Integral Notions of Space and Time
Q Can you briefly recap on the integral notions of space and time covered in the last Chapter?
PC Integral notions of space and time correspond exactly to the mathematical concept of dimensions when given their dynamic holistic meaning.
Thus in the two-dimensional notion, both space and time have complementary opposite positive and negative polarities.
Once again linear understanding, which serves as the basis for standard dualistic communication is asymmetric and literally one-dimensional, where the positive aspect of experience is solely recognised (in formal terms).
In other words with linear understanding, interpretation is based solely on (consciously) posited phenomena in experience.
Two-dimensional understanding strictly points to a nondual ineffable state where opposite polarities are fully reconciled and thereby have no separate existence. However in terms of structure, this state is interpreted in a reduced linear manner by identifying such polarities as positive and negative with respect to each other.
This corresponds in holistic terms therefore with the mathematical notion of a square root whereby a number that is initially two-dimensional (i.e. raised to the power of 2) is now given a reduced interpretation that is either positive or negative in a linear one-dimensional fashion.
Therefore if x2 (as the two-dimensional number) = 1 (strictly 12) then x1 (i.e. its reduced one-dimensional expression as x) = + 1 or - 1.
Thus from an analytic perspective, the reduced linear interpretation of a two-dimensional expression has either positive or negative aspects (which are equal in static absolute terms).
However from the corresponding holistic perspective, the reduced linear interpretation of a two-dimensional expression has both positive and negative aspects (which are now equal in dynamic relative terms). In other words the very means of realising the pure nondual state (where opposite polarities are fully identical) is through keen appreciation of the relative paradoxical nature of their reduced phenomenal expressions (i.e. where positive and negative aspects are initially identified as separate).
So once more I am illustrating the crucially important fact that every analytic mathematical notion has a corresponding holistic interpretation which is inherently dynamic. This latter interpretation of mathematical symbols then serves as the basis for an appropriate integral appreciation of reality that is truly scientific.
These two dimensions (with respect to both space and time) serve as the very means through which dynamic switching as between the horizontal polarities i.e. exterior and interior (and interior and exterior) takes place.
Thus from a human perspective in order to consciously experience - and thereby posit - a phenomenon in physical terms (with respect to both space and time), we must dynamically negate its corresponding existence in a psychological manner.
Likewise in order to consciously experience and posit a phenomenon in psychological terms we must dynamically negate its corresponding existence in a physical manner.
Thus the dual and nondual aspects of experience are inextricably linked. Without a pre-existing nondual state (where both positive and negative are identical) it would not be possible to separate polarities and thereby posit phenomena in dualistic terms. Likewise the dynamic negation of what has been (consciously) posited enhances the nondual spiritual aspect of experience.
When the nondual aspect is not sufficiently developed, the positing of phenomena takes place in a somewhat rigid manner so that little dynamic negation as between opposite polarities occurs. This in turn sets severe limits to the possible growth in nondual spiritual awareness.
It is important to remember that integral notions of space and time, which enable the appropriate dynamic interaction as between the dual and nondual aspects of experience, are symmetrical.
However asymmetric analytic notions of these dimensions dominate conventional scientific understanding.
I associate the mature specialised development of this dynamic two-dimensional symmetrical appreciation of space and time with H1 (i.e. the circular or subtle level). Therefore at H1 the physical and psychological aspects of understanding - which are relatively exterior and interior with respect to each other - become increasingly complementary.
In other words horizontal complementarity (Type 1) exists as between these two poles.
Vertical complementarity also exists as between the "higher" level of H1 (subtle) and its corresponding lower level of L1 (mythic) in that the mature interaction of exterior and interior aspects at H1 is replicated by the corresponding confused interaction of the same polarities - where these aspects are not yet properly differentiated - at L1.
Q Does this two-dimensional integral interpretation of reality apply to all holons?
PC Yes! All holons - at whatever level of existence - are characterised by the interaction of exterior and interior (and interior and exterior) polarities.
Thus all holons, by their very existence, possess both differentiated (dual) and integral (nondual) aspects.
However at the "lower" levels of existence, the interior remains greatly undeveloped so that self-conscious appreciation of such dynamics remains negligible being confined to largely automatic instinctive type interaction. However this is never entirely the case thus enabling evolution in creation to occur.
Q Briefly what is the limitation of two-dimensional understanding?
PC From a psychological perspective such two-dimensional understanding is suitable for the reconciliation of opposite polarities that are interpreted in a merely conscious manner.
However in the dynamics of understanding both conscious and unconscious interactions are interdependent. Therefore lack of development in the unconscious sets limits to the pure development of conscious understanding. Though a considerable amount of spiritual refinement in conscious understanding can indeed take place at H1, a degree of remaining rigidity is inevitable due to lack of sufficient corresponding refinement in the unconscious.
Q Presumably because physical and psychological aspects are complementary this interpretation must equally apply to the physical aspect. What again is the corresponding relationship to the interaction of conscious and unconscious in physical terms?
PC This indeed is a very important issue with considerable implications for physical science.
What are defined as real phenomena in conventional scientific terms apply directly to the conscious aspect of understanding. However such science does not recognise any corresponding physical notion with respect to the unconscious aspect.
So from a holistic perspective we can see how the reduced psychological interpretation of phenomena (i.e. based on a merely conscious aspect of understanding) necessarily implies a corresponding reduced physical identification of such phenomena solely in real terms.
However from a dynamic holistic perspective, material phenomena that are identified in real terms, necessarily interact with an immaterial ground (and can have no meaning in the absence of this context).
Problems with Development at H1
Q Can you go into more detail as to how problems arise with development at H1 due to the lack of sufficient refinement in unconscious understanding?
PC Spiritual development can take many different forms so it is not possible to give just one definitive account of such problems.
However I will attempt here to give a coherent explanation from the perspective of just one type of unfolding that is related to personal experience and which provides the basis for the holistic mathematical type understanding that I am here portraying.
The main emphasis at H1 is on the reconciliation of the horizontal polarities i.e. exterior and interior (and interior and interior). One thereby attempts to spiritually integrate both the physical and psychological aspects of experience within a given level.
However a certain imbalance still remains with respect to the vertical polarities i.e. whole and part (and part and whole) so that it is not yet possible to properly integrate physical and psychological aspects between levels.
In other words the vertical relationship as between whole and part is still interpreted in a somewhat asymmetric fashion.
One typical example of this problem is the ranking of the psychological as superior to the physical aspect with development moving from matter to mind to Spirit. Progress then requires the overcoming of the "lower" physical body so as to engage with one’s "higher" spiritual self.
Thus quite frequently H1 (the subtle level) is approached in an unduly holarchical manner with a strong transcendent direction. Here the intellect is used in a refined matter to control the emotional impulses of the body and Spirit is experienced in an increasingly collective holistic manner.
Though such discipline is indeed very necessary it can easily lead to a distorted relationship as between body and mind through the build up of significant emotional repression. This often reaches a crisis during the "dark night of the soul". Though this stage indeed represents a significant growth in authentic contemplative development, equally it can coincide with pathological depressive elements relating to the blockage of natural physical impulses (that are deemed incompatible with this new life of the Spirit).
When properly understood this depression therefore can serve as a valuable signal pointing to the need for better balance as between nature and Spirit through an increased focus on the immanent aspect of development. The next major stage of H2 (the point or causal level) is then concerned with the reconciliation of the vertical polarities which enable interaction as between "higher" and "lower" levels. This in turn requires refined appreciation of the nature of the two-way interplay as between holarchical (whole/part) and onarchical (part/whole) aspects of development.
Nature of Unconscious Repression
Q Can you go more deeply into the nature of unconscious repression with a view to demonstrating the explanatory value of the holistic mathematical approach?
PC As we have seen conscious understanding, which corresponds with the identification of real phenomena, is based on linear (i.e. one-dimensional) experience relating solely to its positive direction.
However in dynamic terms the positive has no meaning in the absence of the negative direction. Thus when a phenomenon is consciously posited, thereby splitting it off from its complementary pole, then the (unrecognised) negative aspect is repressed in an unconscious manner. So in this sense unconscious repression arises from an unbalanced form of understanding where meaning is identified with the merely conscious aspect of phenomena.
From a volitional perspective such understanding is always likewise associated with an excessive degree of (conscious) attachment.
The root of possessive attachment - which is the key barrier to authentic spiritual development - lies directly in the will. However indirectly it is associated with a distorted form of understanding that is biased towards its positive (conscious) direction.
This leads to phenomenal rigidity whereby the unconscious remains stunted and weakly developed.
Put simply therefore whereas conscious interpretation - in direct terms - is based on the positive direction of experience (+), the unconscious is based on its corresponding negative direction (-).
Furthermore the conscious approach - by its very nature - leads to the dualistic splitting of opposite polarities, where again the positive aspect is solely recognised (with the negative repressed in unconscious terms).
This is why conscious understanding is inherently linear (one-dimensional) as it is based unambiguously on just one direction of experience (+).
By contrast unconscious development leads to the nondual reconciliation of both positive and negative polarities (i.e. through dynamic negation of what has been posited).
So unconscious experience - through appropriation of the negative aspect - is inherently two-dimensional and leads to the (nondual) incorporation of both directions (+ and -).
Balanced spiritual development which properly recognises the complementary role of conscious and unconscious requires therefore the equal development of both directions (+ and -).
Q Can this unrecognised unconscious express itself?
PC Yes, but always in an involuntary manner whereby it is confused with conscious understanding!
The unconscious points ultimately to holistic meaning which is spiritually ineffable. However when not properly recognised (through repression) it is projected involuntarily into consciousness whereby it becomes rigidly attached to phenomenal objects.
The (unconscious) projection is then wrongly identified with the (consciously) perceived objects.
Q What precisely do you mean by negation?
PC We need here to distinguish two types. The one we have been discussing relates to rigid negation. Here the holistic role of the unconscious is not recognised and in effect is confused with conscious symbols.
However the type associated with authentic spiritual development (especially during its most purgative stages) relates to dynamic negation, that enables phenomena to be gradually dissolved in the unfolding of pure nondual awareness.
Such dynamic negation entails a lessening of possessive attachment (associated with the positing of phenomena in an unduly rigid manner).
Dynamic negation is thereby facilitated through a state of growing spiritual awareness where opposite polarities are reconciled. Indirectly this is associated with the recognition that the dualistic positing of phenomena is inherently paradoxical based on arbitrary reference frames.
Thus - as we saw in our discussions on Heraclitus - we can posit a phenomenon in an (exterior) physical manner (reflecting one arbitrary reference frame). However equally we can posit the "same" phenomenon in an (interior) psychological manner (reflecting the opposite reference frame). Though consistency can be maintained within each (as separate) the interpretations of both reference frames - when considered as interdependent - are paradoxical and thereby incompatible with each other.
With growing spiritual development dynamic switching as between horizontal poles (exterior and interior) greatly accelerates, therefore highlighting the paradox associated with dualistic interpretation (where opposite poles are separately posited).
The very recognition of such paradox then serves as the indirect means of relinquishing rigid attachment to understanding (based on separate poles) thus enabling growth in nondual contemplative awareness.
Imaginary Numbers and the Unconscious
Q Can you explain again the extremely important relationship that exists as between the unconscious and the holistic mathematical interpretation of an "imaginary" number?
PC As we have seen, unconscious experience is directly associated with the negative direction (i.e. whereby phenomena that have already been consciously posited are thereby negated). This leads to a certain nondual fusion of both polarities in two-dimensional fashion (i.e. as both + and -) and the quality of this interaction depends greatly on the degree of spiritual development obtained.
However the unconscious can only express itself indirectly in a conscious manner through reduction to a linear (one-dimensional) form.
We have here the exact holistic counterpart to the analytic notion of an imaginary number.
Thus if x2 = - 1, then x i.e. x1 = + i or - i (where i = the square root of - 1).
Thus the two-dimensional expression (i.e. x2) which is equal to a quantity with a negative sign or direction (- 1) is thereby given a one-dimensional result (as imaginary) in either positive or negative terms.
In like manner in dynamic holistic terms, the two-dimensional psychological expression where opposite polarities are identical - which represents the negative direction of a quality i.e. qualitative experience - is given a one-dimensional result (as imaginary) in both positive and negative terms.
I cannot over-estimate the importance of this connection, which is one of the most fundamental that can be made. Quite simply it establishes a fully precise - and yet unrecognised - relationship as between the nature of unconscious experience and the notion of an imaginary number (when given its appropriate holistic interpretation).
In analytic terms number is synonymous with quantitative ordering.
Though not properly appreciated, in holistic terms number is also synonymous with qualitative - more strictly quantitative/qualitative - ordering of all dynamic transformation processes.
Therefore the most precise means we have for structuring development lies in the holistic interpretation of number symbols.
As we know the comprehensive number system (in analytic terms) combines both real and imaginary numbers.
The holistic counterpart to this relates to both the conscious (real) and unconscious (imaginary) processes of development.
This finding is extraordinarily significant with the power to revolutionise the very way we look at "reality".
For in scientific terms "reality" is no longer just "real" (i.e. the product of merely conscious processes of interpretation). "Reality" is now in holistic mathematical terms "complex" with both "real" (conscious) and "imaginary" (unconscious) processes of interpretation.
So we have now made the vital connection which enables us to incorporate the dynamic relationship of the unconscious with the conscious in a coherent scientific integral manner.
Q What is the counterpart in physical terms of "complex" reality?
PC As we have seen physical "reality" represents the dynamic interaction of "real" phenomena (corresponding to consciously observed phenomena) and a fundamental ground (corresponding to unconscious understanding). Now this ground (as with the unconscious) can only express itself in a reduced one-dimensional manner.
Virtual particles provide a fine example of this (reduced) phenomenal activity which temporarily emerge from the ground to interact with "real" particles.
Therefore virtual particles represent the "imaginary" aspect of phenomena.
As all physical "reality" entails the interaction of sub-atomic processes where "real" and "imaginary" (i.e. virtual) particles interact, such "reality" is thereby "complex" in holistic mathematical terms.
And just as "imaginary" (i.e.) particles are associated with the very "low" levels of physical reality, "imaginary" understanding of phenomena is associated with the corresponding "high" levels. So there is vertical complementarity here, as the "high" spiritual is required to give a philosophically coherent interpretation of the corresponding "low" physical level.
Q Are there connections as between the holistic mathematical notion of the "imaginary" and the word as used in its common literary sense as "relating to the imagination"?
PC Remarkably yes! Though mathematicians might deny any intention, there is in fact a very close connection as both relate in direct terms to the unconscious aspect of understanding. However though the literary notion of the "imaginary" may be appropriate for artistic endeavour, the holistic mathematical interpretation is necessary for proper scientific enquiry.
Holistic Mathematical Notion of Four Dimensions
Q Can you elaborate now on your holistic mathematical treatment of the four dimensions?
PC Once more we can establish a direct correspondence as between the mathematical notion of roots and the (reduced) linear interpretation of the "higher" integral levels.
The structural integral understanding of H1 (circular or subtle level) is defined in holistic mathematical terms as the (reduced) linear interpretation of two dimensions (combining both positive and negative polarities).
The structural integral understanding of H2 (point or causal level) is now defined in holistic mathematical terms as the (reduced) linear interpretation of four dimensions.
In analytic terms if x4 = 1 (strictly 14) then x (i.e. x1 = + 1, - 1, + i or - i ( where i = the square root of - 1).
As we have seen there is a direct correspondence as between the holistic mathematical notion of dimensions and the structure of space and time.
Therefore, both space and time have now four dimensions.
Two of these are "real" with a positive and negative direction (+ and -); the other two are "imaginary" (again with a positive and negative direction).
This again is an incredible integral finding as it demomstrates that the inherent structure of space and time corresponds exactly with the (holistic) mathematical interpretation of four dimensions.
Thus in the most fundamental dynamic interactive sense, space and time has a precise mathematical nature.
Q Can you now explain both the psychological and physical significance of these four dimensions?
PC We start with a spatial phenomenon which now - in reduced linear terms - has four directions (two of which are "real" and two of which are "imaginary").
From a psychological perspective, the two "real" directions correspond to the conscious positing and negating respectively of the spatial object.
Through positing the object is given a dualistic location in space; through negation this specific location is eroded leading to a degree of nondual awareness.
As mentioned before the quality of such psychological negation can vary considerably.
In the absence of considerable spiritual development it is likely to be somewhat rigid with little nondual awareness generated. Interaction between opposite poles then tends to be static and mechanical - literally without imagination - serving to confirm the rigid nature of phenomena experienced.
However given a sufficient degree of authentic spiritual development, negation will take place in a very smooth and dynamic manner leading to considerable erosion of phenomenal rigidity. This in turn will be associated with substantial growth in nondual awareness.
Now because of this refinement of phenomena, interaction between opposite poles (exterior and interior) can take place in a very flexible manner. This in turn releases the creative power of the (unconscious) imagination continually transforming the nature of the phenomena experienced.
Directions of Space Objects (Positive and Negative)
From a physical perspective the "two" real directions relate to the actual objects experienced (that correspond to the psychological act of perception) again with both positive and negative directions.
In dynamic experiential terms the psychological and physical aspects are positive and negative with respect to each other.
Thus to switch from the (interior) psychological act of perception to the recognition of the (exterior) physical object observed, we must negate the perception (i.e. that has been posited).
Likewise - in reverse manner - to switch to the (interior) perception we must negate the observation (that has been posited).
Once again therefore, rigid negation - whereby recognition of the complementary pole is repressed - sets severe limits to the quality of interaction as between the self and its environment.
However we now have in addition two "imaginary" directions in space (one positive, one negative) with respect to experience of the spatial object.
The first "imaginary" direction corresponds to the (reduced) indirect conscious observation of what - in direct terms - pertains to the unconscious.
In other words nondual holistic meaning is indirectly projected into consciousness and posited, thereby attaching to the object phenomenon in space.
Because such projections possess an indirect conscious identity, they too must be negated in the process of attaining pure spiritual awareness.
So the negative "imaginary" direction pertains to the erosion of unconscious projections that have been indirectly posited in a reduced manner in consciousness.
Again the quality of such negation can vary considerably. When projections are involuntarily projected, considerable confusion pertains whereby the unconscious is identified with its conscious aspect.
In other words desire - which in root terms is holistic and spiritual - is confused with attachment to material objects.
In this case one has very limited power to achieve successful erosion of such projections.
However with appropriate spiritual development - pertaining to H2 (causal level) - a very refined and flexible form of negation develops whereby phenomena, both with respect to their conscious and unconscious aspects, become considerably eroded thus facilitating a very pure form of nondual awareness.
Then insofar as unconscious projections occur they do so in a very transparent manner as radiant archetypes of pure Spirit so that very little confusion with the local (conscious) perception of phenomena remains.
The physical counterpart then relates to the "imaginary" aspect of the exterior object manifesting itself in space (that corresponds to the interior unconscious projection) leading - in reverse manner - to a similar set of dynamics to those listed above.
Directions of Space Dimensions (Positive and Negative)
Now we can trace out similar dynamics for experience of the general dimensions in space (to which objects are related).
Again these will have four directions two of which are "real" (one positive, one negative) and two "imaginary" (again one positive and one negative).
From a psychological perspective, concepts correspond to dimensions (just as perceptions correspond to object phenomena).
These have two "real" directions in spatial terms. The positive direction relates to the conscious positing of a concept in space in a dualistic manner. The negative direction relates to the erosion of the concept (in nondual awareness).
From the physical perspective these "real" directions are reversed corresponding to the dimension (to which the concept relates).
For example the concept of "computer" which is of psychological origin corresponds to the general physical dimension (that embraces all specific objects within its class).
Again such a dimension has a positive direction, through which it is dualistically posited and a negative direction enabling its erosion in a nondual manner.
In the dynamics of experience to switch to the physical aspect of understanding, the psychological must be negated; likewise to switch to the psychological aspect, the physical must be negated.
Rigidity with regard to negation, where the role of the unconscious is not properly recognised, sets severe limits to the flexibility of interaction possible as between both physical and psychological aspects.
Such general spatial directions also have two "imaginary" aspects relating to a deeper rooted holistic projection of unconscious meaning that now attaches to general concepts (and their corresponding dimensions). Again these aspects are both positive (+) and negative (-) with respect to each other.
The experience of these is generally of a largely involuntary nature. In the absence of quite an advanced level of causal level development at H2, it is not possible to dynamically negate - and thereby successfully erode - remaining rigidity with respect to "imaginary" (i.e. unconsciously projected) concepts and dimensions.
Directions of Time
An exactly similar set of dynamics can be traced out representing the four time directions where every phenomenon has two "real" and two "imaginary directions (with both positive and negative aspects in each case).
Suffice it to say at this stage that the full interdependence of space and time cannot be yet properly achieved at H2. This requires even greater refinement of spiritual insight (which unfolds at H3).
However it may be helpful to state a little here regarding the important dynamics through which experience switches from time to space and - in reverse manner - space to time.
The physical manifestations of phenomena in space and time correspond closely to the interaction as between the affective and cognitive aspects of experience.
Therefore for example if one become aware from an affective perspective of a sense object in space, then cognitive mental intervention will lead to a switch to its experience in time. In like manner if the cognitive aspect is associated with time, then affective appreciation will cause a reverse switch to its experience in space.
We have to be careful to avoid exclusively identifying either space with the affective aspect, or time with the cognitive aspect.
Space and time have both cognitive and affective aspects. So which dimension is associated with each aspect can vary and depends on context.
However it is true that the switch in experience from space to time (or time to space) always requires a switch in modes i.e. from affective to cognitive (or alternatively cognitive to affective).
Therefore integration with respect to experience of space and time (in an exterior manner) requires corresponding integration with respect to the affective and cognitive modes (in a - relatively - interior manner).
Real and Imaginary Aspects of Science
Q As you have defined it there are four directions with respect to objects (in space and time) and four directions with respect to the dimensions (to which these objects are related). Does this not give eight directions in all?
PC No! and the answer to this leads to another crucial insight.
What is "real" with respect to objects is "imaginary" with respect to dimensions; also what is "real" with respect to "dimensions" is "imaginary" with respect to objects.
Therefore - in dynamic holistic mathematical terms - "real" quantities are "imaginary" qualities (or dimensions); likewise "real" qualities are "imaginary" dimensions.
Therefore whenever there is a switch in experience from quantitative to qualitative (or qualitative to quantitative) recognition, there is likewise a switch from the "real" to the "imaginary" (or "imaginary" to the "real") aspect of experience.
Put another way the interaction of quantitative and qualitative in experience, mirrors the corresponding interaction of conscious and unconscious, which are "real" and "imaginary" with respect to each other.
"Reality" is therefore truly "complex" in this precise holistic mathematical sense.
The implications of this are truly enormous.
The essence of the holistic binary approach is that all processes are governed by two interacting logical systems.
The linear (1) system is the dualistic logic of form; the circular (0) system is the nondual logic of emptiness (0) with an indirect dualistic structural expression in terms of paradoxical polar opposites.
However we can now see how these two systems can be phenomenally interpreted in terms of each other. For what is "real" in terms of one system (say the linear) is "imaginary" in terms of the other (i.e. circular).
Likewise if we shift the frame of reference to identify the circular system as "real", then the linear system is now - relatively - "imaginary".
Thus the mathematical notion of the "imaginary" - when given its correct dynamic holistic interpretation - has a vitally important role to play in all development processes.
Let us trace out the implications of this further. Using holistic mathematical language the conventional scientific approach - and by extension conventional modes of intellectual discourse - are heavily based on the linear logical system that deals solely with the "real" aspect of reality. This relates formally to conscious modes of understanding.
In this way, what we know as science is synonymous with "real" science.
However a vital complementary aspect of science remains greatly undeveloped relating to the unconscious (which indirectly can be given a coherent conscious interpretation). This is the "imaginary" component which is directly suited for integral type understanding.
Thus a comprehensive scientific approach to reality is necessarily "complex" combining both "real" (analytic) and "imaginary" (holistic) approaches.
I refer to such a "complex" scientific approach as radial science.
Q From what you say therefore Holistic Mathematics represents the "imaginary" aspect of "complex" mathematical understanding and is thereby directly suited to an integral interpretation of reality?
PC This is correct though such language in itself reflects the holistic mathematical approach. Clearly the "imaginary" aspect of a "complex" expression has a very distinctive meaning in analytical terms.
However ultimately my perspective is intended to be radial where the analytical and holistic aspects of understanding are both properly differentiated and integrated.
Indeed both the analytic (differentiated) and holistic (integral) aspects can only find their proper context in the radial approach.
Therefore - though in direct terms - the analytical ("real") approach is based on a rational (conscious) manner of understanding, indirectly it requires (unconscious) intuition to operate. Indeed for truly creative analytical work, considerable intuition may be required.
Likewise - in direct terms - the holistic ("imaginary") approach is based on the refined spiritual intuition of the unconscious. However, indirectly it requires a rational (conscious) mode of interpretation to become coherent. In the context of building a scientific integral approach that is truly consistent, a considerable rational element is required!
So once again - when properly appreciated - both the analytic and holistic aspects are interdependent and find their proper context within the radial approach.
Q Presumably the "imaginary" aspect can be extended to several other disciplines?
PC In principle this "imaginary" (i.e. holistic) approach can be applied to any discipline thereby providing a uniquely distinctive scientific integral approach.
In my own work I have concentrated chiefly on Holistic Philosophy (or Holophilosophy), Holistic Physics (or Holophysics), Holistic Psychology (or Holopsychology) and Holistic Economics (or Holoeconomics).
However Holistic Mathematics (or Holomath) remains the most important in the sense that it provides the fundamental structural patterns that are common to all these disciplines.
Complex Mathematical Nature of Holons
Q In terms of a holon can you now explain the dynamics by which "real" and "imaginary" interactions take place?
PC Firstly we have to understand a holon properly i.e. in its dynamic bi-directional context.
From a dynamic perspective, the whole and part aspect of all holons are interdependent. The whole has no meaning wihout the context provided by the parts; likewise the parts have no meaning without the context provided by the whole.
Therefore a holon can be defined asymmetrically from a "real" perspective in two ways (which are the reverse of each other).
We can refer to it as a whole/part where every (lower) whole is also part of a (higher) whole.
However we can also refer to it as a part/whole where every (lower) part is also a whole - in the context - of a (higher) part.
If we refer to the former definition (i.e. of a whole/part) as a holon then the latter definition will be in reverse and should - relatively - be referred to as an onhol.
Therefore in the dynamics of experience, holons are continually transformed into onhols; equally onhols are continually transformed into holons. And both of these aspects are identified in "real" conscious terms.
Thus though holons (and onhols) are always either whole/parts (or part/wholes) we differentiate them in reduced conscious terms through either their whole or part aspects.
Thus when a holon is identified with its whole, an onhol is identified - in reverse manner - with its part aspect.
Now strictly speaking the whole aspect of a phenomenon is identified with its concept and the part aspect with its perception respectively.
For example the concept of "computer" is a dimensional whole in that it embraces all part perceptions of actual specific computers within its class.
Again strictly speaking the concept here is of a potential nature, relating to the unconscious aspect of understanding, in that it has the capacity - when given a reduced conscious interpretation - to apply to any actual computer.
Now of course, when experience is interpreted in a solely conscious manner, the distinction as between potential (unconscious) and actual understanding is lost and everything is interpreted in a reduced conscious manner.
Indeed this confusion lies at the very root of conventional mathematical proof.
A "proven" hypothesis e.g. The Pythagorean Theorem is supposed to apply to all actual cases. However strictly speaking, the "proof" has only a potential validity with an infinite application (i.e. to "all" cases). However - by definition - an actual specific case is finite.
So there is a basic confusion of infinite with finite notions lying at the heart of all conventional mathematical proof.
This is not to say that such "proof" has no value but rather to suggest that mathematical truth is of a relative rather than an absolute nature.
So in the dynamics of experience if a perception is experienced in actual (conscious) terms then its concept is unconscious (and of a potential nature).
In other words the conscious "real" recognition of the perception is always associated with an "imaginary" recognition of the concept (which is of an unconscious nature).
Now when the unconscious is not properly developed, little recognition of the "imaginary" aspect takes place and the experience is collapsed to the conscious aspect.
One typical manifestation of this is the belief that object phenomena have an existence independent of the observer (and are simply registered through conscious observation).
However - by contrast - when unconscious recognition is strongly developed, though object phenomena can still be given a "real" conscious identity, they are equally associated with a holistic archetypal quality that radiates something of their ultimate nondual spiritual identity. Because such unconscious is of a qualitatively distinct nature from conscious recognition it always leads to a certain conflict in experience causing a switch to the opposite (vertical) pole.
Thus the "imaginary" aspect fuels the dynamics enabling the continual switching in the "real" conscious experience of wholes and parts (and - in reverse manner - parts and wholes).
From a psychological perspective the interaction between perceptions and concepts thereby is greatly accelerated; from the corresponding physical perspective the interaction between object phenomena and their corresponding dimensions is accelerated.
Relationship of Whole and Part
Q Can you clarify this for me. I think I understand the rationale behind this bi-directional approach to whole and part. Now I can see why the part is contained in the whole. However how can the whole be contained in the part?
PC Conventional notions of the relation between whole and part are very reduced where the dynamic interaction as between quantitative and qualitative is collapsed in merely quantitative (or alternatively qualitative) terms.
Thus the view that the atom for example is contained in the molecule (but not vice versa) represents quantitative reductionism where the complex relationship as between conceptual and perceptual recognition of such phenomena is ignored.
The specific perception of an atom (as "part") has no meaning in the absence of its corresponding general concept (as "whole").
(In this context perception and concept are quantitative and qualitative with respect to each other!)
In like manner the specific perception of a molecule (as "part") has no meaning in the absence of its corresponding general concept (as "whole").
So when we conclude that the atom is part of the molecule (in quantitative terms) we are thereby reducing - in both cases - conceptual to perceptual interpretation.
As we have seen the correct dynamic relationship - using holistic mathematical terminology - is not between atoms and molecules identified in "real"(i.e. quantitative) but rather as between atoms and molecules identified in "complex"(i.e. both quantitative and qualitative) terms.
In other words the "real" perceptions of both atom and molecule are associated with the - relatively - "imaginary" concepts of atom and molecule respectively. Likewise the "real" concepts (in both cases) are associated with the "imaginary" perceptions.
In this way we can give expression to the interaction of both conscious and unconscious aspects of experience (which keeps switching as between both the quantitative and qualitative aspects of these phenomena).
Now actual phenomenal identification of phenomena as holons (or onhols) must necessarily take place in reduced form. However with sufficient spiritual development, such activity becomes increasingly refined and transparent. When this is the case holons (whole/parts) continually switch smoothly to onhols (part/wholes) and - in reverse manner - onhols to holons.
Q Can you now briefly give a more refined interpretation of how the part is contained in the whole (and the whole contained in the part).
PC Though correct appreciation of the interaction of conscious and unconscious, considerable refinement takes place with respect to the experience of phenomena whereby they become increasingly transparent radiating a pure spiritual light.
In this way the "lower" holons (as whole/parts) are transcended in a "higher" spiritual appreciation of the universal collective wholeness of all phenomena. This in turn is facilitated through the dynamic interaction of both the "real" (conscious) and "imaginary" (unconscious) aspect of holons.
Likewise - in reverse manner - the "higher" spiritual appreciation of the whole is made immanent in each "lower" onhol (revealing its special unique quality).
However through experiencing in alternative reference frames, "lower" onhols as (part/wholes) are transcended in the spiritual appreciation of the collective wholeness of nature and - in reverse - this collective wholeness is made immanent in each specific holon.
Thus the transcendent and immanent aspect of spirituality apply equally both to the experience of holons and onhols.
In this way atoms can be spiritually contained in molecules and molecules contained in atoms (where such phenomena serve essentially as mediators of spiritual light).
Q Does this four-directional interpretation have any relevance - say - for the interpretation of conventional mathematical theorems?
PC Very much so, though its significance is lost at the level of linear (i.e. one directional) understanding!
So if we take a theorem e.g. the Pythagorean, it has both positive and negative directions in "real" (conscious) terms.
We can therefore interpret the theorem from a linear asymmetrical perspectiveIn the former case the proposition i.e. that the square on the hypotenuse of a right-angled triangle = the sum of squares on the other two sides, is considered to have an objective validity (unaffected by psychological interaction).
in exterior terms as pointing to an objective truth (independent of mind) in interior terms as pointing to a subjective construct that is true (independent of material circumstances).
In the latter case the same proposition is considered to have a subjective validity expressing the relation between well-defined mental constructs (that are unaffected through interaction with the physical world).
So all mathematical theorems can equally be given either an exterior or interior interpretation.
In dynamic relative terms (which is the nature of experience) these are complementary and opposite to each other i.e. relatively positive and negative with respect to each other.
However in absolute linear terms (as literally with an absolute number) this distinction does not apply.
This entails in practice the principle of double correspondence whereby exterior (objective) truths are interpreted as corresponding directly to interior (subjective) constructs and likewise that interior constructs correspond directly to exterior truths.
In other words with linear (one-directional) understanding both exterior and interior aspects of understanding are reduced in terms of each other.
Likewise all mathematical theorems can be given a perceptual (empirical) and conceptual (theoretical) interpretation.
As we have seen when we properly allow for both conscious and unconscious interactions in understanding these are "real" and "imaginary" with respect to each other (in relation to both exterior and interior aspects).
For example the "real" interpretation of a hypothesis applies to each actual case within its class.
However - more subtly - the "imaginary" hypothesis applies potentially or hypothetically to all cases (and therefore does not strictly apply in any actual case).
Now because conventional interpretation is of a reduced nature, finite and infinite notions are directly identified with each other (and this confusion underlines all mathematical proof).
In other words the "imaginary" aspect of understanding (in both exterior and interior terms) is reduced to the "real" aspect. Thus what is hypothetically true in infinite terms i.e. for "all" (potential) cases is thereby deemed to be also true when applied in finite terms i.e. to specific (actual) cases.
However once we allow correctly for the dynamic interaction of all directions, the nature of mathematics subtly changes with a merely relative truth-value.
Though the reduced interpretation will still take place in an absolute manner, the relative (i.e. integral) interpretation will be in dynamic holistic terms.
All mathematical symbols are now seen clearly to have both complementary (exterior) physical and (interior) psychological interpretations (with corresponding complementary applications).
Also all symbols have both "real" (i.e. local) and "imaginary" (i.e. universal) interpretations and applications.
Q You believe that the reductionist nature of standard mathematics is actually implicit in the interpretation of its own procedures. Can you explain?
PC One issue that puzzled me greatly when I studied (conventional) mathematics related to the asymmetrical nature of powers and roots.
For example let us take the number "1" that is defined in the default manner in linear one-dimensional terms i.e. with respect to the power of 1 i.e. strictly 11!
To "transform" this number to four dimensions we raise it to the power of 4 i.e. 14.
Now to "reduce" this number again in a linear one-dimensional manner we obtain the 4th root.
However we now obtain 4 answers + 1, - 1, + i and – i. Thus what, from the "lower" perspective has just one direction, from the "higher" perspective is now seen to have four.
Indeed this has very interesting implications as it means that the interpretation of each stage continually changes throughout development.
So (conventional) mathematical understanding relates largely to the (linear) conop and formop understanding of the middle rational level of development. This is what I refer to as H0.
However the interpretation of the nature of such conop and formop understanding is altered through the more holistic perspective provided by "higher" stages.
Thus we do not really have just one conop (and formop) stage. Rather we have distinct interpretations of such a stage from the perspective of every other stage in development, with the interpretations of the "higher" stages especially relevant.
Put another way, we can provide a more coherent philosophical explanation of the nature of conop and formop - that is compatible with its integration with other levels - from a "higher" stage (such as H2).
Likewise this applies to other stages. So for example, we can have causal level interpretation of causal understanding (H2) from the perspective of H2. Equally however we can have more integral interpretations of H2 from the perspective of H3 and the radial levels (R1, R2 and R3).
Now my own stated aim here is in providing a holistic mathematical treatment of development that is consistent with a - preliminary - Radial 1 level of experience.
Therefore it is capable of giving a better perspective (in holistic mathematical terms) of the understanding of the "lower" middle and "higher" levels of development.
However though it can indeed recognise the need for the more advanced forms of understanding of R2 and R3, it cannot - by its nature - yet provide adequate perspective on these levels.