I do not remember many details of my life from the early to mid 90's. However in general terms it proved to a period of growing stress, that I only began to properly understand much later.

I was aware however that my development had taken a pronounced vertical character, with the continual unfolding of more advanced spiritual states and structures.

However as I ascended "higher" in a transcendent direction ever closer to pure spiritual intuition (without phenomenal structure), this was complemented by the opposite descent with respect to sensation into the physical realms of the unconscious (repressed from earliest childhood).

Thus increasingly my energy was given over to this dynamic interaction as between "higher" and "lower" (unconscious) bands, as attempted integration of the levels of these bands continued.

I was becoming aware however of a growing problem in that the (conscious) middle levels were being continually bypassed in the process.

And as the more horizontal extensive nature of the middle levels is properly suited for conscious day to day responsibilities, this began to create a considerable amount of stress with respect to the conduct of such activity.

I felt - quite literally - as if my very body was being increasingly stretched in a vertical manner, with less and less oxygen, as it were, available for daily work and social engagements.

Once again the comparisons with a steep mountain climb such as Mount Everest are very apt.

As a climber approaches the summit of Everest, available oxygen in the physical atmosphere becomes steadily depleted. Therefore one can only proceed ever more painfully, as one strives to minimise on undue physical exertion.

Equally as one approaches the transcendent peak in spiritual terms, the psychological oxygen, enabling one to carry out normal conscious activity can likewise become greatly depleted. Therefore the carrying out of day to day activities can prove very burdensome, where no relief can be experienced. One is thereby forced to continually economise on undue conscious effort, which requires that any attachment (positive or negative) be relinquished.

In fact one's breathing in physiological terms becomes almost entirely suspended.

This is compatible with a deep sleep state, where unfolding events leave remarkably little trace in memory. However in the hidden depths, all sorts of physical fantasies of a primitive kind can emerge threatening to disturb this peace.

When I look back now a distinctive form of stress was associated with each of the three major levels (of Band 3).

Level 1 was associated directly with the thinking mental function (connected largely to the expression of my male identity).

The main problem here in terms of stress was associated directly with what would be identified as mental illness.

So as I have stated clinical depression (e.g. endogenous) - even psychosis - is quite likely as one attempts to survive a lengthy "dark night of the soul" experience.

This was associated in my own case with a severe shedding of a masculine ego identity.

It is in the very nature of things that this problem tends to be reinforced through outward events.

So as stated before, at this time my father (who had always provided unconditional affirmation) died. and I also ran into serious problems with respect to employment.

Level 2 was then associated with the affective feeling function (in the hidden expression of a female identity),

The stress that I experienced now was more directly of an emotional kind in the fading of a significant emotional relationship and several social activities that for a while had given me support. Finally and most significantly, my mother died.

So I interpreted this turmoil as a testing of a newly acquired feminine identity. In other words I was forced to develop the feminine within myself to compensate for its outside loss.

Level 3 was associated directly with the volitional function (in the attempt to reconcile both masculine and feminine identities)

However, complementing the pure (empty) activity of will of the "higher" self were the still involuntary promptings of the "lower" physical self (where cognitive and affective functions still remained confused).

So stress now started to manifest itself increasingly in a psycho physiological manner.

In other words, psychological stress - which previously manifested itself in mental and emotional terms - now directly affected the body.

However because I had enjoyed remarkably good health in physical terms prior to this, I was not yet attuned to recognise the new problem.

Then in mid 1996, one night, I began to feel somewhat dizzy and lightheaded. When I developed what seemed like a routine virus the following day, I was relieved as seemingly this was the explanation for the previous night's problem.

Though the virus cleared up, I still seemed unusually weak at times and even passed out one night. Eventually I discovered that I had an internal bleed (which was due to a painless ulcer).

By the time I got to hospital the blood count was dangerously low.

So here was a new crisis, where my own life for a while seemed to be hanging in the balance.

Just as the earlier levels had culminated with the death of my father and mother respectively, this was associated with the threat of death with respect to my personal self, now spiritually experienced in androgynous terms (as both masculine and feminine). Only through such an event could the fear of death be faced in such a way, as to enable the transcendent task of that period of development to be realised.

The person who finally climbs on to the summit of Everest though understandably highly elated, is likely to have already suffered a lot and perhaps be in a fragile physical condition.

In like manner, achieving the transcendent peak of my own journey was associated for a while (in hospital and later outside) with several outpourings of mystical joy. However in many ways with the real descent now beckoning, I was still potentially in a fragile physical state and subject to continuing psychological stress.

## Tuesday, August 25, 2015

## Monday, August 24, 2015

### Exploring the Feminine

As I have stated, the attempt to journey through this "highest" level (i.e. Band 3, Level 3) of transcendent spiritual development, was associated with the corresponding exposure to the "lowest" level of primitive instinctive response.

In this way, I was to learn the direct complementary nature as between the "highest" level (in transcendent spiritual terms) and the corresponding "lowest" level (in an immanent physical manner).

In normal infant development, we are conditioned to continually repress earliest primitive responses as the price for enabling the successful differentiation of conscious structures.

This is still very much the case even for development that would be considered healthy and successful from a conventional social perspective!

Though the unconscious is necessarily involved with such development, it is mainly in a reduced manner, whereby it serves the dominant need of conscious type understanding.

Thus true integration - compatible with the attainment of pure spiritual awareness - is not possible without the later return to the lower levels in the sustained quest to disentangle this earliest primitive repression.

It was at this time that I began to see more deeply into the nature of sexual desire.

And for a heterosexual man, this entails coming to terms with the deep roots of unconscious attraction to the feminine, experienced as somehow separated from one's conscious male identity. In other words, the feminine aspect feels thereby as in large part fundamentally missing with respect to one's unconscious.

Therefore it is in the very nature of sexual attraction to project this unconscious element on to women who one in various ways then consciously desires.

In my own case, I realised how much my former relationships with women typically entailed a slow and often painful learning process. I gradually disassociated from initial romantic attraction to find that the feminine aspect that I was searching for was in fact still lacking with respect to my unconscious development. And with this realisation the attraction invariably completely faded.

However, it was only now that I started to properly realise the true depths of this unconscious longing. This could ultimately only cease through unconscious acceptance of myself physically as a woman!

In other words complementing my conscious identity as a man (which role I could always happily associate with in social terms) was a still unrecognised unconscious identity as a woman (as my other self).

And in the quest for true psychological integration - inseparable from the balanced experience of spiritual union - was the requirement to fully accept this (unconscious) feminine identity. And again this did not just not apply in emotional terms but crucially and most intimately also in a physical manner!

In earliest development, the male foetus is initially inseparable from the female mother. Then as conscious life emerges one learns to slowly separate from the mother. Also civilisation rightly requires that the initial physical longing for the mother (of a primitive sexual nature) be unconsciously repressed from an early stage. This then frees up the conscious aspect of personality to become differentiated from the wider environment.

However fully recovering one's unconscious identity requires returning in psychological terms to this early stage to properly disentangle and thereby accept the incestuous nature of this desire.

And there are huge barriers in our culture to properly embarking on this process which is considered taboo and unhealthy.

Certainly I would agree that one should not attempt to explore such repressive barriers before one is properly ready from a psychological perspective. However for anyone truly embarking on the quest for contemplative mystical union, such psycho sexual development constitutes an extremely important element. However it has been greatly censored with respect to the accepted accounts of those who have attained union in the various traditions (especially Western).

So I realised that the conscious acceptance of myself as a male needed to be balanced by the corresponding unconscious acceptance of myself as female. Only then could the involuntary nature of sexual attraction cease through deep realisation that the feminine aspect that had hitherto been sought for without was in fact to be obtained through the unconscious realisation of this aspect in my own personality.

It is not that attraction to the feminine would then cease, but rather arise more freely as an integral expression of my spiritual identity.

However, I discovered that the barriers to such free acceptance (of my unconscious feminine self) were indeed greatly reinforced at every turn by enforced stereotypes of what constitutes an acceptable male identity in Western culture.

Even when one grasps at a conscious level that these stereotypes are highly unbalanced, it still can take a considerable time to properly realise this also at an unconscious level. For the unconscious mind has its own way of working which does not respond to conscious commands, no matter how urgent!

So it occurs slowly by a growing form of acceptance, where "higher" super ego control put in place from earliest childhood is gradually dissolved and unconscious realisation eventually attained.

However though the nature of the problem is indeed experienced at Level 3 (Band 3) it cannot be properly resolved at this level.

As we have seen though a degree of correction took place and Level 2 in terms of restoring an immanent direction, the emphasis on this outward journey of ascent is firmly on the transcendent aspect. This leads to a mainly top-down approach to integration where one attempts to integrate complementary "lower" stages from the perspective of the "highest" attained.

However this needs to be balanced by a corresponding emphasis on bottom-up integration, where one attempts to integrate the "higher" from the perspective of the revisited "lower" stages (now freed from involuntary repressive elements).

Though I did not clearly realise it at the time, this would require a further descent through the same three levels traversed on the outward journey.

I later began to see this in a Jungian manner with respect to the proper differentiation (and integration) of the four functions.

My strongest function was undoubtedly intuition. Then on the outward journey of ascent, I had sought to integrate this with the thinking function (Band 3, Level 1), then later the feeling function (Band 3, Level 2) and finally the sensation function. So in Jungian terms sensation for me represented the inferior function. However proper integration of the personality would require that this function be developed to a sufficient extent in immanent terms so that it would properly counterbalance intuition (associated with the transcendent direction).

Thus this process of developing the inferior function - so that it could serve in complementary fashion as the basis for bottom-up integration of the psyche - would require a further band (Band 5) entailing the descent though the same three levels (already traversed on the spiritual ascent).

Christian mystics often refer to the attainment of spiritual union as "Spiritual Marriage". In psychological terms, I now understood this to entail the integration of both the masculine and feminine aspects within the personality. This ultimately required the full acceptance of my unconscious physical identity as woman (as the mother archetype with which my identity was initially fused).

Without this full acceptance, sexual desire would always inevitably be involuntarily projected to a degree on to women in the mistaken belief that this could bring conscious fulfilment.

So the paradox arises that to freely experience the full universal nature of sexual desire, one must become detached from any more limited (involuntary) expression of such desire.

In this way, I was to learn the direct complementary nature as between the "highest" level (in transcendent spiritual terms) and the corresponding "lowest" level (in an immanent physical manner).

In normal infant development, we are conditioned to continually repress earliest primitive responses as the price for enabling the successful differentiation of conscious structures.

This is still very much the case even for development that would be considered healthy and successful from a conventional social perspective!

Though the unconscious is necessarily involved with such development, it is mainly in a reduced manner, whereby it serves the dominant need of conscious type understanding.

Thus true integration - compatible with the attainment of pure spiritual awareness - is not possible without the later return to the lower levels in the sustained quest to disentangle this earliest primitive repression.

It was at this time that I began to see more deeply into the nature of sexual desire.

And for a heterosexual man, this entails coming to terms with the deep roots of unconscious attraction to the feminine, experienced as somehow separated from one's conscious male identity. In other words, the feminine aspect feels thereby as in large part fundamentally missing with respect to one's unconscious.

Therefore it is in the very nature of sexual attraction to project this unconscious element on to women who one in various ways then consciously desires.

In my own case, I realised how much my former relationships with women typically entailed a slow and often painful learning process. I gradually disassociated from initial romantic attraction to find that the feminine aspect that I was searching for was in fact still lacking with respect to my unconscious development. And with this realisation the attraction invariably completely faded.

However, it was only now that I started to properly realise the true depths of this unconscious longing. This could ultimately only cease through unconscious acceptance of myself physically as a woman!

In other words complementing my conscious identity as a man (which role I could always happily associate with in social terms) was a still unrecognised unconscious identity as a woman (as my other self).

And in the quest for true psychological integration - inseparable from the balanced experience of spiritual union - was the requirement to fully accept this (unconscious) feminine identity. And again this did not just not apply in emotional terms but crucially and most intimately also in a physical manner!

In earliest development, the male foetus is initially inseparable from the female mother. Then as conscious life emerges one learns to slowly separate from the mother. Also civilisation rightly requires that the initial physical longing for the mother (of a primitive sexual nature) be unconsciously repressed from an early stage. This then frees up the conscious aspect of personality to become differentiated from the wider environment.

However fully recovering one's unconscious identity requires returning in psychological terms to this early stage to properly disentangle and thereby accept the incestuous nature of this desire.

And there are huge barriers in our culture to properly embarking on this process which is considered taboo and unhealthy.

Certainly I would agree that one should not attempt to explore such repressive barriers before one is properly ready from a psychological perspective. However for anyone truly embarking on the quest for contemplative mystical union, such psycho sexual development constitutes an extremely important element. However it has been greatly censored with respect to the accepted accounts of those who have attained union in the various traditions (especially Western).

So I realised that the conscious acceptance of myself as a male needed to be balanced by the corresponding unconscious acceptance of myself as female. Only then could the involuntary nature of sexual attraction cease through deep realisation that the feminine aspect that had hitherto been sought for without was in fact to be obtained through the unconscious realisation of this aspect in my own personality.

It is not that attraction to the feminine would then cease, but rather arise more freely as an integral expression of my spiritual identity.

However, I discovered that the barriers to such free acceptance (of my unconscious feminine self) were indeed greatly reinforced at every turn by enforced stereotypes of what constitutes an acceptable male identity in Western culture.

Even when one grasps at a conscious level that these stereotypes are highly unbalanced, it still can take a considerable time to properly realise this also at an unconscious level. For the unconscious mind has its own way of working which does not respond to conscious commands, no matter how urgent!

So it occurs slowly by a growing form of acceptance, where "higher" super ego control put in place from earliest childhood is gradually dissolved and unconscious realisation eventually attained.

However though the nature of the problem is indeed experienced at Level 3 (Band 3) it cannot be properly resolved at this level.

As we have seen though a degree of correction took place and Level 2 in terms of restoring an immanent direction, the emphasis on this outward journey of ascent is firmly on the transcendent aspect. This leads to a mainly top-down approach to integration where one attempts to integrate complementary "lower" stages from the perspective of the "highest" attained.

However this needs to be balanced by a corresponding emphasis on bottom-up integration, where one attempts to integrate the "higher" from the perspective of the revisited "lower" stages (now freed from involuntary repressive elements).

Though I did not clearly realise it at the time, this would require a further descent through the same three levels traversed on the outward journey.

I later began to see this in a Jungian manner with respect to the proper differentiation (and integration) of the four functions.

My strongest function was undoubtedly intuition. Then on the outward journey of ascent, I had sought to integrate this with the thinking function (Band 3, Level 1), then later the feeling function (Band 3, Level 2) and finally the sensation function. So in Jungian terms sensation for me represented the inferior function. However proper integration of the personality would require that this function be developed to a sufficient extent in immanent terms so that it would properly counterbalance intuition (associated with the transcendent direction).

Thus this process of developing the inferior function - so that it could serve in complementary fashion as the basis for bottom-up integration of the psyche - would require a further band (Band 5) entailing the descent though the same three levels (already traversed on the spiritual ascent).

Christian mystics often refer to the attainment of spiritual union as "Spiritual Marriage". In psychological terms, I now understood this to entail the integration of both the masculine and feminine aspects within the personality. This ultimately required the full acceptance of my unconscious physical identity as woman (as the mother archetype with which my identity was initially fused).

Without this full acceptance, sexual desire would always inevitably be involuntarily projected to a degree on to women in the mistaken belief that this could bring conscious fulfilment.

So the paradox arises that to freely experience the full universal nature of sexual desire, one must become detached from any more limited (involuntary) expression of such desire.

## Friday, August 21, 2015

### The Internet

As the 90's slowly unfolded, I realised that I was becoming subject to a persistent form of stress that was directly related to the nature of personal development that was now unfolding.

Though to some extent a correcting immanent balance had been introduced, the transcendent direction was still very much dominant. And the continual attempt to deal with any form of undue attachment - which my continued contemplative development required - of either a "real" (conscious) or "imaginary" (indirect conscious) nature was leading to considerable strain.

It seemed, that through the desire for an ever purer spiritual experience, I was now required to stifle ever ounce of self expression. First this had related mainly to the intellectual, then later to the emotional and now finally directly to the volitional aspect of personality.

So in my own personal quest to attain the summit of spiritual Mount Everest, the physical body was becoming increasingly sensitive to the continual psychological strain being place on it, which from the conventional ego point of view was quite unnatural.

So even before this latest phase (Level 3) reached its peak, I was already looking for some acceptable means, by which I could allow myself once again communicate with the world and share the many insights that I had developed on the journey. For deep down, I had never lost the conviction that in my holistic mathematical approach, which had unfolded in an entirely original manner, I had truly discovered a pearl of great price that required to be shared with the world.

However the frustrating fact was that the contemplative journey, had so far required that I continually detach myself from any ego involvement associated with this discovery.

And as the unwinding of one layer of attachment always led to a deeper - as yet unrecognised - layer, it often seemed that this situation would last indefinitely whereby I could never hope to share with anyone, the riches with which I had been entrusted.

Then with the rapid development of personal computing and the growth of the Internet, my perspective slowly started to change, as I could see a new world opening up that was perhaps inherently suited to my purposes.

Though I had been lecturing in an educational institution for nearly all my working life, I never identified with an academic ethos, which I always found unduly restrictive and elitist.

Also in relation to my own research, I could see little value in the accepted process of peer review as the requisite means of accepting material for publication. By its very nature, if work is truly original, then the academic peer group for its proper assessment is highly unlikely to exist. In fact this system of peer group review works best for the assessment of marginal progress within a broadly accepted existing paradigm!

However when one's basic position is that the accepted paradigm i.e. Conventional Mathematics itself is quite inadequate to assess a whole new range of discoveries, then there is no hope for the acceptance of such views through the peer review process.

Put again in more general terms, the existing peer group approach is properly suited for the assessment of marginal progress in analytic terms within increasingly specialised scientific disciplines. However it is entirely unsuited as it stands for the assessment of original progress in holistic terms (applying to a wide range of disciplines).

And in the quest for increasing integration in society it is progress in such holistic terms that is now urgently required!

Fortunately however, through the growing influence of the Internet, I could see that this situation would gradually change. So instead of seeking the narrowly defined academic approval of a highly specialised peer group, one can now directly seek to publish one's ideas on the Internet.

So one's peer group in this sense is likely to consist of a much more varied group of those who, perhaps already thinking along the same lines, are therefore far more likely to properly resonate with one's views.

Of course we still have the considerable problem of distinguishing truly enduring ideas from their many crackpot counterparts!

But in time, I can see a new appropriate peer group - not directly identified with academic institutions - of respected like minded pioneers who will gradually decide what is of true value.

Just like the task of picking out successful new businesses, this task will always be imperfect with many mistakes arising.

However the present academic system - with disciplines increasingly defined within narrower specialised areas - is far too conservative and entirely unsuited for the promotion of holistic, as opposed to analytic type developments.

Thus once I had sufficiently mastered the new computer technology with its word processing, e-mail and website facilities, I decided to put some of my ideas in order in the form of the two books "Transforming Voyage"and "The Number Paradigms".

Though to some extent a correcting immanent balance had been introduced, the transcendent direction was still very much dominant. And the continual attempt to deal with any form of undue attachment - which my continued contemplative development required - of either a "real" (conscious) or "imaginary" (indirect conscious) nature was leading to considerable strain.

It seemed, that through the desire for an ever purer spiritual experience, I was now required to stifle ever ounce of self expression. First this had related mainly to the intellectual, then later to the emotional and now finally directly to the volitional aspect of personality.

So in my own personal quest to attain the summit of spiritual Mount Everest, the physical body was becoming increasingly sensitive to the continual psychological strain being place on it, which from the conventional ego point of view was quite unnatural.

So even before this latest phase (Level 3) reached its peak, I was already looking for some acceptable means, by which I could allow myself once again communicate with the world and share the many insights that I had developed on the journey. For deep down, I had never lost the conviction that in my holistic mathematical approach, which had unfolded in an entirely original manner, I had truly discovered a pearl of great price that required to be shared with the world.

However the frustrating fact was that the contemplative journey, had so far required that I continually detach myself from any ego involvement associated with this discovery.

And as the unwinding of one layer of attachment always led to a deeper - as yet unrecognised - layer, it often seemed that this situation would last indefinitely whereby I could never hope to share with anyone, the riches with which I had been entrusted.

Then with the rapid development of personal computing and the growth of the Internet, my perspective slowly started to change, as I could see a new world opening up that was perhaps inherently suited to my purposes.

Though I had been lecturing in an educational institution for nearly all my working life, I never identified with an academic ethos, which I always found unduly restrictive and elitist.

Also in relation to my own research, I could see little value in the accepted process of peer review as the requisite means of accepting material for publication. By its very nature, if work is truly original, then the academic peer group for its proper assessment is highly unlikely to exist. In fact this system of peer group review works best for the assessment of marginal progress within a broadly accepted existing paradigm!

However when one's basic position is that the accepted paradigm i.e. Conventional Mathematics itself is quite inadequate to assess a whole new range of discoveries, then there is no hope for the acceptance of such views through the peer review process.

Put again in more general terms, the existing peer group approach is properly suited for the assessment of marginal progress in analytic terms within increasingly specialised scientific disciplines. However it is entirely unsuited as it stands for the assessment of original progress in holistic terms (applying to a wide range of disciplines).

And in the quest for increasing integration in society it is progress in such holistic terms that is now urgently required!

Fortunately however, through the growing influence of the Internet, I could see that this situation would gradually change. So instead of seeking the narrowly defined academic approval of a highly specialised peer group, one can now directly seek to publish one's ideas on the Internet.

So one's peer group in this sense is likely to consist of a much more varied group of those who, perhaps already thinking along the same lines, are therefore far more likely to properly resonate with one's views.

Of course we still have the considerable problem of distinguishing truly enduring ideas from their many crackpot counterparts!

But in time, I can see a new appropriate peer group - not directly identified with academic institutions - of respected like minded pioneers who will gradually decide what is of true value.

Just like the task of picking out successful new businesses, this task will always be imperfect with many mistakes arising.

However the present academic system - with disciplines increasingly defined within narrower specialised areas - is far too conservative and entirely unsuited for the promotion of holistic, as opposed to analytic type developments.

Thus once I had sufficiently mastered the new computer technology with its word processing, e-mail and website facilities, I decided to put some of my ideas in order in the form of the two books "Transforming Voyage"and "The Number Paradigms".

## Thursday, August 20, 2015

### Holistic Number Types

All the number types - binary, prime, natural, integers, rational, (algebraic) irrational, transcendental, complex and transfinite (in both positive and negative and real and imaginary forms) have well defined meanings from the conventional mathematical perspective (in the accepted analytic sense). Equally they have (unrecognised) holistic meanings with an intimate relevance for the scientific integral understanding of all possible stages of development.

I will just briefly clarify the alternative holistic meaning of the number types (which was the subject of "The Number Paradigms").

We start with the binary numbers 1 and 0.

In holistic terms these numbers are ultimately identical representing the equality of form (1) and emptiness (0). Development commences from the position where they are identical in a totally confused manner (i.e. where neither differentiation nor integration of structures has yet taken place).

So the journey is from this starting confused state to the ultimate mature state of unity (where the differentiation of phenomena has become so refined that they no longer even to appear to arise). In this way this highly dynamic experience of form as relative, becomes inseparable from the absolute present as emptiness.

Perhaps the most exciting breakthrough at this time was the clarity I achieved with respect to the holistic notion of prime numbers. This bears close relationship with the notion of primitive in psychological terms where both conscious and unconscious aspects are still largely confused.

This was to later lead me to the realisation that the study of the ultimate nature of the primes (e.g. with respect to the Riemann Hypothesis) could not be properly undertaken in the absence of both distinctive conscious and unconscious modes of understanding. And as Conventional Mathematics is formally defined in a merely rational (conscious) manner, this poses insuperable difficulties for successfully tackling this issue in the standard manner.

In conventional terms, the natural numbers represent unique combinations of prime factors. In a corresponding holistic sense, natural type experience arises from an underlying psychological base (of primitive instinctive behaviour). However as the infant slowly learns to relate these instincts in an organised manner, the conscious element of experience gradually becomes consolidated.

Thus highly organised experience of objects would require ever greater degrees of conscious specialisation!

The integers involve negative as well as positive natural numbers. In like manner the holistic equivalent entails both positive and negative elements in the growing ability to clearly separate the (external) world from the (internal) self.

So development so far has taken us through the "lower" levels of Band 1, where the conscious becomes ever more clearly separated from the unconscious aspect.

Then at Band 2, full specialisation of rational ability now unfolds. This corresponds in holistic terms with the growing tendency to fragment phenomena into parts through analytic ability, which coincides with the analytic equivalent of rational fractions.

The "higher" levels (Band 3) then leads to the holistic notion of irrational numbers.

In fact the great problem that the early Pythagoreans faced was their inability to satisfactorily provide a holistic explanation of the notion of an irrational number!

Thus a decisive breakdown occurred with respect to the historical development of Mathematics, which increasingly came to abandon holistic notions in favour of a merely reduced analytic meaning.

So in my approach, I showed how the holistic notion of the (algebraic) irrational numbers is closely associated with the nature of Level 1 (Band 3).

I then showed how the notion of a transcendental number (which is also irrational) leads to a greater degree of subtlety. In fact just like π as the most famous transcendental number entails the relationship between a line diameter and its circular circumference, equally the holistic equivalent entails the corresponding dynamic relationship as between both linear (dualistic) and circular (nondual) meaning (as the centre which is common to both).

I also dealt in some detail in this book with the holistic meaning of the directions i.e. positive and negative, real and imaginary and the special limiting complex directions (that are represented as null lines).

In fact this book even went further in showing how the nature of the sub-levels (of each level) could themselves be encoded in a holistic mathematical manner.

In particular, I was anxious here to explain why development in our present culture (especially in mathematical and scientific terms) tends to plateau so strongly with the specialised expression of mere rational understanding i.e. Band 2.

Generally speaking in development at each level, concrete (part) perceptions are first to emerge; then all going well, the next stage will lead to the corresponding emergence of the more general formal (whole) concepts. Finally the interaction of both formal and concrete will take place paving the way for transformation to the next level.

Now the key to understanding the dynamics here is that - properly speaking - part and whole are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other. So in distinguishing concrete and formal sub-levels, I used the Type 1 and Type 2 notions of number respectively to represent both!

This in turn implies that a certain creative fusion of conscious and unconscious aspects is necessary (i.e. of the Type 1 and Type 2 aspects of number) at every level to enable the successful transition to the next level.

Now with the various levels of Band 1, though conscious and unconscious aspects are still confused to a degree with each other, sufficient mutual interaction takes place to enable successful transition between its main levels to take place.

However at Band 2, understanding now becomes so consciously specialised in a reduced manner with only the Type 1 aspect of number recognised, that for most individuals, insufficient interaction as between conscious and unconscious occurs that can enable the radical transition to the more intuitively inspired "higher" levels.

Therefore in our culture, though the unconscious is still necessarily involved, its role is largely reduced to merely serving the needs of conscious type appreciation.

Now there were still many flaws with my approach at that time (especially in terms of adequately explaining the most advanced levels).

However the basic premise - and indeed all the basic insights - were well founded.

In other words the holistic mathematical approach serves as the appropriate basis for an integral scientific approach to development (which potentially entails the entire spectrum of stages).

Also this approach highlights the truly complementary nature of both physical and psychological processes in horizontal, vertical and diagonal terms).

This entails therefore that just as we have many possible levels of psychological understanding, equally we have many possible levels of understanding with respect to the physical world.

So there is not just one valid scientific approach (that equates with Band 2 of the Spectrum)!

There are in fact many other more refined scientific approaches, that apply to the more advanced Bands. Therefore comprehensive understanding requires a truly multi-layered (i.e. multi-level) scientific approach.

So for simplification, I outline three major approaches to Mathematics and Science; 1) analytic 2) holistic and 3) radial, entailing the interpenetration of analytic and holistic.

I will just briefly clarify the alternative holistic meaning of the number types (which was the subject of "The Number Paradigms").

We start with the binary numbers 1 and 0.

In holistic terms these numbers are ultimately identical representing the equality of form (1) and emptiness (0). Development commences from the position where they are identical in a totally confused manner (i.e. where neither differentiation nor integration of structures has yet taken place).

So the journey is from this starting confused state to the ultimate mature state of unity (where the differentiation of phenomena has become so refined that they no longer even to appear to arise). In this way this highly dynamic experience of form as relative, becomes inseparable from the absolute present as emptiness.

Perhaps the most exciting breakthrough at this time was the clarity I achieved with respect to the holistic notion of prime numbers. This bears close relationship with the notion of primitive in psychological terms where both conscious and unconscious aspects are still largely confused.

This was to later lead me to the realisation that the study of the ultimate nature of the primes (e.g. with respect to the Riemann Hypothesis) could not be properly undertaken in the absence of both distinctive conscious and unconscious modes of understanding. And as Conventional Mathematics is formally defined in a merely rational (conscious) manner, this poses insuperable difficulties for successfully tackling this issue in the standard manner.

In conventional terms, the natural numbers represent unique combinations of prime factors. In a corresponding holistic sense, natural type experience arises from an underlying psychological base (of primitive instinctive behaviour). However as the infant slowly learns to relate these instincts in an organised manner, the conscious element of experience gradually becomes consolidated.

Thus highly organised experience of objects would require ever greater degrees of conscious specialisation!

The integers involve negative as well as positive natural numbers. In like manner the holistic equivalent entails both positive and negative elements in the growing ability to clearly separate the (external) world from the (internal) self.

So development so far has taken us through the "lower" levels of Band 1, where the conscious becomes ever more clearly separated from the unconscious aspect.

Then at Band 2, full specialisation of rational ability now unfolds. This corresponds in holistic terms with the growing tendency to fragment phenomena into parts through analytic ability, which coincides with the analytic equivalent of rational fractions.

The "higher" levels (Band 3) then leads to the holistic notion of irrational numbers.

In fact the great problem that the early Pythagoreans faced was their inability to satisfactorily provide a holistic explanation of the notion of an irrational number!

Thus a decisive breakdown occurred with respect to the historical development of Mathematics, which increasingly came to abandon holistic notions in favour of a merely reduced analytic meaning.

So in my approach, I showed how the holistic notion of the (algebraic) irrational numbers is closely associated with the nature of Level 1 (Band 3).

I then showed how the notion of a transcendental number (which is also irrational) leads to a greater degree of subtlety. In fact just like π as the most famous transcendental number entails the relationship between a line diameter and its circular circumference, equally the holistic equivalent entails the corresponding dynamic relationship as between both linear (dualistic) and circular (nondual) meaning (as the centre which is common to both).

I also dealt in some detail in this book with the holistic meaning of the directions i.e. positive and negative, real and imaginary and the special limiting complex directions (that are represented as null lines).

In fact this book even went further in showing how the nature of the sub-levels (of each level) could themselves be encoded in a holistic mathematical manner.

In particular, I was anxious here to explain why development in our present culture (especially in mathematical and scientific terms) tends to plateau so strongly with the specialised expression of mere rational understanding i.e. Band 2.

Generally speaking in development at each level, concrete (part) perceptions are first to emerge; then all going well, the next stage will lead to the corresponding emergence of the more general formal (whole) concepts. Finally the interaction of both formal and concrete will take place paving the way for transformation to the next level.

Now the key to understanding the dynamics here is that - properly speaking - part and whole are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other. So in distinguishing concrete and formal sub-levels, I used the Type 1 and Type 2 notions of number respectively to represent both!

This in turn implies that a certain creative fusion of conscious and unconscious aspects is necessary (i.e. of the Type 1 and Type 2 aspects of number) at every level to enable the successful transition to the next level.

Now with the various levels of Band 1, though conscious and unconscious aspects are still confused to a degree with each other, sufficient mutual interaction takes place to enable successful transition between its main levels to take place.

However at Band 2, understanding now becomes so consciously specialised in a reduced manner with only the Type 1 aspect of number recognised, that for most individuals, insufficient interaction as between conscious and unconscious occurs that can enable the radical transition to the more intuitively inspired "higher" levels.

Therefore in our culture, though the unconscious is still necessarily involved, its role is largely reduced to merely serving the needs of conscious type appreciation.

Now there were still many flaws with my approach at that time (especially in terms of adequately explaining the most advanced levels).

However the basic premise - and indeed all the basic insights - were well founded.

In other words the holistic mathematical approach serves as the appropriate basis for an integral scientific approach to development (which potentially entails the entire spectrum of stages).

Also this approach highlights the truly complementary nature of both physical and psychological processes in horizontal, vertical and diagonal terms).

This entails therefore that just as we have many possible levels of psychological understanding, equally we have many possible levels of understanding with respect to the physical world.

So there is not just one valid scientific approach (that equates with Band 2 of the Spectrum)!

There are in fact many other more refined scientific approaches, that apply to the more advanced Bands. Therefore comprehensive understanding requires a truly multi-layered (i.e. multi-level) scientific approach.

So for simplification, I outline three major approaches to Mathematics and Science; 1) analytic 2) holistic and 3) radial, entailing the interpenetration of analytic and holistic.

## Wednesday, August 19, 2015

### Even and Odd Dimensions

I have highlighted the importance of the holistic mathematical interpretation of 2, 4 and 8 dimensions respectively, for all development processes (psychological and physical).

Basically these relate to the manner in which the principle polarities, that govern all phenomena, dynamically interact.

Jung in his writings drew frequent attention to the importance of mandalas as pictorial images with a strong integrative value for the psyche.

The most common mandalas closely resemble symmetrical 4-fold, or alternatively 8-fold, circular geometrical patterns.

In fact though often presented in an extremely artistic ornate fashion, these pictures fundamentally represent the same structures that can be used to represent both the 4 roots and 8 roots of 1 respectively.

So the deeper explanation as to importance of these mandalas lies in the holistic mathematical nature of all development processes. Again these can most fruitfully be explained in terms of either two (internal/external and whole part) or three (internal/external, whole/part and form/emptiness) polarity sets respectively.

One might further ask why such importance attaches precisely to these polarities!

Again this can be explained with respect to the nature of number.

If we obtain the 1st root of a number, no change takes place and we remain in the world of real positive numbers.

So the first root of 1 = + 1.

However when we now obtain the two roots of a number, we require both the notion of real positive and negative numbers. So the 2 roots of 1 are + 1 and – 1.

Then we obtain the four roots of a number, we additionally require the notion of both imaginary and real numbers (in positive and negative terms). So the 4 roots of 1 are + 1, – 1, + i and – i respectively.

Now universally, all numbers must be either positive or negative in real or imaginary terms.

So in moving from just 1 through 2 to 4 dimensions, we open up vital new notions with respect to all relationships in a holistic mathematical manner.

Now with 8 dimensions we additionally open up four new complex numbers (that have a unique limiting significance).

So the four additional roots (of the 8 roots of 1) are k(1 + i), k(1 – i), k(– 1 + i) and k(– 1 – i) respectively, where k = 1/√2.

The special importance of these roots is that geometrically, they can be represented as null lines.

Thus the ultimate equality of form and emptiness (which can only be experientially approximated in a dynamic relative manner), is provided in a holistic mathematical manner through these additional 4 polarities.

For many years, I attached considerable importance to 2, 4 and 8 dimensions as the basis for integration in experience.

In particular I looked at the three levels of the "higher" band (Band 3) as representing progressively higher levels of integration in a spiritual contemplative manner.

So I referred to Level 1 (where external and internal polarities are interactively fused) as Integral 1: then I referred to Level 2 (where whole and part polarities are interactively fused) as Integral 2 and the Level 3 (where now form and emptiness polarities are fused) as Integral 3.

Each level here in fact literally related to a dimensional number as a power of 2.

So Level 1 = 2

One might raise the possible significance then of higher powers of 2! Would these not imply in turn higher levels of integration?

The answer I would give is yes! However this would not require the discovery of new polarity sets, with which to structure phenomena, but rather increasing refinement with respect to the existing sets.

Indeed in a certain valid sense, only the first two sets of polarities (internal/external and whole/part) are required to structure experience, for as we have seen the 3rd set simply represents a special limiting case with respect to the combined use of the first two sets!

In fact it is remarkably like directions on a compass.

In basic terms we can start with just two polar sets of directions i.e. N and S together with E and W which we can represent as 4 equal equidistant points on a circle.

Then we can introduce 4 new directions (that in a sense divide up the existing directions in an equal manner). So we can have NE and NW and also SE and SW.

And of course we needed end here. We could then divide up the 8 directions obtained in an equal manner to now obtain 16 more precise directions.

In fact, no matter how we divide up the circle, we can always attempt to represent directions obtained as some combination of the starting 4 coordinates.

In this way we can give a more universal significance to the holistic mathematical notion of dimension.

For example, 100 dimensional reality would be indirectly expressed in terms of the 100 roots of 1, which would give a much greater range of varying combinations in which the fundamental "directions" of external/internal and whole/part are configured.

Put another way, to interpret in 100 dimensions would require the ability to experience reality in terms of 100 relatively distinct perspectives, that yet all can all be seamlessly combined together in an integrated nondual fashion.

However despite all this progress with respect to the dynamic understanding of the holistic mathematical nature of the dimensions of experience, I only gradually come to the clear realisation of a crucial imbalance with respect to my understanding.

Putting it simply, I had concentrated largely on the even integral dimensions and then on the simplest form (that represented multiples of 2).

It only slowly dawned on me that the odd dimensions, relating to differentiation, would be equally important in experience (though not directly accessible in the same manner).

Now in fairness to myself, I had earlier in fact given considerable attention to the nature of 3-dimensional experience.

Basically I came to understand it like this!

The culmination of Level 1 (Band 3) represents a new form of temporary equilibrium (where both external and internal polarities are united in a nondual fashion).

This by its nature represents a passive contemplative state of spiritual intuitive awareness.

However in moving to the "higher" contemplative state of Level 2, a breakdown of the previous passive state is required where one get actively involved in reality (through this advancing spiritual perspective).

In many ways this represents an asymmetrical state, where one's existing spiritual vision, initially tends to clash with one's day to day experiences. Understandably this leads to considerable conflict.

Then all going well, when one eventually reaches 4-dimensional experience, a new state of passive contemplative integration is attained.

So experience therefore correctly should keep alternating as between even dimensions (representing an existing passive state of spiritual integration) and odd dimensions (representing a new stage of differentiation with respect to one's active involvements). And ideally, one should keep alternating as between contemplation and activity in a balanced manner.

However I came to realise in my own case that the emphasis was very much more on the passive even dimensions rather than active odd dimensions.

Therefore though I had indeed made considerable progress in spiritual contemplative terms, it had been far too narrowly based with insufficient exposure to every day activities and concerns.

Though in a sense, from a transcendent perspective, I was now approaching the summit of the spiritual mountain, there was still a vast period of adjustment required, before I could hope to place my feet firmly again on the plains below,

So in many ways, this proved a very disillusioning time that quickly led to unexpected health difficulties.

Basically these relate to the manner in which the principle polarities, that govern all phenomena, dynamically interact.

Jung in his writings drew frequent attention to the importance of mandalas as pictorial images with a strong integrative value for the psyche.

The most common mandalas closely resemble symmetrical 4-fold, or alternatively 8-fold, circular geometrical patterns.

In fact though often presented in an extremely artistic ornate fashion, these pictures fundamentally represent the same structures that can be used to represent both the 4 roots and 8 roots of 1 respectively.

So the deeper explanation as to importance of these mandalas lies in the holistic mathematical nature of all development processes. Again these can most fruitfully be explained in terms of either two (internal/external and whole part) or three (internal/external, whole/part and form/emptiness) polarity sets respectively.

One might further ask why such importance attaches precisely to these polarities!

Again this can be explained with respect to the nature of number.

If we obtain the 1st root of a number, no change takes place and we remain in the world of real positive numbers.

So the first root of 1 = + 1.

However when we now obtain the two roots of a number, we require both the notion of real positive and negative numbers. So the 2 roots of 1 are + 1 and – 1.

Then we obtain the four roots of a number, we additionally require the notion of both imaginary and real numbers (in positive and negative terms). So the 4 roots of 1 are + 1, – 1, + i and – i respectively.

Now universally, all numbers must be either positive or negative in real or imaginary terms.

So in moving from just 1 through 2 to 4 dimensions, we open up vital new notions with respect to all relationships in a holistic mathematical manner.

Now with 8 dimensions we additionally open up four new complex numbers (that have a unique limiting significance).

So the four additional roots (of the 8 roots of 1) are k(1 + i), k(1 – i), k(– 1 + i) and k(– 1 – i) respectively, where k = 1/√2.

The special importance of these roots is that geometrically, they can be represented as null lines.

Thus the ultimate equality of form and emptiness (which can only be experientially approximated in a dynamic relative manner), is provided in a holistic mathematical manner through these additional 4 polarities.

For many years, I attached considerable importance to 2, 4 and 8 dimensions as the basis for integration in experience.

In particular I looked at the three levels of the "higher" band (Band 3) as representing progressively higher levels of integration in a spiritual contemplative manner.

So I referred to Level 1 (where external and internal polarities are interactively fused) as Integral 1: then I referred to Level 2 (where whole and part polarities are interactively fused) as Integral 2 and the Level 3 (where now form and emptiness polarities are fused) as Integral 3.

Each level here in fact literally related to a dimensional number as a power of 2.

So Level 1 = 2

^{1}; Level 2 = 2^{2}; Level 3 = 2^{3}.One might raise the possible significance then of higher powers of 2! Would these not imply in turn higher levels of integration?

The answer I would give is yes! However this would not require the discovery of new polarity sets, with which to structure phenomena, but rather increasing refinement with respect to the existing sets.

Indeed in a certain valid sense, only the first two sets of polarities (internal/external and whole/part) are required to structure experience, for as we have seen the 3rd set simply represents a special limiting case with respect to the combined use of the first two sets!

In fact it is remarkably like directions on a compass.

In basic terms we can start with just two polar sets of directions i.e. N and S together with E and W which we can represent as 4 equal equidistant points on a circle.

Then we can introduce 4 new directions (that in a sense divide up the existing directions in an equal manner). So we can have NE and NW and also SE and SW.

And of course we needed end here. We could then divide up the 8 directions obtained in an equal manner to now obtain 16 more precise directions.

In fact, no matter how we divide up the circle, we can always attempt to represent directions obtained as some combination of the starting 4 coordinates.

In this way we can give a more universal significance to the holistic mathematical notion of dimension.

For example, 100 dimensional reality would be indirectly expressed in terms of the 100 roots of 1, which would give a much greater range of varying combinations in which the fundamental "directions" of external/internal and whole/part are configured.

Put another way, to interpret in 100 dimensions would require the ability to experience reality in terms of 100 relatively distinct perspectives, that yet all can all be seamlessly combined together in an integrated nondual fashion.

However despite all this progress with respect to the dynamic understanding of the holistic mathematical nature of the dimensions of experience, I only gradually come to the clear realisation of a crucial imbalance with respect to my understanding.

Putting it simply, I had concentrated largely on the even integral dimensions and then on the simplest form (that represented multiples of 2).

It only slowly dawned on me that the odd dimensions, relating to differentiation, would be equally important in experience (though not directly accessible in the same manner).

Now in fairness to myself, I had earlier in fact given considerable attention to the nature of 3-dimensional experience.

Basically I came to understand it like this!

The culmination of Level 1 (Band 3) represents a new form of temporary equilibrium (where both external and internal polarities are united in a nondual fashion).

This by its nature represents a passive contemplative state of spiritual intuitive awareness.

However in moving to the "higher" contemplative state of Level 2, a breakdown of the previous passive state is required where one get actively involved in reality (through this advancing spiritual perspective).

In many ways this represents an asymmetrical state, where one's existing spiritual vision, initially tends to clash with one's day to day experiences. Understandably this leads to considerable conflict.

Then all going well, when one eventually reaches 4-dimensional experience, a new state of passive contemplative integration is attained.

So experience therefore correctly should keep alternating as between even dimensions (representing an existing passive state of spiritual integration) and odd dimensions (representing a new stage of differentiation with respect to one's active involvements). And ideally, one should keep alternating as between contemplation and activity in a balanced manner.

However I came to realise in my own case that the emphasis was very much more on the passive even dimensions rather than active odd dimensions.

Therefore though I had indeed made considerable progress in spiritual contemplative terms, it had been far too narrowly based with insufficient exposure to every day activities and concerns.

Though in a sense, from a transcendent perspective, I was now approaching the summit of the spiritual mountain, there was still a vast period of adjustment required, before I could hope to place my feet firmly again on the plains below,

So in many ways, this proved a very disillusioning time that quickly led to unexpected health difficulties.

## Tuesday, August 18, 2015

### The Spectrum and the Dimensions

A key aspect of my holistic mathematical approach at this time (early 90's) was the manner in which the major levels of each band was defined with respect to the 3 sets of dimensions that have been already outlined.

In general terms, the "lower" band (i.e. Band 1) relates to the gradual differentiation of these three sets, starting from an initial state where total confusion is in evidence.

So with the commencement of development, all three sets are greatly confused - initially with respect to the embryonic foetus - in the life of the baby infant. Because no meaningful differentiation has yet taken place, meaningful integration can likewise not exist.

The first task then is the gradual differentiation of the fundamental diagonal polarity set of form and emptiness. So eventually with the culmination of Level 3 (which represents the lowest level), form can be distinguished from emptiness. What this entails is that through sensori-motor activity, the infant gradually learns to distinguish the self from the phenomenal world, that is now bounded by more permanent notions of form. Initially the recognition of form is so fleeting that it cannot be distinguished from the (empty) immediate present moment!

While ongoing differentiation with respect to the diagonal polarities necessarily continues, the next task is the differentiation of the vertical whole/part polarities. Prior to this, the infant engages in magical thinking, where holistic (unconscious) reality remains greatly confused with specific (conscious) symbols. However with the culmination of Level 2, substantial progress has been made in the separation of the emotional self from the environment, thereby eroding the magical nature of experience.

The final major task of the "lower" band is then to achieve substantial differentiation with respect to the remaining horizontal polarity set of external and internal.

This then leads to the growing ability to successfully abstract (external) objective events from the (internal) self.

So the main emphasis in earliest development is on the successful differentiation of each of the 3 main polarity sets. Though a certain amount of supporting integration is also necessary, this remains secondary to successful differentiation at this time.

The next band - which I refer to as Band 2 - then relates to growing specialisation with respect to the differentiated development that has already occured.

The first concrete level relates directly to specialisation with respect to the horizontal polarities in the generation of detached objective data in experience.

The second formal level then relates to the growing specialisation with respect to "whole" concepts and the ability to formulate abstract theories.

The final vision level then relates to the two-way interplay of both previous levels, in an increasingly creative manner that implicitly requires the generation of substantial intuition.

However though once again a certain degree of supporting integration also necessarily takes place, in our Western culture this tends to be of a somewhat reduced nature, designed to accommodate the primary emphasis on successful differentiation with respect to experience.

If this is to be properly addressed, a dramatic change in emphasis is required in development with substantial erosion of dualistic activity (expressive of differentiation) required to accommodate true nondual integration).

This relates to the "higher" band (Band 3) in my approach.

The first "higher" level (Level 1) then relates to the paradoxical circular appreciation of the nature of external and internal polarities paving the way for a purer nondual spiritual awareness at this level.

The second "higher" level (Level 2) then relates to the additional paradoxical interplay as to the nature of whole and part polarities that are "real" and "imaginary" with respect to each other.

In turn this leads to substantial recognition of the complementary vertical links that relate Band 1 with Band 3 (especially with respect to Level 2 in each case). Thus the mature integral experience with respect to the "higher" is necessarily with respect to the remaining confused experience with respect to the corresponding "lower" level (with ultimately pure nondual appreciation "higher and "lower" attaining a merely relative secondary meaning).

Finally the third "higher" level (Level 3) leads to to most refined paradoxical interplay as between the form and emptiness polarities. This in turn leads to the greatest degree of interaction both within and between levels with respect to the "lower" band (Band 1) and the corresponding"higher" Band (Band 3).

Thus, though Level 3 - both "higher and "lower" - maintained a certain discrete meaning (as restricted to just one level in each case), in truth their true nature is of a highly dynamic continuous level linking all "higher" and "lower" levels!.

Thus the baby infant in a sense at "lower" Level 3 accesses all levels on the spectrum (in a highly confused manner).

Likewise the advanced contemplative at "higher" Level 3 accesses all levels on the spectrum (in a corresponding mature manner). And both of these Bands remain intimately connected with each other throughout development!

However the major weakness of my approach at that time - which I was only later to clearly realise - is that the middle levels (associated with specialised linear type understanding) became significantly bypassed. What I was really describing was the integration of Bands 1 with Band 3. However, though I indeed recognised the importance of Band 2 (with respect to the specialised differentiation of activity), I was not yet able to show how this Band could then be successfully integrated with both "higher" and "lower" bands.

At that time, I mistakenly believed in a "happy ever after" scenario, whereas the full radial structures, where both dual and nondual would successfully interpenetrate with each other, would now quickly occur in development.

However I was to slowly realise that in balanced development, equal attention needs to be given to both the "ascent" and the "descent". So I had only really yet outlined the "ascent" with respect to development, with the tricky task of successful "descent" to be yet negotiated.

However despite this, the subsequent years have only deepened my conviction in the value of the holistic mathematical approach to development.

My understanding of how it operates has indeed become more refined. However the basic insights that I formulated in the early 90's were certainly all on the right track.

In general terms, the "lower" band (i.e. Band 1) relates to the gradual differentiation of these three sets, starting from an initial state where total confusion is in evidence.

So with the commencement of development, all three sets are greatly confused - initially with respect to the embryonic foetus - in the life of the baby infant. Because no meaningful differentiation has yet taken place, meaningful integration can likewise not exist.

The first task then is the gradual differentiation of the fundamental diagonal polarity set of form and emptiness. So eventually with the culmination of Level 3 (which represents the lowest level), form can be distinguished from emptiness. What this entails is that through sensori-motor activity, the infant gradually learns to distinguish the self from the phenomenal world, that is now bounded by more permanent notions of form. Initially the recognition of form is so fleeting that it cannot be distinguished from the (empty) immediate present moment!

While ongoing differentiation with respect to the diagonal polarities necessarily continues, the next task is the differentiation of the vertical whole/part polarities. Prior to this, the infant engages in magical thinking, where holistic (unconscious) reality remains greatly confused with specific (conscious) symbols. However with the culmination of Level 2, substantial progress has been made in the separation of the emotional self from the environment, thereby eroding the magical nature of experience.

The final major task of the "lower" band is then to achieve substantial differentiation with respect to the remaining horizontal polarity set of external and internal.

This then leads to the growing ability to successfully abstract (external) objective events from the (internal) self.

So the main emphasis in earliest development is on the successful differentiation of each of the 3 main polarity sets. Though a certain amount of supporting integration is also necessary, this remains secondary to successful differentiation at this time.

The next band - which I refer to as Band 2 - then relates to growing specialisation with respect to the differentiated development that has already occured.

The first concrete level relates directly to specialisation with respect to the horizontal polarities in the generation of detached objective data in experience.

The second formal level then relates to the growing specialisation with respect to "whole" concepts and the ability to formulate abstract theories.

The final vision level then relates to the two-way interplay of both previous levels, in an increasingly creative manner that implicitly requires the generation of substantial intuition.

However though once again a certain degree of supporting integration also necessarily takes place, in our Western culture this tends to be of a somewhat reduced nature, designed to accommodate the primary emphasis on successful differentiation with respect to experience.

If this is to be properly addressed, a dramatic change in emphasis is required in development with substantial erosion of dualistic activity (expressive of differentiation) required to accommodate true nondual integration).

This relates to the "higher" band (Band 3) in my approach.

The first "higher" level (Level 1) then relates to the paradoxical circular appreciation of the nature of external and internal polarities paving the way for a purer nondual spiritual awareness at this level.

The second "higher" level (Level 2) then relates to the additional paradoxical interplay as to the nature of whole and part polarities that are "real" and "imaginary" with respect to each other.

In turn this leads to substantial recognition of the complementary vertical links that relate Band 1 with Band 3 (especially with respect to Level 2 in each case). Thus the mature integral experience with respect to the "higher" is necessarily with respect to the remaining confused experience with respect to the corresponding "lower" level (with ultimately pure nondual appreciation "higher and "lower" attaining a merely relative secondary meaning).

Finally the third "higher" level (Level 3) leads to to most refined paradoxical interplay as between the form and emptiness polarities. This in turn leads to the greatest degree of interaction both within and between levels with respect to the "lower" band (Band 1) and the corresponding"higher" Band (Band 3).

Thus, though Level 3 - both "higher and "lower" - maintained a certain discrete meaning (as restricted to just one level in each case), in truth their true nature is of a highly dynamic continuous level linking all "higher" and "lower" levels!.

Thus the baby infant in a sense at "lower" Level 3 accesses all levels on the spectrum (in a highly confused manner).

Likewise the advanced contemplative at "higher" Level 3 accesses all levels on the spectrum (in a corresponding mature manner). And both of these Bands remain intimately connected with each other throughout development!

However the major weakness of my approach at that time - which I was only later to clearly realise - is that the middle levels (associated with specialised linear type understanding) became significantly bypassed. What I was really describing was the integration of Bands 1 with Band 3. However, though I indeed recognised the importance of Band 2 (with respect to the specialised differentiation of activity), I was not yet able to show how this Band could then be successfully integrated with both "higher" and "lower" bands.

At that time, I mistakenly believed in a "happy ever after" scenario, whereas the full radial structures, where both dual and nondual would successfully interpenetrate with each other, would now quickly occur in development.

However I was to slowly realise that in balanced development, equal attention needs to be given to both the "ascent" and the "descent". So I had only really yet outlined the "ascent" with respect to development, with the tricky task of successful "descent" to be yet negotiated.

However despite this, the subsequent years have only deepened my conviction in the value of the holistic mathematical approach to development.

My understanding of how it operates has indeed become more refined. However the basic insights that I formulated in the early 90's were certainly all on the right track.

## Sunday, August 16, 2015

### The Dimensions (Directions) of Development (4)

The 4 additional diagonal polarities are capable of several interpretations that highlights their respective importance.

(In fact I already have dealt with their use in clarifying the nature of the physical forces, such as light, together with their complementary spiritual manifestations).

I use these polarities to translate the - literal - complex nature of the structures involved at Level 3 (Band 3) where real and imaginary parts are equal. This is then used to explain the dynamic nature of experience at this level, whereby phenomena of form become so refined that they become fully consistent with the permanent realisation of emptiness (as nondual spiritual awareness).

However we can look at the complex nature of these polarities from a slightly different perspective, that highlights well the extreme contrast that is experienced as between "higher" spiritual desire and "lower" physical instincts at this time.

The real and imaginary parts (with respect to the polarities) can be viewed as the representing the need to maintain appropriate balance as between the cognitive and affective aspects of personality.

Now the preparation for such eventual harmony as between both aspects, was initially experienced (in my case) as an acute conflict in personality. This conflict represented the clash as between the "higher" cognitive aspect that was being used in an ever more refined fashion to mediate an - ultimately ineffable - transcendent spiritual knowledge and a "lower" affective aspect, that was increasingly associated with the most intimate projections of the physical self.

Thus any remaining over-identification with the transcendent aspect of spirit (as beyond all form) was immediately compensated for in experience by the involuntary promptings of the physical self (as prior to identifiable form).

In other words, this extreme clash of the spiritual and physical was necessary so that by the gradual surrender of all attachment to the "higher" aspect of self, the involuntary nature of "lower" physical projections could be slowly eroded.

However in truth, this task of fully balancing the transcendent and immanent aspects of spirit (which was equivalent to the equal balancing of the cognitive and affective functions) was to continue for some considerable time.

Thus like the climber that eventually scrambles on to the peak of Mount Everest (with the successful descent of the mountain still to be negotiated) the culmination of Level 3 (Band 3) represents but the transcendent peak of the spiritual journey with, the successful journey to the counterbalancing immanence of the plains below still to be negotiated.

However I cannot stress strongly enough the truly complementary nature of pure spiritual desire and the most intimate forms of physical fantasy respectively.

Just as the approach to the highest peak (i.e. Mount Everest) places the greatest stress on the physical body, likewise the approach to the most transcendent spiritual peak likewise places the greatest stress on the body (from a psychological perspective) whereby it craves that its needs be instantly met through the most intimate forms of erotic desire.

Just as continual spiritual attainment requires the full integration of cognitive and affective functions, the very nature of bodily instinct is that these two functions are confused directly with each other.

And therefore because of such confusion their promptings are thereby involuntary (whereby the will can exercise no freedom).

Now we can never fully eliminate involuntary instinctive behaviour. However the nature of such behaviour can indeed be considerably refined.

And the refinement of physical instincts is inseparable from the task of purifying spiritual desire so that they always remain complementary with each other, with remaining confusion of the complex polarities at the instinctive level always counterbalancing their greater integration in spiritual terms.

Once again the complex polarities are k(1 + i), k(1 – i), k (– 1 + i) and k( – 1 – i) respectively, where k = 1/√2.

Thus if 1 is used here to represent the cognitive, i - relatively - then represents the imaginary aspect (and vice versa).

Thus at the "higher" spiritual level where these are largely integrated, they still necessarily will remain confused to a degree at the corresponding "lower" physical level (representing involuntary instinctive behaviour). For the continual refinement with respect to integration of the former depends on the corresponding refinement with respect to the remaining confusion with respect to the latter and vice versa.

(In fact I already have dealt with their use in clarifying the nature of the physical forces, such as light, together with their complementary spiritual manifestations).

I use these polarities to translate the - literal - complex nature of the structures involved at Level 3 (Band 3) where real and imaginary parts are equal. This is then used to explain the dynamic nature of experience at this level, whereby phenomena of form become so refined that they become fully consistent with the permanent realisation of emptiness (as nondual spiritual awareness).

However we can look at the complex nature of these polarities from a slightly different perspective, that highlights well the extreme contrast that is experienced as between "higher" spiritual desire and "lower" physical instincts at this time.

The real and imaginary parts (with respect to the polarities) can be viewed as the representing the need to maintain appropriate balance as between the cognitive and affective aspects of personality.

Now the preparation for such eventual harmony as between both aspects, was initially experienced (in my case) as an acute conflict in personality. This conflict represented the clash as between the "higher" cognitive aspect that was being used in an ever more refined fashion to mediate an - ultimately ineffable - transcendent spiritual knowledge and a "lower" affective aspect, that was increasingly associated with the most intimate projections of the physical self.

Thus any remaining over-identification with the transcendent aspect of spirit (as beyond all form) was immediately compensated for in experience by the involuntary promptings of the physical self (as prior to identifiable form).

In other words, this extreme clash of the spiritual and physical was necessary so that by the gradual surrender of all attachment to the "higher" aspect of self, the involuntary nature of "lower" physical projections could be slowly eroded.

However in truth, this task of fully balancing the transcendent and immanent aspects of spirit (which was equivalent to the equal balancing of the cognitive and affective functions) was to continue for some considerable time.

Thus like the climber that eventually scrambles on to the peak of Mount Everest (with the successful descent of the mountain still to be negotiated) the culmination of Level 3 (Band 3) represents but the transcendent peak of the spiritual journey with, the successful journey to the counterbalancing immanence of the plains below still to be negotiated.

However I cannot stress strongly enough the truly complementary nature of pure spiritual desire and the most intimate forms of physical fantasy respectively.

Just as the approach to the highest peak (i.e. Mount Everest) places the greatest stress on the physical body, likewise the approach to the most transcendent spiritual peak likewise places the greatest stress on the body (from a psychological perspective) whereby it craves that its needs be instantly met through the most intimate forms of erotic desire.

Just as continual spiritual attainment requires the full integration of cognitive and affective functions, the very nature of bodily instinct is that these two functions are confused directly with each other.

And therefore because of such confusion their promptings are thereby involuntary (whereby the will can exercise no freedom).

Now we can never fully eliminate involuntary instinctive behaviour. However the nature of such behaviour can indeed be considerably refined.

And the refinement of physical instincts is inseparable from the task of purifying spiritual desire so that they always remain complementary with each other, with remaining confusion of the complex polarities at the instinctive level always counterbalancing their greater integration in spiritual terms.

Once again the complex polarities are k(1 + i), k(1 – i), k (– 1 + i) and k( – 1 – i) respectively, where k = 1/√2.

Thus if 1 is used here to represent the cognitive, i - relatively - then represents the imaginary aspect (and vice versa).

Thus at the "higher" spiritual level where these are largely integrated, they still necessarily will remain confused to a degree at the corresponding "lower" physical level (representing involuntary instinctive behaviour). For the continual refinement with respect to integration of the former depends on the corresponding refinement with respect to the remaining confusion with respect to the latter and vice versa.

## Friday, August 14, 2015

### The Dimensions (Directions) of Development (3)

In yesterday's blog entry, I showed how the holistic mathematical interpretation of the 4 dimensions is intimately related to the corresponding holistic interpretation of the four roots of 1 (expressed in the conventional linear manner).

So space and time are now understood as possessing both real and imaginary aspects with positive and negative directions respectively.

From this perspective, the very dynamics of evolution (physical and psychological) entail that both space and time continually switch as between real and imaginary aspects. This reflects in turn the two-way interaction as between whole and part polarities.

And the positive and negative directions then relate to the complementary physical and psychological manifestations of both respectively, relating to the corresponding two-way interaction as between external and internal polarities.

So "real" space is always accompanied by "imaginary" time and "real" time by "imaginary" space.

Therefore through mutual interaction, time and space keep switching therefore as between their real and imaginary manifestations in positive and negative directions. Once again "real" relates to quantitative (analytic), whereas "imaginary" relates to corresponding qualitative (holistic) notions respectively!

The massive problem for science is that within its 1-dimensional paradigm, it has no way of recognising the qualitative (holistic) aspect of relationships except in a grossly reduced manner!

From a psychological perspective "real" and "imaginary" equally reflect the manner in which cognitive (rational) and affective (emotional) functions interact.

Therefore whenever the cognitive function, for example, is made conscious i.e. in "real" fashion, the affective function is thereby unconsciously present i.e. in an "imaginary" manner. This thereby causes a switch in experience with the affective now conscious in "real" and the affective unconscious in "imaginary" fashion respectively. Thus both cognitive and affective functions thereby, through mutual interaction of conscious and unconscious, keep alternating as between real (analytic) and imaginary (holistic) manifestations.

Of course when the unconscious aspect is not properly developed (as is typical in our Western culture) this sets severe limits on the manner in which both functions can interact, with just one function tending to consciously dominate (in either an external or internal manner).

Combining both sets of insights (relating to dimensions and functions) the very way in which space and time arise in psychological experience is determined by the manner in which the cognitive and affective functions interact. Then in complementary physical terms, the manner in which space and time arise, is directly linked to control and response patterns as the whole and part interactions of the different particles and organisms.

In fact, I was later to develop this holistic mathematical 4-dimensional framework into a system of 24 Personality Types. While enlarging on the original Jungian and Myers-Briggs profiles, this also simplifies it in crucial respects, while also suggesting extraordinary links with the physical world of strings.

I shall return to this at a later stage!

However during the early 90's, I moved on to a new 8-dimensional appreciation of reality.

Thus as well as the horizontal internal/external and the - relatively- vertical whole/part polarities, we now also include the most fundamental diagonal set of form/emptiness polarities.

In many ways, this last set represents the combined integration of the two earlier sets.

However it is in the nature of development that when the sustained two-way paradoxical interplay of the first internal/external set unfolds (Band 3, Level 1), that one tends to specialise in a somewhat restricted fashion with this set.

Then later when the sustained two-way paradoxical interplay of the second whole/part set unfolds (Band 3, Level 2) that again one tends to specialise in somewhat restricted fashion with this set.

This leaves therefore (Band 3, Level 3) the important task of simultaneously seeking to integrate both the horizontal interplay of polarities within a given level, with vertical interplay of polarities between different levels.

And just as the two diagonal lines representing these polarities intersect at the centre point of the unit circle, success in dynamically reconciling both sets, in psychological terms, leads one to the centre point of the personality (i.e. the will).

What happens in effect is that detachment (of both the "real" conscious and indirect "imaginary" conscious) becomes so refined, that phenomena of form no longer even appear to arise in experience.

One thereby can now continually approach the pure nondual spiritual centre of being.

As I have written before, the diagonal lines (representing these polarities) can be interpreted in two ways (that are dynamically equivalent).

Thus from the standpoint of form, we can represent them as showing the equality of both real and imaginary aspects (which in turn symbolise the pure reconciliation of both conscious and unconscious in experience). Now once again - I hasten to add - that from a human perspective, such a pure state can only ever be approximated rather than fully attained!

However from the standpoint of emptiness, we can equally represent the diagonals as null lines with a magnitude = 0. In other words, the refined unattached relationship to all phenomena of form (real or imaginary) is fully consistent with sustained nondual spiritual awareness.

In this way the purely relative experience of form approaches identity with the absolute experience of emptiness (as spirit).

So space and time are now understood as possessing both real and imaginary aspects with positive and negative directions respectively.

From this perspective, the very dynamics of evolution (physical and psychological) entail that both space and time continually switch as between real and imaginary aspects. This reflects in turn the two-way interaction as between whole and part polarities.

And the positive and negative directions then relate to the complementary physical and psychological manifestations of both respectively, relating to the corresponding two-way interaction as between external and internal polarities.

So "real" space is always accompanied by "imaginary" time and "real" time by "imaginary" space.

Therefore through mutual interaction, time and space keep switching therefore as between their real and imaginary manifestations in positive and negative directions. Once again "real" relates to quantitative (analytic), whereas "imaginary" relates to corresponding qualitative (holistic) notions respectively!

The massive problem for science is that within its 1-dimensional paradigm, it has no way of recognising the qualitative (holistic) aspect of relationships except in a grossly reduced manner!

From a psychological perspective "real" and "imaginary" equally reflect the manner in which cognitive (rational) and affective (emotional) functions interact.

Therefore whenever the cognitive function, for example, is made conscious i.e. in "real" fashion, the affective function is thereby unconsciously present i.e. in an "imaginary" manner. This thereby causes a switch in experience with the affective now conscious in "real" and the affective unconscious in "imaginary" fashion respectively. Thus both cognitive and affective functions thereby, through mutual interaction of conscious and unconscious, keep alternating as between real (analytic) and imaginary (holistic) manifestations.

Of course when the unconscious aspect is not properly developed (as is typical in our Western culture) this sets severe limits on the manner in which both functions can interact, with just one function tending to consciously dominate (in either an external or internal manner).

Combining both sets of insights (relating to dimensions and functions) the very way in which space and time arise in psychological experience is determined by the manner in which the cognitive and affective functions interact. Then in complementary physical terms, the manner in which space and time arise, is directly linked to control and response patterns as the whole and part interactions of the different particles and organisms.

In fact, I was later to develop this holistic mathematical 4-dimensional framework into a system of 24 Personality Types. While enlarging on the original Jungian and Myers-Briggs profiles, this also simplifies it in crucial respects, while also suggesting extraordinary links with the physical world of strings.

I shall return to this at a later stage!

However during the early 90's, I moved on to a new 8-dimensional appreciation of reality.

Thus as well as the horizontal internal/external and the - relatively- vertical whole/part polarities, we now also include the most fundamental diagonal set of form/emptiness polarities.

In many ways, this last set represents the combined integration of the two earlier sets.

However it is in the nature of development that when the sustained two-way paradoxical interplay of the first internal/external set unfolds (Band 3, Level 1), that one tends to specialise in a somewhat restricted fashion with this set.

Then later when the sustained two-way paradoxical interplay of the second whole/part set unfolds (Band 3, Level 2) that again one tends to specialise in somewhat restricted fashion with this set.

This leaves therefore (Band 3, Level 3) the important task of simultaneously seeking to integrate both the horizontal interplay of polarities within a given level, with vertical interplay of polarities between different levels.

And just as the two diagonal lines representing these polarities intersect at the centre point of the unit circle, success in dynamically reconciling both sets, in psychological terms, leads one to the centre point of the personality (i.e. the will).

What happens in effect is that detachment (of both the "real" conscious and indirect "imaginary" conscious) becomes so refined, that phenomena of form no longer even appear to arise in experience.

One thereby can now continually approach the pure nondual spiritual centre of being.

As I have written before, the diagonal lines (representing these polarities) can be interpreted in two ways (that are dynamically equivalent).

Thus from the standpoint of form, we can represent them as showing the equality of both real and imaginary aspects (which in turn symbolise the pure reconciliation of both conscious and unconscious in experience). Now once again - I hasten to add - that from a human perspective, such a pure state can only ever be approximated rather than fully attained!

However from the standpoint of emptiness, we can equally represent the diagonals as null lines with a magnitude = 0. In other words, the refined unattached relationship to all phenomena of form (real or imaginary) is fully consistent with sustained nondual spiritual awareness.

In this way the purely relative experience of form approaches identity with the absolute experience of emptiness (as spirit).

## Thursday, August 13, 2015

### The Dimensions (Directions) of Development (2)

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!

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!

## Tuesday, August 11, 2015

### The Dimensions (Directions) of Development (1)

A major feature of the holistic mathematical approach is an
alternative circular notion of number that gives rise to the various dimensions
of development. As each dimension - when expressed in a reduced linear
manner - leads to a distinctive direction (drawn from the centre of
the unit circle in the complex plane to its circumference), I also refer to
these distinctive dimensions as directions.

The deeper meaning of this statement relates to the fact that all phenomenal experience is conditioned by opposite sets of polarities that dynamically interact with each other in experience. This means that opposite poles are clearly separated, with meaning in any context identified with just one pole.

In this regard, I identify 3 key sets of such polarities.

Firstly, we have the horizontal set of external/internal polarities that operate within a given stage of development.

Secondly, we have the vertical set of whole/part (or alternatively collective/individual) polarities that operate between different levels of development.

Thirdly, we have the most fundamental diagonal set of form/emptiness polarities that operate simultaneously within and between all levels of development.

When these three sets operate in a truly dynamically interactive refined manner (without dualistic restrictions) all hierarchical distinctions in development are ultimately clearly seen to have but an arbitrary relative meaning.

So internal is no longer separate (except in a merely relative manner) from external (and vice versa); wholes are no longer separate from parts (and vice versa); and finally form is no longer separate from emptiness (and vice versa).

The big breakthrough that I made - initially in the late 60's - was the realisation that these fundamental polarities possess a unique holistic mathematical numerical structure. In fact they are associated directly with - what I now refer to as - the Type 2 appreciation of number.

The relationship between internal and external was the earliest polarity set to reveal itself in this manner.

Largely under the influence of Hegelian philosophy, at the time I began to realise that whenever an object is made conscious in an external manner in experience, that it is thereby posited in that manner (where positing now represents the holistic mathematical notion of addition i.e. + 1).

Then in order to switch to corresponding internal recognition, the object must be dynamically negated - at least to some degree - in an unconscious manner, before again being consciously posited. So here we have the corresponding holistic mathematical notion of subtraction (as negation) i.e. – 1.

In reverse fashion, to switch back from conscious internal recognition (as mental construct) to its external manifestation as object, once again we must dynamically negate in some measure (at an unconscious level).

Therefore like left and right turns at a crossroads, when properly understood in an interactive manner, the external and internal polarities of experience are understood as positive and negative with respect to each other Expressed in an equivalent manner, external and internal are dynamic complements of each other!

This then directly expresses the Type 2 understanding of the number 2.

In conventional Type 1 terms, 2 is understood - literally in 1-dimensional manner - where it is more fully expressed as 2

^{1}.

However in holistic Type 2 terms, 2 is now understood directly in terms of the dimension (power, or exponent) of 2 that is expressed in an inverse manner as 1

^{2}.

Then to indirectly express in the accepted linear manner this notion of 2, we obtain the 2 roots of 1, to obtain + 1 and – 1 respectively.

In other words when we attempt to express the nature of 2-dimensional understanding in the accepted linear (1-dimensional) fashion, it appears as deeply paradoxical (or circular).

And this is literally because number is now represented in a circular fashion, where the two roots of 1 are equidistant points on the circumference (of the unit circle in the complex plane).

Mathematical interpretation especially is solely defined in a limited 1-dimensional manner.

For example if we take the notion of any number - e.g. 2 - we can posit it externally as a unitary number object in conscious terms (+ 1).

However equally we can posit it internally in conscious terms as the corresponding mental perception of number (+ 1).

So in 1-dimensional interpretation - where the conscious aspects of understanding are solely recognised - both external and internal poles are assumed to directly correspond with each other in an absolute fashion.

However in 2-dimensional interpretation (in holistic terms) external and internal are now clearly seen as opposites of each other (from a dynamic interactive perspective). In other words such understanding crucially incorporates the unconscious aspect of understanding which is vitally necessary to enable switching as between opposite polarities to take place.

Therefore, the deeper realisation that accompanied this breakthrough in terms of understanding the holistic basis of polarity interaction, is that Mathematics itself can no longer be understood in a merely conscious rational fashion, but must also explicitly incorporate its (unrecognised) unconscious aspect.

And when one properly reflects on the matter, this finding should be seen as indisputable, for the very nature of all mathematical understanding necessarily entails the dynamic interaction of objects with mental constructs (that are dynamically positive and negative with respect to each other).

We can easily appreciate, for example in the context of left and right turns at a crossroads, is that when the direction of approach switches as between N and S directions, that these turns then have a purely relative meaning.

However in the dynamics of mathematical experience, polarities (such as external and internal) are continually switching. Therefore, properly understood from this perspective, all mathematical truth is rendered strictly relative. However once again the full realisation of this will require the explicit incorporation of the unconscious aspect in all interpretation!

## Friday, August 7, 2015

### The Spectrum of Number

During this period (in the early 90's) I attempted to draw on my inner experience to portray the various levels of development (especially post-conventional stages) from the unique perspective that I had developed relating to Holistic Mathematics.

The first book, which I termed "Transforming Voyage", did not - at least explicitly - draw so much on this mathematical framework, . The second, "The Number Paradigms", that was written largely in parallel with the first, then sought to interpret this development with reference to the holistic interpretation of the various number types.

In many ways this was a very ambitious undertaking, the significance of which I feel was not really appreciated.

Basically, the starting premise was very simple. In conventional mathematical terms, number provides the appropriate means for ordering reality in a quantitative manner.

However, when appreciated from the corresponding holistic perspective, number likewise provides the appropriate means for providing order in a qualitative fashion to all transformation processes (psychological and physical). Therefore when understood in this scientific integral manner, all possible stages of psychological development are encoded in a holistic mathematical fashion (based on the major number types).

Though the conventional analytic approach to science (and especially mathematics) - qualitatively speaking - is decidedly of a marked rational nature, in quantitative terms, the rational numbers represent just one limited subset of the overall set of numbers.

Therefore, along with rational - we also have irrational (both algebraic and transcendental), imaginary, complex and transfinite numbers. So just as all these numbers have a well defined meaning in quantitative terms, this is equally true from a qualitative perspective.

So associated with all the main number types is a characteristic paradigm, through which the understanding of the appropriate stages on the spectrum is interpreted.

Clearly we already recognise the rational paradigm, which is appropriate for scientific interpretation with respect to the middle band on the spectrum.

However when we attempt to apply this paradigm to the "higher"stages, it leads to a reduced - and indeed - distorted interpretation.

In a certain sense this is equally true of the "lower" stages.

So before the full unfolding of rational numbers (with the fractions), we have the original or fundamental numbers 1 and 0, the primes, the natural numbers and integers (positive and negative).

In like manner, the lower stages can be successively understood in terms of the holistic appreciation of the binary, primes, natural numbers and integers respectively.

Thus the remarkable - and still unrecognised - truth is, that from an integral scientific perspective, all the stages of human development can be encoded in a precise holistic mathematical manner. Indeed this equally applies to all physical processes, which in dynamic terms complement the psychological!

Therefore corresponding to the accepted analytic meaning of the various number types, there is equally a coherent holistic meaning with a remarkable relevance for the scientific understanding of all transformation processes in an authentic integral fashion.

However to its great shame, the Mathematics profession has remained steadfastly blind to this vitally important holistic side of Mathematics (which requires a uniquely distinctive form of intuitive appreciation for its proper comprehension).

Later I was to come to the clear realisation that even the analytic side of Mathematics cannot be coherently understood in the absence of its complementary holistic aspect.

Indeed this has now crystalised in a truly dynamic appreciation of the intrinsic nature of the number system, within which the Riemann Hypothesis acquires a completely new significance.

However in the early 90's I was not yet ready to deal with the proper harmonisation of the analytic (Type 1) and holistic (Type 2) aspects in - what I now refer to as - Radial (Type 3) Mathematics, as I was still in the process of unravelling key holistic notions.

When I look back at "Transforming Voyage", it is with somewhat mixed feelings. I would classify it as representing, in preliminary form, the understanding associated with the advanced contemplative levels of Band 3. However because Band 3 level is heavily associated with the sustained breakdown of phenomenal experience, it becomes progressively difficult therefore to communicate in accessible manner, while traversing its stages, their true nature (based as they are on profound immersion in the unconscious mind).

Though perhaps understandable, I can see that I then attempted to base the universal nature of the "higher" stages to an unacceptable extent on my personal experience (with its many idiosyncratic characteristics). And I am surprised now at the intense psychological nature of much of the written material!

However it is still perhaps valuable in sincerely representing an authentic account of the stages traversed from a distinctive modern perspective.

And very decidedly, I was not concentrating solely on the unfolding of cognitive intellectual type understanding through the higher levels, but rather attempted to portray the close relationship as between cognitive, affective and volitional aspects for all their stages.

There is also the obvious limitation that I was not yet in a position to outline - except in a somewhat sketchy fashion - the more advanced Bands on the Spectrum. For example I now delineate 4 further Bands (following the 3rd "higher" Band). However at that time, there was an attempt to briefly lump all these Bands (4 - 7) into just one Band (which I termed Radial).

The same limitation in some respect also applies to "The Number Paradigms".

However the intervening years have only served to greatly strengthen in my mind its basic premise i.e. that all stages on the spectrum can be scientifically encoded in a holistic mathematical fashion.

It is just that the later Bands additionally require a continuing refinement in the manner through which their respective stages can be appropriately encoded!

The first book, which I termed "Transforming Voyage", did not - at least explicitly - draw so much on this mathematical framework, . The second, "The Number Paradigms", that was written largely in parallel with the first, then sought to interpret this development with reference to the holistic interpretation of the various number types.

In many ways this was a very ambitious undertaking, the significance of which I feel was not really appreciated.

Basically, the starting premise was very simple. In conventional mathematical terms, number provides the appropriate means for ordering reality in a quantitative manner.

However, when appreciated from the corresponding holistic perspective, number likewise provides the appropriate means for providing order in a qualitative fashion to all transformation processes (psychological and physical). Therefore when understood in this scientific integral manner, all possible stages of psychological development are encoded in a holistic mathematical fashion (based on the major number types).

Though the conventional analytic approach to science (and especially mathematics) - qualitatively speaking - is decidedly of a marked rational nature, in quantitative terms, the rational numbers represent just one limited subset of the overall set of numbers.

Therefore, along with rational - we also have irrational (both algebraic and transcendental), imaginary, complex and transfinite numbers. So just as all these numbers have a well defined meaning in quantitative terms, this is equally true from a qualitative perspective.

So associated with all the main number types is a characteristic paradigm, through which the understanding of the appropriate stages on the spectrum is interpreted.

Clearly we already recognise the rational paradigm, which is appropriate for scientific interpretation with respect to the middle band on the spectrum.

However when we attempt to apply this paradigm to the "higher"stages, it leads to a reduced - and indeed - distorted interpretation.

In a certain sense this is equally true of the "lower" stages.

So before the full unfolding of rational numbers (with the fractions), we have the original or fundamental numbers 1 and 0, the primes, the natural numbers and integers (positive and negative).

In like manner, the lower stages can be successively understood in terms of the holistic appreciation of the binary, primes, natural numbers and integers respectively.

Thus the remarkable - and still unrecognised - truth is, that from an integral scientific perspective, all the stages of human development can be encoded in a precise holistic mathematical manner. Indeed this equally applies to all physical processes, which in dynamic terms complement the psychological!

Therefore corresponding to the accepted analytic meaning of the various number types, there is equally a coherent holistic meaning with a remarkable relevance for the scientific understanding of all transformation processes in an authentic integral fashion.

However to its great shame, the Mathematics profession has remained steadfastly blind to this vitally important holistic side of Mathematics (which requires a uniquely distinctive form of intuitive appreciation for its proper comprehension).

Later I was to come to the clear realisation that even the analytic side of Mathematics cannot be coherently understood in the absence of its complementary holistic aspect.

Indeed this has now crystalised in a truly dynamic appreciation of the intrinsic nature of the number system, within which the Riemann Hypothesis acquires a completely new significance.

However in the early 90's I was not yet ready to deal with the proper harmonisation of the analytic (Type 1) and holistic (Type 2) aspects in - what I now refer to as - Radial (Type 3) Mathematics, as I was still in the process of unravelling key holistic notions.

When I look back at "Transforming Voyage", it is with somewhat mixed feelings. I would classify it as representing, in preliminary form, the understanding associated with the advanced contemplative levels of Band 3. However because Band 3 level is heavily associated with the sustained breakdown of phenomenal experience, it becomes progressively difficult therefore to communicate in accessible manner, while traversing its stages, their true nature (based as they are on profound immersion in the unconscious mind).

Though perhaps understandable, I can see that I then attempted to base the universal nature of the "higher" stages to an unacceptable extent on my personal experience (with its many idiosyncratic characteristics). And I am surprised now at the intense psychological nature of much of the written material!

However it is still perhaps valuable in sincerely representing an authentic account of the stages traversed from a distinctive modern perspective.

And very decidedly, I was not concentrating solely on the unfolding of cognitive intellectual type understanding through the higher levels, but rather attempted to portray the close relationship as between cognitive, affective and volitional aspects for all their stages.

There is also the obvious limitation that I was not yet in a position to outline - except in a somewhat sketchy fashion - the more advanced Bands on the Spectrum. For example I now delineate 4 further Bands (following the 3rd "higher" Band). However at that time, there was an attempt to briefly lump all these Bands (4 - 7) into just one Band (which I termed Radial).

The same limitation in some respect also applies to "The Number Paradigms".

However the intervening years have only served to greatly strengthen in my mind its basic premise i.e. that all stages on the spectrum can be scientifically encoded in a holistic mathematical fashion.

It is just that the later Bands additionally require a continuing refinement in the manner through which their respective stages can be appropriately encoded!

Subscribe to:
Posts (Atom)