Integral World: Exploring Theories of Everything
An independent forum for a critical discussion of the integral philosophy of Ken Wilber

Peter CollinsPeter 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.

Clarifying Perspectives 4

Higher Order Perspectives and Integral Mathematics

Peter Collins

Ken Wilber has made a significant contribution on the nature of distinct perspectives. However paradoxically his approach is greatly lacking in the integral aspect of overall perspective. This is due to the fact that it is largely confined to the interpretation of just one stage of development i.e. vision-logic (that is not designed to consistently interpret the integral dimension of development).

Ken's "Integral Mathematics" represents a very interesting contribution in its own right. However as it likewise reflects such vision-logic understanding, it is not capable of adequately interpreting the integral nature of perspectives.

An integral approach to perspectives should be stage-specific explaining the manner by which the distinct perspectives are configured (thereby defining a unique overall perspective for each stage).

An integral mathematical approach would then interpret the nature of this configuration (for each stage) in an appropriate scientific manner.


In previous contributions I focussed largely on the formation of primary perspectives in a dynamic interactive manner (designed to more closely approximate the very nature of experience). I firmly argued that such an interactive interpretation - which I refer to as radial - is necessary to consistently combine the differentiation of distinct perspectives with overall integral perspective (which ultimately is without perspectives).

While recognising the valuable contribution that Ken is making with his current work on perspectives, I would see it as somewhat inconsistent due to a continued failure to properly reconcile the integral with the differentiated aspect of development. [1]

The deeper reason for this failing - which I will explore further in this article - is due to an attempt to predominantly use the intellectual translation that is appropriate for just one stage (i.e. vision-logic) as a general means for interpreting all development. [2]

However when viewed from the radial perspective, associated with each major stage of development is a unique intellectual method of interpretation. Therefore any translation e.g. of the nature of perspectives is thereby stage specific and strictly only has validity within that limited context.

In my recent approach I define 12 key stages of development (i.e. levels) grouped into four major bands. So as to properly distinguish the differentiated from the integral aspects these are all defined with respect to both linear (discrete) and circular (continuous) aspects. Furthermore what is truly remarkable is that a holistic mathematical interpretation for the precise dynamic configuration of both linear and circular aspects can be given for each stage (thereby scientifically defining the very nature of the stage). The overall intellectual translations - appropriate for each stage - are in turn related directly to these (holistic) mathematical formulations. So we have here I believe - perhaps for the very first time - the key to a truly integral scientific interpretation of all stages of development. [3]

Indeed the implications are more far reaching in intellectual terms. Not alone - in terms of this approach - can we now give 12 distinct intellectual stage interpretations (in principle for every concept in development) but we can also define a potentially unlimited range of further interpretations greatly enriching the nature of overall appreciation.

Because of the importance of this point I will dwell just a little longer with the nature of these interpretations. The Lower Band - which in discrete linear terms would be defined as prepersonal - contains three major levels. These are L3 (archaic), L2 (magic) and L1 (mythical). Now clearly in early development we cannot formulate an intellectually coherent understanding. However because in dynamic terms "lower" levels are complementary with "higher" (and "higher" with "lower"), the interpretations appropriate for these "lower" structures are based on the corresponding "higher" levels. Thus we need the "highest" mature (H3) to interpret the dynamic nature of the "lowest" confused level (L3). In like manner we need H2 to interpret L2 and H1 to interpret L1. [4]

The Middle (personal) Band represents the specialised development of linear (asymmetrical) understanding. Intellectual methods of discourse are predominately based - at least formally - on this type of analytic interpretation. However when used outside its own band it tends to substantially distort the true interactive nature of relationships. This band again contains three major levels L0 (concrete), L0,H0 (formal) and H0 (vision-logic). The concrete level is largely confined to the local empirical interpretation of specific relationships. The formal can then deal with linear relationships in a universal abstract manner. Vision-logic can then more creatively flexibly combine both of these aspects with respect to vast networks of differing relationships. [5]

Now Ken Wilber's intellectual mode of interpretation is mainly based on the use of vision-logic. Though it does indeed provide the most multifaceted use of asymmetrical type understanding (which in Ken's case is based on strong spiritual vision) in its explicit expressions it is formally suited - merely - for the interpretation of the differentiated (rather than the integral) aspect of development.

Therefore the problem I persistently point out in Ken's work (i.e. the lack of an appropriate dynamic interactive treatment of relationships) is rooted in this adherence to vision-logic. By its very nature it - as it is merely represents the most advanced understanding of the Middle Band - it is not suited for interpretation of the increasingly dynamic nature of either the "Higher" or "Lower" Bands (which are complementary). Ken does indeed recognise the need for further spiritual development at the "higher" levels. However - as I have frequently maintained - there is a marked discontinuity evident in his understanding where such development is largely expressed as spiritual states (rather than phenomenal structures). However because spiritual states increasingly interpenetrate in refined manner with the phenomena of these levels, their very structures undergo continual transformation. Therefore the task of intellectual translation at the "higher" levels is to appropriately demonstrate the increasingly paradoxical nature of dual interpretation for all aspects of development (in the light of growing nondual awareness). This requires a new kind of subtle bi-directional understanding (with varying layers of intricacy) which I find almost entirely missing from Ken Wilber's work.

So the "Higher" Band has again three major levels H1 (psychic/subtle), H2 (causal) and H3 (nondual). From the perspective of structural form (which interpenetrates with spiritual states) this leads to three new levels of bi-directional circular understanding of an increasingly dynamic interactive nature.

At H1 such understanding is largely confined to the bi-directional conscious understanding of complementary opposite polarities (in heterarchical terms). In other words one attempts to successfully integrate interior and exterior (and exterior and interior) aspects of development.

At H2 this understanding becomes much more spiritually refined to also include the bi-directional appreciation of indirect projected phenomena (that properly pertains to the unconscious).

At H3 both conscious and unconscious aspects are simultaneously reconciled through the purest development of nondual spiritual awareness. So here phenomena are so refined that they immediately pass from memory as soon as they arise, enabling the closest approximation of the identity of form and emptiness (and emptiness and form). [6]

However this does not represent the end of the story. Because "higher" and "lower" stages are complementary this means that the important middle stages can be substantially by-passed through the attainment of contemplative nondual awareness.

So the final stages of development lead to substantial re-immersion in the dualistic world of form (this time in a largely unattached manner).

The middle levels by definition are neither "higher" nor "lower". Thus what is neither "higher" nor "lower" is complementary with what is both "higher" and "lower". Thus the full integration of the middle stages with "higher" and "lower" can logically only take place when these have themselves been largely integrated (in nondual contemplative terms). This enables the most comprehensive expressions of development where both specialised differentiated and integral aspects can be combined to a remarkable extent.

So in intellectual terms, the interpretations of the three radial levels increasingly combine both linear (asymmetrical) and circular (bi-directional) modes of appreciation in a truly balanced manner, thus enabling understanding that can ultimately be extremely detailed in analytic terms yet also fully holistic from an integral perspective.

R1 (Radial 1) - where true nondual awareness finds its most universal dualistic expression is best suited for the kind of work that I see myself currently engaged with (at a preliminary stage) i.e. the integral mathematical interpretation of all stages of development.

R2 (Radial 2) for example would then enable in intellectual terms much more detailed analytic work with respect to various disciplines (where overall integral consistency is properly preserved).

R3 (Radial 3) would lead in intellectual terms to the most complete expression of such activity entailing a significant outward-transforming dimension through "conversion" to new forms of thought that is utterly motivated by non-possessive spiritual desire. Of course at this stage cognitive, affective and volitional modes would necessarily be so closely harmonised that the cognitive desire with respect to the world for transforming wisdom would be inseparable from the affective desire for transforming compassion which in turn would be inseparable from - what is really most central - i.e. the volitional desire (uniting compassion and wisdom) for transforming love.

Thus each of the major stages (i.e. levels) of development is characterised by a unique overall binary (linear and circular) configuration that can be precisely defined in integral mathematical terms. Associated with each stage in turn is a distinctive intellectual interpretation of development (that is appropriate to the understanding of that stage).

Therefore - in the context of our present discussion - if we are to truly deal with the nature of perspectives in integral terms, we must define the configuration that unfolds (with the characteristic relationship as between distinct perspectives and overall perspective) for each of the major stages of development.

Then in a truly integral scientific approach we must define how the structure of all these dynamic configurations is inherently mathematical (in the most fundamental manner). In this way we can understand how the very nature of development through and through is precisely mathematical (in this integral sense).

(I hope to briefly demonstrate the nature of this integral mathematical formulation of the dynamic nature of perspectives for each of the 12 major levels of development in a future contribution!)

Enhanced Interpretations of Development

However an appropriate integral mathematical approach opens up other fascinating possibilities.

Not alone does it provide us with the means for scientifically defining the intellectual understanding appropriate for each stage, but also it provides us with the means for precise clarification of a whole range of enhanced interpretations of such understanding. In this radial context therefore we can appreciate how the cognitive interpretations that we apply to reality are in a state of continual transition (whereby their very nature significantly changes).

For example let me illustrate this point with respect to the nature of vision-logic. Now this initially will unfold during the centaur stage (as the understanding appropriate to that stage), which represents H0 in my approach. However when development moves on to the next stage H1 (psychic/subtle), not only does a new form of circular bi-directional understanding unfold (which is now more appropriate to that stage) but also a new enhanced interpretation of the earlier stage becomes possible. So in addition to its own characteristic mode of bi-directional appreciation at H1, we have an enhanced interpretation of the earlier stage H0 (and indeed by extension all earlier stages).

Then when development moves into H2 (causal) again this is associated with the unfolding of its own characteristic understanding (with new enhanced interpretations of H1 and H0). So by now we have 3 expressions of vision-logic i.e. its characteristic expression at H0 (and enhanced expressions from the new overall perspectives provided through H1 and H2).

Now perhaps we can see that when development proceeds through the full range of stages that we have 7 distinct expressions of vision-logic. We would then have the characteristic expression of H0 (and enhanced interpretations from the perspectives of H1, H2, H3, R1, R2 and R3). [7]

Again - what is truly remarkable - is that all these expressions of vision-logic have their own distinct intellectual interpretations (which can be precisely defined in integral mathematical terms). [8]

When we can then appreciate that each of these interpretations in turn has potential applicability to the entire range of disciplines (providing a distinct means of overall translation) perhaps we can begin to recognise the truly vast potential of an appropriate integral mathematical approach.

When we then further consider how all these modes interpenetrate in experience in various ways we can perhaps appreciate how potentially the understanding of any issue is infinitely nuanced.

The problem with Ken Wilber's approach is that he so frequently gives the impression that intellectual translation ends with the vision-logic (characteristic of the centaur stage).

However properly understood, appropriate (explicit) integral interpretation of reality (that is truly consistent in holistic terms) only starts to begin after this stage. So there is a vast range of further intellectual interpretations - both characteristic and enhanced - of an increasingly dynamic interactive nature (which are not to found in his work).

This is why I have consistently maintained that Ken, despite the undoubted benefits of his enormous contribution is - in terms of intellectual interpretation - offering a substantially reduced integral approach to development.

The truth of this observation clearly cannot be appreciated merely from within a Wilberian perspective (in the mistaken belief that it is already comprehensive) but rather entails substantial engagement with the alternative perspective (from which the observation is made).

Stage Development of Perspectives

Before dealing with Ken's contribution to Perspectives (as outlined in Appendix 2 of Excerpt C "The Way we are in This Together" on his Shambhala web-site), I will say a little here about my own stage model of perspectives (so as to provide context for subsequent remarks).

The formation of human perspectives coincides with the commencement of development. [9] Initially such experience - which is associated with primitive instinctive behaviour - is characterised by considerable confusion whereby distinct (differentiated) perspectives cannot be properly distinguished from an overall (integral) perspective. Gradually however, as development proceeds with the differentiation of various stages e.g. bodyself and emotional self, distinct natural perspectives do indeed emerge, though confusion still remains as regards the "real" (conscious) and "imaginary" (unconscious) aspects of these perspectives. [10]

Likewise with the gradual consolidation of distinct perspectives, the ability to reflect on and thereby change experience grows. Then by the middle stages - where the specialisation of linear (differentiated) understanding takes place - the ability to form higher order perspectives (of an increasingly complex nature) develops. This reaches its zenith with the most advanced of these stages i.e. the centaur (where vision-logic unfolds). So Ken Wilber's work on perspectives in his Appendix properly relates to the clarification of increasingly higher order distinct perspectives in development. And in this respect it is only fair to say that he is making a very significant contribution.

However the emphasis here is very much on the differentiated (rather than the integral) aspect of perspectives. So to consider this work as an integral contribution on perspectives I would consider very misleading.

Indeed by their very nature the formation of higher order perspectives is necessarily associated with increasing (rigid) definition with respect to the world of form, that inevitably leads away from true integral appreciation (which is inherently formless). Therefore before authentic spiritual development of the more advanced stages can unfold, the radical undoing of such higher order perspectives is required to gradually lead to a pure nondual overall perspective (which is without distinct perspectives).

In an earlier contribution "Clarifying Perspectives 1" I showed how - associated with the four quadrants - we have 16 primary or direct 1st order perspectives. I also stressed how integration is then based on maintaining complementary interdependence as between the perspectives (in opposite quadrants).

However it is not possible to operate successfully in our dualistic world based on the formation of these - merely primary - perspectives, as they literally do not provide sufficient definition for necessary activities. I demonstrated in "Clarifying Perspectives 1" the dynamic manner in which distinct perspectives are formed where for example in a four-quadrant model 16 (i.e. 4 X 4) arise. Now higher-order perspectives are derived from combining such 1st order perspectives with each other two or more at a time thereby resulting in their compound nature.

However it has to be realised that the nature of experiential perspectives is incredibly complex as they can overlap and interact with each other in extremely detailed and varying patterns. So we can only go so far in fruitfully attempting to dissect perspectives. As my approach is ultimately based on the dynamic interpretation of the binary system, perhaps an analogy with early computer experience could be instructive in this regard.

I remember when I bought my first PC back in the early 90's that the graphics card was only capable of providing 16 distinct colours (i.e. 4-bit colour). [11]

Thus when I attempted to view photographic images with this card the definition was extremely fuzzy.

In similar terms when we attempt to deal with reality based solely on the use of primary 1st order perspectives, experience will be somewhat vague and diffuse so that is unable to provide sufficient definition for our activity.

Shortly afterwards I upgraded my graphics card to 256 colours (i.e. 8-bit colour). Though there was still some noticeable loss of resolution with respect to images this was a big improvement on the earlier card (based on 16 colours).

Now of we take our initial 16 primary distinct perspectives - because each of these can in turn be associated with each other - in this way we can create 256 (16 X 16) 2nd order perspectives (from the primary 1st order versions).

However the minimum that would be required to provide proper definition for images would be 65536 colours (i.e. 16-bit). In like manner by associating these 256 2nd order (distinct) perspectives with each other we can potentially generate 65536 (256 X 256) 4th order perspectives. [12]

In present day computers most video cards can produce 32-bit colour (65536 X 65536). Likewise in terms of perspectives if we were to associate all our 4th order versions with each other we would acquire in analogous terms 32-bit definition (at the level of 8th order perspectives).

So as we move into higher order perspectives the ability to provide ever-greater degrees of (static) definition to our phenomenal experience steadily increases (and modern living greatly depends on this ability). [13] However because such higher order perspectives depend so much on the specialisation of the differentiated aspect of development, true integral capacity can greatly suffer. Thus the demands of society for such (distinct) perspectives (leading to corresponding greater degrees of phenomenal rigidity) is equally associated with the loss in the capacity to preserve true overall integral perspective.

So once again the stages of authentic spiritual development initially require a radical undoing of higher order perspectives formed at earlier stages thus enabling a strengthening of the integral - as opposed to the merely differentiated aspect - of experience.

In many ways this requires tracing - in reverse direction - the perspectives of earlier development.

So contemplative spiritual progress is generally associated with a great simplification in living whereby one gradually dismantles secondary perspectives. [14] Then as experience becomes more authentic, one increasingly relies on refined 1st order (direct) perspectives, which can then be more readily integrated in a nondual manner. [15]

However it has to be admitted that this can lead to significant temporary problems as regards one's ability to adapt to the world (defined as it is largely by higher order distinct perspectives). [16]

However with sufficient strengthening in the integral aspect (through the development of nondual contemplative capacity), in the final stages i.e. radial one is ready to embrace the world this time in a non-attached manner (where the differentiated and integral aspects of perspectives can unfold in equal balance).

This enables one therefore to become intensely engaged in dualistic activity demanding very complex higher order perspectives, while equally maintaining the capacity for higher order integration of these same perspectives.

Ken's Higher Order Perspectives

I have spent some time trying to provide some necessary context for my remarks (which is so important here in viewing Ken's contribution).

Let us be clear where Ken is coming from! He is using the vision-logic understanding of the centaur stage to provide much needed clarification on the world of higher order distinct perspectives. Here differentiated primary can be related in various ways to other primary perspectives in an increasingly complex and indirect manner.

However it represents very reduced thinking to equate such clarification of higher order distinct perspectives with overall integral perspective. Indeed it is utterly misleading!

The problem again relates to the very nature of vision-logic. Though it can be inspired - and certainly is in Ken's case - by genuine integral spiritual vision, in formal terms it is explicitly designed to deal in multifaceted detail with the merely linear (asymmetrical) aspects of development.

Put another way, if vision-logic were sufficient for an integral approach there would be no need for the many arduous "higher" stages of development. Ken would certainly admit that such spiritual development is necessary for true integration. However because he identifies this understanding very much with mere states, he thereby does not adequately recognise that we need the refined cognitive structures of the higher levels (which are quite distinct from vision-logic) to consistently interpret the integral aspect of development.

Indeed putting it bluntly, true integral interpretation of reality properly starts with the bi-directional cognitive structures of the "higher" spiritual levels. Vision-logic at best can offer a very reduced and inconsistent view of integration (i.e. where it is substantially identified with the differentiated aspect).

So in the context of what Ken is actually providing in Appendix 2 i.e. valuable clarification of the nature of higher order distinct perspectives in development, his treatment has considerable merit.

However even here I would make the following criticisms.

  1. He does not provide sufficient clarity on the distinction as between 1st and higher order perspectives. Just as when we combine prime numbers together they become composite, likewise when we combine distinct prime (or primordial) perspectives together they become composite. In this way therefore we can have 1st order primary and higher order composite perspectives.
    Ken attempts to deal with higher order (i.e. composite) perspectives through extension of his pronoun language to 4th, 5th, 6th ,7th , person. However this represents a confusing use of such language where really two notions are involved. So 1st, 2nd and 3rd person would relate in this context to different perspectives (of the same order) whereas 4th, 5th and 6th etc. would relate to different orders of perspectives. Thus when we don't clearly establish the correct mathematical structure of 1st order, then it becomes increasingly more difficult to do so with respect to higher order perspectives.
  2. Likewise the relationship is not clear as between this treatment of higher order perspectives in the Appendix and earlier viewpoints (in Excerpt C). Here he alternated between 4 fundamental perspectives (based on the quadrants), 6 perspectives (based on singular and plural versions of 1st person, 2nd person and 3rd person) and then 8 primordial perspectives (based on the delineation of the inside and outside of each holonic quadrant).
  3. Even in the context of explaining distinct (differentiated) perspectives, a dynamic interactive treatment is required so as to enable the integral aspect to be subsequently dealt with in a consistent manner. However because Ken abstracts his treatment from such a dynamic setting, I see no way of properly incorporating the integral aspect within his own set of definitions.
  4. Ken does not clarify the distinction as between explicitly and implicitly recognised perspectives. In actual experience the implicit way in which we organise perspectives is more complex than portrayed by Ken (involving orders of a very high number). It has to be remembered that we always approach perspectives based on a complex pattern of previous organisation. So in this sense we are building up ever more complex perspectives based on past experience.
    However our ability to explicitly clarify the nature of these complex perspectives operates on a far lower level. Indeed this mismatch as between what implicitly happens in experience and what we can explicitly clarify is associated with the customary phenomenal rigidity of our perspectives. Therefore if we are to regain the ability to truly renew our perspectives then such rigidity must be considerably eroded through high level spiritual development.
  5. Ken continues to use the language of personal perspectives (1st person, 2nd person and 3rd person) in the Appendix. However as I demonstrated in "Clarifying Perspectives 3" such use of language for perspectives is not inherently suited for revealing their true mathematical structure.
    Also he continues to use a merely reduced notion of "collective" by identifying it as the plural of "individual".
    Though Ken subsequently uses his own ingenious mathematical notation in an attempt to clarify the nature of distinct higher order perspectives, he does not actually show how the inherent structure of such perspectives is mathematical. [17]
  6. Though Ken's notation has considerable value - once one accepts his somewhat static assumptions - it suffers from certain limitations in a dynamic interactive context.
    For example Ken uses 1p to refer to 1st person and the 1-p to refer to 1st person perspective.
    However in dynamic terms, the very recognition of 1st person already constitutes a perspective. So in dynamic interactive terms we can only recognise 1st person (as revealed perspective) with reference to what is not 1st person i.e. 1st impersonal (as - in this context - hidden perspective).
    However because Ken is attempting to abstract his understanding of perspectives from a true experiential (i.e. interactive) context it leads to a degree of redundancy in his distinctions. Therefore 1st person (recognition) automatically implies 1st person perspective.
    Also in truth the nature of a 1st person perspective is much more intricate than represented by Ken.
    Indeed though 1st person (and in Ken's terms 1st person perspective) are represented in very static terms, in reality they would continually change (through interaction with other perspectives). So the "I" that is in relationship undergoes continual subtle change and Ken's treatment is not geared to deal with this.
    The problem is that 1st, 2nd 3rd (and 4th) person distinctions ultimately have no validity from an integral nondual perspective (as they necessarily all have meaning only in dynamic relation to each other).
    Put another way, "I" for example has meanings associated with all four quadrants (and likewise "we", "you", "he, she it and its").
    Therefore it is strictly meaningless to talk of a 1st person perspective without designating (in this arbitrary context) which quadrant aspect is implied.
    Indeed as I showed in the eight-sector model, we can associate 16 different meanings with "I" (8 of form and 8 of emptiness). So therefore we would have to designate in any arbitrary fixed context which of these meanings applies.
    Again however because Ken's treatment abstracts meanings from the proper interactive setting, we are left with merely reduced notions of 1st, 2nd and 3rd person (where distinctive quadrant and sector interpretations are ignored).
  7. As I mentioned before, Ken concentrates strongly on actual perspectives (pertaining directly to conscious interpretation) while greatly ignoring potential perspectives (corresponding to unconscious appreciation). Thus associated with every actual is a corresponding potential perspective (containing within it the possibility of change with respect to the existing actual perspective).
    Now the proper relation as between both "real" (actual) and "imaginary" (potential) perspectives must necessarily take place in a dynamic interactive context. However even in a more static treatment it is possible to distinguish potential perspectives, which literally enable us to imagine how reality might appear from a variety of alternative perspectives (that are not present in actual experience). [18]
    However when the "imaginary" aspect of perspectives is not recognised, the unconscious (potential) dimension is involuntarily projected into experience where it becomes misleadingly confused with the conscious (actual) aspects.
  8. This leads to yet another significant problem. Again because Ken tends to abstract from the dynamic interactive context concentrating merely on actual perspectives, he has no means of defining their qualitative nature (which significantly depends on the interaction of conscious and unconscious).
    Though we can have a wide spectrum of qualitative perspectives varying from unhealthy to healthy extremes, Ken's approach is somewhat neutral with respect to this point (which would then be of vital importance in an integral context).

Thus even in differentiated terms there are significant problems with Ken's approach to dealing with higher order distinct perspectives.

However when we attempt to view his treatment from an integral perspective, its limitations especially show where it becomes readily apparent that he is in fact offering a very reduced interpretation.

This for me is amply illustrated by Ken's attempt to demonstrate how mutual understanding arises in a relationship. Using his new mathematical notation he comes up with the identity.

1p(1p) x 1p(1-p) x 2p(1p) = 2p(1p) x 2p(1-p) x 2p(1p)

Now translating the Left-Hand side (in Ken's terms) this means.

My first person (aware of 1st person realities) i.e. 1p(1p) has a 1st person perspective i.e. 1p(1-p) of your 2nd person (that is also aware of 1st person realities) i.e. 2p(1p).

The Right-Hand then means.

You as 1st person (that from my context is 2nd person) i.e. 2p(1p) has a 1st person perspective (that from my context is again 2nd person) i.e. 2p(1-p) of your 1st person (that once more from my context is 2nd person) i.e. 2p(1p).

Ken then concludes that if these two sides of the equation match (i.e. if my 1st person view of you matches your own 1st person view of yourself), then we have mutual understanding.

However by abstracting from its dynamic interactive context, Ken has in effect completely reduced the integral aspect of understanding to mere differentiated appreciation. In other words he has given an interpretation of mutual understanding that is frankly absurd.

Because in experiential terms, perspectives necessarily arise in dynamic interactive terms, we cannot properly define them in an absolute manner (as Ken attempts to do). In other words understanding of perspectives is always relative, where what is known (as revealed) interacts with what is not known (as hidden). In other words we do not strictly enter relationships with pre-defined perspectives (in absolute terms) but rather seek to grow in our knowledge of necessarily limited perspectives (through relationship). Also the very nature of relationship requires that there is always an inevitable tension as between our separateness (related to the differentiated aspects of distinct perspectives) and our common identity (as mutual mediators of a nondual spiritual light).

Therefore though we must necessarily approach relationships through our separateness, to the extent that mutual understanding takes place we are enabled to embrace our common identity (where separateness no longer remains).

Thus the mystery of true communication especially in 1st person and 2nd person relationships is how we are led to dissolve our separate distinct perspectives so as to share an overall perspective of what we have truly in common (as ultimately nondual Spirit). In other words mutual understanding directly relates to the integral (rather than the differentiated aspect) of perspectives.

However when we examine Ken Wilber's formulation carefully we can recognise its severe shortcomings.

Though mutual understanding in any meaningful context is two-way i.e. where I relate satisfactorily to your perspectives and you in turn to mine, Ken here appears to represent mutual understanding in a one-way fashion i.e. where my perspective of you agrees with your perspective of yourself. However as we have seen, in experiential terms we do not enter relationships with pre-defined perspectives (though we may in reduced terms attempt to represent perspectives in a misleading absolute fashion!)

Indeed there are obvious worrying parallels here with Ken's - equally one-way - approach to criticism - as demonstrated in a recent letter to this Forum - where it would appear that for mutual understanding what is required is that the critic's perspective on Ken's work should agree with his own view. (The requirement for Ken in this context to agree with the perspective of his critic does not seem to enter his equation!)

Now Ken could perhaps reply that he actually intends the equation to be read from both perspectives. So if my view of, you agrees with your view of yourself and in turn your view of me, agrees with my view of myself then we would indeed be giving it two directions.

However this would only serve to highlight the reduced nature of Ken's interpretation where we would be demonstrating the recognition of what actually separates us (i.e. in our distinct perspectives).

However once again the true integral aspect of communication is in the mysterious process by which we are enabled to move from separate dual identities (in our distinct perspectives) to our common spiritual meaning (which is nondual and strictly without distinct perspectives). So mutual understanding - especially in an meaningful intimate context - does not take place directly just through affirmation of another's distinct perspectives (which emphasises separateness) but rather to the extent that we can let go of such perspectives so as to share what is truly common (where 1st person and 2nd person eventually lose any distinct meaning).

If in communication, another person is only able to see what one already sees in oneself then growth cannot take place. If for example two people have poor self-images and continually mirror these perspectives through communication, this would represent mutual understanding of a sort but of a very alienating kind (confirming both in the poverty of their separate existences).

However for healthy communication there must exist the potential in the relationship to lift one another to some degree out of unhealthy separateness towards the spiritual light of true union. [19] Thus true mutual understanding may mean seriously challenging and perhaps misinterpreting distinct perspectives but in a Spirit that ultimately seeks realisation of our shared spiritual identity.

Ken Wilber's Integral Mathematics

In Appendix 2 (to Excerpt C) Ken develops his Integral Mathematics which is designed as a notational system to represent all indigenous perspectives (rather than the more limited notion of variables as used in the customary mathematical sense). This is certainly a welcome initiative, which in its appropriate context I would see as having considerable potential merit.

However once again we need to place this initiative in perspective i.e. in the sense of providing an overall integral context for what Ken is attempting.

Mathematics it its accustomed analytic sense is very much based - in formal terms - on a distinctive specialised use of the cognitive linear understanding associated with the earlier of the two middle levels (conop and formop in Piagetian terms).

However - while fully recognising its immense contribution - the accepted formal interpretation of mathematical activity represents a gross reduction of what is required to do mathematics in a dynamic experiential manner.

For example in four quadrant terms, mathematics entails the interaction of both Right-Hand (exterior) and Left-Hand (interior) quadrants. However in the purely static interpretation that characterises conventional understanding, this relative interaction (between quadrants) is completely frozen leading to absolute type appreciation.

In conventional analytic terms when we ignore the sign of a number we define it in absolute terms. Therefore for example + 2 and - 2 are identical in absolute terms.

Likewise in holistic terms when we ignore the sign of quadrants (which in dynamic terms interact in a complementary manner to each other) we obtain an absolute understanding.

What this means in effect is that both interior and exterior aspects are reduced in terms of each other leading to the same absolute interpretation. Therefore though one may well concede that mathematical activity necessarily entails interior constructs as a means of interpreting exterior relationships, in absolute terms it makes no difference to the interpretation involved (when we ignore this dynamic context). Likewise Upper and Lower quadrants are conventionally treated in absolute terms and thereby reduced to each other where for example the whole is treated as the sum of the parts (e.g. 2 + 2 = 4). Even more seriously though mathematical activity - especially of the more creative kind - requires the interaction of reason and intuition, in formal terms it is expressed in merely rational terms. I will just give one more example at this stage. Though mathematics requires the use of symbols and signs (which must be sensibly verified) again in conventional terms a reduced - merely rational - interpretation is given of this interaction as between sense and reason.

So in dynamic experiential terms the activity of mathematics entails the interaction of the 4 quadrants (and indeed the eight sectors) [20] However in conventional terms this interaction as between these quadrants (and sectors) is ignored leading to the standard absolute interpretation of its nature.

However when we attempt to appreciate the nature of mathematics more accurately in terms of its inherent dynamic context then it opens up the prospect of remarkable new domains of understanding (with special relevance for a scientific integral interpretation of reality).

Now in fairness to Ken he has indeed accurately seen into some of the limitations of conventional mathematical appreciation. However once again I would see him as offering a vision-logic perspective on the limitations of conventional mathematics. Likewise in turn his attempts to reformulate mathematical understanding - though very interesting in their own right - again reflects a merely vision-logic understanding. However the point that I repeatedly make is that there are more advanced "higher" stages beyond the vision-logic of the centaur and it is the interpretation of these stages that is necessary for a true integral mathematical approach.

Again though the four quadrants are necessarily entailed (with their corresponding perspectives) in the experience of the earlier middle stages, with respect to formal cognitive interpretation, these quadrants will be necessarily significantly reduced in terms of each other (at these stages).

The great advance of vision-logic therefore is the clear recognition that all these quadrants (with their corresponding perspectives) have their own unique validity with distinct applications to reality and that therefore in a balanced approach that they need to be equally honoured. And we could look at Ken Wilber's writings in recent years as a wonderful testament to such vision-logic recognition. Ken has realised, that if we are to consistently apply vision-logic understanding to mathematical activity, that its very symbols should equally signify the unique nature of the quadrants (allowing for both unreduced personal and impersonal interpretations). Indeed because the primordial perspectives are based on such quadrant understanding thereby mathematical symbols should - in a vision-logic interpretation - represent all of these perspectives.

Indeed Ken has also provided in his Appendix a very interesting enhanced interpretation of the nature of conventional mathematics (from the standpoint of vision-logic). [21]

However the problem with this approach is that vision-logic is not in itself capable of interpreting the integral aspect of experience in a consistent manner. As we have seen time and time again, because vision-logic still formally relies on the unambiguous validity of linear (asymmetrical) type distinctions, it inevitably reduces the integral aspect (which is ultimately nondual) to mere differentiated interpretation.

Therefore though I would see Ken as making a valuable contribution in terms of his attempt to provide an alternative mathematics that can honour all perspectives, it inevitably demonstrates the limitations of the stage of understanding on which it is based (i.e. vision-logic). While providing a novel notational system for the classification of higher order distinct perspectives, he does not really clarify in integral terms the mathematical nature of perspectives. He maintains that the personal language of perspectives i.e. 1st person, 2nd person and 3rd person already embeds a universal mathematics. However while it certainly is true that the fundamental integral mathematical structure of reality is deeply implicit in perspectives, this structure is not directly revealed through the personal language of perspectives (which is full of anomalies) but rather through a more balanced dynamic interactive formulation based on complementary opposites.

Indeed the clue to the integral mathematical nature of perspectives lies in the holistic geometrical interpretation of the four quadrants and by extension eight sectors (from which all perspectives can ultimately be derived). However Ken does not yet properly appreciate this vitally important truth which provides the appropriate key for the consistent scientific reconciliation of distinct (differentiated) perspectives with overall (integral) perspective.

A Stage Map of Mathematical Interpretations

However let us now enlarge this vision of mathematics by recognising that associated with every major stage of development is a unique interpretation of mathematical symbols. This means therefore that associated with H1 (psychic/subtle), H2 (causal) and H3 (nondual) we have increasingly refined spiritually based interpretations of mathematical symbols, that offer remarkable new possibilities for the precise scientific interpretation of development that is truly integral.

Then beyond this we have even more comprehensive interpretations R1 (Radial 1), R2 (Radial 2) and R3 (Radial 3) that increasingly can combine both holistic (integral) and analytic (differentiated) aspects of mathematical understanding in a balanced creative fashion.

In general terms. I would define 3 major types of mature mathematical understanding (with 3 sub-categories in each case).

Analytic Interpretations

This mathematical understanding is based on the three middle levels (relating to the unambiguous interpretation of mathematical symbols).

Analytic 1: this relates to the merely concrete empirical use of symbols as in arithmetic (relating to conop understanding).

Analytic 2: this relates to the more abstract theoretical universal use of symbols as for example in algebra (relating to formop understanding).

(Conventional mathematics is formally based on the Analytic 1 and Analytic 2 stages).

Analytic 3: this relates to the vision-logic attempt to use mathematical symbols to represent all aspects of development while still operating within the confines of unambiguous interpretation. (I would see Ken's Integral Mathematics of primordial perspectives as belonging to this category).

Holistic Interpretations

This mathematical understanding is based on the three "higher" levels (relating to the dynamic paradoxical interpretation of mathematical symbols). In other words mathematical symbols now lose their residual unambiguous meaning to become paradoxical symbols of transformation (both as a means of mediating the nondual spiritual light and of likewise attracting such light). In principle every mathematical symbol e.g. number, operation, equation with a well-defined meaning in analytic terms has a corresponding interpretation in a holistic manner (with a direct integral application to development). However such holistic interpretation - through which the inherent interactive meaning of symbols is revealed - cannot replace the need for valid authentic experience of these levels (and indeed is in no way guaranteed through such experience).

Properly understood this understanding is directly intuitive (and only indirectly rational) and unfolds as knowledge of the inherent dynamic structure of the level in question. I would see the integral mathematical understanding of symbols as both the most demanding and rewarding of any meditation practice, coming as close as is humanly possible in the phenomenal realm to pure Spirit.

Indeed the basic interface as between the dual and nondual realms is mathematical in this holistic sense. Inherent in form and emptiness are the holistic notions of two most fundamental number symbols (1 and 0 as unity and nothingness respectively). Likewise implicit in the very notions of existence and non-existence are the two holistic notions of the most fundamental mathematical operations of addition (i.e. as posited) and subtraction (as negated) respectively.

Therefore when we come most closely in meditation to this interface as between form and emptiness the universal meaning of the most simple symbols is wonderfully revealed (so that ultimately all development can be seen to be encoded in binary digital terms).

Put another way holistic mathematical understanding provides the most scientific interpretation possible of the philosophy of the "higher" levels.

Just as there is universal agreement as to nature of the truths of conventional science (in their appropriate context), likewise in future the same kind of agreement with respect to the mystical core of the various religious traditions can take place through holistic mathematical understanding applied to the "higher" levels. In other words substantial agreement in scientific terms as to the dynamic structure of all the key stages of development will thereby be possible.

Integral 1: this is based on the cognitive understanding of the psychic/subtle level (where spiritual intuition and bi-directional circular understanding increasingly interpenetrate in experience). Mathematical symbols are now understood in an utterly distinctive manner where their meanings are derived directly from the dynamic interactions through which they emerge. In other words when appropriately understood at this stage, the inherent integral structure of reality is understood as mathematical (in terms of the corresponding understanding of the stage). [22]

In more formal terms it is based on Type 1 complementarity (relating to the interdependence of complementary relationships understood in "real" conscious terms).

Integral 2: this is based on the more refined cognitive understanding of the causal level. Here the dynamic nature of phenomena can be increasingly understood both with respect to their conscious and unconscious aspects. Holistic mathematical appreciation now becomes truly "complex" (where symbols are understood with respect to their "real" and "imaginary" aspects). In formal terms this interpretation of mathematics also includes Type 2 complementarity relating to interdependent relationships with respect to both their "real" (conscious) and "imaginary" (unconscious) aspects.

Integral 3: this is based on the most refined appreciation (approaching nondual awareness). Here Spirit is so closely related to form that the dynamic nature of phenomena can be simultaneously understood with respect to both their "real" (conscious) and "imaginary" (unconscious) aspects so that they dissolve (without undue attachment) as soon as they arise.

However paradoxically - when used in an unattached manner - mathematical symbols now possess the most universal qualities possible for successful encoding of all stages of development. In formal terms this interpretation of mathematics is based on Type 3 complementarity enabling all relationships in the comprehensive eight-sector model of reality to be successfully encoded.

Radial Interpretations

The radial understanding of mathematical symbols is most comprehensive gradually combining both holistic and analytic interpretation in a balanced manner.

As I have often stated I would see my own recent work as the attempt to provide a preliminary Radial 1 mathematical interpretation of reality. Though I would see such an approach as scientifically consistent in integral terms - by its very nature - it is still confined analytically to a somewhat general outline of development (though it has inherently the capacity for considerable further elaboration). More advanced radial approaches would lead to detailed science of both the holistic and analytic kind (that are consistently related to each other).

The most advanced would lead to a significant transforming dimension through the non-possessive desire to "convert" the world to a new form of advanced scientific thinking (that is equally analytic and holistic). Indeed this would be no different in principle from the desire of the great religious missionaries of the past in various traditions to convert people to God. This would indeed be attempting the same conversion (but in a distinctive manner through creating a new more universally acceptable spiritual belief system).

Radial 1: the emphasis switches back from specialisation of nondual contemplative capacity to reincorporating this in stable terms with dual phenomena.

In radial mathematical terms this enables the presentation of a comprehensive integral viewpoint in broad linear terms. It also enables greater appreciation of both the analytic and holistic interpretation of mathematical symbols where one can switch quickly from one meaning to the other (often leading to highly creative linkages). However it still lies closer to integral rather than conventional interpretation.

Radial 2: this would enable much more detailed analytic work with respect to the various disciplines while maintaining overall consistency in integral terms. For example a Radial 2 approach to string theory would entail not only an ability to appreciate conventional developments in analytic terms (while perhaps contributing to them) but also the capacity to appreciate the meaning of such findings in an integral manner (through a philosophically coherent overall mathematical interpretation). Remember that in principle every mathematical relationship with a well-defined meaning in analytic terms has a corresponding dynamic holistic counterpart with an integral scientific application to development. For example Riemann's Last Hypothesis is an important (as yet unproven) assertion in conventional analytic terms. By definition the holistic interpretation of this hypothesis has an equally important - as yet unknown - integral application to development (with complementary physical and psychological applications). I believe that the true holistic meaning of a great number of - even highly abstruse - mathematical relationships can unfold at this level of development. The most creative scientific approach would require that both analytic and philosophic integral appreciation develop in tandem with each other. In this way new analytic findings would be associated with corresponding integral appreciation. Likewise integral appreciation would suggest new creative analytic lines of investigation.

Radial 3: this would entail a great increase in facility to readily intuit the integral meaning of any analytic mathematical relationship while in turn greatly enhancing creative development with respect to analytic investigation. However it would also be likely to involve a significant exterior transforming dimension in the desire to bring the "good news" of such a new vision of mathematics to a wider world (with the capacity and energy to do so).

Though I would see mathematics in all its expressions as formally relating to a specialised cognitive capacity (relating to a limited range of perspectives), it is important to recognise that at the "higher" and radial levels this capacity can only properly flourish when all perspectives (personal and impersonal) achieve sufficient development. So here - though mathematics - in direct expression - is largely limited to certain perspectives (of an impersonal kind) indirectly it requires for its successful development a significant balance to be maintained as between all perspectives.

So were we have the basis for 9 mature mathematical interpretations of reality (that are directly related in each case to the understanding appropriate to the corresponding stage of reality).

We could then associate 3 more interpretations with the earliest three levels ("lower") relating to the confused understanding of the relationships corresponding to the three ("higher") integral levels. [23] So the structure of reality through and through is mathematical in the most fundamental sense.

Now Ken Wilber indeed obtains a clear sense of this realisation in rooting the nature of mathematics in the sentient relationships as between holons.

However because he is using vision-logic understanding (pertaining merely to the most advanced of the middle levels) his interpretation of mathematics is subject to the limitations of vision-logic (which translates the integral aspect of development in a somewhat reduced manner).

For truly integral appreciation of reality we require the interpretations of the "higher" levels where all phenomenal symbols can be understood in an inherently dynamic manner thereby enabling their continual transformation through growing immersion in nondual spiritual awareness.

Likewise for integral mathematical appreciation we require an interpretation of mathematical symbols, that is directly based on the cognitive understanding associated with the "higher" levels (i.e. psychic/subtle, causal and nondual).


Though I admire Ken's new appreciation of mathematics (as rooted in the relationship between holonic perspectives) I would not consider it - as with his intellectual interpretation of reality generally - as properly integral.

In fact I would be greatly concerned at the way the notion of "integral" is being bandied about - as I see it - in such a reduced manner by Ken. And I feel qualified to speak in this regard as I have been engaged for more than 35 years now in developing an integral mathematical approach (which can deal consistently with many of the limitations that I see in Ken's developmental system). I would also be greatly concerned that the "Integral Institute" is being used to promote Ken Wilber's reduced view of integration as if synonymous with the integral approach generally (while shutting itself off from external criticism of his view).

I have no objection to a "Ken Wilber Institute" as designed to promote Ken's very considerable contribution to development studies. However an "Integral Institute" worthy of the name should promote differing overall perspectives of what is "integral" and not prematurely identify itself with any one - necessarily limited - view. So there is even perhaps a danger if this trend continues that Ken Wilber's "integral approach" could develop into an unhealthy new ideology (with the "Integral Institute" serving as its main propaganda vehicle). [24]

I firmly believe that worthwhile criticism of Ken's thinking is much more likely to come from outside the Institute (and not inside as Ken would have us believe) However whether proper recognition of such criticism can be given by insiders remains another matter! Therefore I greatly welcome Frank's willingness to promote alternative integral viewpoints on his web-site.

So finally while recognising Ken's considerable contribution on perspectives, paradoxically the great weakness of his treatment - in intellectual interpretation - is the lack of any genuine integral dimension.

In other words we need to distinguish carefully the differentiated notion of distinct perspectives (both primary and composite) form the integral notion of overall perspective (relating to the manner in which distinct perspectives are configured in experience).

Whereas Ken has a great deal to offer with respect to the clarification of distinct perspectives (i.e. the differentiated aspect), he has precious little to say regarding overall perspective (i.e. the integral manner in which perspectives are related).

Furthermore Ken's approach is lacking in overall perspective in that the configuration of perspectives is necessarily stage specific with a distinctive understanding corresponding therefore to each of the major levels of development.

However - as we have seen - Ken's treatment is firmly rooted in the vision-logic understanding of the centaur stage (which by its nature is not properly capable of consistently interpreting the integral aspect of development).

Furthermore his "integral" mathematical notation - designed as a way of representing his higher order perspectives - is likewise rooted in vision-logic - and therefore is not designed to translate the integral aspect of overall perspective.

Therefore to add further clarification to these comments, I hope in my final contribution to provide an integral mathematical mapping of the 12 major levels of development, so as to scientifically demonstrate the precise configuration of perspectives (integral perspective) that applies in each case.


1. The differentiated aspect relates to the formation of distinct perspectives in experience. The integral aspect however relates to overall perspective (as the manner through which distinct perspectives are related to each other). As a distinct configuration of perspectives arises for each stage, therefore we can define a unique integral perspective for each of the major levels of development.

2. As the centaur stage (on which vision-logic is based) bridges the middle and "higher" levels of development it partakes of elements of both. However the spiritual intuitive element (facilitating an integral viewpoint) remains merely implicit whereas the actual explicit formulation of relationships is still based on unambiguous asymmetrical appreciation. Therefore vision-logic inevitably reduces - in intellectual interpretation - the integral aspect of experience to the differentiated. However if one looks at development from a - merely - vision-logic perspective this problem of reduced interpretation will not of course be apparent.

3. I am aware that the most original ideas very often do not come to light (as they cannot be properly appreciated in terms of conventional notions of meaning). Almost - by definition - anything that is truly original will not strike a ready chord with accepted wisdom. It is quite probable therefore that others - who I am presently unaware of - are uncovering the same basic truths (without confirmation from the intellectual community). So I see myself as representative of important truth possibly shared by other "outsiders" in the culture (that is not yet generally recognised).

4. More correctly as - in dynamic terms - there is two-way interaction as between "higher" and "lower" levels, one should say that the mature interaction of "higher" and "lower" levels is necessary to explain the structures pertaining to the confused interaction of the same levels.

5. The deductive approach relates to the application of formop to conop understanding whereas the inductive relates in reverse manner using conop to suggest formop relationships.
So vision-logic entails the more ready use of both deductive and inductive understanding. When this is considerably inspired by spiritual intuition, vision-logic can be used in a very flexible multifaceted manner that is implicitly suggestive of integration (though explicitly in a somewhat reduced manner).

In terms of notation I refer to the lowest of the middle levels as L0 (conop) as it still participates in general terms with "lower" level mythical type understanding. So L0 in this context represents the "highest" of the "lowest" levels that is now becoming middle. The middle of the middle levels (formop) most closely bridges the division between "higher" and "lower". Thus I call it L0, H0. The "highest" of the middle levels then implicitly is already sharing the understanding of "the higher" levels. Therefore I refer to it as H0 (the ground floor of the "higher" levels).

6. This entails in practice the ability to dynamically negate mathematical symbols as soon as they arise in experience so that undue possessive attachment does not arise. This equally entails that successful balance be maintained as between the interior and exterior perspectives (with respect to both personal and impersonal aspects). Therefore, though mathematical understanding at this stage explicitly relies on the impersonal use of perspectives, implicitly it requires the appropriate counterbalancing development of personal perspectives.
In intellectual terms, the three higher levels are defined in an integral manner by distinct notions of complementarity. So H1 is defines by Type 1 complementarity that relates to the bi-directional relationship of opposites that are considered in "real" terms (geometrically by the lines representing polarities at a 180 degree angle to each other) . H2 is defined by Type 2 complementarity that relates to the bi-directional relationship of opposites that are relatively "real" and "imaginary" (geometrically at a 90 degree angle to each other) . H3 is defined by Type 3 complementarity that relates to the bi-directional relationship of opposites that are both "real" and "imaginary" (simultaneously) with respect to those that are either "real" or "imaginary" (separately). In geometrical terms these are at 45 degree angles to each other.

7. Just as we can have enhanced interpretations of "lower" from corresponding "higher" levels, equally we have diminished interpretations of "higher" (strictly more advanced) levels from corresponding "lower" levels.
In dynamic interactive terms the experience of all levels is available in some measure (as they are all necessarily related to each other). However when such experience becomes predominantly identified with certain levels (as in our culture the middle) this entails that in practice only a very diminished understanding of the more advanced levels ("higher" and "radial") can unfold.
As my own approach is specifically designed as a Radial 1 cognitive interpretation (in mathematical terms) of reality, this means that I can necessarily only offer a diminished interpretation (in these terms) of R2 and R3. However this still enables the overall integral aspects of these levels to be interpreted (though without detailed applicability to different situations).

8. I will just briefly elaborate on the nature of these enhanced perspectives here.
For example we could take Ken Wilber's holarchic approach to development as representative of vision-logic interpretation (from a vision-logic perspective). Now in holistic mathematical terms this is characterised by giving the relationship as between whole and part (i.e. where every whole is part of another whole) just one "real" direction. However with the enhanced vision-logic perspective from H1, development can now be given two "real" directions in hierarchical terms (i.e. holarchic where every whole is part of another whole) and onarchic (where every part is also whole - in the context - of other parts).
With the enhanced interpretation from H2, development can now be given two "real" (pertaining to conscious understanding) and two "imaginary" asymmetrical interpretations.
Then with the enhanced interpretation from H3, development can be given additionally four "complex" linear interpretations (combining both "real" and "imaginary" elements).
However the very nature of these levels is that opposite directions are more closely related. Therefore overall understanding tends more to nondual rather than dual understanding.
With Radial 1 an even more enhanced appreciation becomes available, where the complex linear interpretations can be used increasingly in a more separate fashion enhancing dualistic appreciation of reality. With Radial 2, the "imaginary" can be likewise separated from the "real" interpretations in dualistic fashion (while also being united in contemplative terms). Finally with Radial 3 the two "real" interpretations can be fully separated in dualistic terms (and correspondingly united in nondual terms).

9. As soon as development commences with respect to any phenomena, likewise the unfolding of perspectives takes place. However though clearly much more complex development can take place in human terms, the nature of perspectives (at the beginning) is the same in all cases.

10. We have here the transition from prime (or primitive) to natural perspectives, which as I hope to demonstrate in the next contribution has a direct relationship to the holistic interpretation of number.
With primitive perspectives the nature of distinct perspectives cannot be distinguished from overall perspective. However though confusion still remains as between the "real" (conscious) and "imaginary" (unconscious) aspects, with natural perspectives more composite understanding (enabling distinct perspectives to be stabilised in experience) emerges.
However in nature the actual configuration as between overall and distinct perspectives remains extremely rigid. What this in effect means is that perspectives can evolve very slowly (perhaps over millions and even billions of years). Thus the making of nests by birds represents a very complex configuration of perspectives. However - for most species - the basic process is likely to remain substantially unaltered for some considerable time.

11. 4-bit is simply 2 X 2 X 2 X 2 = 16 i.e. 2 (raised to the power of 4). 8-bit, 16-bit and 32-bit therefore represent 2 (raised to the power of 8), 2 (raised to the power of 16) and 2 (raised to the power of 32) respectively.
Another interesting analogy can be made as between computer images and perspectives. The higher the resolution of an image the more memory it requires to store. Likewise in human terms (rigid) higher-order perspectives make increasing demands on memory thereby greatly reducing the capacity for creative experience. With truly spiritual experience, perspectives are constantly recreated anew (as the need arise). Therefore as so little needs to be committed to memory, considerable creative freedom can be enjoyed.

12. Other combinations are also possible. 3rd order perspectives (12-bit) would arise from combining the 16 1st order with the 256 2nd order perspectives. Then - for example - if these 3rd order were to be all combined with each other we would obtain 6th order (24-bit) perspectives.

13. It has to be stated however that in general there is a considerable bias in terms of our higher-order perspectives in favour of the merely impersonal aspect (dictated largely by advances in modern technology). This is somewhat ironical given the personal language (1st person, 2nd person and 3rd person) to describe such perspectives. Already with respect to many services simulated has replaced actual human contact (and this trend will undoubtedly increase). Indeed it raises important issues. If I carry out a transaction on the phone and am directed through each step by a simulated human voice can we maintain that any personal contact has taken place?

14. The spiritual movement from higher order rational to lower order natural perspectives not surprisingly leads to a new found appreciation of nature. Thus nature mysticism is often associated with the earliest of the "higher" stages.

15. This explains my own strong preference for a system of thinking that is intimately based on direct personal (and impersonal) experience. I have no objection to research in its rightful place. However this necessarily entails increasingly more indirect perspectives (2nd hand, 3rd hand and so forth) that cannot possibly substitute for direct 1st hand experience.
So the reason I am so confident in my own perspective is because of the manner in which every key idea has been intimately derived from 1st hand experience. For me this provides much more valuable confirmation of its validity than any amount of 2nd hand research.
However I have repeatedly found throughout my career that the academic community in general is far more impressed with secondary research-based ideas (rather than those deriving from direct experience).

16. This perhaps explains why in the past contemplative spiritual development generally entailed - at least temporary - retirement from the world through entering monastic communities. However though clearly appropriate for some, this can accentuate the merely contemplative (nondual) aspect of spiritual development. The attainment of genuine contemplative capacity while remaining in the world - though it may indeed entail severe short-term problems of adjustment - ultimately serves as a better preparation for radial spiritual development (where contemplation and activity are combined to a marked degree).

17. Number notation is in fact used in many contexts as a means of providing order for data. For example a book will be numbered in pages and divided into several Chapters with perhaps a variety of further subheadings (that are also numbered in various ways). Extending this example further the organisation of vast library systems entails the ingenious use of number notation. However while recognising the value of such notation we could hardly claim that it throws much light on the inherent mathematical nature of the underlying structures of reality.
In like manner though Ken Wilber's notation is indeed valuable as a means of classification of higher-order distinct perspectives, in itself it does not provide an interpretation of the inherent mathematical nature of such perspectives.

18. I would like to draw attention once again to the excellent article by Mark Edwards (one of many) on this forum "Through AQAL Eyes Part 7: "I" and "Me" and "We" and "Us" and "You" and "Yous": Sorting out Ken's Holon of Mixed Perspectives" where he deals with the issue of notional perspectives (which relate to the "imaginary" aspect in terms of my own treatment).

19. In dynamic terms this requires that both the actual ("real") and potential ("imaginary)" aspects interact. So the spiritual light - in a direct sense - literally giving a person the potential to change the quality of their perspectives is mediated through the "imaginary" aspect.

20. The four quadrants are not sufficiently comprehensive to explain the full mathematical process. For example the important interaction as between reason and intuition - which characterises all creative work - relates directly to the diagonal polarities of form and emptiness (which requires the eight-sector model).

21. For example this section from Appendix B (to Excerpt C) amply describes the enhanced perspective of conventional mathematical activity from the perspective of vision-logic. Though I would use different language that is inherently more mathematical - to describe the same activity, I would have no real disagreement with Ken on this issue.

"Equations in the real world of sentient beings are thus equations of mutual resonance. Even a mathematician, who writes (x = 3y), and shows it to another mathematician, who agrees that in that case, x does indeed equal 3y, is actually asserting the following: my first person has a first-person perception of a third-person abstraction [(x = 3y)], and I believe that this third-person abstraction is, or would be, true for all other persons who looked at it. Therefore, I am asserting that this abstraction is not merely true for me (or my first person), but is true for all other first persons; which means, if I take a third-person view of my third-person abstractions, I still believe that you will agree with me if you look at them in a third-person way yourself--and not only you, but all others who look at this dispassionately or objectively or rationally, will agree with me. I am actually claiming, then, that my first-person perception of my third-person abstractions is really a third-person (plural) perception [which is represented as (3-p*pl)] of this third-person abstraction: 1p(1p) x 1p(3-p*pl) x 1p(3p), which means, my first person has a third-person plural view of my third-person algebraic assertion. If you look at my algebra and attempt to take up a third-person (plural) view of it, then: 2p(1p) x 2p(3-p*pl) x 1p(3p), which means, your first person has your third-person (plural) view of my first person's assertion (which is the algebra, the third person we are considering, where "3p" in this case means "the assertion x = 3y," which is the third-person "it" we are discussing). The heart of the matter is that you might indeed agree with me that the algebraic equation is correct. If so, we have: 1p(1p) x 1p(3-p*pl) x 1p(3p) = 2p(1p) x 2p(3-p*pl) x 1p(3p) That is what an equation in the real world looks like, even among mathematicians. Equations in the real world equate interiors (as well as exteriors), and thus they are built not just of exteriors that can be "equal" but of interiors that can be "equal," which is to say, can equal each other in mutual understanding or mutual resonance. That is what the equal sign means in the real world. The Kosmos vibrates with those equations of souls touching each other. That is what the Kosmos is made of."

22. Just as at the analytical level, mathematical ability is a special talent that is not equally shared by all, likewise this is true at the holistic level of "higher" level spiritual experience. Indeed it is even more so here as we have as yet no tradition of a properly developed system of integral mathematics. So in the past the great mystics who traversed the "higher" levels did so in the main without any reference to the advanced mathematical structures of these levels. There have been some exceptions. For example in the Western tradition Plato, the Pythagoreans, Nicholas of Cusa, Leibniz, Pascal and in our present day Jung, Merrell-Wolff and Nalimov have in various ways shown some insight into these higher level structures (though falling well short of any developed system)!

Equally it is possible that some may have a latent holistic mathematical ability which - if developed - could act as a considerable catalyst for authentic mystical experience. In this case much of what I am saying could resonate to some degree with their understanding.

23. Again the "lower" appreciation refers to confused dynamic interpretations that structurally complement the corresponding "higher" levels. Thus the interpretation of L1 (mythic) is provided through H1 (where mature interaction between opposites is now interpreted as confused). In like manner the interpretation of L2 (magic) is provided by H2 (causal) and the interpretation of L3 (archaic) by H3.

24. Though I have enjoyed to a degree many of the discussions on "Integral Naked", there has been remarkably little real cutting edge about them. Everyone seems to be singing from the same hymn sheet (i.e. Ken Wilber's) which thereby leads to an inevitable blandness regarding proceedings.


Collins, P (2002) Introduction to Holistic Mathematics (Chap 3 of ongoing work "Development - the Radial Approach").

Edwards, M (2002) Through AQAL Eyes Part 7: "I" and "Me" and "We" and "Us" and "You" and "Yous": Sorting out Ken's Holon of Mixed Perspectives

Mc Dermott, M. (1998) Knowledge and the Knower; Complexity and the Self. This monograph is available for download at the following web address.

Nalimov V.V. (1982) Realms of the unconscious: the enchanted frontier. (Robert G. Colodny, Ed.). Philadelphia: ISI Press.

Penrose R. (1989) The Emperor's New Mind; concerning computers, minds and the laws of physics. Oxford University Press.

Wilber K. (2003) Appendix B (An Integral Mathematics of Primordial Perspectives) to Excerpt C The Ways we are in this Together: Intersubjectivity and Interobjectivity in the Holonic Kosmos from Volume 2 of Kosmos Trilogy available at Ken Wilber online with web address.

Wilber, K. (2003) The Ways we are in this Together: Intersubjectivity and Interobjectivity in the Holonic Kosmos, excerpt from Volume 2 of Kosmos Trilogy.

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