality necessarily entails an internal (subjective) observer in relation to what is observed in external (objective) terms.
Now when we view this relationship in a linear manners, both external and internal frames can be given a merely positive identity (that correspond with each other).
For example I can view the "sun" as in objective terms as an external object.
However equally I can view the "sun" in - relative - subjective terms as relating to its internal perception.
Fortunately for Conventional Science - based on a linear approach - both external object and internal constructs are understood to correspond directly with each other. In this way it does not matter if we designate science in conscious (positive) terms as interpretation of the relationship between object phenomena (external) or alternatively between subject constructs (internal).
However when we attempt to view this relationship in a holistic integral manner, these two frames of reference, external and internal - that seemingly correspond with each other (in linear terms) - are diametrically opposite (in a complementary circular manner).
So now both external and internal are understood as positive and negative with respect to each other (in complementary fashion). Though the rational interpretation of this relationship is paradoxical, the actual reconciliation of both aspects takes place in a directly intuitive (unconscious) manner.
So one clear implication of the Integral Approach is that scientific understanding cannot be conducted in (mere) rational linear terms but rather combines both (linear) sequential and (circular) paradoxical understanding (implying the explicit combination of both reason and intuition).
The refined 2-dimensional appreciation of the role of external and internal aspects of understanding (n both linear and circular terms) corresponds to - what I refer to as - the Integral 1 approach.
The second set of polarities relates to the hugely important distinction as between whole and part.
Once again in linear terms (where both are treated as independent) these can be given a rational interpretation in merely conscious terms (i.e. as positive). For example research data would be identified with the part and theory with the whole aspect respectively.
However in circular integral terms (where both are treated as interdependent) the relationship is as real to imaginary. So if we give the part (in any context) a real identity, the corresponding whole aspect is - relatively - imaginary.
So we have the vital insight here that in order to preserve the appropriate qualitative distinction as between part and whole, we must interpret reality in complex rational terms (with both real and imaginary aspects).
This has several important applications (just a few of which I will mention).
The relationship in physical terms as between object phenomena and dimensions (of space and time) is real to imaginary.
In corresponding psychological terms the relationship between perceptions and concepts is also real to imaginary. Furthermore both of these aspects are fully complementary. This - as we shall see - has huge implications in terms of making sense of many of the findings of string theory which really points to the need for an entirely new understanding of the dimensions of space and time.
The relationship as between cognitive and affective in understanding is also real to imaginary. In other words, in Jungian terms, when the cognitive aspect of understanding is real (conscious), the corresponding aspect is imaginary (as the projection of the holistic unconscious); in like manner when the affective aspect is now real the cognitive is then imaginary. So in the dynamics of experience, we keep switching as between real and imaginary (with the revealed aspect conscious and the alternative hidden aspect unconscious).
Finally, a correct appreciation of such complex interpretation exposes the limitations of conventional mathematical proof (which entails a basic form of reductionism where the whole is confused with the part). So, the general proof that applies holistically to all cases cannot be directly identified with any specific instance (applying to the part).
This leads to need for reformulation of the notion of proof in dynamic relative terms where he it is seen as representing but a special form of social consensus.
The third set of polarities relates to fundamental form and emptiness. Again in linear terms these can be treated in an independent manner. However in corresponding integral terms they require 8-dimensional understanding for appropriate interpretation.
In physical terms this leads to the appropriate holistic mathematical way of understanding the nature of the physical forces; in corresponding terms it provides the corresponding means to translate the volitional psycho spiritual forces; and once again both sets of forces are complementary in experiential terms.
We will look at this again in the next contribution.