4. Journey through the Irrational and Imaginary

4. Journey through the Irrational and Imaginary

Higher Band

We have already entered a strange world where - when appropriately interpreted - experience takes place in negative as well as fractional dimensions (with a sound mathematical rationale that is directly related to the holistic understanding of basic number behaviour).

Before proceeding further therefore, it perhaps would be useful to attempt to clarify once more what this holistic mathematical interpretation of dimensions actually entails.

Perhaps the best way of looking at this notion of dimension is in terms of direction.

Thus when experience is linear there is just one direction involved which we identify as positive. An unambiguous identity is given here which we can identify as time. In other words with linear understanding the direction of time is taken to be positive so that we understand all events as unfolding in a forward moving time framework. The other three “dimensions” which in this understanding are spatial are then identified - in reduced manner - with the quantitative characteristics of object phenomena.  

So in maintaining that dimensional experience is linear with respect to conventional understanding, we are directly implying that the qualitative dimension of time is linear with a positive i.e. forward moving direction.

However once we allow for the negation - as well as the positing - of objects and dimensions - a new understanding unfolds.

Implicitly the idea of negation of time relates to the dynamic fact that experience must necessarily switch as between opposite polarities (such as internal and external) which are complementary opposites with respect to each other. So we have the old paradox as illustrated so many times with respect to road directions that what is a “right” or a “left” turn depends on context. If I am heading “up” the road, left and right have unambiguous meanings. Likewise, if I am heading “down” the road, again left and right have an unambiguous meaning. However what is “right” in terms of the first frame is “left” in terms of the second; also what is “left” in terms of the first is “right” in terms of the second.

So it is exactly similar in dynamic interactive terms with respect to the experience of time. If we identify the frame of reference with respect to the external world (in relation to the self), time moves forward with respect to this aspect; equally when we identify the frame with the internal self (in relation to the world), time likewise moves forward with respect to the self. However relative to each other time is moving in both positive (forward) and negative (backward) directions.

Thus to very ability to dynamically switch as between both poles (internal and external) in experience requires implicitly the ability to negate - as well as posit -each aspect. Thus to switch from experience of (external) reality to that of the (internal) self we must negate, to a degree, external reality. Likewise, in reverse manner, to switch from experience of the (internal) self to that of the (external) world we must also negate the (internal) self.  In this way actual experience implicitly requires the negation - as well as positing - of time. Therefore the dimension of time has now a negative as well as positive direction.

Then - from the perspective - when we recognise these two directions, then each considered as separate represents a fraction (i.e. ½) of the linear (that now is bi-directional).

So in this way we experience in fractional dimensions. 

However though once again these dynamics are implicit - even with respect to linear understanding – they do not actually become explicit, till interpretation in accordance with the Higher Band becomes possible. And this is what we now move to consider.

H1 - (Algebraic) Irrational Numbers 

The discovery by the Pythagoreans - ironically from the famed right angled triangle named in their honour - that the square root of 2 is irrational, created a fundamental problem with respect to their whole approach.

Implicitly, the Pythagoreans were seeking to maintain a radial approach to Mathematics (where analytic and holistic understanding would complement each other).

Initially all seemed to be well. In qualitative scientific terms, a rational approach was adopted. This seemed to coincide with the (sole) existence of rational number quantities in mathematical terms.

However with the discovery that the √2 is irrational, this correspondence broke down. Thus, in philosophical terms, the Pythagoreans realised that they lacked a holistic explanation as to why irrational numbers might arise.

Subsequently Western Mathematics has been conducted in a greatly reduced manner where qualitative philosophical considerations have been considered secondary - and even quite irrelevant - with respect to further developments. 

However it is important to remember that the Pythagoreans were in fact seeking a more integral approach to Mathematics (than currently exists). With this integral approach, mere quantitative interpretation of results was not sufficient but also required a corresponding deeper philosophical explanation (of a qualitative nature) 

Indeed substantial progress - even in quantitative terms - will not become possible with respect to key outstanding problems until this greatly neglected qualitative aspect is restored. Finally appropriate philosophical understanding itself reveals a radical new interpretation of mathematical symbols that has the potential to completely revolutionise the way we understand reality.

Now in outlining the holistic mathematical nature of number symbols appropriate to the first of the higher levels H1 - in what is sometimes referred to as the psychic/subtle realm - we can in fact provide a convincing solution to the Pythagorean dilemma.         

  

                 

Entry to this stage is often preceded by a profound existential crisis. What this entails in effect is the explicit negation of already posited phenomena in experience. In other words as one searches for a deeper spiritual meaning to life, one comes increasingly into conflict with the rigid dualistic experience (based on the mere positing of phenomena).  

So the discovery of a truly nondual spiritual meaning requires corresponding detachment from phenomena (which implies their dynamic negation). In this way - through such negation of conscious activity - the unconscious aspect of experience is enabled to grow and deepen.

With sufficient development taking place in this manner, when conscious phenomena arise, they can be experienced in non-attached fashion consistent with the maintenance of spiritual peace. 

We now have the beginning of the mature holistic experience of our original identity (+) 1 – 1 = 0.

Thus through both the positing and negating of phenomenal form, spiritual emptiness is attained.

This in fact represents experience that now properly takes place in two - rather than one - dimensions.

Indeed the clues to this lies in the reverse mathematical procedure of taking roots.

So the square root of a number can either have a positive (+) or negative (-) value that are equal to each other.

What this means in effect is that when we try to express a relationship according to a circular both/and logic (where positive and negative values are combined) in a reduced linear either/or logic, the positive and negative aspects are thereby separated. So what is both positive and negative (according to two-dimensional understanding) is then expressed as either positive or negative in linear (one-dimensional) terms. 

So in algebraic terms (which we are now explaining holistically), we can as an illustration, take the simple equation x² = 2.

The value of x here is expressed in two-dimensional terms (and is rational)!

However because conventional mathematics is conducted in linear one-dimensional terms, we express this value in terms of the reduced linear value for x (i.e. x¹).  

So x¹  = + √2 or - √2.

Thus in either/or logic we have now two opposite equal values for x¹ (which are clearly separated from each other).

However what seems to be completely missing in conventional mathematics is the realisation that when a number is raised to any power (≠ 1) that a qualitative transformation in the very logic of interpretation is strictly involved.

However once again because of the linear bias of conventional interpretation, this necessarily means that such a qualitative transformation is ignored with the result expressed in (reduced) quantitative terms (according to either/or logic).

Now - what I refer to as two-dimensions - involving the complementarity of opposites - in holistic terms is really one continuous dimension. In other words 2 here really entails a higher form of unity (embracing opposite polarities)

So from this perspective (as reference point) we can equally express the reduced relationship in this manner

X¹ (where 1 now represents a holistic notion of “higher” oneness) = 2.

Thus X^1/2   = + √2 or - √2

Therefore the fractional dimension here (= ½) really represents the fact that we are trying to express what holistically is (one) “higher” dimension (where now opposite polarities are unified in experience) in a reduced linear manner (where they are necessarily separated).

So fractional dimensions therefore are involved whenever we try to “convert” higher level experience - entailing considerable spiritual intuitive refinement of experience -in a reduced linear manner (which conventional dualistic interpretation requires).

The fascinating aspect about (algebraic) irrational numbers such as √2 is that they contain both discrete (analytic) and holistic (continuous) aspects.

Thus the value of √2 can be approximated to any required degree of accuracy in discrete finite terms. For example correct to four decimal placed the value is 1.4142.

However it also has a continuous (irreducible) aspect that its decimal sequence continues infinitely (with no discernible pattern).

In holistic experiential terms, this means that the higher levels (such as H1) entail experience of reality that contains both dualistic (rational) and nondual (spiritually intuitive) features. 

So, a necessary interaction exists as between phenomenal form and spiritual emptiness (at this level of development).

However the deeper problem (for conventional mathematical interpretation) is that the irrational behaviour of numbers (in containing an irreducible infinite aspect) really points to the need for a more comprehensive philosophical manner of interpretation (which goes greatly beyond the confines of linear logic).

Indeed this is the very message posed by the Pythagorean dilemma some 2,500 years ago. However because of the greatly subsequent reduced nature of mathematical interpretation, we are now much less aware of the true significance of this issue than our wise forbears.

It is also worth relating that what seems irrational (from a one-dimensional perspective) is in fact fully rational from the corresponding two-dimensional perspective.

Thus when x² = 2, x in terms of the both/and logic of complementary opposites has a rational value.

However when x¹ = + √2  or - √2, the value is irrational (because it is now expressed in the either/or logic of separate opposites).

So what is rational or irrational in logical terms depends on the perspective we take.

From the two-dimensional perspective - which properly from its own perspective is one-dimensional as a unified “higher” dimension - linear (one-dimensional) understanding, as a fractional dimension - is irrational.

However from the linear i.e. one-dimensional perspective (of separate opposites), the two-dimensional understanding (of complementary opposites) is irrational. 

We will now summarise the three sub-levels of H1.

It is important to bear in mind that these sub-levels are now defined with respect to continual development with respect to two directions (positive and negative) which ultimately are fully complementary. However while the process is unfolding there is a mixture in experience of both dual and nondual aspects.

In many ways SL1 can once again be identified with the more concrete phases of development where a heightened form of sensory perception tends to predominate.

During the first period of dualistic negation (corresponding to an existential crisis preceding the stage) spiritual intuition incubates in the unconscious (representing the fusion of both positive and negative poles). Then when this is eventually released the natural world becomes transformed through a new brilliant illumination.

So in holistic mathematical terms, such experience represents the positive sign with respect to “irrational” perceptions, where (finite) discrete elements as identifiable phenomena and (infinite) continuous elements as spiritual intuition interact.

Then in order to become sufficiently cleansed of secondary attachment with respect to phenomena, a substantial period of dynamic negation is required. During this period of purgation which is really a period of internal illumination (referred to by St. John of the Cross as the “passive night of sense”) dualistic standards of moral conduct become gradually redundant highlighting the growing problem of existential choice where true freedom (conforming truly to spiritual conscience) must be exercised with respect to decision-making.

So this represents in terms the negative (internal) sign with respect to “irrational” perception.

Thus in holistic number terms SL1 (of H1) represents the unfolding of irrational number quantities (positive and negative) as perceptions. Once again these arise through the experience of perceptual phenomena with respect to the fractional dimension (of ½) i.e. where a “higher” unified experience of complementary opposites is interpreted in the reduced dualistic terms of either/or logic.

SL2 can in turn be identified with the more formal phases of the H1 level where a profound and very refined form of universal type conceptual awareness tends now to dominate.

And again this has both positive and negative directions.

During the first stage - which is associated with a deeper form of spiritual illumination - the positive direction with respect to irrational concepts (i.e. qualitative dimensions) unfolds. Here a very refined universal appreciation of the nature of reality, inspired by spiritual contemplation, is experienced. So once again though the general phenomenal form is necessarily dualistic, it considerably interacts with nondual spiritual intuition.

During the next stage - again die to secondary attachment to conceptual symbols - the negative direction unfolds in what is referred to by St. John as the “passive night of spirit”. Here there is a deep erosion of the very dimensional framework on which phenomenal understanding is based. Not surprisingly this leads to a very distressing implosion, where, quite literally, one feels as if one is being squeezed psychologically out of space and time. This is the famous “dark night” which - not surprisingly - has close parallels with the physical phenomenon of black holes.

So in holistic number terms, we have here the unfolding of irrational number dimensions (with respect to both positive and negative signs).

As we have seen at SL1 irrational perceptions (as holistic number quantities) are developed (with both positive and negative directions). Then at SL2, irrational concepts (as holistic number dimensions) are developed.

Then at SL3 one now tries to relate simultaneously both irrational perceptions and irrational concepts in “higher” level equivalent of the vision-logic of the previous stage.

Though - as we shall see - this should lead naturally to a remarkable new type of psychological experience (with a parallel holistic number interpretation) initially because of misinterpretation of experience a dramatic conflict culminating in a terrible crisis can occur.

So though in one sense, one has now reached the deepest level of contemplative (empty) experience, one may also have substantially lost contact (through the continual negation of phenomena) with the world of form. So another switch is required before the fruits of this stage can unfold.

H2 - Imaginary and Transcendental Numbers  

The continual negation of conscious form - which is especially characteristic of the “dark night of the soul” - leads a growing problem of repression of the unconscious.

Ultimately this problem can only be relieved through a switch in emphasis from the conscious to unconscious (indirectly expressed in a conscious manner). As I have repeatedly demonstrated on other occasions, the indirect expression (i.e. in a conscious manner), of what inherently pertains to unconscious meaning, provides us with the holistic interpretation of an imaginary number.

So once again in conventional mathematical terms, the imaginary number  i = √-1.

In corresponding holistic terms 1 corresponds to unity (as inherent in phenomenal form).

The dynamic negation then of conscious form leads to a fusion in unconscious meaning where both positive and negative polarities combine.

Then the indirect expression of this (two-dimensional) unconscious meaning in a linear conscious manner requires the holistic equivalent of obtaining the square root.

So therefore, in this sense, i refers to what is inherent in projected form (i.e. as the indirect expression of unconscious meaning).

As experience necessarily entails both conscious and unconscious aspects, properly understand it is holistically of a complex nature containing both real (conscious) and imaginary (conscious) elements.

For example one may have aspirations to own a “dream” house. So the “house” in this context has a real aspect (though conscious identification) and an imaginary aspect (as the mediator of the unconscious desire for holistic meaning). And this ultimately is true of all phenomena in experience.

However because understanding is heavily dominated - especially with respect to scientific interpretation - by what is merely conscious (and “real”) this leads to considerable confusion, where holistic desire, which is ultimately spiritual and infinite, is misleadingly identified with (finite) specific phenomena. In other words the imaginary (unconscious) is thereby reduced in terms of the real (conscious) elements. In this way “reality” becomes identified with what is real (in terms of conscious understanding).

However in the dynamics of development, conscious understanding of reality undergoes considerable erosion (through dynamic negation) during the stages of H1. So when the unconscious is then allowed to indirectly express itself, it can do so in a much purer form, where refined symbols become mediators of a spiritual infinite meaning.

Thus when the imaginary aspect is present one does not interpret a symbol in terms of its conscious manifestation but rather in terms of an unconscious holistic meaning which is expressed through that symbol.

As we have seen, H1 - as two-dimensional - relates to the positing and negating of conscious phenomena as real (required by increasingly refined interplay as between the horizontal polarities of internal and external).   

Now in like manner H2 - as four-dimensional - relates to additional positing and negating of unconscious phenomena as imaginary (required by the increasingly refined interplay as between the vertical polarities of whole and part).

Again linear understanding, which is geared to real (conscious) interpretation inevitably, reduces the true relation existing as between whole and part.

Here both wholes and parts are given a mere finite existence with the (reduced) whole in any context interpreted as the sum of its parts.

However properly understood the (collective) whole has also an imaginary aspect through which - in holistic terms - infinite spiritual meaning is mediated; likewise this is true of each (unique) part which again has the capacity to mediate such meaning.

And this is all central to mathematics.

Because (conventional) mathematics is based on rational linear interpretation (geared to what is finite and real), it inevitably reduces the meaning of what is infinite treating it misleadingly as an unlimited linear extension of what is finite.

Also - though it recognises the quantitative existence of imaginary numbers - it still greatly lacks any qualitative philosophical appreciation of their true nature.

So once again the imaginary essentially relates in psychological terms to the manner in which unconscious holistic meaning (which in its pure nature is spiritual) is indirectly mediated through conscious symbols. And because psychological and physical aspects of reality are complementary, the imaginary also has an important “lower” physical meaning where at the deepest sub-atomic levels of reality, the ultimate holistic nature of reality (as its fundamental ground) becomes indirectly expressed through dynamic particle interactions.

     

At the previous stage H1, one developed considerably in appreciation of circular (both/and) understanding based on the complementarity of real opposites.

However a continuing problem existed due to an ability to properly distinguish such circular from earlier linear type understanding. So in the attempts to “convert” from one to the other, confusion necessarily remained. In converting from the “higher” circular to the “lower” linear, linear forms still tended to dominate leading to residual attachment to such forms; likewise in converting from “lower” linear to “higher” circular, the circular (paradoxical) forms tended to dominate leading to corresponding residual attachment to these forms.

Only at H2, can these two types of understanding be properly unravelled, where the existence of the imaginary aspect of every phenomenon is properly recognised (as well as the real). Thus the proper separation (and differentiation) of what is conscious and unconscious is necessary as a prerequisite for their eventual union (and integration).

So at H2 we have now four directions, corresponding to the four roots of unity, resulting from the expression of what is inherently 4-dimensional reality in reduced (linear) one-dimensional terms.

Two of these directions are real corresponding to the - relative - positive and negative aspects of conscious understanding.

The other two are imaginary corresponding in turn to the - relative - positive and negative directions of unconscious understanding (that is indirectly expressed through projection in a conscious manner). In this way we attempt to come to terms with the holistic - as opposed to specific - meaning of phenomena as expressive ultimately of truth that is properly spiritual.

This now leads to the second important holistic number feature of the level which has a remarkable parallel with established conventional understanding.

In 1900, Hilbert posed as one of his 23 problems the task of proving that 2^√2 is transcendental.

Now a transcendental number is one that is irrational but however - unlike √2 - not a solution to any algebraic equation.

Much to Hilbert’s surprise a proof was found to this problem shortly afterwards which proved that any rational (or irrational) number when raised to an irrational power is transcendental.

However this has a distinct - and very important parallel - in holistic terms.

As we have already seen the linear levels lead - holistically - to the formation of rational numbers (as quantities) and rational numbers (as dimensions). Again this corresponds to the ability to analytically to break up wholes into parts and then recompose wholes from constituent parts with respect to both perceptions and concepts respectively.

Then with the flexible two-way interaction of both rational perceptions and concepts in this manner, a transformation in experience takes place that is irrational.

Through such repeated dynamic interaction the very assumption on which dualistic understanding (that is based on the independence of separate poles) 

begins to break down and one enters a paradoxical world where truth is understood - in any context - as the complementarity of opposite poles.

Firstly such irrational understanding unfolds with respect to perceptions (as holistic quantities); later it unfolds with respect to concepts (as holistic dimensions). Finally the flexible two-way interaction of both aspects can now take place.

As we have seen at H1 a two-way conversion process continually takes place. From one perspective one keeps trying to reduce the “higher” two-dimensional appreciation of complementary opposites in (reduced) linear terms (amenable to rational interpretation). Equally from the reverse perspective one tries to “convert” the dualistic understanding of separate opposites to the “higher” two-dimensional appreciation (which is ultimately spiritual and nondual).

However because one still tries to maintain - in relative terms - the separate identity of both types of understanding, a conflict inevitably remains.

However at H2 - due to the growing interaction of both modes of understanding - a more refined nature of interpretation begins to unfold.

Thus at this stage one no longer tries to understand reality in dualistic or nondual terms (as somehow separate) but rather as the interaction of both what is dual and nondual.

So phenomenal understanding is now so refined that one no longer seeks interpretation that is linear or circular (as separate) but rather as the interaction of what is both linear and circular. And this holistically is precisely what is entailed by a transcendental number.

If we take the most famous transcendental number i.e. p, to illustrate, we can see that its numerical value directly represses the relationship between line and circle.

So p expresses the value we obtain from dividing the circumference of any circle by its line diameter.   

This feature is also deeply inherent in that other extremely important transcendental number e.

Imagine again an investment with simple interest of 100% due at the end of one year. So the investment doubles in that time i.e. 1 +1 = 2.

However now imagine that compound interest applies. So if we split the time period in two, we now double the dimensional power with the investment now worth 2.25

i.e. (1 + ½)² = 2.25

Now if we keep differentiating the time periods involved into ever smaller discrete intervals the continuous integral compounding (as dimensional power) grows in inverse fashion.

Ultimately - in the limit - the (differentiated) discrete intervals become so small and the (integral) continuous dimensions so frequent that both are indistinguishable from each other.

So here what is discrete (in linear terms) is ultimately indistinguishable from what in continuous (in circular terms). And this limiting value, expresses the numerical value of e (which is transcendental).

Likewise the holistic equivalent of e corresponds to dynamic psychological growth that is so refined that the process of differentiation (in linear terms) becomes indistinguishable from integration (in circular terms).

Thus the very refined process of holistic process of transcendental understanding - that precisely corresponds to the holistic mathematical interpretation of what is transcendental - unfolds at H2.

This experience arises from the process of relating either rational (or irrational perceptions) to irrational concepts.

Once again an irrational perceptions – corresponding to holistic number quantities – relate to phenomenal interpretation according to the dynamic principle of the complementarity of opposites. Likewise irrational concepts – corresponding likewise to holistic number powers or dimensions – relate likewise to general interpretation of reality according to the same dynamic principle of complementarity.

And because perceptions (as quantities) and concepts (as dimensions) are linear and circular with respect to each other, this continual switching in experience requires the ability to dualistically experience in an extremely refined manner (where linear attachment is significantly eroded).

Thus experience is now significantly more of a contemplative than rational nature. 

The great importance of this stage for the interpretation of prime numbers arises from the fact that H2 is directly complementary with L2 (where prime number experience originates).

Thus the “higher“ level interpretations of H2 are necessary to properly unravel what is inherent in behaviour at L2.

Put another way both imaginary and transcendental number behaviour are necessary to properly understand the nature of the primes in both (conventional) quantitative and (holistic) qualitative terms.

We will now briefly sum up this holistic number behaviour with respect to the three sub-levels of H2.

At HL1 we have mainly the unfolding of transcendental perceptions. Here phenomena are understood in very refined manner as the relationship between both the linear (either/or) and circular (both/and) manner of interpretation.

Such understanding unfolds in real conscious terms (where opposite positive and negative polarities are still understood in a relatively separate manner). It also unfolds in an imaginary unconscious manner - indirectly expressed in conscious terms - again in both a positive and negative manner.

Whereas the real understanding of opposite polarities relates to (horizontal) external and internal poles (within each level), the imaginary understanding relates to the (vertical) whole and part poles between levels (H2 and L2).

At SL2 the focus switches more to the unfolding of transcendental concepts. Here phenomena of a more abstract universal nature are now likewise understood as the subtle interplay as between both linear and circular understanding.

Again such understanding takes place in both real (conscious) and imaginary (unconscious) terms with respect to both positive and negative polarities in the manner expressed for the previous sub-level.

Finally at SL3 we now have the growing two-way interplay of transcendental understanding with respect to both perceptions (as holistic quantities) and concepts (as holistic dimensions) with again real and imaginary interpretations with respect to positive and negative poles in operation.

H3 - Transfinite Numbers

As development proceeds through the higher stages, it steadily becomes more refined and spiritualised. Though the holistic number types associated with these stages maintain - to a degree - a discrete finite identity, their true nature becomes increasingly more infinite and continuous.

The extreme in term of this tendency now culminates with the highest level of this band (i.e. H3) where numbers apparently lose their discrete finite identity to become infinite.

In conventional mathematical terms such “infinite” quantities are referred to as transfinite numbers.

Though it was originally believed that only one class of such numbers existed, Cantor demonstrated in the latter part of the 19^th century, that there are many distinct categories of transfinite numbers and that indeed potentially an infinite number of such transfinite sets could be shown to exist.

In particular Cantor showed that the transfinite sets pertaining to all (discrete) rational numbers and that of all real numbers respectively were in fact distinct.

However from a radial perspective this finding seems quite obvious.

As we have seen two logical systems are necessarily involved in terms of a comprehensive understanding of number behaviour. Whereas the linear (1) system is directly geared to the understanding of the discrete number behaviour that typifies the rational numbers, the circular (0) system by contrast is more appropriate for the continuous holistic nature that characterises both the irrational numbers (algebraic and transcendental) and the imaginary.

Thus when we come to the understanding of the real number system - containing both discrete (finite) and continuous (infinite) elements - we properly require the interaction of two distinct logical systems i.e. the linear (1) and circular (0) respectively.

So the important finding of Cantor that the transfinite sets pertaining to the rational and real numbers respectively are of a distinct nature, really pertains to the deeper philosophical conclusion that the linear system of logic – that typifies conventional mathematical understanding - is inadequate to properly comprehend real number behaviour.

And just as in analytic quantitative terms the binary digits (1 and 0) can be used to represent a potentially infinite number of specific number quantities, likewise in holistic quantitative terms, the same two digits can be used to represent a potentially infinite number of generalised number quantity sets (as transfinite numbers).

So from this perspective it should not at all seem surprising that we can have an infinite number of transfinite number sets!

However the more important task for holistic mathematics is then to explain intuitively in an appropriate philosophical manner how this can apply to actual experience of reality.

Now in direct spiritual terms the task of H3 is to properly reconcile the two remaining (diagonal) polarities of form and emptiness.

Putting it simply full contemplative absorption in spiritual emptiness requires an extremely refined interaction with phenomena of form (so that any residual dualistic attachment is greatly eroded). 

There are two aspects to this relationship as between form and emptiness which are referred to as transcendence and immanence respectively.  

Transcendence entails the drive to experience spiritual (infinite) reality as beyond all created form; immanence then entails the reverse complementary drive to then experience - without attachment - the infinite as inherent within all created forms.

So in the light of spiritual immanence, each created (finite) form can mediate the holistic nature of the infinite in a unique manner. In this way we can have an endless variety of transfinite experiences (where the whole infinite is reflected in a uniquely distinct manner).

Thus when properly understood - Cantor’s demonstrations - point to the limitations of conventional mathematical logic that is predominantly of a unary (linear) rather than a binary (both linear and circular) nature.

However the deeper implications of this finding in direct terms have not yet been grasped, as conventional mathematics proceeds - despite its inherent paradoxes - to pursue truth in the same limited logical manner. 

Putting it in my own terms, the specialised rational development with respect to the predominant use of linear logic leads to the analytic - merely quantitative - appreciation of mathematical symbols that typifies Conventional Mathematics.

The corresponding specialised intuitive development with respect to the additional use of circular logic leads to the integral - merely qualitative - interpretation of mathematic symbols that typifies, what I refer to as, Holistic Mathematics

Finally the comprehensive interaction with respect to both logical systems leads to a coherent understanding - both quantitative and qualitative - of mathematical symbols, i.e. Radial Mathematics.

Ultimately, mathematical understanding must be radial for neither the quantitative nor qualitative aspects of mathematical relationships can be consistently explained in isolation from each other.  

We will now briefly look at the three sub-levels of H3 (though in truth there is very considerable overlap between them at this stage).

SL1 Once again this relates more to perceptual understanding (of specific phenomena). However they are now so refined through increasing dynamic interaction of opposite polarities, that they do not even seem to arise in experience.

Thus the positive direction with respect to both real (conscious) and imaginary (unconscious) experience of any phenomenon that arises is quickly negated and passes without attachment immediately from memory. Thus with appearances of form so fleetingly transparent, one remains continually absorbed in the infinite Spirit (as emptiness).

In corresponding holistic mathematical terms one no longer concentrates on the discrete finite aspects of numbers but rather on the infinite nature of the overall number system. But even here some duality remains with respect to both transcendence (emptiness beyond form) and immanence (emptiness within form) respectively. This in turn entails remaining residual problems with respect to full holistic reconciliation of both the linear (1) and circular (0) modes of understanding.

Thus in terms of quantities, a distinction is now made as between discrete quantities (which define the rational numbers) and continuous quantities (which are also included among the real).

Therefore because the linear (1) and circular (0) number systems relating to discrete and continuous understanding respectively are so distinct, we must likewise distinguish the transfinite nature of both sets of number quantities.

SL2 This now relates to the more generalised conceptual understanding of phenomena with a stronger intellectual philosophical appreciation emerging with respect to the fundamental nature of the two logical systems. 

In corresponding holistic mathematical terms it relates to a clearer understanding of the transfinite nature of dimensional quantities with again a distinction necessary with respect to both discrete and continuous understanding.

As is well known the very first of Hilbert’s famous unsolved problems related to the issue as to whether other distinct transfinite sets of numbers could be found to exist as between the rational and real.

Then in the 1960’s, Paul Cohen was able to show that this question could not be decided within the conventional logical framework of mathematics.

However from the holistic mathematical appreciation that characterises this sub-level such a finding is again quite obvious.

For the philosophical issue of whether another distinct type of transfinite number can lie between those corresponding to both linear and circular understanding respectively cannot be decided within the confines of either type of understanding (taken separately).

Thus when we approach this issue from a linear (either/or) perspective we can equally affirm or reject the hypothesis. Thus within one set of linear exams the proposition is true; within another it is false.

However when we approach the issue from the alternative (both/and) perspective we can simultaneously affirm and reject the proposition. In other words within this frame of logic - where neither linear system of axioms is distinguished - the answer is paradoxical. 

SL3 Once again the third sub-level relates to the combined interaction of both perceptual and conceptual understanding (that are both appropriate to the level - in this case H3 - in question).

So we have now the very refined interaction with respect to phenomena of form that are transparent with respect to both perceptions and concepts. This is then associated with the continual experience of Spirit (as emptiness).

In corresponding holistic mathematical terms we now have the very refined understanding with respect to the interaction of numbers both as quantities and dimensions against the background of pure intuitive awareness.

So in a sense we have now journeyed full circle. When development began we had the totally confused relationship of form and emptiness. In corresponding holistic mathematical terms this was equally associated with the confusion of Oneness (1) with Nothingness (0).

Now we have reached the stage where both Oneness (1) and Emptiness (0) can now be holistically united with each other.

Though in many ways conventional mathematics is able to demonstrate the limitations of its own logical approach (with respect to the treatment of infinite notions) because of the lack of the necessary binary logical system it is not able to properly reconcile finite and infinite notions.

However, though its great holistic significance is not yet appreciated, there does exist a remarkable mathematical identity that - when suitably interpreted - can throw light on the mathematical understanding associated with the next (Upper Middle) Band of development.

This in short is the famous Euler Identity!