Indeed if we go back to our example of the crossroads, the very manner by which one recognises that a turn can be simultaneously "left" and "right" illustrates the qualitative notion of "twoness".
Once again, when we use just one isolated polar reference frames (i.e. 1-dimensional interpretation) a turn at a crossroads is unambiguously either "left"or "right". So if one is heading N for example towards the crossroads, a left turn can be unambiguously identified (or posited). So one can refer then to this left turn as + 1. This in turn means that the right turn is thereby excluded (or negated) in this context as – 1.
Now if one now changes the polar reference frame and heads S from the opposite direction towards the crossroads, once again the left turn can be unambiguously identified. So again this can be posited as + 1, with the excluded (or negated) right turn as – 1.
However what was unambiguously a left turn, when heading N is likewise unambiguously a right turn heading S. And what was unambiguously a right turn heading N is now unambiguously a left turn heading S.
Therefore when when we simultaneously combine both polar reference frames (N and S) in what represents 2- dimensional interpretation, each turn can be represented as both + 1 and – 1.
Now this situation is completely paradoxical from a rational dualistic perspective. Such rational understanding by its very nature is linear i.e. 1-dimensional, where opposite polar reference frames are clearly separated.
However the very essence of 2-dimensional interpretation is that it is directly intuitive in nature. Thus through one cannot rationally understand how in this context + 1 can also be – 1 (where only isolated polar reference frameworks are considered) one can however intuitively grasp how this can be so, through intuitively combining two opposite polar reference frames simultaneously).
The great importance of this illustration is that the very notion of "twoness" which again relates to the qualitative notion of "2" occurs directly in experience through the intuitive realisation that occurs from simultaneously combining opposite polar reference frames.
And once again this is fundamental with respect to the dynamic nature in which experience takes place.
In our crossroads illustration we used the opposite polar directions of North and South and Left and Right respectively.
It is then very similar with respect to actual experience where interaction continually takes place with respect to external and internal and also whole and part polarities. The latter set in turn directly relates to the corresponding distinction as between the quantitative and qualitative aspects of number!
So therefore put succinctly, the quantitative aspect of number appreciation relates directly to rational interpretation (that is 1-dimensional in nature).
However the qualitative aspect, by contrast, relates directly to intuitive appreciation (representing the simultaneous recognition of multiple "higher" dimensions i.e. reference frames).
The simplest - and in many ways the most important - example of qualitative understanding relates to the 2-dimensional notion of "twoness".
What happens here is that two polar reference frames - that in rational terms are considered as independent - are now intuitively seen as dynamically interdependent with each other.
This is very similar to the physical level where both a matter and anti-matter particle are brought together. Though + 1 and – 1 with respect to each other as separate particles in their independent states, when brought together as interdependent they fuse together as physical energy.
Likewise when the intuitive realisation takes place in experience of the complementary nature of opposite poles (e.g. external and internal) a corresponding fusion takes place in an intuitive manner (as psychological energy).
Thus in experience, the qualitative nature of number relates directly to intuitive appreciation (of "higher" dimensions).
Once again the simplest case represents the qualitative appreciation of "2" which implies intuitive appreciation of 2 dimensions (i.e. the simultaneous recognition with respect to two opposite reference frames that are equidistant in a circular manner).
In like manner the qualitative appreciation of "3" implies direct intuitive realisation of 3 dimensions (as the simultaneous interdependent recognition with respect to three circular equidistant reference frames).
In general the qualitative appreciation of "n" implies direct intuitive realisation with respect to n dimensions (as the simultaneous interdependent recognition with respect to n circular equidistant reference frames).
In psychological terms, this qualitative recognition - which in some measure must be implicitly present to enable meaningful interpretation even at a quantitative level - is directly of an unconscious nature.
However because Conventional Mathematics is formally geared to the merely quantitative aspect of understanding, it remains completely blind to the unconscious dynamics by which understanding of number takes place.
Put another way, a coherent interpretation of even the simplest number operations requires that both conscious and unconscious type understanding be dynamically integrated in a balanced coherent manner.
We will see some of the basic implications of this in future blog entries!