It was relatively easy for me to work out the holistic mathematical structure with respect to external and internal polarities (with both pertaining to their conscious understanding).
Thus, whereas knowledge with respect to external and internal poles, as separate, is posited (in conscious terms), crucially the dynamic switching between both, requires a degree of negation (in an unconscious manner).
Therefore when one explicitly recognises the importance of such unconscious switching in experience, one realises that both external and internal poles are - relatively - positive and negative with respect to each other.
Put another way, dualistic experience is now rendered deeply paradoxical with respect to the dynamic interaction of both poles. This in turn is consistent with the growth of a deepening intuitive awareness (that expresses the holistic role of the unconscious).
In normal experience, though dynamic negation as between polarities still necessarily operates, it remains at an implicit (unrecognised) level.
This is especially so with respect to conventional interpretation in Mathematics and the Sciences, where truth is formally expressed merely in terms of rational conscious notions.
One interesting example of this "conventional fallacy" is the customary linear way in which we view the arrow of time as moving unambiguously from past through the present on to future events.
However when one deeply reflects on the manner in which the (internal) self interacts with the (external) environment, notions of time are rendered circular and paradoxical. For if time moves forward with respect to the external aspect i.e. the world (in relation to the self) then relatively - it must thereby move backward with respect to the self (in relation to the world). And of course it works in reverse also so that if time moves forward for the self (in relation to the world) then - relatively - it thereby moves backward for the world (in relation to the self).
Now, most people do not recognise this paradox, due to insufficient development of spiritual intuitive awareness (directly expressive of the unconscious).
However, for contemplatives, such recognition - though perhaps expressed in a different manner - is common place leading to the fundamental recognition that phenomenal notions of time possess a merely secondary relative value. Also the acceptance of dualistic paradox with respect to both poles, continually serves to deepen the primary spiritual notion of the present moment (from which secondary notions of time arise).
So in holistic mathematical terms, we have the standard linear (1-dimensional) notion of time as moving positively forward (in just one absolute direction).
However we can equally express this new 2-dimensional notion, where time is merely relative, with two directions of movement (positive and negative) that are paradoxical in terms of each other.
And to express this 2-dimensional notion in the standard linear manner, we obtain the two roots of 1, i.e. + 1 and – 1 (which again appear paradoxical from a dualistic perspective).
However once we have given the number 2 a distinctive new holistic mathematical meaning - relating to the manner in which the polarities of experience interact - this begs the question as to the possible meanings associated with all other numbers.
In this regard, the number 4 is of special importance. Again in conventional linear terms we understand that we live in a world of four dimensions (3 of space and 1 of time).
So the question that now arises is how to give holistic mathematical expression to the notion of 4 dimensions!
The clue here is to include - along with external and internal - the other polarity set of whole and part.
It took me a comparatively long time to properly understand the holistic mathematical meaning of the important imaginary number i (i.e. the square root of – 1).
Now, once again, when we become externally conscious of a (unit) object in experience, it is thereby posited (+ 1). Then, when we switch polarities (from external to internal recognition) the unconscious is involved through a degree of dynamic negation of the object (strictly of the dualistic awareness of the object), So this unconscious negation is represented as – 1.
However from a holistic perspective, this is understood in a dynamic fashion (which requires its initial positing before unconscious negation can occur). So, like matter and anti-matter in physics, unconscious negation in this sense causes a degree of fusion of opposite poles in the generation of spiritual energy (i.e. intuition).
However this takes us in the direction of 2-dimensional understanding (where both poles are involved).
So the taking of the square root simply entails the attempt to express this unconscious holistic fusion of opposites in a reduced rational manner (as separate).
In psychological terms, this relates directly to the nature of projection in experience, where what is of a holistic unconscious origin is identified - and generally confused - with secondary conscious objects.
Then in a mature spiritual fashion, the imaginary is expressed through refined archetypal symbols (which convey their deeply holistic spiritual nature) .
Thus though we may attempt to express knowledge in a "real" conscious manner, in truth all experience entails the interaction of both "real" and "imaginary" aspects where phenomenal symbols play both a conscious - and often unrecognised - unconscious role.
And this is intimately true of Mathematics and the Sciences, for the very notion of interdependence intimately implies the unconscious aspect.
Perhaps the greatest problem with conventional mathematical (and scientific) understanding is the gross reductionism of whole and part notions where both are given a merely quantitative identity (in a real conscious manner).
However the very process by which we are enabled to switch from whole to part notions (and in reverse part to whole notions) requires the intervention of the "imaginary" aspect (directly expressive of qualitative appreciation).
In this sense whole and part (and part and whole) are properly "real" and "imaginary" with respect to each other
I will now explain this briefly with respect to the very manner in which number is understood in a 4-dimensional holistic manner.
We have already dealt with external and internal aspects (in the dynamic interaction of number "objects" with corresponding mental constructs).
Then the real and imaginary aspects basically relate to the manner in which authentic qualitative notions of interdependence can be properly incorporated with quantitative notions of independence.
For example when one is aware of a specific number - say 3 - it is thereby given a quantitative independent identity. However before this can be related with other numbers, one must recognise the shared interdependence of all numbers. This is achieved through the number concept that potentially applies to all numbers.
This potential recognition - that is infinite - is strictly qualitative and imaginary in nature. However one then likewise can switch to the concept of number as actually applying to all specific finite numbers (in a quantitative manner). So the "whole" number concept has now both a "real" (quantitative) and "imaginary" (qualitative) meaning.
It is similar with each (part) number which can be looked on in real actual terms as independent or as potentially having the capacity to be related to every other number.
So the four roots of 1 (as indirectly expressive of the holistic notion of 4) now assume an amazing importance.
For all reality, which inevitably entails internal/external and whole/part aspects in dynamic interaction with each other, now acquires a new 4-dimensional meaning with poles that are alternatively real and imaginary in both a positive and negative manner.
This is true in psychological terms; however it is equally true in physical terms, with both physical and psychological aspects themselves now understood in a complementary fashion.
It was in the early 80's - at a time when I was deeply interested in Jungian Psychology - that I properly unravelled this holistic understanding of 4-dimensions. This relates to the dynamic interaction of the fundamental poles of experience i.e. external/internal and whole/part, in both a physical and psychological manner.
From this perspective, time for example is now revealed as possessing 4 dimensions (i.e. 4 distinctive directions of movement).
So as well as the real directions - positive and negative - which we have already discussed, we now have in addition two imaginary directions.
The real directions basically equate to quantitative notions of time (as representing independent events).
The imaginary directions equate however to qualitative notions of time as representing interdependent events (indirectly expressed in a dualistic type fashion).
The failure to distinguish (conscious) real from (indirectly conscious) imaginary notions, equates with the huge failure in both Mathematics and the Sciences to properly distinguish quantitative notions (of independence) from qualitative notions (of interdependence).
Thus the present scientific world view, which is accepted uncritically by its many proponents, is built on the most massive form of reductionism!