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INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
Publication dates of essays (month/year) can be found under "Essays".
Peter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.
An Integral Mathematical Stage Model of Perspectives
Part 3: Radial Levels
We continue now with the integral mathematical interpretation of perspectives into the radial levels. Though such an interpretation can by no means replace the need for authentic experience of these more advanced levels it does however open up the potential for a radical new type of scientific appreciation of development that is truly unique to this approach.
We now are moving on to the radial levels, which offers the most comprehensive type of mature development, where dynamic balance is successfully maintained as between both the differentiated and integral aspects of experience.
It might help to place the radial levels in context by briefly indicating again the nature of preceding development.
The lower levels entail the (dualistic) movement away from totally confused (circular) experience of integration towards gradual (linear) differentiation with respect to the three fundamental sets of polarities. Separation of the diagonal polarities (form and emptiness) at L3 relates to differentiation of the bodyself; separation of vertical (whole and part) at L2 relates to differentiation of the emotional self; finally separation of horizontal (exterior and interior) at L1 relates to differentiation of the mental self.
The middle levels then entail the specialised (linear) differentiation of these polarities. At L0, this is mainly in terms of local concrete understanding; then at L0, H0 it is with respect to more formal abstract appreciation; finally at H0 both of these aspects are creatively combined and imbued with spiritual intuitive awareness. However this specialisation of (linear) differentiated is associated with a significant reduction in terms of (circular) integral understanding.
So the higher levels gradually address this imbalance with respect to the three sets of polarities in the movement towards (nondual) spiritual integration.
However the middle levels - where differentiated understanding achieves its specialised development - can be largely bypassed in this (integral) spiritual process.
So further specialised development with respect to nondual awareness now requires that "higher" and "lower" (and "lower" and "higher") stages be themselves incorporated in mature fashion with the middle levels (entailing therefore both the balanced specialised differentiation and integration of all stages).
So we now look at these radial stages in more depth.
The previous stage (H3) culminates with approximation to the most purely contemplative type of spiritual experience where immanent and transcendent aspects are united. The transcendent aspect is attained through a psychological death (of possessive attachment to all phenomena) culminating in an entombment (or deep sleep of the unconscious). The corresponding immanent aspect is attained through a complementary deep psychological process that journeys through the womb to the very point where life begins. This culminates in an "enwombment" or merging with the original universal mother (representing the potential for rebirth of all creation). So simultaneously here we have in the marriage of the (masculine) transcendent and (feminine) immanent principles, the identity of both death and life (in spiritual terms). And indeed the experience of mystical ecstasy is most accurately expressed as the simultaneous death and rebirth in a pure life of the Spirit, which is continually renewed in the present moment. However such peak moments of ecstasy cannot be permanently sustained. So further development now entails the gradual rebirth of the phenomenal world with a view to its transformation in Spirit. 
Now it might seem puzzling that such radial development (entailing both dual and nondual aspects) represents an advance on pure (nondual) contemplative awareness. So it is necessary therefore to explain this important point.
We have seen that the key barrier to the pure life of the Spirit is possessive attachment to (rigid) phenomena. It has long been my belief that the mystical personality proves - even in early life - unusually sensitive to such attachment. Therefore the desire to overcome this psychological "allergy" can drive a select group to extraordinary lengths in an attempt to achieve a place of true spiritual rest. So therefore an unusual degree of detachment from the world may prove for some a necessary step in their spiritual training.
However when this task has been sufficiently mastered, healthy psychological dynamics entail that one should attempt to re-engage with the world of phenomena (in a free unattached manner). So paradoxically the truest measure of pure detachment is an ability to intensely engage in phenomenal activity (while maintaining full freedom of Spirit). So it is this immersion in the world of form with a view to its radical transformation (from a pure contemplative perspective) that defines the radial levels.
Having said this, I would consider that the most advanced expressions of radial development (Radial 3) are extremely rare.
Indeed I would broadly define three types of radial involvement.
The first - which is the most common - is characterised by considerable development of the differentiated middle, without however the same degree of contemplative attainment at the "higher" levels. This leads predominantly to an active type of mysticism where spiritual motivation is largely expressed through practical involvement in the world. Though going considerably beyond vision-logic awareness, in some respects this would resemble an enhanced form of centaur type integration.
The second - which is much less common - is characterised by considerable development of the "higher" levels leading to a marked degree of contemplative awareness. However because sufficient specialisation of the middle stages has not equally taken place, such a type may subsequently experience difficulties in terms of worldly adjustment and remain at the Radial 1 (or perhaps Radial 2) stage. Again some personalities are so sensitive to psychological attachment - in the same manner that others are to physical infection - that even with considerable contemplative attainment they quickly prove "allergic" when exposed to active involvement. Therefore though able to contribute in significant ways they tend to remain somewhat "passive" with respect to the world.
The third - which is least common - is characterised by remarkable dynamic balance as between dualistic development of the middle and nondual attainment of the "higher" levels. Individuals of this type - who often are extremely gifted - can then become intensely committed to the world while maintaining a deep spiritual peace (due to an extraordinary degree of sustained contemplative awareness). Such spiritual heroes e.g. the great saints and spiritual adepts of the mystical traditions would best exemplify Radial 3 type development.
However it is important not to take an over-idealised view. Even the greatest exponents will inevitably suffer from human frailties and be limited in many ways.
Also because the stages of development are necessarily continuous to a degree with each other, everyone has some access to these levels. In this sense we all participate in the radial levels though for most of us - largely because of insufficient development at the "higher" levels - this participation remains somewhat attenuated. Even so, in our more enhanced moments we can perhaps attain a better insight into their true nature, thereby inspiring us with the desire for more complete fulfilment.
Radial 1 (R1) - Spiritual Reawakening
As we have seen the "lowest" stage (L3) relates to the differentiation of the bodyself (where a stable bodily form is separated from emptiness). Likewise the "highest" stage (H3) relates to the corresponding mature integration of the bodyself (where the psychophysical interactions of the body/mind are properly integrated in nondual manner with Spirit). Now Radial 1 combines both mature differentiation with mature integration in the emergence of a new transformed bodyself.
This radial bodyself comes as a remarkable revelation. During H3, one remains in a deep sleep of the dark unconscious (largely dead of recognition of the body or surroundings). Associated with this is a suspension of one's natural breathing pattern. So in this contemplative deep sleep state - quite literally - one scarcely breathes (which symbolises well the death of the old self). However now with the first dawning of new light at R1, there is an amazing reawakening where one's physical body is experienced as inseparable from the rest of phenomenal creation. Therefore in the first consciousness of light, one is likewise awakened to the fact that one is now gently breathing in and breathing out the very life of the cosmos (with which one's body is experientially merged). So we have here the gradual reawakening of (conscious) life in the rebirth of the physical world (with which one now stably identifies in direct bodily form).
This discovery of the (personal) radial body would represent the immanent aspect of the stage. However associated with this is a more (impersonal) transcendent awakening of form, which can be more directly identified with the spiritually inspired intellectual understanding of the level.
Thus the Radial 1 level - as it is linked so closely with the deep emergence of Spirit from the unconscious - is best suited for original universal appreciation of the fundamental structures of all development. And mathematics (in its enhanced expressions) provides the most scientific tool possible for the expression of such structures. The radial levels - by definition - relate to balanced incorporation of both the differentiated and integral aspects of experience. Likewise the mathematical approach that I am outlining here, is designed to express in a universal manner the precise binary configuration for all stages (through linear and circular interpretation) of this relationship as between integration and differentiation.
In itself such a radial mathematical appreciation - though arising directly from spiritual contemplation - is of an abstract (impersonal) transcendent form. However in the dynamics of experience it can only properly flourish if it is balanced by complementary appreciation of an (immediate) universal bodyself that is experienced as personal and immanent. So at this radial level we still have the balancing of extremes, however now in a manner where stable form can progressively emerge.
However differentiation not only occurs at the universal level of form (lying very close to the pure contemplative vision) but also at the local level (where it is closer to linear reason). This is likewise associated with the realisation of a limited bodyself. So we have therefore the refined interplay as between an ego identity expressed through the form of a contingent body and a universal spiritual identity mediated though a cosmic body. And the enhanced awareness of the inevitable conflict as between the ego and a universal sense of spiritual identity leads to renewed existential suffering. However this is now accepted with spiritual peace as it reflects the essence of true authentic existence. However differentiated activity still operates in a very limited manner for a simple reason. When it is dark one's vision is restricted to what is immediately present. In like manner at this stage, where the unconscious is still dominant in experience, one can only deal effectively with activities and relationships immediately present in one's local environment. Though very restrictive, in some respects it also helps to focus attention only on tasks that are deemed essential. In this way spiritual energy is not unduly dissipated. 
However it can create a problem in that a considerable gap can remain as between a contemplative awareness (that is truly universal) and practical activity that is somewhat limited in scope. In other words one can experience difficulty in finding an adequate phenomenal expression for the depths of creative spiritual energy already accumulated.
In holistic mathematical terms, R1 represents the mature circular integration of horizontal, vertical and diagonal polarities, with likewise mature linear differentiation with respect to the diagonal polarities. However insufficient differentiation in terms of both vertical and horizontal polarities will have yet taken place.
Perspectives of Stage
Though one has by now attained a mature overall (integral) perspective through the permanent attainment of a pure level of nondual spiritual awareness, only limited development of distinct (differentiated) perspectives take place. Also insofar as these unfold they still are mainly of a 1st order primary nature (directly prompted by Spirit). As we have seen these mainly relate to radial notions of the bodyself and intellect (with respect to universal and local expressions). The key consideration now is to preserve proper authenticity with respect to the unfolding of new distinct perspectives (so that they can serve as an adequate phenomenal expression of Spirit). In this way rigidity due to secondary attachment is thereby avoided.
In terms of mathematical number types, we now gradually move from holistic appreciation (suited for integral interpretation of development) to radial appreciation (suited for both differentiated and integral interpretation). So once again at this stage the key number type relates to the binary digits which find a new enhanced appreciation in terms of both their holistic and analytic applications. Thus from the holistic side there is continued growing awareness of how all transformation processes can most scientifically be understood in terms of the interaction of both linear (1) and circular (0) understanding. However equally from the analytic standpoint there is a renewed appreciation of how all information processes can potentially be likewise encoded in terms of the interaction of both 1 and 0. And because of the increased awareness of both aspects, creative linkages can be more freely made. 3 Thus in terms of full scientific appreciation we really have a double binary system (with analytic and holistic aspects) that can encode both information and transformation processes. Thus throughout R1, one remains directly attuned to the basic binary interface relating overall (integral) spiritual perspective to distinct (differentiated) phenomenal perspectives.
Radial 2 (R2) - Mature Vision
Once again earlier stages can be helpful in clarifying the nature of this level. At L2 we had the separation of vertical polarities (whole and part) associated with the differentiation of the emotional self (i.e. where the ego is placed in a stable dimensional context). Then at H2 we had the corresponding reconciliation of vertical polarities associated with the mature integration of the emotional self. Now at R2 from this standpoint of the mature integration already attained, we also have the gradual mature separation of vertical polarities enabling both the combined radial integration and differentiation of the emotional self.
Because the "higher" stages entail such intense dynamic negation leading to substantial "death" of the old self (defined by a somewhat rigid ego), this in turn can lead to considerable death of affective response. So one indeed may appear uncaring when in truth one now cares so deeply that one's former natural emotions are no longer adequate to convey such feeling.
Therefore just as the return of spiritual light to the bodyself leads to a remarkable new revelation (at R1), now the gradual return of light to one's emotional self, likewise leads to a remarkable new revelation in genuine deep compassion that is universally felt for all created beings. However once again one's perspective gradually changes to the realisation of how in a local setting the contingent ego can remain somewhat limited in expressing such universal feelings of identification. Indeed I often feel a lack of realism permeates accounts of the advanced spiritual stages where exponents are viewed as superhuman (as if not subject to limitations like the rest of us). In fact the opposite is really the truth. Such spiritual people are invariably more keenly aware of their human limitations and failings. However because they can now honestly and openly accept these limitations through deep immersion in Spirit, they thereby become superconductors of that same Spirit which is expressed through them in an enhanced manner.
The counteracting development on the intellectual side is a significant growth in wisdom. The very nature of wisdom is that the cognitive function remains closely related to Spirit. This therefore gives one the power to discern in any given situation how to use reason in an appropriate manner.
The intellectual expression of R2 would relate to the significant development of "radial" disciplines. This would enable therefore both detailed analytic investigation and holistic integral understanding to be combined in a highly creative yet productive manner.
Now I would see radial mathematics as the most fundamental of all disciplines. Therefore from an analytic perspective, it would remain as "queen of the sciences" i.e. an indispensable tool of conventional scientific investigation. However from a holistic integral perspective it would be viewed - perhaps in complementary manner - as "king of the sciences" i.e. as the indispensable tool of deeper philosophical investigation of all kinds. (So I have been illustrating this integral use of mathematics with respect to the basic structures of all stages of development).
In other words the basic structure of all "radial" disciplines would be provided by "radial" mathematics. So let me briefly illustrate with respect to "radial" physics. At the moment physics is largely identified with its merely conventional (analytic) expression. Therefore for example in the search for a TOE in string theory and more recently M-theory, great theoretical advances have been made (largely through the use of mathematical techniques). However remarkably little development in the philosophical understanding of the nature of these findings has emerged. This is due to the fact that the holistic (integral) dimension of science has been all but neglected.
However in an appropriate "radial" physics of strings, all mathematical relationships would have dual linear and circular interpretations (i.e. both analytic and holistic). Likewise then when applied to the physical world, all string relationships would have both coherent analytic and holistic (i.e. philosophical) interpretations. Furthermore because these two aspects are complementary they would mutually assist each other. Therefore analytic developments would suggest new integral philosophical interpretations; equally developments with respect to holistic integral interpretation would suggest new creative lines of analytic investigation.
As I have said before, though I can perhaps suggest in a general way the nature of R2 scientific understanding, I would not consider my overall approach as belonging to this level. 4 Once again I would classify my ideas as mainly representative of a (preliminary) R1 scientific approach. Therefore I can only offer but limited insight into the true nature of this science of a more advanced level.
However I know that when its full development eventually finds expression in our society, it will be much more wonderful, more versatile, more life enhancing than I could ever have imagined.
I also refer to this stage above as "mature vision" the reason for which I will now explain. Corresponding to the emergence of new structures at each stage is a new state representing a special type of spiritual illumination (which characterises the stage). So I would refer to the state especially associated with this level as the mature dream state.
Even in conventional terms the dream state is very important. For example we will hear many athletes at the Olympic Games who have just won a gold medal speak of "a dream come true." In other words actual (conscious) achievement is invariably based on the potential (unconscious) desire for such achievement which we typically refer to "as having a dream". Now this is equally true of this stage though in a more refined spiritual manner. We had seen that at R1 a considerable gap divides universal type understanding from practical attainment in local terms. A gap still exist at R2 (though to a lesser degree). However now because of the presence of the mature dream state, one can form a realistic way of expressing a universal spiritual vision through the limited nature of the contingent self (with its various strengths and talents but also significant limitations and frailties).
So we often see for example in the great spiritual figures during the radial levels, the crystallisation of the need for commitment to some special project of work (which later best defines their contribution to society). The gradual formulation of such a vision also acts as a considerable catalyst in accelerating the further human development that will be required for its fulfilment in the final stage.
From a holistic mathematical perspective, R2 is defined by mature circular integration with respect to horizontal, vertical and diagonal polarities, with corresponding mature differentiation with respect to diagonal and vertical. However restrictions as to the full expression of differentiated activity in terms of the horizontal polarities still remains.
Perspectives of Stage
Again the key task of the radial levels in terms of perspectives is the balanced reconciliation of overall integral perspective (relating ultimately to nondual spiritual awareness) with increasingly complex distinct perspectives (that thereby enable greater active participation in the world). Considerable progress is made at this stage is terms of the strengthening of distinct perspectives (with indeed the development of flexible higher order perspectives now emerging). So personal distinct perspectives (relating directly to the affective function) undergo further development. Likewise impersonal distinct perspectives (relating directly to the cognitive function) also undergo such development. Also highly refined potential perspectives (largely reflecting states) now can interact freely with actual perspectives (reflecting structures). Then because of growing interaction as between the collective (universal) and individual (local) aspects of understanding, higher order distinct perspectives can be increasingly restored (without undue rigid attachment). However the major problem remaining is that personal and impersonal type perspectives remain to a degree somewhat compartmentalised from each other. This in turn reflects a lack of sufficient (radial) differentiation in terms of the remaining horizontal polarities.
From a number standpoint the holistic appreciation of both binary and prime numbers and their complementary "higher" expressions continues to develop. Such understanding is of special assistance in further clarifying the dynamic structure of perspectives associated with L3 and L2 together with H3 and H2 respectively.
However now associated with this holistic appreciation is a renewed analytic appreciation of the same notions (relating to the 3 middle levels).
It would be my contention that the importance of the main number types - even in analytic terms - is not yet properly recognised. For example we have already seen the role of the binary digits as a (potential) means of encoding all information. However very important - and mainly yet undiscovered uses - exist for example for prime numbers.
Indeed it is fascinating - given the role of the binary digits in computers - that due to computer technology, an increasingly important new use for prime numbers has been found in recent years (with respect to data encryption for security purposes). However I would believe that further important uses will be discovered. For example the analytic interpretation of prime numbers could find a special role in the future development of string theory. (This follows from the philosophical importance of the holistic notion of prime numbers at the level of string reality).
Thus the creative marriage of holistic (associated with "higher" and "lower" levels) and analytic mathematical notions (related to the middle stages) then serves to highlight the structures of the radial levels (through which these other stages are incorporated).
Radial 3 (R3) - Creative Transformation
We now reach the final stage, which offers the most enhanced expression of totally committed experience.
Again we will initially approach this stage in the same manner as the two previous levels. Remember how at L1 the separation of horizontal polarities took place with the corresponding differentiation of the mental self! Then at the complementary "higher" level H1, the integration of this mental self took place.
So just like bodyself and emotional self at the previous levels, we now have the mature differentiation of the radial mental self. Now it may seem strange to be stating this given the attention I have placed on intellectual development at the two earlier levels. However this differentiation leads to a new distinct experience of cognitive activity that is charged with a living force which had not previously existed. Put another way, this relates to a more immanent expression of the intellect (where it resonates to a degree with other aspects of personality i.e. affective and volitional).
From another perspective we have reached the stage where an inevitable interdependence applies to all aspects of personality. In this way intellectual activity is now longer impersonal as it is significantly imbued with affective (personal) qualities. Likewise emotional response is no longer merely personal as it is properly disciplined through cognitive (impersonal) factors. At the centre of personality the volitional aspect of will remains primary as the means of co-ordinating both cognitive and affective aspects in accordance with the designs of Spirit.
This remaining differentiation of the mental self brings about a marked change. Because one is now increasingly able to use the cognitive function in a practical yet creative nondual manner, this greatly enhances the capacity for involvement in an increasingly wide range of activities. So the last stage often manifests itself through intense levels of committed involvement that from a merely conventional perspective can seem extraordinary.
For example when we look at some of the great spiritual figures of the Christian tradition e.g. St. Paul, St. Augustine, St. Francis Assisi, St. Catherine of Sienna, St Teresa of Avila, St. Ignatius of Loyola and in more contemporary times Mother Teresa and I believe Pope John Paul, we are dealing with truly exceptional individuals who have combined deep spiritual commitment with intense transforming worldly activity to a very marked - and often heroic - degree.
Though personalities here differ in many ways so that they do not completely fit the ideal blueprint for Radial 3, yet I believe they can all be included. Indeed one of the great mysteries of true spiritual development is how, what may seem as severe limitations in terms of talents at the natural level, can somehow serve through authentic personality transformation to actually enhance the true mediation of Spirit. 5
However I did specifically approach these levels from the standpoint of one initially seeking - through spiritual attainment - an enhanced overall intellectual appreciation in scientific terms of the nature of development. However it has to be said that there is very little recorded evidence of such intellectual development of the radial levels in our culture.
It is possible that evidence does exist of which I am unaware. It is also perhaps possible that a certain select group of individuals operating at the radial levels have indeed dealt deeply with the intellectual issues but that their efforts have not been recognised (due to a clash with conventional norms). However the most likely explanation is that mystical development in the past - while not ignoring the intellectual aspect - did not specifically focus on its advanced expressions.
However we are living in very different times where we are now witnessing the rapid decline of traditional religious institutions in many countries of the West. Therefore it seems to me that a new belief system is urgently required which can successfully marry the best of authentic spirituality with more comprehensive notions of scientific validity (that properly incorporate an integral interpretation of reality). So perhaps the "spiritual missionary" of the near future, operating at this enhanced Radial 3 level of development, will not alone have attained to a pure level of contemplative awareness - together with deep intellectual insight into the nature of this comprehensive science - but will likewise possess the capacity to actively "convert" the world to the "new" belief system.
This does not of course replace the need for other much needed styles of communication. However it seems to me that this extremely important avenue (i.e. integral scientific communication) has been all but neglected.
However though we may approach spiritual growth from a broad intellectual perspective, ultimately its proper attainment requires fulfilment with respect to all significant aspects of development. Therefore, though in direct terms the major contribution of the enlightened intellect at R3 may be in terms of true wisdom, this cannot really be separated from the affective goal of true compassion. And both of these aspects intimately depend on what is most primary i.e. the volitional goal of true love. So in the end the most heightened expression of Radial 3 development is a burning love for all humanity that inevitably entails both compassion and wisdom.
In mathematical terms, R3 entails mature circular integration with respect to all fundamental polarities (i.e. horizontal, vertical and diagonal) with corresponding mature linear differentiation in terms of the same polarities.
Perspectives of Stage
Not surprisingly we have here the purest expression of an overall (integral) perspective that can be effectively combined in true freedom with incredibly multifaceted configurations of both primary and higher-order distinct perspectives. So personal and impersonal perspectives of various orders can be differentiated in a distinct manner and then satisfactorily integrated in overall fashion as (horizontally) complementary. This in turn entails significant balance being maintained as between the revealed and hidden aspects of perspectives. Likewise detailed (individual) and universal (collective) perspectives can be again differentiated as distinct and then integrated in an overall manner as (vertically) complementary. This entails significant balance being maintained as between actual and potential perspectives. Finally one can now readily switch as between the transcendent and the immanent formulation of distinct perspectives while once more achieving successful integration in (diagonal) complementary terms.
From a number standpoint, we now have the combination of both the holistic and analytic interpretation of the various number types providing the fundamental basis for both the qualitative and quantitative structuring of these perspectives. For example, we have now the inclusion in understanding of both combined holistic and analytic appreciation of natural numbers. Though we take it for granted, the quantitative notion of natural numbers is exceptionally important with respect to everyday living. For example personal information such as addresses and telephone numbers, commercial activity, scientific measurements and classification systems of all kinds make extensive use of natural numbers, which are indeed natural to living. Natural numbers always retain their wholeness. So if a cake is divided into five slices each slice in natural terms is still a whole unit. It is only the analytic interpretation of the rational middle levels that interprets phenomena as mere parts thus breeding fragmentation in experience. However now by properly merging analytic and holistic interpretation, reason can be used in an unreduced fashion avoiding such fragmentation. So in the end with Radial 3 understanding one has the capacity to be highly rational with respect to various activities subdividing them into many parts, without suffering any loss however with respect to their integral wholeness. And this is due to achieving the perfect marriage as between the line (conscious) and circle (unconscious) with respect to both their analytic and holistic expressions.
Other Stages of Development
We have dealt with the dynamic configuration of the 12 major levels from an integral mathematical perspective (though in truth the final three levels require the more refined radial mathematical approach). However it is important not to interpret such levels in an idealised manner. So development must be approached in a dynamic relative fashion to which actual experience approximates in varying degrees. Even for the great exponents of Radial 3 development (which represents the most enhanced available), many limitations - perhaps major - will remain. Because abilities and gifts are not equally provided in personality, certain capacities will inevitably be better developed. In other words some perspectives will tend to be much more pronounced than others. What is truly primary however is the degree of spiritual desire, which if authentic can lead to a remarkably creative amalgam of personality traits where even significant limitations play an integral role.
Also though the presentation of levels necessarily follows a somewhat linear path, it is important to bear in mind the circular aspect where all stages are in fact continuous with each other. Thus, though it may well be true that at a certain stage, one's experience is predominantly identified with a particular level (or closely related grouping of levels), in the dynamics of experience all levels remain present (at least as potential for development). Also in peak moments one can of course enjoy a very meaningful temporary experience of a much more advanced level than would be customary. 6
Therefore, though significant development of the more advanced levels generally remains somewhat attenuated in normal experience, we do not remain complete strangers to these stages. So when we hear of the "superstars" of spiritual development who have realised both their human and divine potential to an extraordinary degree there is perhaps a resonance within us that somehow we too can share in this realisation as the eternal heritage of who we truly are.
However to indicate how a more comprehensive integral mathematical map can be constructed, I will say here a little about the other stages.
Transitions between Levels
After the 12 major levels, I define transitions as between levels. The transition levels bridge each of the main stages and relate to a major psychological transformation, which then enables development of the next stage to take place.
Basically with lower levels, transitions are defined in terms of various types of differentiation, whereas with higher, their integral counterparts are more relevant. 
For example the transition from the first level L3 (archaic) to L2 (magic) is marked by the separation of diagonal polarities associated with the corresponding psychological differentiation of the bodyself (through which it is thereby posited in a stable manner). So the further emotional development of the next stage (which leads to another significant differentiation) depends on the success of this first crucial transformation.
Then with respect to the higher levels, the important transition from H0 (vision-logic) to H1 is associated with a corresponding important integration in terms of the horizontal polarities (associated with corresponding integration of the mental self). This relates in holistic mathematical terms to dynamic negation with respect to rigid (i.e. merely posited) differentiation of such polarities. 8 So the spiritually based bi-directional activity of H1 therefore depends on the success of the initial integral process.
I then define three sub-levels (within each level) though at the "lowest" and "highest" levels, these cannot be fully distinguished from each other.
Again the structure of the sub-levels is mathematically based in holistic terms on the same fundamental polarities as used for configuring the levels.
From one perspective this represents another layer of the distinction as between whole and part (and part and whole). Thus from a psychological aspect, we need to distinguish as between (specific) perceptions and (general) concepts; from the complementary physical perspective we need to distinguish as between (empirical) objects and (universal) dimensions. Now properly understood, these two aspects are qualitatively distinct from each other. So the means for example by which perceptions are formed in experience relates directly to the (linear) conscious. However the means by which concepts are dynamically formed - in this context - relates directly to the (circular) unconscious. So in dynamic terms the two activities are linear and circular (or equally "real" and "imaginary") with respect to each other. 
Now the first sub-level of a level (which I refer to as S1) relates largely to the more specific generation of perceptions (as object phenomena) within - as yet - largely undeveloped dimensions (associated with the overall level). This sub-level level is defined by the horizontal polarities (interior and exterior) operating heterarchically within this given sub-level.
The second sub-level (S2) relates to the more general development of concepts (as a new dimensional framework for perceptions generated). However as these are whole (with respect to earlier parts) this bears a vertical relationship to the earlier sub-level. Finally the third sub-level (S3) relates to the combined two-way diagonal interaction of both S1 and S2 (where perceptions are related to concepts and concepts to perceptions in terms of both exterior and interior aspects of understanding).
Now with respect to these three sub-levels, we can demonstrate something truly fundamental about development through holistic interpretation of the inherent nature of the number system. This offers a radically new perspective on development that I believe is utterly unique to this kind of mathematical understanding.
So in number terms, the horizontal aspect relates to number as quantity (as defined with respect to a dimension).
The vertical aspect relates to the number as quality i.e. the power or dimension (to which the number quantity is defined). 10
Finally the diagonal relates to the combination of both aspects i.e. where a number is expressed with respect to a power (greater than 1).
So applying these three uses of number to development as a means of holistic interpretation of the corresponding three sub-levels (of each level), we can show how a remarkable transformation in number terms (at the third diagonal stage) is associated with a corresponding transformation to a new level of understanding.
The interpretation of how initial binary appreciation of L3 (archaic) moves eventually to prime understanding - that characterises in holistic terms the nature of L2 - is the most difficult to neatly explain. (I have however dealt with this process in Note 22 to "An Integral Mathematical Stage Map of Perspectives, Part One").
So what happens basically is this! At S1 (of L3) the holistic binary digits (representing form and emptiness) cannot be properly separated with respect to perceptions; at S2 the same problem arises with respect to concepts (though in truth these two-sub-levels very much overlap at this stage). Then by S3 sufficient phenomenal interaction occurs to enable initial diagonal duality (in terms of both perceptions and concepts that are still closely entangled). Such duality represents the first prime number (in holistic terms). Then in the manner I demonstrated in that footnote, other prime numbers can likewise be generated which leads to the transformation in the level of understanding to L2 (defined by holistic prime notions). So once again development at L2 (magic) evolves through S1 (where primitive perceptions are formed) and then S2 (where primitive concepts emerge). Then at the diagonal sub-level of S3, a new transformation takes place leading to natural understanding (which defines L1). When a prime number is raised to the power of another prime number, this leads to a transformation to a natural number (that is not prime). Likewise in holistic terms, when prime (i.e. primitive) perceptions properly interact with prime concepts we have the transformation to a new form of natural understanding (which in holistic number terms defines L1).
Then development again at L1 (mythic) evolves through S1 (where natural perceptions are formed), then S2 (where corresponding natural concepts emerge). Now implicitly the formation of these natural perceptions and concepts leads to the development of their negative direction (so that one can thereby detach oneself from the products of understanding). When we raise any natural number to a negative (natural) dimension we obtain a rational fraction. For example when we raise the number 2 to the power of - 1 we get 1/2.
Likewise in holistic terms when one learns to properly relate natural perceptions with their corresponding concepts (through implicit negation) it leads to an important new transformation to rational development (which then defines the middle levels).
Therefore we can precisely define the manner in which the rational understanding of the middle levels evolves, from the understanding of the sub-levels of the previous stage.
However it does not end here. For if development successfully unfolds at the rational levels, then one will attain the capacity to develop rational ability with respect to both perceptions and concepts. So L0 (rule/role) will lead to the specialised development of rational perceptions; L0.H0 (formal reflexive) will then lead to the corresponding specialised development of rational concepts. Then finally H0 (vision-logic) will lead to the dynamic interaction of such (rational) perceptions and concepts. 
Now in number terms when we raise a rational number e.g. 2 to a fractional dimension such as 1/2 we obtain an irrational number.  Likewise in development the rational interaction of perceptions with respect to concepts should logically lead to the "irrational" experience that mathematically defines H1 (psychic/subtle level).
This transformation is central for development of the "higher" stages relating to advanced spiritual attainment, yet it rarely takes place (to any significant extent). In other words though human development in our society generally proceeds predictably up to the middle (rational) levels, for the vast majority, it then largely plateaus at these stages with remarkably little further progress at the "higher" spiritual levels taking place.
Now I can offer - based directly on this holistic mathematical demonstration - two very interesting clues.
It is important to remember that the horizontal and vertical aspects of experience - relating to specific (perceptual) phenomena and their corresponding (conceptual) dimensions - are qualitatively distinct. The reason why development does normally proceed successfully through the lower levels is because the interaction as between conscious and unconscious is maintained (though necessarily in a somewhat confused manner). However with specialised understanding of the middle levels, its role (specifically with respect to cognitive interpretation) is substantially reduced to that of the conscious. Therefore because further transformation in development is based on maintaining the qualitative distinction of these two forms of understanding, it tends to level off when such a distinction is lost. Admittedly when some of this distinction is implicitly preserved, we do get the more limited spiritual transformation (to vision-logic). However full transformation - especially in dynamic structural terms - requires that the distinction of conscious and unconscious interpretation be explicitly preserved with respect to rational understanding. In other words, rational concepts (as general dimensions) are qualitatively distinct from rational perceptions (as object phenomena). It is vitally important that the customary reduced nature of interpretation that is unquestioned in conventional terms be fully appreciated before we can properly prepare understanding for further transformation (especially in cognitive terms).
There is another fascinating related point. Though psychologically the middle levels are indeed characterised by increasing rational ability with respect to conceptual understanding, science preserves an earlier more basic notion of natural dimensions (i.e. 3 of space and 1 of time) which is not actually in correspondence with such psychological understanding. Indeed I have long maintained that our very way of looking at dimensions in science is faulty from a true experiential perspective. Now interestingly when we raise any rational number to a natural power, we will obtain a rational number (without transformation). So the way that conventional science trains us to interpret space-time dimensions, is a major factor in preventing radical transformation to cognitive understanding of the higher levels.
When we properly understand the nature of dimensions we realise that all the transformations that apply to numerical quantities equally can apply to dimensional qualities. Thus for example we can have negative dimensions, fractional dimensions, imaginary dimensions, irrational dimensions etc. which all can be given coherent matching physical and psychological explanations thus preparing the way for truly integral appreciation of reality.
Let us take this transformation process one step further. When we raise a quantity to an (algebraic) irrational dimension e.g. 2 (raised to the power of the square root of 2) we get a transcendental number. Likewise with development at H1, the first sub-level (S1) relates largely to irrational appreciation of perceptions (i.e. that are understood in a bi-directional paradoxical manner). Then at S2 the corresponding irrational appreciation of concepts unfolds. Finally at S3 the interaction of such irrational (i.e. paradoxical) perceptions and concepts takes place. Thus the unreduced interaction of such understanding leads to a further transformation which defines H2 (the causal level) and which also demonstrates the remarkable power of the holistic mathematical approach. 13
So just briefly to summarise!
Appropriate unfolding of the various sub-levels (of each level) is extremely important in terms of the dynamics of transformation to a more advanced level. And we have used holistic mathematical number notions to show the precise nature of these structural transformations. Once again we have seen how the three fundamental polarities precisely define the sub-levels. Most importantly, the qualitative distinction as between horizontal and vertical sub-levels - which are linear and circular with respect to each other - is vital for explaining the manner of transformation to the more advanced levels.
Each sub-level is further defined in terms of directions. As we have seen, the very nature of the linear (middle levels) is that they are one-directional, thus enabling unambiguous asymmetrical interpretation of development. In this way the dynamic interaction of quadrants is greatly confused.  For example from a linear standpoint both exterior and interior development would unfold in the same forward direction. However from a bi-directional perspective (in the spirit of Heraclitus) the process is much subtler. Thus if development unfolds in a forward direction for the exterior aspect - then - relatively it unfolds in a backward manner for the interior; likewise if development unfolds in a forward direction for the interior, it unfolds in a backward manner for the exterior aspect. Now at H1 the two horizontal directions are clearly defined. At H2 four directions (horizontal and vertical) and at H3 eight directions (horizontal, vertical and diagonal) are defined. If we use H1 to illustrate, the exterior direction typically unfolds first, relating to an enhanced spiritual appreciation of the physical world. However due to secondary attachment, dynamic negation must take place with respect to this direction. This in turn enables the switch to its opposite direction relating to psychological reality (thereby enabling it to be likewise differentiated). Then later - again due to secondary phenomenal attachment - a new round of negation is required which thereby paves the way for the movement to the next sub-level. So the successful unfolding of the directions, which is a prerequisite of authentic integral understanding, requires that both positive and negative aspects be explicitly recognised.
Now complementarity exists in a confused fashion with respect to directions at the corresponding "lower" levels.
Modes (Primary and Secondary)
Then within each direction of a sub-level, I define modes. The three primary modes are cognitive and affective - which are "real" and "imaginary" with respect to each other - and volitional (which co-ordinates both). In this way exactly the same basic dynamics in terms of horizontal, vertical and diagonal poles takes place with respect to modes.
I also define secondary modes, which entail varying configurations of the primary. These are often misleadingly referred to as lines of development (which relates solely to their differentiated aspect). However when looked at from an integral (interdependent) perspective they are equally circular. However as with perspectives, integration of modes takes place initially in terms of primary (and not secondary) expressions.
I also define personality types 24 (in all) which once more are mathematically derived from the same horizontal, vertical and diagonal relationships.  So we are now able to deal with development in a potentially extremely rich and diversified fashion. Thus we can trace the unique path for each personality type (while recognising that its fundamental binary structure is extraordinarily simple (again being literally a reduced expression of oneness).
Extended Description of H1 (Psychic/subtle)
To illustrate the extended approach, I will briefly apply it to just one important level i.e. H1 The transition to this level usually entails an existential crisis, resulting in considerable dynamic negation of phenomena (reflecting rigid differentiation). This negation largely relates to horizontal (exterior and interior) polarities. So this initial negation represents the dynamic movement away from conscious to the more unconscious aspect of personality (which serves as the direct basis for integration).
Now H1 itself is composed of three sub-levels (S1, S2 and S3) respectively.
The first of these relates to the more concrete (supersensory) bi-directional experience of phenomena (as structure) that interpenetrates with refined spiritual intuition (as state). At H1, this sub-level is defined by two horizontal (heterarchical) directions. Typically the exterior aspect unfolds first with respect to the physical world (which is now bathed in a new light). This corresponds to a form of nature mysticism that typifies the earlier part of the stage.
However as each direction is likewise characterised by (primary) modes, this direction has affective, cognitive and volitional expressions. The affective expression is usually associated with this new spiritualised appreciation of nature. It can also be developed through a special form of romantic relationship (which operates in a profoundly archetypal fashion). 16 The cognitive aspect is then expressed through a new-found bi-directional appreciation of relationships (with growing recognition of the paradox associated with dualistic understanding). The volitional aspect - which really is central - is directly related to purity of spiritual intent.17 Now various secondary modes may also develop. The key (secondary) modes in this context would be artistic, scientific and moral development (relating to affective, cognitive and volitional aspects respectively).
However secondary attachment to symbols (mediating the spiritual light) which is inevitably involved, can gradually cause considerable conflict, which then requires a period of purgation in the dynamic negation of such attachment. This in turn enables a switch to the opposite (interior) direction of S1. Thus the proper differentiation of the directions in experience, requires the balanced negation (as well as positing) of phenomena.
Considerable development with respect to the interior psychological aspect now occurs which again is expressed through the three primary modes.
In affective terms it is initially associated with renewed deep feeling of a profoundly existential kind. Then with growing detachment, the emphasis can switch to a more philosophical appreciation of existentialism. In volitional terms it is best expressed through (blind) faith especially when the renewed purgative phases are present. 18
So with sufficient secondary detachment from concrete symbols the next formal (suprarational) sub-level emerges which is often associated with more powerful spiritual illumination.
Once more this will have two directions with the (exterior) physical phase first to unfold. In affective terms this is associated with very refined aesthetic feelings e.g. a pure experience of spiritual joy. From a cognitive perspective, it can lead to a deeper universally based philosophical appreciation that is based on the complementarity of opposites. In volitional terms it is associated with a deeper more authentic attitude of spiritual intent.
Again however secondary attachment to phenomenal symbols mediating this new light can cause considerable conflict requiring further deep purgation. This then leads into the interior aspect with which a more profound existential experience is associated. Then with the inevitable dynamic negation of this phase, one becomes largely stripped of any conscious attachment to phenomenal symbols (of either a primary or secondary nature). This is associated with the "dark night of the soul". So the 3rd sub-level then relates to an attempt to deal interactively with the world while remaining deeply denuded of contact with phenomena (of either a primary or secondary conscious nature). Though leading to truly contemplative experience, typically of a very transcendent kind, it can also be associated with a severe psychological crisis (due to growing imbalance with the unconscious aspect of personality).
So the resolution of this problem then sets the stage for transition to the next major level.
I have been at pains here to try and present a more authentic existential view of the process of spiritual development from a Western perspective, which is inherently more dynamic (due to the interaction of both structures and states). Though considerable spiritual progress can indeed occur, it always remains a deeply human experience, where one's many limitations are exposed to the nth degree.
There is for me a healthy aspect about this kind of mysticism that is not always reflected in secondary Eastern accounts. In other words the extent that one can rise "higher" in experience of the divine is directly related to the extent to which one can descend "lower" in recognising and then accepting one's many human weaknesses.
I will now conclude by voicing some of my key concerns regarding integral development.
Thus I have briefly sought to demonstrate in these articles how the inherent structure of development at all levels is mathematical. Gradual realisation of this important fact can, I firmly believe, eventually pave the way for a truly massive revolution in our intellectual understanding of reality.
Appendix - Integral Mathematical Summary of 12 Levels of Development
Perhaps it would be helpful to summarise this integral mathematical interpretation of the 12 levels.
As we have seen each major level is defined by a unique configuration of the relationship as between the three fundamental polarities with respect to both their linear and circular expressions. This thereby enables the dynamic relationship of differentiation and integration for each stage to be defined. We have also seen that associated with each of these levels is a unique holistic interpretation of one of the major number types, which scientifically defines the dimensional framework of the stage.
The Lower Levels
We started with the lower levels. The first task of development, which defines these levels, is the gradual movement from confused integration to properly differentiated experience.
Lower 3 (L3) - Archaic.
This commences with the confused circular integration of diagonal, vertical and horizontal polarities eventually culminating with the linear differentiation of the diagonal (with respect to form and emptiness). In psychological terms this diagonal separation is associated with differentiation of the bodyself; physically terms it represents the first separation of forces from matter. From a number perspective the level is defined by the holistic interpretation of the binary digits (oneness and nothingness) where initially both are largely entangled with each other. Thus in scientific terms L3 is defined by a rudimentary binary space-time framework with respect to both physical and psychological reality where matter as form first becomes distinguished from emptiness.
Lower 2 (L2) - Magical
This commences with confused circular integration with respect to vertical and horizontal polarities with linear differentiation in terms of diagonal. However eventually the vertical (whole and part) polarities also separate in linear terms. From a psychological standpoint this vertical separation is associated with differentiation of the emotional self; physically it represents the first separation of dimensions from matter. From a number perspective the level is defined by the holistic interpretation of prime digits. Psychologically this relates to primitive instinctive behaviour where conscious and unconscious aspects have not yet separated satisfactorily. Physically it relates to the corresponding situation where object phenomena cannot yet be clearly distinguished from dimensions. Thus from a scientific perspective, L2 is defined by a prime space-time framework in physical and psychological terms i.e. where objects (perceptions) are not clearly distinguishable from dimensions (concepts).
Lower 1 (L1) - Mythical
This commences with confused circular integration in terms of remaining horizontal polarities and linear differentiation with respect to diagonal and vertical polarities. However by the completion of the stage, the horizontal (interior and exterior) polarities will also have separated. In psychological terms this horizontal separation is associated with differentiation of the mental self; in physical terms it is associated with the growing independence of (observed) objects from the observer. From a number perspective the level is defined by the holistic interpretation of natural digits. Psychologically this relates to behaviour defined by general custom or tradition; physically it relates to a natural world - made up of constituent wholes - as distinct from the self.
Thus in scientific terms, L1 is defined by a natural space-time framework (i.e. 3 of space and 1 of time) in physical and psychological terms, where objects however (as observed) are not fully independent from subject (as observer)
The Middle Levels
The middle levels are characterised by the specialised development of linear (differentiated) understanding. However this is equally associated with a considerable reduction of integral to (mere) differentiated appreciation.
Lower 0 (L0) - Rule/role
This explicitly entails specialised linear differentiation with respect to diagonal, vertical and horizontal polarities in concrete terms (with implicit circular integral confusion remaining at a general universal level). In concrete terms, full independence therefore of the physical from the psychological aspect can be maintained. From a number perspective the level is defined by the holistic interpretation of rational quantities (with respect to concrete understanding) with - as we have seen -physical and psychological aspects of understanding clearly separated.
Thus in scientific terms, L0 is defined by a rational space-time framework in physical and psychological terms i.e. where object phenomena and perceptions can be progressively sub-divided into parts.
Lower 0, Higher 0 (L0,H0) - Formal Reflexive
This entails specialised linear differentiation with respect to diagonal, vertical and horizontal polarities in formal terms. So now full independence of physical from psychological reality is maintained (in terms of both concrete and more abstract theoretical understanding) From a number perspective, the level is defined by the holistic interpretation of rational quantities (with respect to formal generalised understanding) with again physical and psychological aspects clearly separated. In scientific terms L0,H0 is defined by a rational space-time framework in psychological terms i.e. where general concepts can be progressively sub-divided into parts. However very significantly, though a corresponding interpretation should apply in physical terms i.e. where dimensions are likewise sub-divided into parts, this in fact does not happen. So though rational science does indeed allow for the formal division of concepts it does not allow for the corresponding division of dimensions (in scientific terms). Instead it continues to use a - merely - natural interpretation of dimensions (i.e. where dimensions are treated as whole units with 3 of space and 1 of time). And we have seen, when looking at the major sub-levels, a further demonstration of the power of holistic mathematical understanding, indicating how this important misinterpretation can explain why development rarely proceeds in our scientific culture into the "higher" levels (especially with respect to understanding of the dynamic structural nature of these stages).
Higher 0 (H0) - Vision-logic
This explicitly entails specialised linear differentiation of diagonal, vertical and horizontal polarities. However implicitly it also combines in varying degrees the intuitive understanding associated with the "higher" spiritual levels (without radical transformation however in the nature of associated structures). So again in explicit structural terms, physical and psychological aspects are still separated. From a number perspective, we now interactively combine both the concrete holistic notion of a rational number with its formal dimensional characteristic. When appropriately understood, this should lead to a transformed understanding of the nature of all structures (physical and psychological). However because of the conventional scientific misinterpretation of the true rational nature of space-time, the possibilities for such transformation are greatly restricted.
Thus in scientific terms, vision-logic is explicitly defined by a rational space-time framework psychologically (though not physically) while implicitly imbued with spiritual intuition (of the "higher" levels).
The Higher Levels
Because of the significant reduction of the integral aspect to differentiated understanding at the middle stages (associated with the corresponding fragmentation of experience), the "higher" levels are increasingly concerned with the specialised development of the (nondual) integral aspect. This takes place in a reverse direction to that of differentiation.
Higher 1 (H1) - Psychic/subtle
This entails the mature circular (bi-directional) integration of horizontal (interior and exterior) polarities with linear differentiation remaining with respect to both vertical and diagonal. In psychological terms this is associated with integration of the mental self; in corresponding physical terms it entails the (complementary) integration of (observed) object phenomena with (observing) subject.
From a number perspective the level is defined by the holistic interpretation of (algebraic) irrational quantities. In bi-directional terms at H1, such understanding is still rational (though of a higher dimension). However from the linear (one-directional) perspective of the middle levels it appears "irrational" (i.e. paradoxical).
Thus in scientific terms, H1 is defined structurally by an "irrational" space-time framework in both physical and psychological terms (where both aspects - that interpenetrate with spiritual awareness - are understood as complementary in "real" conscious terms). Furthermore from a vertical standpoint, this "irrational" dimensional framework of H1 is complementary with the natural perspective of L1.
Higher 2 (H2) - Causal
This entails the mature circular integration of both horizontal and vertical polarities with linear differentiation still - rigidly - remaining with respect to diagonal. In psychological terms this is associated with integration of the emotional self (requiring the balanced interaction of spiritual states with phenomenal structures). Physically it is associated with the corresponding integration of object phenomena with their corresponding dimensions (where both are understood as purely relative).
From a number perspective the level is defined by the holistic interpretation of transcendental quantities (as the relationship of both linear and circular aspects of understanding). However as "real" circular understanding can be indirectly translated in "imaginary" linear terms we can equally express this holistic interpretation as the relationship of both the "real" and "imaginary" aspects of linear understanding.
Thus in scientific terms, H2 is structurally defined by a transcendental space-time framework (in both physical and psychological terms) where "real" (conscious) and "imaginary" (unconscious) aspects are understood as complementary. Furthermore in vertical terms, this transcendental framework of H2 is complementary with the prime correspondent of L2.
Higher 3 (H3) - Nondual
This entails the mature circular integration of horizontal, vertical and diagonal polarities culminating in pure (nondual) spiritual awareness. In psychological terms this is associated with integration of the bodyself requiring the balanced psychophysical interaction of body and mind with Spirit. Physically, this is associated with the corresponding integration of objects dimensions and forces (in a pure physical symmetry where phenomenal distinction no longer remains). From a number perspective the level is defined by the holistic interpretation of transfinite quantities (where phenomena lose any residual rigid finite meaning). The closest we can approximate to this empty nondual state in finite terms is where reality is simultaneously both linear and circular i.e. "real" and "imaginary" (which defies logical interpretation). In scientific terms H3, is defined by a transfinite space-time where physical and psychological aspects are indistinguishable through mature spiritual integration. It is vertically complementary with the binary space-time framework of L3 (where again physical and psychological aspects are indistinguishable in a totally confused manner).
The Radial Levels
Though development of the "higher" levels - because of complementarity with corresponding "lower" - considerable two-way integration of both sets of stages takes place. However the middle levels (which initially are complementary with each other) can be significantly bypassed in this process.
Thus the radial levels are concerned with the combined integration of both "higher" and "lower" (and "lower" and "higher") with the middle levels. And because - by their very nature - the middle levels are most differentiated - this means that the radial ultimately entails the balanced mutual interpenetration of all levels in terms of both dual (differentiated) and nondual (integrated) aspects.
Radial 1(R1) - Reawakening
This entails the mature circular integration of horizontal, vertical and diagonal polarities, with the mature differentiation of diagonal; however vertical and horizontal are not yet differentiated in a radial manner (i.e. where differentiation takes place against a background of mature integration).
At this level both physical and psychological aspects are very closely identified. So the differentiation of the radial bodyself with which the level is associated, entails the experience of one's body as inseparable from the whole physical world of form.
From a number perspective the level is defined by the holistic interpretation of transfinite numbers (based on the pure interaction of the holistic binary digits) and equally by renewed appreciation of the corresponding analytic interpretation of the same binary digits. So a double binary system (with both holistic and analytic aspects) is clearly understood as the basic means of encoding all transformation and information processes.
So in scientific terms R1 is best interpreted in terms of a binary space-time framework interpreted with respect to both holistic and analytic aspects. However because of the necessary interdependence of all aspects, at the local level rational activity is restored (though in a very limited manner).
Radial 2 (R2) - Mature Vision
This entails the mature circular integration of all sets of polarities with mature differentiation now taking place with respect to diagonal and vertical.
It is associated psychologically with the radial differentiation of the emotional self, i.e. as the felt compassion for all creation. Psychological and physical aspects, though still close, separate somewhat so that we have the freer interplay of states and structures (which are "real" and "imaginary" with respect to each other). Therefore the universal psychological experience of the radial dream state interacts with the universal appreciation of physical reality (as structure) and vice versa. At a local level - though still restricted - a greater facility in rational linear understanding is experienced. So overall there is greater vertical interpenetration of universal with local understanding. This culminates in the mature dream of how best to express universal understanding in limited (local) circumstances.
From a number perspective, the level is defined by the growing appreciation of all types from both the qualitative (holistic) and quantitative (analytic) perspective. So now the holistic interpretation of transfinite (and binary) with transcendent (and prime) is combined with the growing rational appreciation of the middle levels (enabling the enhanced analytic appreciation of these number types also).
So with distinct perspectives once more restored in ever more complex fashion, the space-time framework likewise becomes more complex combining both the holistic and analytic interpretation of the various number types.
Radial 3(R3) - Creative Transformation
This is defined by the mature circular integration of all polarities with likewise the mature linear differentiation with respect to the same polarities.
It is associated with the full radial differentiation of the rational self that can now operate with equal facility directly in terms of analytic and indirectly with respect to holistic type appreciation.
We now have full dynamic interaction as between the universal and local aspects of experience. Put another way from a psychophysical perspective, body can be fully integrated (yet also separated) from mind. In physical and psychological terms states and structures can be fully integrated (yet also separated when the need arises). Finally with respect to interior and exterior aspects, stages (as self) can be integrated with stages (as reality) yet also separated in the required context.
From a number perspective we now have the full appreciation of all types (both quantitatively and qualitatively). Thus though overall perspective is utterly simple (as pure Spirit) the configuration of this with distinct perspectives is now incredibly complex entailing the space-time framework associated with all number types (in terms of both holistic and analytic appreciation).
1. I have often used the example of the fire to illustrate this point. Peak contemplation resembles the flames from the fire. However if material is not added to the fire these flames cannot be sustained. In like manner contemplation cannot be properly sustained if it is not fed through phenomenal involvement. Thus when the fire is continually renewed through the constant adding of a material source, the flames can be sustained - and even increased - on a permanent basis. In like manner, contemplative awareness is sustained and indeed deepened through intense (unattached) active engagement in phenomenal affairs.
2. Monastic communities can represent this situation well. Of course it would be naοve to think that commitment to monastic life necessarily entails advanced contemplative development. However some clearly do advance in this way adapting to life that can entail very localised responsibilities directly defined by community rules. Though this may indeed suit some personalities, in general I would see it as too restrictive. Though obedience may be easily defined through adherence to externally imposed rules, in the end it may prove inauthentic in terms of spiritual growth. True obedience comes from following the enlightened conscience, which - because of the uncertain demands it can entail - may prove much more difficult. In this regard the life of Thomas Merton - which has been very well documented - raises very challenging questions for the traditional ideal of the monastic contemplative life.
It should furthermore be clearly stated however that I am approaching the experience of the radial levels from the contemplative (passive) standpoint. However a distinctive interpretation would apply coming from a more active perspective.
3. I will just give one example here. I have good reason to be thankful to Mike Mc Dermott (who is also a contributor to the Reading Room) for a very fruitful correspondence on some fascinating numerical and geometrical relationships (with due appreciation of the holistic context).
Much of this relates to the very interesting Fibonacci sequence.
The Fibonacci sequence is basically very simple. One starts with the two digits 0 and 1 and then by continually adding the last to the previous term, keeps generating new terms.
Now, Mike drew my attention to a remarkable result. If we start with the first Fibonacci term and divide by 10 and then add the next digit dividing by 100 and the next dividing by 1000 and so on in this manner that the sum of this sequence, i.e. 0 + .01 + .001 + .0002 +. 00003 + .000005 + .0000008 + . = 1/89
Very interesting indeed, especially in view of the fact that 89 is one of the Fibonacci terms.
However - due largely to the holistic binary approach - I have discovered in a complementary analytic manner that many numerical relationships reveal especially interesting patterns when converted to binary form.
So if now express these decimal terms in binary form and divide by 10, 100, 1000 etc (again in binary terms) and then add the terms (to infinity) a more remarkable result emerges where the sum of the terms (to infinity) = 1
In base 10, 100 - 10 - 1 = 89
In binary 1 = 1, 2 = 10, 3 = 11, 5 = 101, 8 = 1000 and so on.
Using more familiar denary notation (base 10) this result can equally be expressed as
It is notable how this sum of decimal terms (to infinity) based on the Fibonacci sequence gives the same result as simpler infinite sequence
1 (in binary) = .1 + .01 +.001 +.0001 +
which in conventional denary (base 10) is
1 = 1/2 + 1/4 + 1/8 +1/16 .
It is important to remember that such analytic results (which are fascinating in their own right) equally have holistic interpretations that reveal a deeper philosophical meaning to such behaviour.
I would attempt to explain the latter result holistically in this manner. The successive division by powers of 2 in the formulae relates directly in holistic terms to successive dualistic division of the whole (unit). Imagine a cake that is divided in half. We then set one half aside and divide the remaining part in half. Again we put one part away and divide the remaining part in half (again putting one part of it away). Now if we continued in this manner, accumulating all these parts put away the total sum of parts would approach the (original) whole cake. So we are demonstrating a reduced notion of wholeness (oneness) here (where it is identified with the sum of these ever-smaller dualistically obtained parts).
Now in holistic terms, the Fibonacci type relationship represents a more complex version of the same reduced notion of wholeness (approximated as the sum of parts obtained through repeated dualistic sub-division).
4. In practice things are never so clear cut. Because dynamic interaction necessarily exists with respect to all levels, someone operating mainly at H1 will sometimes have insights - even significant - that are representative of "higher" approaches.
5. The reason for this can perhaps be explained. As we have seen the quality of genuine detachment is paramount with respect to authentic spiritual development.
Keen awareness of one's limitations in many areas therefore can be extremely helpful in developing the essential quality of humility (where one goes "lower" into self) thereby enabling one to successfully transmit the "higher" spiritual power.
6. Peak experience of more advanced levels in dynamic terms necessarily entails temporary access to both the structures and states of these levels (though typically either the state or the structure will dominate). Also it is important to distinguish the type of access which would fall into at least three categories. Firstly, we have the normal confused access at the very low levels (where they are still strongly complementary with corresponding "higher" levels). Thus the baby infant at the early "prepersonal" has continued confused access to the "highest" transpersonal stage (so that both are not really distinguishable). Then given sufficient differentiated development, we can have more authentic temporary access from a "lower" to a "higher" level. Here the experience is representative of the (mature) "higher" level while it lasts, but then is quickly dissipated and reduced on return to one's customary level. However there is also the especially lucid childhood experience of a "higher" spiritual level, which can characterise those with a latent mystical potential. This is so powerful while it lasts, that one recognises in some measure on return to the customary level that it does not serve as an adequate expression of what one has experienced. Therefore it can act as a considerable catalyst for further advanced development (more appropriate to the true nature of the experience). However in a balanced account we should also stress the equal importance of "valley" experiences (i.e. where one at a "higher" level has temporary access to the corresponding "lower" stage) which broadly can be classified in the three same ways.
7. Where transition to each of the radial levels is concerned, we have the interaction of both differentiation and integration.
8. In simple terms differentiation is associated with positing of polarities (in a conscious fashion). Integration is associated with their corresponding negation (in an unconscious manner).
9. When appropriately understood in holistic mathematical terms, perceptions and concepts are "real" and "imaginary" with respect to each other.
Of course this distinction is lost at the level of conventional rational understanding where both are interpreted in a merely reduced (conscious) manner. However if we are to understand the vital dynamics by which transformation to new levels of development takes place we must preserve this distinction. Interestingly at the "lower" levels, where conscious and unconscious are still closely associated, the distinction is preserved (though in a confused manner). It is only lost at the middle levels where the conscious becomes largely separated - especially in cognitive interpretation - from the unconscious.
10. Though number representing quantity and number representing dimension are truly distinct (referring to different logical systems that are linear and circular with respect to each other), the conventional mathematical interpretation is highly reduced with both numbers interpreted in a merely (linear) rational fashion.
Again this points to the deeper problem, that though mathematics in dynamic experiential terms entails the interaction of both (linear) reason and (circular) intuition, it is explicitly interpreted in a merely rational manner.
11. I am aware that I am treating - what are defined as levels - i.e. L0. L0.H0 and H0 as sub-levels in this context. Indeed for many years I tended to look at the middle band as comprising just one major level with three sub-levels (which I now define as levels). There are good reasons why I made this change though it is not necessary to go into them in detail in this context. Indeed the newly defined levels can also be defined in terms of sub-levels. For example at the L0 (concrete) sub-level, initially understanding will be based on local perceptions. However later localised concepts will develop so that interaction between both can then take place in various ways. Of course growth in rational activity implicitly requires a growing ability for negation (thereby achieving greater detachment from the phenomena investigated). This negation in turn helps to screen out mere sensory features thus enabling the transition to the more universally based stage of L0,H0 (formop). Here again perceptions first emerge (of a more a universal abstract nature) then concepts and the interaction between both. H0 (vision-logic) then brings about dynamic interaction with respect to both local and universal use of both perceptions and concepts.
12. In fact this is how we obtain the square root of 2. So the highly interesting point about this formulation is that it entails the use of a fractional dimension (1/2). This is not science fiction! All rational understanding - especially of the more developed kind - entails the use of fractional dimensions. However because of the reduced nature of scientific interpretation (confined to merely natural dimensions) we do not recognise this remarkable fact. So a fractional dimension, from the psychological perspective, simply relates to the ability to analytically subdivide concepts into smaller units. Therefore when we apply such understanding to the physical world we are thereby experiencing reality through fractional dimensions.
13. 2 (raised to the power of the square root of 2) formed the basis for one of Hilbert's famous important unsolved mathematical problems at the turn of the 20th century. In other words Hilbert was seeking a general proof to demonstrate that any such rational (or irrational) number raised to an irrational power is thereby transcendental. The proof was provided - sooner that Hilbert anticipated - in 1934 and is known as Gelfond's theorem. However what is truly remarkable - coming from a purely holistic perspective when the same mathematical symbols are now understood in a dynamically interactive fashion - is that we have demonstrated the truth of the same theorem (from an entirely distinctive perspective). In other words the holistic significance of Gelfond's theorem relates to the manner by which the transformation in development from H1 (psychic/subtle) to H2 (causal) takes place.
So this demonstrates two types of "proof". Firstly we have the analytic type of proof (on which conventional mathematics solely concentrates). However we also have a holistic type of "proof" which is entirely neglected. The holistic "proof" relates to a fully coherent philosophical demonstration of how mathematical symbols (as holistically interpreted) apply to development. In future, in a properly developed radial mathematics, analytic proof will prompt the search for their holistic equivalents. Equally - and perhaps even more remarkably - holistic "proof" in some areas may come first which will then prompt the search for complementary analytic type proof.
Even if Gelfond's theorem had not been solved, we would be able to say from a holistic perspective that the theorem is analytically true from such holistic understanding. Now this assertion would not directly constitute a proof (in analytic terms) but would however suggest that a proof could indeed be found.
14. Though vision-logic is certainly capable of distinguishing the four quadrants - with their unique perspectives on reality - it is only able to do so in a reduced (unambiguous) manner.
In other words from a vision-logic perspective, development with respect to the four quadrants takes place in the same forward direction. In other words such understanding is precisely one-directional (as I define it). However when the directions of development are properly differentiated, the very notion of direction becomes paradoxical. For example at H1 where horizontal (heterarchical) directions are differentiated exterior and interior (and interior and exterior) aspects of development take place in opposite directions to each other. So if the exterior aspect develops in a forward (positive) direction, then - relatively - the interior develops in a backward (negative) manner; likewise if the interior develops in a forward, then the exterior develops - relatively - in a backward manner.
15. I also define phases, which are used to describe the pattern of unfolding with respect to a group of stages. Each major level is characterised by a distinctive phase pattern. For example at H1, we have a marked cyclical or wave like pattern which relates to the combination of linear and circular type understanding that typifies the level. So the upward phases of the cycle are characterised by illumination and the downward phases by purgation respectively.
16. Dante's relationship with Beatrice would typify such archetypal romantic attraction (in this case enabling Dante to experience God with a feminine face).
17. Using my own experience to illustrate, I remember that intellectual inspiration at this time came chiefly from Einstein and Teilhard de Chardin (both interpreting the physical world in a more holistic creative fashion). The interest in Einstein reflected an early conviction regarding the complementarity of physical and psychological relationships (in that I was convinced that his main findings could be given coherent psychological interpretations). De Chardin provided a more dynamic view in spiritual terms of the evolution of both aspects in creation (which paved the way later for a deep interest in Hegelian philosophy).
18. Because of genuine conviction regarding the need for authentic 1st hand accounts, I generally write with respect to my own experience, while recognising that very distinctive spiritual paths can exist for other personalities. Though any one account is necessarily limited, I believe that it serves the integral process much better to accurately reflect one's own experience (with which one is most familiar) than to attempt to report 2nd hand the experience of others (with whom one is necessarily much less familiar).
So my account here reflects an intensive phase (associated with this development) where I was deeply attracted to existential type literature gradually taking a more philosophical turn especially with respect to an identification with the work of Kierkegaard.
P. Collins (2003) "Development: The Radial Approach" - Chapter 16 http://indigo.ie/~peter/Radial16.htm