Tuesday, December 16, 2014

Reflections on Experience (11)

In yesterday's blog, I dealt briefly with the holistic interpretation of the prime number notion.

This leads directly to a dramatic change in perspective regarding the nature of prime numbers, whereby one views them inherently in dynamic terms, as entailing the interaction as between both quantitative and qualitative aspects (which are complementary).

This equally entails that one cannot hope to properly understand the primes in merely conscious rational terms, as this gives but a reduced - and thereby distorted - interpretation.

So the implications could not be greater for Mathematics, as it directly implies that we must therefore move to a newer more comprehensive appreciation that formally includes both conscious and unconscious aspects.

It might be useful to also look briefly at the holistic nature of some of the other number types which have a special relationship with the primes.

The natural numbers are in this context very interesting and in holistic terms emerge in experience through a psychological process (that is akin to multiplication).

Once again at the earliest stage of infant development, experience is highly primitive so that object phenomena are of a very fleeting nature. Quite literally, at this stage, the infant is still unable to meaningfully organise the objective (quantitative) with the dimensional (qualitative) aspect, so that both are directly confused with each other.

Put another way, the infant is unable to distinguish cardinal (as independent) from ordinal number notions (as interdependent).

However, through the gradual differentiation of consciousness, a stabler organisation of the child's experience emerges, with natural phenomena thereby achieving greater consolidation. From this perspective, natural phenomena can be seen to represent unique configurations with respect to earlier - merely primitive - i.e. prime impulses.

However this raises a deep question with respect to the conventional mathematical treatment of multiplication, which is interpreted in a merely reduced quantitative manner. For, properly speaking, such quantitative transformation is not possible without a corresponding qualitative transformation in the nature of space and time! So there is a crucial dimensional aspect literally missing from the conventional treatment of multiplication.

Interestingly, primitive type behaviour coincides for the infant with the growth in development of natural type phenomena. However just as with the distribution of the primes among the natural numbers, the primitive aspect tends to then die out with ongoing further consolidation of natural type phenomena. So in general terms the movement from primitive (prime) to natural experience represents the corresponding move from unconsciousness - more correctly confusion of conscious with unconscious - to more mature conscious development.

Transcendental numbers also play a huge role in the subsequent understanding of the primes. For example the famous number e, on which the natural log system is based, is transcendental in nature.

The important question then arises as to the holistic meaning of the transcendental notion.

Perhaps the best way of appreciating this is with reference to the best known transcendental number π.

Now with respect to any circle, π represents the relationship between the (circular) circumference and its (line) diameter.

In corresponding holistic terms, the line is associated with the conventional quantitative approach to Mathematics (in a linear rational manner). The circle by contrast is associated with the corresponding qualitative approach (in an intuitive fashion that seems paradoxical in linear rational terms).

 So transcendental in its holistic meaning, pertains very subtly - not to quantitative or qualitative as separate - but rather to the relationship as between both quantitative and qualitative aspects. And this is basically true not just with respect to π, but indeed all transcendental numbers.

Now when one reflects on it, the holistic transcendental is directly complementary with the corresponding prime notion (which entails the confused relationship of quantitative and qualitative).

Therefore it requires the integrated holistic notion of the transcendental (in the relationship of quantitative to qualitative) to unravel - as it were - the corresponding confused prime notion.

Needless to say however the holistic concept of the transcendental cannot be understood in conventional mathematical manner as this would simply reduce the transcendental to merely rational terms!

Thus once again this suggests the true magnitude of what is requires in Mathematics in the need to open up completely new means of understanding mathematical relationships!

The imaginary is another key notion with a vitally important holistic - as well as analytic - meaning.

Basically in holistic terms, the unconscious is dynamically activated through negation of unitary poles of reference.

For example in conventional mathematical terms, interpretation always takes place with respect to such unitary poles of reference i.e. in a 1-dimensional (linear) fashion.

So this is the case for example when we attempt to interpret mathematical relationships from a merely objective perspective (that formally takes place in a conscious rational manner).

However the very activation of the unconscious in understanding requires - literally - the dynamic  negation of what has been posited in unitary fashion.

So in holistic terms we represent this unconscious aspect understanding as – 1.

Now just as matter and anti-matter particles fuse in energy, likewise the psychological negation (of what has been posited) results in a 2-dimensional fusion of both positive and negative poles.

Thus  the imaginary notion represents the indirect attempt to express in a conventional analytic (i.e. linear) manner the holistic notion of qualitative meaning (that directly represents an energy state in psychological and physical terms).

This is of vital significance for example for appropriate interpretation of the Riemann Hypothesis.

The postulated existence of the Riemann zeros (as all lying on the imaginary line through .5) relates directly therefore (in complementary fashion to the customary analytic interpretation of the primes) to the indirect representation of  number meaning that is properly holistic i.e. as energy states.

Thus in approaching the Riemann Hypothesis from the holistic perspective (with which I had immersed myself for several decades) I was confident that I could bring a radical different perspective to bear that would substantially alter the very interpretation of what the Hypothesis is fundamentally about!

Monday, December 15, 2014

Reflections on Experience (10)

I mentioned that my initial foray into Type 3 (i.e. Radial) Mathematics related to the Euler Identity.

Now the Euler Identity - as mentioned - is generally considered the most remarkable formula in Mathematics (with an extremely important holistic significance that is not formally recognised).

So shortly afterwards, I began to give serious attention to the Riemann Hypothesis, which is considered by many as the most important unsolved problem in Mathematics.

Thus I was attempting to see if the holistic vision could help to understand this famous problem in a new light. And what subsequently unfolded greatly exceeded my wildest expectations.

Now, I must stress once more that I would see myself as attempting to offer with respect to these two mathematical issues only the most preliminary version of Type 3 Mathematics (where both analytic and holistic aspects of mathematical symbols are dynamically integrated with each other with respect to understanding).

In fact I have outlined before 3 sub-classes with respect to this Type 3 approach.

The first of these - while fully mindful of analytic type interpretation - is still mainly geared to holistic type clarification.  Now this is ably suited to a more philosophical appreciation of the precise nature and significance of the Riemann Hypothesis offering a much more comprehensive vision than is possible in (Type 1) terms. It is also ably suited for the provision of an enlarged general mathematical framework (combining both analytic and holistic interpretation)  that can convincingly demonstrate for example why the Riemann Hypothesis cannot be proven (or dis-proven) in a conventional manner.

I would see my own approach as very much fitting into this first category.

The second - while providing a background holistic appreciation - is mainly geared to the opening up of new highly creative analytic means of interpretation.
To a degree, mathematicians such as Riemann and Ramanujan would at least implicitly have belonged to this category (though they did not yet explicitly acknowledge the holistic aspect of Mathematics).

The third - which still represents very much an ideal yet to be realised - would combine both the analytic and holistic aspects of interpretation in equal balance. So this would open up the greatest possibilities for immensely productive mathematical work in the most creative manner.
Indeed with such understanding, both a highly specialised form of reason would be fully integrated with the contemplative vision in the purest exercise of refined intuition.

I mentioned in earlier entries a long immersion in holistic mathematical notions that culminated in viewing number as a spectrum of energy.

Intrinsic to this approach was the attempt to provide a corresponding holistic meaning for all the major number types.

Because of their fundamental importance, in this context I thereby gave considerable attention to the holistic nature of prime numbers.

I gradually began to realise that the holistic notion of "primeness" was intimately related to what in psychological terms would be classified as "primitive" instinctive behaviour.

Central to such behaviour is the interaction as between both conscious and unconscious aspects of meaning (where both aspects however considerably confused with each other).

Indeed this comprises an inevitable part of earliest infant experience where the conscious mind has not yet been differentiated from the unconscious. So the infant therefore confuses the holistic desire for meaning (which is unconscious in origin) directly with immediate specific phenomena of an embryonic nature.

Permanent object constancy is therefore not yet  possible due to this direct confusion of conscious with unconscious meaning.

Indeed - quite literally - the dimensional aspect of experience through space and time structures (unconsciously recognised) cannot yet properly exist, due to its immediate identification with specific - necessarily fleeting - object phenomena (of a conscious nature).

Then gradually as development takes place, all going well, the infant successfully learns to differentiate the conscious from the unconscious aspect of personality with objects assuming greater permanence within a more stable environment of space and time.

Indeed I then quickly realised that complementary type behaviour likewise exits at the "earliest" physical levels with respect to the sub-atomic nature of phenomena. This led to the vitally important realisation that we cannot hope to properly understand the nature of particle behaviour at the quantum level in the absence of a holistic - as well as analytic - interpretation of events.

Indeed, the very reason why sub-atomic particles prove so short-lived at greater depths of investigation is because their (analytic) object identity is still so closely tied to their dimensional (holistic) existence. Indeed this suggested to me that the conventional understanding of space and time represents but a reduced analytic type perspective, that fails to deal with its true intrinsic nature!

So it is only at the macro level of experience, which coincides with developed conscious awareness that physical objects assume a rigid stable identity.

Both Mathematics and Science - as commonly accepted - represents therefore the rational understanding associated with the specialised development of the conscious mind.

Unfortunately however this is equally associated with a grossly reduced treatment of the unconscious (as inseparable from conscious type interpretation)

If we apply this to the nature of prime numbers we must start with the clear realisation that they in fact - in their intrinsic nature - represent the dynamic interaction of both analytic and holistic aspects of behaviour.

Now clearly in earliest development, both of these aspects are present but however greatly confused with each other.

Therefore the clear implication is that appropriate understanding of the primes ultimately requires the specialised development of both analytic (conscious) and holistic (unconscious) modes of interpretation.

However this poses a considerable dilemma for conventional mathematical interpretation of primes which is defined in merely reduced conscious notions!

Therefore because of decades of immersion in this neglected holistic dimension of Mathematics, my hope was that I could ultimately bring a radical different - and much more coherent - perspective to the understanding of the primes and their relationship with the natural number system.

Tuesday, December 9, 2014

Reflections on Experience (9)

For several decades then, I had devoted my energies to the clarification of basic holistic mathematical notions in - what I refer to as - Type 2 Mathematics.

Now, once again, when appropriately understood, Type 2 (qualitative) is of equal importance to Type 1 (quantitative) Mathematics.

So from this perspective, I had barely scratched the surface of a uniquely distinctive form of Mathematics, which has a truly unlimited potential for further development.

However I recognised that the most comprehensive form of Mathematics (Type 3) would combine both Type 1 and Type 2 aspects in dynamic interaction with each other.

Indeed ultimately both the Type 1 and Type 2 aspects properly only achieve proper balance with respect to the integrated Type 3 approach (i.e. Radial Mathematics).

Therefore around the turn of the new millennium, I slowly began to attempt some initial  attempts at developing a - most preliminary -  Type 3 appreciation with respect to some key mathematical issues.

The first problem that I seriously tackled related to the famous Euler Identity, which is generally accepted as perhaps the most remarkable formula in all of Mathematics.

This is usually given as e= – 1, which equally be expressed as,

e2iπ =  1 (which I refer to as the fundamental Euler Identity). 

What initially attracted my attention is that the Euler Identity provides a direct connection with the (hidden) Type 2 aspect of the number system!

So once again I had defined the natural number system in terms of two aspects

1) where each (base) number, which varies, is defined in terms of a corresponding default dimensional number (or power) that remains fixed as 1,

11, 21, 31, 41,......

2) where each (dimensional) number, which varies, is defined in terms of a corresponding default base number that remains fixed as 1,

11, 12, 13, 14,...... 

As I devoted greater attention to the mysterious nature of the Euler Identity, I began to appreciate that it could not be properly understood in - mere - conventional mathematical terms,

Rather each of its symbols required to be given both an analytic (Type 1) and holistic (Type 2) interpretation.

Then when seen in this light, the Euler Identity provided a truly marvellous means of converting between both aspects of the number system.

Now e2iπ  1 i.e. 11.

Therefore 11, 12, 13, 14,......have corresponding unique expressions in terms of e, as e2iπ,  e4iπ, e6iπ, e8iπ, ......

In reduced quantitative terms,

e2iπ=  e4iπ= e6iπ= e8iπ= 1

However from a qualitative perspective these represent uniquely distinct expressions relating to varying dimensions (or degrees) of (circular) interdependence.

Now, when we have one independent reference frame (as dimension), interdependence is thereby reduced to independence (as this independent polar frame is necessarily interdependent with itself!).

However with two poles (as dimensions) genuine interdependence arises from the relationship of these opposite poles.

Then as the number of reference frames increases, the corresponding level of interdependence likewise increases.

So the Type 2 aspect of number is properly geared to the holistic notion of natural numbers as representing increasing levels of interdependence (with respect to the corresponding increase of distinct reference frames involved). The reference frames in turn relate to varying interactions with respect to the fundamental poles which condition all experience i.e. external/internal and whole/part).

These are then intimately related to appropriate qualitative interpretation of numbers .

With just one number - say - 3, the qualitative nature of multiplication does not yet properly arise with the result strictly 1-dimensional (i.e. measured in linear units).

However with two numbers to be multiplied - say - 3 * 5 - , a true qualitative issue arises as we can see in geometrical terms that 3 * 5 now represents square (i.e. 2-dimensional) units. So a qualitative change in the nature of units has taken place with the result properly relating to 2-dimensional, rather than 1-dimensional format!

Now if we treat the numbers in solely independent terms then multiplication is not strictly separable from addition (representing just a short hand expression of addition).

So from this perspective 3 * 5 = 5 +  5 + 5 = 15 (i.e. 151).

However, this is not satisfactory, as a dimensional change has also clearly taken place in the units.

So what is involved - treating it geometrically as 3 rows of 5 - is that notions of number interdependence (as well as independence)  are now required. Therefore the recognition of 5 items implies a recognition of the common identity of the individual units in each row, with each other (thereby enabling them to be placed in a mutual correspondence).

Thus properly understood, multiplication in this case entails both the notion of independence (as the initial 1st dimension) and the notion of interdependence (in the mutual identity of the 5 items in each row (representing the 2nd dimension).

Now if we extended further - say to the multiplication of 3 numbers - a further layer of interdependence would be involved in the recognition of the mutual identity of 2-dimensional rows and columns with each other (now representing the 3rd dimension).

So each further dimension entails a higher level of interdependence (to which the Type 2 system directly relates).

However the truly incredible feature of the Euler Identity is how it then facilitates (indirect) conversion from Type 2 (holistic) to Type 1 (analytic) interpretation.

Therefore to convert for example the simplest 2-dimensional case in a quantitative manner,

Here we set x= 1and 1respectively.

For 1and 12 we use e2iπ  and e4iπ.

Through De Moivre's Theorem, e2iπ = cos 2π + i sin 2π 

So in the first case x = cos π + i sin π = cos 180  + i sin 180 = – 1

In the second case x = cos 2π + i sin 2π = + 1

Therefore, when expressed indirectly in Type 1 terms, the Type 2 notion of 1 (as representing the 1st dimension) remains unchanged as 1). This again means that that 1-dimensional understanding reduces  qualitative to quantitative interpretation.

However the indirect quantitative expression of the 2nd dimension is – 1.
In qualitative terms, – 1 implies the (temporary) negation of posited, i.e. conscious, understanding (which provides the very means of moving from the appreciation of independence to the complementary notion of interdependence).

In fact this new appreciation of the Euler Identity was eventually to provide my appreciation of the vital role of the (unrecognised) Zeta 2 zeros in connection with the Riemann Hypothesis.

Friday, December 5, 2014

Reflections on Experience (8)

I have already mentioned the physical notion of a spectrum of energy levels (i.e. electromagnetic energy).

The most common expression of such energy is natural light!  Then there is the complementary notion of spiritual light (as representing intuitive type energy).

So just as we can identify many distinct bands with respect to electromagnetic radiation, equally we can identify complementary type bands in psycho spiritual terms! So each band thereby represents a distinct quality with respect to the development of intuitive energy in experience.

However as I slowly developed the holistic mathematical approach, I began to realise that a much more precise spectrum exists with respect to number.

So every number for example has already a well-defined analytic type meaning in quantitative terms.
Likewise every number potentially has an equally important holistic type meaning in a qualitative manner!

And when we consider number with respect to this latter holistic aspect, it can then be seen to represent a spectrum of psycho spiritual energy states.

I also realised - though it was not the direct focus of my interest at the time - that because of the holistic complementarity of both the psychological and physical aspects of development, that number should likewise represent a spectrum with respect to physical energy states (e.g. at the sub-atomic level of matter).

From this holistic perspective, the natural number system in fact represents the most fundamental spectrum.

Just as the natural numbers, in analytic terms, are commonly identified with specific quantities, from a holistic perspective, they can be seen to represent more general qualitative categories (through which overall experience is organised). Unfortunately however, in conventional mathematical terms, the understanding of numbers - as representing dimensional notions - is likewise reduced in a mere quantitative manner, so that their crucial qualitative meaning is thereby completely overlooked.

Indeed though intuition implicitly is necessary for all mathematical operations at the conventional level, explicitly its role is reduced in a mere rational manner. This indeed is what can accurately be represented as 1-dimensional interpretation (where - literally - understanding takes place unambiguously with respect to just one polar reference frame).

However, 2-dimensional understanding for example. entails the interaction of 2 opposite polar reference frames (in what can be referred to as the complementarity of opposites).

Thus the very ability to literally "see" at this dimensional level requires a newly refined form of intuition (representing thereby a distinctive energy state in psycho spiritual terms).

"Higher" dimensions then require corresponding "higher" levels of intuitive refinement.

For example 4-dimensional understanding entails the ability to"see" reality as representing the interdependent interaction of 4 distinctive polar reference frames (which are given in terms of the corresponding four roots of 1).

So at this level, reality is seen as representing the interaction of two"real" polarities  (positive and negative) in conscious terms, as well as two "imaginary" polarities (again positive and negative) in an unconscious fashion (which is then indirectly expressed in conscious terms).

This in fact gives an entirely distinctive interpretation to the 4 dimensions of space and time (from a holistic perspective) which again simply means that in dynamic interactive terms, all relationships in space and time, are governed by the interaction of two real and two imaginary polarities respectively (both with positive and negative directions).

However we do not have to stop here as every natural number can now be seen to represent a unique dynamic configuration with respect to interacting frames of reference.

So for example 100 in this holistic sense entails the appreciation to "see"  reality in terms of 100 relatively distinct interacting reference frames.

Thus associated with each number is now a unique quality of intuitive recognition (representing  a special form of psycho spiritual light) corresponding to that number on the spectrum.

Needless to say the "higher" numbers require an ever more refined quality of intuition.
So just as it has long been recognised in the contemplative traditions that the process of authentic enlightenment requires a distinct type of psycho spiritual training, equally this is true in terms of the holistic appreciation of mathematical symbols.

Now, conventional mathematicians may indeed concede that implicitly appreciation at the recognised 1-dimensional level of interpretation requires a degree of intuition (as well as reason) especially where more creative work is concerned.

However explicitly interpretation is then reduced in a merely static and somewhat absolute manner.

However if we are to look at mathematics in a properly balanced fashion, it should lay equal emphasis on the development of both reason and intuition (which are utterly distinct from one another).

Thus specialisation with respect to holistic mathematical understanding entails ever further refinement with respect to pure intuitive understanding (indirectly expressed in a paradoxical rational manner).

In fact just as specialisation with respect to Conventional Mathematics represents the (linear) rational extreme of interpretation (as fixed symbols of form) likewise specialisation with respect to Holistic Mathematics represents the corresponding (circular) intuitive extreme of interpretation (where understanding is now so dynamically relative, that symbols approach pure psycho spiritual energy states). And once again such understanding has a complementary relevance at the physical level!

However it is not just the natural numbers that can represent a spectrum (of distinctive energy states).
All numbers in holistic terms possess the same important potential.

In this regard, the famed "Riemann zeros" simply represent one especially important spectrum (that is uniquely related to the working of the natural number system).        

Thursday, December 4, 2014

Reflections on Experience (7)

A major consideration for me during the late 80's was the attempt to map out the main stages on the full spectrum of development from a holistic mathematical perspective.

This then culminated in my first expression  as the online book "The Number Paradigms" which was completed in the early 90's.

This sought to provide the holistic mathematical interpretation of all the main number types e.g. the binary numbers (1 and 0) primes, natural (counting) numbers, integers, rational numbers, negative and imaginary numbers, irrational (algebraic and transcendental) and transfinite numbers.

These then from a scientific perspective potentially could play a major role in providing distinctive paradigms providing various unique modes of possible interpretation

For example the conventional scientific approach is based on what holistically can be referred to as the rational paradigm. This leads to analytic type interpretation in a quantitative manner. However we can equally define in a coherent manner an "irrational paradigm" (with various sub classes) that provides for an utterly distinctive holistic type appreciation of a qualitative nature.

And then in the most comprehensive scientific understanding, analytic and holistic aspects would increasingly interpenetrate. This would be modelled in turn by a complex paradigm (with real and imaginary aspects) that would include both rational and irrational type numbers as paradigms.

An even simpler model of this approach can be given through the holistic expression of the binary numbers (1 and 0).

As is well known potentially all information can be encoded in a binary digital manner (based on the clear analytic separation of both digits).

However equally all transformation processes can be encoded in a corresponding binary digital manner (now based on the holistic interdependence of both digits).

So every stage of development can be shown to represent a unique configuration of linear (1) and circular (0) type logic, corresponding to both analytic and holistic understanding respectively.

Basically the story of development can be outlined simply in these terms.

With initial infant development both conscious (linear) and unconscious (circular) type understanding are greatly confused with each other.

Earlier childhood development is then largely concerned with the gradual differentiation of the conscious from the unconscious aspect of understanding.

Then by early adulthood - certainly in Western culture - this culminates in the specialised development of linear (1) understanding. This is especially true in terms of conventional appreciation of mathematics and science and in a more general way of intellectual life generally.

However for the unconscious to attain corresponding substantial development, many "higher" stages of refined intuitive type appreciation are required. Traditionally this has been largely confined to the esoteric mystical traditions (East and West) in the appropriate attainment of contemplative type spiritual awareness.

My own particular insight was the realisation that these stages - when appropriately understood - have a fundamental importance with respect to a new qualitative appreciation of Mathematics and science (in what I refer to as Holistic Mathematics and Holistic Science respectively.

So just as we can have the specialisation of linear reason (i.e. the digit 1 in holistic terms) likewise we can have the specialisation of (circular) intuition (i.e. the digit 0 in holistic terms). "Circular" in this context is identified indirectly with the paradoxical rational type manner (based on the complementarity of opposite poles) through which such intuitive holistic awareness is expressed.

So we now potentially in development have identified the two extremes (analytic and holistic) to which both mathematical (and associated scientific) understanding can attain.
And remarkably in our culture only the analytic aspect is formally recognised!

Then the most comprehensive development is potentially associated with the two-way interpenetration of both linear (1) and circular (0) type appreciation. From this perspective, I then customarily refer to  Radial Mathematics (and Radial Science) respectively.

Therefore we have 3 main levels of potential development with respect to both Mathematics (and Science).

Firstly we have customary quantitative appreciation (in the holistic specialisation of the digit 1).

Then we have the largely unrecognised qualitative appreciation (in the holistic specialisation of the digit 0).

Then we have the most comprehensive type appreciation combining interactively both quantitative and qualitative aspects (in the holistic specialisation of the combined two digits 1 and 0).

Not surprisingly, my overall appreciation of this general model has become much more refined in the 20 years since its first enunciation.

I now outline 7 major bands (with 3 main levels in each band) on the spectrum of development.

The first two relate principally to linear (conscious) development; the second two relate to circular (unconscious) development with the final three relating to the mature interpenetration of both linear (1) and circular (0) aspects. A brief outline is given in the two updates (from 2008) which still are largely representative of my present thinking.

Wednesday, December 3, 2014

Reflections on Experience (6)

Following some 10 years or so in mental darkness (enabling the holistic development of the unconscious), I gradually returned to intellectual type work, which inevitably led back to my initial interest in Mathematics.

During this time (in the early 80's) I began to resonate strongly (as formerly in relation to Hegel and St. John of the Cross) with the writings of C.G. Jung. Though not a mathematician himself in the accepted sense, I found his writings however remarkably compatible with holistic mathematical notions.

Indeed indirectly, growing familiarity with Jung enabled me to properly clarify the holistic meaning of the imaginary mathematical notion (which has immensely important consequences).

I particularly resonated at the time with Jung's work on "Personality Types" which directly suggested to me a holistic mathematical interpretation.

Indeed, from one perspective, Jung's delineation of the four functions of personality correspond from a psychological basis with the four dimensions of space and time. So different personality types (entailing unique configurations with respect to the four functions) would thereby tend to understand space and time relationships in a corresponding distinctive manner.

The very way that Jung described the functions with two rational (thinking and feeling) and two irrational (sense and intuition) already suggested holistic mathematical type notions.

However, on reflection I gradually realised that the more appropriate designation was in terms of two real (conscious) and two imaginary (unconscious) functions.

In this way the four dimensions of space and time could now be directly related in holistic qualitative manner with the four coordinates of the complex number plane.

In fact the holistic imaginary notion can be expressed in precisely the same manner as the accepted quantitative analytic notion, with however each symbol now given a corresponding holistic mathematical interpretation.

So i represents. the square root of – 1.

Thus i= – 1. In this way i can be seen as the attempt to express the notion of – 1 (initially expressed in 2-dimensional terms with respect to the power of 2) in a reduced linear (1-dimensional) fashion.

In corresponding holistic terms, – 1 represents in psychological terms the negation of  the conscious pole of understanding, This dynamically complements therefore + 1, where by contrast this pole (e.g. objective) is literally posited in a real conscious manner.

So for example in science "real" phenomena correspond directly with those that can be consciously observed.

Now in analytic terms, symbols are given a fixed static meaning; however in corresponding holistic  terms their meaning is of a relative dynamic nature.

In this context,  – 1 relates to the dynamic negation in experience (of what has been consciously posited). And just as matter and anti-matter fuse as physical energy, likewise through the negative direction of understanding, an unconscious fusion to a degree of psycho spiritual energy necessarily takes place. Intuition in fact represents such psychological energy, which is directly generated through the negation of linear reason (in an unconscious manner).

Thus if we are to properly understand the interaction of reason and intuition with respect to mathematical interpretation, we must first realise that they represent distinct modes of understanding that are quantitative (analytic) and qualitative (holistic) with respect to each other.

This equally implies that we must explicitly incorporate the (hidden) unconscious with the (recognised) conscious aspect with respect to all mathematical understanding.

Now the imaginary notion arises in a holistic qualitative manner through the indirect attempt to express - what  inherently pertains to - the unconscious in a linear rational manner.

So from this perspective the extension of the number system in a complex manner, with both real and imaginary parts simply implies the recognition that reality itself contains both analytic and holistic aspects, which continually interact with each other.And from a psychological perspective this entails the interaction of both conscious and unconscious. However as the unconscious aspect cannot be directly accessed in a real (i.e. phenomenal) manner it must thereby be indirectly represented in an imaginary fashion (where its symbols properly relate to holistic type meaning).

Now this realisation is extremely important in both physical and psychological terms.

For example, it is well recognised that complex numbers play an especially important role in understanding the behavior of sub-atomic particles.
Now the reason for this is that such particles do not strictly enjoy an independent identity at this level but can only be understood through their interaction with other particles.

So in philosophical terms there is an inescapable holistic dimension to the behaviour of physical reality (which cannot be properly encapsulated in the accepted analytic fashion).

However what I am saying here is much more fundamental in that there is equally an inescapable holistic dimension to the behavior of mathematical reality. So for example, all numbers can be given both analytic (quantitative) and holistic (qualitative) type interpretations. And properly understood, both of these aspects are of equal importance. Remarkably however, at present the holistic aspect of mathematical understanding is not even recognised in formal terms.

So during the 80's, I finally was making substantial progress with respect to the fundamental nature of  this (unrecognised) holistic aspect of understanding and devoted a great deal of my energy to its further clarification.

I was especially interested in psychological development at this time. So like the electromagnetic spectrum, there are several distinctive bands that can potentially unfold. Now I mainly concentrated on the scientific clarification of the "higher" contemplative stages of development (which generally are solely associated with the esoteric mystical traditions). And then beyond these I could see the possibility for more more advanced stages (which I referred to as radial).

And the amazing discovery that I made at the time is that all of these stages find their most precise scientific expression through the holistic interpretation of mathematical symbols.