STAGES OF DEVELOPMENT

(Notes can be directly accessed by clicking on the numerical links throughout the text)

A stage is a sequence of development that is characterised by a coherent overall pattern. 1
All stages can be encoded in a precise holistic mathematical fashion (which intimately reflects the Theory of Everything). 2
So the basis of this integral scientific approach is that all stages of development can be precisely encoded in a dynamic binary digital fashion.

We have a wide variety of means for classifying stages varying from the broad encompassing an extensive range of development to the narrow where the range is very limited.
However there is nothing arbitrary however regarding the manner of this classification from the holistic mathematical perspective. Here - in the context of the approach - stages are always precisely defined (i.e. in terms of their underlying holistic mathematical structure).
In my treatment the Bands represent the broadest and the Modes the narrowest classification of stages respectively. 3

Linear and Circular Aspects

All stages can be defined in a linear (1) and a circular (0) manner reflecting both differentiation and integration respectively. In dynamic terms these two aspects necessarily interact with the precise configuration of both aspects varying throughout development. The very essence of the holistic mathematical approach is that it can demonstrate the fundamental nature of this configuration for every stage.

From the linear (differentiated) perspective (1) stages are viewed in a discrete manner where they are considered relatively independent of each other. The relationship between them then appears asymmetric e.g. proceeding unambiguously from the lowest to the highest. 4

From the corresponding (integrated) perspective (0) stages are viewed in a continuous manner where they are considered relatively interdependent (with all other stages). The relationship between stages is then characterised by dynamic symmetry where polar opposites (in dualistic terms) are directly complementary with each other. Therefore the relationship between stages - by contrast - now appears paradoxical e.g. where "lowest" and "highest" stages are directly related. 5

The greatest problem with conventional approaches is that theses two crucial aspects of development are not properly distinguished from each other (with integration invariably reduced to differentiation). 6
However as they are based on distinctive logical systems it is vital to clearly deal with both aspects throughout development.

To maintain consistency as between the linear (differentiated) and circular (integrated) treatment of stages, it is vital to demonstrate the relationship as between both approaches.

Just as in analytic terms 1 - 1 = 0 likewise it is true in holistic mathematical terms.
Therefore to move to integral appreciation all (positive) asymmetric interpretations must be dynamically negated. This then leads to appreciation of an equally valid opposite mirror interpretation for every asymmetric relationship..
Thus to move properly from a linear (differentiated) to circular (integral) appreciation of development, every asymmetric must be paired with its opposite (mirror) interpretation creating a bi-directional linear approach. 7
Therefore it is the pairing of these opposite interpretations (which are mirror images of each other) that is the basis for the (circular) complementary symmetry that characterises the true integral approach.

Horizontal, Vertical and Diagonal Polarities

Horizontal: Stages of Self; Stages of Reality

Again in accordance with the holistic mathematical Theory of Everything, all stages can be defined with respect to horizontal, vertical and diagonal polarities having - as we have seen - both linear (1) and circular (0) interpretations. 8

Every stage of development is characterised by a dynamic interaction between the self and reality (i.e. the world).
Thus from one perspective, we can only know the self (with reference to the world).
From the equally valid opposite perspective we can only know the world (with reference to the self).
Thus the self always has a mirror interpretation in terms of the world; equally the world has a mirror interpretation in terms of the self.

When we view the stages of development from the linear (asymmetrical) perspective, these polar reference frames will necessarily be treated in isolated fashion with - typically - the self being viewed as the interior and reality as the exterior pole respectively.

The Spectrum of Development then becomes split - as it were - into two distinctive halves with Stages of Self listed on one side (conventionally the Left-Hand side) and Stages (or Planes) of Reality on the other (Right-Hand side).

Thus in terms of this linear perspective, we can view development either as progressive stages of self (interior) or alternatively as progressive stages of reality (exterior) both of which correspond with each other (i.e. move in a forward direction).

However from a dynamic perspective this is very misleading. As in the often repeated example of the two drivers when A moves forward (relative to B), then B thereby moves backwards (relative to A). Also when B moves forward (relative to A), A thereby moves backward (relative to B).
Thus - in dynamic terms - forward movement with respect to the Stages of Self (in relation to Reality) implies backward movement in the Stages of Reality (with respect to Self).

Likewise forward movement in the Stages of Reality (in relation to Self) implies backward movement in the Stages of Self (in relation to Reality).
Thus in mature dynamic terms progression (with respect to one pole) always implies regression (with respect to the other) and vice versa.

This is an extremely important point. Because by its very nature, a linear (asymmetric) treatment cannot properly handle the dynamics of development, it always creates a tendency in such an approach to avoid such dynamics through over-identification with one arbitrary pole of development.
Thus the Spectrum of Development is invariably treated as a largely (interior) psychological process where the emphasis is heavily on the Stages of Self. Though it may well be admitted that it can equally have an interpretation as Stages of Reality, it will be typically assumed that these - misleadingly - directly correspond to the Stages of Self (i.e. both unfolding in the same progressive forward manner).

So let me make this point strongly. Psychological development - especially in relation to advanced mystical stages - does not strictly relate to the self (as somehow in isolation from the world). Rather - more accurately - it relates to the development (and of course envelopment) of the dynamic interaction of the self with the world (and the world with the self). So for proper balance we must equally emphasise each aspect. And if we are to understand this relationship properly we must attempt to do so it in dynamic terms.

Thus what has been already posited (and thereby differentiated) as a stage must to a degree be also negated before integration can successfully take place (the precise degree of negation required depending on the respective stage).
Thus a Stage of Reality can only unfold (revealing a certain understanding of the world) through the dynamic negation of the corrresponding Stage of Self . Likewise a Stage of Self can only unfold (revealing a certain psychological understanding of the self) through the negation of the corresponding Stage of Reality. 9

Failure to properly negate at any stage (on either side) sets severe limits on the possibility of successful further progression in development (either as Stages of Self or Stages of Reality).

When the negative direction of development is properly recognised, the opposite mirror interpretation with respect to both Stages of Self and Stages of Reality can unfold. One can now understand in bi-directional terms, clearly recognising that for any linear (asymmetrical) interpretation of development - by definition - a corresponding opposite interpretation always exists which is equally valid.

So now one realises that the Stages of Self have opposite mirror interpretations (as Stages of Reality). Likewise Stages of Reality have opposite mirror interpretations
(as Stages of Self). 10

Integration in development then depends on the extent to which these opposite interpretations can be successfully reconciled.

Thus from an integral perspective the Stages of Self (throughout development) and Stages of Reality are complementary with each other. When properly understood therefore - from this integral perspective - the Self and Reality are mirrors of each other (and ultimately perfect mirrors). Therefore the structure - at any level - of a stage corresponding to the (psychological) self equally has a complementary interpretation in relation to (physical) reality. Likewise the structure of any stage applying to (physical) reality has a complementary interpretation in relation to (psychological) reality.
So in an integral approach there is no ultimate division as between (personal) psychology and (impersonal) science for they are now properly seen as mirrors of each other always revealing complementary structures.
And the very basis of this scientific approach is that these structures can be precisely encoded in holistic mathematical format (with opposite signs for both aspects).
Thus the holistic mathematical approach provides a unique means for revealing the deep complementary horizontal structures (i.e. exterior and interior) that underlie development at all levels (psychological and physical).

So to summarise

The horizontal poles of development relate to exterior and interior aspects (and interior and exterior) which always dynamically interact in experience.

In linear (asymmetrical) terms - suited to differentiation of experience - these are separated; in circular (dynamically symmetrical) terms - suited to integration - these are treated as complementary (and ultimately identical).

Thus in linear (asymmetrical) terms, development can be posited in a horizontal manner both as Stages of Self (interior) and Stages of Reality (exterior) which unfold in an unambiguous progressive fashion.

In circular (symmetrical) terms - at every level - Stages of Self and Stages of Reality are complementary with each other. Stages are now - to a degree - both posited and negated revealing their complementary nature i.e. as unfolding in opposite directions from each other.
It is the reconciliation of such paradoxical notions (in dualistic terms) through spiritually intuitive nondual awareness that constitutes the very essence of integration (where separate polarity does not strictly exist). 11

Every stage has a coherent structure, which can be precisely encoded in holistic mathematical format.
Integral Science 1 relates to the attempt to reveal these complementary structures (interior and exterior) for both psychological and physical reality at each level. This process - by its very nature - greatly facilitates the integration of experience.

Vertical: Stages as States; Stages as Structures

Every stage is likewise characterised by a dynamic interaction as between a basic state (emptiness) and a fundamental structure (form). The state represents an overall spiritual manner of "seeing" (enabling interpretation of the stage). The corresponding structure characterises the holistic pattern of the stage that is thereby interpreted. 12

Cleanly the manner in which we see reality (state) will influence the interpretation of phenomenal relationships (structure). Likewise the nature of such an interpretation (structure) will in turn influence the manner of its seeing (state).

This dynamic interaction as between states and structures (and structures and states) is deeply pertinent to understanding the second set of vertical polarities (relating to the interaction of wholes and parts).

With conventional (conscious) understanding, wholes and parts are necessarily reduced in terms of each other. So in relation to perceptions, the (qualitative) whole aspect is reduced to the (quantitative) part. By contrast with concepts the (quantitative) part aspect is reduced to the (qualitative) whole. The very recognition of holons (in conscious terms) entails this very reductionism.
Alternatively we could say that with such holons the manner of its "seeing" and the corresponding phenomenal interpretation are not properly distinguished i.e. states are confused with structures (and structures with states).

This confusion is especially apparent in conventional science where the waking state through which reality is viewed is assumed to be neutral with respect to phenomenal interpretation. Thus the important dynamic interaction as between states and structures (and structures and states) is ignored. Not surprisingly such scientific interpretation - which we literally identify as reality - leads inevitably to considerable reductionism in terms of our appreciation of both wholes and parts.

So let us now attempt to unravel the truly subtle nature of this dynamic interaction between whole and part (and part and whole).

Say, for example, I form a perception of a phenomenon such as a house (part). Now to move to the corresponding conceptual recognition of the house, I must be able to see it as sharing a common spiritual quality (i.e. of "houseness") that potentially applies to all phenomena within its class.
Thus the actual insight which enables this transition from perception to concept (quantity to quality) is inherently spiritual representing a state of "seeing" (transcendence). However in conventional understanding, once this switch has been made, the subsequent concept is quickly reduced in "real" conscious terms.
So with conventional science we deal with both perceptions and concepts merely as phenomenal structures. Though the spiritual state of "seeing" is deeply necessary for making connections as between both perceptions and concepts (facts and theories), it remains merely implicit and in formal terms is ignored.
Not surprisingly therefore the ability to creatively "see" is greatly diminished and our scientific interpretations become largely robbed of their inherent mystery.

I have dealt briefly with the manner in which we move from perceptions (parts) to concepts (wholes) indicating the crucial role played by one's spiritual state in making this change.

However the opposite direction is equally important, whereby we move from concepts (wholes) to perceptions (parts). Once again an opposite spiritual state (immanence) is required to enable this switch to be made. Now the collective quality of "houseness" is seen to be embodied in each individual house.
So even when unrecognised, switching from perceptions (parts) to concepts (wholes) and from concept (wholes) to perceptions (parts) implicitly requires the intervention of spiritual states before the corresponding (phenomenal) structures can be identified.

Once again, when we identify reality - as with conventional science - in solely "real" terms, states are reduced to structures.

A great value of the holistic mathematical approach is that it can now enable us to precisely identify the true dynamic relationship as between wholes and parts.

Quite simply, when structures are identified as "real", then the corresponding states are "imaginary" (in a precise holistic mathematical fashion).

We have already demonstrated in a previous Chapter that the "imaginary" is the means through which unconscious spiritual recognition becomes indirectly embodied in phenomena.
Thus when we explicitly recognise the necessary interaction (without gross reductionism) of states with structures (and structures with states) then reality is viewed in dynamic "complex" terms (i.e. with both "real" and "imaginary" aspects).
The direct conscious recognition of phenomena (as reduced structures) provides the "real" aspect of this understanding.
However the indirect recognition - through which spiritual states become expressed in phenomena as holistic symbols or archetypes - provides the corresponding "imaginary" aspect.

So once again from the vertical aspect of development where we express the interaction of wholes and parts (and parts and wholes) we can attempt to view it in both a linear (differentiated) and circular (integrated) fashion.

Thus from one perspective we can view development in linear (asymmetrical) terms as the progressive unfolding of (phenomenal) structures (which are unique for each stage).

Likewise from the alternative perspective we can view development - again in linear (asymmetrical) terms - as the progressive unfolding of spiritual states of seeing (which again are unique for each stage).

However because structures and states (and states and structures) dynamically interact in experience, these directions become paradoxical (when viewed in relationship to each other).
Thus for each asymmetrical interpretation, a corresponding opposite (mirror) version can be given which is equally valid for both structures and states. 13

From an integral perspective, structures and states are complementary. Once again we move to this integral position through appreciation of bi-directional linear (asymmetrical) interpretations for both structures and states. 14
It is the interaction between both that enables vertical mobility as between "higher" and "lower" (and "lower" and "higher") stages to take place. Thus when little interaction is evident, vertical mobility becomes very restricted with experience largely conforming to a very limited range of the overall spectrum. 15

Thus in vertical terms "higher" and "lower" (and "lower" and "higher") stages are complementary (with the "highest" directly complementary with the "lowest").

As we shall see the fundamental dynamic interaction as between states and structures for corresponding "lower" and "higher" levels is exactly similar. However whereas for "lower" levels these bi-directional dynamics are greatly confused, at the corresponding "higher" levels they are appropriately unravelled and thereby properly integrated.

This entails that it requires the corresponding "higher" structure to interpret a complementary "lower" structure.

One important implication with relation to the stages (planes) of reality is that if we wish to properly understand the fundamental relationships governing the "lowest" sub-atomic physical particles, we must use the corresponding "highest" structures (associated with the most advanced stages of spiritual development). 16

So again to summarise

The vertical poles of development relate to whole and part aspects (and part and whole) which always dynamically interact in experience.

In linear (asymmetrical) terms - suited to differentiation - these are separated; in circular (dynamically symmetrical) terms - suited to integration - these are treated as complementary (and ultimately identical).

From a linear (asymmetrical) perspective, stages of development can be posited in vertical terms as States (wholes) or as Structures (parts) which unfold in an unambiguous progressive fashion.

However when these stages are appropriately negated they reveal equally valid opposite linear interpretations (where States have now a part and Structures a whole interpretation respectively).

Proper (circular) integration of states with structures (and structures with states) requires significant bi-directional understanding of their complementary dynamics. This in turn greatly facilitates increased vertical mobility through the various stages of the Spectrum (from "high" to "low" and from "low" to high" respectively).

Typically - with conventional understanding - states are significantly reduced to structures with the associated phenomena identified in solely "real" terms (i.e. literally as "reality").

To properly preserve however the subtle interaction as between wholes and parts and parts and wholes (alternatively states with structures and structures with states) without gross reductionism all phenomena must be defined dynamically in holistic mathematical "complex" terms (i.e. with both "real" and "imaginary" aspects).

Thus if we identify the "real" aspect in direct conscious terms (as structure) then the corresponding state is - relatively - "imaginary". In other words this "imaginary" aspect represents the indirect manner though which phenomena serve as symbols or archetypes of spiritual "seeing".
However because structures and states (and states and structures) keep dynamically switching, both have "real" and "imaginary" aspects.

Alternatively in proper dynamic context, wholes and parts are always "real" and "imaginary" (and "imaginary" and "real") with respect to each other.
Finally this equally applies that to avoid reducing the qualitative aspect to the quantitative (and the quantitative to the qualitative) all phenomena must be defined in dynamic "complex" fashion (with - relatively - "real" and "imaginary" aspects). 17

Integral Science 2 relates to the attempt to reveal these complementary structures (whole and part) incorporating the dynamic interaction of states and structures for both psychological and physical reality at each level. This requires the explicit adoption of a "complex" holistic mathematical approach (with both "real" and "imaginary" aspects)

Diagonal (Stages as Mind; Stages as Body)

It might help to put this section in context by summarising briefly the nature in which "higher" spiritual integral development takes place.

The first main task is the attempted reconciliation of the horizontal polarities (exterior and interior) within a given level (which would correspond broadly with the subtle realm).
However there are significant limits on the degree to which this task can be fully achieved. This arises from the inability - as yet - to properly distinguish structures from states. Thus typically at the subtle level, though there may indeed be significant realisation of an enhanced spiritual state, undue attachment to symbols as spiritual archetypes will tend to remain. So to a degree we still have the confusion of states with structures.

So the next major task - corresponding to the causal realm - is the attempted unravelling of this confusion of states and structures (and structures and states) which equally entails the proper dynamic interaction (without reductionism) of the vertical polarities of whole and part (and parts and whole). This leads to an enhanced ability to flexibly switch between the "higher" and "lower" (and "lower" and "higher") stages of the Spectrum greatly enhancing integration.

However a further type of confusion now becomes especially evident. So far we have not distinguished the role of cognitive and affective understanding in development (which vitally affects this switching ability). So typically with causal type development one of these modes (largely depending on personality) will still tend to dominate. So for example we may well have someone at this level, who relies unduly on the refined use of the intellect to guide development (leading to repression of physical instinctive behaviour).
Also in an attempt to limit such troublesome dynamics, there may be still an imbalance in terms of development with too much attention to just one side of the Spectrum (typically the Stages of Self).

So the final main task of integration is the simultaneous reconciliation of both the horizontal and vertical polarities (i.e. the diagonal).
So one now simultaneously tries to achieve equal balance as between the horizontal polarities (Stages of Self and Stages of Reality) and also the vertical polarities (Stages as States and Stages as Structures).

However to achieve this simultaneous balance successfully, it is now necessary to pay intimate attention to the fundamental psychophysical interactions which underlie experience. This in turn requires achieving a successful balance as between the cognitive and affective modes.

It may be helpful initially to look on the cognitive mode as the means through which the self exercises independence gaining a degree of control over reality.

However for successful integration, one must be equally able to respond effectively to reality (accepting one's dependence and vulnerability). So in a direct sense it is - relatively - through the corresponding affective mode that this response is made.
Putting it another way, now at the most fundamental level body and mind must be properly integrated (which in turn requires integration of both the cognitive and affective modes).

As I say, intimate focus on fundamental psychophysical interactions leads to an appreciation of what are in effect the diagonal polarities.

Thus when we combine the horizontal polarities (exterior and interior) with the vertical polarities (higher and lower) we have the following diagonal dynamics.

Thus the "higher" stages (with respect to Reality) are complementary with the "lower" stages (with respect to Self); equally the "lower" stages (with respect to Self) are complementary with the "higher" stages (with respect to Reality).

Also the "higher" stages (with respect to Self) are complementary with the "lower" stages (with respect to Reality); equally the "lower" stages (with respect to Reality) are complementary with the "higher" stages (with respect to Self).

Now what is very important to appreciate is that these complementary diagonal dynamics are themselves intimately dependent on continual successful switching as between cognitive and affective (and affective and cognitive) modes respectively.

This development - which depends very much on pure volitional intent - is extremely demanding in terms of the maintenance of any kind of phenomenal attachment.

Thus, for example a momentary "high" stage cognitive attachment can arise with respect to the Stages of Reality. So in contemplating the wonders of reality one may become momentarily attached to the particular intellectual manner through which this experience is interpreted.
This then will be quickly followed by a troublesome instinctive impulse relating to a complementary "low" stage affective attachment of the Self. So in dealing successfully with this attachment, one has to simultaneously switch direction in both horizontal (from exterior to interior) and vertical ("high" to "low") terms which requires significant lessening of the rigidity of the attachment. Therefore through continual refining of such attachment - which with phenomenal awareness must necessarily to some degree always exist - one can become extremely proficient at making such switches thereby continually residing in the emptiness of Spirit.

By combining States and Structures with Stages of Self and Reality we can give four separate interpretations of development in linear (asymmetrical) terms.

1) It is viewed as a progressive movement through Stages of Self (with respect to their corresponding Structures).
2) It is viewed as a progressive movement through Stages of Self (with respect to their corresponding States).
3) It is viewed as a progressive movement through Stages of Reality (with respect to their corresponding Structures).
4) It is viewed as a progressive movement through Stages of Reality (with respect to their corresponding States).

Now once again when the positive direction (leading to the asymmetrical understanding) is appropriately negated, understanding then becomes properly bi-directional with opposite (mirror) interpretations in each case. 18

Circular Integration then tales place through the nondual spiritual reconciliation of complementary bi-directional diagonal interpretations (which are paradoxical in dualistic terms).

The diagonal polarities - which combine both horizontal and vertical notions of direction - have therefore (simultaneously) two aspects (i.e. "real" and "imaginary") in formal structural terms.
Equally, as the successful integration of these polarities requires a remarkable degree of non-attachment (of a rigid kind) as the TOE demonstrates they have an alternative interpretation as null lines = 0, pointing to the deep empty state of spiritual awareness that results from this dynamic process.
Thus, quite remarkably, the holistic mathematical approach can demonstrate the precise manner through which states and structures (with respect to Self and Reality) are ultimately harmonised.

From another equally valid perspective, the diagonal polarities relate to the proper balancing in experience of the cognitive and affective modes (which are "real" and "imaginary" with respect to each other).
When these two modes are properly integrated in experience, both the transcendent and immanent aspects of Spirit can reach their purest expression becoming identical in empty Spirit.

So finally once more to summarise

The diagonal poles of development relate to most fundamental manifestations of form and emptiness (emptiness and form). These point ultimately to the most intimate psychophysical interactions (where mind and body simultaneously interact).

Such interactions in turn relate directly to the subtle interaction of cognitive and affective modes in experience (representing the most basic patterns of control and response with respect to self and the environment).
Quite clearly body and mind cannot be properly integrated without corresponding integration of the cognitive and affective modes respectively.

These diagonal polarities can be expressed (combining both horizontal and vertical aspects) as the pairing of either States or Structures with Stages of Self or Reality leading to four distinct formulations.

In linear (asymmetrical) terms these are again separated; in circular (dynamically symmetrical) terms entailing bi-directional understanding, diagonally opposite pairings are related and understood as complementary (and ultimately identical) representing the most profound degree of spiritual integration.

The holistic mathematical approach with its dual (equal) interpretation of diagonal polarities demonstrates clearly in dynamic terms how Structures and States with respect to both Self and Reality are ultimately perfectly harmonised.

Integral Science 3 relates to the attempt to reveal these complementary structures (fundamental form and emptiness) incorporating the dynamic interaction of body and mind for both psychological and physical reality at each level. This requires the explicit adoption of an approach with twin aspects (that are ultimately identical) i.e. in terms of structure as simultaneously "complex" (with "real" and "imaginary" aspects) and also as nulll ineas (= 0) in terms of corresponding states.

Conclusion

This treatment of stages - without reference as yet to their distinct methods of classification - has been necessarily abstract.
However what I have been at pains to demonstrate how the holistic mathematical approach enshrining in dynamic binary format the Theory of Everything, can be precisely applied to the key features that are applicable to all stages.

Thus we established that such stages can be defined in a linear (1) and circular manner (0) pertaining directly to their differentiation and integration respectively. Not alone this, but the holistic binary approach establishes the precise relationship between the linear and circular enabling the appropriate interface (through bi-directional understanding) to be consistently maintained.

We then established that the horizontal polarities (interior and exterior) with a holistic mathematical "real" explanation, can be directly applied to understanding the Spectrum in terms of Stages of Self and Stages of Reality.

In linear (asymmetrical) terms - suited to differentiation - these are defined with respect to their merely positive (+) directions i.e. progressive evolution with respect to both aspects (treated in a relatively independent manner).
In circular (symmetrical) terms - suited to integration - these are defined in dynamic bi-directional fashion with respect to both their positive (+) and negative (-) directions. It is the very reconciliation of the dualistic paradox that results that leads to the integration of both directions in a transformed nondual spiritual awareness.

We next established that the vertical polarities (whole and part) that have a holistic mathematical "imaginary" explanation, can be directly applied to understanding the Spectrum in terms of States and Structures (with States and Structures dynamically "real" and imaginary" with respect to each other).

Again in linear (asymmetrical) terms - suited to their differentiation - these are defined with respect to their merely positive (+) directions i.e. progressive evolution of stages with respect to both States and Structures (treated in a relatively independent manner).

In circular (symmetrical) terms - suited to their integration - these are again defined in a dynamic bi-directional fashion with respect to both positive (+) and negative (-) directions in "real" and "imaginary" terms.
Again it is the reconciliation of the dualistic paradox that results that leads to the integration of both aspects in an enhanced nondual spiritual awareness.

Finally the diagonal polarities (the most fundamental manifestations of form and emptiness) have a double holistic mathematical interpretation as simultaneously "complex" (with "real" and "imaginary" aspects alternating between positive and negative expressions) providing the most refined phenomenal explanation of structures, and also null lines = 0 (the most refined expression of an empty spiritual state).

In linear (asymmetrical) terms - suited to differentiation - these are defined with respect to two variables (which can be positive or negative) leading to four possible linear "complex" directions.

In circular (symmetrical) terms - suited to integration - these are defined in a dynamic bi-directional fashion where both "real" and "imaginary" aspects of the "complex" formulation is paired with a mirror formulation that is opposite in both respects.

Notes

2. TOE