APPENDIX - GENESIS OF HOLISTIC MATHEMATICS
Q When did your Holistic Mathematics first originate?
PC When at Secondary School in Ireland (High School) in the 60's, Mathematics (i.e. conventional Mathematics) was my strongest interest. Not surprisingly therefore I entered college with the intention of continuing my studies in the discipline. However very early on, I lost interest in the conventional syllabus while simultaneously becoming intensely interested in the deeper philosophical issues underling Mathematics. (Indeed I have already related one such instance in an earlier discussion on the reduced notion of the infinite used in Mathematics).
I would say the first real breakthrough came in early 1969 - what I would describe as a powerful mystical illumination conveyed through the intellect - when I obtained a clear vision of an integral approach to science (which equally would be supported by a new form of integral mathematical understanding).
However there is a considerable gap as between a vision that is deeply suggestive of a new approach and its actual development and it took many years of struggle before I could properly realise this earlier dream (to my own satisfaction).
Q You seem utterly confident in the validity of this vision. Is this the case?
PC Yes! Though we may be unable to adequately explain its nature, most of us can recognise when we have been touched by truth in a profound fashion. In such cases you do not seek confirmation from others that what you have experienced has indeed been meaningful. You just know!
So I simply know (through what I have experienced) that Holistic Mathematics is an extremely important revelation regarding the deeper integral nature of reality (which is universal). I see myself therefore as one who has stumbled on a great treasure, which is everyone's heritage, and ardently wish therefore that others can likewise discover it for themselves.
Q I know you have done so before but can you place Holistic Mathematics in context?
PC It is in fact easy to suggest the general basis for Holistic (i.e. Integral) Mathematics.
The Spectrum of Development is composed of many distinct levels (or stages). Now associated with each of these levels is likewise a distinctive type of intellectual understanding (that is unique for that stage).
Conventional science and mathematics are largely defined in terms of the intellectual understanding of the (rational) middle level, which is strongly analytic in nature (and suited to the differentiated interpretation of reality).
However associated with the "higher" levels is a more refined dynamic type of cognitive understanding that is suited to a proper integral appreciation of reality.
Therefore when we apply this type of understanding to the interpretation of mathematical symbols, a unique form of integral (rather than analytic) meaning is revealed.
In principle every mathematical symbol (relating to numbers, operations, equations etc.) can be given an alternative distinctive holistic interpretation (that complements its analytic counterpart).
So the entire nature of mathematics can be readily transformed through consistently interpreting its symbols in an appropriate dynamic holistic manner (reflecting the integral appreciation of the "higher" spiritual levels). And this is no vague wishy-washy exercise. Indeed mathematical symbols are used in a much more precise - and philosophically consistent manner - in integral terms (than at the analytic level).
Furthermore, because the physical and psychological aspects of reality are complementary at the "higher" levels (and ultimately identical), this means that in integral terms, every symbol has equal validity with respect to both aspects.
Indeed - in an integral sense - all the fundamental structures of reality are inherently mathematical (with complementary physical and psychological interpretations).
It is vital to appreciate that we are not talking here about the conventional use of mathematical symbols as a form of notation to convey more integral understanding.
No! we are dealing with an entirely distinctive type of mathematical appreciation where symbols are understood in terms of their original dynamic holistic meaning. Unfortunately such appreciation has been so greatly lost through the dominance of the static conventional approach that most people at this stage find it even difficult to conceive that there could be any alternative.
Q You lay special importance on the holistic interpretation of number in your approach. Why is this the case?
PC In the quantitative realm, number is the best means we have for coherently ordering data and indeed has become synonymous with this task.
One particularly powerful expression of this power of number to provide order is provided by the binary digital system, where all information can be potentially encoded through use of just two numbers (1 and 0). So our digital age, which is leading to dramatic changes in communications technology, ultimately is related to the power provided by just two numbers.
Now when we look at the binary digits in an analytic context they are characterised by clear asymmetrical interpretation.
So 1 is not equal to 0 and 0 is not equal to 1. The relation between both symbols is defined in terms of an either/or logic of separation (i.e. a digit can be either 1 or 0 but not both simultaneously).
However an entirely distinctive holistic meaning can be given to these same two digits based on an alternative both/and logic of complementarity (and ultimate identity).
So in our alternative system 1 = 0 and 0 = 1.
Just as the analytic interpretation of these two digits provides the basis for encoding all information processes, the holistic interpretation has the potential power to encode all transformation processes (in a scientific integral manner).
Q What do you actually mean when you say that in holistic terms 1 = 0 and 0 = 1?
PC It has always struck me as significant how the major Western religions (Judaism, Christianity and Islam) are based on the belief in one God.
Ultimate spiritual attainment is then characterised as unity (i.e. oneness). Indeed in the Christian tradition the most advanced stage of development is referred to as the Unitive life.
So here we have in fact the holistic expression of 1 (i.e. oneness) implicit in the ultimate attainment of form (which is without specific form).
By contrast Eastern mystical traditions e.g. Buddhism tend more characteristically to look at ultimate attainment as emptiness in the voiding of all - merely relative - phenomenal constructs.
We have here therefore the holistic expression of 0 (nothingness).
Of course ultimately it is recognised that these two expressions are identical. This is stated for example (in a part of) the famous Buddhist Sutra
"Form is not other than emptiness;
Emptiness is not other than Form"
Therefore in holistic mathematical terms (using the binary digits)
1 (i.e. oneness) = 0 (nothingness)
0 (i.e. nothingness) = 1 (oneness).
However though this is a first step we have not yet properly established equivalence between the holistic digits.
In analytic terms we establish equivalence through using both the positive and negative sign of 1, i.e. + 1 - 1 = 0. Because normally the + sign is assumed by default this is written 1 - 1 = 0.
Equally we can say from the other direction 0 = 1 - 1.
So the next step then is to establish the holistic mathematical interpretation of the operations of addition (+) and subtraction (-). Equally these could be expressed as positing (+) and negating (-).
It is here that I found one important expression of Taoism especially helpful.
This can be stated by saying that the primal indivisible unity of reality (which is without specific form) splits at a phenomenal level into opposite polarities (i.e. Yin and Yang) which are positive and negative with respect to each other.
Now I found here that there was an important link with the mathematical notion of the square root of 1 (more correctly 12) .
So if we take the square root of 1 (i.e. 12) we get two results i.e. + 1 and - 1. In other words the original primal unity (1) splits into two opposite poles (at a reduced level).
When we represent this geometrically we obtain a circle bisected by a (diagonal) straight-line that is drawn in opposite directions from the centre point. (Note the close association as between the binary digit "1" and the (vertical) geometrical representation of a straight-line and also the binary digit "0" and the corresponding geometrical representation of a circle!)
Furthermore we can give an extremely significant holistic interpretation of such geometrical symbols.
So therefore in holistic terms ultimate reality i.e. the (primal) unity is always conditioned in phenomenal terms by pairs of polar opposites.
Also the relation between these opposite poles involves both linear and circular understanding reflecting two distinct logical systems.
So we (dualistically) differentiate in experience through the (conscious) positing of phenomena. We integrate (in nondual terms) through the (unconscious) negating of these same phenomena. So when all phenomena (as form) have been successfully negated (in unconscious terms) emptiness results i.e. 1 - 1 = 0 (in holistic terms). In other words the mystical notion of oneness entails a totally dynamic interpretation of 1, where all unitary phenomenal form (that is posited) is simultaneously fully negated (as emptiness).
Likewise from the alternative perspective 0 = 1 - 1.
In other words (nondual) emptiness is the continual basis for the dynamic creation of a world of form (both with respect to positing and negating).
I owe a great debt to the philosopher Hegel whose writings thought me a great deal about the dynamic circular process of negation. I owe much more however to the Spanish mystic St. John of the Cross who provides in his writing an altogether more profound appreciation of this process of dynamic negation (i.e. dark nights) from an authentic mystical perspective.
So the mystical purgation that St. John describes in such an intensive manner has a direct mathematical expression (in holistic terms) as the pure operation of subtraction (i.e. dynamic negation).
And when all phenomena of form have been successfully negated, so that one no longer possessively identifies with them, nada (nothingness) results in the pure nondual awareness of ultimate reality.
Q So you made these discoveries early on. Briefly can you describe any major later insights?
PC Yes, most of the early discoveries I have mentioned - at least in crude expression - were made quickly in the early 70's and would be consistent with an Integral 1 approach with Type 1 complementarity (based on Type I differentiation and Type 1 integration).
Type 1 differentiation refers to linear asymmetrical understanding between directly opposite poles that is bi-directional. (I illustrated this with respect to the holarchical and partarchical approaches to development earlier in the discussion). Type 1 integration (again with respect to directly opposite poles) refers to circular paradoxical understanding that is also bi-directional where opposite reference frames are considered in relation to each other (as both forward and backward).
The Integral 1 approach is beautifully illustrated as circular duality (i.e. the holistic interpretation of the two roots of unity).
Thus we have a circle that is equally divided by its line diameter. Now the linear direction of the line can be taken from the centre in two opposite directions from each other. So in this sense the line is bi-directional. Also the circle can be taken in two directions (either clockwise or anti-clockwise).
So the Integral 1 approach represents the corresponding holistic interpretation of these geometrical notions.
However then there was a significant lull in terms of any further progress for more than 10 years.
The reason for this is interesting. The holistic mathematical understanding of Integral 1 represents an authentic refined cognitive expression of H1 (the subtle realm).
Likewise Integral 2 - which is directly based on the holistic interpretation of the four roots of unity - represents an authentic expression of H2 (the causal realm). So I found that I was unable to make the critical connections necessary until my spiritual understanding had become sufficiently refined for the task.
So Integral 2 leads to a fully comprehensive holistic interpretation of - what literally is geometrical terms - the four quadrants.
I had in fact been using a reduced version of the four quadrants much earlier in relation to my work as an economist. Here I was seeking a dynamic methodology that would incorporate both horizontal poles i.e. external and internal and vertical poles i.e. - what I referred to as - general and particular. In other words I was attempting to propose a way of integrating macro and micro economic activity (with respect to both objective and subjective interpretation).
However again I realised that I had not satisfactorily solved these issues from a holistic mathematical perspective. For if we take the four roots of unity, two are real i.e. + 1 and - 1. However the two others are imaginary + i and - i.
So I needed to explain the nature of imaginary numbers in their original dynamic context. And I was unable to do so for some time.
Q How did you eventually solve this problem?
PC The holistic interpretation of the imaginary numbers, intimately relates to the role of the unconscious in phenomenal experience.
Science is based on "real" i.e. merely conscious interpretation of phenomena, which it - quite literally - refers to as reality.
However the comprehensive number system (i.e. complex) is composed of both real and imaginary quantities.
Therefore because - by definition - all analytic mathematical symbols can be given a corresponding holistic interpretation, then this necessarily applies to the complex number system, giving it potentially an extremely important qualitative role. (More correctly it gives it an extremely important role in explaining the dynamic interaction of quantitative and qualitative aspects of reality).
Now when we reflect on the matter it is not possible - in dynamic terms - to have a merely conscious experience of phenomena, as the unconscious - which is ultimately expressive of the spiritual desire for meaning - is inevitably involved.
Thus our actual experience of phenomena necessarily embodies both conscious and unconscious meaning.
Thus symbols possess a dual identity where in conscious terms they are given a local (unambiguous) identity and in (pure) unconscious terms an archetypal (paradoxical) meaning.
Sometimes for example this unconscious element is directly reflected in conversation. A person may for example refer to a "dream house".
So though here the symbol "house" conveys a conscious localised meaning (that is identifiable) it equally conveys an unconscious meaning that is expressive of holistic desire.
Remarkably the unconscious aspect of this experience is "imaginary" in a precise holistic mathematical sense.
As we have seen the conscious direction of experience is positive (i.e. where objects are literally posited). However the unconscious direction of experience by contrast is negative (i.e. where we dynamically attempt to remove such phenomena from experience).
Now whereas the conscious is based on the either/or logical system (where opposite polarities are clearly separated), the unconscious - by contrast - is based on the both/and system (where both polarities are combined).
However, for unconscious meaning to enter experience in a phenomenally meaning, it must be indirectly converted to the one-dimensional linear system.
As we have see because the unconscious is two-dimensional (combining both polarities) this "conversion" to one-dimensional form therefore requires the holistic equivalent of taking a square root.
We are thereby enabled to explain the negative direction of form (inherently relating to a circular two-dimensional system) in linear one-dimensional terms.
So in other words insofar as the unconscious becomes embodied in phenomena, the experience is "imaginary" rather than "real".
However because both aspects are necessarily involved the experience of all phenomena is actually "complex" in a precise holistic mathematical sense (with both "real" and "imaginary" aspects).
Now it was mainly through reading Carl Jung - especially his manner of dealing with the personality types - that led me to this crucially important insight.
With Jung whenever a conscious function is too dominant, its complementary (inferior) function will be projected into experience in an unconscious manner.
In fact Jung implicitly had a holistic mathematical interpretation of his four functions two of which he termed rational (conscious) and two irrational (unconscious). Indeed this is often presented in terms of a four-quadrant diagram.
However if we now rather identify what is conscious with "real" and what is unconscious with "imaginary", then we can make the required holistic mathematical interpretation of the four quadrants. Therefore the two dominant functions (in Jungian terms) which achieve substantial conscious development are thereby "real". The two weaker functions, which indirectly express themselves through (unrecognised) unconscious projections are thereby "imaginary"(in a confused manner).
Q Why do you consider this discovery of the integral interpretation of the imaginary notion so significant?
PC Firstly it points to a hidden shadow dimension in conventional mathematical understanding. Though imaginary numbers are used, a great lack of appreciation exists with respect to their true philosophical nature. In fact they represent meaning that properly belongs to the circular (both/and) logical system indirectly expressed in a linear either/or form.
The important implication of this is that we can always indirectly express interpretation according to one logical system indirectly in terms of the other.
Therefore what is "real" (i.e. directly understood) in the circular logical system is (indirectly) "imaginary" in linear terms.
Likewise what is directly "real" in terms of the linear system is indirectly "imaginary" in circular terms.
Indeed one valid way of directly expressing the real result of the square root of -1 is as the number whose value is + 1 and - 1 (or - 1 and + 1) simultaneously. However this circular expression has no meaning in a system geared to linear (either/or) logic. So we can only express the result therefore - through a linear format - indirectly in imaginary terms (where its meaning remains concealed).
Secondly it is vitally important in scientific terms, as it points to a more comprehensive manner of interpreting - what is referred to as - "reality". So all phenomena are truly "complex" with both "real" and "imaginary" aspects in continual dynamic interaction. Again though in some areas of physics e.g. quantum mechanics, complex mathematical notions are used, again there is a great lack of intuitive insight into the philosophical meaning of such results.
So a philosophically coherent integral interpretation of nature (that incorporates both "real" and "imaginary" aspects) would greatly help to provide greater coherence for the emerging worldview of modern physics.
Thirdly, the holistic mathematical interpretation of the imaginary notion plays a crucial role in providing a coherent scientific integral view of the dynamic interaction of holons (whole/parts) and onhols (part/wholes). In other words the fundamental manner by which whole switch to part notions (and part to whole) intimately requires the intervention of the "imaginary" notion.
Fourthly the holistic mathematical notion of an imaginary notion brings a radical new perspective to bear on the phenomenal activity of the "higher" spiritual level of H2 (causal realm).
Though it is largely absent of rigid "real" phenomena (i.e. that are directly conscious) it is intensely active with respect to short-lived virtual (i.e. "imaginary" phenomena). It is through such instantly dissolving phenomena (as pure transparent archetypal images) that the Spirit is largely mediated at this level. Also because intense virtual phenomenal activity greatly accelerates the process of switching as between opposite poles, "real" phenomena eventually do not even appear to arise in experience.
Of course because of the vertical complementarity of levels, the imaginary concept is equally valuable in terms of providing a coherent philosophical interpretation of the virtual particle activity of the "lower" physical level L2. So through appreciating the complementary phenomenal structures of both domains (physical and spiritual) we are enabled to move towards a truly integral scientific appreciation.
Q Can you briefly outline the nature of the Integral 2 approach (associated with H2)?
PC The Integral 2 approach in some respects is similar to Integral 1 in that it combines (linear) bi-directional differentiation with (circular) bi-directional integration. However it is of a much more intricate nature.
We now have bi-directional activity with respect to both horizontal and vertical poles.
Type I differentiation and Type 1 integration again applies with respect to both sets of poles separately (with one understood in "real" and the other in "imaginary" terms). However Type 2 differentiation and Type 2 integration applies when we try to relate horizontal and vertical poles i.e. the dynamic manner in which holons switch to onhols (and onhols to holons).
I do not expect that the implications - of what is necessarily an extremely refined spiritually inspired cognitive process - will be properly appreciated at this stage. However it will perhaps clearly suggest that there are indeed several levels of integral understanding (with each in turn the explicit expression of the appropriate dynamic interactivity of the corresponding stage to which it is related).
Also it may be apparent how we are enabled to precisely explain the nature of this integration through direct application of holistic notions (here using the precise mathematical interpretation of the four-quadrants which directly results from the four roots of unity).
Q Very briefly can you comment on the Integral 3 approach?
Well this carried the holistic mathematical explanation to the eight roots of unity. So now in geometrical terms as well as horizontal and vertical, we also have twin diagonal polarities.
Remarkably the diagonal lines can be given dual interpretations in terms of the most fundamental relationship as between form and emptiness.
So basically the Integral 3 approach enables us to provide a fully scientific means of explaining not only dynamic relationships at a conscious and unconscious level but also the interactions between these levels and also with Spirit.
So a comprehensive integral approach arises at H3 (nondual reality)
However this is not the end. We have provided a scientific inetrpretation of an integration that is expressive of pure contemplative awareness. However we now need to show how one-directional linear activity (which characterises dualistic differentiated understanding) can be fully incorporated in a refined manner with nondual integral awareness.
So in my scientific approach, the middle stages (of H0) represent a specialisation in (mere) differentiated understanding (characterised by one-directional linear understanding).
The higher stages (H1, H2 and H3) represent a specialisation in (mere) integral understanding of pure contemplative awareness (characterised by both bi-directional linear and circular understanding).
The radial stages (R1, R2 and R3) then represent a corresponding specialisation of the mature interaction of both differentiated and integral understanding (characterised by both one-directional and bi-directional linear and circular interpretation)
I would characterise my present work as a preliminary attempt to outline from a consistent scientific intellectual perspective the nature of the worldview associated with Radial 1. I am not in a position therefore to develop these levels (especially Radial 2 and Radial 3) in any great detail.
However it will still be possible to make valid general remarks with respect both to their structure and nature.
Q In conclusion why do you think that people should be impressed with your claims?
PC As I have stated before - whether others do or do not recognise - personally I have no doubt whatever, that what is being uncovered here is something of truly great original value (which has immense potential implications).
Firstly I believe that I am offering a radical new vision of mathematics (in principle applying to every symbol and relationship) with a directly integral significance.
Secondly, with the powerful help provided through the more important basic mathematical symbols it has already been possible to precisely encode all stages of development - to a considerable level of detail - in a precise holistic mathematical fashion. Indeed in this approach such encoding uniquely defines the nature of each stage.
Thus the structure of reality is, through and through, mathematical in this holistic sense.
Put another way the Operating System governing reality is inherently mathematical and deeply implicit in all relationships. Though always contained therefore within phenomena, Mathematics cannot be directly identified with them, being of a more fundamental nature. Indeed in one very important sense, mathematical relationships - in their original dynamic context - provide the important intermediary means through which both dual and nondual realms can interact.
Thus though we can have many distinctive phenomenal interpretations of reality, the most fundamental is necessarily of a mathematical nature, which implicitly - as it were - provides the Operating System which enables the interpretations - encoded through other software - find meaning.
So correctly understood i.e. in its original integral sense, Mathematics is more basic than any other form of inquiry lying closest in the phenomenal realm to pure Spirit.
Though I would not however wish for a holistic mathematical approach to remove the need for other valid approaches, clearly it can play a very special role in the development of exciting new interpretations of integral science (e.g. physics, psychology, economics, philosophy etc).
Thirdly I would suggest that - given the importance of digital information age at present - that special consideration should be given to this alternative holistic interpretation of the digital system with the potential power to encode all transformation processes.
And this is far from a merely speculative notion - which however would be very interesting in its own right - for I am already employing this binary system (to a considerable level of detail) in my interpretation of development.
Finally, a dynamic model which literally is a reduced root expression of unity and can yet lead through just three layers of dualistic splitting to all structures of development strikes me as remarkable yet wonderfully reassuring, in that the nature of reality is indeed truly revealed as being utterly simple, yet holding in that beautiful simplicity all the complexity and mystery that the human heart can embrace. So one in the contemplation of this holistic mathematical vision of reality comes as close as is possible in the phenomenal realm to pure Spirit.
Yes, I do fully believe that this indeed provides - quite literally - the most original and thereby most powerful manner for the interpretation of phenomenal reality.
When others can likewise "see" for themselves, they will believe so too.