In yesterday's blog entry, I attempted to outline succinctly the holistic mathematical nature of "2" (as representing a dimensional number).
This in turn is directly related to the unfolding of the various stages of Level 1 (Band 3).
What happens here is that one's former dualistic vision of reality becomes steadily eroded, through a marked acceleration in unconscious development, where phenomena become - literally - negated to a considerable extent. This then leads to a more spiritually refined intuitive worldview, where the dynamic relative nature of all phenomena can be readily appreciated.
In my own account, Level 1 is largely concerned with - what I refer to as - the horizontal polarities of external and internal (which are relatively + 1 and – 1 with respect to each other), which necessarily condition the experience of all phenomena.
And in holistic mathematical terms, this culminates in the coherent vision of the nature of 2-dimensional - as opposed to 1-dimensional - reality.
However, when one recognises that the number "2" can be given a distinctive holistic mathematical meaning, then this likewise implies that, in principle, the same should apply to all numbers.
In particular as the notion of 4 dimensions - as understood in conventional physical terms - is directly relevant to the manner we understand space and time, this implies that "4" should likewise carry an immense significance from a holistic mathematical significance.
I have mentioned on many occasions how I experienced an immediate affinity with Jung's writings (when I seriously studied them in the early 80's).
I could see holistic mathematical understanding was strongly implicit in his work.
For example his four functions can be illustrated as 4 equidistant points on the unit circle.
Likewise his mandalas, used as symbols of integration, are then often depicted in the same manner as dividing the circle equally in the same four-fold - or alternatively - 8-fold manner).
So when one of Jung's great followers Marie-Louise Von Franz stated "Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four" it is this holistic mathematical "circular" notion of "4" that implicitly she had in mind.
In fact Jung came close to a more explicit holistic mathematical expression of "4" when he termed 2 of his functions as rational and 2 as irrational, setting up dynamic complementary links between them with respect to their respective conscious and unconscious usage.
However, after much reflection, I came to the firm conclusion that the true holistic mathematical nature of "4" is given by the four respective roots of 1 i.e. + 1, – 1, + i and – i respectively.
This in turn raises the all important issue of the holistic mathematical meaning of the imaginary number i (i.e. the square root of – 1).
Now we have already seen, in holistic mathematical terms that – 1 relates to the (unconscious) negation, in a dynamic manner, of - formerly - consciously posited ("real") phenomena, now entailing 2 dimensions of understanding!
If one then attempts to express such (unconscious) holistic understanding indirectly in a conscious manner, this entails expressing what is inherently of a 2-dimensional nature, in a reduced 1-dimensional manner, which is the equivalent of obtaining the square root.
So the all important holistic mathematical conclusion is that the imaginary notion relates to holistic unconscious meaning that is indirectly conveyed in a conscious manner.
When one understands this, one realises that reality is necessarily of a complex nature (entailing both real and imaginary aspects).
The "real" aspect relates to the (local) recognition of phenomena in a directly conscious manner.
The "imaginary" aspect is then related to recognition of the global nature of phenomena in an unconscious fashion. In fact without this imaginary aspect, it would be impossible to relate phenomena with each other!
And we have already seen how this problem lies at the heart of conventional mathematical understanding.
Though of course in standard analytic terms, such Mathematics recognises the importance of both real and imaginary numbers (as quantities), from a qualitative perspective it attempts to conduct all interpretation from within a solely "real" i.e. conscious framework.
So as we have already seen, in every context, it effectively reduces the distinctive holistic notion of number interdependence in a reduced independent fashion.
Thus symbols in Mathematics never possess a solely conscious meaning, for implicitly the very requirement for relating these symbols requires unconscious meaning (of a holistic nature).
However we can then indirectly attempt to relate this unconscious aspect in a conscious fashion through the adoption of the imaginary notion (in a qualitative manner).
Now when the unconscious aspect is nor properly recognised - as is this case in present accepted Mathematics - it is blindly projected on to objects (without of course its nature being recognised).
Therefore, though - certainly from my perspective - an obvious fundamental problem lies at the heart of all Mathematics, this can never be addressed while interpretation remains rigidly stuck within real (i.e. rational conscious) modes of expression.
So the clarification of the imaginary notion, which represented a significant further breakthrough in my holistic mathematical understanding, related very much to the on-going development of Level 2 (Band 3).
So just as all numbers in analytic terms can be expressed within a complex framework (allowing for real and imaginary quantities with positive and negative values), equally I now was in a position to elaborate on a similar complex framework in a holistic qualitative manner.
I will return to some of the enormous implications of this in the next few blog entries.