Once again it consists of a circular component (0) – based on complementarity of polar opposites - and a bi-directional linear component (1) – based on (dualistic) separation of these same opposites.

Just as in conventional mathematics 0 can be statically expressed as 1 – 1, likewise in holistic mathematics 0 (i.e. the intuition underlying circular logic) can be dynamically expressed in terms of the combination of both the positive and negative directions of linear understanding (1 – 1).

Thus it is the very use of the bi-directional (rational) linear approach that naturally generates the non-directional (circular) intuition (i.e. 1 – 1 = 0 in holistic terms). Likewise this circular (non-dualistic) approach leads back naturally to a bi-directional linear approach (0 = 1 – 1 in holistic terms).

Thus in this qualitative binary approach the very dynamics of experience are expressed through the mutual interaction of linear and circular aspects (and circular and linear). Through the linear aspect experience is differentiated; through the circular aspect experience is integrated.

So experience once again is a continual process of differentiation and integration (integration and differentiation).

By contrast in a unary approach, linear and circular elements are separated.

Thus when the (rational) linear approach is employed it tends to be used in a one-directional fashion.

Even when (circular) intuitive experience is recognised it remains divorced from (linear) rational understanding.

Ken Wilber’s approach – though magnificent in its detail and sophistication – is basically a unary one.

The root problem of Ken’s work – in the terms that I define it – is methodological (i.e. the use of a unary rather than a binary system). All the other difficulties (e.g. his treatment of the pre/trans fallacy and evolution) spring from this root problem.

I am now going to illustrate the above with reference to Ken’s popular treatment of the four quadrants (which has been mentioned in recent posts on this Forum). I hope to demonstrate clearly the nature and limitations of Ken’s use of a one-directional linear approach.

Now Ken defines his four quadrants horizontally in terms of exterior and interior aspects and vertically in terms of individual and collective aspects. All holons have these four aspects (or facets).

UL (Upper Left) quadrant represents the individual-interior and the UR (Upper Right) the individual-exterior aspects. (Ken also refers to these quadrants as Intentional and Behavioural respectively).

Now when we examine Ken’s treatment closely we can see that he is using a strictly one-directional approach. Thus he defines UL in terms of psychological experience and UR as one's (exterior) physical body.

Thus in polarised terms UL is personal and UR is impersonal.

Ken frequently associates this (interior) psychological aspect with depth and (exterior) physical aspect with surface.

However we can reverse these poles and see UL as (interior) physical and UR as (exterior) psychological.

Insofar as the self is simply responding emotionally to surface stimuli, it is (interior) physical (as a body-self). Equally insofar as the mind creates a conceptual context for the registering of scientific empirical data it is (exterior) psychological.

This leads directly on to the next point where I believe Ken makes a naïve distinction.

Once again Ken identifies (interior) UL with (hermeneutic) interpretation and (exterior) UR with (scientific) perception. From an experiential perspective this is simply not tenable.

In order to interpret subjective (interior) data we must – as I have stated - register these data.

For example if someone greets me in an unpleasant manner I first respond emotionally. Through this experience I perceive subjective phenomena. This is the immediate surface reaction. I may then try to interpret this experience (e.g. understand why I feel so upset). Thus in experiential terms both the perception and the interpretation are complementary. We cannot have one without the other.

Equally – from an experiential point of view – there is no such thing as simple perception of scientific data. All empirical data are meaningless until given some coherence by the organising mind. Thus we always interpret scientific data (through the conceptual frameworks we impose on that data). Scientists frequently forget this important point but it does not alter the experiential reality. I am surprised however that Ken makes such a fundamental mistake!

Once again we cannot have (exterior) perception without corresponding conceptual interpretation.

Thus expressing this experiential reality in polarised language requires a bi-directional approach.

If we now move to the bottom (horizontal) quadrants, we have LL which is Interior- Collective and LR Exterior-Collective. There are also referred to as Cultural and Social.

Now once again we have the same problem.

In Ken's treatment LL has a direct psychological meaning (representing a shared group consciousness). LR has a physical meaning - representing the material structures of society (through which shared values are institutionalised).

However the relationship can also be taken in the other direction so that LL has a physical and LR a psychological aspect. Again the physical represents the (immediate) emotional reaction to material institutions. The psychological represents the contextual framework through which institutions are viewed.

So again LL is not confined to interpretation; it equally involves perception.

Likewise LR is not confined to perception; such perception would have no meaning in the absence of corresponding interpretation.

The vertical direction of the quadrants involves the relationship of individual to collective. These would be UL and LL and UR and LR.

This can be read in bi-directional fashion. Now, both individual and collective have quantitative and qualitative aspects. In the former case we emphasise specific facts; in the latter we emphasis holistic form.

We can start by identifying the (physical) self directly as (affective) emotional experience and the (psychological) self as (cognitive) rational experience.

Thus if UL represents the individual (psychological) self, then the opposite pole represents the collective (physical) self. Also the former involves a qualitative (holistic) and the latter a quantitative (specific) interpretation.

When we take it in the other direction we have the individual (physical) self and the opposite pole of the collective (psychological) self. The former involves a quantitative (specific) and the latter a qualitative (holistic) meaning.

Thus in experiential terms, because of the interaction of reason and intuition, understanding acquires both qualitative (intuitive) and quantitative (rational) aspects. These apply to both the individual UL and the collective LL.

This also applies on the opposite right-hand vertical side.

Thus UR and LR i.e. exterior physical reality (individual) and exterior physical reality (collective) also have both quantitative and qualitative aspects.

Thus if UR represents a quantitative analysis of the physical body (rational) then LR represents a qualitative synthesis of physical reality (intuitive).

Finally taking it in the other direction if UR represents a qualitative analysis of the physical body (i.e. intuitive) then LR represents a quantitative collective synthesis of psychological reality (rational).

Once again each quadrant (in horizontal terms) can be given both a concrete (perceptive) and formal (interpretative) understanding; furthermore each quadrant (in vertical terms) can be given a rational (quantitative) and intuitive (qualitative) understanding.

Thus using a bi-directional linear approach we can show that each of the four quadrants in fact displays four distinct aspects. So we now have 16 rather than 4 distinct ways of viewing reality.

There is a remarkably close relationship here with the Jungian Theory of Personality Types as extended by the Myers-Briggs Type Indicator.

The MBTI is based on four clear polar distinctions.

Thus we have the distinction as between extrovert and introvert (E and I).

This corresponds very closely to Ken's distinction as between exterior and interior.

We then have the distinction as between sense and intuition (S and N).

This corresponds quite closely to Ken's distinction as between individual and collective.

So we now draw a direct parallel between the MBTI and Ken's four quadrants.

ES types (extrovert and sense oriented) will operate out of UR (exterior-individual).

IS types (introvert and sense oriented) will operate out of UL (interior-individual).

EN types (extrovert and intuitive) will operate out of LR (exterior-collective).

Finally IN types (introvert and intuitive) will operate out of LL (interior-collective).

So basically one's tendency to identify with a particular quadrant is a function of personality type.

However the MBTI involves a considerably more detailed breakdown than the four quadrants. In fact it corresponds directly with the 16 fold breakdown that I have demonstrated (using the two-directional linear approach).

I will now attempt to flesh out the four facets of each quadrant in terms of the full MBTI descriptions.

The UR (exterior-individual) quadrant relates to the extrovert rational personality. Its four aspects are:

ESFP (extrovert, rational, personal, concrete)

ESTP (extrovert, rational, impersonal, concrete)

ESFJ (extrovert, rational, personal, formal)

ESTJ (extrovert, rational, impersonal, formal)

The UL (interior-individual) quadrant relates to the introvert rational personality. Its four aspects are:

ISFP (introvert, rational, personal, concrete)

ISTP (introvert, rational, impersonal, concrete)

ISFJ (introvert, rational, personal, formal)

ISTJ (introvert, rational, impersonal, formal)

The LR (exterior-collective) quadrant relates to the extrovert intuitive personality. Its four aspects are:

ENFP (extrovert, intuitive, personal, concrete)

ENTP (extrovert, intuitive, impersonal, concrete)

ENFJ (extrovert, intuitive, personal, formal)

ENTJ (extrovert, intuitive, impersonal, formal)

The LL (interior-collective) quadrant relates to the introvert rational personality. Its four aspects are:

INFP (introvert, intuitive, personal, concrete)

INTP (introvert, intuitive, impersonal, concrete)

INFJ (introvert, intuitive, personal, formal)

INTJ (introvert, intuitive, impersonal, formal)

Of course these 16 personality types should not be interpreted too rigidly. However the very fact that they are so widely used indicate their general validity.

Now if take these personality types seriously, then not alone will a person tend to operate typically out of one quadrant but from one aspect of this quadrant.

The problem with Ken's four quadrant approach (and the MBTI) is that they are both conceived in static terms. Now by definition this leads to a one-directional tendency whereby reality is interpreted from a fixed frame of reference.

Thus whereas it is perfectly reasonable for Ken to encourage four quadrant understanding, there are grave practical limitations when employing a static linear approach.

So we now need to consider the corresponding circular approach (where polar opposites) are reconciled.

This intimately related to the two-directional approach.

Though in dynamic poles opposite poles are relative, the essence of the linear approach is to understand them dualistically (in unambiguous terms).

So when we fix one pole as positive, then the (opposite) pole is negative (i.e. no rational ambiguity arises).

Now if we fix the second pole as positive, then the first pole is negative; (again in terms of this direction, no rational ambiguity arises).

However if we now bring both together (simultaneously), then each pole is both positive and negative. This can be (indirectly) expressed in rational terms (with a paradoxical meaning); however it can be only be understood directly in intuitive terms. So when we use reason to express intuition we have a (circular) both/and logic (in relation to opposite poles).

Thus what is positive is negative; what is negative is positive.

What is physical is psychological; what is psychological is physical.

What is interior is exterior; what is exterior is interior etc.

Without circular understanding one will tend to understand rigidly identifying unduly with just one quadrant.

The "higher" levels lead to a great growth in intuitive type understanding.

The circular level (the subtle realm) involves the specialisation of intuitive understanding (in relation to horizontal polarities).

The interior and exterior aspects of understanding are intuitively unified during this level.

I refer to this simply as the (horizontal) centre. Thus whereas all holons in (linear) dualistic terms have interior and exterior facets, in circular (intuitive) terms they have a centre (without location). The centre is nowhere, yet everywhere.

In relation to horizontal opposites one now acquires binary understanding.

Thus at an intuitive level on has a circular understanding of opposites while at a rational horizontal level a (bi-directional) linear understanding.

For example if one formerly identified with the UL quadrant, one can now equally identify with the UR quadrant. (Proper vertical understanding however is not yet possible).

The essence of the point level (causal realm) is the additional specialisation of intuitive understanding in relation to vertical opposites. Individual and collective aspects can now be properly unified. I refer to this as simply the (vertical) centre.

Whereas all holons, in (linear) rational terms have both individual and collective facets, in (circular) intuitive terms they have a (spiritual) centre.

Thus all four quadrants can now be both properly differentiated in rational (linear) terms and integrated in intuitive (circular) terms.

In holistic mathematical terms dynamic evolution can be expressed in terms of the interaction of linear (1) and circular (0) aspects.

Thus all holons continually emanate - horizontally and vertically - from an ever present centre that is absolute. They express themselves in (relative) space and time as phenomena (conditioned by polar opposites). They continually return (horizontally and vertically) to the same centre.

Thus in binary terms, the correct basis for evolution is the present moment (continually renewed). All linear expressions of evolution (while maintaining a certain limited validity) are then presented in bi-directional terms, leading back to this ever-present source. Creation - in all respects - can then be truly appreciated as a spontaneous expression of the present moment.

The problem again briefly with Ken's treatment is that the linear aspect and the circular aspect (nondual reality) remain separated. Ken's approach to evolution is predominantly one-directional and holarchical. So he winds up trying to impose the nondual perspective on his evolutionary framework at the end of a linear trend. In dynamic binary terms however it should form the centre of evolutionary movement at all stages.

The full integration of the Myers-Briggs personality types goes beyond the quadrant approach.

Again each personality is presented in terms of four sets of dualities.

Thus one is differentiated either as an extrovert (E) or introvert (I).

With (circular) integration, one is primarily a centrovert (C). One may still remain be - at a secondary level - an extrovert or introvert.

Secondly, one is either sense-oriented (S) or an intuitive (N).

Again with (circular) integration one is primarily spiritual in a mystical sense (M).

Thirdly, one is either a thinking type (T) or feeling type (F).

Again with integration one is primarily volitional i.e. will-oriented (V). In Christian mysticism union is essentially a union of will.

Finally one can rely on perception (P) or judgement (J) in making decisions.

With integration one relies primarily on discernment (D) which represents a form of mature wisdom.

Whereas we have 16 personality types (in linear dualistic terms) we have an additional 8 in circular terms (all with the primary CMVD orientation).

Those belonging to the CMVD group may not necessarily be well integrated. Rather - more than other personality types - they must aim at this integration to realise their innate potential.

Again in the (ideal) case of full integration one will belong in primary terms to the circular category of CMVD.

In dualistic terms one's personality will be bi-directional. In Jungian terms this simply means that one's shadow will be very well differentiated.

Thus someone for example - originally identified as an ENFP - will now equally identify with the opposite INTJ (i.e. shadow characteristics are fully differentiated).

The purpose of this posting - in one important respect - has been to illuminate certain aspects of Ken's four-quadrant approach while showing its close integration with Jungian Personality Types.

More importantly I have been at pains to illustrate again the value of a coherent qualitative mathematical approach.

I have confined myself throughout to just one important concept i.e. the qualitative binary system.

In terms of the potential of Holistic Mathematics, this barely scratches the surface.