We saw yesterday how the two key polar pairings can be holistically interpreted in terms of the coordinates of the both the real and imaginary axes (in positive and negative directions) in the complex plane.
In particular the unit circle (drawn in the complex) plane will have coordinates on the (horizontal) real axis (x) of + 1 and – 1 respectively and on the (vertical) imaginary axis (y) of + i and – i respectively.
And once again, the former reflect the holistic mathematical interpretation of the external and internal polarities and the latter the corresponding interpretation of the two directions with respect to the qualitative aspect (relating to interdependence). And these are both "imaginary" with respect to the quantitative aspect as "real" (relating to independence).
The dramatic importance of this new holistic mathematical mapping is that all the various roots of 1 can now be expressed as representing unique dynamic combinations with respect to the interaction of whole and part in both physical and psychological terms.
And the importance of these roots in turn is that they enable us to uniquely express the qualitative holistic nature of each number (indirectly in a 1-dimensional manner). And remember again that such 1-dimensional interpretation informs the normal dualistic nature of rational discourse!
So I will illustrate such holistic interpretation with respect to one of the the simplest - and many ways most important - numbers i.e. "2".
Thus once more in an indirect linear (1-dimensional) manner, we can express the holistic qualitative nature of "2" through obtaining the 2 roots of 1 (and interpreting the results in the corresponding holistic manner).
Now the roots of 1 are + 1 and – 1. Therefore these directly relate to the complementary nature of external and internal polarities.
Once again conventional dualistic understanding is strictly 1-dimensional based on just one positive pole of understanding (where the qualitative holistic aspect of understanding is reduced in a quantitative manner). Therefore conventionally the two roots of 1 (i.e. + 1 and – 1 ) are interpreted in a merely quantitative unambiguous fashion (as separate opposites in absolute terms).
However the essence of 2-dimensional understanding is that we now understand relationships more subtly as necessarily entailing the interaction of both external and internal polarities, that are - relatively - positive (+) and negative (– 1) with respect to each other. Therefore they are independent in only a relative sense. This implies that the recognition of their complementary nature implies the new appreciation of qualitative holistic interdependence.
So now, the two poles are understood as relatively independent to a degree (implying quantitative appreciation) and also relatively interdependent (implying corresponding qualitative appreciation).
This can easily be illustrated with reference to the common place example of a crossroads.
If one is heading N towards a crossroads both left and right turns can be given an unambiguous meaning (i.e. in 1-dimensional terms).
In one now from the opposite direction heads S towards the crossroads, again left and right turns can be given an unambiguous meaning (i.e. in 1-dimensional terms).
However, if one now tries to understand the two turns at the crossroads, when simultaneously combining N and S directions, then the notion of direction is rendered paradoxical (i.e. circular). For what is a left turn (heading N) is right (heading S). And what is right (heading N) is left (heading S).
So this latter paradoxical appreciation implies 2-dimensional interpretation (where 2 polar reference frames are simultaneously combined).
And this quite simply represents the qualitative holistic appreciation of 2!
Therefore, by extension the qualitative holistic appreciation of 3 would imply the ability to simultaneously combine 3 reference frames (as represented by the 3 roots of 1).
And the qualitative holistic appreciation of n, would imply the ability to simultaneously combine n reference frames (as represented by the n roots of 1).
This is fairly easy to state, but the implications are truly enormous. I will return to this in the next entry.