5. Circular Dimensions

5. Circular Dimensions

Upper Middle Band

In my approach the Upper Middle spans the higher and radial Bands.

In one respect it represents the specialisation of spiritually intuitive circular type understanding (which ultimately is formless). However equally it represents the turning point in understanding, whereby one prepares to coordinate in mature fashion this circular with the linear mode already developed.

We have seen how - using holistic mathematical language - development commences from a confused binary position where form (1) is directly confused with emptiness (0).

The emphasis then in earlier development over the Lower, which culminates in the specialised linear understanding of the Lower Middle Band, is on the differentiation of conscious understanding. In effect this entails the reduction of emptiness (0) i.e. unconscious spiritual modes of understanding to what is conscious as form (1).

And it is important to stress once again that conventional mathematics - which in my terminology refers to Analytic Mathematics with a well-defined quantitative bias - is strongly based on this type of understanding.  

Then with the Higher Band, emphasis switches from linear to circular type understanding. Though ultimately such understanding is of a purely intuitive spiritual nature (as emptiness), indirectly it remains associated with ever more refined paradoxical phenomena of form.

Thus in our survey of the Higher Band we carefully traced again in holistic mathematical fashion the true structure of H1 as irrational (algebraic) of H2 as imaginary and transcendental and finally of H3 as transfinite numbers.

Though the transfinite as number sets initially appear to transcend what is merely finite and in that sense appear empty, paradoxically a finite quantitative element then arises through recognition of the existence of many such sets.

So though we search for the one infinite set as it were that transcends all natural members, we find that the infinite is in fact immanent in the finite. Due however to the limitations of conventional reason this finding can only be indirectly demonstrated through the linear logic (in conventional mathematics) which ultimately is unsatisfactory.

So experientially the contemplative drive that carries one through the higher levels of development is the search for pure spiritual transcendence as emptiness (0) beyond all form (1). However because form and emptiness are interdependent, with balanced development, this drive must eventually give way to the complementary realisation of spiritual immanence (as emptiness within all form).

So the task of reuniting the circular with linear logic now commences and this is the task of the radial bands.

However for many years, I felt that there was still something missing in my holistic mathematical approach. I could – at least in basic terms - map all the levels of the Lower, Lower Middle and Higher Bands in holistic terms to corresponding number types. However there the trail seemed to dry up as it were with only somewhat vague connections as between the remaining Bands and appropriate number types seemingly possible.

However through a closer study of the Euler Identity, I gradually came to discover that - when suitably interpreted - this in fact had the capacity to provide the precise mapping I required to explain in appropriate holistic mathematical fashion the nature of the three levels of the Upper Middle Band. Then later I discovered that the Riemann Zeta Function had a similar capacity to provide – again when appropriately interpreted - a suitable mapping for the Radial Band. However we will be confining ourselves here to the Euler Identity    

H3-L3 - Euler Identity 1

We have now come to the specialisation of the spiritual intuitive worldview. Though there is still scope here for the very refined use of cognition (of a circular paradoxical nature) it is much closer to contemplation than reason.  However this still leaves room for a very important form of mathematical appreciation which - as we shall see - is directly related to the holistic interpretation of the famous Euler Identity, generally recognized as the most remarkable in all mathematics.

I have termed this level as H3-L3 in contrast to the previous level of H3. Because understanding is now very close to pure nondual awareness, remaining discrete hierarchical distinctions as between stages lose their relevance. So here the higher is no longer separated from the complementary lower, but closely integrated with its complementary lower level in a continuous ever-present experience.

However as spiritual writers in various traditions testify, a subtle problem can still remain, whereby the transcendent drive may still be affirmed to some degree at the expense of the complementary immanent spiritual direction.

The strong motivation that carries one as it were through the arduous development of the higher stages is the great desire to experience nondual Spirit as beyond all the limitations of finite phenomena.

At first, one confuses Spirit directly with the linear manifestations of phenomena.  Later in a more refined manner, one still indirectly confuses Spirit with the circular manifestations of form. So lingering attachment remains with respect to the paradoxical symbols through which Spirit is mediated.

The next stage then is to seek that central spiritual point - as it were - that unites both linear and circular understanding. So now Spirit, though hidden, is identified with a non-dimensional point as the essence or pure potential for all created being.

However a subtle problem of attachment can still exist in that one may still affirm that point as the negation of all form thus identifying too strongly with transparent Spirit (as beyond all created phenomena).

And this is precisely the situation which defines the present level H3-L3.

In continually seeking the ever-present Sprit (that cannot be confused with manifest phenomena) one still thereby still unconsciously manifests an aversion to form.

Indeed, as I have sought to demonstrate in another context, this is well summed up in the well-known Chinese Ox-herding pictures.

These pictures are designed to represent the stages of phenomenal illusion that must be negated before ultimate unity with spiritual reality can take place.

The spiritual goal, which initially appears separate from the unenlightened ego, is represented by the ox. This ox must be found and then tamed (so that Spirit is experienced as identical with one’s true self).  

In Picture 7 - which is directly relevant here – the ox is forgotten leaving the man alone.

In other words in the realisation of Spirit, subject-object duality is transcended to such a degree that the (self-conscious) search for Spirit is no longer necessary.

However self-conscious awareness still remains due to a subtle duality, which still separates (spiritual) emptiness from the world of phenomenal form.

Put another way, in identifying ultimate meaning with the (intensive) non-dimensional point of being (where phenomenal identification ceases) one thereby to a degree remains attached to the aspect of emptiness (as opposed to form).

You might wonder what has t his to do with Mathematics!

Well the surprising answer is that it is intimately tied up with the holistic appreciation of the Euler Identity (which in turn has profound implications generally for all mathematical interpretation).

You will recall how we demonstrated progression through the higher stages in holistic mathematical fashion.

By raising a rational quantity to a rational power (dimension) we obtained an (algebraic) irrational quantity. In a corresponding qualitative fashion, understanding at H1 thereby turned from rational to irrational (in the appreciation of circular paradox) with both positive and negative directions.

Then by raising a rational (or irrational) quantity to an irrational power (or dimension) we thereby derive a transcendental quantity. Again in corresponding holistic qualitative fashion, we showed at H2 how understanding turns from (algebraic) irrational to transcendental (in appreciation of the subtle interplay of both linear and circular) with both real and imaginary directions.

The logical next step would be to ask what happens when we raise a transcendental quantity to a power (dimension) that is both imaginary and transcendental.

Well in a very unique manner the Euler Identity demonstrates such a quantitative transformation where the transcendental quantity is e and the power pi.

And the remarkable result is that when we do this i.e. raising e to pi, we get as our result the simplest of all numbers i.e. 1 (in its negative form).

So e^pi  = -1

As I have demonstrated before the more fundamental version of this Identity is given by squaring both sides as,

e^2pi  = 1

The dimension here i.e. 2pi,  is the circumference of that imaginary circle which in fact is the pure non-dimensional point (so celebrated in mystical literature).

Now e in holistic terms is deeply symbolic of highly refined balanced growth (where differentiation and integration in experience both coincide). In other words the differentiation of phenomena of form is here so dynamic and refined as to be continuous with continual absorption of Spirit.

We can see this unity of form (1) and emptiness (0) coming from such organic growth in another way

e⁰  = 1 (as any number raised to 0 is 1)

So therefore in a special sense from this context 2pi = 0

However what we are really encountering here is a decisive change in the very meaning of a dimension.

In conventional mathematics we use linear numbers to represent dimensions and then in quantitative calculations give them a merely reduced interpretation.

However here we have magically switched to the authentic circular meaning of a dimension. So though we can say in linear terms that e⁰  = 1 (like any number raised to 0), the fundamental Euler Identity - in maintaining that e^2pi  = 1 - really entails a circular meaning of 0.

However this is not all.

For the fundamental Euler Identity can then be used to directly derive the qualitative vertical number system that I mentioned earlier.

I postulated earlier that we really have two natural number systems.

The first again is the (conventional) horizontal system geared directly to quantitative interpretation (which is defined with a default dimensional quality of 1).

  

1¹, 2¹, 3¹, 4¹,………

The second is the holistic vertical system geared directly to qualitative interpretation (which is defined with a default quantity base of 1)

1¹, 1², 1³, 1⁴,………

Again, the real significance of these numbers as dimensions is that they define distinct logical metaparadigms with which to interpret the entire mathematical system.

Conventional mathematics is defined by the number 1. In other words the logical system of interpretation is qualitatively one-dimensional (i.e. linear).

However a variety of (indeed potentially infinite) other logical interpretations are possible.

For example the interpretation appropriate to H1 understanding, is two-dimensional based on the complementarity of polar opposites (internal and external). So we have two directions of understanding here (both positive and negative).

The structure of the dimensions is obtained by obtaining the corresponding roots of unity (by raising to corresponding reciprocal of the number in question).

So for example, √1 = 1^1/2 = + 1 or – 1 (Strictly 1^1/2 = – 1 and the other root is obtained as 1^2/2 = + 1).

Then the holistic logical interpretation switches to both/and where all interpretation has both positive and negative polarities (which are relative).

Now the Euler Identity leads directly to the vertical number system I postulated in the following manner.

e^2pi  = 1¹

e^2pi/2  = 1^1/2 with holistic counterpart (1² )

e^2pi/3 = 1^1/3 with holistic counterpart (1³ )

e^2pi/4 = 1^1/4 with holistic counterpart (1⁴ )

etc.

Thus I would see the main significance of the Euler Identity as relating to Holistic Mathematics, with an indirect quantitative interpretation applicable in conventional mathematical terms.

However the deeper philosophical relevance of the Euler Identity for the very foundation of mathematics is not yet appreciated.

For even in quantitative terms, it ultimately entails that the conventional manner of mathematical interpretation represents just one of a set of potentially infinite possible varieties.

And this is a problem lying at the very centre of quantitative interpretation with respect to the Riemann Zeta Function.

Indeed the ultimate importance of the Riemann Hypothesis is that it is necessary to enable a certain crucial consistency to be maintained in quantitative calculation when operating from the linear perspective.

However again the deeper philosophical explanation of why this is the case is of a qualitative mathematical nature. And I would maintain that ultimately the coherent interaction of both types of interpretation (quantitative and qualitative) in interpreting the Riemann Zeta Function is vitally necessary.

We will now sum up the sub-levels involved of H3-L3 (though in truth they are highly interdependent at this stage).

SL1 Once again this would relate more to the transcendence of phenomena (as perceptions). In corresponding mathematical terms this would relate to the quantitative aspect of experience. However in the desire to fully transcend, so that pure Spirit is no longer identified with these phenomena, one still suffers to a degree from an unconscious aversion to form.

In terms of the appropriate holistic mathematical Euler relationship it could be expressed as (e^2pi)¹

SL2 This would now relate more directly to the transcendence of concepts (as qualitative dimensions). Again however in the slightly unbalanced desire to continually transcend in pure spiritual awareness again one unconsciously suffers from an aversion to dimensional form.

In terms of the appropriate holistic mathematical relationship this could be expressed as 1 raised to the power of e^2pi

SL3 Once again this entails the dynamic interaction of both aspects. Though successful switching here between refined perceptions and concepts of form require a diminution with respect to remaining unconscious resistance, it still remains to a degree.

H3-L3, R0 – Euler 2

This is the second of the transitional levels. The rather inelegant terminology implies that it bridges both the previous level H3-L3 and the succeeding level R0.

As we have seen, though spiritual understanding - in its pure transcendent focus - has condensed as it were to a central non-dimensional point (as the pure potential for all created phenomena) a subtle problem still remain in the lingering attachment to this point. So once again in seeking pure emptiness, one unconsciously resists, as it were, equal acceptance of the world of form. If this state were to continue therefore the immense spiritual energy already attained (incubating in the unconscious) would remain trapped and unable to express itself in an appropriate phenomenal manner.

So for the rebirth with respect to the world of freely created form to occur, another negation is required (i.e. to the lingering attachment of the transcendent as opposed to the immanent expression of Spirit).

Then with both polarities of Spirit in equal balance one can purely transcend all phenomena in Spirit (as ultimate Goal) while equally allowing such Spirit to become immanent within all phenomena (as their creative Source).

The holistic mathematical implications of this experience therefore relates to the negation of the dimensional aspect of e.

So instead of   e^2pi  = 1 (fundamental Euler Identity 1)

We now have e^-2pi  = 1 (fundamental Euler Identity 2).

The value of this second formulation - in terms of the qualitative (natural) vertical number system - is that we can now establish a direct correspondence as between the Euler Identity and the negative numbers in this system.

So 

   

e^-2pi  = 1^-1

e^-2pi/2  = 1^-1/2 with holistic counterpart (1^-2 )

e^-2pi/3 = 1^-1/3 with holistic counterpart (1^-3 )

e^-2pi/4 = 1^-1/4 with holistic counterpart (1^-4 )

etc.

The significance of these negative holistic dimensions is that they relate directly to the purely intuitive – and empty - spiritual understanding (whereas the positive relate to the corresponding circular paradoxical intellectual appreciation).

Though it is already looking ahead to the Riemann Zeta Function, in purely intuitive terms, complementary opposites, as negative even roots, provide solutions for the Function  = 0. (Likewise the corresponding spiritual experience of these negative holistic dimensions is in phenomenal terms empty).

What is also implied here, is that the “higher” transcendent attachment to Spirit as emptiness leads to a corresponding attachment - though admittedly very refined - to the paradoxical phenomenal representation of such emptiness. Equally it leads to an involuntary “lower” physical attachment (e.g. erotic fantasy).

So again looking ahead to the Riemann Zeta function, we can see how “higher” and “lower” imaginary phenomena (and corresponding mathematical dimensions) are related in complementary terms. Thus an involuntary imaginary attachment (i.e. as the indirect conscious expression of what in unconscious) to what is “higher” in spiritual terms, inevitably is associated with a corresponding involuntary imaginary attachment to what is - relatively - “lower” in physical terms.

So through the negation of such imaginary attachment at this level, pure spiritual experience as emptiness (0) can emanate from the “higher” unconscious, while equally pure physical fantasy can freely emanate from the “lower” unconscious again as emptiness (0) without undue attachment.  

In other words - in qualitative mathematical terms - through such “smoothing out” of experience, primitive attachment is gradually overcome. Thus though primitive instincts must necessarily remain they are now perfectly coordinated with natural experience.

In like manner - in quantitative mathematical terms - again through such “smoothing out” the completely balanced relationship of prime to natural numbers (in the unique distribution of primes among the natural) can be obtained.

Again we will now quickly summarise the sub-levels involved.

SL1 this is associated with the negation of lingering the most refined imaginary attachment (which is transcendental i.e. as the subtle relationship of what is linear and circular) with respect to quantitative perceptions.

In terms of the appropriate holistic mathematical Euler relationship it could be expressed as (e^-2pi)¹

SL2 is then associated with the negation of corresponding refined attachment with respect to even deeper qualitative concepts (i.e. dimensions).

In terms of the appropriate holistic mathematical relationship this could be expressed as 1 raised to the power of e^-2pi

SL3 if finally associated with the growing interaction of both pure spiritual understanding with respect to all phenomenal experience (perceptual and conceptual).

So this sub-level would represent the culmination of experience as pure emptiness.

R0 – Euler 3

Just to recap! In the first of our Upper Middle Levels (H3-L3) the emphasis was on spiritual transcendence with lingering attachment to the highly refined imaginary expressions of form associated.

Then with the second level (H3-L3,R0), the emphasis was on negation of this attachment corresponding in attainment of purely empty spiritual experience.

Then with R0 - with such attachment negated - a decisive turning point takes place where Spirit can now begin to flow extensively (in a conscious manner) as well as intensively (in unconscious fashion).

So the rebirth of the world of form can now take place in a refined manner.

What this entails - in the most fundamental terms - is that both linear (1) and circular (0) understanding are now sufficiently refined specialized to operate together in a creative and productive manner.

And the combined interaction of such developed appreciation - that is both linear and circular - I refer to as radial understanding.

Indeed I often refer to this as the use of the double binary system. So from a linear perspective 1 and 0 can be used potentially in a quantitative manner to encode all information processes; however equally from a circular perspective, 1 and 0 can be potentially used in a qualitative holistic manner to encode all transformation processes.

In terms of the Euler Identity this leads to a further refinement whereby we combine the (fundamental Euler Identity 1 and 2) together.  

So e^2pi  = 1¹

and e^-2pi  = 1^-1

Therefore  e^2pi * e^-2p = e⁰ = 1¹* 1^-1 = 1⁰

In other words we have at last managed to fully unite linear form (1) with circular emptiness (0).

Also we have shown the remarkable relationship as between e and 1.

For in this relationship e⁰ = 1⁰; therefore in a sense e = 1. However what this really means is that e raised to 0 (as interpreted in linear terms) = 1 (as interpreted in circular terms).

And indeed e does represent a continuous - as opposed - to discrete version of 1.

For e = (1 + 1/n)ⁿ where n → ∞ 

So as n becomes larger 1 becomes almost entirely continuous as 1 + 1/n → 1.

Therefore this value → 1. is now continually compounded.

 

From another perspective we now have fully completed the integral with respect to the vertical number system (with 0 the final number to be inserted).

As we have seen in the middle levels, dimensions are used in a merely reduced linear fashion (as quantitative measurement).

Then in the higher level a continuous element steadily grows as they proceed from rational to (algebraic) irrational, imaginary, transcendental. Then with the Euler Identity the dimension approaches a mere dimensional point.

Then finally even that point disappears and we have the purely empty spiritual notion of dimension.

So 1⁰ symbolises in holistic mathematical terms the specialised attainment of the linear aspect of understanding as 1 (quantitative) and the circular dimensional understanding as 0 (qualitative) respectively.

So analytic and holistic mathematics can now be combined in the commencement of radial mathematical appreciation.

Now we will again briefly summarise the sub-levels

SL1 relates to the combined purely transcendent and immanent expression of perceptions as their goal and source (where phenomena momentarily are negated unconsciously and are momentarily posited extensively in experience).

Put another way one is now in a position to affirm perceptions intuitively and rationally in a balanced manner in experience.

SL2 relates to the similar combined expression of concepts as their goal and source where again they momentarily arise and disappear in experience.

So the balanced appreciation likewise of concepts (in a refined interactive manner combining reason and intuition) can take place.

SL3 relates to the growing interaction of both perceptions and concepts (though still of a momentary fleeting nature in experience).