1. Setting the Scene

1. Setting the Scene

Need for Radial Approach

I have commented often before on a fundamental problem of conventional mathematics where - in formal terms - the interpretation of its relationships is viewed in a significantly reduced manner.

One way of getting a better perspective on this problem is by considering the spectrum of possible development which can - for convenience - be viewed as comprising a number of levels (i.e. major stages) each of which is characterised by a distinctive type of cognitive understanding.

In my own approach I distinguish several bands e.g. lower, middle, higher and radial in this overall spectrum. Each of these bands in turn comprises three major levels.

Indeed in my most revision of this approach we have six major bands, the lower (L), lower middle (LM), higher (H), upper middle (UM), radial (R) and advanced radial (AR) respectively.

Each of these bands comprises in turn three major levels (giving eighteen levels in all).

Putting it briefly, we can for our purposes here ignore the lower stages, as cognitive development is insufficient here to support mature mathematical interpretation. 

The lower middle stages are then characterised by - what I refer to as - linear understanding.

This is based on the separation of opposite polarities and uses a clear unambiguous type of either/or logic leading to asymmetric type distinction.

For example in conventional mathematical terms a number is either positive or negative, a hypothesis is either true or false, one number quantity is unambiguously greater or less than another (assuming numbers are not equal) etc.

Though implicitly, as for example with creative work, other forms of understanding can be involved, formal conventional mathematical interpretation is characterised by the dominance of linear logic (so much so in fact is that mathematics has become synonymous with this type of interpretation).

However an utterly distinctive - and much more refined - type of understanding characterises the higher and upper middle levels of the spectrum.

Though these levels historically have been largely associated with advanced spiritual development associated with pure contemplative states, equally they entail a paradoxical circular both/and type of logic that is quite distinct from that which characterises the middle levels (i.e. linear).

When this logic is combined with linear understanding and coherently applied to mathematical symbols, it leads to a remarkable new form of mathematical appreciation, which I refer to as Holistic Mathematics.

Most of my own intellectual work has been devoted to the development of this type of Mathematics and its application to the integral interpretation of development. 

It is important to remember however that while Holistic Mathematics uses the same symbols as its conventional (analytical) counterpart, it is not directly concerned with the derivation of quantitative type relationships. Rather in strives to provide, through the use of the same mathematical symbols, a uniquely qualitative (or philosophical) type appreciation of such relationships.

So in simple terms, Analytic (Conventional) Mathematics is strongly based on linear (either/or) logic and associated with the direct quantitative interpretation of relationships.

By contrast, Holistic Mathematics additionally combines circular (both/and) logic and associated with the direct qualitative interpretation of these same relationships.

Analytic Mathematics is thereby suited as the special tool for the detailed scientific examination of the distinctive parts of reality; Holistic Mathematics - though not properly recognised - is likewise suited as the special tool for coherent overall scientific integration of these parts. 

In fact we can briefly show here how these two types of logic are characterised in holistic mathematical terms.

Linear logic relates to the line which is literally one-dimensional. So in binary digital terms this logic corresponds to the holistic interpretation of 1 (as the logic of form).

Though it will require a little more clarification later, circular logic relates to the symbol of the void in the holistic interpretation of 0 (as the logic of emptiness).

However whereas both Analytic and Holistic Mathematics relate respectively to the specialised use of these logical systems (in relative isolation from each other) they cannot yet be effectively combined with each other.

However in the spectrum of development there are even more advanced stages possible, through - what I refer to as - the radial levels. Again, historically these have been mainly referred to in a spiritual context where deep contemplative awareness is wedded to sustained active involvement in reality.

The important counterpart here in cognitive terms is the coherent interaction of both the linear and circular forms of logic in understanding.

This then leads to a new form of Radial Mathematics where both the quantitative and qualitative aspects of mathematical symbols can be properly related to each other in a new type of understanding that is both greatly productive and immensely creative.

To briefly illustrate, present day computers and other devices - which are based on the (merely) quantitative interpretation of the binary digits (1 and 0) - provide us with significant sources of new information. So this is a product of analytical interpretation.

However new integral forms of understanding based on the corresponding qualitative interpretation of these same binary digits (1 and 0), have the capacity to provide remarkable means of encoding dynamic transformation processes (of which human development is a prime example).  Indeed though still operating as a very crude prototype of this understanding, I have couched my overall interpretation of such development on the qualitative use of the binary digits! So I have found - at least to my own satisfaction - that the fundamental dynamic structures of development can be interpreted in a holistic binary digital fashion.

However all this pales in comparison with a developed radial form of mathematics. Here the binary digits (1 and 0) can be used coherently in relation to each other with both the quantitative and qualitative aspects of interpretation preserved. This leads to both extraordinary new possibilities for information processing combined with the required transformation in understanding required to interpret such processing. In other words it opens the possibilities for an extremely rich yet balanced form of scientific understanding.

My basic contention is that comprehensive mathematical understanding needs to be radial. Because of the tremendous achievements of conventional (Analytic) mathematics it is not realised how limited and reduced such understanding remains in many important respects and that vast largely unexplored territories await investigation. In particular the proper unravelling of many long standing prime number problems will require radial mathematical appreciation.

When we look presently at the nature of prime numbers we will see that in a very special way they embody, in their inherent nature, the two types of logic (linear and circular) that I have mentioned.

Thus the proper understanding - not alone solution - of many outstanding problems with respect to prime numbers (e.g. the Riemann Hypothesis) will ultimately require the radial approach.

Though I can only hope here to scratch at the surface of the great mysteries enshrouded in the Riemann Zeta Function and its famous related Hypothesis, at least I can yet attempt to bring a much needed new perspective for viewing this problem.

Especially as I would see that my main ability relates to the holistic - rather than the analytic - aspect of Mathematics, in filtering this long standing problem through its unused lens, I hope to be able to provide in philosophical terms a clearer picture of the true nature of this extremely important problem.   

Numbers as Dimensions

The limitations with respect to the conventional (analytic) approach to mathematics can be clearly seen with respect to the (reduced) manner in which numbers as dimensions are represented.

If we look at the natural number system 1, 2, 3, 4,……., these are defined with respect to the default dimension (power or exponent) of 1.

In other words we could more comprehensively outline this number system as

1¹, 2¹, 3¹, 4¹,……..

In Conventional Mathematics, when a number is initially expressed to another power, its (eventual) quantitative value is given with respect to the (default) 1^st dimension.

So 2² = 4 (i.e. 4¹)

Thus though an important qualitative change is involved here through changing the dimension of the number, this is not reflected in the reduced mathematical result.

For example we could illustrate 2² in geometrical terms as a square with side 2.

Thus the area of this square is 4 square units (rather than 4 linear units).

So once again in attempting to give a standard quantitative value to 2² we thereby reduce the qualitative nature of the dimension in merely one-dimensional terms.

And as a line is one-dimensional, I thereby refer to the logical form of Conventional (i.e. Analytic) Mathematics as linear (where number quantities are represented as lying on a straight line).

In deriving the significance of Holistic Mathematics I will now define another type of natural number system based directly on the definition of number as dimension. In this case we use “1” as the default number quantity which is successively raised to the natural number powers 1, 2, 3, 4,….

So here the natural numbers 1, 2, 3, 4,…..(as dimension) have a more comprehensive expression as

1¹, 1², 1³, 1⁴,……  

In conventional terms, this alternative number system is of little interest as the reduced quantitative value remains unchanged (i.e. 1). In other words though we are using different numbers as qualitative dimensions, the reduced quantitative value in each case does not alter. 

However we can now define a fascinating alternative circular number system by extracting the natural number roots of 1 (i.e. the 1 root, 2 roots, 3 roots, 4 roots etc).

Alternatively we could attempt to derive this system by raising 1 to the reciprocal of each of these natural numbers in turn i.e. 1^1/1, 1^1/2, 1^1/3, 1^1/4,…..

Though this procedure, as for example with 1^1/4 leads to the extraction of just one root (the 4th root) the other 3 roots in this case (1^2/4, 1^3/4 and 1^4/4) can be then obtained from this value by raising it to the power of 2, 3 and 4 respectively.

All these roots will then lie as points on the unit circle (drawn in the complex plane).

So for example the four roots of unity i.e. 1, - 1, i and – i lie as four equidistant points on this unit circle.

In like manner if we took the hundred roots of unity they would lie as 100 equidistant points on the same unit circle.

So by treating the dimensions (powers) in this fragmented manner as reciprocals of their corresponding natural numbers we define a fascinating new circular number system.

Admittedly, the existence of this quantitative number system is indeed recognised in conventional terms (though its true importance remains largely hidden).

Crucially however, the identity of Holistic Mathematics comes from the realisation that these circular numbers (derived indirectly as quantities) have - in direct terms - an extremely important qualitative (philosophical) role in describing the fundamental nature of logical systems of interpretation with far reaching implications for the interpretation of reality. 

Just try and reflect for a moment on the significance of this statement! Though the logical system that informs standard mathematical understanding is linear (one-dimensional), potentially an infinite number of possible alternative logical systems exist, corresponding to numbers as dimensions (with extremely important applications to reality).

In other words associated with each number as dimension (power) is a unique logical system which - when applied to mathematical symbols - defines a distinctive metaparadigm applying to the understanding of all its symbols and relationships.

So there is not just one metaparadigm - based on the logical system conforming to the 1^st dimension - for valid mathematical understanding (as currently believed) but potentially an infinite range.

Because of the importance of this statement, I will briefly illustrate it with respect to the dimensional number 2. In my writings this informs the intellectual understanding that characterises H1, as the first level of the higher band (often referred to as the psychic/subtle realm).

The indirect quantitative interpretation of the fragmented dimension of 2 (i.e. as the reciprocal of 2) is – 1. In other words when we raise 1 to the power of ½ (i.e. obtain the square root of 1) in quantitative either/or logic, the answer is – 1. Then we obtain this with the other root 1^2/2 (= 1), the two roots of unity are + 1 and – 1 respectively.

When we move to the direct qualitative interpretation as the holistic (whole) dimension of 2 we now switch to both/and logic so that answer is + 1 and – 1 simultaneously.

To illustrate what this two dimensional interpretation actually means, I will use the often repeated example of directions on a straight road.

For example when I point in one direction along the road and designate it as “up” and then move in that direction, right and left turns have a clear unambiguous meaning.

When I now move in the opposite direction “down” the road, again right and left turns have an unambiguous meaning. So in terms of a partial polar reference frame (i.e. where “up” and “down” are considered in isolation from each other), right and left turns along the road have an unambiguous meaning. In other words a turn is either right or left.

However if we now consider both reference frames (“up” and “down”) as interdependent (in relation to each other), right and left have a paradoxical meaning. For what is right in terms of the “up” direction is left in terms of the “down”; likewise what is left in terms of “up” is right in terms of “down”.

Thus when the reference frames are considered simultaneously right = left and left = right.

The significance of this illustration lies in the fact that actual experience is necessarily comprised of opposite polarities in dynamic relationship with each other. For example all manifest phenomena involve both internal (subjective) and external (objective) aspects. This equally applies to all mathematical experience where in experiential terms, we can never divorce the subjective mental constructs necessary for understanding from the objective symbols used. 

Conventional mathematical interpretation thereby entails the freezing of both poles (internal and external) thus enabling unambiguous either/or interpretation (within isolated frames).

So we can interpret a mathematical result in this sense (a) as the unambiguous relationship between external (objective) symbols or (b) as the equally unambiguous relationship as between the internal (subjective) constructs used. Thus, from this static perspective, the results from both frames seemingly confirm each other in identical fashion.

However when we allow for the actual dynamic interaction as between both poles - which inevitably is true of actual experience - then a degree of paradox is entailed with respect to any dualistic finding (when considered from a holistic integral perspective). Then when we approach the dynamic limit where the interchange as between opposite polarities approaches simultaneity (as with pure spiritual contemplation) the utterly paradoxical nature of all dualistic findings becomes clearly evident. In other words opposite phenomenal polarities then cancel out in the experience of nondual spiritual emptiness.

This clearly applies also to mathematical experience when seen from the refined intellectual perspective of a spiritually contemplative worldview. In other words

+ 1 and – 1, as inherent in both the positing and negating of phenomenal form, comprise a dynamic unity resulting in nothingness or spiritual emptiness (in this holistic sense). In other words the successful dynamic negation of form (as holistic oneness), which requires a radical spirit of non-attachment with respect to phenomena, results in the spiritual experience of emptiness (as holistic nothingness). Alternatively we could equally say where opposite polarities are properly balanced in experience, their rigid phenomenal manifestations are eroded resulting in a purely intuitive spiritual experience.   

Raising mathematics to the level of contemplative experience was once the admirable goal of the Pythagoreans. I would greatly support this goal but simply add that it requires new methods of logical interpretation to properly accommodate it. 

So if we are to reflect mathematical experience through the qualitative understanding associated with the holistic dimension 2, it requires a new refined type of paradoxical both/and logic that is inherently circular in nature.

Broadly this requires in terms of mathematical interpretation that external (objective) and internal (subjective) aspects of experience can no longer be considered in isolation from each other but rather in dynamic terms (where frames are interdependent and continually influence each other).

Properly understand however two-dimensional understanding combines both rational and intuitive elements serving as a necessary bridge as between linear and circular understanding. Thus in appreciating circular paradox (where opposite reference frames are viewed as interdependent we initially view them in a partial linear (i.e. independent) manner.

Higher dimensional numbers (> 2) likewise entail a mix of linear (rational) and circular (intuitive) understanding. However through greater refinement, the rational becomes less discrete until ultimately it is indistinguishable from the continuous intuitive element. In spiritual terms this state of pure contemplation can be equated with the identity of form and emptiness (where neither circular nor linear aspects separately phenomenally exist).

Thus in qualitative holistic terms, the higher dimensional numbers relate to dynamic logical structures. These coherently map out in a scientific integral manner the ever more refined nature of the interaction of reason and intuition that characterise the higher levels of understanding (ultimately attaining pure spiritual contemplation of reality).

Indeed in this holistic mathematical approach, such levels of understanding are inseparable from numbers as dimensions (when given their appropriate qualitative interpretation).

And once again the key to obtaining these structures is through obtaining the corresponding roots of 1 (as reciprocals of the corresponding whole numbers).

So for example to obtain the more refined dynamic logical structure of 4 (as qualitative dimension) we extract the four roots of unity i.e. + 1, - 1, + i and – i.

So in linear terms we now have opposite poles in both real and imaginary terms.

The key to unravelling the corresponding holistic meaning of this result is the recognition that all phenomenal experience has both conscious (real) and unconscious (imaginary) aspects. So four dimensional qualitative interpretation of reality entails the ability to see all reality as governed by the interaction of complementary polar reference frames (with both real and imaginary aspects).

In my own work on the stages of development, I have concentrated mainly on the logical intellectual systems that are properly associated with each of the higher levels corresponding to H1 (subtle), H2 (causal) and H3 (nondual). These are associated in turn with the qualitative interpretation of the dimensional numbers 2, 4 and 8 respectively. In general I would consider that these - at a minimum - are necessary for a satisfactory scientific integral interpretation of reality.

Though I have not yet penetrated in depth other dimensional interpretations, I would associate the other even number dimensions with the first of the radial levels (R1).

The second radial level R2, which is especially relevant for comprehensive appreciation of the Riemann Zeta Function (and Hypothesis), would then be associated with a special case of both real and imaginary dimensions and finally R3 with the full range of all dimensional interpretations.

However to sum up this stage I will once again reiterate this crucial point.

Associated with each number (as dimension) is a unique qualitative interpretation that leads to an utterly distinct form of logical interpretation of reality.

The structural form of each logical system is related to the corresponding quantitative roots of unity (where these numbers are used). However whereas the quantitative interpretation results in a circular number system based on either/or logic the corresponding holistic interpretation is based on both/and logic.

So Holistic Mathematics relates directly to qualitative - as opposed to quantitative - interpretation of the circular number system. (However - as we have seen - in dynamic interactive terms, it is always initially associated with the linear quantitative interpretation!)

The task of Radial Mathematics is then to coherently fuse both analytic (quantitative) and holistic (qualitative) understanding in a new creative synthesis.

Though in a certain sense, comprehensive interpretation of any mathematical problem ultimately requires a radial approach, it is especially relevant for the understanding of important unsolved problems such as the Riemann Hypothesis.

Relationship of Real and Imaginary

Though imaginary numbers are of relatively recent origin in conventional mathematics, they are now fully accepted as an integral component of the overall number system leading to considerable advances with respect to the understanding of many areas.

However an extremely important philosophical problem exists (which has remained greatly hidden). This relates to the fact that imaginary numbers - though now incorporated within the linear either/or logically approach - are really the indirect expression of the alternative circular logical system. Thus to properly understand the nature - and indeed use - of imaginary numbers we must incorporate circular logic in mathematical understanding.

Using psychological terminology, formal mathematical presentation is based on the merely conscious (rational) interpretation of its symbols and relationships.

However in truth, actual mathematical understanding represents a dynamic interactive experience involving the relationship of both conscious (rational) and unconscious (intuitive) aspects.

What is “real” in mathematical terms corresponds to what is rationally verifiable in corresponding to our conscious experience of reality. Thus it is easy for example to form a concept of what 1, 2 or 3 means because we readily associate these numbers with everyday experience!

However what is “imaginary” in mathematical terms as the square root of – 1 seems distinctly an abstraction with no correspondence in everyday living.  

However properly understood, the imaginary notion in psychological terms relates to the role of the unconscious (projected into experience in an indirect conscious manner).

Again in holistic mathematical terms, 1 is inherent in all form. To posit (+) is to make conscious; to negate in a dynamic manner (-) is then to render unconscious (what was formerly conscious).

Now this dynamic negation process leads to a fusion of both positive and negative polarities (which is two-dimensional) Thus for the unconscious to enter conscious experience (indirectly through projection) it must be expressed in a reduced linear (one-dimensional) manner. So this entails that we obtain the square root of the negated form. So in a precise holistic mathematical manner we can see that the imaginary notion relates to the role of the unconscious (as it indirectly manifests itself in all experience).

Thus when seen in this holistic mathematical sense, all experience is properly complex (with both real and imaginary aspects). However, once again standard mathematical interpretation is qualitatively of a reduced nature. In other words though the existence - in indirect quantitative terms - of imaginary numbers is indeed recognised, the actual metaparadigm used for qualitative interpretation is solely real (i.e. based on conscious rational interpretation).

So standard (Analytic) Mathematics makes use of both real and imaginary quantities within a reduced - merely real - qualitative manner of logical interpretation (i.e. the metaparadigm corresponding to 1 as dimensional number).

Holistic Mathematics by contrast makes use of both real and imaginary qualities within a reduced manner of quantitative interpretation. In other words Holistic Mathematics, though potentially of great use in qualitative (philosophical) terms is not directly geared to obtaining quantitative results!

Radial Mathematics however is equipped to make full use of both real and imaginary quantities that are balanced with corresponding real and imaginary qualitative interpretations. Though it is looking a good deal forward in our treatment this point is of special relevance when interpreting the Riemann Zeta function. Not alone is it necessary for proper understanding, to interpret the use of the real and imaginary number quantities used, but equally to provide an appropriate qualitative interpretation (combining the interaction of both the linear and circular logical systems). 

   

Now the next crucial point - which fits in well with Jungian psychology - is that in actual experience the relationship as between object phenomena and dimensions is real and circular (i.e. conscious and unconscious) with respect to each other.

Thus insofar as we are directly aware (i.e. conscious) of an object phenomenon as real, its background dimension remains unconscious (as imaginary); then in turn when the dimension becomes conscious (as real), the object now remains to that extent hidden and unconscious (as imaginary).

So, when we allow for both conscious and unconscious interaction, actual experience - when appropriately interpreted in qualitative holistic mathematical terms - is always complex (with real and imaginary aspects). 

The same relationship applies between numbers as quantities and numbers as dimensions. When the number quantity is linear the corresponding dimension (in qualitative terms) is of a circular nature. As we have already seen, we indirectly can translate dimensions in quantitative terms through raising 1 to the reciprocal of the dimensional number in question. Thus when we consider for example 1⁴, its indirect circular quantitative expression is given by extracting the fourth root of 1, i.e. 1^1/4  = i . (The other three roots -1, -i and 1 can then be obtained through raising 1 to the power of 2/4, 3/4 and 4/4 respectively). These four roots will all then lie as equidistant points on the circle of unit radius (in the complex plane).

The direct qualitative interpretation then of 4 (as dimension) relates to the holistic manner by which these four extreme values are related as complementary opposites (using paradoxical circular logic). So the crucial point is that the value 4, which as a quantity would lie on the real straight line (and thereby be linear) when considered as dimension is now - relatively - qualitative and circular (corresponding to the four equidistant points on the unit circle with a holistic logical interpretation).

Now i does not lie of course on the real straight line. However it does lie as a point on the unit circle. Therefore when ever i is used as a dimension, in conjunction with the default value of 1 (as quantity), the result will convert back to a real quantitative value.

In particular therefore whereas 1^1/4  = i (a circular number i.e. that lies on the unit circle in the complex plane ), 1^(1/4)i   = .207879… (a linear number i.e. that lies on the real number line).

In fact 1^(1/4)i  is the well known case of the value of  i ⁱ . However the philosophical reason why this numerically has a real quantitative value can be satisfactorily  explained in holistic mathematical terms i.e. where the inherent relationship between number as quantity and number as dimension - in relative terms - is properly linear  as to circular.

Now of course when we raise non-unitary real quantities to imaginary dimensions, (as a special limiting case of complex numbers to complex dimensions), a hybrid mix of both linear and circular logic is involved resulting in both real and imaginary quantities, which in radial terms will be balanced qualitatively with real (analytic) and imaginary (holistic) interpretation.  Again this finding is crucial when we come to grapple with the Riemann Zeta Function.

Thus, because the imaginary notion properly belongs to a distinctive logical system (i.e. circular) its behaviour when used as a dimensional value is utterly distinct from that of a real dimension. Again this really points to the need for a comprehensive radial approach so as to properly interpret complex number behaviour in both quantitative and qualitative terms.

To properly grasp this point requires looking more closely at what is perhaps the most mysterious - yet most important - identity in Mathematics.

This of course is the Euler Identity commonly represented as

e^pi = - 1.

However I have argued (from a holistic mathematical perspective) that a more fundamental version of this relationship arises through squaring both sides.

So we then have

e^2pi =  1

Now this embodies in a remarkable way the relationship between the two (extreme) logical approaches that I have mentioned i.e. linear and circular.

As

e^2pi =  1, thereby because e⁰ = 1, then in a certain sense 2pi = 0.

Now when we consider the circumference of a circle the formula (from a linear logical perspective is 2pr (where r is the radius). Therefore in the case of the unit circle (where r =1) then the circumference is 2p.

However the limitation here is that we are using a linear method of interpretation to relate the notions of line and circular circumference.

So if we are to switch to a truly circular notion of interpretation, we use the imaginary rather than the real notion of the radius. So the circumference is now 2pi. In other words both line and circle have now vanished as it were to be represented by the same non-dimensional point. The importance of this is that we thereby get the reconciliation of both the linear and circular methods of interpretation.

We can attempt to demonstrate this logical interpretation of the imaginary circle as the complete cancellation of movement in either direction. In other words if I stand at a point and attempt to move forward but immediately have the movement cancelled by an equal opposite movement, I remain at the same point.

And this is what happens in the case of the mystical circle (much used in contemplative literature). Its symbol is O which literally represents the holistic void (that is represented as by the slightly modified symbol of 0).

However the important fact is that even though we are not dealing directly with spiritual literature here, the most important identity - perhaps in all mathematics - pertains to the same great mystery.

So though we can represent from a linear perspective the value of any number raised to the power of 0 = 1, the unique feature of the fundamental Euler Identity is that we have in the dimension the alternative circular interpretation of 0 (i.e. as 2pi).

And uniquely e is the only number that when raised to 0 (with both linear and circular interpretations) = 1. So when understood in this way, e serves a unique role in terms of providing the interface as between the linear and circular logical systems.

Now - though it is again moving ahead in our story - inherent in the very nature of a prime number is a fascinating interaction as between the extreme versions of both linear and circular logic.

Not surprisingly therefore the appropriate use of e (for example in natural log transformations) plays an extremely important role in the interpretation of prime number behaviour.

Thus from a holistic mathematical perspective, I would strongly contend that considerable confusion exists in contemporary mathematical usage (especially where complex dimensions are employed) from results pertaining to what are inherently distinctive logical systems. Again this particularly applies to interpretation of the Riemann Zeta Function which is a veritable minefield in terms of this kind of confusion.

Indeed remarkably I believe a satisfactory holistic solution of the Riemann Hypothesis can be provided through unravelling such confusion, where it can then be understood as a self obvious axiom intrinsic to all prime number behaviour.

In other words the Riemann Hypothesis is so fundamental in terms of the relationship as between the linear and circular modes of interpretation (inherent in the very nature of prime numbers) that it may well be the case that a satisfactory proof - in conventional reduced analytic terms - is not even possible.

Before we leave this section, I wish to demonstrate once more this failure in conventional mathematical interpretation to recognise the qualitative distinct nature (i.e. in terms of the form of logical interpretation) of the various numbers as dimension.

So is standard terms  1¹ = 1² = 1³ = 1⁴…………= 1ⁿ    = 1.

Here the qualitative dimension to which the number 1 is raised (i.e. 1, 2, 3, 4,….n) in each case does not affect the (reduced) quantitative interpretation of the result and is effectively ignored.

However the opposite problem exists when we now raise 1 successively to imaginary powers or dimensions e.g.

1ⁱ, 1²ⁱ, 1³ⁱ ,1⁴ⁱ ,…… 1ⁿⁱ

As we have seen because

e^2pi =  1

Then we raise both sides to the power of i we get

e^-2p =  1ⁱ

Therefore 1ⁱ  = .0018644..

However in conventional interpretation this represents just one possible value

Because again e^2pi =  1, then we can keep multiplying by e^2pi while maintaining the same value of one.

Thus e^2pi = e^4pi = e^6pi = ……. e^2kpi   = 1 where k = 1, 2, 3, 4,…..

Therefore when we raise each of these expressions to the power of i we have

e^-2p = e^-4p = e^-6p = ……. e^-2kp   = 1ⁱ where k = 1, 2, 3, 4,…..

Therefore by this logic, associated with 1ⁱ (i.e. e^2pi) is a supposedly infinite set of quantitative values for 1ⁱ of which 1ⁱ  = .0018644…. is just the first (principal) value.

For example the next value corresponding to k = 2 is .0000034873….

However this again represents the failure to properly recognise that each quantitative value is in fact associated with a unique imaginary dimension of 1.

So e^2pi = 1ⁱ  = .0018644….

However e^4pi = 1²ⁱ = .0000034873…

So just as I earlier demonstrated how each real natural number dimension of 1 is associated with a holistic qualitative interpretation (i.e. corresponding to a unique logical system), in reverse manner I have now demonstrated how each imaginary natural number dimension of 1 is likewise associated with a unique (analytic) quantitative result.

So implicit within the treatment of standard mathematics are results pertaining to two distinct logical systems (which arise when 1 is raised to real and imaginary exponents respectively).

However a proper explicit treatment of such behaviour requires the full incorporation of these two logical systems in both quantitative and qualitative terms. And just as in analytic binary terms, a potentially infinite set of numbers can be uniquely represented through the use of the two digits 1 and 0, likewise in corresponding holistic binary terms a potentially infinite set of holistic number dimensions (representing distinctive logical metasystems) can be represented through the same two digits.

This would then enable the derivation of consistent numerical results in complex number terms combined with appropriate philosophical interpretation. And in a very special way this intimately applies to interpretation of the Riemann Zeta Function (and its associated Riemann Hypothesis) where satisfactory interpretation requires an appropriate satisfactory interface of both quantitative and qualitative number behaviour.

In other words satisfactory explanation here is as much (if not more) of a philosophical as of a (conventional) mathematical nature.