tag:blogger.com,1999:blog-83984493096497320392017-08-19T11:00:37.643-07:00Spectrum of DevelopmentPeter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.comBlogger158125tag:blogger.com,1999:blog-8398449309649732039.post-82484824390539879372017-07-07T07:52:00.000-07:002017-07-12T14:11:55.751-07:00Dual and Nondual RealityIn my latest model of development, I now include 8 bands - where a band represents the most general classification of stages - in total. And as each of these bands then contains three main levels, this leads to a model containing 24 major levels.<br /><br />Then in my holistic binary approach to development, the nature of all these levels is dynamically defined in both linear (1) and circular (0) terms, allowing for unique configurations with respect to differentiation and integration respectively.<br /><br />I also define sub-levels (usually 3 within each level).<br /><br />Then a key distinction is made as between the 3 main polarities of experience, external/internal whole/part and form/emptiness. And again in dynamic interactive terms, these polarities combine notions of relative independence (where the poles are separated) and relative interdependence (where the poles are viewed in a complementary manner) respectively.<br /><br />The first polarity set (external/internal) leads to the view of stages as relating internally to the psychological self and externally to the physical world respectively.<br /><br />This means for example that at the higher levels one realises that each new stage not only represents a new understanding with respect to internal self but equally with respect to the external world.<br /><br />So for example - as I have repeatedly emphasised, we do not have the existence of just one type of Mathematics, as is commonly believed (that is equated with the absolute nature of Band 2 understanding) but potentially infinite types of Mathematics (with each containing a certain limited relative validity).<br /><br />The second polarity set (whole/part) leads to the view of stages as representing a dynamic relationship as between states and structures. In the "<a href="https://sites.google.com/a/paradig.net/www/thestagesofdevelopment">Stages of Development</a>" I detail for the 21 levels (of the 7 bands that I defined at the time) the dynamic behaviour of both external/internal (as stages of self and reality) and likewise at even greater length the dynamic behaviour of whole/part (as states and structures).<br /><br />The third polarity set (form/emptiness) leads to the recognition of the deepest psycho-physical interactions operating throughout development. Though I have delayed consideration of the various ways in which interactions take place through the major levels (because of other concerns) I may return to this important topic in future.<br /><br />I have also highlighted the importance of mirror stages of development, the explicit recognition of which occurs at the more advanced stages of development.<br /><br />The mirror stages - referred to in Christian mysticism as purgation or "dark nights"- provide the essential link as between the differentiation and corresponding integration of stages.<br /><br />Thus each stage of development must initially be posited in conscious terms for differentiation to take place. However it must then be equally negated (in an unconscious manner) for corresponding integration to occur.<br /><br />I also place great emphasis on the importance of - what I refer to as - the primary modes, which are the cognitive, affective and volitional aspects of experience.<br /><br />I then distinguish these from the secondary modes, such as linguistic, logico-mathematical, musical, kinisthetic, interpersonal abilities and so on, which all represent certain configurations with respect to the "primary colours" (i.e. primary modes).<br /><br />So for successful integration - especially at the more advanced stages of development - a satisfactory balance must be maintained as between the three primary modes.<br /><br />I also place emphasis on the distinctive personality types, which can have a considerable bearing on the precise dynamics through which development unfolds.<br /><br />In all, I define 24 such types, based on a holistic mathematical rationale with each representing a unique configuration with respect to the 4 qualitative dimensions of experience (that indirectly can be expressed by the four roots of 1).<br /><br />And I have been very conscious that my own particular development represents just one unique example (with all its inevitable idiosyncracies) of a particular personality type.<br /><br />So I certainly would not wish to offer this as a blueprint for all possible development. However its special value lies I believe in providing the appropriate framework from which the holistic mathematical approach - which does indeed have a universal value - itself naturally evolved.<br /><br /><br />However, though recognising the extraordinary level of detail that must necessarily inform a comprehensive approach, it has never been my intention to attempt to pin down development through rigidly imposed definitions or classifications<br /><br />Rather my key motivation has been to bring a truly dynamic rationale to the the understanding of development that intimately affects the very way we intellectually attempt to deal with the matter.<br /><br />And certainly from my perspective, I would see deep issues with respect to the manner in which integration - as opposed to differentiation - is handled.<br /><br />To put it bluntly, I see the very nature of integration - certainly in terms of intellectual translation - repeatedly reduced in terms of differentiated type understanding.<br /><br />In fact to avoid such reductionism, the recognition of three distinct processes with respect to development are required.<br /><br />Firstly, we have the differentiated aspect (conforming to linear asymmetric type understanding); secondly we have the integrated aspect (conforming to circular paradoxical type understanding that entails the dynamic complementarity of opposite poles); thirdly we have - what I refer to as - the radial aspect (conforming to the mature interpenetration of both linear and circular type understanding).<br /><br />And my very approach with respect to an overall model of development reflects these distinctions.<br /><br /><br />So the first great task in development is the successful unfolding of the differentiated aspect (with dualistic understanding).<br /><br />This task relates to Band 1 with specialisation of this differentiated aspect then taking place at Band 2. Though of course integration must also necessarily take place at this time, it is largely of an implicit nature, used to support the more dominant differentiated aspect.<br /><br />The next great task is the radical unfolding of the holistic integral aspect. This is associated with spiritual contemplative type development and rarely takes place in a substantial manner in Western culture.<br /><br />This mainly relates to Band 3, with specialisation of the integral aspect taking place at Band 4 (with nondual awareness).<br /><br />However the final great task is the radical unfolding of both the differentiated (analytic) and the integral (holistic) aspects simultaneously in mutual interpenetration with each other.<br /><br />Now this begins to commence with Band 5 (in what I refer to as pre-radial development).<br /><br />Then it attains its mature expression during Bands 6, 7 and 8 (with the radial stages).<br /><br />Now the great weakness as I see it with conventional approaches is that an attempt is made to graft on an Eastern style treatment of nondual awareness with a Western style treatment of dualistic understanding.<br /><br />This leads to a dichotomy as between the conventional stages of Western psychology (that emphasise phenomenal structures) and the esoteric stages of Eastern mysticism (that emphasise spiritual states).<br /><br />However, whereas Western psychology is very weak on successfully integrating spiritual states with phenomenal structures, equally Eastern mysticism is equally somewhat weak in integrating phenomenal structures (at the advanced stages of development) with spiritual contemplative states.<br /><br />So for example - as I repeatedly emphasise - the advanced stages do not lead solely to the emergence of increasingly refined spiritual states (though indeed these constitute an element of primary importance) but also increasingly dynamic phenomenal structures in both cognitive and affective terms.<br /><br />Therefore when one emphasises this latter aspect - say with respect to cognitive structures - then one quickly realises that associated with the advanced contemplative stages are new forms of holistic mathematical understanding (with potentially immense implications for the scientific understanding of all transformation processes).<br /><br />Therefore though in "integral" accounts of development the importance of advanced spiritual development for true integration should indeed be emphasised, because of an unduly Eastern emphasis on states, insufficient recognition still exists that the very nature of integration requires a distinctive type of intellectual interpretation from that used to deal with differentiation.<br /><br />In other words with respect to the translation of development the integral aspect is repeatedly reduced in a differentiated fashion.<br /><br />So integration is typically dealt with by emphasising the multiple aspects in which differentiated development takes place, with integration supposedly then arising from the combination of these separate elements. However this is simply to confuse integration with the multi-differentiated aspect of development.<br /><br />However in properly moving from the dual emphasis (that characterises differentiation) to the nondual emphasis (that characterises integration), one must coherently show how any dualistic asymmetric distinction, with a partial limited validity in relative differentiated terms, is rendered fully paradoxical in a holistic integral context.<br /><br />And then this equally requires in the proper preservation of the unique aspects of differentiation and integration with respect to development, that the simultaneous mix of both (where they mutually interpenetrate with each other in a harmonious manner) requires a new distinctive terminology, which I refer to as radial.<br /><br />And properly speaking in a comprehensive model of development, allowing for the full potential of human understanding, pride of place must then be given to radial development, for it is only from this perspective that one can see both the differentiated and integral aspects (as considered in a relative separate manner) in the appropriate context. <br /><br /><br />So Western understanding places far too much emphasis on dualistic understanding (though strictly one cannot have such understanding without a nondual element).<br /><br />However in the Eastern mystical traditions there is in turn too much emphasis on nondual awareness (though again strictly speaking one cannot have nondual awareness without a dualistic component).<br /><br />Thus even at the most advanced levels of contemplative awareness, we do not just have the experience of emptiness, but rather the highly refined interaction of both dual and nondual. So what in fact happens is that experience now becomes so dynamic that phenomena of form do not even appear to arise. A similar situation is now recognised in physics whereby the "quantum vacuum" is in fact associated with the highly dynamic activity of virtual particles!<br /><br />And radial development is certainly not just an expression of nondual awareness.<br /><br />Now it might be maintained in the texts of - say a Zen master - that one now sees the world exactly as it is i.e. in a nondual fashion.<br /><br />The simple point remains however that one cannot even register phenomena of form without a dualistic element arising.<br /><br />So what happens is this! Say one looks at a flower with the most developed form of awareness!<br /><br />Then to recognise the distinct phenomenal identity of the flower, one must momentarily separate the external from the internal pole of experience (with a dualistic element therefore involved).<br /><br />However one then immediately recognises the complementary aspect of the self in the experience (thereby cancelling out this dualistic recognition). Therefore one is again momentarily aware in dualistic fashion of the internal self (in the experience of the flower) which is immediately cancelled out through the complementary recognition of the alternative external aspect.<br /><br />So one keeps dynamically switching as between momentary dualistic recognition of both the external and internal aspects of flower recognition through the mutual complementary fusion of both as nondual spiritual awareness.<br /><br />Thus the true experience of the flower now properly entails a highly refined supremely balanced dynamic interplay of both dual (differentiated) and nondual (integrated) aspects i.e. radial understanding.<br /><br />Therefore the - strictly - misleading emphasis on nondual awareness in Eastern mystical traditions has led I believe to an important imbalance where the value of pure contemplative awareness is often emphasised at the expense of commited dualistic action with respect to the world of form.<br /><br />So properly understood, during the radial levels, nondual awareness is not only designed to interact with refined dualistic activity, but even the most rigid of such activity.<br /><br />I often use the analogy of a fire to emphasise this point.<br /><br />Eastern mysticism often reminds me of a fire where the material is already ablaze (with the flames representing nondual awareness).<br /><br />However if we were content to leave it at this, the fire would eventually die out (reflecting a comparative lack of dualistic activity).<br /><br />Therefore the most healthy situation reflects a fire where some material is already ablaze but where other material has not yet been set alight (reflecting a mixture of both nondual awareness and dualistic activity). And as the strength of the fire grows, this would enable the addition of ever more material (even that which is least flammable).<br /><br />And this is exactly how I see radial development i.e. as the situation where both dual activity (entailing engagement at varying levels of refinement) and nondual awareness grow in a dynamically balanced manner with each other.<br /><br />So it represents a continual on-going process with remarkable potential for dramatic evolutionary development.<br /><br />However most Eastern accounts greatly ignore this dynamic evolutionary aspect through an undue emphasis on (nondual) spiritual awarenesss. Western mystical accounts by contrast sometimes do deal with the (dualistic) dynamic aspect but without sufficient emphasis on the pure nondual awareness properly required.<br /><br />In particular from my own perspective I realise that the radial levels have potentially startling implications for a true revolution in understanding with respect to Mathematics and all the sciences.<br /><br />However I can see scarcely any recognition whatsoever of such implications in either Eastern or Western traditions. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-90683650367197934922017-07-06T10:08:00.003-07:002017-07-07T02:12:53.473-07:00Personal or Social Development!When one looks at the stages of development from a social perspective, one would have to conclude that "average" attainment remains quite low (in terms of the full potential spectrum).<br /><br />Therefore from one perspective, one could conclude for most individuals - certainly in Western society - that development plateaus significantly with the stages of Band 2.<br /><br />This of course is not to deny that such individuals may in many respects be highly capable and talented and occasionally reach (at least with respect to some abilities) more advanced bands.<br /><br />However one would have to conclude that very few indeed achieve sustained experience of the contemplative or radial stages.<br /><br /><br />This in turn creates an important difficulty for the rare few individuals who feel called - even from an early age - to significantly advance to these advanced stages.<br /><br />For in attempting to do so, one is placing oneself outside the norms of conventional society, where perhaps for the remainder of one's life, one must be willing to swim significantly against the tide of conventional expectations on a lone and often treacherous journey with no assurance of eventual success.<br /><br />And in a nutshell, this is why I believe so few individuals ever seriously embark on this spiritual journey.<br />It is not that they necessarily lack the ability to do so, but rather that the desire to be recognised by "normal" society is so strong that very few indeed are prepared to willingly let go its comforting support and approval.<br /><br />I believe that one does not voluntarily choose to pursue a lone spiritual route. Rather one sometimes finds through personality and life circumstances, that the very attempt to preserve one's sanity requires a deeper spiritual meaning than provided through the norms of conventional society. In other words, it is really an underlying keen sensitivity, springing from the unconscious, to the lack of any true coherence with respect to conventional social expectations, that for a small minority, creates an unquenchable desire for true spiritual integration.<br /><br /><br />Though determined form the age of 17 to pursue my own spiritual quest, irrespective of the social consequences, I often wondered if this would necessarily doom me to a life of great isolation. This was due to the intense nature of experience associated with the more advanced contemplative stages, which by definition could not be properly shared with others. And for a very considerable period of time, this was indeed the case.<br /><br />Therefore though I was certainly convinced regarding the authentic spiritual nature of what I had experienced, in many ways this seemed to create ever deeper barriers to the prospect of any <br />form of social intimacy.<br /><br />And then I slowly began to realise that I was looking at the spiritual journey from the wrong perspective, and that ultimately my own quest for spiritual integration was inseparable from that of society at large.<br /><br />In other words, when one achieves significant personal development, one's relationships with society undergo dramatic change. However this does not happen in just one direction.<br /><br />Rather, in reverse, society's relationships with the person thereby likewise undergo significant change.<br /><br />So from one perspective, one's relationships with society have changed; however from the reverse perspective, society's relationships with the person equally have changed.<br /><br />Thus far from being a passive participant, all of wider society is necessarily involved in any one person's spiritual transformation.<br /><br />However I was to slowly realise that when the emphasis in development was primarily on the "higher" super-conscious stages, that I was especially prone to see development in merely personal terms, with the - ultimately - mistaken view that I was thereby advancing beyond others with respect to these "higher" stages. And it was in such moments that I would then most strongly feel the lack of any recognition from others regarding what I had achieved.<br /><br />However this was then reversed when the emphasis turned - especially during Band 5 - to the "lower" sub-conscious stages, where through constant exposure to the must primitive "prepersonal" fantasies, it was easy to believe that others, living at the normal conscious level, were now at a more advanced stage.<br /><br />So therefore when the emphasis in development becomes equally balanced as between both "higher" super-conscious and "lower" sub-conscious stages, both individual and social development come at last into dynamic equilibrium. Then one feels no longer better or worse than others but rather fully at home as equals with everyone in society.<br /><br /><br />And this returning home coincides with the onset of the radial stages.<br /><br />However one must be careful not to be too romantic regarding the process. In truth, though many may set out with the best of intentions on this arduous spiritual journey, few truly arrive home, with the majority still struggling somewhere on the mountain slopes, with little hope of early arrival. <br /><br />And even for those who genuinely make it home, as always life then brings a host of new challenges, with no time to rest on one's laurels.<br /><br />This is why I customarily see the radial stages as most accurately representing an enhancement of normal life with the same joys and sorrows that everyone experiences, but perhaps in a much more wide-ranging and intensive manner due to a keenly felt spiritual desire to serve society to the best of one's abilities.<br /><br />So in this respect - from the mature radial perspective - there is no clear distinction as between personal and social development.<br /><br />Though early on, one might have seen spiritual progress in personal terms as relating to the unfolding of more advanced stages with respect to one's individual development, one gradually comes to later see such stages as equally reflecting spiritual progress with respect to wider society.<br /><br />Thus in the former case one sees oneself as choosing an arduous spiritual path; however in the latter case one equally realises that this path has been chosen for one by society!<br /><br />So one's own development (in relation to society) then equally reflects the corresponding development of society (in relation to oneself).<br /><br /><br />Therefore though the average level of social development may be based on the dominant influence of Band 2 stages, for a small minority, this will not be the case, with the stages of the more advanced bands now determining behaviour.<br /><br />So again, one can say that development for this small group of individuals goes therefore well beyond social norms; however equally one can say that society chooses the group for this special kind of development.<br /><br />Indeed in an even deeper sense, when such individual and social aspects are finely balanced, one can validly say - though perhaps expressed in a variety of different ways - that God chooses such individuals to act as superconductors of spirit for the greater benefit of the society in which they operate. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-52078773168479441052017-07-05T06:45:00.000-07:002017-07-14T02:13:02.647-07:00All Stages RevisitedIn yesterday's entry I was at pains to portray that with Band 5 development both ascent and descent become equally associated with "higher" super-conscious and "lower" sub-conscious development respectively.<br /><br />And as the complementary dynamic nature of both aspects becomes ever better appreciated, the very notion of "higher" and "lower" with respect to development eventually melts away (with both terms understood in a merely secondary relative fashion).<br /><br />This paves the way for true radial development, where all experience emanates directly from a spiritual centre of being, which equally is understood as the physical centre of the entire universe.<br /><br /><br />And there are remarkable links here with the true nature of the number system.<br /><br />Thus, corresponding to super-conscious ascent, I discovered the great importance of the qualitative holistic notion of number as dimension (associated with the Zeta 2 zeros). In other words "higher" level of development is literally related to such numbers, each of which represents an ever more refined configuration with respect to how the fundamental polarities of experience (external/internal and whole/part) are dynamically configured.<br /><br />Then with super-conscious descent, I was to realise that these holistic dimensions (of unconscious origin) are implicitly involved in our every-day understanding of ordinal number notions such as 1st, 2nd, 3rd and so on.<br /><br />Later, with respect to the sub-conscious descent, I discovered the corresponding great importance of the qualitative counterpart of the more common notion of number as representing quantity (associated with the Zeta 1 i.e. Riemann zeros). And these in turn now represented the highly important ways in which primitive instinctive behaviour itself can be configured in an integrated fashion. <br /><br />Then finally with the sub-conscious ascent, I discovered in turn how these new numbers (i.e. the Zeta 1 zeros) are likewise implicitly involved in our every-day cardinal understanding of the natural numbers 1, 2, 3 and so on.<br /><br />In other words, I had now discovered the truly relative nature of the number system in both cardinal and ordinal terms where analytic (quantitative) notions were dynamically inseparable from their corresponding holistic (qualitative) meanings. Or to put it another way, I witnessed the unveiling of the hidden (unconscious) aspect of number and its fundamental relationship with accepted conscious notions.<br /><br />Thus in the most mysterious manner, the relative independence of each quantitative number (in cardinal or ordinal terms) is remarkably synchronised with the relative interdependence of all numbers in qualitative terms.<br /><br />And this then provides the means for the encoding of created phenomena. In other words, all phenomena of form are ultimately mathematically encoded in a dynamic number format; alternatively such observed phenomena in nature represent the subsequent decoding of hidden number patterns.<br /><br />So mathematical language in this enhanced dynamic sense (where both quantitative and qualitative aspects are explicitly recognised) serves as the essential intermediary as between the ineffable nature of spiritual emptiness and the phenomenal world of observed form. <br /><br />Thus when we can eventually read phenomenal reality as it is truly structured (i.e. in this dynamic mathematical manner), it will become fully transparent to spirit, where the relative momentary existence of each separate phenomenon in space and time will be instantly understood to be fully compatible with the relative interdependence of all phenomena (as the present moment continually renewed).<br /><br /><br />However in acknowledging yesterday how I had long given notions of the ascent with respect to development undue attention - if even at an unconscious level - I also realised that the same applied to my customary use of "revisited stages" as applying only to "lower" development.<br /><br />In fact properly understood, once it gets under way, then in an important sense all stages - whether "higher" or "lower" - take on the status of "revisited stages" throughout remaining development.<br /><br />Before development commences, the stages of the spectrum are totally confused with respect to each other, whereby no differentiation has yet taken place and correspondingly no integration.<br /><br />Then when it unfolds, development never takes place with respect to just one stage (in a sequential manner) but rather in a qualified sense with respect to all stages.<br /><br />However to properly recognise this fact, we have to keep looking at development in a dynamic interactive manner, where notions of differentiation and integration are mutually involved.<br /><br />Now, whereas differentiation (of states and structures) takes place sequentially in a relatively discrete manner, corresponding integration takes place in a complementary continuous fashion. <br /><br />Therefore when we talk about the initial unfolding of the body-self at the first level of Band 1, this relates merely to the default (i.e. first substantial) differentiation of this stage.<br /><br />Therefore with respect to such differentiation, it does indeed appear to constitute the earliest lower stage in development.<br /><br />However there is also a much overlooked integral aspect to this stage development, where the "lowest" is complementary with the "highest". In other words what is implicit in the earliest stage (as mere potential for development) is then made fully explicit as actualised development at the corresponding "highest" stage. And the three "higher" levels of Band 3 have well-defined complementary links with the corresponding three "lower" levels of Band 1.<br /><br /><br />In my own binary approach - which is deliberately designed to take account of such complementarity - the first level commences from a state of total confusion with respect to the three main polarity sets (which all have a well-defined holistic mathematical rationale).<br /><br />Thus in the most fundamental sense, there is total confusion with respect to the diagonal polarities of form and emptiness; there is equally total confusion with respect to the vertical polarities of whole and part; finally there is total confusion with respect to the horizontal polarities of external and internal.<br /><br />So with the default differentiation of the first major level, the infant gradually learns to separate in experience the diagonal polarities of form and emptiness. In other words the infant becomes aware, through recognition of the body self, of the distinction between form and emptiness (i.e. a phenomenal world as distinct from nothingness).<br /><br />However from an integral perspective, because of the complementarity of the "lowest" with the "highest" level, this leads to a still greatly confused identification of the "lowest" with the "highest" level.<br /><br />This in seeking integration, the infant at this stage inevitably confuses the transcendent notion of the (spiritual) emptiness beyond all form, with the corresponding immanent notion of this emptiness as already contained within all such form.<br />In other words, there is a direct confusion of pre and trans notions (with the "highest" level identified with the "lowest" (and the "lowest" with the "highest").<br /><br />Thus from this confused integral perspective, the infant has a greatly inflated sense of his/her omnipotence (immature transcendence) which is then directly identified with every fleeting phenomenon of form that enters experience (confused immanence).<br /><br />Therefore whereas it is indeed true that the differentiated aspect of stage development appears to unfold in an unambiguous sequential fashion, the integrated aspect by contrast unfolds in a paradoxical manner. Thus whereas linear notions of development characterise the differentiated aspect, circular notions characterise the corresponding integral aspect.<br /><br />And whereas from the differentiated perspective, polarities are separated in a relatively independent manner, from the corresponding integral perspective, polarities are united (as complementary opposites) in a relatively interdependent fashion.<br /><br /><br />We need to bear this in mind for example in understanding the relationship as between pre and trans throughout development.<br /><br />In earliest development, a high level of confusion characterises the complementary relationship as between "lower" and "higher" and "higher" and "lower" levels of development.<br /><br /><br />Thus properly understood, earliest development is not prepersonal (as opposed to transpersonal) but rather represents the confused experience of both prepersonal and transpersonal aspects of experience.<br /><br />Then as the default sequential differentiation of stages progresses the confusion as between pre and trans lessens (though still exists to a degree).<br /><br /><br />However, this explains well how peak experiences with respect to "higher" stages can take place from complementary "lower" stages (especially subtle from mythic) and equally - though not emphasised sufficiently - how valley experiences with respect to "lower" stages can take place from complementary "higher" stages.<br /><br />In particular it is extremely common - especially in Western society - for those with established contemplative experience of the subtle realm (Band 3, Level 1), to have frequent "valley" experiences with respect to the mythic level (Band 1, Level 3) as the major religions heavily rely on the use of myths in their teaching.<br /><br /><br />Therefore pre and trans (and trans and pre) elements remain to an extent still enmeshed with other, with respect to both their confused and mature expressions, throughout development. However the key distinction to be made is that whereas in earliest development a largely confused relationship exists as between pre and trans (with the possibility of occasional peak experiences of a more mature nature), with established contemplative development a mature relationship now exists as between both aspects (with the possibility however of occasional valley experiences of a confused nature).<br /><br />There is also another extremely important point, which is very relevant to our age.<br /><br />Because of extreme specialisation with respect to Band 2 (at least with respect to rational cognitive type understanding) trans and pre can thereby become separated from each other to an extreme degree.<br /><br />So we refer to Band 2 (the middle band) as representing personal development (i.e. neither pre nor trans).<br /><br />However as pre and trans (and trans and pre) are dynamically complementary with each other with respect to "lower" and "higher" bands, such extreme specialisation can thereby greatly cut off access to all other bands of the spectrum. <br /><br />We see this especially evident with respect to the conventional approach to Mathematics, which is identified solely with the rational understanding of Band 2.<br /><br />Therefore the huge importance of the other bands for mathematical understanding - which I have been illustrating in my blog entries - are thereby completely missed.<br />In fact there is no way to even begin accessing these other forms of understanding (holistic and radial) from within the present accepted paradigm!<br /><br />Therefore I would see this extreme specialisation (which is consistent with the present scientific worldview) as perhaps the greatest enemy to true experience of both trans (transcendent) and pre (immanent) aspects of spirituality. So we are certainly now living in an information age, but unfortunately one which is deeply inimical to the corresponding need for authentic transformation!<br /><br /><br />So the true goal should be to properly preserve the dynamic context in which both pre and trans can find their true expression.<br /><br />Thus the proper differentiation of structures (where pre is separated in a relative manner from trans) should be balanced with the corresponding integration of structures (where both pre and trans are now understood as relatively interdependent with each other).<br /><br /><br />Therefore throughout development, it is not just the "lower" stages that are revisited but also the "higher".<br /><br />So once again when the "lowest" level (Band 1, Level) is initially differentiated in a relatively discrete manner in development, this likewise entails the corresponding continuous integration with all other levels on the spectrum. However so often this shadow integral side of development remains hidden with emphasis merely on the differentiated unfolding of individual stage structures.<br /><br />Therefore when we consider that only "lower" stages (that have already been differentiated) can be revisited we are ignoring the integral aspect of stage development. <br /><br />In fact we are thereby reducing the integral aspect of stage development (which is utterly distinct) in mere differentiated terms.<br /><br />And to put it bluntly, this reduced treatment of development still remains the norm with respect to "Integral Studies" where no proper methodology yet exists to properly distinguish the integral from the differentiated aspect of stage development.<br /><br /><br />Once again from a differentiated perspective, asymmetric distinctions are indeed important with respect to development (though merely in a relative manner).<br /><br />Therefore with respect to one's integral methodology, one must then be able to demonstrate how all such asymmetric distinctions, that enjoy a certain validity within a limited relative context are thereby rendered paradoxical in overall holistic terms. And this is the very means by which one <br />is ultimately enabled to properly preserve phenomenal dual distinctions with a nondual spiritual vision.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-68100254766445789932017-07-04T04:21:00.000-07:002017-07-07T02:14:40.158-07:00Ascent and Descent RevisitedThe proper dynamics of both ascent and descent with respect to spiritual development are subtler than commonly realised.<br /><br />For example, I have characterised Band 3 as a model of the spiritual ascent towards a formless type of transcendence.<br /><br />However even during Band 3, I had become deeply aware of a corresponding form of descent that was taking place.<br /><br />So whereas the ascent was then associated with the unfolding of the "higher" stages of super-conscious development, the descent was associated with the corresponding unravelling of "lower" immanent stages of sub-conscious behaviour.<br /><br /><br />Thus I was already strongly aware then of the complementary nature of both the "ascent" and "descent", assuming that full spiritual union would gradually emerge with the completion of Band 3 (where transcendent and immanent directions of spirituality would then be fully reconciled).<br /><br />However whereas I was aware of this requirement at a conscious level of understanding, I was only later to realise that considerable further development would be required before this realisation could fully take hold likewise at an unconscious level.<br /><br />Also, as Band 3 unfolded - especially with respect to Level 3 - I noticed that a significant imbalance had opened up with respect to development, in that the continual attempt to vertically integrate in bi-directional fashion both the "higher" new stages of Band 3 with the "lower" revisited stages of Band 1, led to a considerable by-passing of the rational linear stages of Band 2.<br /><br />And as normal everyday activity is deeply grounded in these stages of Band 2, this this was to lead to growing difficulties with respect to involvement with conventional social and work responsibilities.<br /><br />Therefore though Band 4 in my account is designed to outline the specialised development of the holistic contemplative stages of development, I realised that considerable on-going development was required before this could be effectively achieved.<br /><br />In fact this situation is not at all unusual in development. As each stage of development is necessarily linked to all other stages, then the full range of stages must thereby unfold - which in truth is a never-ending task - before any individual stage can achieve its full mature developed expression.<br /><br />Therefore though in earlier development, Band 2 is associated with the specialised unfolding of the linear rational stages, without further on-going contemplative development, these will inevitably be understood in an unduly rigid manner.<br /><br />So long before the specialised development of such conscious stages can be complete, one must already embark in the contemplative direction of Band 3, before the unconscious energy needed for this task becomes significantly eroded.<br /><br />Thus as I have repeatedly stated, the specialised default rational development of Band 2 i.e. the understanding associated with the first significant unfolding of Band 2 stages, can itself create a major block in terms of the subsequent unfolding of Band 3 (and the more advanced bands of development).<br /><br />Likewise the attempted specialisation of default development of the contemplative awareness of Band 4, could likewise create a major problem with respect to the subsequent successful grounding of this awareness in the rational understanding of Band 2.<br /><br /><br />Therefore, though for simplicity I have tended to characterise Band 5 with the spiritual descent i.e. in the attempt to integrate spiritual contemplative awareness with everyday linear understanding, the vertical dynamics of this Band are in fact more complex.<br /><br />Thus with respect to Band 5, the relative descent properly relatives firstly to the super-conscious awareness of the "higher" stages (initially associated with Band 3). However rather than now being directly associated with the "lower" sub-conscious stages (initially associated with Band 1), these are directly integrated with the linear "middle" stages (of Band 2). In this way "higher" contemplative awareness can become properly grounded in everyday normal reality.<br /><br />However there is an important complementary type process that equally unfolds with respect to the (revisited) sub-conscious stages (initially associated with Band 1).<br /><br />Because development here - due to unconscious repression from earliest childhood - tends to lag behind the unfolding of super-conscious awareness, one must continue with the descent into the lowest regions of instinctive behaviour, before all repressed primitive material can be fully unearthed from the unconscious. So until this process is completed (which depends on the depth of integration involved) remaining repressed material will be projected in a somewhat involuntary manner into consciousness. <br /><br />Now with respect to personal experience, even though both the ascent and the descent with respect to super-conscious awareness had been largely completed with the culmination of Level 2 (of Band 5) only the corresponding descent with respect to sub-conscious behaviour - away from Band 2 - likewise had become completed with the culmination of Level 2.<br /><br />However overall integration of both (intuitive) super-conscious awareness and (instinctive) sub-conscious behaviour, finally requires the ascent of instinctive sub-conscious activity (now suitably free of involuntary type projections) from the primitive subterranean depths back to Band 2, where in like manner it can be also successfully integrated with everyday conscious activity.<br /><br />Now this I would see as the task of Level 3 (Band 5). <br /><br />Thus when duly completed we have the situation where both spiritual intuitive awareness and physical instinctive behaviour - both now suitably refined so as to be free of any inordinate attachment of either a conscious or unconscious nature - can both be truly integrated (from above and below as it were) with everyday conscious life.<br /><br />And when this integration is finally achieved, the notion of "higher" and "lower" stages in development loses any residual meaning (except in a merely relative sense).<br /><br />Thus the "lowest" stage in development - relating to the physical instinctive behaviour of the baby infant - is just as important as the "highest" stage relating to the most advanced type of spiritual contemplative awareness.<br /><br />Indeed in a certain sense, because its significance tends to be so devalued, this "lowest" is often the most crucial stage and generally the last that can be successfully integrated with overall development.<br /><br />So with the onset of true radial development (Band 6) the very tendency to rank "higher" and "lower" stages, except in a merely relative sense, melts away, with all experience now emanating from a spiritual centre of being, which is equally experienced as the centre of the entire physical universe of form.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-27338027884550189882017-06-26T11:10:00.001-07:002017-07-19T02:06:57.422-07:00Revision of Bands (4)In the context of radial development, it can be perhaps fruitful to distinguish as between three mystical types.<br /><br />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.<br /><br />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.<br /><br />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). <br /><br />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.<br /><br />Therefore though exemplifying radial development in many ways, its full expression remains limited in important respects. <br /><br /><br />The passive type, where contemplative type development is now likely to be dominant, is much less common, certainly in Western culture.<br /><br />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.<br /><br />However the greatest difficulty for this type is to then subsequently ground such development in the temporal world of form.<br /><br />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.<br /><br />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).<br /><br /><br />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.<br /><br />Here, both the conscious (dualistic) and unconscious (nondual) aspects are harmonised with each other to an extraordinary degree.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />Thus paradoxically true immersion in nondual reality enables one to embrace duality in all its fullness, when deemed appropriate.<br /><br />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).<br /><br />So true spiritual (nondual) enlightenment should thereby be combined with total realism with respect to one's (dual) practical activity.<br /><br /><br />We will now move on to briefly outline the three levels of Band 7 (which represent the specialised expression of radial development).<br /><br />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. <br /><br />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.<br /><br />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.<br /><br />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.<br />Now there still is likely to be a marked distinction as between the active and passive mystical types.<br /><br />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).<br /><br />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.<br /><br />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.<br /><br /> 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.<br /><br />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.<br /><br />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).<br /><br />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.<br /><br />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).<br /><br />This then results in the fullest expression of divine love.<br />Love is a much misunderstood word in our culture being often identified with mere romantic feelings (often of a very self-centred nature).<br /><br />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.<br /><br />I have always referred to the volitional mode as the most important of the three primary modes.<br />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.<br /><br />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.<br /><br />So the innate sense of our eternal destiny (in God) is provided through the will (and its consequent volitional intent).<br /><br />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).<br /><br />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).<br /><br />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).<br /><br />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.<br /><br />And human life inevitably ends in physical death. So ultimately one has to prepare well for one's own death.<br /><br />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).<br /><br />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,<br />politics, the arts etc. likewise we tend to celebrate spiritual superstars who likewise have become famous through their activities and writings.<br /><br />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).<br /><br />I have mentioned already how even at the radial stages we find (depending on personality) active and passive expressions.<br /><br />Now with the final band there may a conscientious attempt to go against type.<br /><br />This would mean that the active type - possibly due to unavoidable personal circumstances - may be required to move in a strongly contemplative direction.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />Thus in brief the three levels of Band 8 are:<br /><br />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.<br /><br />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.<br /><br />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).<br /><br />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.<br /><br />And this unselfish example has exercised an immense impact on subsequent followers. <br /><br />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!Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-12739215987188733042017-06-23T11:20:00.002-07:002017-07-19T02:00:07.003-07:00Revision 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.<br /><br />And whereas the discrete relates directly to the differentiated aspect of stage development, the continuous relates to the integral aspect.<br /><br />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.<br /><br />The radial stages in effect represent the interpenetration of all stages with respect to their specialised differentiation and integration respectively.<br /><br />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.<br /><br />However in practice, this is likely to represent a severely attenuated experience due to insufficient development with respect to both aspects.<br /><br />However it is important to emphasise the dynamic interactive nature of stage development (where differentiation and integration are both necessarily involved).<br /><br />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.<br /><br />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).<br /><br />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).<br /><br />Thus I refer to the typical understanding of a stage at the time of its initial unfolding as the default understanding.<br /><br />Thus the understanding associated with Band 2 when it first properly unfolds in development represents its default understanding.<br /><br />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).<br /><br /><br />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.<br /><br />However as we have seen, this is strictly only true in a qualified sense.<br /><br />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.<br /><br />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. <br /><br />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).<br /><br />And it is in the bi-directional establishment of complementary linkages that the true integral aspect of stage development is expressed.<br /><br />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.<br /><br />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.<br /><br />This probably remains the most widely misunderstood aspect of development, with merely the "top-down" aspect of vertical integration typically emphasised.<br /><br />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. <br />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. <br /><br />Therefore, one needs to equally emphasise in reverse "onhols", where "higher" development represents a corresponding increase with respect to individual uniqueness.<br /><br />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.<br /><br />Finally, we have bi-directional diagonal linkages that simultaneously entail both horizontal integration within each stage and vertical integration between appropriate stages.<br /><br />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.<br /><br />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).<br /><br />And this enlightened understanding is eventually achieved with radial development.<br /><br />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.<br /><br /><br />In my model, I outline three relatively distinct bands, which would now be designated as Bands 6, 7 and 8 respectively.<br /><br />I caution to add however that I can only hope to offer a diminished perspective on the radial bands.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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).<br /><br />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.<br /><br />As I see it, great changes are already taking place which will force us to substantially re-assess traditional approaches to the spiritual life.<br />A cultural revolution is now underway whereby young people are rejecting the right of the institutional churches to arbitrate on any aspect of morality.<br /><br />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.<br /><br />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.<br /><br />In particular there are special problems in properly integrating religion and science.<br />I have expressed the view elsewhere in my blogs "<a href="http://integralscienceetalia.blogspot.ie/2010/07/integrating-science-and-religion.html">Integrating Science and Religion</a>" and "<a href="http://integralscienceetalia.blogspot.ie/2010/07/more-on-science-art-and-religion.html">More on Science, Art and Religion</a>" that two major developments are required to facilitate this task.<br /><br />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.<br /><br />Secondly the very nature of science will need to dramatically change.<br />In present science, the qualitative aspect of relationships is simply reduced in a quantitative manner.<br /><br />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).<br /><br />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. <br /><br />So the integration of science and religion therefore properly awaits a much more advanced form of psychological development.<br /><br /><br />Now with much caution, I will to tentatively outline the various levels of Band 6 (which represent the first of the 3 radial bands)..<br />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.<br /><br />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.<br /><br />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).<br /><br />So the first level in early childhood development is concerned with the unfolding of the body-self.<br />Likewise Level 1 (Band 6) is directly related to the emergence of a radial body-self.<br /><br />In other words one now identifies with a universal cosmic body that embraces everything in nature.<br /><br />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.<br /><br />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.<br /><br />For in the experience of this new universal identity is likewise born a responsibility for all creation.<br /><br />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.<br /><br />There is also a parallel here to the sensori-motor activity of infant development that is largely associated with instinctive physical reactions.<br /><br />During level 1, this implies that one only acts, when inspired to do so directly by the spirit.<br /><br />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.<br /><br />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.<br /><br /><br />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).<br /><br />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.<br /><br />So again this existential dilemma will be keenly experienced.<br /><br />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.<br /><br />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!<br /><br />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.<br /><br />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).<br /><br />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. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-14818943225544796422017-06-22T11:01:00.002-07:002017-07-13T09:05:53.550-07:00Revision 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.<br /><br />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.<br /><br />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.<br />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.<br /><br />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.<br /><br />So with respect to this radical new vision of the true nature of Mathematics, I believe I have something especially important to offer.<br /><br />I have mentioned before how I gradually became aware of a very significant problem as development moved to the "highest" level of Band 3.<br /><br />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.<br /><br />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.<br /><br />And this culminated in the bursting of an undiagnosed duodenal ulcer which reduced my blood count to a dangerously low level.<br /><br />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).<br /><br /><br />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.<br /><br />And this likewise entails that the middle levels (which had formerly been significantly by-passed) need now to become the central focus. <br /><br />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.<br /><br />This then leads inevitably to a degree of super-ego control serving to repress the "lower" instinctive capacities of the personality.<br /><br />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.<br /><br /><br />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.<br /><br />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.<br /><br />Thus the first level on the descent related largely to the horizontal internal/external polarities.<br /><br />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.<br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /><br />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).<br /><br />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.<br /><br />As I have recounted in detail on other blogs esp. "<a href="http://resolutionofriemannhypothesis.blogspot.ie/">The Riemann Hypothesis</a>", 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).<br /><br />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.<br /><br />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).<br /><br />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. <br /><br /><br />Another remarkable mathematical transformation was then set to take place - initially with respect to the "lower" primitive levels.<br /><br />In conventional terms, the primes are considered the basic "building blocks" of the natural number system (in cardinal terms).<br /><br />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.<br />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.<br /><br />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).<br /><br />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.<br /><br />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.<br /><br /><br />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!<br /><br />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. <br /><br />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. <br /><br />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.<br /><br />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. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-54924666315392621192017-06-21T12:03:00.001-07:002017-06-27T08:00:10.342-07:00Revision of Bands (1)Since the last revisions "<a href="https://sites.google.com/a/paradig.net/www/thestagesofdevelopment">Update on Classification of Stages</a>" and "<a href="https://sites.google.com/a/paradig.net/www/thestagesofdevelopment">Brief further Update</a>" on my overall model of development (made in 2008), I have been operating with 7 major bands (each containing 3 major levels).<br /><br />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.<br /><br />So with this extra addition, this now gives 8 bands in my overall model (with 24 major levels).<br /><br />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.<br /><br /><br />So perhaps in the light of this latest revision, it would be instructive to outline briefly the 8 major bands.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /><br />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.<br /><br />Scientific specialisation with respect to the horizontal polarities is associated with a concrete stage (in the amassing of "objective" data). This is Level 1.<br /><br />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.<br />Then in practice these two stages are often combined, where empirical research is wedded to testable hypotheses. And this constitutes Level 2.<br /><br />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).<br /><br />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).<br /><br />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).<br /><br /><br />However there are always exceptions with some individuals remaining especially sensitive to unconscious promptings from an early age.<br /><br />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.<br /><br />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.<br /><br />Level 1 (Circular) is based on growing interdependence of conscious type phenomena with respect to external and internal polarities.<br />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.<br />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.<br /><br />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.<br /><br />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!<br /><br />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).<br /><br />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.<br /><br />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!<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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. <br /><br />And the direct clash as between "higher" super-conscious and "lower" sub-conscious regions of personality becomes much more pronounced at this stage.<br /><br />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).<br /><br />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.<br /><br />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.<br /><br />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.<br />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.<br /><br />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.<br /><br /><br />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.<br /><br />This in turn can be associated with three relatively distinctive levels.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-18972764338627501472017-06-13T08:12:00.002-07:002017-07-16T02:50:52.844-07:00Band 5 - The Three LevelsA distinct form of mathematical understanding is associated with each of the 3 levels of Band 5.<br /><br />Now even before we can consider the primes we must already admit the existence of the two original numbers i.e. 1 and 0.<br /><br />And the importance of these two numbers i.e. the binary digits cannot be overstated.<br /><br />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.<br /><br />And in this this context we are referring to the quantitative (analytic) use of these two digits where 1 is clearly separated from 0.<br /><br />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. <br /><br />This latter meaning is reflected very well in some mystical traditions where (phenomenal) form becomes ultimately identical with (spiritual) emptiness.<br /><br />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. <br /><br />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.<br /><br />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.<br /><br />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).<br /><br />And whereas differentiation in development is related directly to the linear, integration by contrast is directly related to the circular aspect.<br /><br />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.<br /><br />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.<br /><br />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!<br /><br />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).<br /><br />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). <br /><br />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.<br /><br />This task chiefly occupied me from the years 2001 to 2008 (approx.). Some of the fruits of this labour can be found at "<a href="https://sites.google.com/a/paradig.net/www/thestagesofdevelopment">The Stages of Development</a>".<br /><br /><br />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.<br /><br />Indeed from one very important perspective, all reality can then be viewed as representing in fact such a dynamic twin-based binary system.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />So what was eventually revealed was the hidden qualitative basis of both the ordinal and cardinal number systems respectively.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /><br />Further development then takes place at Level 3 (Band 5).<br /><br />One of the tasks here is to achieve better integration of the earlier binary approach with the new prime/natural number based approach.<br /><br />For it has to be acknowledged that 1 and 0 in a very important sense precede the primes.<br /><br />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.<br /><br />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).<br /><br />Even with respect to the Zeta 1 and Zeta 2 zeros, we can see clearly the influence of 1 and 0.<br /><br />First of all it is the zero solutions to the equations (= 0) that are especially relevant in both cases.<br /><br />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.<br /><br />Also whereas the sum of roots = 0, the product of roots = 1 (+ 1 for an odd number and <span style="font-family: "times new roman"; font-size: 12pt;">– 1 for an even number) </span> <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><br />Then in relation to the Zeta 2 solutions, these come in conjugate pairs as .5 + it and .5 <span style="font-family: "times new roman"; font-size: 12pt;">–</span> it respectively. So when one adds the two, the result = 1.<br /><br />Another advance at Level 3 is in relation to one's understanding of how the cardinal and ordinal systems interact.<br /><br />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).<br /><br />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. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-69369125559703840272017-06-12T03:15:00.001-07:002017-07-16T05:51:18.636-07:00Band 5 - Psychological Development (7)Another major transformation emerged in my thinking relating to Band 5 (level) 2 development.<br /><br />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!<br /><br />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.<br /><br />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.<br /><br />However this was all to change with respect to increasing understanding of the nature of the Zeta 1 (Riemann) zeros.<br /><br />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).<br /><br />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).<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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).<br /><br />However for some considerable time, I experienced much more difficulty in "seeing" what the Zeta 1 (Riemann) zeros properly represented.<br /><br />And then it eventually dawned on me as to the nature of the problem.<br />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. <br /><br />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.<br /><br />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.<br /><br />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.<br /><br />In fact another remarkable observation can now be made.<br />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).<br /><br />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).<br /><br />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.<br /><br />In conventional terms, we identify mathematical understanding consciously solely with the cognitive mode of reason.<br /><br />This can be directly related - as with the number line - to the "real" aspect of understanding.<br /><br />However, as we know, we equally can have imaginary as well as real numbers.<br />So from the holistic mathematical perspective, the imaginary aspect in this context relates to the hidden (unrecognised) affective aspect of mathematical understanding.<br /><br />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.<br /><br />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.<br /><br /><br />In conclusion, I found a remarkable vindication of my new thinking when I now returned to interpret the Riemann Hypothesis.<br /><br />From the cognitive rational perspective, it is readily understood that all cardinal numbers lie on the "real" number line (as the product of primes).<br /><br />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.<br /><br />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.<br /><br />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.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-7909841887712889732017-06-11T12:18:00.001-07:002017-07-21T02:41:00.651-07:00Band 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).<br /><br />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!<br /><br />It is similar with primitive instinctive behaviour, where both conscious (specific) and unconscious (holistic) aspects are necessarily intertwined.<br /><br />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. <br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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)!<br /><br />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). <br /><br />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.<br /><br />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.<br /><br />And in truth both psychological and mathematical tasks are themselves - when properly understood - fully complementary.<br /><br />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.<br /><br /><br />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.<br /><br />However this requires sustained intimate exposure on a daily basis to intimate projections such as sexual fantasies, that spontaneously emerge from the hidden depths.<br /><br />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).<br /><br />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.<br /><br />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).<br /><br />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.<br /><br /><br />However I now came to realise during this time that exactly the same dynamics applied to the appropriate dynamic understanding of prime number behaviour.<br /><br />In fact both mathematical and psychological aspects themselves act in complementary fashion - as a two-sided coin - of human experience.<br /><br />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). <br /><br />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.<br /><br />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.<br /><br />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.<br /><br />However intrinsically there is an equally important holistic qualitative aspect to the primes (both in individual and collective terms) relating to number interdependence.<br /><br />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.<br /><br />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.<br /><br />So we have therefore three main stages of mathematical development:<br /><br />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.<br /><br />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. <br /><br />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. <br /><br />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).<br /><br />At Band 2 one forms the view that the prime numbers are absolute in a quantitative manner.<br /><br />Then with Band 3 and 4 one gradually realises that they equally possess a qualitative holistic identity of an utterly distinct nature.<br /><br />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.<br /><br />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).<br /><br />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).<br /><br />Thus with each zero, both the quantitative and qualitative identity of every prime approaches perfect identity (in a dynamic relative manner).<br /><br />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).<br /><br /><br />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!<br /><br />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.<br /><br />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.<br /><br />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! Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-2167153724301513082017-06-09T08:01:00.002-07:002017-06-15T04:11:55.338-07:00Band 5 - Psychological Development (5)It would be instructive once again to attempt to convey the dynamic meaning of the Zeta 1 (Riemann) zeros.<br /><br />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).<br /><br />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 <span style="font-family: "times new roman"; font-size: 12pt;">– 1 respectively.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">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. </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">However the relative independence is expressed through each individual root (i.e. + 1 and </span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">– 1 respectively).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">So the key point here is that a perfect dynamic balance is maintained as between independent and interdependent aspects (in this interactive context).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">I have explained this many times in relation to our understanding of left and right turns at a crossroads.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"> – 1</span></span> (i.e. not a left turn).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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 </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"> – 1</span></span></span></span> (i.e. not a left turn).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">So what is left (i.e. + 1) when approached in a N direction, is right (</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">i.e. – 1) when approached </span></span></span></span></span></span>in a S direction; and what is right (i.e. </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">– 1) when approached in a N directions is left (i.e. + 1) when approached is a corresponding S direction.</span></span></span></span> </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">Now the former unambiguous understanding of relative independence (where the designation of number remains fixed) corresponds to rational understanding of a quantitative (analytic) kind.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">The quantitative (analytic) aspect - relating to relative independence - corresponds to the default root of the 2 roots of 1 (= <complete id="goog_1495077510">+ </complete>1); the qualitative (holistic) aspect relating to relative interdependence - corresponds to other root (= </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">– 1). In dynamic interactive terms this implies paradox in that the sign of 1 continually switches depending on context. </span></span></span></span></span></span> </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">However the qualitative (holistic) aspect is completely reduced in conventional mathematical interpretation (and identified in a merely quantitative manner).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">Further more the quantitative aspect is identified in an absolute manner, though from an appropriate dynamic interactive perspective it is strictly relative.</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><br /></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">Though perhaps somewhat more difficult to appreciate, the Zeta 1 (Riemann) zeros, likewise operate in a dynamic complementary manner.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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 </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"> in cardinal terms</span></span>.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">From the standard quantitative, perspective we can attempt to measure for example the frequency of prime numbers up to a given natural number.</span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><br /></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">How from an alternative quantitative perspective we could attempt to measure the frequency of all natural number to (distinct) prime factors.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">In both cases one will obtain relatively independent unambiguous answers in a quantitative manner.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"></span></span><br /><span style="font-family: "times new roman";">So one common approximation for the first case is given by n/log n.</span><br /><br /><span style="font-family: "times new roman";">And a corresponding approximation in the second case is given as log n/loglog n.</span><br /><span style="font-family: "times new roman";">Then if we denote log n as </span><span style="font-family: "times new roman";"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">n<sub>1</sub></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">, then this second approximation is given as </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman";"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">n<sub>1</sub></span></span></span></span><span style="font-family: "times new roman";">/log </span><span style="font-family: "times new roman";"> </span><span style="font-family: "times new roman";"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">n<sub>1</sub></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">. </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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). </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">So in fact there are two complementary relationships entailing the relationship as between primes and natural numbers.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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) .</span></span><br /><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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). </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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.</span></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">So we always can only hope to approximate to such ultimate understanding in a dynamic experiential manner. </span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">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. </span></span>Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-79493093008572249222017-06-08T09:53:00.000-07:002017-06-19T01:58:43.681-07:00Band 5 - Psychological Development (4)We have briefly described in the previous blog entry the truly relative nature of the ordinal number system, entailing both quantitative (analytic) and qualitative (holistic) aspects in complementary relationship with each other.<br /><br />Then the special limiting case of Conventional Mathematics, where number is treated in an absolute manner, represents the reduction of the qualitative aspect in a merely quantitative manner (in what represents 1-dimensional interpretation).<br /><br />Both the relative and absolute interpretations of the ordinal nature of number are indirectly expressed in a quantitative fashion, through the circular number system (in obtaining successive roots of 1).<br /><br />And as we have seen in previous entries, when we remove the one absolute case to obtain all the remaining truly relative interpretations, we are left with the Zeta 2 zeros as solutions to the finite equation,<br /><br />1 + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>1</sup></span> + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>2</sup></span> + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>3</sup></span> + ... <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>t </sup></span><sup>– 1 </sup>= 0 (for different values of t)<br /><br />So for example when t = 3, we get,<br /><br />1 + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>1</sup></span> + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">x<sup>2</sup></span> = 0, so that x = <span style="font-family: "times new roman"; font-size: 12pt;">– .5</span> + .866i and <span style="font-family: "times new roman"; font-size: 12pt;">– </span>.5 <span style="font-family: "times new roman"; font-size: 12pt;">– </span> .866i respectively.<br /><br />And these two values then express - in an indirect quantitative manner - the relative qualitative ordinal notions of 1st and 2nd (in the context of a group of 3). As we have seen 3rd (in the context of a group of 3) equates with the absolute case of the ordinal notion, where it is always fixed in meaning as the last member of its respective group.<br /><br /><br />This recognition of everyday ordinal notions as - literally rooted - in "higher" dimensions of understanding (relating to the unconscious realm) is a product of Band 5 understanding (Level 2).<br /><br />And these - formerly hidden - unconscious dimensions of qualitative mathematical understanding are uncovered directly in an appropriate holistic intuitive manner. However, they then can be given an indirect rational type explanation in a circular (paradoxical) logical manner. <br /><br />However the very nature of Band 5 understanding is that an ever closer complementarity is established as between the "higher" super-conscious levels and corresponding "lower" sub-conscious (primitive) levels.<br /><br />As we have seen with respect to the "higher" levels (in the descent to integration with the middle levels of Band 2), direct appreciation of the truly relative nature of the ordinal number system is revealed. In a complementary fashion, in the attempt to raise up - as it were - in a corresponding ascent, the "lower" levels to integration with the middle levels, direct appreciation of the truly relative nature of the cardinal number system likewise gradually unfolds.<br /><br />Now, once more in the absolute quantitative manner of Conventional Mathematics, the primes are viewed as the "building blocks" of the natural number system.<br /><br />However the primes represent (in their individual identities) but a special limiting component of the natural number system.<br /><br />As we have seen, when the primes are used in conjunction with each other (as the unique factor combinations of composite natural numbers) they thereby attain a truly relative qualitative (holistic) status (through the interdependence of such factors).<br /><br />Therefore, whereas the primes in their independent identity express the quantitative (analytic) aspect, the primes in their interdependent identity (as constituent factors of the composites) - relatively - express the qualitative (holistic) aspect of number.<br /><br />Of course, as always reference frames can switch. So each individual prime likewise has an interdependent identity (as a group of ordinal members). And the multiplication of the primes (as factors) has a corresponding quantitative identity. The important point to remember is that complementarity always applies in dynamic terms as quantitative to qualitative (and qualitative to quantitative) respectively.<br /><br />Thus when we start by viewing the primes in conventional quantitative terms, the collective relationship between primes (expressed through the unique factor composition of the composite natural numbers) is then - relatively - of a qualitative (holistic) nature.<br /><br />And once again this qualitative relationship - now with respect to the collective relationship of the primes with each other - can indirectly be expressed in a quantitative fashion.<br /><br />And this is what precisely the famed Zeta 1 (Riemann) zeros represent i.e. the indirect quantitative expression of the qualitative (holistic) nature of the primes (in their collective relationship with the natural numbers).<br /><br />In fact, using more psychological terminology that would be especially familiar to Jungian followers, the Riemann (non-trivial) zeros represent the perfect shadow number system of the primes.<br /><br />So when we rationally view the primes in a (conscious) quantitative manner, appreciation of the Riemann zeros should properly operate in a complementary intuitive (unconscious) qualitative manner; however when from the reverse perspective we now look on the relationship between primes in an intuitive (unconscious) manner, then the Riemann zeros should properly operate rationally in a complementary (conscious) quantitative manner.<br /><br />Therefore in the dynamics of experience, continual switching takes place as between the primes and Zeta 1 (Riemann) zeros with respect, in both cases, to their quantitative (analytic) and qualitative (holistic) aspects.<br /><br />However because in conventional mathematical understanding the unconscious element is so reduced in mere conscious terms, we thereby remain blind to the necessary dynamic interaction as between these two key aspects of the number system. Though the interaction still necessarily takes place implicitly (without which the number system would have no meaning) the unconscious aspect remains completely hidden (through being reduced in a conscious rational manner).<br /><br />To put it more simply, if we look on the (cardinal) natural number system as a two-sided coin, if on one side we have the natural number system, then on the other we necessarily have the Zeta 1 zeros (and vice versa). However this can only be appropriately understood in a dynamic interactive manner entailing the balanced interaction of both conscious and unconscious appreciation. <br /><br />So in order for the assumption to hold that all real numbers lie on a straight line (in a conscious rational manner), it is necessary for all the Zeta 1 zeros to likewise lie on a straight line (as the indirect quantitative expression of the holistic qualitative basis of the number system).<br /><br />For if we think about it for a moment, it is not only necessary that all numbers (as independent entities) lie on a straight line, but equally that the relationships as between those entities (as interdependent) equally lie on a straight line.<br /><br />Because of conventional quantitative reductionism we assume that both aspects (number independence and number interdependence respectively) relate to the same straight line.<br /><br />However when we properly distinguish both quantitative and qualitative aspects, then if the (independent) quantitative entities lie on a real line, then the (interdependent) qualitative relationship between them must lie - relatively - on an imaginary line (as quantitative and qualitative are real and imaginary with respect to each other).<br /><br />And this indeed is precisely inferred by the Riemann Hypothesis!<br />However clearly this cannot be proven in conventional mathematical terms. Rather it is a necessary condition that must necessarily hold so that our faith in the subsequent consistency of the entire mathematical enterprise is justified. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-88786132177358019032017-06-07T06:05:00.001-07:002017-06-27T07:40:30.639-07:00Band 5 - Psychological Development (3)As we seen at Band 3, the movement is towards an increasingly holistic type understanding that expresses the interdependent nature of all reality.<br /><br />And this applies equally to the number system.<br /><br />So whereas formerly one understood the number system (at Band 2) in an absolute (1-dimensional) manner, now at Band 3 one realises that one can have an unlimited set of natural number dimensional understandings (with an increasing relative validity).<br /><br />Thus in terms of psychological development, one moves - in the journey to "higher" super-conscious levels - from rigid rational notions to a refined intuitive contemplative worldview. In like manner, one moves from the static notion of number representing an absolute unchanging form to the opposite dynamic notion of number representing a pure energy state.<br /><br />Then in the corresponding journey at Band 3 into the "lower" sub-conscious levels of personality, one equally moves from the absolute notion of prime numbers as fixed quantitative entities to the dynamic notion of primes as entailing both quantitative (conscious) and qualitative (unconscious) aspects.<br /><br />And as one probes ever more deeply in psychological terms the primitive depths of personality (in the rapid but short-lived projection of instinctive phenomena), one likewise comes closer to appreciation of the truly relative nature of the primes.<br /><br /><br />However it is only at Band 5 that this emerging new mathematical understanding can start to reach maturity.<br /><br />Thus again the super-conscious understanding of Band 3 represents the transcendent ascent away from the dualistic understanding of Band 2 towards a more rarefied intuitive contemplative understanding. However the danger here is that one can thereby lose significant touch with Band 2.<br /><br />So Band 5 represents the corresponding immanent descent, in the reverse direction, in the attempt to properly integrate this newly developed holistic intuition with the former rational linear understanding (of Band 2).<br /><br />And this in turn has dramatic implications for the understanding of number.<br /><br />As we have seen, the default conventional understanding of number is 1-dimensional, where it is understood in an absolute quantitative manner.<br /><br />However associated with all the other dimensional numbers are relative interpretations (corresponding to qualitative notions of interdependence).<br /><br />Now one can indirectly express in a quantitative manner the structure of all these dimensions through obtaining the various roots of 1.<br /><br />So for example the qualitative structure of 5-dimensional reality is indirectly expressed through the 5 roots of 1.<br /><br />Now the key insight that then emerged is that these roots in fact express - again in an indirect quantitative manner - the ordinal nature of number.<br /><br />Thus in the case of our example of the 5 roots of 1, these express - in indirect quantitative fashion - the qualitative ordinal notion of 1st, 2nd, 3rd, 4th and 5th respectively (in the context of 5 members of a number group).<br /><br />These would be written as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>1/5</sup></span>, <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>2/5</sup></span>,<span lang="EN-IE" style="mso-ansi-language: EN-IE;"> 1<sup>3/5</sup></span>, <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>4/5 </sup></span>and <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>5/5 </sup></span>respectively <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><div class="MsoNormal"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><br /></span></div><br />Now as I have explained before the last of these i.e. the 5th of 5 necessarily reduces to the accepted 1-dimensional notion of number (i.e. 1<sup>5/5 </sup>= <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>1</sup></span>).<br /><br />And this is how ordinal notions - which are inherently of a distinctive qualitative nature - are successfully reduced in a cardinal quantitative manner.<br /><br />So each ordinal unit is fixed as the last of its corresponding group. So 1st is the last of 1 member, 2nd the last of 2 members, 3rd the last of 3 members and so on.<br /><br />However properly understood each ordinal member can enjoy an unlimited number of relative identities. So 2nd in the context of 3, 2nd in the context of 4 and 2nd in the context of 5 members respectively all represent distinctive relative definitions of the ordinal notion of 2nd.<br /><br />So the remarkable discovery here is that implicit in our everyday understanding of the ordinal notions of 1st, 2nd, 3rd, 4th and so on is corresponding higher dimensional interpretation (that is of an unconscious nature).<br /><br />Thus again when we place 5 members of a group for example in ordinal relationship with each other, implicit in this understanding is a corresponding 5-dimensional notion (expressing the interdependence of these 5 dimensions). However though of necessity such understanding implicitly underlines interpretation of ordinal notions, for the most part it remains completely blind and undeveloped in conventional understanding.<br /><br />So it is only with the intuitive type development corresponding to Band 3 that the holistic unconscious basis of number can be properly brought to light.<br /><br />And then in the recognition that all these qualitative dimensions are intimately involved in the ordinal nature of number, the holistic (qualitative) then can become properly grounded with accepted analytic (quantitative) appreciation.<br /><br />So in psychological terms, this represents the integration of both (conscious) rational and (unconscious) intuitive aspects. And this corresponds to Band 5 understanding.<br /><br /><br />A further refinement of this understanding was then to occur later, as I once more looked at the basic nature of multiplication.<br /><br />I was considering the simple example of 2 * 3 and imagined a concrete example of two rows with 3 coins in each row.<br /><br />Therefore in order to use 2 here as the initial operator, we must recognise that the 3 coins in one row match the 3 coins in the other. In other words we must recognise that the two rows are interdependent with each other.<br /><br />So to recognise the 3 coins (in each row) we must treat them as - relatively - independent of each other. However to recognise that the 2 rows are similar (thereby enabling us to treat the relationship in a multiplicative fashion) we must then recognise the - relative - interdependence of each row.<br /><br />So 2 in this multiplication example thereby applies to the qualitative notion of interdependence.<br /><br />However we are using 2 here as a base - rather than dimensional - number.<br /><br />Now once again when 2 is written in Type 1 terms as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">2<sup>1</sup></span>, 2 represents the base number (with 1 the default dimensional number).<br />However when 2 is written as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>2</sup></span>, 2 represents the dimensional number (with 1 the default base number).<br /><br />Therefore holistic interdependence can apply to both base and dimensional numbers (depending on relative context).<br /><br />Thus whereas formerly, I had derived the ordinal notion as applying to base with respect to corresponding dimensional numbers, I realised that we have here the complementary opposite situation (where reference frames have switched). So here in reverse, the notion of ordinal numbers as applying to dimensions is derived with respect to their corresponding base numbers. So for example the notions of 1st and 2nd (in the context of two dimensions) is derived from the holistic interdependent notion of 2 (as applying to the base number).<br /><br />So the key point in this example is that the number 2 - whether representing a base or dimensional number - can be given both quantitative (analytic) and qualitative (holistic) interpretations. And the quantitative corresponds with cardinal and the qualitative with ordinal meaning respectively. <br /><br />And in the dynamics of experience, these keep switching as between cardinal and ordinal (and ordinal and cardinal) meaning respectively in a complementary manner.<br /><br />And just as this applies to 2, it equally applies to every natural number.<br /><br />However to properly understand, we must recognise that all numbers - indeed all mathematical symbols - have twin analytic and holistic meanings (which in the dynamics of appropriate understanding are fully complementary).Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-77451187867055248482017-06-06T07:27:00.002-07:002017-06-27T07:37:00.575-07:00Band 5 - Psychological Development (2)From a dynamic perspective the very notion of "higher" development represents a certain imbalance.<br /><br />Therefore the journey toward "higher" super conscious states and structures should properly entail an equally important journey into the "lower" sub conscious region of primitive instincts and in personality terms both of these are fully complementary with each other.<br /><br />Or as I have put it on many occasions, integration of the psyche properly entails both a top-down aspect (where one attempts to integrate earlier stages from the perspective of the latest superconscious stage attained) and a bottom-up aspect (where one in reverse fashion attempts to integrate later stages from the perspective of the earliest subconscious stage, which has now become considerably refined through continuous revisiting).<br /><br />Certainly in my case, superconscious development became strongly identified with the unfolding of an increasingly holistic type of intuition mediated through the refined use of reason.<br /><br />Then in complementary fashion, subconscious development began identified with primitive sensations of an increasingly intimate nature that expressed the still repressed affective aspect of personality.<br /><br />So just as in scientific physical terms, we have a major conflict as between General Relativity (in its depiction of the universal cosmic nature of reality and Quantum Mechanics in the behaviour of sub-atomic particles of matter, equally a conflict emerged in my experience as between the refined super-conscious and - as yet unreformed - primitive aspect of personality. And this fundamentally represented a clash as between cognitive and affective modes (each with its own distinctive means of operation). <br /><br /><br />And indeed there is a deep lesson here for physics in its attempt to find a Theory of Everything (understood in a reduced cognitive manner).<br /><br />For just as in psychological terms the conflict is only solved through the realisation of the deep complementary nature of both cognitive and affective domains, ultimately leading to their full reconciliation in an ineffable spiritual experience, it is likewise similar in physical terms. So properly understood - what we call - physical reality in truth is a complex reality with two interacting aspects, which are real and imaginary with respect to each other. And the ultimate reconciliation of these two distinctive aspects once again leads to an ineffable state (that is spiritual).<br /><br />So ultimately a truly integrated worldview in physical terms, is dynamically inseparable in experience from the attainment of true integration in a psychological manner<br />Put another way, the subjective "knower" and what is objectively "known" ultimately merge fully together in a mutual embrace which is spiritual. <br /><br />The earliest holistic mathematical breakthrough I made - again during "lower" stage development Band 3 - was the important realisation that the nature of prime numbers is very different from what is conventionally understood.<br />I had become deeply fascinated with the nature of primitive response and realised that it represented a direct confusion of both conscious and unconscious aspects of personality. So with primitive response the holistic desire for meaning (in unconscious terms) is directly identified with some specific phenomenon (in a conscious manner).<br /><br />And it is this central confusion as between both conscious and unconscious meaning that leads to the involuntary nature of such a response!<br /><br />Indeed when this confusion is especially severe (as in earliest infant development) an immediate collapse with respect to the dimensions of space and time take place so that the object of response can maintain only a transitory existence.<br /><br />In fact it is very similar at the physical level with respect to sub-atomic particles which often enjoy but an extremely short-lived existence. So in a very true sense these thereby represent primitive particles that operate in a highly dynamic interactive manner. <br /><br />So it is only when in psychological terms a certain appropriate separation of conscious from unconscious has taken place that the object phenomena of experience can attain a degree of stability (through being placed in a dimensional framework of space and time). Likewise it is only when sub-atomic physical particles have obtained a certain separation from the holistic ground from which they have emerged that they can begin to enjoy a more stable existence in space and time.<br /><br />It then struck me that it was no accident that what we identify as "prime" with respect to numbers bears a close etymological relationship with the word "primitive" with respect to psychological behaviour.<br /><br />In other words if we are to understand the primes in an appropriate manner (i.e. that is dynamically interactive in nature) we must recognise that they contain both quantitative (conscious) and qualitative (unconscious) aspects.<br /><br />However the huge problem with accepted mathematical interpretation is that it attempts to understand primes in conscious terms with respect solely to their quantitative aspect.<br /><br />Now if one were to suggest in today's climate that primitive instinctive behaviour can be adequately understood in a merely conscious manner, people would quickly dismiss such a reduced notion.<br />Understanding of psychological behaviour has advanced so much over the past few centuries that we readily accept the dual role of both conscious and unconscious <br /><br />However when it comes to our understanding of prime number behaviour, our understanding, as it were, is still very much in the dark ages. For, in truth, this equally requires recognition of the dual role of both quantitative (conscious) and qualitative (unconscious) aspects.<br /><br />So I now saw clearly the great importance of uncovering the hidden qualitative aspect of prime number behaviour.<br /><br />However it was only with the unfolding of the Band 5 levels that I was able to give proper mathematical expression to the true nature of the primes.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-71370789800569824502017-06-05T09:22:00.002-07:002017-06-27T07:33:47.021-07:00Band 5 - Psychological Development (1)I have dealt at length in the previous entries with the unfolding appreciation of the nature of number, when viewed in appropriate dynamic interactive manner from complementary Band 3 and Band 5 perspectives.<br /><br />However this development - in my approach - is intimately tied in turn to the unfolding psychological development of the various levels of these Bands.<br /><br />So the deeper realisation slowly revealed to me during this time was that all physical and psychological development is precisely encoded in a dynamic interactive mathematical manner. And as I was to eventually discover, this encoding applies equally from this new mathematical perspective to every aspect of development (of a cognitive, affective and volitional nature).<br /><br />Now it is important to keep emphasising - lest there be any misunderstanding on this matter - that the ultimate nature of reality is ineffable (and therefore not capable of direct representation in phenomenal terms). However human experience always necessarily entails the interaction of this ultimate spiritual reality of an empty nature and more accessible phenomena of form, which then indirectly can serve to mediate this reality.<br /><br />So what I was discovering is that appropriate interpretation of mathematical symbols - that allows for the balanced interplay of both quantitative (analytic) and qualitative (holistic) aspects - provides the purest possible means of mediating as between these two fundamental domains i.e. spiritual emptiness and phenomenal form.<br /><br />So to put it more succinctly, phenomenal reality is fundamentally encoded in a mathematical manner (that is ultimately ineffable); equally from the alternative perspective, phenomenal reality represents the corresponding decoding of its inherent mathematical nature.<br />And once again this does not just apply in a more limited manner to what we would recognise as the scientific (cognitive) nature of reality, but equally for example to artistic (affective) and religious (volitional) domains.<br /><br />However once again it is vital to emphasise that I am not referring to Mathematics here in the merely reduced conventional quantitative manner, but with respect to an utterly distinctive vision (which is directly compatible with the unfolding of these more advanced bands of understanding).<br /><br />So conventional mathematical understanding is strictly a product of Band 2 type understanding (where rational conscious type interpretation is separated to an extreme degree from the unconscious).<br /><br />Therefore appreciation of this new mathematical vision requires firstly the recovery of the hidden unconscious basis of Mathematics in direct appreciation of the qualitative (holistic) nature of all its symbols, before eventually then achieving the balanced integration of both quantitative and qualitative aspects (in what I refer to as Radial Mathematics).<br /><br />From this perspective, the 3 major levels of Band 3 relate to development of the qualitative (holistic) aspect of Mathematics.<br /><br />Then then just as Band 2 represents the specialised development of the quantitative (analytic) aspect, Band 4 can likewise be seen as representing the corresponding specialised development of the qualitative (holistic) aspect.<br /><br /><br />It has to be stated clearly that at the present stage of human evolution, remarkably little sustained development has yet taken place beyond Band 2.<br /><br />However over the past few millennia, a very small minority of gifted human beings have indeed managed to traverse these bands and in some cases have left detailed accounts of the nature of their experience.<br /><br />However even here, this has generally taken place within specific religious traditions, with major emphasis placed on the more advanced intuitive contemplative states corresponding to such development.<br /><br />And in a primary sense these states indeed constitute a key feature of such development.<br /><br />However, properly understand, associated with the "higher" levels of the more advanced bands are corresponding cognitive, affective and volitional structures (of an increasingly dynamic interactive nature). These then provide the basis for new distinctive types of appreciation with respect to science, the arts and morality respectively (with ultimately these three domains seen as truly integrated in a complementary manner).<br /><br />In particular in this context, I have been at pains to explore the profound implications of such development for our understanding of the very nature of Mathematics.<br /><br />And this quite literally constitutes a new radical dimension with respect to the more advanced bands, which has not been explored in any sustained manner to this point.<br /><br />Furthermore, I believe that this new type of understanding will prove vital in coming years in helping our cultures adapt to ill-understood major social transformations indirectly resulting from a slavish adherence to the present reduced model of science.<br /><br /><br />So I would simply characterise this present model that so dominates accepted notions of Mathematics and Science as 1-dimensional.<br /><br />This is manifest for example in the attempt to view observed objects as existing in an abstract manner (as external to a passive observer).<br /><br />An object therefore is thereby given just one unambiguous external direction i.e. dimension. <br /><br />This is especially true with respect to the conventional way of interpreting numbers that are viewed as abstract quantitative objects (that exist independent of the inquiring mind).<br /><br />Therefore any consideration of how the (internal) mind interacts with the (external) objects or perhaps even more crucially how the quantitative (independent) objects can thereby be related in a qualitative (interdependent) manner, is thereby avoided.<br /><br />So all real numbers are rationally represented as points on a (1-dimensional) line.<br /><br />However with Band 3 development, one gradually begins to realise - rather than just one absolute interpretation (i.e. 1-dimensional) of mathematical symbols - that an unlimited number of other possible dimensional interpretations exist (all with a certain limited relative validity).<br /><br />The simplest of these is 2-dimensional, where - using standard dualistic language - all interpretation has both positive and negative aspects (that are dynamically complementary).<br /><br />So an object for example such as number has both external (objective) and internal (subjective) aspects which interact in dynamic fashion.<br /><br />However every number also has potentially a dynamic dimensional significance.<br /><br />Therefore for example, with the highly important 4-dimensional case, all objects now have both real and imaginary aspects (with positive and negative poles). So both quantitative (real) and qualitative (imaginary) aspects are dynamically interdependent in relative fashion, with both having interdependent twin poles (as objective "reality" and subjective interpretation respectively).<br /><br />So this increasingly relative understanding of the nature of mathematical truth is a product of Band 3 development. Here dualistic understanding of a rational kind continually breaks down as one moves to an increasingly holistic intuitive appreciation of the nature of phenomena.<br /><br />More precisely it is a product of super conscious development, which is associated with the firm belief that one is now traversing "higher" levels of development.<br /><br />And in my own case this was likewise directly associated with continual refinement of the cognitive mode. So once again rigid rational notions were eroded in the unfolding of a contemplative intuitive worldview, where I sought to directly experience the interdependence of all reality (in a transcendent spiritual fashion).<br /><br /><br />This in turn led to a dramatic change in my appreciation of the nature of number.<br /><br />At the conventional levels of Band 2, number is interpreted in a static rigid manner.<br />Indeed one of the great attractions of Mathematics for so many is the belief in the absolute nature of its symbols. This leads to the notion for example of a prime number as representing an unchanging universal form, frozen as it were in time and space. Though we may have reluctantly conceded with the advent of quantum physics that the apparent rigid nature of physical forms is but an illusion, we cling to the mistaken belief that we can still safely take refuge in the absolute nature of mathematical forms!.<br /><br />However Band 3 development leads to the gradual erosion of this belief as one discovers the truly relative nature of all mathematical symbols.<br /><br />This then culminates in the appreciation of number as representing a pure energy state (in contrast to earlier understanding as a rigid absolute form).<br /><br />So again its is all a matter of interpretation! If one approaches number from a limited linear rational perspective (as at Band 2) it will of course appear to possess an absolute form.<br /><br />If however one now approaches number from a refined intuitive perspective (in keeping with the most advanced level of Band 3) it will now equally appear to represent a purely relative existence as an energy state.<br /><br />So these represent the two extreme positions of quantitative (analytic) in the former and qualitative (holistic) interpretation in the latter case, respectively.<br /><br />And though the holistic understanding is directly intuitive - representing a psycho spiritual state - this properly is associated with an increasingly refined rational appreciation of a circular i.e. paradoxical nature.<br /><br /><br />However the problem with Band 3 development is that it too can become quite unbalanced. Therefore as I attempted to gain an increasingly specialised appreciation with respect to the true qualitative nature of mathematical symbols, I started to lose touch with former quantitative understanding.<br /><br />So as well as the ascent to "higher" super conscious levels at Band 3, one must also undergo - for balanced development - a corresponding "lower" descent into the subconscious regions of personality, to eventually discover that these too unexpectedly possess an important mathematical significance. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-31044608349988122312017-05-31T02:56:00.004-07:002017-06-01T07:49:59.972-07:00Number and Development (14)There are in fact two complementary relationships as between the primes and the natural numbers.<br /><br />From the standard well-known perspective, this relates to the quantitative (base) aspect of such numbers.<br /><br />However from the little regarded alternative perspective it relates to qualitative (dimensional) aspect of such numbers i.e. as represented by factors.<br /><br />So for example, from the former external perspective we might seek to calculate the frequency of primes up to a given natural number.<br /><br />From the latter internal perspective, we might then seek to calculate the ratio of natural to prime factors within this given number.<br /><br />In both contexts, log n is of special importance.<br /><br />In the former external case, it measures the average gap as between prime numbers. In the latter internal case, it measures the average frequency of natural number factors.<br /><br />Thus again in the former case, n/log measure the average frequency of primes to n. In the latter case n/log n measures the average gap as between the natural number factors of n.<br /><br />So one can see clearly, even in these simple illustrations, how two complementary measurements with respect to the primes and natural numbers, are at play which are - relatively - quantitative and qualitative with respect to each other.<br /><br />So there is a (base) quantitative aspect which relates to the fundamental notion of such numbers as points on the (1-dimensional) line. However there is also a qualitative dimensional aspect aspect, which relates to the equally important aspect establishing the spatial - and indeed temporal - relationship as between such numbers.<br />And this is vitally important to emphasise because numbers (like physical matter) strictly can have no meaning in the absence of space and time dimensional characteristics.<br /><br />It is therefore only the gross fallacy of the continual reduction of the qualitative (relational) aspects of number in quantitative terms that has led to the utterly mistaken assumption that numbers can have an absolute abstract existence (independent of space and time).<br /><br />So without a qualitative (relational) aspect, numbers can enjoy no meaningful existence as quantities. And without a quantitative (independent) aspect, no meaningful relationship can be established as between numbers.<br /><br />However clearly both of these aspects can only be appropriately understood in a dynamic interactive manner, where both (quantitative) independence and (qualitative) interdependence <br />are understood in a truly relative fashion.<br /><br /><br />This is all very pertinent to interpretation of the true meaning of Riemann's Hypothesis.<br /><br />The assumption here is that all the Zeta 1 (non-trivial) zeros lie on an imaginary line drawn through 1/2.<br /><br />Now bear in mind what I have been repeatedly saying in these blog entries regarding the truly complementary nature of both quantitative and qualitative aspects of the number system!<br /><br />And remember that I have also strongly maintained that the Riemann zeros in fact indirectly represent the hidden qualitative aspect of this system (in cardinal terms)!<br /><br />However because of the reduced nature of Conventional Mathematics, where qualitative considerations are reduced in an absolute quantitative manner, the misleading rational assumption is made that all real number quantities already lie on the (1-dimensional) number line.<br /><br />However when we view the number system appropriately in a truly dynamic interactive manner, then we are no longer entitled to make this assumption regarding the horizontal linear nature of the quantitative aspect independent of the qualitative aspect (represented by the non-trivial zeros).<br /><br />Equally we are not entitled to make the assumption regarding the vertical linear nature of the qualitative aspect (represented by the non-trivial zeros) independent of the quantitative aspect (represented by the real line).<br /><br />In particular there is no way of proving the truth regarding the vertical nature (on an imaginary line) of the non-trivial zeros i.e. of proving Riemann's Hypothesis in conventional mathematical terms, as this very assumption is already implicit in the acceptance of the horizontal nature of the real numbers (on the real line).<br /><br />In other words, in making the assumption that all real numbers can be consistently expressed as lying on a (1-dimensional) line, we already assume total consistency as between the independent aspect of number (in quantitative terms) and the interdependent aspect of number (in qualitative terms) whereby numbers can be consistently related with each other.<br /><br />So once again the truth regarding the qualitative aspect of number, which the non-trivial zeros indirectly express, is already unwittingly assumed in the very assumptions regrading the real number line.<br /><br /><br />So the truly fundamental issue which the Riemann Hypothesis - when appropriately interpreted - raises, is the ultimate consistency with respect to both the quantitative (analytic) and qualitative (holistic) use of mathematical symbols.<br /><br />In more psychological terms, it relates to the ultimate consistency of both the conscious (rational) and unconscious (intuitive) interpretation of these same symbols.<br /><br />And once again this issue cannot remotely be solved in an absolute conventional mathematical manner, as it already blindly assumes that the qualitative aspect is consistent with the quantitative.<br /><br />So properly understood, ultimate belief in such underlying consistency represents a giant act of faith in the entire subsequent mathematical enterprise.<br /><br />What we can therefore say in a dynamic relative manner, is that if our quantitative assumptions regarding the number line are to be valid, then equally the assumption that all the non-trivial zeros lie on an imaginary line (through 1/2) must also be true.<br /><br />Equally, if the assumption that all the non-trivial zeros must lie on an imaginary line (through 1/2) is to be valid then the assumption that all real numbers lie on the number line must also be true.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-56198576614979569882017-05-29T06:35:00.002-07:002017-06-06T15:44:34.816-07:00Number and Development (13)I ended the last blog entry by attempting to succinctly explain the true significance of the famed Riemann (i.e. Zeta 1) zeros. <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><br />In fact what I said there requires just a little more clarification.<br />Remember the fruits of this understanding arise from a dynamic interactive manner of understanding number relationships (which always involves complementary opposite poles)!<br /><br />So therefore when we start with the customary analytic view of the natural number system i.e. as independent cardinal numbers in quantitative terms, the Zeta 1 (Riemann) zeros then operate as the qualitative (holistic) counterpart of this system i.e. where the interdependence of these numbers, through their unique prime factor combinations, can be indirectly represented in a numerical fashion.<br /><br />However because in dynamic terms reference frames continually switch, we can equally start with the (unrecognised) holistic view of the natural number system (where one is directly aware in an intuitive manner of the interdependence of prime factors). Then, from this perspective the Zeta 1 (Riemann) zeros operate as the quantitative (analytic) counterpart to this understanding, in providing an independent set of numbers on which the holistic understanding is necessarily grounded.<br /><br />So in this interactive sense, both the natural numbers and Zeta 1 (Riemann) zeros can be seen to contain both quantitative (analytic) and qualitative (holistic) meanings, which necessarily keep switching with each other in the dynamics of experience.<br /><br />And of course similar dynamics relating to continually switching reference frames likewise apply to the Zeta 2 zeros with respect to understanding of the true ordinal nature of the number system.<br /><br />However it is possible to now probe more closely the exact nature of the Zeta 1 (Riemann) zeros and the clue to this again lies in the appreciation of the meaning of complementary opposite relationships.<br /><br />And in this important sense, these zeros directly complement - in dynamic interactive fashion - the primes!<br /><br />So from one valid perspective, we have seen that the primes and natural numbers operate in a complementary opposite fashion.<br /><br />Likewise the primes as numbers without constituent factors complement those composite numbers (with factors).<br /><br />Therefore in looking for the complementary opposite of the primes we should be attempting to determine all of the factors (or divisors) of the natural numbers.<br /><br />So it is in this way - though the interaction of such factors - that the qualitative interdependent nature of the primes is expressed.<br /><br />Now as always a lot depends on how we precisely define factors.<br /><br />In conventional terms even the primes have factors with 1 being a constituent factor and the prime number itself. Thus from this perspective, each prime has 2 factors (which represents the minimum that a number can contain).<br /><br />However because 1 is necessarily a factor of all numbers, just as we treat 1 as a trivial root of the number 1, we likewise treat 1 as a trivial factor of every number.<br /><br />Therefore from this perspective, each prime has just 1 factor, which directly concurs with its 1-dimensional nature.<br /><br />So for example if we start with 2, the independent quantitative nature of 2 as a prime is expressed through the fact that this is the only factor of 2.<br /><br />Likewise with 3, the independent quantitative nature of 3 as a prime is likewise expressed through the fact that this is the only factor of 3.<br /><br />However with 4 a new situation arises in that 2 and 4 are now constituent factors..<br /><br />Therefore 2 now acquires a new interdependent qualitative status as a constituent factor of 4. This can equally be expressed by the fact that 2 as a prime must be now combined with another prime 2 to generate 4.<br /><br />So the interdependence here arises directly through the fact that the number 2 can be expressed as part of a multiplication operation (which directly implies a qualitative transformation).<br /><br />However 4 in this single context - though a composite number - acquires a relatively independent quantitative status (i.e. as a number that can be placed on the number line).<br /><br />However when 4 then subsequently exists as a sub-factor of a larger number e.g. 8, then it too now acquires a qualitative interdependent status.<br /><br />So each new natural number - when initially uniquely generated by prime factor combinations - carries a relative independent quantitative status. However when this number then exists as a factor of a larger number, its qualitative interdependent status is revealed.<br /><br />Of course ultimately all natural number factors can be expressed as combinations of primes.<br /><br />So we can say for example that for the number 8, as well as the default 8 as a factor (in a relatively independent quantitative sense), 2 and 4 are also factors (in a relatively interdependent qualitative manner).<br /><br />So again though 2 is indeed a prime number, as a factor of 8 it obtains a unique qualitative status. Likewise 4 also obtains a unique qualitative status in this context (as a sub-factor of 8).<br /><br />However this can also be expressed by saying that the prime combination of 2 * 2 thereby obtains a unique qualitative status (as a sub-factor of 8).<br /><br /><br />Therefore what we are saying here is that the qualitative (holistic) nature of the number system - to which the Zeta 1 (Riemann) zeros are intimately associated - directly relate to the natural factors of each member of the number system.<br /><br />In other words when a number exists as a factor of another number, this implies that it is directly connected to that number through a (non-trivial) multiplication operation, and because multiplication always - when appropriately understood - entails a qualitative type transformation, this essentially therefore is what defines the qualitative (holistic) nature of such factors. <br /><br />Now if we attempt to calculate the frequency of such factor combinations a surprising link exists to the harmonic series.<br /><br />I have already mentioned that 2 is a factor of every 2nd number. Therefore 1/2 of all numbers contain 2 as a factor. Likewise 1/3 of all numbers contain 3 as a factor and 1/4 of all numbers contain 4 as a factor and so on.<br /><br />Therefore in this way one might conclude that the average no of factors of the number n = 1/2 + 1/3 + 1/4 +...+ 1/n.<br /><br />Seeing as we are leaving out 1 this would = log n <span style="font-family: "\22 times new roman\22 ";">– 1 + γ.</span><br /><br />However there is a slight problem that arises with this logic in relation to discrete numbers. For example if we are counting to 10, this might suggest that 3 occurs 3 + 1/3 times, 4, 2 + 1/2 times, 6, 1 + 2/3 times, 7, 1 + 3/7 times, 8, 1 + 1/4 and 9, 1 + 1/9 times. However clearly each of these will occur just a whole number of times. Therefore to eliminate these fractions we need to make an adjustment by subtracting (1 <span style="font-family: "\22 times new roman\22 ";">– γ). See "<a href="http://spectrumofmathematics.blogspot.ie/2016/04/surprising-result.html">Surprising Result</a>". So this would give us log n – 2 + 2γ (or in the case where 1 is included as a factor log n – 1 + 2γ). </span><br /><br /><span style="font-family: "\22 times new roman\22 ";">However if we ignore the Euler-Mascheroni constant (which arises in adjusting for discrete values) the simple formula for the average no. o factors in the number n = log n – 1. </span><br /><br /><span style="font-family: "\22 times new roman\22 ";">Then the total accumulated factors to n = n(log n – 1).</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">This then bears a remarkable similarity with the formula for calculating the frequency of Zeta (Riemann) non-trivial zeros to t which is given as t/2π.</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">Thus where n = t/2π, the two formulae are identical.</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">Now it must be remembered that qualitative (holistic) notions of number relating to dimensions i.e. factors properly relate to the Type 2 notion of number (based on the unit circle).</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">And as we have seen such circular notions of number can then be converted in an imaginary linear manner (i.e. as points on an imaginary axis).</span><br /><span style="font-family: "\22 times new roman\22 ";"><br />However to convert from circular to linear units we divide by 2π.</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">Therefore if we want to approximate the accumulated sum of factors to n, we count the frequency of non-trivial zeros on the imaginary scale to n * 2π.</span><br /><br /><span style="font-family: "\22 times new roman\22 ";">So for example the accumulated sum of natural number factors to 100 will match very closely the corresponding frequency of non-trivial zeros to 628.138 (approx). </span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--></span></span></span></span></span></span></span></span><br />Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-22322327130675149762017-05-27T08:42:00.002-07:002017-06-09T13:36:56.800-07:00Number and Development (12)We are accustomed through conventional mathematical training to view the primes in quantitative terms as the independent "building blocks" of the natural number system.<br /><br />However what is not all realised is that very nature of the primes changes when they exist - not individually - but rather as factor components of the unique product combinations that generate the composite numbers.<br /><br />It is in this manner therefore that the qualitative nature of the primes arises i.e. through their interdependence with other prime factors.<br /><br />So for example both 2 and 3 (as separate individual primes) can in a valid - though strictly relative - sense be viewed as independent "building blocks" in quantitative terms.<br /><br />However when 2 and 3 are then combined through multiplication to uniquely generate the composite number 6, i.e. 2 * 3 = 6, both 2 and 3 now acquire a relative interdependent meaning in this context, which is thereby of a qualitative (holistic) nature.<br /><br />And of course there is ultimately no limit to all the relative contexts in which each of the primes can be used (with respect to unique factor combinations with other primes). So 2 for example must necessarily exist as a factor with respect to every even composite number!<br /><br /><br />Just as we saw earlier that there is an inherent paradox in terms of the definition of each individual prime (with complementary quantitative and qualitative aspects), equally this is true with respect to the collective relationship of primes with respect to the natural number system.<br /><br />Again from the conventional quantitative perspective, we are automatically trained to see the relationship between the primes and natural numbers unambiguously in a one-way manner.<br /><br />So clearly from this perspective, the natural numbers appear to depend for their existence on the primes.<br />However implicit in this view is an unexpected problem which is rarely recognised, as our very understanding of the primes already requires the natural numbers for their proper comprehension.<br /><br />In other words, the very ability to spatially separate in a meaningful fashion the primes from each other already implies a notion of order that applies to the natural numbers.<br /><br />So the positioning of each prime already depends on the composite ordering of prime factors.<br /><br />And if we cannot meaningfully assign a position to each prime, then equally we cannot meaningfully provide it with a definite numerical identity!<br /><br />Thus again we have two complementary perspectives.<br /><br />From the quantitative perspective, the natural numbers appear to depend on the independent primes as "building blocks".<br /><br />However, from the qualitative perspective, the positioning of each prime appears to depend on the interdependence of prime factors that uniquely generate the natural numbers.<br /><br />Therefore to properly appreciate this paradox, we must once again move to a dynamic interactive appreciation of number behaviour, entailing both quantitative and qualitative aspects as equal partners.<br /><br />This then leads inevitably to the realisation that - just as with micro - the macro behaviour of the number system entails the synchronistic behaviour of both quantitative (analytic) and qualitative (holistic) aspects, which is ultimately ineffable.<br /><br />And of course as micro behaviour (associated with the Zeta 2 function) and macro behaviour (associated with the Zeta 1 function) are themselves dynamically complementary, ultimately neither has a meaning independent of the other <br /><br />Thus properly understood, in dynamic interactive terms, both the primes and natural numbers (and natural numbers and primes) mutually co-determine each other. both with respect to micro and macro aspects, in a synchronistic manner. <br /><br />So again the key fallacy with respect to conventional understanding is the attempt to view the number system in an absolute - merely quantitative - manner where the relationship as between primes and natural numbers is misleadingly viewed as one-way and unambiguous. <br /><br /><br /><br />We have already seen how the solutions to the Zeta 2 function provide an indirect quantitative means of expressing the qualitative nature of each individual natural number member of a prime number group (at the micro level).<br /><br />Likewise - again in true complementary fashion - the solutions to the Zeta 1 (Riemann) function, provide an indirect quantitative means of expressing the qualitative nature of the collective interdependence of prime factors with respect to the natural number system (at a macro level).<br /><br />So this is the key revelation that can now be made with respect to the Riemann zeros.<br /><br />Remember how Hilbert, when once queried as to most important problem in Mathematics replied,<br />" the problem of the zeros of the zeta function, not only in Mathematics but absolutely the most important!"<br /><br />And the true reason why these zeta zeros are indeed so important is that indirectly they express the hidden qualitative nature of the natural number system.<br /><br /><br />And in even more precise terms we can say that the Zeta 2 zeros indirectly express the qualitative (holistic) nature of the ordinal natural number system, whereas the Zeta 1 (Riemann) zeros express the corresponding qualitative nature of the cardinal natural number system, And ultimately, both ordinal and cardinal aspects are completely interdependent with each other.Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-23914021808963722532017-05-26T04:09:00.001-07:002017-05-27T03:40:00.952-07:00Number and Development (11)It may be useful again at this point to emphasise the key significance of what I have been articulating in these blog entries.<br /><br />We are accustomed to thing of Mathematics - especially in its treatment of number - in an absolute unambiguous manner (where the meaning of symbols remains fixed).<br /><br />However in truth an unlimited number of relative type interpretations can potentially apply, with the standard conventional approach representing just one special limiting case.<br /><br />Put another way the standard interpretation is of a 1-dimensional nature, whereby qualitative type considerations with respect to mathematical symbols are reduced in a merely quantitative manner (within a rigidly fixed framework).<br /><br />However associated with every mathematical symbol is a unique qualitative manner of interpretation.<br />Thus Analytic i.e. Conventional Mathematics relates to the quantitative interpretation of mathematical symbols whereas Holistic Mathematics relates to their qualitative inetrpretation. And the integration of both approaches - in what I refer to - as Radial Mathematics, requires a dynamic interactive approach, whereby both quantitative (analytic) and qualitative (holistic) interpretations are inherently understood in a complementary manner.<br /><br />So the huge unacknowledged problem is that just one special limited interpretation (where the qualitative meaning of symbols is reduced in a merely quantitative manner) has become synonymous with "all Mathematics", thereby completely blinding us to the infinite array of riches that would readily unfold through recognition of its hidden qualitative aspect as an equal partner.<br /><br />So in psychological terms, the vitally important unconscious aspect of understanding has been completely blotted out in the misleading attempt to formally portray Mathematics as merely a rational (conscious) discipline.<br /><br /><br />However so far we have been looking at merely the "micro" aspect of number behaviour.<br /><br />In other words we have started from the notion of an individual prime as a quantitative "building block" of each natural number to discover its hidden qualitative meaning as a number group whose individual ordinal members are uniquely defined in a natural number fashion.<br /><br />So the inherent paradox of the nature of each prime is thereby clearly revealed. Thus from the quantitative perspective, the primes appear to unambiguously generate the natural numbers (in a cardinal manner); however from the (hidden) qualitative perspective each individual prime already is defined by an unbroken sequence of natural numbers (in an ordinal manner).<br /><br />Thus from a dynamic interactive perspective - entailing both cardinal and ordinal aspects - it becomes clear that both the primes and natural numbers are ultimately co-determined in a synchronistic manner (which is ultimately ineffable).<br /><br />However this realisation can have no meaning within the conventional mathematical perspective that misleadingly insists on viewing the nature of number in a merely quantitative manner!<br /><br /><br />However as well as the "micro" aspect of each individual prime, we likewise have the "macro" aspect of the collective behaviour of the primes with respect to overall natural number system.<br /><br />And of course both individual "micro" and collective "macro" aspects of prime (and natural number) behaviour are themselves complementary in a dynamic interactive manner.<br /><br />So strictly speaking we neither start with each individual prime (as somehow pre-defined) or the entire natural number system (as likewise somehow pre-defined).<br /><br />Rather they both arise through mysterious dynamic interaction patterns that then subsequently enable their separate identities to be abstracted in a fixed manner. <br /><br /><br />In now looking at the "macro" behaviour of the number system, we make direct contact with Riemann's zeta function and the famed (non-trivial) zeros.<br /><br />However, it was intense investigation over many decades relating to the less investigated ordinal nature of number (briefly outlined in the 10 previous blog entries) that eventually provided me with the appropriate framework to understand the true nature of these Riemann zeros.<br /><br />However the starting point goes back to a classroom revelation when attending primary school in Ireland, regarding an unacknowledged problem regarding the nature of multiplication.<br /><br />As we know in quantitative terms all natural numbers can be uniquely expressed as the product of primes.<br /><br />So if we take the number "6" to illustrate this is uniquely expressed as 2 * 3.<br /><br />Now the number "6" is represented as a point on the number line. In other words 6 is represented in a 1-dimensional manner as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">6<sup>1</sup></span>.<br /><br />However if we represent 2 and 3 here in geometrical fashion - say as two sides of a rectangular tables (measured) in metres - then the area represented by 2 * 3 is given in square (i.e. 2-dimensional) terms. In other words, though each side relates to a measurement in 1-dimensional, the resulting area relates to a corresponding measurement in 2-dimensional units.<br /><br />However with respect to the standard treatment of multiplication, this qualitative transformation in the nature of units is simply edited completely out of consideration.<br /><br />Thus the result of 2 * 3 (and by extension every factor combination of primes) is given in a merely reduced (i.e. 1-dimensional) quantitative manner.<br /><br /><br />However though this early insight still remained fully valid, it was only at a later date that I realised an additional - perhaps even more fundamental - problem with the conventional nature of multiplication.<br /><br />When we speak of the area of a table (as for example in my illustration) we are still operating at the analytic level of understanding dimensional numbers.<br /><br />However as we have seen, all such numbers have a corresponding holistic interpretation and this too is intimately involved with the very nature of multiplication.<br /><br />If for these purposes we have two rows with 3 coins in each row, each row would represent in analytic terms a dimension (e.g.. length) and each column (of 2 coins) another dimension (e.g. height).<br /><br />Now without specific reference to rows or columns, we could attempt to treat each coin in an independent manner and obtain the result by adding up each of the (independent) units.<br /><br />However the key point about multiplication that in order to use the operator of 2 we must recognise the mutual interdependence of the two rows (with 3 coins).<br />Thus crucially whereas with addition, we can proceed by recognising the independence of each individual item, for multiplication we must also recognise the mutual interdependence of rows and columns.<br /><br />So when we recognise the two rows as interdependent we use 2 as operator to obtain 2 * 3.<br /><br />Equally when we recognise the three columns as interdependent we use 3 as operator to obtain 3 * 2.<br /><br />So conventional multiplication attempts to represent mutiplication - very misleadingly - as a short-hand form of addition.<br /><br />So with respect to addition when we add the two rows we get 3 + 3 = 6. And is each row is defined in a 1-dimensional manner the result through addition is likewise 1-dimensional in nature.<br /><br />Thus <span lang="EN-IE" style="mso-ansi-language: EN-IE;">3<sup>1</sup></span> + <span lang="EN-IE" style="mso-ansi-language: EN-IE;">3<sup>1</sup></span> = <span lang="EN-IE" style="mso-ansi-language: EN-IE;">6<sup>1</sup></span>.<br /><br />Conventional multiplication then attempts to "speed up" this process. Because 3 is repeated 2 times with respect to addition, we now multiply 3 two times i.e. 2 * 3 to apparently get the same result.<br /><br />Now this "speeding up"is of course not so evident when the operator is 2. However imagine if 3 was to be added to 3 one hundred times, then the expression of this through multiplication as 100 * 3 would indeed be much more efficient. <br /><br />However the crucial unrecognised problem is that the very switch from addition to multiplication requires the recognition of the rows and columns (representing the two dimensions) not only as each containing independent items but also that these dimensions themselves as mutually interdependent (and thereby freely interchangeable with each other).<br /><br />And this latter recognition requires the qualitative (holistic) interpretation of the nature of a dimension.<br /><br />So I was already well primed - to excuse the pun - to see that there was a distinct qualitative aspect to the nature of multiplication, which was completely unrecognised in conventional mathematical terms. And I was already confident that - when appropriately understood - the Riemann (Zeta 1) zeros directly related to this hidden qualitative aspect of the cardinal number system. <br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-55649774846427421842017-05-25T06:46:00.001-07:002017-05-26T09:55:17.802-07:00Number and Development (10)I will start this entry by contrasting the respective meanings in Type 1 and Type 2 terms of the fractions 1/3, 2/3 and 3/3 respectively.<br /><br />In Type 1 terms these would be given as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">(1/3)<sup>1</sup></span>, <span lang="EN-IE" style="mso-ansi-language: EN-IE;">(2/3)<sup>1</sup></span> and <span lang="EN-IE" style="mso-ansi-language: EN-IE;">(3/3)<sup>1</sup></span> respectively.<br /><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><div class="MsoNormal"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">So from the standard quantitative (analytic) perspective </span><span lang="EN-IE" style="mso-ansi-language: EN-IE;">(1/3)<sup>1</sup></span> represents 1 of 3 (equal) parts i.e. one third;</div><div class="MsoNormal"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"></span><span lang="EN-IE" style="mso-ansi-language: EN-IE;">(2/3)<sup>1 </sup></span><span lang="EN-IE" style="mso-ansi-language: EN-IE;">represents 2 of 3 (equal) parts i.e. two thirds.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">(3/3)<sup>1</sup> represents 3 of 3 (equal) parts i.e. 1 as a whole unit, </span> </div><br />However in Type 2 term, these fractions would be given as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>1/3</sup></span>,<span lang="EN-IE" style="mso-ansi-language: EN-IE;"> 1<sup>2/3</sup></span> and 1<span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>3/3</sup></span> respectively.<br /><br />Now in standard quantitative terms, these represent the 3 roots of 1 i.e. <span style="font-family: "times new roman"; font-size: 12pt;">– </span> .5 + .866i, <span style="font-family: "times new roman"; font-size: 12pt;">– </span> .5 <span style="font-family: "times new roman"; font-size: 12pt;">–</span> .866i and 1 respectively.<br /><br />However these also have an important qualitative (holistic) interpretation. And in dynamic interactive terms, when the Type 1 interpretation relates to the quantitative (analytic) aspect, then the corresponding Type 2 interpretation relates to the corresponding qualitative (holistic) aspect.<br /><br />And the qualitative (holistic) interpretation of <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1/3, </span><span lang="EN-IE" style="mso-ansi-language: EN-IE;">2/3</span> and 3/3 (i.e. <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>1/3</sup></span>,<span lang="EN-IE" style="mso-ansi-language: EN-IE;"> 1<sup>2/3</sup></span> and 1<span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>3/3</sup></span>) would be expressed as 1st of 3 (related) dimensional units, 2nd of 3 (related) dimensional units and 3rd of 3 (related) dimensional units respectively.<br /><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--><span style="font-family: "times new roman"; font-size: 12pt;">Of course reference frames continually switch with respect to experience. So if we now start with the Type 2 aspect defined in the standard analytic manner (as the 3 quantitative roots of 1), then in Type 1 terms 1/3, 2/3 and 3/3 now take on a complementary holistic qualitative meaning as the 1st of 3 (related) base units, the 2nd of 3 (related) base units and the 3rd of 3 related base units respectively.</span><br /><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Now the key significance of a prime from the qualitative (holistic) perspective is that each of its ordinal members (with the exception of the last) is always uniquely defined.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">In other words, when we omit consideration of the last unit (i.e. given in general terms as the nth of n units), the indirect numerical designation of all other ordinal positions for each prime, by definition, cannot be replicated with respect to any other prime.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So again from the quantitative (analytic) perspective, the primes collectively are unique as the fundamental factors or "building blocks" of the natural numbers.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">However, from the qualitative (holistic) perspective, each individual prime group is unique in that all its natural number ordinal members (except the last) -indirectly represented in turn by the various roots of 1 (except the last) - are unique for that prime group.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So from this perspective, 5 (representing a prime group of individual members) is unique in that its 1st, 2nd, 3rd and 4th members - indirectly represented in turn in a qualitative manner by the 1st, 2nd 3rd and 4th roots cannot - by definition, be replicated with respect to the ordinal members of any other prime group.</span><br /><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">It was at this time that I gave intense consideration as to the precise significance of the "trivial" last root i.e. 1 (which exists for every prime group).</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Then it slowly dawned on me that it was the interpretation with respect to this root that naturally occurs in the standard ordinal interpretation of number (where ordinal notions are reduced in a cardinal manner).</span><br /><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So the number system starts with 1 (which or course is not prime). Then when we move on to consideration of 2, in standard analytic terms the 1st unit is unambiguously fixed as 1 with 2nd unit now in likewise manner fixed with the last remaining unit of 2.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Then when we move on to consideration of 3 the 1st and 2nd units have already been unambiguously fixed with the 1st two units so that the 3rd unit is now likewise unambiguously fixed with the last unit (of 3).</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">We started with the quantitative definition of 3 (as cardinal number) = 1 + 1 + 1.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">If we now define this in ordinal terms, 3 = 1st + 2nd + 3rd units.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">However because in conventional mathematical terms, each ordinal position is unambiguously fixed in an absolute manner with the last unit of its corresponding cardinal number group (= 1), from this perspective, </span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><br /></span><span style="font-family: "times new roman"; font-size: 12pt;">1st + 2nd + 3rd = 1 + 1 <complete id="goog_947312883">+ 1 = 3.</complete></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883"><br /></complete></span><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883">So there seems therefore from the quantitative (analytic) perspective that there exists a perfect correspondence as between cardinal and ordinal notions!</complete></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883"><br /></complete></span><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883">However from the corresponding qualitative (holistic) perspective, all ordinal positions are merely relative and can be interchanged with each other.</complete></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883"><br /></complete></span><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883">And as these are indirectly given in quantitative as the corresponding roots of 1, from this perspective,</complete></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883">1st + 2nd + 3rd = </complete></span><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883"><span style="font-family: "times new roman"; font-size: 12pt;">– </span> .5 + .866i <span style="font-family: "times new roman"; font-size: 12pt;">– </span> .5 <span style="font-family: "times new roman"; font-size: 12pt;">–</span> .866i + 1= 0.</complete></span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883"><br /></complete></span><span style="font-family: "times new roman"; font-size: 12pt;"><complete id="goog_947312883">In others, strictly speaking from the holistic qualittaive perspective, the pure interdependence of all ordinal positions has no cardinal meaning!</complete></span><br /><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So therefore the important point to grasp is that in standard quantitative (analytic) terms, each new ordinal unit is unambiguously fixed with the last unit of the number group in question. </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">However the key point regarding the qualitative (holistic) interpretation of number is that ordinal positions can be undercharged with each other.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Thus in a group of 3 for example what is 1st from one relative perspective can equally be 2nd and 3rd from two other equally valid perspectives. Likewise what is 2nd from the first perspective, can be equally 1st and 3rd from the other perspectives, and finally what is 3rd from the first can equally be 1st and 2nd from the other perspectives.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So the holistic appreciation of ordinal positions implies the interdependence of each individual member with each other member of the respective group.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">The analytic appreciation implies by contrast the independence of each individual member, whereby the ordinal position is unambiguously fixed with this member.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So as we have seen from this latter perspective, 1st is unambiguously identified with the last member (of a group of 1) , 2nd with the last member (of a group of 2), 3rd with the last member (of a group of 3) and so on.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">It was at this stage that I suddenly saw how a striking complementarity in fact existed as between all this work on the holistic nature of ordinal numbers and the famed Riemann zeros. </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">In fact I could see now that were in fact two sides as it were of the same coin.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">This insight arose from the attempt to isolate the truly unique holistic solutions indirectly implied in general terms by the t roots of 1 (except the default root of 1).</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So the t roots of 1 are obtained from the equation, </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">x</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>t</sup></span></span> </span><span style="font-family: "times new roman"; font-size: 12pt;"> <span style="font-family: "times new roman"; font-size: 12pt;">– 1 = 0, or alternatively as better suits our purposes 1</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"> –</span></span> </span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">x</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>t</sup></span></span> = 0.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Thus to eliminate the default root (where x = 1), we divide by 1 </span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">– x, to obtain</span></span></span></span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">1 + </span></span></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">x</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>1</sup></span></span></span> </span></span></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">+ </span></span></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">x</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>2</sup></span></span> </span></span></span></span></span></span> </span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">+ </span></span></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;">x</span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><sup>3</sup></span></span> </span></span></span></span></span></span>+ ... +</span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"> x<sup>t </sup></span><sup>– 1 </sup></span></span><span style="font-family: "times new roman"; font-size: 12pt;"> = 0.</span><span style="font-family: "times new roman"; font-size: 12pt;"> </span><span style="font-family: "times new roman"; font-size: 12pt;"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> </span><br /><div class="MsoNormal"><br /></div><span style="font-family: "times new roman"; font-size: 12pt;"> So I started to refer to this as the Zeta 2 in contrast to the complementary type Riemann zeta function (i.e Zeta 1) where</span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><br /></span><span style="font-family: "times new roman"; font-size: 12pt;">1</span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><sup>– s</sup></span></span> + </span><span style="font-family: "times new roman"; font-size: 12pt;">2</span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><sup>– s </sup></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">+</span> 3</span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><sup>– s </sup></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">+</span> 4</span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;"><sup>– s </sup></span></span></span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">+</span> .... = 0.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">So the famed non-trivial zeta zeros represent the solutions for s to this equation. </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">But note the complementarity as between both functions! </span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Whereas the former (Zeta 2) is of of a finite nature that be extended without limit, the latter (Zeta 1) is infinite in nature.</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Likewise whereas the sequence of natural numbers appear as dimensional powers (with respect to the Zeta 2), they appear as a sequence of base numbers (with respect to the Zeta 1).</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">Also, whereas the unknown to be solved from the equation is a base number (with respect to the Zeta 2), it is a dimensional power (with respect to the Zeta 2).</span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;">The realisation of the complementary nature of these two functions was then to prove invaluable in "unearthing" the true significance of the Riemann (Zeta 1) zeros. </span><span style="font-family: "times new roman"; font-size: 12pt;"> </span><span style="font-family: "times new roman"; font-size: 12pt;"> </span>Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-90690249259406126722017-05-24T03:24:00.001-07:002017-05-24T08:48:37.813-07:00Number and Development (9)As we have seen, there are two aspects with respect to the understanding of all numbers - and indeed by extension all mathematical relationships - that are quantitative (analytic) and qualitative (holistic) with respect to each other.<br /><br />Both of these aspects can only be properly understood in a dynamic relative context, where each type of understanding implies the other in a complementary manner.<br /><br />Now, from a psychological perspective, the analytic aspect is directly related to rational type appreciation (of a conscious kind); by contrast the holistic aspect is directly related to intuitive type appreciation (of an unconscious nature).<br /><br />Therefore, when understood appropriately, the role of intuition with respect to mathematical understanding is utterly distinct and cannot be confused with reason.<br /><br />However, because in effect conventional mathematical understanding entails the reduction of holistic type meaning (in an analytic manner), equally this entails the corresponding reduction of intuitive (directly related to the unconscious) with rational type appreciation (of a distinctive conscious nature).<br /><br />So once again the truly central issue with respect to all mathematical interpretation is thereby missed.<br /><br />From the external physical perspective, this relates to consistency with respect to both the quantitative and qualitative interpretation of its symbols; from the corresponding psychological perspective - which in dynamic terms complements the physical - this equally relates to consistency with respect to the rational and intuitive interpretation of these same symbols.<br /><br />In fact as I have repeatedly stated in my blog entries, from the appropriate dynamic interactive perspective, the Riemann Hypothesis can be seen as a key statement with respect to this central issue.<br /><br />And by the same token, because in conventional mathematical terms the qualitative aspect is not formally recognised (as distinct from the quantitative) this implies that such attempted "proofs" of the Riemann Hypothesis are rendered futile!<br /><br /><br />When one looks at the nature of the primes from this dynamic interactive perspective, appreciation of their very nature is thereby transformed.<br /><br />Once again using "3" to illustrate this of course represents a prime number!<br /><br />Now from the conventional analytic perspective in cardinal terms, this prime thereby represents a constituent "building block" of the natural number system.<br /><br />However from the (unrecognised) holistic perspective, in ordinal terms, the position is reversed with each prime group representing a unique configuration of its constituent individual members.<br /><br />So "3" for example, thereby represents a unique configuration with respect to its 1st, 2nd and 3rd members.<br /><br />The next prime "5" would then represent a unique configuration with respect to its 1st, 2nd, 3rd, 4th and 5th members.<br /><br />However this begs the significant question as to the derivation of the 4th member (which implies the number "4"). Therefore though from the cardinal perspective "5" is already viewed as an independent "building block", clearly from the ordinal perspective the composite natural number "4" is directly implied with respect to its 4th member!<br /><br />In other words, from a dynamic interactive perspective it is quite untenable to maintain this absolute stance with respect to the primes as representing the independent "building blocks" of the natural number system!<br /><br />Certainly from a relative perspective, the primes appear as the "building blocks" of the natural number system (in cardinal terms). However from an equally valid alternative relative perspective, each constituent prime appears as representing a unique configuration of its individual natural number members (in an ordinal manner).<br /><br />So again in cardinal terms, the natural numbers appear to be determined by the primes; however from the ordinal perspective, each prime appears to be determined by its natural number members.<br /><br />The key implication therefore is that from a dynamic interactive perspective - which represents the true nature of the number system - both the primes and natural numbers are co-determined in a synchronistic manner (that is ultimately ineffable).So the primes and natural numbers ultimately mirror each other in a perfect manner.<br />And through right understanding one can experientially approach, to an ever closer degree, true appreciation of this perfect mirroring.<br /><br /><br />So in terms of Band 5 development, a new appreciation of "dimensional" numbers started to open up, whereby they could now become fully grounded in the linear (1-dimensional) levels of Band 2.<br /><br />So I now came to the clear realisation that the very means of "converting" the qualitative notions of 1st, 2nd, 3rd, ... entailed the holistic appreciation of the corresponding roots of 1.<br /><br />So again for example in Type 2 terms, we can represent 3 as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>3</sup></span>.<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><div class="MsoNormal"><span lang="EN-IE" style="mso-ansi-language: EN-IE;"><br /></span></div>Thus is holistic terms 3 represents the notion of 3 dimensions as fully interdependent with each other (which is directly grasped in an intuitive manner).<br /><br />Thus with each number is associated a distinctive "quality" of intuition. Thus associated with 2 is the quality of appreciating the interdependence of 2 related dimensions, with 3, the quality of appreciating the interdependence of 3 related dimensions, with 4 the quality of appreciating the interdependence of 4 related dimesnions and so on.<br /><br />However we can indirectly convert such qualitative notions, in a quantitative (1-dimensional) rational manner, by taking the corresponding roots of 1 (associated with each number).<br /><br />Thus for example the 2 roots of 1, i.e. + 1 and <span style="font-family: "times new roman"; font-size: 12pt;">– 1</span> express - in a necessarily paradoxical "circular" manner - the linear rational notion of the interdependence of two objects.<br /><br />As I have repeated many times before this naturally arises in our appreciation of the paradoxical nature of turns at a crossroads.<br /><br />So in approaching the crossroads from one direction one can unambiguously define left and right turns. So if "left" is designated as + 1 (as 1st), then "right" in this context is designated as <span style="font-family: "times new roman"; font-size: 12pt;">– 1 (as 2nd).</span><br /><br />However when the crossroads is approached approached from the opposite direction, what was formerly a left turn is now right and what was a right turn is now left. So what was + 1 (as 1st) is now <span style="font-family: "times new roman"; font-size: 12pt;">– 1</span> (as 2nd), and what was <span style="font-family: "times new roman"; font-size: 12pt;">– 1 (as 2nd) is now + 1 (as 1st). </span><br /><br />So in this context of mutual relative interdependence, + 1 and <span style="font-family: "times new roman"; font-size: 12pt;">– 1 can switch between each other (with each possessing a merely relative validity). In fact the interdependence of the two numbers is expressed through the requirement that their sum = 0.</span><br /><span style="font-family: "times new roman"; font-size: 12pt;"><br /></span><span style="font-family: "times new roman"; font-size: 12pt;">Likewise the 3 roots of 1 i.e. + 1,.5 + .866i and .5 </span><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">–.866i express (in an indirect linear rational manner) the interdependence of 3 numbers with respect to 1st, 2nd and 3rd positions (which can mutually switch in a relative manner as between each other).</span></span><br /><br /><span style="font-family: "times new roman"; font-size: 12pt;"><span style="font-family: "times new roman"; font-size: 12pt;">And in more general terms the n roots of 1 likewise express (in an indirect linear rational manner) the interdependence of n numbers with respect to 1st, 2nd, 3rd,..., nth positions (which can all mutually switch in a relative manner as between each other)..</span></span>Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-52075272943767454122017-05-22T09:38:00.000-07:002017-05-22T12:31:18.398-07:00Number and Development (8)There is an important (unappreciated) paradox with respect to the quantitative definition of any number.<br /><br />For example, if we take the cardinal number "3"to illustrate, it can be defined in the conventional mathematical manner as,<br /><br />3 = 1 + 1 + 1.<br /><br />This represents - what I term - analytic interpretation, whereby the (whole) sum i.e. 3, is treated in an actual quantitative manner as the sum of its independent (part) units. <br /><br />So here again, each of its three (sub) units is defined in an independent homogeneous manner i.e. without qualitative distinction.<br /><br />Therefore, from an ordinal perspective, there is no way to distinguish (with respect to dimensions of space and time) which units are 1st, 2nd and 3rd respectively.<br /><br />So in this ordinal context, each unit can potentially qualify as both 1st, 2nd and 3rd respectively.<br /><br />In other words in - what I term - holistic interpretation, each (part) unit is treated in a qualitative manner as potentially representing the interdependence of all the ordinal elements of its corresponding group.<br /><br />Thus from this holistic qualitative perspective, 1st, 2nd and 3rd (as ordinal positions) can equally be identified with each of the individual units of 3.<br /><br />Therefore when one one appreciates this central paradox i.e. that a number that is defined from an extreme quantitative perspective (in analytic terms), gives rise to the opposite extreme qualitative perspective (in a corresponding holistic manner), then one must accept that the conventional attempt to define number in absolute fixed terms must itself be abandoned.<br /><br />In other words, to resolve this paradox, one must move to a dynamic interactive interpretation of number, defined in a balanced relative manner, equally containing both quantitative (analytic) and qualitative (holistic) aspects.<br /><br />So now from this new dynamic perspective, the quantitative aspect of number is viewed analytically in relatively independent manner; in complementary terms, the qualitative aspect is viewed holistically in a relatively interdependent fashion.<br /><br />Therefore, again from this dynamic perspective - which concurs directly with the human experience of number - quantitative independence (in analytic terms) always implies qualitative interdependence (in a holistic manner); likewise qualitative independence necessarily implies quantitative independence, with both interpreted in a - necessarily - relative manner.<br /><br /><br />Thus with respect to the number "3" in our illustration, this number is now understood to entail the dynamic interaction of both the quantitative notion of 3 (understood in an analytic manner) and the qualitative notion of 3 i.e. as "threeness" (understood in a corresponding holistic fashion).<br /><br />Now, when one properly appreciates what is stated here, then it should become apparent that the standard accepted interpretation of number is simply not fit for purpose. It reduces the distinctive qualitative aspect (which can only be properly understood in a holistic manner) in an absolute quantitative manner (that is interpreted in a merely analytic fashion).<br /><br /><br />I have explained on many occasions how I have sought to remedy this deficiency in the standard interpretation of number by employing a truly dynamic appreciation, which entails the complementary interaction of both Type 1 and Type 2 aspects.<br /><br />Thus when we interpret "3" - now understood appropriately in a relative manner - with respect to its quantitative characteristics, in Type 1 terms, this is written as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">3<sup>1</sup></span>.<br /><br />So the dimensional number (i.e. exponent) of 3 is defined with respect to its default status of 1, which implies that we can concentrate in this context on the relative quantitative nature of 3.<br /><br />Then when we interpret "3" with respect to its corresponding qualitative characteristics, it is now written in Type 2 terms as <span lang="EN-IE" style="mso-ansi-language: EN-IE;">1<sup>3</sup></span> .<br /><br />So 3 now directly represents its dimensional status defined with respect to the default base of 1, which implies that we can now concentrate in this alternative context, on the relative qualitative nature of 3.<br /><br />Thus more simply expressed, when In Type 1 terms we are aware of the quantitative nature of 3 (in analytic terms), then - relatively - in Type 2 terms we are aware of the qualitative nature of 3 (in a holistic manner).<br /><br />However reference frames continually switch in experience.<br /><br />Therefore there is also a valid Type 1 sense in which 3 takes on a qualitative meaning (in a holistic manner), with 3 then - relatively - carrying a quantitative Type 2 meaning in analytic terms.<br /><br />In fact this latter qualitative Type 1 aspect of number is central to proper interpretation of the nature of multiplication, whereas the quantitative Type 2 aspect arises for example in the geometrical appreciation of a cube (with 3 linear dimensions).<br /><br />Therefore, properly understood, the number "3" - and by extension every number - keeps switching in experiential terms as between its quantitative (analytic) and qualitative (holistic) meanings with respect to both Type 1 and Type 2 aspects.<br /><br />And once again, the standard interpretation of number (in absolute quantitative terms) distorts these key dynamics in a grossly reduced manner. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-78318232264018297592017-05-05T05:06:00.002-07:002017-05-07T02:36:50.045-07:00Number and Development (7)In integral terms, the levels of Band 3 properly constitute both the new emerging "higher" stages of that band and the continually revisited "lower" stages of Band 1 (with which they are - in horizontal, vertical and diagonal terms, dynamically complementary).<br /><br />Therefore from a true integral perspective, we do not have here individual stages (in a discrete separate manner) but rather the growing interpenetration of all the stages of Band 1 and Band 3.<br /><br />However because integration is not yet fully balanced, typically more emphasis is placed initially on the differentiation of the new "higher" stages of Band 3 (without full consideration of the consequent need for integration of these with the corresponding complementary stages of Band 1).<br /><br />So in this differentiated sense, it is still correct to give each new stage of Band 3 a relatively distinct independent identity.<br /><br /><br />In terms of my own journey through these stages (of Band 3), I was indeed well aware in one sense of the need for their two-way integration with those of Band 1.<br /><br />I was convinced - even at this time - of the need for both top-down integration (where the "lower" stages of Band 1 would be integrated from the perspective of the "higher" stages of Band 3) and bottom-up integration (where the "higher" stages of Band 3 would be integrated from the revisited stages of Band 1).<br /><br />However in practice, the "higher" levels, based predominantly on the cognitive mode (of reason) were in important respects still repressing the instinctive behaviour of the "lower" levels, based predominantly on the affective mode (of emotion).<br /><br />And such imbalance as between cognitive and affective modes is in many ways inevitable until proper integration is eventually achieved (relating to the radials stages of Band 6 and 7 in my account).<br /><br />And this is why I always emphasise the volitional mode as truly primary, as this needs to be used with ever-greater discernment to eventually bring both cognitive and affective into true harmony.<br /><br />So one's ever more refined sense of something remaining "not quite right" with development, can only be addressed at the appropriate time, thus enabling eventual harmony to be achieved. And this balance - which always is of an approximate nature - is dictated by the volitional mode!<br /><br /><br />In this dynamic account of development, it is necessary to distinguish as between the default, diminished and enhanced experience of each stage.<br /><br />At the very beginning of development the infant literally moves all over the spectrum in a confused manner (where neither the differentiation of distinct stages nor their integration with each other has yet taken place).<br /><br />So from a discrete (differentiated) perspective, earliest development relates to the unfolding of the stages of Band 1. This is what I then refer to as the default understanding of these stages.<br /><br />However, because all stages are necessarily to a degree still related to all other stages, this does enable a diminished perspective on "higher" stages.<br /><br />Then when for example one moves to differentiation of the stages of Band 2, this then becomes the default understanding of those stages.<br /><br />However one is now enabled to form both a diminished perspective on - still - "higher" stages, while, in revisiting the earlier stages from Band 2, form an enhanced view of their features.<br /><br />So therefore we do not have just one experience i.e. default, of each level (of each band) on the spectrum but in fact a whole series of continually changing perspectives on other levels of both a diminished and enhanced nature.<br /><br />Thus whereas, from a differentiated perspective, the earliest level of Band 1 is the first to unfold (and of the most primitive nature) from the opposite integral perspective, this likewise remains the last level to be properly integrated in terms of overall development.<br /><br />So from this latter perspective (in both top-down and bottom-up terms) successful integration of this level (with all other levels) now represents the "highest" goal in development.<br /><br />However because of the standard linear asymmetrical approach, far too-much attention is typically placed on the differentiated aspect of development (in the unfolding of distinct stage structures) which then runs directly counter to what is experientally required to achieve true integration of all stages.<br /><br />So in the default understanding of the levels of Band 3, while I was aware of the need for bi-directional integration of "higher" levels with "lower" (and "lower" with "higher"), in practice attempted integration was still predominantly of the top down variety (where primitive instincts of the earliest levels were unwittingly repressed).<br /><br /><br />This was even evident in my holistic mathematical understanding of the nature of number at the time.<br /><br />So with respect to the "higher" stages of Band 3, there are in fact two directions.<br /><br />One represents the "ascent", where one sees these stages as progressively transcending, in an intuitive contemplative manner, the "middle" dualistic stages of Band 2.<br /><br />Thus my very understanding of the holistic nature of number as representing varying "higher" dimensions of experience fitted in very well with this notion of the "ascent".<br />In particular I associated the holistic notion of "2" with Level 1, the holistic notion of "4" with Level 2 and the holistic notion of "8" with Level 3.<br /><br />And then the larger numbers were associated with even more refined contemplative development beyond these levels.<br /><br />However I was only to later realise that this new holistic mathematical understanding needed to be properly grounded in the analytic levels of Band 2.<br /><br />So in my account Band 5 (Level 1, Level 2 and Level 3) relates to the corresponding "descent" back to the middle levels, which then opened up marvellous new revelations regarding the ordinal nature of number.<br /><br />The other direction at Band 2 is the subterranean "descent" from the middle levels of Band2, towards the earliest primitive understanding of the levels of Band 1.<br /><br />However because my psychological understanding of these levels still remained somewhat repressed (through the predominant influence of "higher" cognitive development) my corresponding holistic mathematical understanding did not readily emerge at this time.<br /><br />I had indeed formed the conviction that somehow the prime numbers (given a new holistic interpretation) would be deeply relevant in terms of the structure of these levels. However I was not yet able to precisely see what such holistic understanding entailed. <br /><br />I was also aware that my "circular" understanding of number as dimension was also relevant, in the sense that the 3 "lower" levels represented in complementary terms the confused understanding of the corresponding 3 "higher" levels of Band 3.<br /><br />So therefore the earliest (most primitive) level represented the confusion of all 8 polar directions (i.e. form with emptiness, wholes with parts and external with internal).<br /><br />With the next level, form could be distinguished from emptiness, but the two other confusions still in large measure remained.<br /><br />Then finally, as the "lower" levels approached the middle level, only the remaining confusion of external with internal directions remained.<br /><br />However, I knew that a deeper holistic mathematical knowledge of the nature of the primes and how this ultimately could be grounded in accepted analytic understanding, was necessary.<br /><br />Indeed this required nothing less than a radical new interpretation of the true dynamic nature of the number system, which was likewise to unfold during Band 5. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0tag:blogger.com,1999:blog-8398449309649732039.post-42432752589382350672017-05-02T06:40:00.003-07:002017-07-06T02:04:14.964-07:00Number and Development (6)I have drawn attention to the great holistic mathematical significance of 2, 4, and 8 (representing dimensional nos.) in terms of the developmental task of integration (with complementary applications in physical and psychological terms).<br /><br />Once again, 2 directly relates to horizontal bi-directional integration (within a given level). <br />4, relates to horizontal and vertical bi-directional integration (within and between levels), where both aspects of integration are still pursued in a - relatively - separate manner.<br />8, then relates to diagonal bi-directional integration simultaneously within and between levels.<br /><br />This is required to fully integrate the "higher" levels of Band 3 with the complementary "lower" levels of Band 1 (within and between levels).<br /><br />However in dynamic experiential terms, the task of integration cannot be properly considered in the absence of the corresponding requirement for sufficient differentiation of the major levels of each band.<br /><br />So we can have two extremes:<br /><br />1. Where differentiation - especially with respect to the specialised development of the linear levels of Band 2 - takes precedence. Here, limited exposure to the "higher" levels of Band 3 is likely to take place with integration significantly reduced and geared to the successful further spread of differentiated type experience, in what typically constitutes the "active" life..<br /><br />2. Where integration - especially with respect to the specialised "higher" levels of Band 3 - takes precedence. However this often leads to the significant by-passing of the conventional everyday experience of the levels of Band 2. In former times this typically constituted the "passive" i.e. contemplative life, where spiritual aspirants left worldly concerns behind to live in confined monastic communities.<br /><br />However the fullest development requires that an equal balance can be given to both differentiation and integration in a mixed approach. Then, when successfully attained, one becomes become fully grounded in the differentiated activities of everyday life, while maintaining in the midst of such activity, a deep contemplative vision (which then serves to properly integrate all things).<br /><br />Thus the greatest exponents of such mixed development - who often in the past were pioneering religious reformers - were thereby enabled to live amazingly productive lives, full of creative endeavour that served to dramatically transform the world in which they lived. <br /><br />However this "full life" combining ever growing differentiation and integration always remains an ideal to which one can only ever roughly approximate.<br /><br />However the lessons to be learnt from each - somewhat limited - attempt to achieve this exalted goal can possibly open up important new features with respect to development that have not been adequately recognised previously.<br /><br /><br />Again I have placed special emphasis on the holistic mathematical relevance of 2, 4 and 8 - which literally represents varying dimensions - for development. Though potentially all numbers possess a distinctive relevance, 2, 4 and 8 (especially 4) still maintain a special importance which can be illustrated as follows.<br /><br />In quantitative terms, every number can be represented on the complex plane. This contains a real axis where a number can be positive or negative and an imaginary axis (where likewise a number can be positive or negative).<br /><br />So all numbers can be shown to represent unique configurations with respect to these 4 aspects.<br /><br />Likewise in qualitative (i.e. holistic) mathematical terms, all numbers can be shown to represent unique configurations with respect to these same four aspects (i.e. positive, negative, real and imaginary).<br /><br />And the holistic mathematical appreciation of 4 opens up this latter realisation.<br /><br />Therefore experience throughout all development entails unique configurations of wholes and parts with respect to their external and internal aspects.<br /><br />Thus when one has mastered the 4-dimensional approach, one has the blueprint as it were to successfully map all experience.<br /><br />Here, ever greater development at the "higher" levels then necessarily entails more refined ways in which one can experience the relationship between wholes and parts with respect to their internal and external directions.<br /><br /><br />However, I gradually came aware of a crucial distinction with respect to even numbered and odd numbered dimensions respectively.<br /><br />As I have stated on many occasions, I was at the time very much influenced by the writings of St. John of the Cross.<br /><br />However I slowly came to the view that he laid too much emphasis on "passive" rather than "active" nights.<br /><br />Now the "passive" nights can be directly related to the deep task of unconscious spiritual integration.<br /><br />However for healthy development - even during the intense purgation of the "dark nights"- passive must be balanced to a degree by active type development (where differentiated activity is necessarily involved).<br /><br />So I saw the healthy unfolding of "higher" development as follows.<br /><br />Each relatively more differentiated stage should then be followed by a subsequent stage where the primary emphasis would be on integration.<br /><br />Thus in holistic mathematical terms, each odd numbered would be followed by an even numbered stage.<br />There is in fact a crucial mathematical distinction as between the even and the odd roots of 1. With the even roots a complementary relationship always exists as between the roots so that every positive root can be matched by a negative counterpart.<br />However with odd numbered roots this is not the case. One of these roots however will always be 1, with the other roots existing as complex conjugates of each other.<br /><br />Therefore, whereas the even roots can always be associated with integration (in the complementarity of opposites), the odd roots can be associated with new forms of differentiation (where linear understanding maintains a degree of independence). <br /><br />So the linear (1-dimensional) stage of highly differentiated dualistic activity is then followed by the 2-dimensional stage where nondual integration of external and internal polarities takes place.<br /><br />However this should then be followed by the 3-dimensional stage, which represents a new "higher" attempt at engaging in differentiated activity, from the perspective of the spiritual integration that has already taken place.<br /><br />However this leads inevitably to a new clash with respect to dual and nondual which then requires an even "higher" 4-dimensional stage to resolve.<br /><br />So each new "higher" stage of differentiation (with an odd number) gives way to a corresponding new "higher"stage of integration (with an even number).<br /><br /><br />However, if development of the middle levels of Band 2 has not been especially prolonged (due to early intense exposure to contemplative experience), a limited basis may thereby exist for differentiation at the subsequent "higher" levels. So development will then be unduly intensive (in an unconscious manner) and insufficiently extensive (in a conscious fashion).<br /><br />Furthermore, the continual need to bi-directionally integrate the 3 levels of Band 3 with the 3 corresponding levels of Band 1, will mean that experience becomes so dynamically interactive (in both directions) that one will then find it extremely difficult to take rest as it were in the more stable phenomena of Band 2.<br /><br />So the levels of Band 2 are likely therefore to become significant by-passed at this time, with their successful integration with both Bands 1 and Band 3 requiring on-going development through further stages of understanding. Peter Collinshttp://www.blogger.com/profile/03702540376694818466noreply@blogger.com0