Q We return to the two number systems that you have defined i.e.
horizontal and vertical. Can you show now how these lead to both a linear
and circular interpretation of number respectively?
PC It is important to remember that even though the same symbols are used in both cases they actually relate to very distinctive interpretations of number.
In the first system though the number (as quantity) can vary the dimension (as quality) remains fixed as 1,
i.e. 11, 21 , 3 1, 41,
In the second system though the number (as dimension) can vary the number (as quantity) remains fixed as 1,
i.e. 11, 12, 13, 14, …
So in the first system, the natural number symbols 1, 2, 3, 4, … refer to quantities that are defined with respect to a fixed (qualitative) dimension of 1.
In the second system, the same natural number symbols 1, 2, 3, 4 … refer to qualities (i.e. dimensions) to which the fixed quantity of 1 is raised.
However as they stand these two systems are not comparable in terms of each other.
What is requires therefore is a means to express the vertical (qualitative)
in terms of the horizontal (quantitative) system and likewise - in reverse
fashion - the horizontal (quantitative) in terms of the vertical (qualitative)
Q So how do you indirectly express the (qualitative) vertical in
terms of the (quantitative) horizontal system? In other words how can you
indirectly express the notion of a dimension in terms of an object phenomenon?
PC We have already dealt with basic mathematical symbols from both a holistic arithmetic and geometric standpoint. We now do so from a more powerful holistic algebraic perspective (which includes the other two interpretations).
As always mathematical operations have both analytic and holistic aspects.
Firstly we will show how to express the vertical (qualitative) number - indirectly - in horizontal (quantitative) terms.
To do this let us introduce a very basic bit of algebra.
Once again in the horizontal system the number as quantity can vary while the number as quality (i.e. dimension) remains fixed as 1.
i.e. 11, 21 , 3 1, 41, …
So if x1 is a number in this system then x can take on any of the values 1, 2, 3, 4, … (as quantities).
Thus when x = 1, x1 = 11; when x = 2, x1 = 21; when x = 3, x1 = 31; when x = 4, x1 = 41 etc.
By contrast in the vertical system the number as quality (dimension) can vary while the number as quantity remains fixed as 1.
i.e. 11, 12, 13, 14, …
So if 1x is a number in this system then x can now take on any of the values 1, 2, 3, 4, … (as dimensions).
Thus when x = 1, 1x = 11; when x = 2, 1x = 12; when x = 3, 1x = 13; when x = 4, 1x = 14 etc.
Now as the first (horizontal) system is expressed with respect to an invariant fixed dimension of 1, to express the second (vertical) system in this fashion requires taking the corresponding root of each number (where x now represents the root number).
Thus in the case where x = 1 and 1x = 11, we obtain the 1st root of 11 which remains the same as 1 (i.e. strictly 11).
In the case where 1x = 12, we thereby obtain the 2nd root (i.e. square root) of 12 , which is + 1 and - 1 (i.e. strictly + 11 and - 11).
Now when we plot these points, we can represent them at opposite ends of the line diameter, that bisects the circle (of unit radius) into two halves.
What is truly remarkable is that we obtain subsequent roots in this fashion - defined in the complex plane - they will always lie as equidistant points on the same circle of unit radius.
Thus through attempting to express the vertical (qualitative) system
- indirectly - in horizontal (quantitative) terms with respect to the invariant
dimension of 1, we create a coherent circular number system i.e. where
all numbers can be expressed as equidistant points on the circle of unit
Of special interest here in the context of my approach are the algebraic (and subsequent geometrical) representations for 2, 4 and 8 roots.
As we have seen for x = 2, when 1x = 12, the 2nd (i.e. square) root of 12 has two possible solutions + 1 and - 1 (defined with respect to a fixed dimension of 1).
These two results (i.e. + 1 and - 1) can then be plotted as opposite ends of the horizontal line (diameter) - drawn in the complex plane - that bisects the circle of unit radius.
Then for x = 4, when 1x = 14, the 4th root of 14 now has four possible solutions two of which are real i.e. + 1 and - 1 and two of which are imaginary i.e. + i and - i (where i is the square root of - 1).
Once again the two real roots + 1 and - 1 can be plotted as opposite end points of the horizontal line diameter (through the circle of unit radius).
The two imaginary roots + i and - i can then be plotted as opposite end points of the corresponding vertical line diameter (through the same circle).
Furthermore these two line diameters literally divide the circle into four quadrants (now with a precise mathematical interpretation).
So here we have the first vital clue as to the all important mathematical relationship as between quantitative and qualitative (which plays such a vital role in the corresponding holistic interpretation of the four quadrants).
Thus if we represent the horizontal line diameter (or axis) in real
terms as representing quantities, then the vertical line diameter (or axis)
as representing dimensions is thereby imaginary (in precise mathematical
Finally for x = 8, when 1x = 18, the 8th root of 18 now has eight possible solutions.
We have already encountered four of these with out two real (i.e. + 1 and - 1) and two imaginary roots (+ i and - i).
We now have in addition four complex roots. These are k(1 + i), k(1 - i), k(- 1 + i), and k(- 1 - i) respectively (where k = 1 divided by the square root of 2).
As we have seen the two real (+ 1 and - 1) and the two imaginary (+ i and - i) roots can be represented as opposite end points of the horizontal and vertical line diameters (axes) respectively of the circle (of unit radius).
The four complex roots can be represented as end points of the two corresponding diagonal line diameters (drawn at an equal distance from both horizontal and vertical).
Thus k(1 + i) and k(- 1 - i) lie at opposite ends of the diagonal that bisects the upper right and lower left quadrants.
k(- 1 + i) and k(1 - i) lie at opposite ends of the other diagonal that bisects the upper left and lower right quadrants.
Now remarkably the diagonal lines (in each quadrant) have an alternative interpretation as null lines = 0. This can be easily shown through using the Pythagorean theorem.
Again the holistic explanation of this dual interpretation in terms
of equal complex co-ordinates and null lines plays a vital role when we
attempt to give a precise scientific explanation of the nature of diagonal
Q So your purpose here is to show how all numbers can be given a
coherent linear and circular interpretation in analytic terms. You then
will presumably move on to showing how we derive the corresponding dynamic
holistic interpretation of the circular system?
PC Yes! That is correct.
Though the circular system is of course recognised in mathematical terms, its great potential significance has been greatly overlooked. A reduced interpretation of this system inevitably results for - as we have seen - ultimately all interpretation is with respect to the linear system (with an invariant dimension of 1).
The true significance however of the circular system comes from holistic interpretation where - when properly understood - it provides the appropriate basis for an integral TOE (indeed a series of integral TOEs paradoxically of both increasing complexity and yet increasing simplicity).
It is certainly true that an implicit holistic understanding of this circular system of number exists in many esoteric traditions.
If modern writers Jung had an especially strong sense of its importance. Thus when Marie Louis von Franz said that
"Jung devoted practically all of his life to the importance of the number 4" she implied the circular interpretation of 4 (in a dynamic holistic sense).
Indeed Jung pointed to the integral significance of mandalas the most important of which entail ornately designed patterns that are fundamentally similar to the geometrical representations above).
However the deep mathematical significance of these representations
- serving as the basis for a truly scientific approach to development -
now needs to be made fully explicit.
Q Before moving on to holistic interpretation can you briefly show
how - in reverse fashion - the horizontal (quantitative) can be indirectly
expressed in terms of the vertical (qualitative) system? In other words
how can one indirectly express a phenomenal object in terms of a general
Whereas in the previous case, numbers as dimensions were expressed in terms of a fixed dimension of 1 (through obtaining successive roots defined by these same numbers), in this instance, numbers as quantities are expressed in terms of a fixed quantity of 1 (though obtaining successive powers defined by these same numbers).
Now this leads directly back to the vertical system.
So when x (as quantity) = 1, 1x = 11.
When x (as quantity) = 2, 1x = 12.
Now the deeper significance of this dimension of 2 is that it really is defined in terms of a both/and logic that combines two directions i.e. combining both positive (+) and negative (-) real directions of unity. This again can be represented in circular terms as end points of the horizontal line diameter that bisects the circle of unit radius (in the complex plane).
When x (as quantity) = 4, 1x = 14.
Again the deeper significance of this dimension of 4 is that it is defined in terms of a both/and logic that combines four directions i.e. both positive (+) and negative (-) directions of unity in real and imaginary terms. This again can be represented in circular terms as end points of both horizontal and vertical line diameters that bisect the circle of unit radius.
Likewise when x (as quantity) = 8, 1x = 18.
Once more the deeper significance of this dimension of 8 is that it is defined in terms of a both/and logic that combines eight directions i.e. both the positive (+) and negative (-) directions of unity in real and imaginary terms and four complex directions (that are simultaneously of equal magnitude in both real and imaginary terms).
This again can be represented in circular terms as end points of both
horizontal, vertical and diagonal line diameters that bisect the circle
of unit radius.
Two Logical Approaches
Q I think I am beginning to see what you are getting at and I will try and summarise.
The true significance of your linear and circular type number systems is that number can be defined in terms of two utterly distinctive logical systems.
The conventional understanding - which is directly suited to linear interpretation - is based on an unambiguous either/or logic.
However there is an alternative system - directly suited to circular interpretation - based on a paradoxical both/and logic. (However because conventional mathematics does not recognise this logic it can only deal with this system in a distorted reduced fashion).
Each system can be expressed indirectly in terms of the other.
Thus the linear (vertical) system has an indirect circular interpretation in horizontal terms.
Likewise the linear (horizontal) system likewise has an indirect circular interpretation in vertical terms.
Equally - starting from the circular viewpoint - we can say that the circular (vertical) system has an indirect linear interpretation (in horizontal terms).
Likewise the circular (horizontal) system has an indirect linear interpretation in vertical terms.
Therefore each number system has both a linear or circular interpretation (depending on the logical system through which it is viewed).
However the great problem with conventional mathematics is that it operates solely in terms of one logical system. Therefore notions - which are properly circular - can only be understood in a reduced linear (i.e. one-dimensional) fashion.
Thus the very notion of dimension (as greater than 1) is inherently circular.
This has immense consequences as it shows that the real meaning of dimension conforms to a circular notion of direction (that directly conforms to the mathematical notion of a root).
Therefore - as we have already seen in the previous discussion - the fundamental structure of the physical (and psychological) dimensions of reality is inherently mathematical in this circular sense.
Finally, when related to each other - which is always necessarily the case - number (as quantity) and number (as dimension) are always - properly - linear and circular (or alternatively circular or linear).
Thus if we interpret number (as quantity) in linear terms, number (as dimension) is thereby circular. Alternatively, if we interpret number (as quantity) in circular terms, number as quality (or dimension) is thereby linear.
Thus by exploring this hidden (circular) aspect of number interpretation - initially in analytic terms - we are led to the brink of the truly dynamic holistic understanding.
I think that does it. However can you briefly clarify again the two ways in which the circular can be interpreted (and the two ways likewise of the linear).
PC When we obtain the various roots of the number 1 (i.e. 2nd and higher), the results conform - even in analytic terms - to a circular pattern (as equidistant points on the circle of unit radius).
However when we interpret these results in terms of the linear (either/or) logic - conforming properly to the horizontal quantitative system - they are separated. So for example the square root of 1 is either + 1 or - 1.
Now - in reverse fashion - when we raise the number 1 to various powers or dimensions (2 or higher), the results this time conform - in direct fashion - to a circular pattern (again as equal points on the circle of unit radius).
This time the interpretation is in terms of the circular (both/and) logic. Therefore the square of 1 (i.e. 12) properly refers to the dimension of 2 (combining the two directions + 1 or - 1 as complementary).
However because conventional mathematics does not recognise this alternative paradoxical both/and logic, this interpretation of dimension is completely lost and all that remains is the indirect linear interpretation (where such numbers now represent dimensions on a vertical linear axis).
Once again however we are on the verge here of the dynamic holistic
interpretation of these symbols to which we should now perhaps return.
Imaginary Number Interpretation
Q This is all of course very subtle and I can see absolutely fundamental. I am beginning to glimpse here a key implication for the interpretation of an imaginary number. Am I correct in assuming that the true meaning of such an imaginary number requires circular understanding (and the corresponding use of paradoxical both/and logic)?
However because conventional mathematics only recognises the either/or
logical system can we thereby only attempt to deal with imaginary numbers
in an indirect linear fashion which hides their true nature!
PC Yes! You are fully correct.
I must confess that I have long had a great fascination with imaginary numbers. Even from an early age I felt that there was an important hidden meaning that conventional mathematics somehow had failed to demonstrate.
Then gradually after several decades the secret at last revealed itself when it dawned on me that whereas a real number corresponds directly to linear logic, an imaginary number corresponds directly to circular interpretation.
Thus for example - in linear terms - a number can be either + 1 or - 1. However it cannot be + 1 or - 1 simultaneously. So either/or logic inherently governs real number interpretation.
However an imaginary number inherently operates according to an alternative logic where opposite polarities are simultaneously combined. Thus i (i.e. the square root of - 1) has a direct interpretation as a number which is simultaneously + 1 and - 1. However this has no meaning in conventional either/or terms and therefore its secret remains locked within. So operations can be only carried out indirectly on imaginary numbers, as if they were real numbers (according to an either/or logic).
This also means that imaginary numbers truly come into their own within
a dynamic holistic interpretation (which properly incorporates circular
Q I can also see once again the importance of the binary approach. Not alone are the binary numbers so fundamental but every number (whether as quantity or dimension) has a binary interpretation as linear (1) and circular (0) and - in reverse terms - circular (0) and linear (1) respectively .
In this context can you explain how for example with respect to 12
the dimension - which properly conforms to circular interpretation - is
equal to 0. How can 2 = 0?
PC Remember conventional mathematics - which is geared to solely to either/or logical interpretation cannot properly represent circular notions. So 12 itself represents an indirect - and thereby reduced - linear interpretation of what is properly circular.
Thus the number representing the dimension in this case (i.e. 2) properly points to the complementarity of opposite positive (+) and negative (-) poles which - in terms of a both/and logic cancel out as 0.
However when we try to represent this in the language of separation we concentrate on the fact that there are two distinct poles. So the number 2 - in linear terms - represents these two distinct poles.
So again 14 (as dimension) in proper circular terms is again = 0. This time the two real poles (+ 1 and - 1) and the two imaginary poles (+ i and - i) using complementary both/and logic in both cases cancel out = 0.
However in linear (separate) terms we again concentrate on the fact
that there are now four separate poles. So the dimension is thereby represented
Q This relates to what you were saying earlier where the very nature
of the linear approach is to see opposite poles from a merely positive
perspective (i.e. where each pole is posited separately). Equally it also
entails seeing such poles in solely real terms. This logically leads to
the four-quadrant approach being equally a four-dimensional approach in
a reduced manner (where understanding takes place in terms of four separated
PC It also of course ties in with the notion that the linear approach is suited directly for differentiation whereas the circular approach is directly suited for integration.
So strictly speaking whereas a differentiated interpretation may be four-quadrant (and thereby four-dimensional), from an integral perspective it is strictly non-quadrant (and thereby non-dimensional).
In other words from an integral perspective the notion of separate quadrants
dissolves in the realisation of nondual understanding.
Holistic Integration: Type 0
Q Can we now move on to the holistic interpretation of these number relationships.
Let is start with the simplest case where there is no difference
as between the horizontal and vertical interpretation of number.
PC When we look at both systems we can see that 11 belongs to both systems.
We will examine here more carefully first what happens when we try to convert the vertical interpretation (with respect to dimension) indirectly in terms of the horizontal system. This requires - in this instance - obtaining the 1st root of 11 which gives an unchanged answer i.e. 11.
The implication is this is that the transformation is strictly linear in this case (and does not lie on a circle).
In holistic terms this means that in the linear transformation circular (paradoxical) notions are not formally recognised and thereby reduced in linear terms. In other words the distinction of qualitative and quantitative is effectively ignored (with the qualitative effectively reduced in quantitative terms).
Now alternatively we can attempt to express the horizontal in terms of the vertical number system (i.e. by indirectly expressing quantitative notions in qualitative terms).
This requires - in this instance obtaining the 1st power of 1 (11) which again gives an unchanged answer i.e. 11.
Again the implication is that the transformation is strictly linear (again not lying on a circle).
So once more in holistic terms through linear transformation (in this case from quantitative to qualitative) the distinction as between linear and circular notions is not formally recognised (with the quantitative now reduced in qualitative terms).
Put another way with linear interpretation relationships are formally interpreted solely through either/or logic. This renders it impossible to thereby preserve the key distinction as between quantitative and qualitative notions which are respectively based on linear (either/or) and circular (both/and) logic respectively.
So in precise holistic mathematical terms linear understanding is one-directional with respect to both (quantitative) objects and (qualitative) dimensions.
Therefore with linear understanding formal understanding with respect to both the quantitative and qualitative aspects of phenomena is merely posited (in conscious terms).
This in turn means that linear understanding is based on the separation of opposite poles (leading to dualistic interpretation).
Such dualistic understanding is associated with - in any given context - unambiguous asymmetrical connections between variables.
By its very nature linear understanding - which is based on unitary form (1) - is not geared to deal with notions of emptiness (0). Such notions require the complementary appreciation of opposite poles (both positive and negative) using circular both/and logic.
Linear logic is thereby geared for differentiated understanding of phenomena. However it is quite unsuited for corresponding integral understanding.
Thus it can only deal with integration through effectively reducing it to differentiation.
So in terms of complementarity, I refer to linear understanding as Type 0 complementarity (i.e. the absence of complementarity).
This is likewise associated with Type 0 differentiation (i.e. unambiguous one-directional differentiation associated with a corresponding lack of complementary understanding).
Such understanding is associated with the middle level of the Spectrum (L0, H0) where the specialisation of rational understanding occurs.
This is the understanding that typifies conventional scientific understanding
and indeed the great bulk of intellectual discourse of all kinds.
Q Can you now briefly illustrate the nature of this understanding
with respect to conventional scientific understanding?
Though quantitative and qualitative aspects of understanding dynamically interact it may be helpful to initially approach the issue where we identify - in any context - perception with the horizontal (quantitative) and the corresponding concept with the vertical (qualitative) aspect.
Now the every nature of such scientific i.e. analytic understanding is that is has no appropriate means to preserve the distinction of quantitative and qualitative (and qualitative and quantitative) and therefore must necessarily reduce one to another.
So if we for example take the perception of an actual (i.e. real) molecule", the corresponding concept of "molecule" will be viewed as applying to all actual molecules. Therefore a direct correspondence is assumed to exist as between perceptions and their corresponding concepts.
Likewise if we start from the concept of an actual molecule again it will be viewed as relating to any particular molecule perception (within its class) so that again a direct correspondence is assumed to exist - in reverse manner - as between concepts and their corresponding perceptions.
In this way therefore in science a (conceptual) theory is interpreted as directly applying to all corresponding empirical (perceptual) facts within its class. Likewise empirical facts are interpreted as applying to corresponding (conceptual) theories.
So in effect this double correspondence as between theory and facts (and facts and theory) implies equally a double form of reductionism whereby the qualitative is reduced to the quantitative (and the quantitative to the qualitative).
Putting it another way - because analytic scientific understanding is
so heavily based on linear (either/or) interpretation - it cannot properly
distinguish as between the differentiation and integration of phenomena.
As we have seen such integration requires an alternative circular (both/and)
Q I know it is your contention that this form of linear reductionism
pervades most forms of intellectual discourse include those that that specifically
aim at a comprehensive integral view of development. Can you attempt to
explain this firther.
PC Yes it is possible to have an extremely comprehensive model of development (based on a genuine spiritual integral vision) which is yet heavily linear in intellectual interpretation (conforming to both Type 0 complementarity and Type 0 separation).
Now perhaps it is unfair to single out Ken Wilber but he is undoubtedly the best known proponent of the integral approach. However, I would find it that his characteristic style of interpreting development is in fact strongly linear and thereby not properly suited as an integral approach.
In other words - again in terms of intellectual interpretation - to my mind he consistently confuses integration with multi-differentiation.
For example it is certainly true that a more comprehensive approach to development requires the various types of differentiation with accompanying sophisticated analysis that Ken has so brilliantly demonstrated throughout the years. So among his many refinements his model carefully distinguishes prepersonal, personal and transpersonal stages, levels of self and levels of reality, structures states and bodies of development, the basic levels (or waves) and transitional levels, many distinct lines (streams) of development moving through the various waves, the interpretation of all these waves and streams in terms of the four quadrants etc.
Now it is also true that he strongly advocates the need for authentic spiritual practice as a prerequisite for genuine integration. However the problem that I find is that such practice leading ultimately to nondual realisation is treated largely separately from his characteristic analytic understanding of development.
In other words there is a considerable discontinuity evident in terms of his somewhat dualistic intellectual treatment of development (strongly based on linear asymmetrical notions) and his admittedly very sincere nondual approach to spiritual development.
However from a dynamic perspective dual and nondual continually interact mutually changing the manner in which both aspects are realised and this is especially true at the "higher" levels of development.
So the very purpose of an integral approach - in dynamic terms - is to properly show how initial rigid notions of asymmetrical connection (based on Type 0 interpretation) gradually give way to a much more refined paradoxical interpretation properly consistent with spiritual nondual awareness. In the terms that I describe it this therefore requires clarifying the nature and role of Type 1, Type 2 and Type 3 understanding (that properly defines the integral understanding of the "higher" spiritual levels of development). Finally we show how these forms of circular understanding are then fully incorporated with Type 0 linear interpretation.
Thus the radial approach is concerned with demonstrating both the relative
independence and interdependence of linear (differentiated) and circular
Q With reference to the stages of development already mentioned can
you briefly explain the nature of an Integral 0 approach (i.e. based on
Type 0 complementarity).
PC Firstly prepersonal, personal and transpersonal are treated in a somewhat discrete asymmetrical manner. So prepersonal unambiguously relate to the lower, personal to the middle and transpersonal to the higher stages of development.
However such distinctions are properly suited for differentiated rather than integral understanding. So from a correct integral perspective prepersonal and transpersonal (and transpersonal and prepersonal) are mutually complementary and ultimately with nondual awareness identical. In other words balanced integration occurs in both a top-down (transpersonal to prepersonal) and bottom-up fashion (Likewise the personal are complementary with both the (integrated) personal and transpersonal stages. In other words what is neither prepersonal not transpersonal (i.e. personal) is complementary with what is both prepersonal and transpersonal.
However as complementary understanding is strictly incompatible with asymmetrical interpretation and undermines its very assumptions.
Likewise the holarchical approach to stages of development is very much a linear asymmetrical approach being based on one-sided interpretation of the dynamic relationship as between whole and part (i.e. where the whole is part of a higher whole).
Whereas holarchy is suitable as one way of differentiating stages, it is not suited as a means of integration which requiring the mutual interdependence of holarchy i.e. where each whole is part of a higher whole with partarchy i.e. where each part is a whole (in the context) of other parts.
Likewise any approach to interpreting the four quadrants based on unambiguous notions of Right-Hand or Left-Hand (and Upper and Lower) is again - by definition - linear as I define it. In effect such unambiguous identification always entails understanding using isolated reference frames (based on separation of polar opposites).
So for example to identify the Right-Hand with exterior reality requires treating the exterior pole as relatively independent of the interior.
However whereas this again is initially suitable as a means of differentiating quadrants it is not appropriate as a means of integration.
Again an integral approach establishing interdependence - requires the two way interaction of complementary poles. All fixed notions of location such as Right or Left (and Upper or Lower) are thereby rendered paradoxical.
Finally the notion of lines of development as relatively independent (again defined as passing through stages in a holarchical manner) is very much linear. (The very use of the terms "lines" clearly implies this!)
Again the identification of such "lines" is suited to the task of differentiating
development. However it is not suited for integration.
Q Do you imply that someone using a linear approach - as you define
it - actually believes that development unfolds in a linear manner?
PC Of course not! Linear has a very precise meaning in my approach implying literally one-dimensional interpretation (i.e. one-directional). So a linear approach is based - in any context - on interpretation of the relationship between development variables in unambiguous sequential terms. This in turn is always associated with asymmetrical understanding.
Thus if one maintains for example that the atom is contained in the molecule (but the molecule not contained in the atom) then this is a clear example of linear understanding.
Circular understanding by contrast is always based on the symmetrical recognition of relationships between variables that simultaneously are bi-directional (moving in opposite directions from each other).
Though such recognition is directly intuitive and spiritual, indirectly it dynamically associated with a paradoxical means of intellectual interpretation.
Likewise I do not suggest that those who use linear approach believe that integration is the same as differentiation (or that the qualitative aspect is the same as the quantitative) . Rather I am pointing out that - in effect - asymmetrical interpretation always leads to a reduction of one aspect in terms of the other.
So in conclusion Integral 0 understanding - using Type 0 complementarity (which is directly identified with Type 0 polar separation) lends itself to linear (asymmetrical) interpretation.