Monday, June 26, 2017

Revision of Bands (4)

In the context of radial development, it can be perhaps fruitful to distinguish as between three mystical types.

The active type, is by far the most common. Though considerable contemplative development at the "higher" levels will necessarily take place, it never removes a strong grounding in the world of dualistic phenomena. Thus for these types contemplation tends to serve the purpose of bringing a greater sense of integration to everyday responsibilities.

Thus following the opening up of new contemplative levels in development, backward integration tends to readily take place, with spiritual capacities integrated broadly with one's many activities.

It is highly unlikely in such cases that the most refined contemplative states e.g. at the causal level can ever thereby undergo sustained development (because of this natural affinity with duality).

So therefore though such a person may be superbly gifted in many ways, enabling much fruitful work to be carried out with untiring energy and dedication, because of the dominance of the conscious aspect of personality, spiritual meaning is likely to become sometimes rigidly identified with the world of form.

Therefore though exemplifying radial development in many ways, its full expression remains limited in important respects.


The passive type, where contemplative type development is now likely to be dominant, is much less common, certainly in Western culture.

Here a person is likely to show from an early age special sensitivity to the promptings of the unconscious mind. Then if later developments proceeds in a successful manner, this leads to growing absorption in intuitive contemplative states (and accompanying dynamically refined structures) ever more refined in nature.

However the greatest difficulty for this type is to then subsequently ground such development in the temporal world of form.

Even when successfully achieved with the transition to radial development, such a person, while now willingly engaging with the world, is likely to operate with a degree of caution and in a much more restricted manner than the active type.

So again this represents a limitation with respect to the full expression of radial development (with the holistic unconscious remaining more dominant than the conscious aspect).


The third type, the mixed type, is I believe extremely rare, representing in some ways an ideal of what human development at its best might represent in the future.

Here, both the conscious (dualistic) and unconscious (nondual) aspects are harmonised with each other to an extraordinary degree.

This then enables radial involvement in the world that is immensely productive and creative, with a continually deepening contemplative attitude, thereby enabling a wide variety of activities to be fully integrated in a highly purposeful manner.

In other words, when one can truly engage with phenomenal symbols (without possessive attachment) then one can thereby act, as it were, as a superconductor of spirit. And in this blessed state, both activity and contemplation mutually support each other, with activity deepening the intensity of one's contemplative orientation, and this orientation in turn facilitating an ever greater breadth with respect to one's active engagement.

Now in general one can perhaps criticise Western mystical  traditions for traditionally placing too much emphasis on (dualistic) form, and Eastern traditions too much emphasis on (nondual) emptiness.

Thus the most complete expression of radial development exquisitely balances both dual and nondual aspects in a greatly refined manner, through a highly dynamic interactive experience.

Thus paradoxically true immersion in nondual reality enables one to embrace duality in all its fullness, when deemed appropriate.

For example imagine someone at this level that becomes involved in supporting the rights of a strongly marginalised group in society! This could involve intense involvement in combating extreme prejudice and resistance to the most basic human demands. So one would require the patience to hold firm in the midst of considerable pressure. Therefore though one's basic attitude may be formed by an enlightened spiritual outlook, this would now need to be wedded to realistic targets for achievement (that can be objectively measured).

So true spiritual (nondual) enlightenment should thereby be combined with total realism with respect to one's (dual) practical activity.


We will now move on to briefly outline the three levels of Band 7 (which represent the specialised expression of radial development).

Indeed it is only at this band that both the specialised stages of linear activity (Band 2) and the specialised stages of contemplative awareness (Band 4) can now achieve their fullest combined expression. 

The key point here is that we now witness the mature and balanced behaviour of the three primary modes (volitional, cognitive and affective) with respect to both their differentiated and integral aspects.

Though there is now necessarily considerable overlap as between all three, we can to an extent distinguish each level with respect to one key mode.

So Level 1 (Radial 1) can best be identified with the radial cognitive mode in the exercise of true wisdom, where both holistic and analytic aspects interpenetrate to their fullest extent.
Now there still is likely to be a marked distinction as between the active and passive mystical types.

In the first (active) case the analytic aspect will be to the forefront in an ability to become fruitfully involved in a wide range of activities (that properly actualise an underlying commitment to the spirit).

However in the second (passive) case the holistic aspect will be to the forefront, where one is likely to be involved in a more restricted range of activities (that in the circumstances offer the best expression of one's spiritual potential). So in the first case activity is seen as the means of actualising the spirit; in the second case activity is seen as the potential means for generating the spirit.

Now of course both of these perspectives are ultimately fully complementary and best harmonised with the mixed type. However, typically - even at the radial level - a greater or lesser degree of imbalance is likely to persist.

 Level 2 (Radial 2) can best be identified with the radial affective mode, where again both holistic and analytic aspects interpenetrate to the fullest degree.

This results then in the expression of a developed form of compassion, which is felt deeply both in a general sense for all creation and in a more active manner for those who one directly serves in society.

Again there are likely to be differences as between the active and passive mystical types. The active type frequently operates best in large groups in the ability to powerfully communicate with a considerable number of people. The passive is often best in a one-to one situation, in truly accepting another person unconditionally (free of preconceived notions).

However the important thing here is that in the case of both types, they will now have found the respective circumstances in which they both can best express their respective gifts.

Level 3 (Radial 3) represents the true high point where the volitional aspect of personality itself reaches its fullest development. In the Christian tradition this is often expressed as acting perfectly in accordance with the will of God. This would then equate in a psychological manner to the situation where both the conscious and unconscious aspects of personality are now perfectly harmonised (though in experiential terms such a situation can only ever be approximated rather than fully attained).

This then results in the fullest expression of divine love.
Love is a much misunderstood word in our culture being often identified with mere romantic feelings (often of a very self-centred nature).

However the true measure of love is how far one can go in identifying one's own being with universal being so that they are experienced as the same essential reality. (Again in Christian terms this is the ability to identify one's own will fully with the will of God). And when one can operate from such enlightened volitional intent, then everything that one does represents the direct expression of a human love, that is equally divine in nature.

I have always referred to the volitional mode as the most important of the three primary modes.
In fact it is the volitional mode that ultimately is required to enable the two other modes (cognitive and affective) to work in perfect harmony with each other.

And when considerable barriers still exist with respect to successful development, the volitional mode is most important in terms of providing the vital sense of what true integration entails, which can then empower the psyche with considerable capacity for the necessary transformation entailed.

So the innate sense of our eternal destiny (in God) is provided through the will (and its consequent volitional intent).

However, though Band 7 is designed to represent in a sense the full flowering of radial development, I also portray a further band (Band 8).

Band 8 is designed to deal with the later stages of radial development, which though often seeming somewhat of a diminishment on what has gone before, can represent in fact its finest ultimate expression (in human terms).

The painful fact is that the full expression of radial abilities is not likely to last a long time. Realistically one may well be middle aged (or older) before human development can reach this point (though its more active expressions can occur somewhat earlier in life).

Physical ailments and serious illness may limit one's active response. Also a certain form of disillusionment can set it as one realises that the problems and frailties of human existence continue to persist despite one's best interventions.

And human life inevitably ends in physical death. So ultimately one has to prepare well for one's own death.

So just as early development alternates between light and darkness, this remains true with respect to the radial stages. However by now one can better embrace the darkness with equanimity (as representing the hidden light).

There is a great danger in applying secular notions of success even to the highest stages of development. So just as we celebrate those with considerable achievements is sport, business,
politics, the arts etc. likewise we tend to celebrate spiritual superstars who likewise have become famous through their activities and writings.

However "spiritual success" by its very nature tends to remain hidden and largely unknown to those who look at the world through material eyes. And here the "most successful" are often already among us dealing heroically with unforeseen difficult circumstances in life (that most would rather avoid).

I have mentioned already how even at the radial stages we find (depending on personality) active and passive expressions.

Now with the final band there may a conscientious attempt to go against type.

This would mean that the active type - possibly due to unavoidable personal circumstances - may be required to move in a strongly contemplative direction.

Likewise the passive type might be required to become fully commited to some active cause e.g. deep solidarity with a group facing social injustice.

Among religious leaders Pope John Paul 11 represents a good example of the former as Parkinson's disease greatly impeded his activity in later years. However rather than seeking to resign the papacy, he chose to act as a witness on behalf of disabled people everywhere, thereby offering hope to others with similar problems.

Perhaps the Dali Lama represents an example of the latter type. Though clearly of a more strictly contemplative disposition, he has been forced for much of his life into a very difficult political role resisting Chinese domination on behalf of the people of Tibet.

Thus in brief the three levels of Band 8 are:

Level 1 (Suffering Involvement 1). This arises when the shadow side of - even enlightened radial activity - inevitably begins to surface in one's life, often accompanied by a decline in one's physical - and perhaps mental - powers. So darkness intermingles increasingly with the spiritual light.

Level 2 (Suffering Involvement 2). This relates to the courageous "going against type" that I have just mentioned. Though done willingly, this is likely to entail an inevitable increase in suffering.

Level 3 (Suffering Involvement 3). This represents the final stages, where one prepares for death with resignation and equanimity (which could entail a prolonged period of physical and/or mental suffering).

Again coming from a Christian background the key symbol that we are offered is of a man undergoing death through a gruesome Crucifixion. And the important significance of this is to show that ultimate meaning can be found even in the midst of the greatest torment.

And this unselfish example has exercised an immense impact on subsequent followers.

It is useful to ponder finally on how the potential value of physical and mental suffering can be properly recognised within our models of psychological development!

Friday, June 23, 2017

Revision of Bands (3)

In the binary approach to development, equal emphasis is given to the discrete aspect of the unfolding of new stages of development and the continuous aspect, whereby one - because all stages are necessarily related in some respect to each other - thereby enjoys access to all stages.

And whereas the discrete relates directly to the differentiated aspect of stage development, the continuous relates to the integral aspect.

So for example, though from a discrete perspective, the radial stages would be the last to properly unfold in development, there is a certain valid sense in which everyone already has access to these stages.

The radial stages in effect represent the interpenetration of all stages with respect to their specialised differentiation and integration respectively.

However, because by definition, in dynamic terms all stage development represents a certain configuration with respect to differentiation and integration, one thereby already enjoys a certain access to the radial levels.

However in practice, this is likely to represent a severely attenuated experience due to insufficient  development with respect to both aspects.

However it is important to emphasise the dynamic interactive nature of stage development (where differentiation and integration are both necessarily involved).

Thus stage development becomes continually modified through integration with other stages. For example one might maintain that Band 2 development typically occurs relatively early in life and is generally consolidated by early adulthood.

However when one returns to Band 2 from the perspective of a higher band (say Band 3) a new enhanced integral perspective is thereby possible (through the additional context provided through this later development).

However equally - as I was to reluctantly discover - a new diminished differentiated perspective is also likely. In other words when the focus is strongly on the development of holistic intuitive awareness (as at Band 3) one can thereby temporarily suffer a considerable diminution in former analytic type ability (associated with Band 2).

Thus I refer to the typical understanding of a stage at the time of its initial unfolding as the default understanding.

Thus the understanding associated with Band 2 when it first properly unfolds in development represents its default understanding.

Thus again when one later views Band 2 from the perspective of the default understanding of  Band 3, one will likely obtain an enhanced appreciation in integral terms (due to the enlarged context); however when one in reverse now views Band 3 (from the default understanding of Band 2) it represents a diminished appreciation in differentiated terms (due to an erosion of analytic ability).


Now this represents one of the key reasons why I have such difficulties with the customary hierarchical view of stage development, where Band 3 somewhat unambiguously would be viewed as higher than Band 2.

However as we have seen, this is strictly only true in a qualified sense.

Ultimately, when one allows for the proper differentiation and integration with respect to all stages (which occurs with the radial bands), one can indeed form an enhanced perspective with respect to previous bands. However this can only be achieved comparatively late in development and is never achieved in a complete manner.

Therefore until the radial stages are reached, which in a mature developed sense occurs but rarely, what seems progress from one perspective (e.g. integration) may seem like a regress with respect to the alternative perspective (i.e. differentiation) and vice versa.

Therefore though once again there is a certain valid sense in specifically marking the differentiation (in default terms) of the various stages on the spectrum of development, these stages then undergo continual change due to constantly emerging new linkages with other stages (relating to both earlier and later periods of development).

And it is in the bi-directional establishment of complementary linkages that the true integral aspect of stage development is expressed.

So once again we have bi-directional horizontal linkages within each stage, where the (external) physical aspect is related to the (internal) psychological aspect and vice versa.

Then we have bi-directional vertical linkages between appropriate stages where the "higher" is related to the "lower" (and the "lower" in turn to the "higher"). So once again balanced integration must be of both a "top down" and "bottom up" variety.

This probably remains the most widely misunderstood aspect of development, with merely the "top-down" aspect of vertical integration typically emphasised.

It ultimately reflects an unbalanced emphasis on the very notion of holons where, in terms of development, "higher" holons are viewed merely with respect to an increase in collective wholeness. However the very notion of "wholeness" entails two complementary meanings.
Thus from a transcendent perspective all "lower" stages are transcended in a "higher" collective whole stage. However equally, the "higher" stage is made uniquely immanent in each "lower" part stage.

Therefore, one needs to equally emphasise in reverse "onhols", where "higher" development represents a corresponding increase with respect to individual uniqueness.

And of course in this dynamic interactive context, both "holons" (whole/parts) and "onhols" (part/wholes) play a directly complementary role as relatively "higher and "lower" (and "lower" and "higher") with respect to each other.

Finally, we have bi-directional diagonal linkages that simultaneously entail both horizontal integration within each stage and vertical integration between appropriate stages.

In other words, when the complementary nature of integration in both horizontal and vertical terms is simultaneously understood, the very notion of distinct (unrelated) stages loses any residual meaning.

Therefore, one can now successfully maintain the relative distinction of each stage (in differentiated terms) with the relative interdependence of all stages (in an integral manner).

And this enlightened understanding is eventually achieved with radial development.

So what we have dealt with so far with respect to band development serves but as a preparation for the radial bands, which entails the mature balanced interaction of both specialised differentiated and integrated aspects of development.


In my model, I outline three relatively distinct bands, which would now be designated as Bands 6, 7 and 8 respectively.

I caution to add however that I can only hope to offer a diminished perspective on the radial bands.

However, having said this there has been for some time a consistency in my position with respect to the main features that I wish to outline.

In general terms, though I have found that the Eastern mystical traditions offer a much purer and more detailed account of the spiritual contemplative states of development, that the Western traditions offer a more satisfying dynamic account of the radial aspects.

For example in my own Roman Catholic tradition, the great exemplars of radial development (in what Evelyn Underhill refers to as "The Unitive life) have been major spiritual reformers such as St. Paul, Francis of Assisi, Catherine of Sienna, Ignatius of Loyola, Teresa of Avila and in more modern times Padre Pio and Mother Teresa.

Typically in these cases, following conversion to a dedicated spiritual life, an intense incubation period follows as the future saint seeks to throw off the old self in a total commitment to God.

Then when, after much struggle, this task has been suitably achieved, our saint can then enter back into society ready to embrace the problems of humanity with a superhuman degree of energy.

Though I suspect that the depiction of the lives of  these saints has been unduly idealistic, there is undoubtedly a certain truth here, for in the most basic sense with radial development, one identifies with a new cosmic body, where everyone and everything in creation is seen as an extension of oneself. Thus in identifying with this new body, one does indeed take on in a sense the problems of the world. However it is the true existential dimensions of this new situation that I find specially interesting, for even the greatest saint can only respond in the most limited way to the enormity of human suffering (in all its multiple dimensions).

Also I think that the time has now come to alter this limited perspective on radial development as somehow identified - as in the past - with religious reformers inspired exclusively by their specific tradition.

As I see it, great changes are already taking place which will force us to substantially re-assess traditional approaches to the spiritual life.
A cultural revolution is now underway whereby young people are rejecting the right of the institutional churches to arbitrate on any aspect of morality.

And it is not that the desire for authentic spiritual truth is dying but rather that new means are now required to foster the quest for such truth.

In the past, I have devoted some considerable time to the belief that there is a great need to coherently integrate the three big cultural pursuits of religion, science and the arts.

In particular there are special problems in properly integrating religion and science.
I have expressed the view elsewhere in my blogs "Integrating Science and Religion" and "More on Science, Art and Religion" that two major developments are required to facilitate this task.

Firstly, established religions will have to free themselves from literal belief in the various myths used to express their teaching. However as this would be likely to severely undermine their authority among followers, I do not realistically see this taking place. Therefore an alternative type of universal religion will need to emerge - at least where the dialogue with science is concerned - not related to the specific teaching of any one tradition, but attempting however to preserve the primary truth that is deeply implicit in all.

Secondly the very nature of science will need to dramatically change.
In present science, the qualitative aspect of relationships is simply reduced in a quantitative manner.

So a new holistic science (that specifically concentrates on this qualitative aspect) requires to be developed, before eventual integration of both quantitative and qualitative aspects can then take place (as radial science).

And as I have been at pains to outline in recent blog entries, the emergence of this "new science" is intimately tied to the unfolding of the more advanced bands on the spectrum of development.   

So the integration of science and religion therefore properly awaits a much more advanced form of psychological development.


Now with much caution, I will to tentatively outline the various levels of Band 6 (which represent the first of the 3 radial bands)..
In the truest sense possible, entry into the radial bands concurs with a "born again" experience. So at last, one has sufficiently weaned oneself from a more narrow ego based identity to successfully awake to the startling realisation of one's true being as eternal.

So all the phenomena of everyday life act like transparent veils that can no longer hide one's essential being as always already existing in the continual present moment.

Now in many ways it may be helpful to recap on the earliest levels of Band 1 for these now again apply (in a greatly transformed sense).

So the first level in early childhood development is concerned with the unfolding of the body-self.
Likewise Level 1 (Band 6) is directly related to the emergence of a radial body-self.

In other words one now identifies with a universal cosmic body that embraces everything in nature.

So one no longer essentially feels the self as separate from the environment but rather experiences both fully merging as the centre of one's being. Then the breath of life (which had become almost entirely suspended during all the earlier privations) slowly returns, so that in breathing in and breathing out one feels truly inspired, as the life of the entire cosmos is mediated through one's very being.

However just as separation begins to threaten the original fusion of the infant foetus with the mother, this initial radial experience of unity quickly gives way - at least with the personality type with which I am identified - to a deepening existential dilemma.

For in the experience of this new universal identity is likewise born a responsibility for all creation.

However one begins to quickly discover the huge gap as between one's limited actions to serve the world and the universal vision inspiring such action. And this is especially likely to be the case with the more passive type, that formerly was confined to lengthy periods of contemplative solitude.

There is also a parallel here to the sensori-motor activity of infant development that is largely associated with instinctive physical reactions.

During level 1, this implies that one only acts, when inspired to do so directly by the spirit.

So one has no preconceived plans of how to act but rather practices discernment through being guided by a mixture of (physical) instinct and (spiritual) intuition i.e. the will of God. In this way, one is mysteriously led into activity which best expresses one's emerging spiritual potential to contribute to the world.

However once again, initially a huge gap is likely to persist as between the adequacy of such action in properly representing the universal holistic vision with which one is inspired. And this represents an existential dilemma that is keenly felt.


Level 2 is then concerned directly with the unfolding of the radial emotional self, which can best be described as a growing general feeling of compassion for everyone and everything in creation (with which one's very being is now directly identified).

However again coming from the passive contemplative direction, a considerable gap again is likely to be experienced with respect to this universal feeling of compassion and one's capacity to actively intervene on behalf of others. In other words, conscious ability to act will not yet properly serve one's unconscious holistic vision.

So again this existential dilemma will be keenly experienced.

Indeed there could well be a form of "radial magic" associated with this stage. Because of a limited ability to actively serve others, one may resort to a form of wish fulfilment in wanting the spirit to directly solve one's problems.

Now, of course it is good to pray for others. However it is unrealistic to expect prayer to resolve problems that properly require committed action. So in attempting at times to substitute prayer for  action, one thereby engages to a degree in magical belief!

Level 3 can then be identified with the unfolding of the radial mental self. This is likely to represent a major leap forward as one becomes much more skilled now in applying analytic type abilities to various tasks in a holistic integrated fashion.

And with the progressive development of this ability, a marked increase in committed action that best expresses an inherent potential to serve, is now likely to emerge.  However as always (based on this particular personality profile) one will remain acutely aware of how little is actually achieved(compared to the breadth of one's potential vision).

So there can be marked differences with respect to the degree of activity involved for the distinctive personality types that finally embrace the radial levels. We will return to this important point in the next entry.   

Thursday, June 22, 2017

Revision of Bands (2)

I am very conscious in writing this account of development that no one template can adequately account for the distinctive features attributable in any individual case.

So my own account not only represents the product of a distinctive personality type (5 with a strong 4 wing in Enneagram terms) but just one example of such a type with all the personal idiosyncracies thereby associated.

However it does seem to me especially suited for the purpose that I have in mind, which is to portray accurately an authentic modern outline of one person's journey through the various stages of development with all the key difficulties that this entailed.
And though necessarily limited, this then provides a valid perspective from which to address many other issues that I find are not adequately dealt with in existing accounts.

Also it is from such development that the holistic mathematical approach naturally evolved and which then in turn acted as the appropriate vehicle to encode all development in a holistic scientific manner.

So with respect to this radical new vision of the true nature of Mathematics, I believe I have something especially important to offer.

I have mentioned before how I gradually became aware of a very significant problem as development moved to the "highest" level of Band 3.

Though this was designed to result in the two-way vertical integration in complementary manner of the "higher" levels of Band 3 with the "lower" levels of Band 1, I realised that the middle levels of Band 2, were thereby significantly by-passed.

This then meant in effect that I suffered increasing difficulties in carrying out my day to day activities (which largely depended on Band 2). So in psychological terms, it felt - literally - as if my body was being continually stretched from toe to foot, so that I could hardly breathe. Thus the inspiration to carry out conscious tasks - which normally one takes for granted - all but vanished creating considerable stress of a psycho-physical nature.

And this culminated in the bursting of an undiagnosed duodenal ulcer which reduced my blood count to a dangerously low level.

This crisis then marked the end of the extreme ascent phase (with respect to the transcendent spiritual journey). So in a somewhat weakened state, I now slowly contemplated the corresponding descent of the spiritual mountain (which had not formed any part of my previous training).


So Band 5 in my account is concerned with the spiritual descent in the attempt to achieve eventual balance as between both the transcendent and immanent directions of development.

And this likewise entails that the middle levels (which had formerly been significantly by-passed) need now to become the central focus.  

The spiritual ascent itself depends, chiefly in the latter stages, on the quality of volitional intent (in the search for pure spiritual meaning). However inevitably one's "higher" rational capacity, even if now of a considerably refined nature, is still likewise involved.

This then leads inevitably to a degree of super-ego control serving to repress the "lower" instinctive capacities of the personality.

So the first step in recovery is a certain relaxation without respect to the transcendent drive, which then gradually allows the "lower" levels to speak more freely for themselves in the projection of - hitherto - repressed material from the sub-conscious depths. However all of this can take some considerable time with once again various levels to be negotiated.


Level 1 (Binary) in my experience was associated with an exciting new phase where, through the wonders of the Internet, I found myself for the first time in fruitful dialogue with several like-minded individuals.

Through the very attempt to successfully communicate my ideas to others, I was led to adopt a more objective stance unfettered by mere personal considerations.

Thus the first level on the descent related largely to the horizontal internal/external polarities.

So though my communication invariably related to deeply holistic notions, in attempting to express them clearly in an objective manner, this required a new increased degree of integration with the linear levels of Band 2.
In other words, I was slowly discovering that a "higher" holistic vision needed to be properly grounded in the middle levels if it was to have any influence on a wider audience.

Now inevitably initial illumination in the discovery of a new interested Internet audience, was followed - as typical in my experience - by a longer period of purgation, with the strong need to wean myself further from the insidious desire for personal recognition.  However though very difficult, this in fact acted to further weaken super-ego control facilitating much freer access to the primitive "lower" levels.

Gradually I then could return to my intellectual work - now without the support of an appreciative external audience - where I spent considerable time on the elaboration of the binary model of development.

So the realisation that the holistic binary digits could be successfully used - even to a very high level of detail - for the scientific encoding of all stages of development, provided for me startling confirmation of the importance of this qualitative mathematical approach.


Level 2 (Prime) entailed  a much closer two-way relationship as between "higher" super-conscious and "lower" sub-conscious levels (now reflecting that a greater balance existed  between both aspects).

As I have stated many times before, this then became associated with continual absorption in the fundamental nature of both the ordinal and cardinal number systems.

As I have recounted in detail on other blogs esp. "The Riemann Hypothesis", I first discovered that the true qualitative nature of the ordinal numbers is expressed, in an indirect quantitative manner, by the solutions to simple equations (related to the n roots of 1) . And these solutions I refer to as the Zeta 2 zeros (as they bear an intimate complementary relationship with the famous Riemann zeta zeros).

So in the common place understanding of ordinal notions of 1st, 2nd, 3rd and so on, we have the unconscious echo of corresponding "higher" holistic dimensions of understanding (that are of a qualitative nature). So these qualitative dimensions are implicitly grounded in the conventional (1-dimensional) understanding of number.

However the great task - that has not yet been remotely faced by the mathematical profession - is to bring these unconscious holistic dimensions fully into the conscious light, before then finally accepting that the very notion of number can only be adequately understood in a dynamic interactive manner (with complementary quantitative and qualitative aspects).

Though I had been long aware (relating to Band 3 understanding ) of these "higher" holistic dimensions, it was only now during Band 5, that I could see clearly how they were related to everyday ordinal notions. So this discovery with respect to number was intimately related to the personal psychological quest to properly ground "higher "super-conscious understanding (of a holistic universal nature) in the everyday linear understanding of the middle levels.


Another remarkable mathematical transformation was then set to take place - initially with respect to the "lower" primitive levels.

In conventional terms, the primes are considered the basic "building blocks" of the natural number system (in cardinal terms).

However once again there is an unrecognised qualitative aspect to such prime behaviour, so that ultimately quantitative notions with respect to cardinal numbers have no strict meaning in the absence of corresponding qualitative connections.
So the famed Zeta 1 (i.e. Riemann) zeros represent the solutions to an infinite equation which indirectly encode in a quantitative manner the qualitative (holistic) basis of the prime numbers.

It is akin to the notion of particles and waves with respect to the behaviour of matter at the sub-atomic level. So if the primes represent the particle aspect of number, the Zeta 1 (Riemann zeros) then represent the wave aspect and vice versa. So ultimately the quantitative and qualitative aspects of the cardinal number systems (represented by the primes and Zeta 1 zeros) are entirely interdependent (in an ineffable manner).

And again both mathematical and psychological behaviour are themselves fully interdependent with the Zeta 1 zeros in turn representing the manner in which primitive instinctive behaviour is unconsciously encoded. Thus when one has experientially unravelled these zeros (through ending their involuntary nature) the ultimate interdependence of conscious and unconscious aspects of development is realised. This requires an intense exposure to the most intimate fantasies continually emitted into consciousness, which then mysteriously disappear when the involuntary aspect has ceased. And this corresponds exactly with the mathematical realisation of the fully interdependent nature of both quantitative and qualitative aspects of the primes.

And this is just another way of saying that ultimately all behaviour is encoded in a mathematical fashion, thereby representing the all-important fundamental bridge connecting phenomena of form with spiritual emptiness.


Level 3 (Binary/Prime) in many ways entails the balanced co-ordination of the earlier two levels. So underlying all primitive activity is a binary capacity where phenomenal  form (1) can be distinguished from spiritual emptiness (0). So the involuntary nature of primitive instinctive behaviour represents the most basic way in which form can be confused with emptiness!

This likewise represents the stage where both "higher" super-conscious and "lower" subconscious activity come into appropriate balance, through each becoming fully integrated with the linear understanding of the middle levels. Thereby super-conscious understanding becomes properly grounded in these levels so that holistic intuitive awareness now seamlessly informs rational understanding, whereas primitive instinctive reactions are raised from the depths to the same middle levels, whereby they now can become fully compatible with conscious understanding.

So here, any remaining dualistic distinction as between the very notion of "higher" and "lower" ends, with all development now understanding as emanating from the spiritual centre of one's being, which equally represents the centre of the entire physical universe. 

However just as Band 4 cannot be properly completed without on-going further development, the same is true of Band 5, which serves as the ideal preparation for the radial development of Bands 6, 7 and 8 respectively.

In fact whereas each stage of development can be given a certain discrete (differentiated) identity, through continual integration with other stages, this identity always changes. So one cannot experience any stage in its fullness until all stages have undergone appropriate development, which is a never-ending task.     

Wednesday, June 21, 2017

Revision of Bands (1)

Since the last revisions "Update on Classification of Stages" and "Brief further Update" on my overall model of development (made in 2008), I have been operating with 7 major bands (each containing 3 major levels).

However I have come to realise that in recent work, I have in fact been making extensive use of a distinctive band - what I refer to as - Band 5, which was not included in the original account.

So with this extra addition, this now gives 8 bands in my overall model (with 24 major levels).

And in the the context of the holistic mathematical approach these can all be scientifically encoded in a dynamic binary digital manner, representing unique configurations with respect to both differentiated and integral aspects of development respectively.


So perhaps in the light of this latest revision, it would be instructive to outline briefly the 8 major bands.

Band 1 starts from a state of total confusion where differentiation and integration have not yet taken place. And the first major task is to achieve substantial differentiation with respect to the structures that then gradually emerge in childhood experience.

So we have Level 1 sensori-motor activity in the differentiation of the diagonal polarities of form and emptiness. When successfully completed this leads to the emergence of a distinctive body-self.

Level 2 is initially associated with magical experience representing the confusion of the vertical polarities of whole and part, i.e. where the (holistic) unconscious is confused with (specific) conscious phenomena. However when successfully negotiated this leads to the corresponding emergence of a distinctive emotional self.

Level 3 is then associated with mythical type understanding due to remaining confusion with respect to the horizontal polarities of external and internal. However once again when successfully negotiated a child will now be able to substantially distinguish the (psychological) self from the (physical) environment.

Though successful integration should accompany the differentiation of each stage, in practice - especially in Western culture - it is of a somewhat reduced nature that is largely geared to supporting the functioning of more dominant differentiated structures.


Band 2 is then heavily geared to increasing linear specialisation of these conscious structures, especially with respect to the rational cognitive mode that so defines modern technological society.

Scientific specialisation with respect to the horizontal polarities is associated with a concrete stage (in the amassing of "objective" data). This is Level 1.

Corresponding specialisation with respect to the vertical polarities (where the whole can be increasingly separated from the parts) is then associated with a formal stage of pure theory.
Then in practice these two stages are often combined, where empirical research is wedded to testable hypotheses. And this constitutes Level 2.

When the two-way interplay of fact and theory is especially dynamic, rational understanding (of form) can then itself become fuelled with creative insight (spiritual emptiness). And this represents the vision stage (Level 3).

Though much has indeed been achieved in our world through the increasing specialisation of consciousness with respect to the cognitive mode, unfortunately it is also hugely unbalanced (where in scientific and mathematical terms the unconscious aspect of true qualitative type appreciation is reduced in a merely conscious manner).

Not surprisingly therefore, psychological development in modern culture significantly plateaus at Band 2 (with access to both the "higher" super-conscious and "lower" sub-conscious aspects of personality greatly impeded).


However there are always exceptions with some individuals remaining especially sensitive to unconscious promptings from an early age.

In such cases a major existential conflict is likely to emerge in late teenage (or early adult) life. Then if successfully resolved through a commitment to seek authentic meaning (irrespective of social pressures), this can lead to the significant unfolding of more advanced spiritual stages of development.

Band 3 is then associated with the continual refinement of a new intuitive mode of nondual contemplative type awareness. However this generally requires - for long periods - substantial erosion with respect to conventional dualistic type understanding.

Level 1 (Circular) is based on growing interdependence of conscious type phenomena with respect to external and internal polarities.
Then, as the rigid dualistic distinction between such phenomena fades, holistic intuitive awareness of a nondual unconscious kind emerges. However indirectly this is expressed in a conscious manner through the medium of circular i.e. paradoxical, logic.
A key emphasis in my own approach is the necessity to appropriately balance both states and structures with respect to development. So each newly refined state of intuitive awareness should then be appropriately balanced through a corresponding structure based on a refined circular type of paradoxical expression.

Thus there are many "dimensions" corresponding to these increasingly refined dynamic structures of form, which then can be given a precise circular numerical expression in holistic mathematical terms. And though - because of this scientific emphasis - I generally emphasise cognitive structures, there are also corresponding indirect ways of translating paradoxical appreciation with respect to affective and volitional structures.

So another key emphasis of my approach is to need to achieve appropriate integration as between cognitive, affective and volitional understanding throughout development, as these constitute the primary modes of understanding!

Level 1 is comprised of many relatively distinct stages. For example, we can have an initial holistic illumination stage with respect to sensory symbols followed by a corresponding purgative stage (of dualistic erosion) before a new deeper  intellectual illumination stage unfolds. Then a major purgation ("dark night") can then occur leading to the substantial erosion of all conscious phenomena culminating in an extremely oppressive volitional stage (of pure passive contemplation).

Though a belief persists during the illumination stages that one has now progressed to "higher" super-conscious type awareness, during purgation one can then become plunged deep in "lower" sub-conscious stages becoming prey to every human weakness.

Thus another key aspect of development that I emphasise is the need to successfully balance "higher" super-conscious with "lower" sub-conscious development. And this is extraordinarily difficult to properly attain!

There is generally a very pronounced transcendent emphasis to Level 1. To some extent this is rectified with Level 2 (Point). For a while one becomes deeply attuned to holistic affective awareness of an immanent kind, where once more one feels grounded in a stable spiritual reality. However rather like a mountaineer that takes a temporary rest to acclimatise when ascending Mount Everest, one soon becomes involved in seeking an even more refined level of transcendent type awareness.

Level 2 is then largely associated with the "imaginary" aspect of consciousness (both in super-conscious and sub-conscious terms) through the indirect conscious projection of phenomena that directly speak of  a deeply holistic unconscious origin.

And once again one journeys through illumination phases of a highly passive dim light, where one becomes attuned to holistic understanding of a universal nature and purgative phases where substantial erosion of such imaginary phenomena takes place. However both extremes can now be experienced with more equanimity.

Whereas Level 1 becomes associated with a prolonged form of intellectual withdrawal (or negation), Level 2 now relates more directly to corresponding emotional withdrawal whereby one can derive very little sensible consolation from phenomena that increasingly are of a fleeting virtual nature (as indirectly expressive in conscious terms of what properly is of a a holistic unconscious nature).. However deep within, one becomes habitually attuned to a permanent form of awareness of the spiritual present.

And the direct clash as between "higher" super-conscious and "lower" sub-conscious regions of personality becomes much more pronounced at this stage.

Level 3 (Null Level) is then associated with the final assault as it were on the transcendent summit. Here development relates more directly to the volitional aspect as one keeps striving to attain a pure transcendent form of awareness (free of inordinate phenomenal attachment of a conscious or unconscious kind).

However this leads to the deepest psycho-physical type of stress as spirit and nature come directly in conflict with each other. So rather like the mountaineer approaching the summit of Everest, becomes greatly fatigued suffering a debt of oxygen, likewise one now feels extremely stretched in psychological terms with scarcely any inspiration remaining to carry out conscious tasks.

Therefore though sincerely attempting to balance the "lower" physical aspects of personality with the "higher" spiritual aspects, there are inevitable limitations in achieving complete success at this stage.

It is my contention that the most difficult task of all in psychological development is fully coming to terms with the primitive instinctive aspects of behaviour.
And so far in the attempt to achieve pure transcendence, the "higher" cognitive tends to dominate the "lower" affective side of personality. Thus primitive feelings still remain unwittingly repressed preventing true integration.

Therefore though in one sense the culmination of Level 3 marks the completion of the ascent in the attainment of the transcendent spiritual quest, it is achieved at a considerable price. Due to the severe conflict of a psycho-physical nature, one is now likely to become especially vulnerable to some debilitating form of illness.


Just as Band 2 is designed to represent the specialisation of dualistic (linear) type understanding, Band 4 is likewise designed to represent the corresponding specialisation of nondual (circular) type awareness.

This in turn can be associated with three relatively distinctive levels.

Level 1 (Transcendent) represents the transcendent extreme in a holistic contemplative awareness that is beyond all phenomenal symbols. However when not properly balanced by the corresponding immanent aspect, this leads to a subtle aversion to form at the "lower" instinctive levels, causing involuntary projections of primitive repressed phenomena into consciousness.

Level 2 (Immanent) represents the complementary extreme in a holistic type awareness of individual uniqueness that is already embodied in all specific phenomena as their spiritual source of being.

Level 3 (Transcendent/Immanent) represents the subsequent balancing of both contemplative extremes in a refined awareness, where Form and Emptiness are experienced as two indispensable sides of the same coin. So transcendence in an experience of (spiritual) emptiness as beyond all form is necessary to support immanence in the complementary experience of emptiness that is already inherent in all form and vice versa.

However it may take a considerable time for these two spiritual poles of transcendence and immanence respectively to become properly harmonised in experience. Thus before this task is completed we are likely to see the emergence of Band 5 in development.

Tuesday, June 13, 2017

Band 5 - The Three Levels

A distinct form of mathematical understanding is associated with each of the 3 levels of Band 5.

Now even before we can consider the primes we must already admit the existence of the two original numbers i.e. 1 and 0.

And the importance of these two numbers i.e. the binary digits cannot be overstated.

As is well-known, the IT revolution which is now having such an impact on so many aspects of modern living is intimately related to the binary digits 1 and 0 as a means of potentially encoding all data. Thus all information can be converted in a binary digital manner.

And in this this context we are referring to the quantitative (analytic) use of these two digits where 1 is clearly separated from 0.

However what is not at all well known is that these same two digits can equally be given a qualitative (holistic) meaning, where ultimately they become entirely interdependent with each other.

This latter meaning is reflected very well in some mystical traditions where (phenomenal) form becomes ultimately identical with (spiritual) emptiness.

Now if we rephrase this slightly it entails that ultimately the unity of all form (1) is indistinguishable from nothingness (0) as the creative void from which phenomena potentially emerge.

And just as all information can be encoded in an (analytic) binary digital, likewise all transformation processes can be likewise potentially encoded in a (holistic) binary digital manner.

So the main task that absorbed my attention during Level 1 (Band 5) was the ambitious attempt to encode the full spectrum with respect to the stages of development (with complementary physical and psychological aspects) in a holistic binary digital fashion.

Now basically 1, which geometrically resembles the line can be used to represent conscious linear type development and 0 which resembles (with small variations) the circle can be used to represent corresponding circular development (as the indirect paradoxical expression of unconscious type development).

And whereas differentiation in development is related directly to the linear, integration by contrast is directly related to the circular aspect.

So in using this binary digital approach, all stages can be shown to represent a certain unique configuration with respect to both 1 (differentiation) and 0 (integration) respectively.

Thus when applied to the 7 major bands (with 21 major levels), development starts from a state of total confusion with respect to both 1 (differentiation) and 0 (integration) respectively.

The first major task (Band 1) is with respect to the successful differentiation of the various structures (i.e. where 1 is separated from 0). Then the specialisation of these differentiated structures (1) takes place at Band 2. Once again it is important to recognise that conventional interpretation of Mathematics is exclusively based on this band!

Then Band 3 is now concerned with the gradual integration of states and structures (earlier differentiated). So this represents the journey from 1 to 0. Band 4 then relates to the corresponding specialisation of these same states and structures. (0).

Band 5 now represents the attempt to gradually combine 0 with 1 in a mature fashion. Finally Band 6 and Band 7 relate to the mature specialisation of both differentiated and integrated structures (1 and 0).

This represented therefore my first clear statement that reality is dynamically encoded in mathematical manner with the binary digital approach representing the most universal holistic means of satisfactorily expressing all development.

This task chiefly occupied me from the years 2001 to 2008 (approx.). Some of the fruits of this labour can be found at "The Stages of Development".


Now if we move on a little, the final breakthrough comes at Band 6, where - what I refer to as - the twin binary system can be jointly as both a means of encoding information and transformation.

Indeed from one very important perspective, all reality can then be viewed as representing in fact such a dynamic twin-based binary system.

Now the great concern I would have regarding our present world is that it is becoming increasingly subject to changes directly or indirectly brought about through the rapid pace of IT developments.

However at root this IT revolution solely relates to information. Though it is certainly bringing about the consequent need for rapid transformation in social political and economic terms, it runs directly counter to the requirements for authentic transformation.

So what we are seeing is adaptation to many changes that is increasingly of a superficial nature. Indeed we are now reaching the stage in developed societies where democracy itself is slowly being undermined due to an inability to achieve the required transformation in an authentic manner.

This is why I believe that a radical change is now urgently required in our perspective on Mathematics, with the hidden holistic aspect of all its symbols at last properly realised.

For in truth though the present approach is certainly highly specialised in a quantitative manner, it is likewise extremely unbalanced. This in turn fosters a mind-set that is deeply inimical with respect to providing a truly integral approach to dealing with so many of our problems.


However before fully completing my binary approach, I became completely absorbed by a new problem, which I already suspected reached into the deepest recesses of unconscious development. This was the famed Riemann Hypothesis and the profound issues it poses with respect to the nature of the primes and their relationship with the natural numbers.

This now constituted Level 2 (Band 5). In fact most of the blog entries I have contributed since 2009, which now approach 1 million words, relate directly to this issue.

Now, because I have dealt with it at considerable length - at least with respect to the issues that I felt especially important to pursue - I will not repeat it all again.

Suffice it to say that my initial expectations were vindicated to a remarkable degree in that proper understanding of the nature of the number system reaches into the deepest level of unconscious understanding both with respect to "higher" super-conscious and "lower" sub-conscious development.

So what was eventually revealed was the hidden qualitative basis of both the ordinal and cardinal number systems respectively.

Properly understood, we cannot hope to understood either system in an absolute quantitative type manner (based on mere rational notions). Rather the number system, both with respect to its ordinal and cardinal aspects must be conceived in a dynamic relative fashion, representing the interaction of quantitative (analytic) and holistic (qualitative) aspects. Thus absolute conventional interpretation represents just one special limiting case.

And whereas the Zeta 2 zeros represent the indirect quantitative means of expressing the qualitative basis of the ordinal aspect, the Zeta 1 (Riemann) zeros represent the indirect quantitative means of expressing the corresponding qualitative basis of the cardinal number system.

Perhaps the biggest discovery here was the realisation that the mathematical task of understanding the true nature of the zeros (Zeta 1 and Zeta 2) and the psychological task of attempting to achieve full integration of conscious with unconscious are really one and the same.

So the zeros (representing different energy states) in both cases, express - again in an indirect quantitative manner - the very manner through which the unconscious achieves integration with the conscious mind. And this in turn - as already discovered with respect to the Zeta 1 zeros - has complementary links with the physical world with respect to the different energy levels of atoms.

Thus when one properly understands the dynamic interaction of the primes with the natural numbers, one recognises that they are ultimately completely interdependent with each other in an ineffable manner. And this likewise applies to the zeta zeros which dynamically mediate as between both aspects. So ultimately the number system - with respect to its cardinal and ordinal aspects - behaves in a fully synchronistic dynamic manner.


Further development then takes place at Level 3 (Band 5).

One of the tasks here is to achieve better integration of the earlier binary approach with the new prime/natural number based approach.

For it has to be acknowledged that 1 and 0 in a very important sense precede the primes.

Though the binary system does indeed provide for the most basic way of encoding all quantitative and qualitative type interactions, it does not enable the same diversity that then follows from the prime/natural number approach.

So ultimately all the rich variety in nature with respect to both quantitative and qualitative characteristics are rooted in the latter number system. However even here the binary numbers fully underlie this diversity. So just as the binary numbers need to be decoded to reveal the rich diversity of information data, likewise the double binary system needs to be decoded, using the two-way relationship as between the primes and the natural numbers, to reveal the rich diversity of all phenomenal reality (in quantitative and qualitative terms).

Even with respect to the Zeta 1 and Zeta 2 zeros, we can see clearly the influence of 1 and 0.

First of all it is the zero solutions to the equations (= 0) that are especially relevant in both cases.

Then with respect to the Zeta 2, the collective sum of quantitative roots of 1 (as the indirect quantitative expression of ordinal members of a number group) = 0. Then, in complementary fashion, each quantitative part of the corresponding cardinal number = 1.

Also whereas the sum of roots = 0, the product of roots = 1 (+ 1 for an odd number and – 1 for an even number)

Then in relation to the Zeta 2 solutions, these come in conjugate pairs as .5 + it and .5 it respectively. So when one adds the two, the result = 1.

Another advance at Level 3 is in relation to one's understanding of how the cardinal and ordinal systems interact.

At Level 2, one tends still to think of both in a relatively separate manner. However two-way dynamic interaction between both aspects takes place at Level 3 so that one clearly realises that the ordinal system has no strict meaning in the absence of the cardinal (and that likewise the cardinal has no meaning in the absence of the ordinal).

And in psychological terms this further entails the ability to appreciate both sets of Zeros (Zeta 1 and Zeta 2) and the primes and natural numbers in cognitive and affective terms.

Monday, June 12, 2017

Band 5 - Psychological Development (7)

Another major transformation emerged in my thinking relating to Band 5 (level) 2 development.

Prior to this I had been accustomed to think of mathematical activity as solely related to the cognitive mode. I was of course already aware that this cognitive mode itself undergoes a considerable amount of refinement throughout the course of development!

So again in broad terms, I distinguished the the linear use of reason (of Band 2) for example from the circular paradoxical use of reason of Band 3 (as the indirect cognitive expression of intuitive type understanding) from the yet more refined interplay of both the linear and circular uses of reason of Band 5.

Though I would have long maintained that both the affective and cognitive modes are themselves complementary, I still tended to identify mathematical activity exclusively with the cognitive.

However this was all to change with respect to increasing understanding of the nature of the Zeta 1 (Riemann) zeros.

Perhaps a key feature of my development with respect to Band 5 was that "higher"superconscious structures became almost entirely associated with the cognitive function (in a refined manner), while in turn the "lower" subconscious structures became equally associated with the affective function (in a primitive manner).

And I came increasingly to realise the close complementary nature of these two modes. Therefore any attachment to phenomena at the "higher" cognitive level (reflecting its dominance as the "superior" mode) would immediately lead to an involuntary reaction though projections with respect to the "lower affective level (still viewed in an "inferior" fashion).

So I gradually came to the realisation that notions of "higher" and "lower" with respect to these two modes were of a purely relative nature. However this required the continual strengthening of the "inferior" affective function with respect to experience so that both modes could eventually be brought into proper balance with each other.

And when proper balance is maintained (with involuntary projections largely ceasing) the will can then more freely operate in a seamless fashion with respect to the integration of both modes.

As I have explained in my understanding, the Zeta 2 zeros (relating to the ordinal nature of number) had earlier become associated with "higher" super-conscious understanding, that eventually become largely grounded in the conventional conscious levels of Band 2.

I was able to now satisfactorily understand, to my own mind, both the relative and absolute basis of the ordinal nature of number. And this conformed to a cognitive appreciation of number (based on refined rational interpretation, fuelled by intuitive insight of a qualitative nature).

However for some considerable time, I experienced much more difficulty in "seeing" what the Zeta 1 (Riemann) zeros properly represented.

And then it eventually dawned on me as to the nature of the problem.
In experiential terms - as described - the Zeta 1 (Riemann) zeros relate to the primitive aspect of experience. However this is directly of an affective rather than cognitive nature  Therefore in order to intuitively "see" what the zeros represented, I needed to understand them in an affective - rather that a cognitive rational - manner. And when I was finally enabled to do this, I was then able to understand them with the clarity that I was seeking.

However the implications here are enormous for what we understand as Mathematics, because ultimately its proper comprehension requires that all the primary modes (cognitive affective and volitional) undergo appropriate development.

And this is fully consistent with my ultimate vision of Mathematics as the encoded basis of all created phenomena. So experience of reality - whether we advert to it or not - is thereby ultimately encoded in a mathematical form. Now we never can see this with total clarity as by their very nature mathematical notions - such as number - are embedded in phenomenal form (in complementary physical and psychological terms).  So there always remains a problem of satisfactorily distinguishing the encoded mathematical nature of reality from the phenomenal veils through which this encoding appears.

However as development approaches Band 6 on the spectrum it is now understood that mathematical understanding entails both cognitive and affective modes. and as the proper integration of these in experience entails the volitional mode, then this too is inseparable from the dynamic experiential appreciation of Mathematics.

In fact another remarkable observation can now be made.
In holistic mathematical terms, both the cognitive and affective modes are real and imaginary with respect to each other. Thus when we understand in "real" terms (in a conscious rational manner), the affective mode is thereby necessarily present  in "imaginary" terms (as affective in a blind unconscious fashion).

Likewise when with reference frames switched, we now understand in "real" terms (in a conscious emotional fashion) the cognitive mode is now necessarily present in "imaginary" terms (as rational in a blind unconscious manner).

So therefore we can have "real" and "imaginary" understanding with respect to both cognitive and affective modes. And these in turn reflect the manner in which both conscious and unconscious aspects with respect to understanding keep switching in the dynamics of experience.

In conventional terms, we identify mathematical understanding consciously solely with the cognitive mode of reason.

This can be directly related - as with the number line - to the "real" aspect of understanding.

However, as we know, we equally can have imaginary as well as real numbers.
So from the holistic mathematical perspective, the imaginary aspect in this context relates to the hidden (unrecognised) affective aspect of mathematical understanding.

Therefore through complex numbers (with both real and imaginary aspects) are used in a merely reduced cognitive rational manner in Mathematics, from a holistic qualitative perspective, real and imaginary entail both cognitive and affective modes.

Thus in true dynamic appreciation - allowing equally for both the quantitative and qualitative aspects of mathematical understanding - cognitive and affective modes must necessarily be involved.


In conclusion, I found a remarkable vindication of my new thinking when I now returned to interpret the Riemann Hypothesis.

From the cognitive rational perspective, it is readily understood that all cardinal numbers lie on the "real" number line (as the product of primes).

Therefore the Zeta 1 zeros - as the indirect quantitative representation of the complementary opposite qualitative interpretation - should thereby likewise reflect, in psychological terms, the affective aspect of understanding. And this should, in relative terms, therefore lie on an "imaginary" line. And this is what the Riemann Hypothesis in fact postulates.

However this assumption, which properly postulates a dynamic interaction as between the quantitative and qualitative aspects of mathematical understanding, cannot be proven from within a system that solely recognises the quantitative aspect.

However if form a dynamic perspective, our assumption that all real numbers lie on the number line is to be justified, then the non-trivial zeros must necessarily all lie in corresponding fashion on an imaginary line (and vice versa). And this dynamically intertwined assumption points directly to the truly relative nature of mathematical truth.

Sunday, June 11, 2017

Band 5 - Psychological Development (6)

It is my contention that all primitive instincts are rooted in prime number behaviour (when properly understood in a dynamic interactive manner).

So we have seen the paradox of prime numbers. Both individually and collective we can view them in quantitative terms as the "building blocks" of the natural numbers. However equally in reverse fashion, we can view the primes as being determined through the qualitative interdependence of the natural numbers!

It is similar with primitive instinctive behaviour, where both conscious (specific) and unconscious (holistic) aspects are necessarily intertwined.

Primitive confusion - as in earliest childhood - arises through a failure to distinguish the specific from the holistic meaning of phenomena. This in turn is due to the fact that as conscious structures have not yet been successfully differentiated in experience, therefore corresponding (unconscious) integration cannot properly take place.

And this is why primitive behaviour is necessarily involuntary in nature, as both quantitative (conscious) and qualitative (unconscious) aspects remain completely intertwined with each other.

In fact, once again there are complementary parallels here with physical nature, where at the sub-atomic quantum level, nature behaves in an increasingly "primitive" fashion.

In other words - though we are not yet accustomed to looking at reality in this manner - specific particles cannot be properly distinguished from the holistic interactive environment in which they emerge.

So the quantitative aspect (as independent phenomena) still remains greatly confused with the qualitative aspect (through holistic interdependence with related phenomena). And this is why particle interactions are so short-lived (as they cannot yet be properly placed within a dimensional framework of space and time)!

So equally it has been my long held contention that at the  quantum level of physical nature, particle interactions are likewise ultimately rooted in prime number behaviour (when understood in dynamic terms).

To put it another way, the problem of primitive instinctive reaction can only be properly solved through firstly successfully differentiating the conscious from the unconscious, before then properly integrating both distinctive aspects.

Now if we apply this in number terms, likewise the problem of the primes can only be properly solved through first successfully differentiating quantitative (analytic) and qualitative (holistic) before then successfully achieving integration of both aspects.

And in truth both psychological and mathematical tasks are themselves - when properly understood - fully complementary.

Therefore to fully unveil the true wonder and mystery of the relationship of the primes with the natural numbers - which requires both highly refined reason and intuition - the full integration of both conscious and unconscious aspects of personality is required.


So with Band 5 (Level 2), subconscious development at the lowest levels becomes so intensive that it can now at last be properly balanced with corresponding super conscious development at the "higher" levels.

However this requires sustained intimate exposure on a daily basis to intimate projections such as sexual fantasies, that spontaneously emerge from the hidden depths.

During this time a gradual learning process can take place whereby one is enabled to erode the involuntary element of such projections to a considerable degree. And this in turn arises from the ability to properly place each primitive projection in an overall holistic context (whereby the impulse to identify it narrowly with any specific phenomenon is removed).

Now the limit in terms of such experience is to reach a stage whereby phenomenal identification with respect to primitive projections is eroded in the very instant of their initial generation.

One thereby attains to the highly dynamic - and purely relative stage - where each projection, while preserving a momentary distinct independent existence, yet can be fully integrated in a holistic interdependent manner with overall experience.  And this approximates a pure energy state (where the notion of form has now all but lost any residual meaning).

And when this state is attained, the involuntary nature of unconscious projection is finally removed, so that in psychological terms both conscious and unconscious aspects of personality can be fully harmonised.


However I now came to realise during this time that exactly the same dynamics applied to the appropriate dynamic understanding of prime number behaviour.

In fact both mathematical and psychological aspects themselves act in complementary fashion - as a two-sided coin - of human experience.

Though we can never see this in a totally clear manner as mathematical notions are always embedded to a degree in physical and psychological phenomena, primitive instinctive reactions, whether in human terms or the natural world are rooted in the primes (when appropriately understand in a dynamic manner).

Now what happens with conscious type experience is that these primes become dynamically arranged in an increasingly organised manner into composite phenomena, which then attain stability at a natural conscious level. This is likewise associated in psychological terms with increasing differentiation of the conscious aspect of personality.

However this can thereby lead to the gradual reduction of the unconscious holistic aspect in a merely conscious manner. So we then misleadingly  believe - as is so much the case with conventional scientific and mathematical understanding - that object phenomena can enjoy an absolute independent existence.

In particular this is the case with the way we view the primes i.e. as essential "building blocks" in quantitative terms of the natural number system.

However intrinsically there is an equally important holistic qualitative aspect to the primes (both in individual and collective terms) relating to number interdependence.

Unfortunately this has been all but lost in conventional mathematical interpretation, based as it is in formal terms on rational specialisation of a merely conscious kind.

So therefore to recover the qualitative aspect of the primes, psychological development itself must proceed through many more advanced stages of development, where firstly refined holistic intuitive capacity (related directly to the unconscious aspect) unfolds and then eventually where this intuitive capacity is successfully integrated with former rational understanding.

So we have therefore three main stages of mathematical development:

Stage 1 conforming to conventional analytic interpretation of mathematical symbols in a quantitative manner. This relates in my map of development to Bands 1 and 2 on the spectrum.

Stage 2 conforming to the largely unrecognised holistic interpretation of mathematical symbols in a qualitative manner. This relates in my map to Bands 3 and 4 on the spectrum.

Stage 3 conforming to - what I term - radial mathematics -where both quantitative and qualitative aspects are integrated with each other in a coherent manner. This relates to Band 5, 6 and 7 on my map of the spectrum.

So what I am describing here, at a very fundamental level, is the emergence of Stage 3 understanding of the primes and natural numbers (corresponding to Band 5 development).

At Band 2 one forms the view that the prime numbers are absolute in a quantitative manner.

Then with Band 3 and 4 one gradually realises that they equally possess a qualitative holistic identity of an utterly distinct nature.

Then here at Band 5 one begins to properly integrate this understanding through recognising both individually and collectively that the primes have both quantitative and qualitative aspects, which are relatively independent and interdependent with respect to each other.

However this requires an increasingly dynamic understanding, where the notion of the primes, as enjoying any absolute identity, dissolves in understanding in the moment of their generation. So one reaches the other extreme from the conventional Band 2 understanding of the primes as representing absolute forms to this emerging Band 5 understanding that approximates pure energy states (with remaining phenomenal notions enjoying but an elusive fleeting identity).

In this context, the qualitative identity of the primes arises through their composite relationship with each other. So the famed non-trivial zeros i.e. the Zeta 1 (Riemann) zeros represent indirect quantitative expressions of the elusive qualitative connections between primes. These enable at one and the same time the relative quantitative independence of the primes to be consistently matched with their corresponding qualitative interdependent nature (in an ability to maintain a coherent interdependent relationship with each other).

Thus with each zero, both the quantitative and qualitative identity of every prime approaches perfect identity (in a dynamic relative manner).

And this represents the opposite extreme to conventional Band 2 understanding where both quantitative and qualitative identities are absolutely separated from each other (with the qualitative notion of relationship reduced altogether in merely quantitative terms).


And this again is why the non-trivial zeros represent the "shadow" system to the primes i.e. in that they represent the opposite extreme in understanding approaching energy states in a purely relative manner, to the conventional understanding of the primes as fixed numbers in an absolute fashion!

And because of the complementarity of mathematical and psychological processes, this means that the non-trivial zeros (as the indirect quantitative expression of holistic unconscious connections) are necessarily involved in experience as the means by which primitive instinctive reactions are coherently integrated with everyday conscious life.

It is important to remember that such instincts always remain a vitally important part of human experience! However there is a big difference between the immature situation, where instinctive reactions remain of a largely involuntary nature, thereby intruding into behaviour in a blind unpredictable manner and the psychologically developed state where such instincts become so refined, that they can be seamlessly integrated with conscious activity.

So just as it is now becoming clear in physical terms that the zeros represent distinctive energy states with respect to atomic behaviour, likewise in psycho-spiritual terms, the zeros represent distinctive intuitive energy states with respect to the qualitative ordering of the primes.  And when one considers that the primes are often referred to as the "atoms" of the natural number system, one can perhaps better appreciate this important complementary connection!

Friday, June 9, 2017

Band 5 - Psychological Development (5)

It would be instructive once again to attempt to convey the dynamic meaning of the Zeta 1 (Riemann) zeros.

And it this respect one can obtain assistance from looking at the corresponding nature of the Zeta 2 zeros (especially with respect to the simplest 2-dimensional case).

So the 2 roots of 1, serving as the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 2 members) are + 1 and – 1 respectively.

Now the relative interdependence of these two members, arises through the fact that the sum of roots = 0. This reflects that ordinal positions are relative and can arbitrarily switch depending on context. 

However the relative independence is expressed through each individual root (i.e. + 1 and – 1 respectively).

So the key point here is that a perfect dynamic balance is maintained as between independent and interdependent aspects (in this interactive context).

I have explained this many times in relation to our understanding of left and right turns at a crossroads.

When one approached the crossroads for example heading in a N direction, left and right turns have a relative independent existence in an unambiguous manner. So if we designate a left turn in this context as + 1, then a right turn is thereby – 1 (i.e. not a left turn).

Again when approaching the crossroads heading in a S direction, left and right turns again have a relative independent existence, in an unambiguous manner. So if again we designate a left turn in this alternative context as + 1, then a right turn is – 1 (i.e. not  a left turn).

However when we now envisage approaching the crossroads from both N and S directions (simultaneously) then left and right have a relatively interdependent existence (in a paradoxical fashion).

So what is left (i.e. + 1) when approached in a N direction, is right (i.e.  – 1) when approached in a S direction; and what is right (i.e. – 1) when approached in a N directions is left (i.e. + 1) when approached is a corresponding S direction. 

Thus we are able to combine relative relative independence (in the interpretation of turns when approached from just one direction with relative interdependence (in the interpretation of turns simultaneously from both directions).

Now the former unambiguous understanding of relative independence (where the designation of number remains fixed) corresponds to rational understanding of a quantitative (analytic) kind.

However the latter understanding of relative interdependence (where the designation of number switches)  corresponds to intuitive understanding of a qualitative (holistic) nature which then indirectly can be given a rational interpretation in a (circular) paradoxical fashion.

Now these two types of understanding - which implicitly are involved in the commonplace understanding of two turns at a crossroads - are utterly distinct from each other.

The quantitative (analytic) aspect - relating to relative independence - corresponds to the default root of the 2 roots of 1 (= + 1); the qualitative (holistic) aspect relating to relative interdependence - corresponds to other root (= – 1). In dynamic interactive terms this implies paradox in that the sign of 1 continually switches depending on context.    

However the qualitative (holistic) aspect is completely reduced in conventional mathematical interpretation (and identified in a merely quantitative manner).

Further more the quantitative aspect is identified in an absolute manner, though from an appropriate dynamic interactive perspective it is strictly relative.








Though perhaps somewhat more difficult to appreciate, the Zeta 1 (Riemann) zeros, likewise operate in a dynamic complementary manner.

However here - instead of focussing on the internal ordinal members of a number group - we concentrate on the collective relationship of the primes to the natural numbers in cardinal terms.

Now just as our approach to the crossroads was taken from two directions, equally we cal look at this relationship as between the primes and natural numbers from two perspectives.

From the standard quantitative, perspective we can attempt to measure for example the frequency of prime numbers up to a given natural number.

How from an alternative quantitative perspective we could attempt to measure the frequency of  all natural number to (distinct) prime factors.

In both cases one will obtain relatively independent unambiguous answers in a quantitative manner.


So one common approximation for the first case is given by n/log n.

And a corresponding approximation in the second case is given as log n/loglog n.
Then if we denote log n as n1, then this second approximation is given as n1/log n1

However - as with our approach to the crossroads from two opposite directions - if we now attempt to view simultaneously the two relationships as between the primes and natural numbers, i.e. in a relatively interdependent manner, then inevitable paradox is involved.

Thus from the standard perspective the relationship appears to be between primes and natural  numbers (considered as base numbers). However from the alternative  perspective, the relationship appears to be between natural numbers and primes (considered as representing the dimensional aspect of number as factors). 

So in fact there are two complementary relationships entailing the relationship as between primes and natural numbers.

Thus when we properly view this twin relationship in a dynamic interactive manner, complete paradox results. From one perspective, the relationship is between primes and natural  numbers (in Type 1 terms) . From the alternative perspective the relationship is between natural number and prime factors (in Type 2 terms).

Therefore from the holistic (qualitative) perspective - which simultaneously combines both complementary reference frames - one realises that the primes and natural numbers are fully interdependent with each other (ultimately in an ineffable manner).

From the analytic (quantitative) perspective - taken from just one partial perspective - one realises that both primes and natural numbers are relatively independent of each other (both as base numbers and dimensional factors) .


Now just as the roots of 1 - as the indirect quantitative expression of ordinal notions - play this role of balancing notions of quantitative independence and qualitative interdependence, the non-trivial zeros play the same role with respect to the primes in cardinal terms.

Thus when we start by viewing the primes and natural numbers in a relatively independent quantitative manner, each Zeta 1 zero expresses  the holistic qualitative nature of this relationship as a point (on an imaginary line through .5) where both of the perspectives we have looked at are simultaneously valid. What this entails is that at each point there is no distinction as between the nature of a prime or natural number (or alternatively the notion of a base of dimensional number). 

This however has no direct meaning in a linear rational manner but rather relates directly to pure intuitive understanding (as a psycho-spiritual energy state).

So the simplest way of explaining each trivial zero is as a pure energy state of pure number interdependence (with complementary physical and psychological interpretations) where the cardinal notion of number (as representing a fixed form) loses any residual meaning.

Now each zero in a sense still has a certain form as a complex number with a transcendental imaginary part. However this represents the most elusive and highly dynamic nature of number possible, as the final bridge as between form and ineffable emptiness.
So we always can only hope to approximate to such ultimate understanding in a dynamic experiential manner.

However the trivial zeros, in their collective identity, equally have a relatively independent status (in a quantitative manner) . And it is this aspect that was so wonderfully exploited by Riemann to zone in precisely on the measurement of the primes up to any given number.