I have read once again your postings on "Holons" and "How Mind and Body Interact is Already Solved" and can see that they are both based on a profound intuitive grasp of the issues involved. I feel we are on the same wavelength here so let me offer my comments as to how holistic mathematics could be utilised.
I see all dynamic interactions in terms of control and response patterns i.e. agency and communion in Kenís terminology. (At the psychological level, reason and intuition represent control and response respectively).
These interactions can then be analysed at three levels:
Horizontal representing INFORMATION
Vertical representing TRANSFORMATION
Diagonal representing both INFORMATION AND TRANSFORMATION (i.e. ESSENCE OF BOTH).
Thus all holon interactions necessarily involve (horizontal) information, (vertical) transformation and (diagonal) essential processes respectively.
The horizontal level in mathematical terms is "real" relating to information states which can be measured rationally in quantitative terms.
The dynamic exchange of information is quite simple. The holon bears a complementary relationship to its environment (with both exercising control and response patterns).
For illustration purposes we will treat this environment as another holon so we have holons A and B. Information is then exchanged by dynamic switching of both holons as between control and response. When control is on in A (and response off), then response is on in B (and control off). In other words A is now active and exchanges information to B. Off course in alternate fashion when control is off in A (and response on) then response is off in B (and control on). B is now active and exchanges information to A.
Thus A and B dynamically switch as between on and off states mutually exchanging information. In other words A and B have both positive and negative aspects (exhibit both on and off states alternately).
In holistic mathematical terms therefore, the horizontal level of INFORMATION is "real" with positive and negative directions.
The vertical level in mathematical terms is "imaginary" relating to transformation states which can be measured intuitively in qualitative terms.
Whereas information switching between holons involves control and response states to be alternatively on and off, transformation switching involves these states to be also simultaneously on and off. This simultaneity of states represents a potential existence which is qualitative (rather than an actual existence which is quantitative). In mathematical terms when we can accurately refer to this potential qualitative existence as "imaginary". Transformation then takes place in a holon through the interaction of "real" (quantitative) and "imaginary" (qualitative) states.
Again this has both positive and negative directions. In other words in interactions A can control and be the agent of transformation in relation to B (positive). Equally B can control and be the agent of transformation in relation to A (negative). Of course, these directions are once again entirely relative.
So now we have the mutual interaction of - in mathematical terms - of "real" information states and "imaginary" transformation states in both positive and negative directions.
Finally we have diagonal switching patterns between holons which represents the intersection of information and transformation states and is in essence spiritual and ineffable. Ultimately it represents the ground and goal of evolution and is formless and empty (in phenomenal terms). This diagonal switching can be represented by null lines (as being purely infinite) or alternatively (in reduced terms) in complex mathematical format (where "real" and "imaginary" components are equal).
So finally now we have the mutual interaction of "real" information, "imaginary" transformation and null (or reduced "complex") essential states - representing intersection of information and transformation - in both positive and negative directions.
All holons - at whatever level - involve the mutual interaction of these three fundamental symmetries.
Translating this in slightly different language every holon has external (positive) and internal (negative) aspects. This enables switching as between (objective) physical and mental (subjective) states.
Likewise every holon has individual (real) and collective (imaginary) aspects. This enables switching as between (quantitative) part and (qualitative) whole states..
Finally every holon has phenomenal (finite) and empty (infinite or transfinite) aspects. This enables switching as between formal and empty states.
Conventional science as - presently
constituted - is fundamentally unsuited to the
understanding of these dynamic holon interactions.
In holistic mathematical terms, science represents a positive real finite approach to reality. Because it is merely positive it can only deal directly with the external (objective) aspect of experience.
It therefore simply reduces the (related) negative aspect to the positive. There is no room for subjective experience in conventional science.
This is perhaps the most characteristic feature of this science. It involves - necessarily - a fundamental mind-matter split. The world (externally) exists in objective terms. Mental representations (internally) - like in a photograph - accurately capture this reality. The objective existence of reality is therefore fundamentally unaffected through (subjective) interactions with it.
Now this comforting view is breaking down rapidly. Much as scientists dislike it, the role of the subjective observer can no longer be ignored. In quantum physics - for example - consciousness is forcing its way into their interpretations. However the basic mind-set is still one directional and linear so that findings seem increasingly strange and paradoxical.
So in correct dynamic terms every holon has both positive (external) and negative (internal) aspects.
In other words every holon - at whatever level - represents a mind-matter interaction. Thus when a scientist studies this reality a very subtle interaction takes place whereby there is mutual interaction of both physical and mental aspects. It is no longer possible to identify the object "out there" (the holon being investigated) and the subject "in here" (the investigating scientist). A dynamic complementary relationship exists so that the object is both "out there" and" in here" (and subject both "in here" and "out there") alternately. As one is no longer able to clearly separate the "outer" object" and "inner" subject", the relationship is now one of radical (horizontal) equality.
Thus in correct dynamic terms (objective) physics and (subjective) psychology bear a complementary relationship to each other. Every physical has a corresponding psychological aspect; likewise every psychological has a corresponding physical aspect.
Because science is merely real it can only deal directly with the actual (quantitative) aspect of experience. Again it simply reduces the potential (qualitative) aspect to the qualitative. It is - in other words - unable to correctly deal with the dynamic relationship as between parts and wholes (and equally wholes and parts) always collapsing - as it were - in any context, the qualitative aspect of reality to the quantitative.
This in turn reflects the dominance in science of cognitive reason (relating directly to the conscious approach). Now affective sense (relating directly to the unconscious) is the true source of potential qualitative understanding. However - even though the senses are necessarily involved in scientific investigations - the data obtained is translated in (reduced) rational terms. This reflects the difference as between the scientific and artistic perspective. For example the (conventional) scientific interpretation of a rose would be as an object belonging to the common class of rose. Roses can then be defined in quantitative terms. The artistic interpretation of a rose however, would be as a unique object. Each rose can then be defined in qualitative terms. Thus the very methodology of (conventional) science leads to the desensitisation of objects in experience and the loss of their inherent qualitative intuitive aspect.
So in correct dynamic terms, every holon has both real (individual) and imaginary (collective) aspects. Every holon represents a two-way interaction of parts (with wholes) and wholes (with parts).
So now when a scientist studies this interaction there is mutual interaction of individual and collective aspects. Again a complementary (vertical) relationship exists between scientist and holon involving both actual (real) and potential (imaginary) aspects. In practical terms this means that the scientist must incorporate both reason (real) and intuition (imaginary) in the approach. This corresponds in turn to the individual and collective aspects of the holon being investigated. In direct terms whereas the rational aspect interprets the (quantitative) information, the intuitive aspect interprets the corresponding qualitative transformation. Again this relationship works two-ways (both down-up and up-down). Thus when (quantitative) parts are transformed in any system to reveal a new (qualitative) whole identity (down-up), there is a corresponding reverse transformation. Now the whole (viewed now in it's related quantitative aspect) reveals a new qualitative identity (up-down) where each part is seen as unique. Thus the process which integrates and transforms (quantitative) parts into new (qualitative) wholes, is the same process which differentiates and transforms (quantitative) wholes into new (qualitative) parts.
As one is now no longer able to clearly separate individual (partial) and collective (whole) identity there is now a new radical (vertical) equality. This means that there is ultimately no difference as between "lower" physical and "higher" psychological reality. The "higher" psychological structure interprets the "lower" physical structure. Indeed more correctly they interpret each other.
This opens up amazing possibilities whereby one can discover the structure of all lower physical structures from a knowledge of the complementary "higher" psychological level. (Superstring Theory for example can be successfully explained in this intuitive manner). Alternatively one can determine the structure of the "higher" psychological levels from a correct knowledge of the corresponding "lower" physical levels.
The dilemma for contemporary physics (esp. particle) is that while "lower" levels are subject to increasing investigation, there is no satisfactory "higher" level paradigm to meaningfully interpret results. Because science clings to the rational linear paradigm - even when inappropriate - there is a deep reluctance to accept the complementarity of all physical and psychological relationships.
As holistic mathematics however is inherently dynamic, it provides a simple means of identifying this complementary structure, and therefore has immense value in terms of the total integration of physics and psychology.
Finally because science is merely finite, it cannot deal directly with the phenomenal (material) aspect of experience. It simply reduces the essential (spiritual) aspect to the phenomenal. It is therefore unable to deal correctly with the dynamic relationship of matter and spirit (and spirit and matter), again reducing in any context the spiritual to the material.
Whereas in psychological terms the "real" corresponds to cognitive reason and the "imaginary" to sense intuition, the "transfinite" corresponds directly to the volitional activity of will. This acts as a central co-ordinator of "real" reason and "imaginary" intuition. However as it is not recognised by the linear paradigm of science, inevitably its activity becomes translated in reduced material fashion.
However in correct dynamic terms, every holon has both finite (phenomenal) and transfinite (formless) aspects. As these aspects embrace both horizontal (real) and vertical (imaginary) directions of experience, each holon represents a four-way interaction of matter and spirit. From the spiritual perspective this interaction is represented by the diagonal null lines (which I have mentioned before) and has no finite interpretation. However from the (reduced) material perspective the interaction is represented by four complex directions i.e. real (+) and imaginary (+), real (-) and imaginary (+), real (+) and imaginary(-), and real (-) and imaginary (-).
In holistic mathematics these have a fascinating interpretation in both psychological and physical terms. The "real" aspect reflects spiritual illumination of the whole (in relation to parts) which is transcendence. The "imaginary" aspect reflects spiritual illumination of parts (in relation to whole) which is immanence. The "positive" aspect then reflects illumination of world (in relation to self). The "negative" aspect reflects illumination of the self (in relation to the world).
The physical interpretation (in holistic mathematics) provides a simple way of unifying the four forces of nature. The equality of "real" and "imaginary" in each of the four quadrants represents the pure intersection of "real" objects and "imaginary" dimensions. Thus a physical force - though neither an object or dimension - yet displays aspects of both. This is well demonstrated by the nature of light with its particle aspect taking an objective form and its wave aspect a dimensional form.
Thus at the diagonal level of interaction it is no longer possible to separate the formal and formless aspects of holons for both now mutually interpenetrate. There is a radical (diagonal) equality where even the smallest part fully expresses the whole (immanence), and the whole fully contains each part (transcendence).
This dynamic interpretation of holons has dramatic implications for the way in which we structure space and time. At the linear level, time is one directional (flowing solely in a forward positive direction). This enables objects to achieve a rigid independent existence (i.e. appear three dimensional).
However there is a fundamental lack of symmetry attaching to this approach.
Holistic mathematics corrects this deficiency.
At the horizontal level of the "real", time has two directions (positive and negative). Time moves in different directions for both world and self. (Space now also has two complementary directions). If time moves forward for the world (in relation to self), then - in relative terms - it moves backwards for the self (in relation to the world). This is why highly spiritual people realise that only the present moment truly exists. The temporal dimensions of phenomenal experience are seen as purely relative flowing out from an ever present centre.
At the vertical level of the "imaginary", time again has two additional directions (positive and negative). The "imaginary" directions of time relate to the potential qualitative nature of phenomena. Thus - correctly interpreted - quantitative changes are measured in "real" time and qualitative changes in "imaginary" time (both of which are relative)..
We now have four directions of time ("real and "imaginary" with positive and negative poles respectively).
When we switch "real" and "imaginary", we get the corresponding four directions of space. Thus space and time are created through dynamic interaction, continually switching as between alternate states.
If the "real" relates to objects and the "imaginary" to dimensions, then we come up with the fascinating insight that - in dynamic terms - "real" objects are "imaginary" dimensions. Alternatively "real" dimensions can be viewed as "imaginary" objects.
In dynamic terms corresponding to each object is a unique complementary dimension, so that reality is truly multidimensional. The reason we cannot see this is - as we have seen - because science deals with the common rather than the unique aspects of objects.
At the diagonal level of pure emptiness (indirectly represented in complex mathematical terms) we now have eight directions to time (and eight to space) The four complex directions however really represent the total harmony of real and imaginary.
In other words one can now appreciate the pure symmetry patterns underlying reality, and it is this refined appreciation which is the basis of ineffable spiritual experience, which now smoothly interacts with its finite relative manifestations.
Thus we can see how the application of the basic symmetries of holistic mathematics leads to a radically different world-view. In one way this new approach is very simple. The difficulty is that we have become so indoctrinated in what is a very limited and - ultimately distorted - linear approach that we find conversion extremely difficult to accept.
I have noted with interest Andi, that you work with computers. I thought therefore that you might appreciate a binary digital explanation of how a holon and indeed all reality operates in dynamic terms.
This indeed is a double binary system as it encodes both (horizontal) information and (vertical) transformation.
Whereas in the (conventional) binary system a state is either on or off in absolute terms, here, these states - due to dynamic interaction - have a relative meaning continually switching alternately as between on and off.
The horizontal real axis represents the (quantitative) binary information system with a positive (external) and negative (internal) direction. When the positive direction is on (1), the negative direction is off ( 0). When alternately, the negative is on (1), the positive is off (0). Both directions are therefore alternately 1 and 0 as they dynamically switch information between each other.
The vertical imaginary axis represents the (qualitative) binary transformation system again with positive (whole) and negative (part) directions. Again when the positive direction is on (1), the negative is off (0). When alternately the negative is (1), the positive is off (0). Both directions again are alternately 1 and 0 as they switch transformation states.
Now when we take either the horizontal and vertical scales separately, states will be alternately on and off. However when we take them together on the diagonal complex axes, they will be simultaneously on and off.
Thus we really have the combination here of two logical states, (quantitative) rational logic (separation of opposites) and qualitative intuitive logic (complementarity of opposites). The former represents the language of (horizontal) information and the latter the language of (vertical) transformation. This the empty essential state.
Thus all holons combine both binary digital systems in mutual interaction, ceaselessly informing and transforming each other.
Indeed at its very simplest, reality can be represented by this binary digital system. Evolution - at all levels - can be seen as the movement from a totally undifferentiated binary state (both information and transformation states confused) to a fully differentiated (and integrated) dynamic binary state (the plenum-void) which represents the pure marriage of information ("real" actuality) with transformation ("imaginary" potentiality).
I hope Andi, that this gives you - in relation to the questions posed - a good example of Holistic Mathematics in action. Whereas well developed language exists for expressing linear reason (West) and spiritual intuition (East), the successful integration of both has been greatly hindered through the lack of an appropriate method of translation.
Holistic Mathematics has been specifically designed to address this problem.
PS As regards your "special" question, I suspect Andi, that you are more informed on the matter than I am. I would therefore be interested in any comments you wish to offer.