We live in an age that is increasingly dominated by digital technology
All data can be potentially encoded though use of the binary digits 1 and 0, which can be readily adapted for use in electronic circuits. This in turn enables the transfer of increasingly complex forms of information e.g. in computers, telecommunications and entertainment.
Number provides the best means for the quantitative ordering of information and is widely synonymous with this role. Indeed in this context the binary numbers 1 and 0 provide the most fundamental symbols and as we have seen with digital technology, they provide an extremely important example of the power of number to order information.
In conventional terms number is understood in static analytic terms.
So the binary digits are based on an either/or logic, which are defined in an unambiguous manner where they are clearly separated from each other.
However what is not yet properly appreciated is that all-mathematical notions such as number have dynamic holistic interpretations incorporating an alternative both/and logical system. 1
The implication of this alternative appreciation is enormous as it directly implies that mathematical symbols can also be used in a precise scientific manner to translate an integral vision of reality. 2
So just as science - as conventionally understood - requires the use of analytic interpretation provided by mathematical symbols, equally - from this perspective - integral science, representing a dynamic universal appreciation of reality, requires a corresponding holistic interpretation of these same symbols.
In this context let me immediately address one key application of this holistic interpretation.
We have just commented on how the (analytic) interpretation of the binary digits can be used to potentially encode all information processes.
Quite remarkably, the dynamic (holistic) interpretation of these same digits can be used in a corresponding manner to potentially encode all transformation processes.
Phenomenal existence in all its varied manifestations, human and natural, represents a dynamic transformation process. So for example, all key structures with respect to the Spectrum of Development - both psychological and physical - can be precisely encoded in a binary digital manner.
I firmly believe that such a translation - based on deep appreciation of the holistic interpretative power of mathematical symbols - is urgently needed to provide a scientific translation of development that is fully integral. Equally, it can greatly facilitate the opening up vast new areas of inquiry that would be virtually impossible to access by other means.
So the purpose of this series of articles is to explain some basic features of the holistic mathematical approach. We will then show how it leads to a dynamic "Theory of Everything", directly based on the binary system, that can be used in a wonderfully coherent - yet extremely detailed - fashion to precisely encode the fundamental structures underlying all stages of the Spectrum.
In order to outline properly the nature of this holistic binary system, we will now introduce a series of related holistic mathematical concepts.
The binary digits
We start with the fundamental binary numbers 1 and 0.
Firstly, let us examine these numbers in analytic terms.
Implicit in the distinction of any (separate) form is this notion of oneness. (If I for example notice a house, I implicitly recognise it as a unit i.e. one object). 3
Likewise generalised notions of form imply the recognition of more holistic notions of oneness.
By contrast 0 in analytic terms implies the absence of form and the notion of nothingness (or emptiness). Thus, if for example a box is empty, there is nothing in it.
So in analytic terms 1 is directly identified with form and 0 with emptiness respectively. Furthermore these notions are clearly separate (i.e. mutually exclude each other).
However from a dynamic holistic perspective form (1) and emptiness (0) become increasingly interdependent (and ultimately identical).
Indeed the holistic use of these digits is deeply implicit in the understanding of the mystical traditions.
Western traditions tend to approach the ultimate mystical experience from the perspective of form. Thus the highest experience is typically identified as oneness i.e. the experience of union. (Here the separation inevitably attached to merely dualistic notions of form is finally eroded). Indeed in Christian mysticism the most advanced stage is commonly referred to as the unitive life (which is another expression of oneness). 4
In the Eastern esoteric traditions the ultimate experience is frequently approached through the corresponding notion of emptiness. Here the highest experience is often identified in terms of a creative void or spiritual emptiness that is absent of phenomena.
Of course ultimately, in balanced mystical terms, these two aspects are themselves identical and the most complete experience is identified as a plenum-void where the pure potential of emptiness continually gives way to a refined world of form.
"Form is not other that Void
Void is not other than Form"
Thus though the binary digits are completely separate in analytic terms, ultimately from a dynamic holistic perspective they are identical.
In dynamic terms, form and emptiness, in both their separate and complementary aspects, continually interact and provide the basis for all development processes.
A key task in integral terms is to show how experience moves from the dualistic identification of form (where it is separate from emptiness) to the nondual experience (where it is dynamically complementary with emptiness).
Alternatively we could say the task is to show how understanding moves from the appreciation of differentiated phenomena to the experience of their corresponding integration.
This requires showing that the binary digits, which are separate in analytic terms, are yet ultimately identical from a dynamic holistic perspective.
Addition and subtraction
To appreciate how the binary digital approach can be applied to this important task of moving from dual to nondual appreciation we now need to introduce the two most familiar operations in mathematics i.e. addition and subtraction.
Once again these operations can be given both analytic interpretations that are unambiguous and holistic interpretations that are paradoxical.
In mathematics from an analytic perspective, addition carries a positive sign whereas subtraction carries a negative sign.
Thus in analytic terms 1 is related to 0 through combining both its positive and negative signs. i.e. 1 - 1 = 0. (1 = + 1)
We have here a static transformation of 1 (through combining both signs) that leads to 0.
Alternatively we could say equally that 0 is related to 1 though reversing the direction of this transformation i.e. 0 = 1 - 1.
So to move in analytic terms from 1 to 0, or 0 to 1, we must negate what has been statically posited (i.e. 1).
Once again, we have exact matching dynamic holistic interpretations of these two mathematical operations, which are fully complementary.
Here the two operations relate to the fundamental manner through which phenomena are posited and negated in existence.
In psychological terms to posit is simply to make conscious thus enabling differentiation of phenomena. 5
However interaction with the unconscious requires corresponding negation. The implicit recognition of the other pole then enables a switch in experience to take place (to this opposite pole). So in terms of exterior and interior poles, recognition keeps switching from exterior to interior (and interior to exterior).
When complementary reference frames are not properly recognised, one thereby identifies with the merely positive pole so that considerable reductionism takes place. Phenomena obtain a rigid unambiguous identity (with the opposite pole becoming largely repressed in the unconscious). Limited dynamic negation can then take place greatly impeding balanced interaction in experience.
However when complementary poles are clearly recognised through growing spiritual awareness, a far greater degree of balanced negation is possible. Phenomena then reveal their inherent mystery and become increasingly transparent. This in turn facilitates a more dynamic flexible interaction between poles.
Ultimately the purest experience of spiritual emptiness results from complete dynamic negation of all phenomena as soon as they are posited.
Because phenomena are now successfully negated the moment they arise they do not even appear to be present in experience. This in turn leads to a total immersion in spiritual emptiness.
A beautifully simple explanation of these dynamics can be given in holistic mathematical terms, where we combine the two binary digits with the fundamental operations of positing and negating.
From one perspective
1 - 1 - 0
In dynamic holistic terms, when the positive direction of form (i.e. 1) is completely negated by its opposite (- 1), emptiness (i.e. 0) results.
So here in this simple relationship we have a proper dynamic balance maintained as between form and emptiness. Emptiness in this context cannot be conceived in the absence of form.
We could equally say from the opposite perspective that
0 = 1 - 1.
In other words a state of emptiness is the very basis for positing and negating form.
So in the highest mystical experience of reality, refined form continually gives way to emptiness and emptiness to refined form.
So when we say that form is not other than emptiness we are referring to a dynamic notion of oneness where the positing of form is fully balanced in experience by its corresponding negation.
Equally to say that emptiness is not other than form refers to the corresponding reality that emptiness (nothingness) represents the pure potential for the creation of a world of form (in both its positive and negative aspects).
However by definition the dual and nondual aspects necessarily interpenetrate at all levels of experience. So the nondual can only be experienced to the extent that the dual is negated. Likewise the very ability to creatively experience the dual world of form depends on the degree of nondual awareness already present.
Putting it in yet another way directly pertinent to my approach, differentiation in experience is always based on the positing of phenomena. Integration by contrast - at least implicitly - requires their corresponding negation.
Therefore when the negative direction is not properly emphasised in experience, integration cannot fully take place. A reduced experience of integration then results where it is unduly identified with differentiation. 6
Thus a proper integral approach must incorporate the negative (as well as the positive) direction of experience.
To appreciate more deeply the precise nature of this integral approach we will now introduce some fundamental geometrical notions.
Line, circle and point
Once again we start with the analytic interpretation of symbols.
Above we have a circle with its diameter represented by a horizontal line. The radius of this circle (i.e. half the diameter is 1 unit).
Now when we examine this line we find that it has two directions.
So starting from the centre point (which is represented by the o (the origin which is 0), movement to the right is defined as positive. So the radius in a right direction is + 1.
However movement in the opposite direction is thereby negative.
So the radius in a left direction is - 1.
However it is important to understand that the identification of sign is merely an arbitrary convention.
We could equally define movement of the line to the left as positive, in which case corresponding movement to the right would be negative.
The very symbol, which is used to represent one, is in fact a straight line. Likewise the symbol used to represent zero (nothing) is - with very minor modification - a circle.
There are intimate connections as between 1 and 0; likewise there are intimate connections as between the line (diameter) and the circle (circumference).
Just as numerically 0 is connected to 1 through combining the positive and negative signs of 1 (i.e. 0 = 1 - 1), likewise the circle (circumference) is connected to the line through combining both the positive and negative directions of the line (diameter).
So both directions of the line (drawn from the centre) are enclosed by the circumference of the circle. Likewise this bi-directional line defines the nature of the corresponding circular circumference.
Now once again there is an extremely important dynamic holistic interpretation of these same symbols - the proper appreciation of - which greatly clarifies the very nature of both differentiation and integration in development.
In relation to the notion of a line we have in holistic terms a linear approach to development.
In relation to the notion of a circle we have a corresponding circular approach.
Positive and negative equally define opposite poles of development.
Quite simply when we look at development asymmetrically (i.e. solely with respect to its positive direction) we thereby adopt a linear approach.
This always entails an arbitrary fixing of the frame of reference with just one pole (whereby it is treated as largely separate from its opposite pole).
This can be made much clearer through a simple illustration.
It has become customary to interpret development in holarchical terms.
For example we might say that the atom is transcended and included in a molecule which is transcended and included in a cell which in turn is transcended and included in an organism and so on in an ascending sequence of more collective wholes.
Now this represents - as I define it - the linear approach.
In dynamic terms our experience of the world entails the self in relation to the environment so that interior and exterior poles necessarily interact.
However with (linear) asymmetrical understanding there is always an attempt to define experience unambiguously with respect to one pole (as reference frame).
So in our example, the reference frame is fixed with the exterior pole and relationships then viewed as if somehow independent of the interacting mind.
Relationships here are defined with respect to the positive pole (in this case the exterior) and viewed as moving solely in one-direction leading to the typical asymmetric interpretation e.g. atoms are included in molecules; thereby molecules are not included in atoms.
Now I am not criticising such linear interpretation. It plays an extremely important role in understanding and is vitally necessary for the proper differentiation of experience.
Rather - with the help of fundamental holistic mathematical concepts - I am defining its true nature.
So (linear) asymmetric understanding always results from the attempt to understand in terms of just one arbitrarily fixed reference pole (though the opposite pole is equally valid).
However as we have seen with our circle, we could equally define understanding in terms of the other pole as reference frame (in this case the interior) leading to an opposite asymmetric interpretation of development which is equally valid.
By definition we cannot hope to reconcile these paradoxes within a linear approach which is necessarily partial. Therefore when we come to translate integration in experience (as opposed to differentiation) a very distinctive approach is required.
In simple terms proper integral translation requires a balanced circular approach (which fully recognises the true complementarily of opposite poles of development).
Now this is beautifully symbolised in the diagram of the line and circle.
So in holistic terms the circular approach includes two linear directions of interpretation that are always opposite in terms of each other.
In other words circular understanding results from linear understanding that is dynamically bi-directional.
It leads in turn to the deep realisation - trough clear recognition of the arbitrary fixing of polar reference frames - that all dualistic understanding can be rendered paradoxical.
Now the actual recognition of such paradox is nondual and arises through deepening spiritual awareness.
However form and emptiness are themselves dynamically complementary. Therefore the appreciation of paradox at a dualistic level acts as a catalyst towards its reconciliation through spiritual intuitive awareness. Equally the deepening of this awareness sharpens recognition of the inevitable paradox at a dualistic level. So bi-directional appreciation of form and spiritual intuition, increasingly interpenetrate mutually enhancing each other.
Duality and the number 2
After 1 and 0, 2 is the next most important number.
In mathematical terms it can be simply derived from 1 in this manner;
1 + 1 = 2.
Therefore whereas 0 derives from combining 1 with 1 using opposite signs (i.e. 1 - 1), 2 by contrast is derived from combining 1 with 1 using the same sign.
As before the holistic interpretation of 2 can be obtained in similar manner where now the interpretation of symbols carries a dynamic rather than a static meaning.
So in dynamic holistic terms 1 + 1 = 2.
What this in fact implies is that duality derives from combining opposite poles that are defined in a positive manner.
In other words duality results from the separation of opposite poles with both now recognised in terms of their merely positive (i.e. phenomenal conscious) aspect.
For example objective and subjective constitute opposite poles. Now from a nondual spiritual perspective there is a dynamic coincidence of opposites in union (which equally constitutes a creative void or emptiness).
However from a dualistic phenomenal perspective, objective and subjective poles are always - to a degree - separated so that literally these poles come into opposition with each other.
Thus a world of (objective) phenomenal form is now experienced in opposition to a (subjective) observing self and equally from the opposite perspective the (subjective) self is experienced in opposition to the (objective) world..
So duality always results from a splitting off or separation of opposite poles e.g. subjective and objective, whole and part, form and formlessness (which from a nondual spiritual perspective are ultimately identical).
In view of these fundamental holistic mathematical relationships, let us explore briefly the dynamics of how the self interacts with the world in terms of the horizontal i.e. exterior/interior poles.
The very interaction of both poles implicitly requires nondual awareness (where to a degree they are seen as identical). So in this awareness the opposite poles of form (1 - 1) cancel out in nondual spiritual awareness (0).
However the phenomenal identification of dualistic phenomena (exterior or interior) requires a degree of separation of both poles (as positive) so that they are experienced in opposition to one another. Here 1 + 1 = 2 (In other words the separation of both polar aspects of form leads to duality).
Now the switching in experience as between both poles results from the conflict as between the nondual awareness (where they are implicitly identical) and their dualistic separation in the experience of conscious phenomena.
If nondual awareness is largely undeveloped, little dynamism in this interaction will be possible.
Therefore phenomena will tend to become rigid and dense with subjective (interior) attitudes and objective (exterior) events mutually confirming each other in a non-creative fashion.
However where a very significant level of nondual awareness exists (as with the mystical stages of development), considerable dynamic interaction can take place. This then leads to very creative understanding that serves to highlight the paradox associated with dualistic understanding based on isolated poles. This in turn acts as a necessary catalyst for the reconciliation of such paradox through a transformation in nondual spiritual awareness. In this way transformed nondual awareness and (reduced) bi-directional paradoxical understanding mutually support and enhance each other.
Horizontal, Vertical and Diagonal Polarities
We can draw the (bi-directional) line diameter of the circle as a horizontal line.
However equally we can draw it as a vertical line (through the centre at right angles to the horizontal).
Finally we can draw it as two diagonal lines (though the centre at an equal distance from both horizontal and vertical).
These lines have immense significance in dynamic holistic terms relating to the most fundamental designation of polar opposites.
The horizontal line can be used to represent polarities that condition experience within a given level of experience.
These relate to the important objective/subjective (and subjective/objective) polarities which can be referred to in various ways e.g. exterior/interior, external/internal, outer/inner etc. In holistic mathematical terms these polarities are positive and negative with respect to each other.
In other words - whatever the level of understanding - phenomenal experience necessarily entails exterior and interior (and interior and exterior) poles that are in dynamic relationship with each other.
It is important to appreciate that the relationship between poles can be interpreted in two ways
1) Differentiation: here opposite poles are relatively separated leading to (asymmetrical) dualistic understanding.
As we have seen in precise terms this means that both poles are understood in terms of their (merely) positive aspect (i.e. 1 + 1 = 2). Understanding is here linear and unambiguous in one-directional (asymmetric) terms.
2) Integration: here opposite poles are understood as relatively complementary (and ultimately identical) leading to bi-directional dualistic understanding. This is inherently paradoxical (in dual terms) and is reconciled through a dynamic transformation to nondual (spiritual) awareness (i.e. 1 - 1 = 0). So in dynamic terms understanding here is both linear and paradoxical - in bi-directional (symmetric) terms - and circular and nondual. So just as the bi-directional line diameter and the circular circumference are intimately related in analytic geometric terms, likewise bi-directional linear understanding and nondual circular understanding are intimately related in a holistic mathematical fashion. 7
In dynamic terms therefore - for all transformation processes in development - we have the ceaseless interaction of the two aspects of differentiation and integration.
Once again in the first case opposite poles separate in development (though of course never completely separate).
In the second case these opposite poles to a degree unite through recognition of their complementary nature (though again never completely).
So development is characterised by the relationship between opposite poles that is always to a degree dual (where they are separated) and nondual (where they are complementary) though the balance can vary greatly.
The vertical line can be used to represent poles that condition experience between various levels of experience.
These relate to the extremely important whole/part (and part/whole) polarities that can again be expressed in various ways e.g. individual/collective, particular/general, quantitative/qualitative etc. Once again in holistic mathematical terms these are positive and negative with respect to each other.
Strictly speaking whole/part transformations in development always entail changes in the level of development. Thus from the perspective of holism, evolution involves the movement of reality to more complex whole levels of (collective) organisation. Equally evolution here entails the movement from lower quantitative (physical) to higher (spiritual) structures. 8
As with the horizontal poles (within a given level), the relationship between vertical poles (between levels) can be given a (linear) differentiated interpretation and a (circular) integrated interpretation respectively.
The Four QuadrantsWhen we combine both horizontal and vertical lines through the mid-point of the circle we literally create four quadrants in geometrical terms.
In analytic terms the first important thing to observe is that these quadrants - by definition - combine both lines and circle.
Therefore in corresponding holistic terms the interpretation of the four quadrants combines both linear and circular notions.
Likewise - as we have seen - the signs of the lines in opposite quadrants are positive and negative with respect to each other respectively. By convention the sign of the horizontal line in the Right-Hand quadrants is positive and the Left-Hand negative; also the sign of the vertical lines in the Upper quadrants is positive and the Lower negative. (However we are equally entitled to fix the sign of the horizontal line in the Left-Hand quadrants as positive with the corresponding sign in the Right-Hand as negative; also we can fix the sign of the vertical line in the Lower quadrants as positive with the sign in the Upper Quadrants negative).
Though in dynamic terms the signs in each quadrant, horizontally and vertically, keep switching, in a rigid linear interpretation the positive direction is solely emphasised (in each of the four quadrants). Thus the direction of development appears similar for each quadrant (moving from lower to higher levels in each case). However this position is untenable from a proper dynamic perspective.
Again in dynamic holistic terms, linear understanding is positive and negative (i.e. complementary) with respect to opposite quadrants.
Thus though the direction of understanding - in horizontal and vertical terms - is unambiguous for each quadrant taken separately, when opposite quadrants are related to each other the directions are complementary and paradoxical
So once again the task of differentiating and integrating quadrants has to be clearly distinguished. When differentiating, the direction of movement (for each isolated quadrant) appears unambiguous; however when integrating the directions for opposite quadrants are complementary and paradoxical. 9
The two diagonal lines can be used to represent poles that condition experience (simultaneously) both within and between the various levels of experience.
These relate to the most fundamental form/emptiness (and emptiness/form) polarities that can be also expressed in various ways i.e. actual/potential, finite/infinite, immanent/transcendent etc.
In holistic mathematical terms these are positive and negative with respect to each other. However as they combine both horizontal and vertical aspects (in balanced fashion) the sign designation is more complex.
Thus in the UR quadrant the designation is +, +, with the corresponding designation in the diagonally opposite LL quadrant -, -.
Also in the UL quadrant the sign designation is -, +, with the corresponding designation in the diagonally opposite LR quadrant +, -.
Of course the very implication of combining polarities both within and between levels simultaneously is that the very notion of a level loses any distinct meaning.
So as pure nondual awareness is approached in mystical terms, these fundamental diagonal polarities are the last to be successfully reconciled. 10
The Eight Sectors
For a comprehensive interpretation of development, reality must be defined in terms of horizontal polarities (within levels), vertical polarities (between levels) and diagonal polarities (simultaneously within and between levels). 11
A differentiated understanding of reality results from the relative separation of these poles yielding a (linear) asymmetrical interpretation.
However though such interpretations appear unambiguous within each sector considered in isolation, when related to each other, dynamic complementarity exists in horizontal, vertical and diagonal terms. 12
This in fact implies that we are dealing with eight different holarchies.
Basic Outline of the Theory of Everything
Already with the aid of some simple mathematical notions - interpreted in a dynamic holistic sense - we have a system that can fundamentally structure the nature of any development process.
So to recap we started with number and in particular the binary digits 1 and 0 which in holistic terms encode form and emptiness respectively.
Next we had the operations of addition and subtraction, which in dynamic holistic terms encode the means by which phenomena are posited (as conscious) and negated (as unconscious) respectively.
We then introduced the basic geometrical symbols of the line, circle and point and their dynamic equivalents in relation to both linear, circular and nondual understanding. We established their close relationship (in both analytic and holistic terms) to the binary digits and the operations of positing and negating. We also gave a compelling dynamic explanation of duality (i.e. the holistic meaning of "2")
For the moment the Theory of Everything can be summed up as follows. 13
All dynamic processes are conditioned by polarities that can be identified - in holistic mathematical terms – as positive and negative.
Differentiation entails the dualistic separation of poles (where both are treated as positive). This leads to an unambiguous linear appreciation of relationships.
Integration entails the nondual complementarity (and ultimate identity) of opposite poles (which are positive and negative with respect to each other).
This leads to a paradoxical circular appreciation of relationships.
There are three fundamental polarities, which operate in a horizontal, vertical and diagonal fashion.
The horizontal polarities operate within a given level (e.g. interior and interior).
The vertical polarities operate between levels (e.g. wholes and parts).
The diagonal polarities operate both within and between levels simultaneously (e.g. form and emptiness).
Remarkably, every process – physical and psychological - can be precisely encoded using these three polarities i.e. horizontal, vertical and diagonal with respect to both their linear and circular interpretations.
I will demonstrate later – in considerable detail – how this holistic mathematical encoding is applied to structuring the entire Spectrum of Development.
Indeed the Theory of Everything represents the holistic counterpart of the (analytic) binary digits that can potentially be used to encode all information processes.
In like manner this inherently dynamic interpretation of these same digits can be used to potentially encode all transformation processes.
However for a deeper appreciation of the nature of the Theory of Everything we need to introduce some further holistic mathematical notions.
1. Nature of Holistic Mathematics
2. Reductionism and Holistic Mathematical Symbols
3. Holistic Mathematics and the dynamic interaction between perceptions and concepts
4. On One God
5. Complementarity of Physical and Psychological Processes
6. Mystical Purgation as Dynamic Negation
7. Holistic Mathematical Interpretation of a Point
8. Holism and Partism
9. Dynamic Nature of Four Quadrants
10. Integration of Body-Self
11. Levels of Development
12. Illustration of Dynamic Eight Sectoral Approach
13. Nature of TOEs