PC Yes. The main point in my approach is that both reason and intuition are involved in all mathematical experience. A vital distinction is made in Holistic Mathematics as between number representing a horizontal quantity and number representing a vertical quality (i.e. number as dimension or power). Every number therefore has both a quantitative and qualitative interpretation. The horizontal (standard) approach which concentrates directly on numbers as quantities - is explicitly rational (and implicitly intuitive). The vertical approach - which is scarcely recognised - concentrates directly on numbers as qualities and is explicitly intuitive (and implicitly rational). Clearly a comprehensive dynamic approach involves both aspects.
Now the standard definition of a number quantity always contains a (hidden) dimensional characteristic, Thus if I refer to the number "4", implicit in this is the hidden dimension of 1. In other words the number could more correctly be written as 41. Now this is precisely what defines the rational (linear) definition of number (and indeed by extension the rational linear paradigm). All number quantities - in this approach - are (implicitly) expressed with respect to the power of 1. If initially a number is expressed with respect to another dimension or power, its value ultimately will be given in (reduced) one dimensional terms.
Thus 22 has a reduced one-dimensional value of 4 (i.e. strictly 41).
In the standard horizontal approach each number quantity (which can vary) is expressed with respect to the same (invariant) unitary dimension.
Thus in the (horizontal) natural number system 1, 2, 3, 4 etc represent
different number quantities all implicitly expressed with respect to the
1st dimension (i.e. raised to the power of 1).
However just as each number can refer directly to a number quantity (with dimension fixed at unity), in complementary fashion each number can refer directly to a number quality (i.e. dimension, with quantity fixed at unity).
Thus in the vertical (linear) definition of number we keep the number quantity fixed as 1, and then raise this invariant unit quantity to differing number qualities (i.e. powers or dimensions).
Thus in the (vertical) natural number system 1, 2, 3, 4 .... (i.e. 11,12,13,14 ....) represent different number qualities (or dimensions) to which 1 (as number quantity) is raised.
Thus to sum up, the number system can be defined in two complementary
In the standard (horizontal) approach differing numbers (representing quantities) are raised to the same invariant unit number (representing a quality or dimension).
In the complementary (vertical) approach the same invariant unit number (representing a quantity) is raised to differing numbers (representing qualities or dimensions).
Now when one views the vertical number system from a rational perspective,
it seems trivial and uninteresting. For when the number 1 is raised to
differing powers (dimensions), its reduced (one dimensional) value remains
unchanged as 1.
Thus from this rational quantitative perspective this number system has little value.
However there is an important problem here which is conveniently ignored in standard mathematics. When a number is raised to another power a qualitative transformation is always involved. Thus 11 is one linear unit (which geometrically can be represented by a one dimensional line). 12 strictly is 1 square unit (which geometrically can be represented by a two dimensional figure). 13 is one cube unit (which can be represented by a three dimensional figure).
Thus quite clearly when the power (dimension) of 1 changes, a distinct qualitative transformation is always involved. However in a reduced rational (merely quantitative interpretation) this qualitative aspect is overlooked.
The (horizontal) linear system is correctly suited to quantitative transformations (against an unchanging qualitative background).
The (vertical) linear system is however correctly suited to qualitative transformations (against an unchanging quantitative background).
It is this latter number system that is especially relevant to holistic mathematics.
Q. Can you explain how you convert
your linear number system (which is qualitative) into a circular number
system (which is quantitative).
PC. Yes, I believe this to be the key insight underlying holistic mathematics. Though it is possible - as I have demonstrated - to show that qualitative transformations are involved when using the vertical linear number system (which is intuitively based), it is difficult initially to see how further progress can be made.
However we have already seen that - relative to reason which is linear - intuition is circular. Thus expressed in (reduced) quantitative terms the vertical linear number system will be circular.
Thus to convert from the (vertical) linear qualitative number system to the corresponding circular quantitative system, we simply extract the corresponding roots of unity.
Thus the circular number corresponding to 11, will of course be the one root of unity and be unchanged as unity. This in itself is highly revealing for it means that there is no difference here as between the qualitative and quantitative definition of number. Likewise with the rational linear (one dimensional) paradigm no distinction is made as between qualitative and quantitative interpretation of phenomena.
The circular number corresponding to 12, will be given by the two roots of unity (i.e. +1 and -1) and will be represented by the two points where the (horizontal) line diameter intersects the circle (of unit radius).
What is fascinating is that all additional roots will lie as equidistant points on the circle of unit radius (defined in the complex plane).
For example if we take the vertical number 100 (i.e. 1100). Then the corresponding quantitative circular number is given by the 100 roots of unity which will lie as 100 equidistant points on this circle of unit radius.
Thus corresponding to our (vertical) linear qualitative number system, we have a fully coherent quantitative circular system.
Now as we have seen standard mathematics is largely defined in terms
quantitative linear system.
However holistic mathematics is by contrast largely defined in terms
of the quantitative circular system.
The (horizontal) linear quantitative number system can be translated - in complementary fashion - as a (vertical) circular qualitative system. This provided the appropriate means of transforming (standard) mathematical understanding in holistic intuitive terms.
Thus we can now precisely translate (qualitative) transpersonal understanding
in (reduced) rational mathematical fashion; equally we can precisely translate
(quantitative) mathematical understanding in (transformed) intuitive transpersonal
Q. Can you now briefly illustrate the importance
of this circular number system.
PC Basically this is quite simple. There are really two aspects to mathematics . The first is the quantitative analytical aspect geared to the partial differentiation of systems which is so highly developed in our culture. This has become so synonymous with mathematics that most people have great difficulty in even considering the possibility of other approaches.
However there is a second - equally important - complementary approach which is the qualitative synthetic aspect geared to the holistic integration of systems.
Though it has the capacity to revolutionise our appreciation of reality, surprisingly this latter aspect has been almost completely neglected throughout the history of mathematics. Indeed it leads simply and naturally to a comprehensive - yet beautifully simple "Theory of Everything". My purpose is to highlight the significance of this extremely important yet neglected aspect of mathematics.
Now as I have demonstrated in my postings on Holistic Mathematics, each of the major levels of the psychological spectrum can be defined precisely in qualitative terms through using appropriate roots of unity. (In complementary reverse vertical fashion each of the major levels of the physical spectrum can be defined in similar terms).
The linear level is defined in terms of the one root of unity and has one (unambiguous) positive direction only. Now to achieve this requires a significant amount of reductionism with the consequent freezing of dynamic interactions.
Thus from a psychological perspective subjective experience is reduced to objective; affective experience is reduced to cognitive and finally infinite reality is reduced to finite terms. From the physical perspective the mental (internal) aspect of reality is reduced to the material (external) aspect; the holistic aspect of reality - though qualitatively different - is reduced to the parts; finally the potential formless aspect of reality is reduced to actual phenomenal form.
It is important to clearly realise that (conventional) mathematics and scientific method is firmly rooted in this linear level. Existing attempts within physics to achieve a Theory of Everything (where fundamental symmetries in nature are realised) are based therefore on a paradigm which is unsuited to the task. Indeed the linear is the only level which by its very nature is fully asymmetrical. (I will comment on this again later).
The circular level is defined in terms of the two roots of unity and - expressed in reduced quantitative terms - has two directions (both positive and negative).
What this entails is that the first of the underlying symmetries of nature (i.e. horizontal symmetry) is now realised.
From a psychological perspective this entails that the dynamic complementarity of positive (objective) and negative (subjective) aspects of all experience is now fully recognised.
From the physical perspective this implies the recognition in physical nature of dynamic interaction involving external (material) and internal (mental) aspects. In other words consciousness is now seen to pervade all physical interactions.
This complementarity of opposites is recognised to some degree in most
spiritual traditions, in philosophy (e.g. Hegelian) and modern physics.
Indeed the fundamental basis of Taoism is expressed directly in this two directional fashion (i.e. the underlying Tao or universal symmetry splits in phenomenal nature into two complementary poles).
This dynamic complementarity pattern solves one of the fundamental problems i.e. the reconciliation - at any given level - of mind and matter. However it is not rich enough to properly deal with the remaining problems of reconciling wholes and parts and form and emptiness.
The point level is defined in terms of the four roots of unity - which in reduced quantitative terms - has four directions (two real and two imaginary).
What this entails is that the second underlying symmetry i.e. vertical symmetry is now realised.
From a psychological perspective this means that cognitive and affective experience - in relative terms - are seen mathematically as "real" and "imaginary" with respect to each other. This allows for experience to be translated in both quantitative and qualitative terms (without confusion).
From the corresponding physical perspective this means that throughout nature, wholes are no longer confused with parts but correctly defined as "real" and "imaginary" with respect to each other.
Thus both psychologically and physically, the world is now interpreted in "complex" rather than "real" terms.
Again this "higher" level symmetry though solving two of the fundamental problems is not yet able to deal satisfactorily with the third i.e. the relationship between finite (form) and (infinite) emptiness.
The radial level is defined in terms of the eight roots of unity - which
in reduced quantitative terms - has eight directions (two "real", two "imaginary"
and four "complex").
What this now entails is that additionally the third underlying symmetry i.e. diagonal symmetry is now realised.
Psychologically this means that actual (formal) and potential (formless) experience of reality are now fully reconciled. They are now seen as finite and transfinite with respect to each other. This implies the full integration of reason and intuition in experience. Physically this entails that (explicit) natural phenomena and the underlying implicit ground from which they emerge are reconciled (again being finite and transfinite with respect to each other).
These eight roots of unity when qualitatively interpreted provide the
three fundamental symmetries of reality and provide a comprehensive yet
beautifully simple Theory of Everything. Put another way all holons (which
in dynamic terms comprise reality) can be fully explained in terms of three
fundamental symmetries (horizontal, vertical and diagonal) derived from
the eight roots of unity.
Q. So you explain your Theory of Everything
in terms of three fundamental symmetries.
Why do you stop with eight roots? Why not 16 or 32? Also what about intermediate roots such as 3 or 5?
PC Yes, holistic mathematics can be accurately described as the mathematics of (dynamic) symmetry.
Standard mathematics by contrast is the mathematics of asymmetry. The
world of visible phenomenal form is a world of broken symmetry. This (standard)
mathematics therefore is built on a paradigm suited to the study of broken
symmetry and is extremely successful in this regard. (Who could fail to
be thrilled for example by the results of the recent Mars probe?) However
when (conventional) science tries to approach the underlying symmetries
of nature - which is more a qualitative than a quantitative task - it is
at a severe disadvantage. The very methods, concepts and mindset entailed
by the linear paradigm are fundamentally unsuited to the recognition of
the true dynamics of nature.
Holistic mathematics however is by definition the mathematics of symmetry. It is inherently dynamic and concentrates at the onset on the underlying complementarity of reality (not manifest in phenomenal form). It starts with the recognition of a hidden qualitative (dimensional) aspect to mathematics (which remarkably has remained secret throughout its history). As I have shown this leads to the definition of a vertical linear number system with a qualitative interpretation. This then has a reduced interpretation as a circular quantitative system (derived from appropriate roots of unity) which provides the mathematical key to understanding dynamic symmetry.
Now the reason I use one, two four and eight roots of unity to describe my system is because ultimately they can be described in binary termsin terms of powers of 2. 1 is 20. 2 is 21; 4 is 22; 8 is 23.
These roots have special symmetry patterns not replicated by other roots.
Also it is fascinating how these roots generate the differing number types.
The one root of unity gives us a positive "real" number. The two roots
gives a negative as well as positive
"real" number. The four roots give "imaginary" as well as "real" numbers. The eight roots in addition give (special) "complex" as well as "real" and "imaginary" numbers. These special "complex" numbers can also be given an interpretation as transfinite numbers.
So all the important number types are generated from these few roots.
Higher roots such as the 16 roots of unity do not possess the same important
symmetry properties and do not generate "new" number types. (The eight
additional roots here will also be "complex").
Intermediate roots such as the three roots of unity have a fascinating
in their own right but are not truly fundamental. I have in fact devoted a considerable amount of time to studying the holistic significance of the three units of unity which has yielded several important insights. They can be used to precisely explain the earlier transitional stages of the point level. Likewise roots 5, 6 and 7 can be used in connection with the transitional stages of the radial level. However as the explanation of these are complicated - and not truly fundamental - I have not concentrated on them here.
Indeed one of the most helpful ways of understanding my three symmetries is in terms of dynamic binary systems. All holons involve interactions which generate (quantitative) information. This can be represented by a horizontal binary system. When - relatively speaking - one holon is in an on state exchanging information, the other is in an off state receiving information; likewise when the second exchanges information it now switches to an on state and the first correspondingly to an off state.
All holons likewise involve dynamic interaction through (qualitative) transformation. This can be represented by a (dynamic) vertical binary system which interacts in similar manner.
Finally all holons involve the intersection of both (quantitative) information
and (qualitative) transformation states. This intersection - representing
essential states comprising both information and transformation - can be
represented by the diagonal binary system (which is really the sum of the
other two systems). We have four possibilities here, information + and
transformation + , information + and transformation -, information - and
transformation +, information - and transformation -.
Thus reality can be fundamentally be seen in terms of these three binary systems.
In static absolute terms all information can be represented in terms of a single binary system. However in dynamic relative terms - as well as "real" information - reality involves "imaginary" transformation and "transfinite essential (empty) states.
Thus to comprehensively interpret reality we need three binary systems. (However as in a certain sense, the third diagonal system can be explained in terms of a perfect balance of the other two, at its simplest we can explain reality in dynamic relative terms by a double binary system).
Q. Does your "Theory of Everything" involve
any mathematical equations?
PC This is an interesting question. Remember, mathematical symbols can be given the customary analytical linear interpretation (standard mathematics) or alternatively a synthetic circular interpretation (holistic mathematics).
Now equations in all forms are largely expressions of the cause-effect logic which characterises the rational linear level.
As we have seen, mathematics in a qualitative sense is based on a different (circular) system i.e. the complementarily of opposites (which ultimately is the coincidence of opposites) which is the very logic of dynamic symmetry. The very purpose of holistic mathematics is to show how reality in a qualitative sense can be precisely explained in terms of these complementary opposites. Once again three fundamental types of symmetries (horizontal, vertical and diagonal) are required to comprehensively explain all holons.
Because this dynamic task of integration - by definition - is a circular approach there is little use here for linear equations.
However because of necessity we have to use quantitative symbols to demonstrate the holistic approach linearity remains (though minimised to an extreme degree).
Thus remarkably "The Theory of Everything" can be explained in terms
of one simple equation i.e. xn = 1.
When n = 1, we get the qualitative structure of the asymmetrical linear level; when n = 2 and 4 we get the lower level symmetry of the circular and point levels respectively; when n = 8 we get the supersymmetry of the radial level which is The Theory of Everything.
Now what I find so immensely appealing about this is that we are literally apply to express "The Theory of Everything" as a reduced expression of unity. Nothing else could satisfy so well the mystical longing for beauty and simplicity. The fundamental structure of reality is so simple and obvious - in mathematical terms - that we have fail to recognise it. So when I first glimpsed the power and comprehensiveness of this approach - which is literally an expression of unity - I knew that it just had to be right!
My equation for The Theory of Everything is so basic in linear mathematical terms (as to be laughable). However the essential key is in interpretation - where the result is given a (dynamic) holistic - rather than an analytical interpretation.
The most difficult problem for standard mathematics to solve is the
"Theory of Everything" This is as I have said before because it is based
on the paradigm of asymmetry.
By contrast the easiest problem for holistic mathematics to solve is a true "Theory of Everything". As it is based on the paradigm of symmetry (complementarity of opposites), it leads naturally and easily to its goal.
Q. Can you stay with your Theory of Everything
and explain its significance.
PC The Theory of Everything can be looked on in several different ways.
We can use the eight roots to provide a description of the ultimate superfinite ground of transpersonal spiritual development. However using horizontal symmetry - which in relative terms involves negation (changing + signs to -) it then becomes a description of the ultimate superfinite ground of physical creation. (In this superfinite ground - the personal self and the impersonal world are identical).
However if we apply the Theory of Everything to the Omega point of spiritual transformation, then in relation to the Omega point of physical transformation we have a Theory of Nothing. Of course these are reversible. If we start with the physical pole as The Theory of Everything, then in relative fashion, the psychological pole represents The Theory of Nothing.
Thus depending on whether we start with psychological or physical poles of reality, The Theory of Everything is equally The Theory of Nothing. This simply results from pushing complementarity to its extremes while using rational language (based on the separation of opposites) for its interpretation. Thus in rational language if we affirm reality as totally quantitative (physical) we must negate what is totally qualitative (psychological).
Likewise if we now affirm reality as totally qualitative (psychological) we must thereby negate what is totally quantitative (physical). Thus in rational language the ultimate Theory of Everything (TOE) must be equally be a Theory of Nothing (TON).
We can equally use the eight roots to provide a description of the ultimate subfinite ground of creation.
Thus starting now with the physical pole, The Theory of Everything represents the Alpha of physical creation.
In horizontal complementary fashion The Theory of Nothing then represents the Alpha point of psychological creation. Again as before in relative terms these positions are reversible.
What this implies is that ultimately there is no horizontal hierarchy as between internal (psychological) and external (physical) levels of creation. In both the (formless) superfinite and subfinite grounds these aspects are identical
We can now apply vertical complementarity to connect the superfinite and subfinite grounds. If we start with the eight roots as a description of the superfinite ground (Omega) of reality then to switch to the subfinite we must change all real quantities (in the roots) to imaginary format (and imaginary to real). Again if the Omega ground represents The Theory of Everything then the Alpha ground represents The Theory of Nothing.
Of course these can be reversed so that the Alpha ground becomes The Theory of Everything and the Omega ground The Theory of Nothing.
What this really implies is that ultimately there is no vertical hierarchy as between the "highest" and "lowest" levels in creation. In the fundamental ground superfinite and subfinite are identical.
Now the polarity that is evident in terms of both horizontal and vertical
symmetry represents a rational approach (represented by lines of unit magnitude)
where ultimate reality can only be explained in terms of total paradox.
So The Theory of Everything is equally The Theory of Nothing.
What this really points to the is the mysterious Experience of Everything-Nothing (i.e. the plenum-void) where these opposite polarities are intuitively reconciled.
This leads us on to diagonal symmetry. From one perspective the four diagonal lines (in each of the quadrants of the circle have a numerical magnitude of zero). This in itself powerfully symbolises the purely intuitive interpretation of these four diagonal roots.
What they represent in fact is the intersection of both horizontal and
Thus here both horizontal internal (psychological) and external (physical) aspects of reality and vertical "higher" (Omega) and "lower" (Alpha) aspects of reality are simultaneously reconciled in terms of the same experience.
So a complete explanation of our qualitative solutions involves the Theory of Everything (in qualitative terms), which is equally The Theory of Nothing (in quantitative terms). Of course these positions are themselves reversible so that The Theory of Everything (in quantitative terms) is The Theory of Nothing (in qualitative terms).
This paradoxical rational interpretation of the ultimate ground of reality is then reconciled in The Experience of Everything-Nothing which is the intuitive realisation of the mystery inherent in the rational paradoxes.
Q. So your holistic mathematical solution
should be interpreted in both rational and intuitive terms?
PC Precisely. This is a strong conviction of mine. Holistic mathematics represents by its very nature a fusion of reason and intuition (of mathematics and transpersonal psychology).
Now the problem with standard mathematics is that as a merely rational pursuit it has become divorced from the intuitive insights of mystical spirituality.
Unfortunately mystical spirituality - especially in Eastern traditions
- is often too intuitive and becomes divorced from any coherent rational
Ultimate reality (non-dual reality) is typically presented as a merely intuitive realisation.
This is unbalanced. Ultimate reality is equally a (purely) rational as well as (purely) intuitive realisation. It is the very keenness of the awareness of paradox thrown up by reason that provides the intense need for the undivided intuitive realisation of the mystery (inherent in such paradox). To use an analogy frequently adopted by mystical writers (e.g. St. John of the Cross) we cannot conceive of the flames of a fire in the absence of the material (e.g. wood) which generate them.
Reason provides the wood in our analogy; intuition provides the energy and the flames. However to keep the fire burning we need both the wood and the flames. We need both reason and intuition.
So properly understood ultimate realisation (where the inherent symmetries of nature are experientially realised) represents an approximate limiting case which can never be fully realised (in this mortal life). So symmetry is inevitably tied up with asymmetry. Indeed it is the very distortions of the broken symmetry of manifest (formal) reality that continually provides the motivation to seek the hidden unbroken symmetry of unmanifest (formless) reality contained within.
So The Theory of Everything is only true as a special limiting case. What is equally important is the reconciliation at various levels of experience of symmetry with asymmetry. In scientific mathematical terms this requires the reconciliation of Holistic Mathematics with Standard Mathematics.
Q. What does your TOE actually describe?
PC I will try to briefly summarise.
Starting from the psychological perspective all phenomena have both a positive (external) and negative (internal) aspect. In other words matter - at all levels - is related to consciousness. Though scientists conveniently forget this point, the (internal) act of observing phenomena can never be divorced from the (external) phenomena observed.
When this inherent (horizontal) identity of matter and consciousness is fully realised as evolution we reach the "highest" superfinite ground of reality which is formless and empty.
From the physical perspective all phenomena equally have positive (external)
and negative (internal) aspects. Again even at the "lower" levels e.g.
sub-atomic particles, matter always entails consciousness.
When this inherent (horizontal) identity of matter and consciousness is fully attained as involution we reach the "lowest" subfinite ground of reality which equally is formless and empty.
This horizontal symmetry is provided by the two "real" roots +1 and -1.
All phenomena equally have a "real" (quantitative) and "imaginary" (qualitative) aspect.
In other words at all levels of reality, parts (quantitative) are related to wholes (qualitative).
Again we can in polarised form approach reality either in "real" (quantitative) terms - as in physics - to discover the ultimate nature of matter, or in "imaginary" (qualitative) terms - as in mystical spirituality - to discover the ultimate nature of mind or spirit.
However when this inherent (vertical) identity of matter and consciousness is fully attained, the "highest" and the "lowest" levels become fully identical. Thus the formless superfinite ground (Omega) is identical with the formless subfinite ground (Alpha).
Now the relation between both is "real" to "imaginary" and "imaginary" to "real". Thus for example if we designate the Alpha ground as "real", then in relative terms the Omega ground is "imaginary".
This is an extremely important point. What it entails is that to understand the "lower" levels of physical nature we require the "higher" levels of psychological understanding. "Higher" and "lower" in vertical terms are purely relative complementary terms. In mystical union they are identical. Thus the arrival at the qualitative goal of creative evolution is equally the arrival at the quantitative origin of evolution (which alternatively is the arrival at the quantitative goal of creative involution).
The implications of this is that in terms of the reconciliation of qualitative and quantitative (wholes and parts), science requires a "complex" rather than a "real" paradigm.
In other words a comprehensive paradigm must have two components i.e. a "real" analytical framework (to differentiate quantitative parts) and an "imaginary" holistic framework (to integrate qualitative wholes). The problems that bedevil the reconciliation of Quantum Mechanics (the parts) and Relativity Theory (the whole) is due to the lack of this comprehensive paradigm.
Thus vertical symmetry requires the addition of the two "imaginary"
roots + i and - i.
Finally all phenomena have a finite (actual) identity and a transfinite (potential) identity. In other words at all levels of reality form is related to emptiness (formlessness).
What this entails is that both horizontal and vertical symmetry are themselves related. In diagonal symmetry horizontal and vertical opposites are identical. We can no longer conceive of finite reality in absence from the infinite. This is easy to appreciate in terms of the "higher" levels of spiritual reality. (Indeed a persistent criticism of mine is the typical overemphasis in mystical spirituality on the infinite). However it is equally true of the "lower" levels of physical reality.
Here there is the opposite problem. Physicists persist in trying to see reality in merely finite terms. However reality is equally transfinite as well as finite.
This diagonal symmetry requires the addition of the four remaining "complex" roots (obtained from the eight roots of unity).
It is important not to conceive of the ultimate ground of reality in linear terms.
This ultimate ground has no finite beginning in time. Rather it exists as the ever present source of all phenomena (physical and psychological). These phenomena serve as secondary relative asymmetrical expressions of what is truly primary and fundamental.
Thus the attempt within physics to trace the "beginning" of reality
back to a linear point in time i.e. "The Big Bang" is very misleading.
Equally the attempt - often exemplified by institutional religion - to
portray eternal life as relating to some point in the future is equally
misleading. Both the "origin" of the universe and the "destination" of
the universe are ever present as the same moment continually renewed.
To conclude the results of my "equation" - from a physical perspective
- provide a qualitative solution to the fundamental problems:
Q. Can you say something about the quest
within physics to find a TOE.
PC My basic position is quite simple on this. Much as I admire the great intelligence and imagination of many leading physicists, from a holistic perspective their very methodology is flawed. (There are of course notable exceptions such as David Bohm who has adopted a coherent philosophical approach to physics). This is because invariably - in their reliance on standard mathematics - they are operating largely from the standpoint of the rational scientific paradigm.
Once more I will recite the fundamental problems with this approach.
This misleading tendency can be traced to the rational paradigm which is inherently built on the notion of an objective reality (independent of mind). Thus, though the findings of modern physics undermine the very basis on which the rational paradigm is built, it has not yet been replaced. Thus the main barrier within all the sciences is the continued dominance of a paradigm which in many ways has lost its relevance.
The implications for physics of facing up to this first problem are truly revolutionary.
It entails indeed that mind and matter - far from being independent of each other - are complementary. Thus physics must be fully integrated with psychology. And as the psychology that recognises this complementarity goes well beyond the linear level, it means that physics (and by extension the other sciences) must be integrated with transpersonal psychology.
To sum up this problem. Conventional science breaks the first symmetry
rule i.e. the fundamental (horizontal) identity of mind and matter.
In other words conventional science is geared to the partial analysis of nature (quantitative); it is not suited to the equally important task of holistic synthesis of nature (qualitative).
Thus again the continuing attempt within physics to find the ultimate constituent particles of nature is philosophically naive. Again, this reflects the attempt to deal with quantities in abstraction from a holistic (qualitative) perspective.
Indeed - as I have mentioned - it is this very tendency that lies at the root of the continuing difficulty in reconciling Quantum Mechanics with the Theory of Relativity.
Quantum Mechanics is a theory of the "parts" and owes much to close
observation of nature. Relativity Theory by contrast is a theory of the
"whole" and owes more to qualitative philosophical speculation (e.g. as
in Einsteinís approach) than empirical investigation.
Now because the "whole" and "parts" are qualitatively different - when related to each other - they cannot be reconciled in terms of any "real" theory (which is geared solely to quantitative reality). What is required is a "complex" theory (involving analytical "real" and holistic "imaginary" components). As my own "Theory of Everything" shows, this reconciliation of "real" and "imaginary" components leads to the infinite void (which transcends both).
Thus I say with full confidence. Any consistent theory that properly
reconciles Quantum Mechanics and Relativity Theory requires going beyond
merely finite categories in a purely qualitative explanation of reality.
(Alternatively in paradoxical fashion this will be a purely quantitative
explanation of reality). Once we deal with the world of phenomena
- regardless of how transient or minute - we are in the world of broken symmetry where quantitative and qualitative interact. When properly understood any theory at this level is by definition inconsistent. Thus, all present attempts to achieve a TOE which do not explain the fundamental transfinite ground of reality are doomed to inconsistency and failure.
Now by its very nature the standard mathematics of the rational paradigm is not suited to dealing with this fundamental ground. However by contrast holistic mathematics leads naturally to this ground and provides the true language for a TOE.
To sum up this problem. Conventional science breaks the second symmetry
rule i.e. the fundamental (vertical) symmetry of wholes and parts.
In some ways this is the most significant problem and most typifies contemporary science. Though scientists may well recognise implicitly the role of intuition, explicitly science is treated as a merely rational discipline. Now reason is properly geared to deal with finite reality (the world of broken symmetry). It is totally unsuited to deal with infinite reality (the world of unbroken symmetry) which pertains directly to intuition.
So understandably because of this bias towards finite reality, present attempts within physics at a TOE never get away from phenomena. In other words even here physics does not go beyond the world of broken symmetry.
For example Superstring Theory represents what is presently considered the most promising approach to a TOE. However it is still dealing with a world of objects (one dimensional strings) even though these are at levels of reality which cannot be observed. It is still dealing with dimensions (even though these are not the conventional dimensions).
It is still dealing with physical forces though admittedly in a novel manner. Also despite the mathematical value of this Theory there seems to be a great lack of intuition into its physical relevance. This raises the interesting point of the validitity of a TOE where we have little intuitive insight into what it actually means. (Of course Superstring Theory cannot claim in any case to be such a TOE!)
However - as I have outlined in my last posting - Holistic Mathematics with its qualitative bias - can provide compelling intuitive insight into the significance of Superstring Theory.
Thus to sum up this problem. Conventional science breaks the third symmetry
rule i.e. the fundamental (diagonal) symmetry of form and emptiness.
Holistic mathematics provides the appropriate basis for a true TOE. This is a mathematical description of the three symmetries (horizontal, vertical and diagonal) of the fundamental (empty) ground of reality which is totally paradoxical.
Standard mathematics - dealing with the asymmetrical behaviour of physical
phenomena - cannot provide a consistent TOE. All such attempts - though
they well provide interesting insights along the way - are ultimately doomed
Q. How do we deal mathematically with the
relationship of the symmetrical and asymmetrical aspects of reality.
PC We require here a comprehensive paradigm for mathematics which is "complex".
Such a paradigm combines the "real" (quantitative) analytical approach of standard mathematics with the "imaginary" (qualitative) synthetic approach of holistic mathematics.
Now in terms of our spectrum if we place the (rational) linear level in the middle, we have three "higher" levels above i.e. the (spiritual) circular, point and radial respectively.
These are complemented by three levels below i.e. the (physical) circular, point and radial levels.
So with the exception of the linear level, (vertical) complementarity applies to all other levels.
The "higher" (spiritual) radial is complementary with the "lower" (physical) radial level;
The "higher" (spiritual) point is complementary with the "lower" (physical) point level;
The "higher" (spiritual) circular is complementary with the "lower" (physical) circular level.
We now approach the understanding of all levels in terms of our "complex"
The use of the "real" analytical (quantitative) aspect of mathematics is maximised at the rational (linear) level. The only role for the "imaginary" holistic aspect here is in terms of providing a qualitative explanation of the philosophical paradigm within which such mathematics operates. The limitations of this paradigm for example would then suggest the validity of other types of mathematical enquiry.
The use of the "imaginary" synthetic (qualitative) aspect of mathematics is maximised as the idealised void of the radial level (where of course both "higher" spiritual and "lower" physical reality are fully complementarity). This purely qualitative approach provides the appropriate context for a truly holistic Theory of Everything.
At the other levels - radial (before completion), point and circular, a mixture of both "real" analytical and "imaginary" holistic mathematics is appropriate.
The rational "real" approach would provide the typical quantitative
mathematical modelling of reality, equations, hypotheses etc.
The intuitive "imaginary" approach would provide a holistic qualitative interpretation of the scope, implications and limitations of such modelling.
As I have already stated Superstring Theory provides a perfect illustration.
In terms of my Spectrum this relates to the "lower" physical point level. Therefore it requires a combination of analytical and holistic mathematics for its interpretation.
Now there is no shortage of the first. Indeed Superstring Theory is primarily a mathematical theory (in the standard sense). However huge difficulties have been experienced in terms of providing a satisfactory intuitive explanation of its contents. This is where Holistic mathematics comes in. Because Superstring Theory relates to the "lower" (physical) point level on the Spectrum, it requires the psychological understanding of the corresponding "higher" (spiritual) point level. Thus the insights of Holistic Mathematics 2 - which relate directly to this level - are especially relevant here in terms of a qualitative intuitive appreciation of what is implied by Superstring Theory. (I will develop this in more detail in a future posting).
Q. Can you know briefly summarise your findings?
PC The key starting point is to distinguish clearly as between the rational (quantitative) and intuitive (qualitative) interpretations of mathematical symbols. Thus I initially distinguish - in linear terms - both quantitative (horizontal) and qualitative (vertical) number systems.
The qualitative (vertical) system in turn has an indirect rational translation as a coherent circular quantitative number system. It is this number system which is of special importance in holistic mathematics. It provides a highly scientific and precise device for dynamically integrating reality at all levels.
The use of this system leads naturally to a simple yet comprehensive Theory of Everything. Here the fundamental ground of reality can be interpreted in terms of three dynamic symmetry patterns.
Standard mathematics - rooted in asymmetrical thinking - by its very nature is unsuited to the task of a TOE. Any such effort will inevitably be limited in its scope and prove inconsistent.
When we come to study the relationship of the (hidden) symmetrical and
manifest asymmetrical aspects of reality we require a "complex" rational
paradigm. In mathematical terms this requires a comprehensive mathematical
approach consisting of "real" analytical quantitative and "imaginary" synthetic
qualitative aspects. The former aspect is provided by Standard Mathematics;
the latter aspect is provided by Holistic Mathematics.
To end on a personal note, I find it truly incredible that despite the
enormous potential value of - what I term - Holistic Mathematics (or Holomatics),
that its relevance is so little appreciated. Hopefully this posting in
its own small way will help to remedy this glaring deficiency.