INTRODUCTION TO HOLISTIC MATHEMATICS
Spectrum of Translation Methods
I would like to start by introducing a simple idea that has enormous potential consequences.
People at this Conference will be already familiar with the notion of a Spectrum of Consciousness with many different stages, representing distinct levels of consciousness.
In my model I have 7 distinct major levels, divided into "lower" (3) middle (1) and "higher" (3).
What is not properly appreciated is, that associated with each level, is a unique method of intellectual translation (leading in turn to a unique interpretation of reality).
Mature translations of reality relate to understanding of both the middle and "higher" levels. The most comprehensive translations (which I refer to as Radial) relate to the interpenetration of all levels.
When we grasp this point we recognize that the interpretation of science is indeed multilevel and that the conventional scientific paradigm simply corresponds to the intellectual interpretation of the middle level of the Spectrum.
Thus there is vast and yet largely unexplored scope for science to be equally translated through the refined cognitive understanding of the "higher" levels (H1, H2 and H3).
Intriguingly this equally applies to Mathematics. Mathematics - as we know it - relates to the interpretation of the middle level (H0 in my approach).
Holistic Mathematics - which represents an integral interpretation of mathematical symbols - is based on the understanding of the three "higher" levels. It is subtle and dynamically interactive in nature providing the appropriate framework for a truly scientific integral appreciation of reality.
Nature of Holistic Mathematics
Every mathematical symbol with a quantitative interpretation in (static) analytic terms, has a corresponding qualitative interpretation that is dynamic and holistic. Thus if we take any number, operation, function, relationship or theory it has dual interpretations in mathematical terms.
The standard analytic interpretation is based on the rational understanding of the middle level (where polar opposites are separated).
The holistic interpretation - by contrast - is based on the subtle rational understanding of the "higher" levels (where polar opposites are complementary).
Mathematical symbols - when appreciated from the dynamic holistic perspective - have a truly remarkable scientific integrative capacity. I will illustrate this briefly in the following sections with respect to some of the key symbols.
Because integral understanding is based on the complementarily of opposites in experience, this means that holistic mathematical interpretations apply equally to both the physical and psychological aspects of reality (which have underlying common structures). Thus for example we can use Holistic Mathematics as a precise means of ordering the psychological levels of the Spectrum; equally we can use Holistic Mathematics to map out the various levels of scientific interpretation of reality.
(I noticed that the five Universal symbols used in the Conference logo are geometrical. Though they have a quantitative analytic interpretation, their deeper significance - as is directly implied - is qualitative and holistic)
In this section I will briefly indicate the relevance of a couple of key symbols i.e. the line and the circle.
The geometrical notions of the line and circle will be familiar to all.
However when we reflect on it, it should be apparent that we also implicitly use them in a holistic sense. Thus we talk of linear understanding. Indeed the conventional stage models of development are based very much on linear notions e.g. the sequential ordering of stages.
We can also talk about circular notions (e.g. the circle of understanding).
From a linear perspective such notions are paradoxical and typically are used to translate the nature of nondual mystical awareness.
Linear understanding is thereby directly associated with dualistic interpretations of reality whereas circular notions are associated with nondual interpretations.
This in turn leads to an extremely important connection with the fundamental notions of differentiation and integration.
Every stage of development - in dynamic terms - entails a degree of differentiation and integration (appropriate to the stage).
From a translation point of view, the differentiated aspect relates (directly) to linear notions of understanding, whereas the integral aspect relates to corresponding circular notions.
Thus the use of (merely) linear analytic methods of translation, results in fundamental mistranslation of development (in dynamic terms).
Once again because differentiation and integration are involved at all stages, proper translation involves the use of both linear and circular logic (with a unique configuration of both applying to each stage).
I will make explicit a remarkable implication of this finding later in the article.
The notion of the line and the circle (and its corresponding holistic equivalents) has already been mentioned.
At the centre of a circle we have a point (which is also the centre of any line diameter through the point). So the point is central to both the line and the circle.
We can also give the point a holistic interpretation as indeed is done in many mystical traditions.
Thus the spiritual centre of personality (the Self) is common to both differentiation and integration of experience (where neither has any distinct meaning).
In geometrical terms, if we gradually compress the size of the circle, both the line and circumference will ultimately coincide at a central point (singularity). (I have deliberately used the term singularity to suggest that such notions equally have physical applications). Likewise at the causal level of understanding, the erosion of all phenomenal notions of understanding of either a linear or circular kind, lead to the arrival at ones pure spiritual centre of personality..
However even the point represents a limitation as it tends to concentrate awareness within (at a concentrated point).
We can use the mathematical notion of null lines,which can be understood simply in terms of the point, line and circular circumference, to give a representation of nondual reality which once again has fundamental physical applications.
The most important numbers in Mathematics are 1 and 0 i.e. the binary digits.
A fully comprehensive number system (the binary system) is based on these two digits. Indeed this system is the basis of our digital revolution. This present Conference would not be possible without it.
All information can (potentially) be successfully encoded with the use of these two digits. The application of this simple insight is rapidly changing the nature of the World we live in.
The actual symbols we use to represent one and zero are in fact the line and the circle (with minor modifications).
Deeply inherent in our very choice of symbols therefore is their prior holistic nature. Appreciation of this simple fact leads to an incredible finding with great possibilities for translating transformation processes.
By our basic premise, every mathematical symbol - as quantitatively understood - has a corresponding dynamic holistic interpretation. Thus the two binary digits 1 (one) and 0 (zero) have corresponding dynamic holistic interpretations as 1 (unity) and 0 (nothingness). Alternatively we could use the terms form and emptiness!
The very recognition of form (1) inherently is based on distinction (i.e. a unique unit).
The recognition of emptiness (0) is based on a dynamic notion of nothingness.
Thus pure emptiness is nothingness (in phenomenal terms), that represents the potential for all created form.
Thus right at the heart of nondual mystical experience we find the pure holistic interpretation of the binary digits.
"Form (1) is not other than Void (0)
Void (0) is not other than Form (1)"
So whereas as the analytic (quantitative) interpretation of the binary digits is based on (linear) separation, the corresponding qualitative interpretation is based on (circular) complementarity.
The deeper implications of this are truly startling. Just as the binary digits (as quantitatively understood) can be used to (potentially) encode all information processes, equally their corresponding qualitative interpretation can be used to (potentially) encode all transformation processes, both in their physical and psychological aspects.
Human development represents a psychological transformation process. Therefore all the levels and stages of the Spectrum can be precisely encoded using the dynamic holistic interpretation of the binary digits.
In other words we need to use a combination of the logic of form (1) and the logic of emptiness (0) to properly translate all levels (with a unique configuration existing for each level).
Another simple finding of Holistic Mathematics as we shall see determines how the configuration for each level is obtained.
So the binary system is really a double binary system.
One aspect with which we are familiar gives a quantitative analytic interpretation of the digits (based on the separation of polar opposites).
This system is ideally suited for the (potential) encoding of all information processes
The other aspect gives a corresponding holistic interpretation of the digits (based on the complementarily of these same opposites).
This is potentially suited for the encoding of all transformation processes.
The present digital information revolution is indeed very exciting but without corresponding recognition of the corresponding need for (digital) transformation is most unbalanced.
I have referred already to the two logical systems of form and emptiness. Once again these are based on the principle of the separation and complementarity of polar opposites etc.
The fundamental mathematical operations of addition and subtraction - when interpreted in a dynamic holistic manner - provide a precise way of translating the two systems.
In order to become conscious of a phenomenon in experience, we literally posit it in a space-time framework.
Before making it unconscious once again, we must dynamically negate the phenomenon.
As actual experience involves the continual interaction of conscious and unconscious, it therefore entails the corresponding dynamic positing and negating of phenomena.
So positing represents the dynamic holistic interpretation of addition (+) whereas negation represents the corresponding dynamic interpretation of subtraction (-).
When we separate polar opposites only the positive direction of experience is formally recognised.
(Typically the unrecognised negative polarity is then repressed in the unconscious).
The logic of form based on such separation is properly suited to the (conscious) differentiation of phenomena. Such logic is asymmetric and one-directional (linear) in nature with connections between variables made in an unambiguous manner.
However integration in experience requires equal recognition of the corresponding negative pole leading to a harmonious fusion of opposites. In its refined form this leads to pure intuitive awareness.
So the logic of emptiness is based on the complementarity of both positive and negative poles and is properly suited to the (unconscious) integration of phenomena. Such logic is symmetric and dynamically bi-directional (circular) and attempts to make connections as between opposite poles in a simultaneous paradoxical manner.
As actual experience involves the interaction of conscious and unconscious (differentiation and integration), substantial dynamic mistranslation takes place when we try to interpret development (solely) through the logic of form.
Quite simply, accurate translation requires both logical systems at all stages of development. When we try to interpret using just one system (typically the logic of form), it leads to considerable imbalance and inconsistency from an overall perspective.
There is in fact a very close relationship as between the two fundamental numbers and the operations (of positing and negating).
In analytic terms 0 = + 1 1. Likewise in dynamic holistic terms 0 (as emptiness) =
+ 1 1 (as form).
In other words the state of emptiness (0) results from the complete dynamic negation of (phenomenal) form (+ 1 1).
In analytic terms the notion of number is synonymous with our (quantitative) notion of ordering.
Therefore in (dynamic) holistic terms, the notion of number should be equally synonymous with our (qualitative) notions of order.
This means in effect that the most precise way for ordering the various levels and stages of the Spectrum is mathematical (in a qualitative holistic sense).
It may help to appreciate this point by reflecting on the nature of the rational paradigm. In fact as a paradigm this represents the holistic qualitative notion of the rational.
Our scientific notions of good order are synonymous with this number type.
Rational numbers for example can be subdivided into fractions. Rational concepts and perceptions can be sub-divided in similar manner. Indeed this is the very hallmark of the analytic approach.
Rational numbers are always finite and absolute in value.
Likewise the rational paradigm relates to the finite absolute understanding of relationships.
The nature of rational numbers also points to a severe weakness in the rational paradigm.
Any rational whole number can be divided into component parts so that the (original) whole is equal to the sum of its parts.
In qualitative terms, this means that the rational paradigm leads to reductionism as it tends by its very nature to see the whole as simply the sum of its constituent parts. Thus for example our scientific view of the Universe is based on the attempt to see it as the sum total of all that it contains. From a dynamic perspective this understanding is very flawed.
However there are many other number types besides the rational numbers.
Starting with the binary digits 1 and 0, we have the prime and natural numbers and then the integers and rational numbers. We also have negative numbers, irrational (algebraic) and transcendental numbers, imaginary and complex numbers and even transfinite numbers.
Now rational understanding relates to just one level of the Spectrum (i.e. the middle).
The implication of what I am saying is that the understanding of al the other bands corresponds to the dynamic holistic interpretation of the other number types.
In this sense the Spectrum is in fact a number Spectrum.
Any major new number type discovered in Mathematics would entail the discovery of a corresponding new qualitative band on the Spectrum.
Equally the discovery of a new band on the Spectrum would entail the existence of a yet (undiscovered) number type in Mathematics.
Just as the science of the middle level is based on the rational paradigm, the appropriate science of all the other levels is bases on the corresponding holistic interpretations of the other number types.
This entails the existence of several meta-paradigms for science rather than just one.
Most of these other meta-paradigms are directly suited to an integral rather than a differentiated understanding of reality.
As every discipline (e.g. Physics, Psychology, Economics Philosophy and of course Mathematics itself) can be translated according to the understanding of these meta-paradigms, there is vast - as yet unrecognized - scope for extension of our scientific understanding of reality.
These Number Paradigms (properly meta-paradigms) in fact provide the appropriate structural basis for all the Methods of Translation mentioned in the first paragraph.
Theory of Everything
Mandalas have been long used as symbols of integration. The most common mandalas include the very notions we have mentioned.
A very important mandala is represented by the simple cross diagram where the cross represented by horizontal and vertical lines, divides a circle into four quadrants.
A more complete pattern - besides horizontal and vertical - includes two diagonal lines through the center of the circle.
This has a fascinating and very important mathematical rationale which relates to a circular interpretation of number.
Conventionally we define numbers in linear terms. Numbers are literally represented as points on a (straight) line. However a fascinating alternative circular interpretation of number results from taking the various roots of a number.
If we take the square root of 1, we get two answers +1 and 1. These lie on the circumference of the circle of unit radius as the two extreme points on the horizontal axis.
If we take the four roots we will get four (equidistant) points on the circumference which lie as the extreme points on the horizontal and vertical axes respectively.
If we now take the eight roots we will get eight (equidistant) points on the circumference which lie as the extreme points on the horizontal, vertical and diagonal axes respectively.
The qualitative mathematical interpretation of theses roots is extremely important, so much so in fact that it truly comprises a holistic "Theory of Everything" (which in mathematical terms is literally a reduced expression of unity!)
Quite simply - in dynamic terms - all relationships can be understood as the interaction of opposite poles that are interpreted in terms of horizontal, vertical and diagonal polarities according to the two logical systems of form and emptiness..
Horizontal polarities relate to exterior/interior, vertical to whole/part and finally diagonal to fundamental finite/infinite aspects of reality respectively.
These notions directly apply to interpretation of all the levels and stages of the Spectrum enabling a fully dynamic treatment that balances linear notions of independence and separation with circular notions of interdependence and complementarity.
I will briefly comment on the latter aspect, which is greatly neglected within models of development.
Within each level we have (in structural terms) horizontal symmetry. This relates to the two-way complementarity of physical and psychological aspects that are - relatively - exterior and interior with respect to each other.
Thus we can use knowledge of physical structure to infer the (complementary) psychological structure; equally we can use the knowledge of the psychological structure to infer the physical.
Between levels we have vertical symmetry. This relates to the two-way complementarity of "higher" and "lower" physical levels, or equally "higher" and "lower" psychological levels. (H3 and L3, H2 and L2 and H1 and L1 are complementary with each other respectively).
If for example we know the psychological structure of a "higher" level - say H2 - then we can infer the (confused) structure of L2. In other words in dynamic terms, pre and trans (and trans and pre) are very much complementary.
Within and between levels we have diagonal symmetry. This entails for example that we can use the structure of a "higher" psychological level to infer the corresponding structure of a "lower" physical level (or alternatively the structure of the "lower" to infer the "higher"). In this way transpersonal psychology can be intimately related to sub-atomic physics.
This offers but the briefest glimpse into what promises to be an amazing new world of scientific integrative understanding
We have only scratched the surface. Yet in tracing out the dynamic holistic implications of a few key mathematical notions we have found
- The Qualitative Binary System (of form and emptiness) that (potentially) can encode all transformation processes.
- The Holistic Number Spectrum. Here the qualitative interpretations of the main number types provide the most precise means of ordering the stages and levels of the Spectrum (in both its physical and psychological manifestations).
- The "Theory of Everything" where all relationships can be understood in terms of three fundamental sets of polarities. These can be equally understood in asymmetric terms (where polarities are separated) and symmetric terms (where polarities are complementary).
Transforming Voyage This is a personalized book length account of the Spectrum where holistic mathematical notions are worked into the text (providing some intial context for their rationale).
The Number Paradigms This is based on the structures outlined in "Transforming Voyage" and uses the qualitative notion of number types to provide a detailed mapping of the Spectrum.
The Science of Integration This is a full length treatment in Q and A format
of some of the main concepts in Holistic Mathematics and their general application.
Original Vision This is a very lengthy introductory chapter which outlines the key ideas of Holistic Mathematics in a condensed fashion.
A Spectrum of Translation Methods This briefly indicates how the qualitative binary number can be used to encode the levels, thereby generating a range of methods for scientifically interpreting reality.
For second article click on
DYNAMIC APPROACH TO TRANSLATING THE FOUR QUADRANTS