Actually this is where I would make another slight modification with respect to the position I outlined some five years ago!.
I referred then to this as representing the first of the radial bands of development, where linear (rational) and circular (intuitive) modes of understanding are comprehensively integrated in a mature manner.
However I would now consider on reflection that this band should be referred to more accurately as pre-radial.
Indeed there is an interesting - and not entirely accidental - connection here with another finding that I outlined yesterday with respect to cyclic primes.
And as this whole episode illustrates the nature of holistic - as opposed to analytic - behaviour well, I will relate it here.
Many years ago when I was developing my holistic approach to number types, I began to realise that there were strong connections as between the inherent dynamic nature of prime numbers and the earliest stages of infant behaviour (i.e. primitive instinctive behaviour).
Now the very nature of such blindly instinctive behaviour is that neither conscious nor unconscious aspects of personality have yet been properly differentiated.
This means in effect that the infant still directly confuses the (holistic) unconscious desire for meaning with (specific) conscious phenomena.
And when extreme confusion is evident (as at the earliest stage of development) this means that phenomena can enjoy no permanency but rather a very fleeting temporary existence (which immediately becomes eroded in memory).
At a deeper level because of the direct confusion of conscious and unconscious, the infant cannot yet provide a dimensional background of space and time within which to experience object perceptions.
In other words we have a direct confusion here of qualitative (unconscious) and quantitative (conscious) aspects of experience.
Indeed this has intimate connections to the nature of sub-atomic particles which enjoy but a fleeting - and ultimately undetectable - existence in space and time.
So once again with such particles, the holistic dimensional context of their existence becomes largely inseparable from their specific identity (as distinct objects).
Thus we could very accurately describe such sub-atomic particles as relating to prime spacetime (where prime is now understood in its inherent holistic dynamic sense).
Now using holistic mathematical notions, I could see that the essence of the prime notion in dynamic terms, entails the (ultimate) identity of both linear (specific) and circular (holistic) notions of identity.
Coming from this holistic perspective, I later became fascinated - as I have stated elsewhere - with the nature of cyclical primes (where successive digits are taken in a circular - rather than linear - manner).
So for example in linear terms, the digits 142857 can only be traced forward in one unambiguous manner and then backward (i.e. 758241) again in an unambiguous manner.
However in a circular fashion we can have 6 distinctive circular orderings of digits i.e.
142857, 428571, 285714, 857142, 571428 and 714285. Also in this circular context forward and backward lose their meaning.
I then examined the possibility of numbers that could uniquely satisfy both linear and circular notions of order.
This entailed that they could only contain two digits.
So for example in base 2, 01 and 10 satisfy these criteria. The two possibilities in terms of linear definition are 01 (forward) and 10 (backward). Likewise these are the same two possibilities in a circular manner (i.e. 01 and 10).
Finally - to make it meaningful - I imposed the condition that in obtaining the difference of the two possible numbers that we arrive back at our starting number.
So 10 – 01 in binary = 01 (which is our original number).
Now interestingly no such 2-digit numbers exist in base 10 that satisfy these requirements.
However I then established the bases where it would be possible and discovered that in all these cases the original number required represented the number 3 (as a cyclic prime).
I then found that when converted into our denary system that these numbers represented the successive terms in the octagonal series,
1, 8, 21, 40, 65,…..
(Indeed 01 is the cyclic prime 3 expressed with respect to base 2 which in denary terms then constitutes the first term in the series!)
So here is an interesting mathematical result (which I have never seen mentioned anywhere) which came about entirely from pursuing a holistic line of enquiry.
And, though it has never been my main concern to do so, I have managed to make many other - for me - original (though very simple) numerical discoveries directly through following holistic notions.
Now the relevance of all this to my “map of development” is briefly as follows.
Basically this map is framed in a holistic mathematical manner to show through the various bands of development the relationship between linear (conscious) and circular (unconscious) development.
Each of the bands is dub-divided into 3 main levels (i.e. principal stages).
Thus I delineate 21 levels in all (for the entire map).
Furthermore the map is based on dynamic linkages within and between levels in a horizontal, vertical and diagonal manner (in accordance with the geometrical nature of the 8 roots of unity).
Now 21 in base 8 is 25 (which represents the cyclic prime 3 in base 8).
And course 21 is the 3rd octagonal (i.e. related to 8) number.
So it is remarkable how the very structure of this map of development fits in so well with subsequent numerical discoveries.
The meaning of holistic is closely related to integrated! And this is why I have so much confidence in the ideas that I express because they all spring naturally from a holistic mathematical manner of approach (which in formal terms is entirely unrecognised within Conventional Mathematics).
Now there are close similarities with the 3 levels of the 3rd band (higher) and the corresponding 3 levels of the 5th band (which I now refer to as pre-radial).
In some ways it can be likened to climbing a mountain.
In spiritual contemplative terms the “higher” band is generally associated with the “ascent” in a transcendent sense.
As we have seen 2nd band is associated with the specialisation of differentiated (conscious) ability. In (linear) rational terms, Conventional Mathematics represents an extreme in such specialisation in what can be referred to as dualistic interpretation (i.e. where polar reference frames are clearly separated).
So the 3rd band represents the gradual breakdown of dualistic type understanding as one moves continually towards nondual (i.e. intuitive) type appreciation of a holistic kind. The very notion of nondual refers to the fact that polar opposites (such as objective and subjective) are now seen as interdependent and ultimately identical.
In holistic terms the 1st level of this band relates to (algebraic) irrational type understanding. This essentially relates to the reduction of (nondual) higher dimensional intuitive type meaning in a (dualistic) 1-dimensional rational manner.
So for example 2-dimensional understanding is characterised by the holistic intuitive fusion, from the (unconscious) recognition of the interdependence of the opposite polarities (that condition all experience). Now (just like the square root of a number) we can attempt to express this in a linear rational manner as the complementarity of opposite poles which - literally - appears paradoxical (i.e. irrational) from this perspective. .
So the main problem with the 1st level is a persistent conflict whereby nondual meaning becomes unduly identified with its reduced rational means of expression.
The 2nd level of the band then relates to (transcendental) irrational understanding which is of a more refined nature.
One no longer attempts to express meaning in a reduced linear fashion but rather in a subtler dynamic manner as the relationship between linear (conscious) and (circular) unconscious aspects.
However this implies a much greater transience with respect to phenomenal experience as rigid identification with objects would imply a (reduced) merely conscious understanding.
So by this level understanding is very holistic indeed with all meaning seen as entailing the interaction of conscious and unconscious.
The 3rd level (transfinite) would be largely concerned with removing remaining rigidities with respect to the dynamic interpretation of phenomena evolving (in a more purely nondual contemplative type of experience.
Then having reached the contemplative mountain top, rather like one on the top of Mount Everest one looks around so as to attain greater perspective from this new vantage point.
So the 4th band (with its three extremely refined “levels”) represents the specialisation of pure holistic (intuitive) consciousness.
Now the 5th band (the pre-radial levels) represents the descent in immanent terms from the top as one gradually tries to maturely integrate dualistic type understanding with a permanent contemplative vision.
In fact I would maintain that most of my own efforts in the last number of years represent a preliminary pre-radial (level 1) interpretation of reality.
This entails in psychological terms the emerging ability to express in a necessarily dualistic manner the overall relationship throughout development as between linear (conscious) and circular (unconscious) type reality (with its corresponding complement in physical terms of the analytic and holistic aspects respectively of reality.)
However I must stress that in formal terms, science and mathematics operate from the specialised development of merely the 2nd band (where no formal recognition of the holistic aspect exists).
So we hear a great deal for example about analytic science (and indeed mathematical analysis). But where is the equal emphasis on holistic science (and mathematical holism)?
And even more subtly, where is the emphasis on mathematics and science as necessarily entailing a dynamic interaction as between its analytic and holistic elements?
However, in certain respects an intense focus on the Riemann Hypothesis has led me on to the next descent (level 2) where the dualistic implications of (algebraic) irrational understanding become more apparent.
Quite simply what this entails - but which only recently has become obvious to me - is that the very means by which we enabled to make ordinal distinctions at a dualistic level - literally - has its roots in higher dimensional understanding.
For example in the case of relating two objects with respect to each other, which involves the simplest type of qualitative distinction, implicitly the holistic interpretation of the two roots of 1 are entailed.
Now of course, we do this automatically in experience (which is another way of saying that we remain totally unconscious of what is holistically involved).
However the very simplest ordinal distinction we can make relates directly to unconscious (rather than conscious) ability. And as ordinal necessarily interpenetrates with cardinal, without such unconscious ability it would likewise not be possible to make any cardinal distinctions (of a quantitative nature).
So once we grasp this, we can no longer hope to conduct Mathematics in a merely (conscious) rational manner, as this greatly distorts its true nature.
Thus the pre-radial stages are largely concerned with now bringing the unconscious to light in showing how it needs to be properly integrated with conscious understanding for meaningful phenomenal interpretation of reality.
Indeed, the famed zeta zeros (which I refer to as Zeta 1) represent the direct unconscious basis of our everyday quantitative interpretation of objects.
However this lies at an even deeper level to that of ordinal understanding. Therefore the task of bringing it successfully to light (which represents the more advanced level 3 pre-radial understanding) is much more difficult to access in an intuitively obvious manner.
In other words when you can intuitively recognise as fully obvious that the zeta zeros represent the necessary unconscious infrastructure for all quantitative distinctions (at a conscious level), then you are operating from level 3 of pre-radial understanding.
As for myself, though I know that this must necessarily be so, I have only yet had a partial rather than full intuitive realisation of its obvious nature.