Mathematical Revolution

We have demonstrated how numbers can be viewed in both an external and an internal manner which in dynamic terms are complementary.

So once again from an external (analytic) perspective, the relationship between the primes and the natural numbers seems one-way, with all natural numbers (except 1) resulting from the unique combination of prime factors in cardinal terms.

Likewise from an internal (holistic) perspective, the relationship between the primes and natural numbers again appear one way - in a reverse manner – with every prime representing a unique combination of natural number members (again except 1) in ordinal terms.

So when one views the two-way relationship of the primes to the natural numbers (and the natural numbers to the primes) it becomes increasingly apparent that ultimately they are fully interdependent with each other.

So the ultimate nature of the primes – or more correctly this two-way relationship of both the primes and natural numbers - is shrouded in deep paradox, which is just an indirect rational manner of saying that it is utterly mysterious.


We therefore have only the appearance of some deterministic relationship as between both (i.e. the primes and the natural numbers) when we attempt to separate to a degree the external (Type 1) analytic from the corresponding internal (Type 2) holistic aspect of understanding.

So from the Type 1 perspective, we have the Riemann Zeta Function and the famed non-trivial zeros (arising as solutions for complex values of s) all seeming to line up on a vertical imaginary line through ½.

Now when we attempt to look at this in isolation it again gives the appearance of some definite unambiguous factor underlying the relationship of the primes to the natural numbers in quantitative terms. And hence the continuing obsession in the Mathematics profession to prove the Riemann Hypothesis!

Then when we look at this problem in relative isolation from the Type 2 perspective, we again have the appearance of an underlying factor explaining this relationship (in a complementary qualitative manner).

However when we realise that true understanding requires the incorporation of both the Type 1 and Type 2 approaches (in – what I refer to as – a Type 3 manner) then it becomes ever more apparent that the quantitative aspect of such explanation entails the qualitative and in like manner the qualitative aspect entails the quantitative!

In other words from the Type 3 comprehensive perspective, both the quantitative (analytic) and qualitative (holistic) aspects of this key relationship - connecting the primes with the natural numbers – are mutually contained in each other in a manner that is ultimately totally ineffable.


The implications of what I am saying here – when properly grasped – could not be more devastating for the conventional approach to Mathematics, for this applies not just to interpretation of the Riemann Hypothesis but intimately to every mathematical symbol and relationship!

So putting it simply, we have attempted to build Mathematics on the illusion that symbols and relationships can be (formally) given a mere quantitative meaning.
However in truth all such symbols and relationships possess an equally important qualitative (holistic) as well as quantitative (analytic) meaning.

So in the terms that I employ, Mathematics can be given a Type 2 – as well as Type 1 – interpretation for every symbol and relationship.


Now we can indeed attempt to obtain specialised knowledge with respect to either aspect (in relative isolation).

However a complete comprehensive mathematical appreciation ultimately requires the growing interaction of both types of understanding (Type 1 and Type 2) in a mutually interdependent fashion (Type 3).

Now contrast this vision with the present state of Mathematics!

Because there is no formal recognition of a Type 2 holistic aspect, equally there can be no recognition of a Type 3 (which entails the appreciation of both Type 1 and Type 2 as ultimately fully complementary).

What is even worse is that Conventional Mathematics, insofar as it deals with the quantitative aspect, is thereby rooted in an absolute - rather than relative – type appreciation that blocks access to the other aspects.

So we can have a closed quantitative system ìn Type 1 terms of an absolute nature, or an open system that is defined in a relative manner. And once again, as it stands, Conventional Mathematics represents the extreme specialisation of the former system.

This is why I have little faith that the necessary revolution that is now so crucially needed can emerge from within the Mathematics profession. However, in fairness there will always remain a small group, who can perhaps remain open to questioning basic assumptions.


When one redefines the Riemann Zeta Function in a Type 3 manner, the true nature of the Function and its associated Riemann Hypothesis become immediately apparent.

It also demonstrates emphatically why Conventional Mathematics is so unsuited to this task!

As I have stated, every mathematical symbol and relationship can be given both a Type 1 (analytic) and Type 2 (holistic) explanation.

So for example from the Type 1 perspective, the one value where the Riemann Zeta Function remains undefined in quantitative terms occurs where s (representing the power or dimension of the Function) = 1. The Function can thereby be only defined in quantitative terms for all other values (except 1).

Likewise from the Type 2 perspective, the one value where the Riemann Zeta Function remains undefined in qualitative terms occurs where s = 1.

In this context s = 1, refers to the linear i.e. 1-dimensional method of interpretation that formally characterises Conventional Mathematics.
So the Riemann Zeta Function remains uniquely undefined from a conventional mathematical perspective!


What this precisely entails becomes clearer when one looks at the nature of other values.

Now again in Type 1 terms we cannot give a linear interpretation to values of the Function for s < 0. What this entails therefore is that such values represent an indirect quantitative representation of a (circular) holistic meaning (i.e. directly relating to Type 2 understanding). So from a Type 3 perspective the true significance of Riemann’s Functional Equation is that it associates values with a direct Type 1 quantitative value (for s > 1), with a direct Type 2 qualitative meaning in complementary fashion for corresponding values of s < 0 (that are indirectly represented in a linear quantitative manner). So the whole point about the Functional Equation from this perspective is that we can thereby match analytic (quantitative) with qualitative (holistic) values throughout the complex plane. Thus a key importance thereby attaches in such a formulation to the points where quantitative and qualitative type interpretations exactly coincide. Therefore the significance of the Riemann Hypothesis in this context is that it sets the condition for a coincidence of interpretations to take place. The Functional Equation associates values for ζ(s) on the RHS with values for ζ(1 – s) on the LHS. Where s > 1 on the RHS and s < 0 on the LHS, a clear distinction as between analytic and holistic meanings applies with values on the RHS directly interpreted in analytic and values on the LHS interpreted in a holistic manner respectively.

Then for values within the critical region 0 < s < 1, both analytic and holistic interpretation apply.

The condition then for the full coincidence of these values, where ζ(s)= ζ(1 – s) = 0, requires that the real part of s = .5.


So from this perspective, the famous Riemann Hypothesis is the condition required for the full coincidence of both quantitative (analytic) and qualitative (holistic) type understanding with respect to the non-trivial zeros. The non-trivial zeros represent the solutions for s to the equation which necessarily occur in pairs of the form .5 + it and .5 – it respectively!


However there is a catch here in that an unlimited number of non-trivial zeros potentially are involved, of which only a finite number can be known.

Thus strictly the truth to which the Riemann Hypothesis points (i.e. the ultimate identification of quantitative with qualitative type meaning) can only be approximated from a phenomenal perspective in a relative finite manner.

So the ultimate nature of this relationship of the primes to the natural numbers (and natural numbers to the primes) pertains in turn to the corresponding nature of quantitative to qualitative (and qualitative to quantitative) meaning, which is totally mysterious and can thereby be only approximated in a finite relative manner.

As the Riemann Hypothesis – again from this more comprehensive perspective – states the condition with respect to number, for the identification of analytic (quantitative) with holistic (qualitative) meaning, not alone can it not be solved from the conventional Type 1 perspective, but its true nature cannot be understood in this manner!


So once again the Riemann Zeta Function remains uniquely undefined for s = 1 (in both quantitative and qualitative terms).

This of course means that in thereby remains uniquely undefined in conventional mathematical terms (defined merely by its 1-dimensional manner of interpretation).

For all other values of s, e.g. in the simplest case where s = 2, a dynamic interaction arises as between two aspects of understanding which are quantitative and qualitative with respect to each other.

This type of understanding uniquely is missing from conventional appreciation based on the attempt to understand relationship in a merely absolute quantitative type manner!


So rightly understood, true interpretation of the Riemann Hypothesis should serve as the requirement for the most important revolution yet in the history of Mathematics where the qualitative aspect of interpretation is recognised as equally important to the quantitative and where both are then mutually incorporated in complementary fashion through a new dynamic interactive form of understanding intimately affecting every mathematical symbol and relationship.

So again in the context of the Riemann Hypothesis, we started with the Euler Zeta Function defined in a real quantitative manner.

This was then extended by Riemann to the complex plane (entailing both real and imaginary aspects) again in a quantitative manner.

What I am now clearly suggesting is that the necessary next evolution in understanding requires that the complex plane itself be defined in both a quantitative (analytic) and qualitative (holistic) fashion.


From a qualitative perspective the "real" relates direct to the analytic and the "imaginary" directly to the holistic aspect – indirectly expressed in a linear manner – respectively.

So, all complex notions therefore can be given both quantitative (analytic) and qualitative (holistic) interpretations. In fact, strictly speaking it is impossible to have quantitative without qualitative (or qualitative without quantitative) meaning!

When one clearly sees this, then the key mathematical issue pertains to the ultimate nature of this dynamic relationship between quantitative and qualitative aspects!

So once again the Riemann Hypothesis – which directly points to this issue – cannot be solved using conventional mathematical axioms as they are solely based on a (reduced) quantitative notion of such meaning.

Subatomic Structure of the Primes

The primes are often referred to as the atoms of the number system.
However once again, this simply reflects a reduced Type 1 (i.e. quantitative) interpretation whereby all natural numbers externally considered can be expressed as the unique product of primes in cardinal terms.

However as we have seen there is an equally important unrecognised Type 2 (i.e. qualitative) interpretation whereby each prime is necessarily comprised internally of a set of natural number members.

Therefore if we take the prime number “5” to illustrate, from an internal perspective this is necessarily composed of a 1st, 2nd, 3rd, 4th and 5th member respectively. And the very essence of ordinal – as opposed to cardinal – interpretation is that it involves qualitative type distinctions!

And because the qualitative aspect strictly relates to the (interdependent) relationship as between numbers, without such distinctions it would not be possible to order number coherently in any relevant context.


Once again because of the extreme quantitative bias of Conventional Mathematics, this key issue is avoided and thereby grossly misinterpreted.

Thereby insofar as an internal structure is recognised it will be portrayed in a merely quantitative manner.

So from this perspective 5 is internally comprised of 5 homogenous units (understood in a merely quantitative manner).

So 5 = 1 + 1 + 1 + 1 + 1.


However the problem in defining number in such a manner is that it provides no means of providing a coherent order in qualitative terms. In other words the ability to rank numbers in an ordinal fashion reflects the qualitative aspect of number and cannot be explained in a quantitative manner!


Therefore properly understood, all mathematical understanding reflects the interaction of quantitative with qualitative type notions. However in formal terms this vitally important qualitative side remains completely unrecognised in the accepted interpretation of number.

The issue as I have repeatedly stated on these blogs could not be more fundamental in that through completely ignoring this qualitative aspect of mathematical understanding, present accepted interpretation is simply not fit for purpose.


We are now reaching a stage where Mathematics is in need of the most fundamental radical overhaul in its history intimately affecting every notion.

I have seen this clearly for many years now with a conviction that has steadily increased. Therefore I suspect that there are many others in our culture ready to reach the same conclusion. Sadly however, the greatest resistance to such fundamental change is likely to be experienced from within the Mathematics profession itself. For it requires a willingness to look at mathematical notions from a completely different perspective, before the reduced and misleading nature of present interpretation becomes fully apparent.


So returning to the primes, when we look at these numbers in an internal manner, their qualitative nature (as it were) is revealed in a highly dynamic interactive manner that closely resembles the quantum nature of physical reality.

And just as we cannot hope to understand the nature of quantum physical reality with reference to Newtonian type physics, even more we cannot hope to understand the internal nature of number with respect to standard quantitative mathematical notions.


The very essence of the conventional approach is that the major poles of understanding i.e. external (objective) and internal (subjective); individual (part) and collective (whole) are abstracted from each other leading to reduced absolute type interpretation. So in effect the internal subjective is reduced to the external objective aspect; likewise in any context the whole is reduced to part notions i.e. where the whole is interpreted as merely a collection of its individual parts in a merely quantitative manner.

However to properly embrace the subatomic nature of number we need to accept as our starting point the dynamic interaction in experience of both external and internal aspects of understanding and likewise whole and part aspects. So in this new understanding both poles maintain a – relatively - independent identity, while also sharing a common interdependence.

So the notion of any truth as absolute in objective terms is surrendered here with a new appreciation that all objective results necessarily reflect a certain type of mental interpretation; equally all quantitative mathematical notions equally can be given a coherent qualitative meaning with a proper integrated understanding thereby equally combining both aspects.


So in this entry I will demonstrate again this new thinking with respect to the number “2”, which in many ways serves as the blueprint for dynamic interpretation of all other numbers.

We are well accustomed to the conventional notion of 2 as a cardinal number (defined merely in quantitative terms).

Now this actually reflects just one limited type of interpretation which is accurately defined as linear (1-dimensional).

So the very essence of such linear appreciation is that in explicit formal terms we treat the objective pole as independent of mental interpretation (thereby creating the illusion of numbers possessing an abstract identity in absolute terms).

Likewise we treat the quantitative aspect as independent of the qualitative creating the further illusion that numbers possess a mere quantitative identity.
So the nature of number in conventional terms simply reflects the 1-dimensional type of (mental) interpretation that is employed.


However when we employ a distinctly different appreciation, the very nature of number changes in line with this new understanding.

So the key characteristic of 2-dimensional interpretation is that external and internal poles are now considered as dynamically interdependent.

This leads to what is often referred to as the complementarity of opposites. So instead of an unambiguous either/or logic suited to linear understanding, we now by contrast employ a paradoxical both/and logic in circular terms.

I have used the example of a crossroads on many occasions to illustrate the subtleties associated with this new interpretation.

We can give two – relatively - independent interpretations to movement along a (vertical) road i.e. “up” or “down”.

So if we are moving up the road an encounter a crossroads then the left turn will have an unambiguous meaning. Then having gone through the crossroads, if we now define direction as “down” the road when we again encounter the crossroads, a left turn will have an unambiguous meaning. However when we simultaneously attempt to embrace both directions “up” and “down” as interdependent, we are faced with paradox as the turns at the crossroad are now necessarily right and left (and left and right) with respect to each other.

So quite simply, the logic associated with independence is unambiguous and linear, whereas the logic associated with interdependence is circular and paradoxical.


Now the huge unrecognised problem with Conventional Mathematics is that it attempts to interpret, in any relevant context, the holistic notion of interdependence in a reduced analytic manner. In other words it attempts to explain what is qualitative (i.e. interdependence) in a merely reduced quantitative manner (as befits independence).


Once again let me demonstrate this problem with respect to the conventional treatment of the primes.

Now as we have seen Conventional Mathematics is based on linear interpretation (where polar reference frames are treated as independent).

So when we fix this frame with the individual nature of primes, unambiguous quantitative results can be obtained. This is directly analogous to fixing the movement along our vertical road so as to unambiguously identify a left turn at the crossroads!

Then when the reference frame is switched to study of the collective behaviour of the primes, again unambiguous quantitative type results can be derived. This is now analogous to again unambiguously identifying a left turn at the crossroads when we approach it from the opposite direction.

However as we can see when we consider “up” and “down” simultaneously as interdependent then the turns at the crossroads must necessarily be “left” and “right” (and “right” and “left”) in relationship to each other.

In like manner when we consider both individual (part) and collective (whole) notions in relationship to each other, such prime notions of behaviour must be both quantitative and qualitative (and qualitative and quantitative) respectively.

In other words the behaviour of the primes in terms of their individual and collective identity necessarily entails the two-way interplay of analytic (quantitative) and holistic (qualitative) interpretation with respect to each other.
However quite remarkably, Conventional Mathematics allows no formal recognition for this holistic (qualitative) dimension of understanding!

Putting it bluntly therefore, the conventional approach to understanding prime behaviour is similar to the attempt to understand two intersecting turns at a crossroads as both having a left direction!

Indeed it is truly futile therefore to attempt to understand the primes merely in a quantitative manner for intrinsic to their behaviour is a qualitative (holistic) aspect that cannot be successfully reduced in a quantitative manner.


Now in mathematical terms, the complementarity of opposites simply represents the Type 2 interpretation of the number “2”.

Here we obtain the 2 roots of 1 to quantitatively express the two ordinal members of 2 in a circular fashion as + 1 and – 1 respectively.

So we posit movement up the road as unambiguous (+ 1) by implicitly negating the corresponding down direction (as – 1) . Equally, we then unambiguously posit the movement down the road ( + 1) by implicitly negating the corresponding up direction (as – 1). So crucially we now define independence in a relative – rather than – absolute manner.

Then we understand the holistic interdependence of these two directions by combining (adding) + 1 and – 1 simultaneously = 0 (which has no quantitative significance).

So the Type 2 (circular) interpretation of each prime number, representing its internal nature, combines the relative quantitative independence of each of its natural number ordinal members (indirectly expressed in a circular quantitative manner) with their overall collective holistic interdependence. And this qualitative interdependent aspect is indirectly represented as the sum of all the individual members = 0!


Such understanding requires the interplay of refined reason with intuition with the independent aspect provided directly by reason and the holistic component by intuition.

As we move on to larger numbers, this interplay of recognition of independent members becomes so seamlessly combined with their collective interdependence that they no longer even appear to arise.

So the full integration of both the quantitative and qualitative aspects leads ultimately to the experience of number in Type 2 terms as representing pure (spiritual) energy states, which complements their existence as equally representing pure energy states in physical terms.


Thus we can now perhaps see here with reference to the simplest example of 2, how both distinctive linear (analytic) and circular (holistic) interpretations can be given to every number.

Thus from the standard linear perspective, prime numbers such as 2 are viewed externally somewhat like indivisible unique atoms from which all natural numbers are quantitatively derived.

However from the new circular perspective, each prime number is viewed internally as a group whose ordinal members (except 1) are uniquely defined in natural number terms (which then can be indirectly expressed in a circular fashion).

When one appreciates this fact, then the ultimate relationship as between the primes and the natural numbers can be clearly seen as paradoxical.

Put another way – when properly understood in a bi-directional fashion - ultimately the primes and natural numbers are fully interdependent with each other i.e. mutually contained in each other in an indivisible manner.

With this realisation the true nature of the Riemann Zeta Function becomes clear as the relationship between analytic (quantitative) and holistic (qualitative) type meaning with the Riemann Hypothesis as the condition required for the mutual identity of both aspects.

Therefore from this perspective, as the Riemann Hypothesis entails the condition
for ultimate identity of both the quantitative and qualitative aspects of number, it cannot be proved or disproved from the standpoint of Conventional Mathematics (as it allows no formal recognition for the qualitative aspect).

Incredible Nature of the Zeta Zeros (25)

We can perhaps throw further light on the nature of the Type 1 non-trivial zeros with respect to calculation through the general formula for their frequency.

So for example using this formula we can calculate how many zeros would be obtained on the imaginary line up to 100!


Riemann himself suggested the formula in his famous 1859 paper on the primes as

t/2π*log(t/2π) - t/ 2π.

Now in a more complete manner it is given as:

t/2π*log(t/2π) - t/2π + O(log t).

However ignoring the last term it still gives – unlike its counterpart for the frequency of primes (t/log t) - surprisingly accurate results (even over a very limited range).

So the formula predicts 28.17 non-trivial zeros up to t = 100 (as against the correct result of 29)!


What is perhaps surprising is that this formula can be derived with reference to my holistic (Type 2) approach, with the added benefit of throwing much greater light on the true nature of these non-trivial zeros.

We start here by considering the general distribution of primes. As we have seen the spread between these primes steadily increases as we ascend the natural number scale and is approximated by log t.

Put another way the frequency of primes up to t is approximated by 1/log t.

Now in an inverse manner as the frequency of prime numbers decreases, the frequency of remaining composite numbers increases. So for example as the general frequency of primes falls to ½ of its previous level the corresponding average unbroken frequency of composite numbers roughly doubles.

However there is not an exact inverse relationship here.

For example if the average frequency of primes is 1 in 4, the average unbroken sequence of composite numbers would thereby be 4 – 1 = 3.

So therefore is we measure the average frequency of primes as 1/log t, the average unbroken sequence of composite numbers is log t – 1.


Now again the key distinction as between the primes and the composite natural numbers is that whereas the primes represent a measure of number independence (with no factors other than the prime in question and 1), the composite natural numbers represent a measure of number interdependence (where each natural number represents a unique combination of prime factors).

So the average number of primes up to t is obtained by multiplying t by the average frequency of primes i.e. t * (1/log t) = t/log t.

In inverse fashion the average frequency of composite (interdependent) numbers
= t * (log t – 1). It must be remembered in this context of interdependence that each (individual) member is counted as whole group. Thus each individual member of a group of 3 (for example) - through such interdependence - counts as 3!

However whereas independent notions properly relate to a linear, interdependent notions properly relate to a circular scale (where again each number through relationship occupies all delineated points on the circle).

So the interdependence of an unbroken group of composite numbers would thereby be represented on the circumference of the circle of unit radius.

So the circular unit of measurement here = 2π.

An unbroken group of 8 composite numbers would thereby be geometrically represented as 8 equidistant points on this unit circle.

Thus expressing interdependence between them by the total circular length 2π there would be 8 different starting points (as - relatively - independent) for connecting with all other points as representing interdependence. So our numerical measurements here necessarily reflect a reduced measurement of this dynamic relationships as between (quantitative) independence and (qualitative) interdependence respectively!


Then to convert to linear units we would divide by 2π.

So therefore the formula for average frequency for such interdependent numbers

= t/2π{log (t/2π) – 1} = t/2π*log (t/2π) – t/2π.

So we can see that the formula for general frequency of the trivial zeros represents an inverted version of the general frequency of primes so that as the frequency of primes for example halves the corresponding unbroken sequence of composite numbers roughly doubles.

Then, whereas primes relates to linear (independent), the composite numbers relate to circular (interdependent) notions.


Therefore we can clearly see here that whereas the primes represent the independent aspect of the number system, the non-trivial zeros represent the corresponding interdependent aspect.

This indeed is why the non-trivial zeros are dual to the primes (and vice versa).
In other words the primes and the non-trivial zeros in this sense represent the two extreme poles with respect to the number system.

So at one extreme we can consider each number as independent; however at the other extreme we can consider all numbers as necessarily interdependent with each other.
In between these two extremes number possesses both independent aspects (as quantitative) and interdependent aspects (as qualitative) in dynamic relationship with each other.


So conventional understanding of number represents but an extreme case where we attempt to understand its nature in an absolute independent manner (as quantitative).

The non-trivial zeros however directly point to the opposite extreme, where understanding of number is so dynamic that we can no longer separate its quantitative aspect (as independent) and its qualitative aspect (as interdependent) respectively. So we are attempting here to understand number simultaneously as both quantitative and qualitative in both analytic and holistic terms respectively!


Now we can only approximate this extreme in phenomenal terms. However this is what the non-trivial zeros represent i.e. the closest approximation in the phenomenal number realm to pure ineffable reality (i.e. where quantitative and qualitative aspects in the combined analytic and holistic appreciation of number, can no longer be distinguished from each other).

So from a physical perspective, the non-trivial zeros represent approximations to pure energy states; in complementary psycho spiritual terms they represent approximations to pure (intuitive) energy states (where reason operates seamlessly in such a refined manner with intuition as to become fully transparent).

And if we are to understand the nature of the non-trivial zeros, without gross reductionism, such highly refined understanding (seamlessly blending both rational and intuitive aspects) will be ultimately required!


And of course once again this explains why the non-trivial zeros line up neatly on an imaginary line.

Again in qualitative terms, the mathematical notion of the imaginary relates to the indirect representation of holistic circular notions in a linear manner!

So even though the non-trivial zeros are given a numerical measurement, these properly represent a (linear) representation of relative type approximations with respect to the (circular) integration of both the (quantitative) analytic and (qualitative) holistic aspects of number (where both are understood in complementary terms as ultimately identical).

Incredible Nature of the Zeta Zeros (24)

Once again we have seen that in conventional terms, the primes and natural numbers are understood as independent number entities (marked by points on a straight line).

Yet at another level we must recognise a unique form of interdependence that connects both types of number.

So from a Type 1 perspective, with respect to the overall number system, the composite natural numbers represent an interdependent relationship among primes so that each composite natural number (other than 1) can be uniquely expressed as the product of primes.

However this interdependence entails both quantitative (analytic) and qualitative (holistic) aspects that cannot be properly expressed in a real independent manner (geared merely to quantitative type understanding).

So therefore when we say that for example that the number 6 is uniquely expressed in cardinal terms as the product of 2 prime numbers i.e. 2 * 3 - as well as the recognised quantitative transformation - a qualitative holistic dimension is involved (which is intimately related to the fundamental nature of the multiplication process).

This indeed indicates the key distinction as between addition and multiplication. In other words, though multiplication can be interpreted in quantitative terms as addition with for example 2 * 3 = 2 + 2 + 2, this essentially reduces its qualitative aspect in a merely quantitative manner!

What is not clearly recognised is that an alternative Type 2 perspective exists, whereby each number (internally) can be viewed as defined by a circle of (ordinal) natural number members.
Once again the primes are central to this approach with this circle of numbers (other than 1) uniquely defined for prime integers (now understanding as representing dimensional rather than base numbers).

So again to illustrate in the expression 1^3, 1 here represents a base number quantity and 3 a dimensional number (which - relatively - is of a qualitative nature).


So again for example the ordinal members (1st, 2nd and 3rd) of the number 3 are defined in quantitative terms by its 3 roots of unity, i.e. 1, – ½ +.866i and – ½ –.866i, respectively. And apart from 1, which is always one of the roots, the remaining roots are uniquely defined for all prime number integers.

Now these 3 roots in quantitative terms bear a complementary relationship with the corresponding ordinal members of 3, considered directly in qualitative terms.

This simply implies that rather than considering members as separate i.e. independent we consider them in holistic terms as interdependent.

Indirectly in quantitative terms, this interdependence can be demonstrated by adding the three roots which always (apart from 1 representing the number dimension) = 0.
So this simply indicates that pure interdependence has no meaning in quantitative terms!

What it also clearly indicates is that the very notion of interdependence can be given no coherent meaning from the conventional mathematical perspective, which is formally defined in a linear (i.e. 1-dimensional) manner.

Thus once again the conventional mathematical approach, that is qualitatively 1-dimensional in nature in formal terms, attempts to understand relationships in a merely quantitative manner.
Therefore, from this perspective the primes and natural numbers are misleadingly viewed in absolute terms as (solely) representing number quantities.


Quite simply the relationship between the primes and the natural numbers (and the natural numbers and the primes) cannot be coherently understood within the conventional mathematical perspective and no amount of abstraction or sophistication with respect to increasingly complex techniques will ever change this situation!

Indeed worse than that the very attempt to approach the key relationship in this manner will only inevitably lead even further away from true appreciation.

So in a nutshell before understanding the two-way relationship of the primes and natural numbers, the unrecognised qualitative dimension of Mathematics must be incorporated in interpretation as a fully equal complementary partner.
And this leads to an inherently dynamic interpretation of all mathematical symbols as representing the interaction of both their quantitative and qualitative aspects.

So once again the composite natural number 6 can be equally expressed as the product of two primes i.e. 2 * 3. However this can be given Type 1 (quantitative) and Type 2 (qualitative) interpretations (which are complementary).

In Type 3 terms we then combine both aspects of interpretation in a simultaneous manner.

So in Type 1 terms we treat number as representing independent entities (where the quantitative pole of understanding is clearly separated from the qualitative).

This approach in qualitative terms is 1-dimensional so that 2 = 2^1 and 3 =3^1

And the resulting product = (2 * 3)^1 = 6^1.

In Type 2 terms, both the quantitative and qualitative aspects of interpretation are considered as dynamically interdependent.

So each number - in reverse manner - relates to an integer as representing a dimensional power (or exponent) defined with respect to a default base quantity of 1.

So 2 = 1^2 and 3 = 1^3 from this perspective.

So both 2 and 3 in this context relate to the (internal) ordinal composition of the whole numbers 2 and 3 respectively

So 2 has a 1st and 2nd member while 3 has a 1st, 2nd and 3rd member.

A unique structure is associated with each dimensional number when prime, which quantitatively is obtained through obtaining its corresponding, roots.

So the unique structure of 2 (through the corresponding 2 roots of unity) in quantitative terms is given by 1 and – 1 and the corresponding unique structure of 3 (through the corresponding 3 roots of unity) in quantitative terms is given by 1, – ½ +.866i and – ½ –.866i.

Then the corresponding qualitative interpretation entails the combination of these ordinal components (represented in a circular manner) as interdependent.

So therefore the dynamic understanding of the numbers “2” and “3” from a Type 2 perspective entails the ability to recognise in both cases their separate ordinal elements in a relatively independent quantitative manner (represented by circular numbers) while equally combining these members in a qualitative holistic manner as interdependent.


In the dynamics of understanding this explicitly requires both rational (analytic) and intuitive (holistic) elements of interpretation!

So in Type 2 terms 6 as a composite dimension arises from the product 2 and 3

i.e. 1^(2 * 3) = 1^6.

So this would then generate six ordinal members (represented in circular quantitative terms) as 1, – ½ +.866i, – ½ –.866i, – 1, ½ –.866i and ½ –.866i in a relatively independent manner combined with the holistic qualitative interdependence of all members as interdependent.

Now as the prime numbers representing dimensions become very large, the dynamic interactive nature of understanding must necessarily increase so that ultimately the independent nature of each member quantitatively can no longer be explicitly distinguished from the overall interdependence of all members in a qualitative manner.

Thus this opposite extreme in understanding, of the pure interdependence of number, relates to the situation where p becomes unlimited in size (where in understanding number as material form cannot be distinguished from number as representing intuitive (psycho-spiritual) energy. The corresponding physical complement of this entails that number as material form cannot be distinguished from number as representing physical energy states!


The non-trivial zeros in Type 2 terms therefore provide the ready solution for this interdependence (with respect to quantitative and qualitative aspects) for each number (as internally conceived in ordinal terms) in a manner where its rationale can be explained in an intuitively accessible manner.
In other words this very notion of interdependence in Type 2 terms is inherently based on a dynamic appreciation of the relationship as between the quantitative and qualitative aspects of number.

By contrast the (recognised) non-trivial zeros in Type 1 terms likewise provide the ready solution for this interdependence (of quantitative and qualitative) with respect to the overall number system externally considered in a cardinal manner.

However as explained in the last blog entry, appreciation of the nature of such interdependence is not directly accessible from a Type 1 quantitative perspective, which is directly based on independent quantitative notions.

Therefore it requires the holistic appreciation (arising from the Type 2 approach) to convey in an indirect manner, how the non-trivial zeros in this case relate to interdependent - rather than independent - number notions.

Now I attempted to convey the nature of such indirect appreciation in the last blog entry.

In addition it should be stated that an inevitable uncertainty attaches to the whole process. Though in principle the set of all non-trivial zeros is of an infinite nature, clearly in phenomenal terms, we can only approximate through the understanding of a finite number of this set.

Thus the very relationship of the primes to the natural numbers is thereby of a relative - rather than absolute - nature.

Likewise, the non-trivial zeros can be only approximated in value. Thus, strictly each non-trivial zero (though given an individual identity) only has meaning in the context of the holistic collection of all zeros (which remains ultimately unknowable).

Likewise the whole set only has meaning in the context of each individual zero in quantitative terms (which likewise ultimately remains unknowable).


Therefore, the Type 3 appreciation simultaneously seeks to combine appreciation of the non-trivial zeros in both Type 1 and Type 2 terms.

This leads to the paradox that the primes and natural numbers are ultimately related to each other in an absolute ineffable manner (which cannot be identified in phenomenal terms).

Thus the two-way relationship between the primes and natural numbers (arising from both Type 1 and Type 2 appreciation) strictly represents relative approximations with respect to their ultimate identity (which is ineffable).

Such ineffable reality is variously referred to in the mystical traditions as an ultimate unity (1) that is equally a void or nothingness (0).
In like manner the very number notions of 1 and 0 are already deeply implicit in the subsequent relationship of the primes to the natural numbers (and the natural numbers to the primes).

So the numbers 1 and 0, are already implicit in all phenomena (as their potential for existence). Then bridging the phenomenal world with ineffable reality, both the primes and (other) natural numbers arise (in relative time and space) as the first dynamic interaction of both the quantitative and qualitative (and internal and external) poles which subsequently govern the course of all created evolution.

Such numbers do not therefore exist as abstract entities but rather as the inherent encoding of created phenomena (in both physical and psychological terms). And the consistent two-way relationship of the primes with the natural numbers is a necessary condition for the subsequent unfolding of phenomenal evolution (with respect to both its quantitative and qualitative aspects).


So ultimately there is a great mystery to the nature of number underlying our - seemingly - most obvious intuitions.

For example, the manner in which we readily see a one-to-one correspondence as between the natural numbers in cardinal and ordinal terms (with 1 being paired with 1st, 2 with 2nd, 3 with 3rd,...) would not be possible but for the two-way interdependence of both the primes and the natural numbers.

So the most inaccessible relationship of all is thereby necessarily already implicit in the - seemingly - most simple!

Ultimately therefore, final knowledge of the nature of number (and indeed of phenomenal reality) can only be obtained through pure experience of its unfathomable mystery.

Incredible Nature of the Zeta Zeros (23)

Yesterday we explained the nature of the non-trivial zeros of the Zeta 1 Function as a set of numbers that represent the interdependence of both its quantitative and qualitative aspects in a linear manner.

Earlier we saw that the corresponding non-trivial zeros of the Zeta 2 Function is likewise a set of numbers, which represents the interdependence of both its quantitative and qualitative aspects in a circular manner (where each integer representing a dimension is associated with its own circular group of individual members).

And just as the size of these circular groups (of non-trivial zeros) steadily increases in Type 2 terms with larger integers, likewise the frequency of non-trivial zeros steadily increases as we ascend the linear number scale in Type 1 terms.


So to put it more simply, whereas the Type 2 zeros express the interdependence of the primes and natural numbers (internally) from a (circular) holistic, the Type 1 zeros express this interdependence (externally) in terms of a linear quantitative perspective.


I now wish to probe more deeply into the philosophical significance of the manner in which the non-trivial zeros are represented.

Firstly - which gives rise to the Riemann Hypothesis - all the Type 1 zeros appear as complex numbers of the form ½ + it and (½ - it) respectively.

½ has a crucial significance in this context.

It must be remembered that the non-trivial zeros relate to numbers (as dimensional powers or exponents).

Now the qualitative significance of ½ can be expressed in terms of the two roots of unity i.e. + 1 and – 1 respectively. These two points lie at opposite ends of the diameter of this unit circle in the complex plane with the midpoint at 0. However if we represent this diameter in a linear fashion (on a positive number scale) with length 1, the midpoint that divides the line will lie at ½.

So the significance of this is that the non-trivial zeros arise only in the context of the opposite polarities (internal and external) which govern all reality, remaining perfectly balanced with each other. Indeed this is required in order to fully preserve the independence of individual quantitative elements with corresponding interdependence with respect to their collective qualitative nature.

Without this equality, an unbalanced emphasis - either on the quantitative or qualitative aspects of the relationship - would arise.

So in the simplest manner, the requirement that all non-trivial zeros lie on the real line through ½, is necessary to ensure the ultimate identity of both the quantitative and qualitative aspects of number.

Once again it is futile trying to prove this in a conventional mathematical manner as this serves as a necessary prior condition for the subsequent consistent use of its very axioms.


At a deeper level, this also implies that appropriate interpretation of the nature of the non-trivial zeros requires the realisation that ultimately the objective nature of mathematical objects (such as numbers) cannot be separated from their subjective means of mental interpretation.

Now this could not be more damaging in terms of the conventional paradigm, which is based on the complete separation of objective from subjective poles leading to the reduced (and ultimately untenable) interpretation of mathematical objects existing in independent terms as abstract unchanging entities!

So we are now at the other extreme, where understanding of mathematical phenomena is so dynamic that they barely seem to appear in experience. So this experience of the pure interdependence of number requires that objective data be fully integrated with their subjective means of interpretation.

This implies a highly subtle appreciation, where rational structures interact in such a refined manner with holistic intuition that they can be seamlessly integrated with each other.


The next interesting fact is that the non-trivial zeros always have an imaginary component.

Now the importance of the imaginary notion can hardly be overstated.
Basically it relates to the means of interpreting - what is properly of a holistic circular nature - indirectly in a linear manner.

So from this perspective, the imaginary aspect relates to the unrecognised qualitative side of Mathematics.

This is obscured in current interpretation by the fact that all imaginary notions can equally be given a merely reduced quantitative interpretation.


Thus the Riemann Zeta Function for example is defined with respect to the complex plane, which necessarily entails real and imaginary number variables.

However in conventional mathematical terms, this is interpreted in a merely quantitative manner!

However, equally the complex plane can be given a qualitative definition, where it relates to the need for both quantitative (analytic) and qualitative (holistic) interpretation.

And once we recognise that all reality (physical and psychological) necessarily entails both differentiated (quantitative) and (integrated) holistic elements, then it is easy to appreciate that we necessarily live in a complex world (from a mathematical perspective). The reason again why we think that it is simply “real”, reflects a scientific paradigm that reduces the qualitative to the quantitative aspect!


However this qualitative appreciation of the imaginary notion is completely overlooked in conventional terms, with no recognition of the subtle vital dynamics that underlie the number system through the interaction of its quantitative and qualitative aspects.

In fact putting it bluntly, conventional mathematical interpretation completely misrepresents the true relationship of the primes to the natural numbers (and the natural numbers to the primes) as this relationship cannot be properly approached in a merely (reduced) quantitative manner!

So the fact that all the non-trivial zeros lie on an imaginary line (through ½), points to the fact that these zeros directly relate to holistic notions of circular type interdependence (that are then indirectly represented on an imaginary number scale).

And earlier in explaining the basic formula for calculating the average gap as between the non-trivial zeros (as we ascend this number scale) I employed this very fact!


Next, we see that the non-trivial zeros always occur in pairs.

Actually there is a very close relationship here with the nature of virtual particles in physics.

As sub-atomic reality becomes increasingly dynamic, extremely short-lived virtual particles tend to arise spontaneously as matter and anti-matter pairs (which then quickly annihilate each other turning into energy).

Virtual in this context is synonymous with imaginary, as these particles serve as but an indirect expression of the ultimate interdependent nature of matter (in what we could refer to as the holistic ground of reality).

We have a similar situation in psychological terms – especially in earliest infancy – before stable phenomenal perception emerges. So any phenomena that arise do so in a spontaneous short-lived virtual manner, as the most primitive expression of the unconscious (which is the corresponding holistic ground of all psychological phenomena).


However these occurrences are of a merely implicit nature in earliest physical and psychological life.

Thus it requires the other extreme of full maturity (where conscious and unconscious are seamlessly integrated in experience) before what is most fundamentally implicit with respect to nature, can finally be made fully explicit with respect to human understanding.

Once again complete experience of the nature of the non-trivial zeros will require a profound level of contemplative awareness (together with the most refined degree of rational understanding). And we are a long way from that yet in our evolution, though remarkably have already reached a position, whereby perhaps we can now at least glimpse the nature of its wonderful secrets.


A further intriguing factor regarding the nature of the imaginary part of these zeros is that they represent transcendental numbers.

Some 20 years ago - long before I seriously looked at the Riemann Hypothesis – I set out to show how all the structures of all various stages of psychological (and complementary physical) evolution can be precisely encoded in a holistic mathematical manner in number terms.

What I concluded at that time was that the most advanced contemplative structures (before full union in mystical contemplative awareness) would have an imaginary transcendental structure.

So it is no surprise therefore in a complementary fashion, that the most fundamental numbers (representing the interdependence of primes and natural) would also implcitly contain the same imaginary transcendental structure.

We have already seen that imaginary in this context represents the indirect linear expression of what is directly of a holistic (circular) nature. In psychological terms this refers to the indirect conscious expression of what is directly of a holistic unconscious nature.


The nature of transcendental in this qualitative context can best be appreciated with respect to the most famous transcendental number π.

Now π in quantitative terms represents the perfect relationship as between the circumference of a circle and its line diameter.

In corresponding qualitative terms π would represent the perfect relationship as between circular (holistic) and linear (analytic) type understanding.

All transcendental numbers entail in dynamic terms the relationship of circular and linear type notions.

Thus to understand in transcendental terms is to interpret relationships in neither an analytic or holistic manner (as separate) but rather as the interdependent relationship of both aspects!.

So imaginary transcendental understanding goes a step further in being able to understand a linear presentation of meaning as the indirect representation of a relationship entailing the interaction of both analytic and holistic aspects.

And this precisely is what the non-trivial zeros represent i.e. the indirect linear presentation of the relationship between the primes and the natural numbers that are understood as analytic (quantitative) and holistic (qualitative) with respect to each other.

The inclusion of ½ as the real part entails the further requirement that both the objective nature and mental interpretation of this reality be perfectly identified with each other (so that no separation exists).


So therefore we can now give a full qualitative explanation of the nature of the non-trivial zeros in this fashion!

The non-trivial zeros represent the simultaneous identity of numbers both as objects and mental interpretation, that express in an indirect linear form, the ultimate complete interdependence as between the primes and natural numbers from a dynamic relative perspective, both of which are understood as possessing analytic (quantitative) and holistic (qualitative) aspects with respect to each other.

So the full explicit understanding of the non-trivial zeros mirrors precisely the corresponding - merely implicit - nature of the non-trivial zeros in mathematical terms, where the most evolved understanding (from a psycho spiritual perspective) is necessary to interpret the least evolved - which is thereby equally the most involved - nature of reality (in a corresponding physical fashion).

The non-trivial zeros therefore represent in number form, complementary facets of the deepest intrinsic and extrinsic nature of both the physical and psychological aspects respectively of the dynamically interactive system, which lies at the very core of all created phenomena.

And this as close as one can get in the phenomenal realm to formless ineffable reality.

So now one can perhaps appreciate a little better what the non-trivial zeros truly represent and why they are so significant!

Incredible Nature of the Zeta Zeros (22)

As we saw yesterday, the Zeta 2 Function provides the means of internally defining the interdependent nature of integers in a consistent manner.

So the individual ordinal members of each integer are thereby designated in a circular fashion through obtaining the corresponding roots of 1. And this serves as the appropriate way of representing such interdependence.

Ultimately as each integer becomes very large, the relationship as between individual quantitative and overall collective components become so dynamic that they approach total identity with each other in a seamless manner.


The primes have a special significance in this context as their individual members are uniquely defined.
So if p is a prime number, the p roots of 1, establish the relationship between its natural number members from 1 to p respectively and the overall relationship between its p members (in a collective qualitative sense).

The relationship here as between each prime and its natural number members is as quantitative to qualitative (and qualitative to quantitative) respectively.


All this is a valuable preparation for understanding the essential role of the non-trivial zeros with respect to the Zeta 1 Function.

Here, we adopt a complementary focus with respect to the relationship between the primes and natural numbers in an external manner where now - in reverse fashion to Zeta 2 - the prime numbers appear as the building blocks of the natural number system (in cardinal terms)

In the Zeta 2 approach each prime number is identified internally with its natural number individual components (in an ordinal manner). Here with the Zeta 1 approach, by contrast, the (entire) natural number system is identified externally with its individual prime number components (in cardinal terms).

So when we understand the relationship between Zeta 1 and Zeta 2 in an appropriate dynamically interactive manner, we can see clearly how they exactly mirror each other as complementary partners.


We saw in earlier blog entries that when the Type 1 approach attempts to “break up” the internal nature of an integer that a crucial problem arises.

For example we attempt to the define the number “3” internally (in terms of its individual number components) as 1 + 1 + 1. However this merely quantitative approach strictly leaves us with no means of providing an ordinal ranking to the numbers. In other words the very ability to clearly distinguish - in any context - a 1st, 2nd and 3rd member, requires giving the numbers some qualitative distinction!


So a new circular number system is required to uniquely define the members of an integer group in a qualitative manner.

A similar problem arises when we try to "break up" the external nature of natural number integers.

For example if we now attempt to define the number 6 externally (in terms of its prime number components) in Type 1 terms, we will be told that it is uniquely defined in terms of its two prime factors i.e. 2 and 3 in a quantitative manner.

So 6 = 2 * 3 in quantitative terms!


However this - apparently simple – multiplication process, once again conceals a crucial difficulty.

The individual prime numbers here i.e. 2 and 3 respectively are originally defined in a linear (1-dimensional) manner. Thus, quite literally we can geometrically represent both numbers as points on the real number line.

However when we now multiply both numbers, an important qualitative - as well as quantitative - transformation takes place.

So in quantitative terms, the answer is indeed 6! However strictly we have now switched from a 1-dimensional to a 2-dimensional number expression!

This can easily be represented by representing the product of the two numbers in terms of a rectangle with side measurements of 2 and 3 (linear) units respectively.

Therefore, whenever we multiply prime numbers (to obtain composite natural numbers) both a quantitative and qualitative transformation is entailed.

However due to the reduced quantitative nature of the Type 1 approach, such qualitative change in the variables is simply ignored.


So in Type 1 terms, 2 * 3 = 6 is interpreted in a reduced quantitative fashion (where implicitly 6 = 6^1). In other words - though strictly a qualitative transformation in the dimensional context has taken place (i.e. 2-dimensional) - the result is misleadingly still interpreted in linear (1-dimensional) terms.


This is why I repeatedly characterise the very nature of conventional Mathematics as 1-dimensional (in qualitative terms).

Of course I recognise that Conventional Mathematics recognises dimensional transformations of number; however it does so merely from a (reduced) quantitative perspective!

Now there is indeed at some level, a recognition of a problem here among professional practitioners.

This is usually expressed in terms of the uneasy relationship that exists as between addition and multiplication.

Thus from the additive perspective we could represent a number such as 6 as

1 + 1+ 1 + 1 + 1 + 1.

However equally we could represent it as 2 * 3.

Unfortunately, there does not appear from this quantitative perspective any obvious way of reconciling the two viewpoints.

However the fundamental issue is so obvious that I am amazed at why it is not clearly seen.

Indeed this very issue caused my earliest disillusionment with the conventional approach when I was 10 years old.
It is that experience that has informed my subsequent development, so that I have never felt unduly bound by conventional mathematical wisdom (which in many ways remains blind to fundamental problems of interpretation).

Putting it simply, wherever two numbers are multiplied (or a number raised to a power or exponent) a qualitative - as well as quantitative - transformation is involved.

So, that in a nutshell is the fundamental issue with respect to reconciling addition with multiplication. It can only be done so in an approach that explicitly recognises both the quantitative and qualitative aspects of mathematical appreciation (which ultimately are fully complementary with each other in a dynamic manner).

Thus the Fundamental Theorem of Arithmetic i.e. that very natural number integer other than 1 is either prime or can be uniquely expressed as the product of prime number factors, is misleadingly portrayed through the (conventional) Type 1 approach in a merely reduced quantitative manner. However properly understood there is also - literally - a qualitative dimension to this Fundamental Theorem (which is continually misrepresented in conventional terms).


So we once again come back to the key issue of interdependence! Because Conventional Mathematics in formal terms is defined strictly within independent polar reference frames (i.e. where objective is abstracted from the subjective pole and quantitative abstracted from the qualitative pole) it has no means within its interpretations of dealing with the key issue of (holistic) interdependence (except in a grossly reduced fashion).

However once we understand clearly that every composite natural number (as representing the unique product two or more primes) involves both quantitative and qualitative transformation then we are led directly into this key notion of holistic interdependence (as the means of reconciling the dynamic relationship of both aspects).

So this is the context in which the non-trivial zeros properly arise.

In the Zeta 2, we saw how the non-trivial zeros (as the unique values for the prime numbered roots of 1), provide the means of solving the internal problem (of the relationship of the primes to the natural numbers).

So once again for example from the (standard) reduced Type 1 perspective

5 = 1 + 1 + 1 + 1 + 1.

Because these units are all homogenous in quantitative terms, we are left with no way of making a qualitative distinction.

However through the 5 roots of 1 we can give natural number member an individual unique distinction (on a circular number scale) in - relative - quantitative terms, while also maintaining a holistic interdependence as between the collective group of prime members.

So the very meaning of interdependence in this context is that it provides the means of reconciling quantitative with qualitative Type interpretation.

And - quite magically - the non-trivial zeros of Zeta 2 (as the solutions to the equation) when appropriately interpreted, provide the ready solution for this internal relationship between the natural numbers and primes (in all possible cases).

So in a direct complementary manner we can now perhaps appreciate that the (famed) non-trivial zeros of the Zeta 1 equation, likewise provide the solution for the external relationship as between the primes and natural numbers (in all possible cases).

In other words - at one fell swoop as it were - these non-trivial zeros provide the ready means of reconciling each individual prime on the one hand with their overall collective relationship with the natural numbers.

In short - though again it is not clearly recognised as such - the non-trivial zeros represent the solution to this problem of interdependence with respect to the entire natural number system.

So just as the prime numbers represent the independent extreme, the composite natural numbers (as the product of prime numbers) entail an interdependent identity with quantitative and qualitative aspects.


The formula for the non-trivial zeros clearly demonstrates this as it represents the circular equivalent of the standard linear formulas with respect to both the gaps as between primes and the general frequency of primes.

So as the composite numbers (in inverse fashion to the primes) become ever more prevalent as we ascend the natural number scale, one would expect their frequency to increase in inverse fashion to the primes (which indeed is what happens).

So in short the non-trivial zeros represent the interdependent relationship of primes to natural numbers with the whole set of non-trivial zeros completely establishing this interdependence.

So at one extreme we have the independent nature of the primes and natural numbers (where quantitative and qualitative notions are clearly separated). Then at the other extreme we have the interdependent nature in the entire set of zeros where the primes and natural numbers are understood as identical.

And in a comprehensive understanding neither of these aspects can be understood in the absence of the other. They are in fact ultimately fully complementary.


We perhaps can put it even more simply.

Both the primes and natural can be defined in both a quantitative and qualitative manner (i.e. in Type 1 and Type 2 terms).

However, in dynamic terms, the relationship between both primes and natural numbers and (natural numbers and primes) is as quantitative to qualitative (and qualitative as to quantitative) respectively.

The non-trivial zeros in each case (from complementary opposite standpoints) establish the means of reconciling both interpretations i.e. through showing that quantitative and qualitative aspects are ultimately identical (in a - necessarily - relative approximate manner).


As I have repeatedly stated, Conventional Mathematics attempts to view quantitative and qualitative poles as completely separate (purely independent).

However quantitative and qualitative are dynamically related in phenomenal terms and ultimately identical (purely interdependent).

And properly understood, we cannot begin to understand the nature of our everyday – seemingly independent - number system in the absence of the interdependent dynamics operating at its very core, as the fundamental requirement for its - relative - consistency.

The non-trivial zeros (from both perspectives) where both its quantitative and qualitative aspects can be understood as ultimately interdependent, represent the opposite polar extreme to current mathematical interpretation of the nature of number.

The non-trivial zeros therefore simply cannot be properly reflected through the conventional mathematical paradigm.

As it formally has no recognition for the qualitative aspect, it therefore can have no proper notion of interdependence (which implies both aspects).


In short, the non-trivial zeros point to the need for a completely new understanding of mathematical - and indeed all scientific - reality (where quantitative and qualitative aspects of understanding can be dynamically related at every level).

Incredible Nature of the Zeta Zeros (21)

Yesterday, we looked at the - as yet unrecognised - significance of the Type 2 non-trivial zeros.

We saw that they provide a ready means of assigning a unique identity to each individual member of a number group in ordinal terms.


So for example we can look on 4 as a collection of individual ordinal members i.e. 1st 2nd, 3rd and 4th respectively.

The 4 roots of 1, obtained through solving the equation 1 – s^4 = 0, then provide a relative quantitative identity (in circular number terms) to each of these individual members i.e. i, – 1 – i and 1 respectively.

Then the combined addition of these four numbers = 0, expresses their qualitative interdependence.


As 1 is always a root with respect to any dimensional value (t), we divide 1 – s^4 = 0 by 1 – s to obtain 1 + s + s^2 + s^3 = 0 which provides the non-trivial solutions.

Then to convert to a complementary form to the Zeta 1 equation we multiply by s
So s^1 + s^2 + s^3 + s^4 = 0.

More generally,

s^1 + s^2 + s^3 + s^4 + …+ s^t = 0.

The key significance of solutions to this equation (where t is a prime number) is that it ensures a unique identity to each ordinal member.

So the non-trivial zeros have a special significance in this context, as the means of providing a unique identity to the (natural number) ordinal members of a prime group.

Not alone has each member a unique individual identity in quantitative terms, but also a perfect holistic collective identity in qualitative terms (illustrated through the sum of roots =0).

Therefore the key (unrecognised) function of these Zeta 2 zeros is that they enable the seamless integration of each individual natural number (in ordinal terms) with their overall collective prime number identity.

For example the 1st, 2nd, 3rd, 4th and 5th members of 5 (as represented by the individual 5 roots of 1) are given a unique identity in quantitative terms; then equally, they have a holistic qualitative identity in their combined relationship with each other (represented as the sum of the 5 roots).

So in this way, the (ordinal) individual natural number members (of 5) in quantitative terms, share an overall qualitative holistic relationship with the cardinal prime number 5 (considered as a whole number set).

As the prime number grouping increases the dynamic interactivity required to reconcile the individual (quantitative) members with their overall collective (qualitative) shared interdependence so greatly increases that both aspects (quantitative and qualitative) can no longer be explicitly distinguished from each other. So in the seamless integration of both, qualitative approaches identity with quantitative meaning.

In this sense, therefore the natural numbers and the primes are likewise identical with each other. So, we can then no longer distinguish the individual members (as ordinal natural numbers) from the overall collective grouping (as a cardinal prime).
In fact any static identification of what is quantitative or qualitative loses meaning as switching between both aspects now occurs so rapidly as to be instantaneous!


However this can only be conceived in a relative rather than absolute sense.

One might be tempted to propose that by allowing the prime number to be infinite that we can thereby embrace all its natural number members (likewise in an infinite manner).

However this is a strictly meaningless proposition, as we would have no means of obtaining the infinite roots of an equation. So therefore there is no way of establishing a unique identity for each individual member or likewise of establishing an overall collective identity (in infinite terms).

But by making the dimensional number t larger and larger in finite terms, we can approach ever more closely to this identity of natural numbers with each prime (in a relative approximate manner).

So again from a very important perspective, this more refined treatment of number (allowing for both quantitative and qualitative aspects) exposes clearly the reductionist nature of the standard approach to infinite notions.


Put simply, ultimately at the interface of finite and infinite notions, we always face inevitable uncertainty. Indeed the uncertainty arising from the interaction of quantitative and qualitative (and qualitative and quantitative) is but a direct expression of this prior relationship as between finite and infinite (and infinite and finite).

And as mathematical activity (implicitly or explicitly) entails in any context the relationship between finite and infinite notions, it too is rooted inevitably in uncertainty.

So mathematical truth, as so graphically demonstrated in the very nature of the non-trivial zeros, is necessarily of a relative approximate nature.

Now, through reductionist procedures, we may certainly create the illusion of an absolute Mathematics; however ultimately this illusion is built on shifting sand without any solid foundation.


I have gone on at some length about the Zeta 2 approach to the non-trivial zeros for two major reasons.

Firstly its true significance (with respect to the relationship as between the primes and the natural numbers) remains totally unrecognised.

Secondly the equally important significance of the Zeta 1 non-trivial zeros can only be properly appreciated with respect to the complementary nature of the Zeta 2.


As we have seen, when one properly allows for both quantitative and qualitative aspects, there are two ways of viewing the relationship between the primes and the natural numbers.

From the standard Type 1 perspective, we can view the primes (in their collective nature) as the building blocks of the natural number system (in a cardinal manner).
However from the unrecognised (shadow) Type 2 perspective, we can equally view each prime (in its individual whole nature) as composed of natural number building blocks (in an ordinal manner).

In my own development - precisely because I have specialised for so many years now in this unrecognised holistic aspect of Mathematics - I had already become aware some time ago of the enormous significance of the Type 2 perspective (which requires very little in the way of abstract mathematical techniques).

However, it is very much the opposite with respect to the standard Type 1 approach where highly specialised complex techniques have been developed to deal with all aspects of the Riemann Zeta Function.

These would be largely inaccessible to all but a small number of professional practitioners. However, in my opinion, intuitive insight into what it is really all about still remains remarkably thin on the ground.

For example it has been patently obvious to me for some time that the Riemann Hypothesis is not capable of proof (within standard mathematical procedures). Put another way, its very nature greatly transcends conventional mathematical interpretation.

No amount of further improvements with respect to sophisticated mathematical procedures are going to change this situation. In fact they will lead even further away from any fundamental intuitive insight into what the problem truly entails.


The repeated failures with respect to attempts to “prove” the Riemann Hypothesis are in fact clearly indicating profound limitations in our very understanding of what Mathematics is about.

So the Riemann Hypothesis is really pointing to the ultimate identical nature of the quantitative and qualitative aspects of Mathematics. But this can never be appreciated while formally approaching interpretation in a mere quantitative manner.


We hear often for example practitioners state that some big new idea is required before real progress towards a proof of the Riemann Hypothesis can be made!

Well that big idea is that the qualitative aspect of interpretation must now be included in Mathematics not only to make sense of the Riemann Hypothesis but ultimately to make sense of all Mathematics!

We have been trading for far too long on the quantitative illusion i.e. that Mathematics can be formally interpreted in a merely quantitative manner.

However that illusion has run straight into the rocks protecting the Riemann Hypothesis and the truths underlying the very nature of the number system.


When one begins to accept the qualitative aspect with respect to all mathematical procedures, a marvellous new sense of mystery accompanies exploration into the deepest recesses of the mathematical system.

What we have then in the Type 1 and Type 2 approaches, two complementary visions of the origins of number. However growing appreciation of such complementarity, eventually leads one to the realisation that in the end we must surrender all phenomenal attempts at understanding its nature.


So reason can cooperate with intuition in drawing one into its sublime secrets, but in the end the final realisation of what it is (where everthing is now understood as interdependent), simply involves a surrender to that very mystery.

And in the ultimate questions regarding the number system - indeed regarding all phenomenal reality - is likewise found the ultimate answers in the pure experience of mystery. And here, the primes and the natural numbers finally melt together in an ineffable embrace.

Incredible Nature of the Zeta Zeros (20)

In yesterday’s blog I simplified the nature of the non-trivial zeros as relating directly to the interdependent aspect of the number system.

Furthermore in line with Zeta 1, Zeta 2 and Zeta 3 approaches, we can provide 3 - relatively - distinct perspectives on what such interdependence entails.

Though its true significance to my mind remains as yet completely unrecognised, it would perhaps be most illuminating to start with the Zeta 2 (non-trivial) zeros.


Once again these zeros arise as the solution to the equation,

s^1 + s^2 + s^3 + s^4 +……+ s^t = 0.


Now one of these solutions is for s = 0.

The others then relate in quantitative terms to the non-trivial roots (excepting the common root 1) of t.

Now if t is a prime number, then all these non-trivial roots will be unique (i.e. cannot occur as the solutions for any other prime value of t).

All this of course is well known in conventional mathematical terms. However it does not convey the true significance of these prime numbered root values.

What is not yet clearly recognised is that associated with all these quantitative values (as relatively independent) is an important complementary meaning of qualitative interdependence in holistic terms.

When seen from this perspective each grouping of prime numbered roots (including the common root 1) provides a unique circle of interdependence from an ordinal qualitative perspective.

For example when t = 3 (where of course t is prime) we have the Zeta 2 equation,

s^1 + s^2 + s^3 = 0.

Once again, s = 0 represents one of these 3 solutions.


The other two solutions (correct to 3 decimal places) relate to the unique (non-trivial) roots of 1,

i.e. – ½ + .866i and – ½ - .866i.

Now in the Type 2 approach (that I have introduced) a dynamic complementary relationship always necessarily exists between such values in both quantitative and qualitative terms.

So if we include the trivial root i.e. t = 1, the three roots of 1 in relative quantitative terms,

are 1, – ½ + .866i and – ½ - .866.


However these now bear a complementary relation to the same 3 values now considered as a circular interdependent group in a qualitative holistic manner.

In other words, the true unrecognised significance of these 3 circular numbers in this context is that they provide a unique ordinal means of identifying a group (containing 3 members).

Again this issue is completely overlooked in conventional mathematical terms where it is - wrongly - assumed that ordinal identification of members can be carried out in an unambiguous fashion.

So therefore from this perspective we can identify unambiguously the 3 members of a group in ordinal terms as the 1st, 2nd and 3rd respectively.


However the key problem is that with the numbers of a group increasing the very meaning of 1st, 2nd and 3rd (given earlier) is no longer relevant in the new changed context.

So for example if the number t now increases to 5, 1st, 2nd and 3rd acquire a new – relatively - distinctive meaning in the context of this larger grouping of ordinal members.


Therefore when one reflects carefully on the matter, ordinal natural number rankings have a merely relative validity depending on the overall size of any group in question.

This raises then the serious issue of providing some means of unambiguously distinguishing such rankings (as the finite size of the grouping changes).

And the key to this is with reference to the new circular number system that is associated with the various roots of 1.

And in this system for any value t, we can unambiguously identify the different ordinal members of the group in a – relatively – quantitative numerical manner, while also allowing for the overall interdependence of the group members in a qualitative holistic fashion.


So once again in the context of 3, we unambiguously identify the 3 members of this group in a quantitative manner through the circular number system (defined with respect to points on the circumference of the unit circle in the complex plane).

So in relative quantitative terms, these 3 members are identified as
1, – ½ + .866i and – ½ - .866i


The qualitative aspect arises through combing these 3 values as a whole.

And as the sum of roots = 0, this means that the qualitative interpretation (with respect to the interdependence of this group) strictly has no meaning from a quantitative perspective.

Now again the key significance of t as representing a prime dimensional number in this context, is that the various internal members of the group (except 1) will be defined in a unique manner.

Now the fact that 1 is not unique (yet a member of the prime group) is necessary so as to provide a link with the complementary linear manner of defining the relationship of primes to natural numbers.

And we have already seen - again in a complementary fashion - that the one value for t for which the sum of roots ≠ 0 is where t = 1!

So we are always interpreting in a relative – rather than absolute – terms.

Thus – in this relative sense – each prime group is defined uniquely in ordinal terms by its natural number constituents (through quantitative numbers in the circular system).

And the corresponding interdependence of such a group - represented as the sum of individual members) strictly has no quantitative meaning! In other word the quantitative value of this sum = 0.


When seen in this light, the true significance of the solutions for t (as the non-trivial zeros) for the Zeta 2 equation is that they provide the means to define number in an interdependent manner.

And as the sum of roots = 0 for all values of t (except 1), group independence with respect to all these numbers is thereby seen to relate to the holistic qualitative nature of number.

And when t becomes very large, dynamic interactivity so increases that the relative independence of each quantitative member becomes inseparable from the relative interdependence collectively of these same members (in qualitative terms).


So therefore any distinction between a collective prime number grouping (in qualitative terms) and its uniquely distinctive ordinal number members (in quantitative terms) thereby ceases. So in this sense the non-trivial zeros of the Zeta 2 (for unlimited t) entail the identity of the prime with the natural numbers.

We will see again that the Zeta 2 provides a directly inverse way of defining the quantitative/qualitative relationship between number to the Zeta 1 approach.
So with the Zeta 2 we have shown how the sum of all the natural numbers to t, as solutions for the base numbers s (in ordinal fashion) of the equation = 0 with t having no upper finite limit!

Then with the Zeta 1 we will show how the sum of all the natural numbers to infinity as solutions for dimensional numbers s (in cardinal fashion) = 0 likewise have no upper limit.

Thus the non-trivial zeros for Zeta 1 and Zeta 2 simply represent two sides of the same coin with respect to demonstrating the interdependent nature of the number system.

Whereas Zeta 2 demonstrates this interdependence with respect to their ordinal nature (in a circular number fashion), Zeta 1 demonstrates such interdependence with respect to their cardinal nature (in an imaginary linear number fashion).

Incredible Nature of the Zeta Zeros (19)

In order to appreciate what is involved, we need to keep placing the non-trivial zeros of the Riemann Zeta Function in a greatly enlarged mathematical context.

As we have seen conventional mathematical appreciation is based on the abstract notion of number as independent entities.

This is especially the case with respect to the treatment of the prime and natural numbers which are customarily represented on the same number line.

However this appearance of independence hides a deep-rooted problem.

For - as it is well known - the natural numbers (except 1) are uniquely derived in cardinal terms as the product of prime number factors. Therefore from this perspective, there is an important sense in which the natural numbers thereby depend for their identity on the primes.


Equally - though unfortunately not yet properly recognised - each prime number in an important manner is derived from its (internal) natural number members in an ordinal fashion.

So for example the prime number 5 necessarily is composed of a 1st, 2nd, 3rd, 4th and 5th member in ordinal terms.
Furthermore whereas the cardinal notion of number lends itself to quantitative type interpretation, strictly the ordinal notion relates directly to a qualitative - rather than quantitative - meaning.


So from one arbitrary perspective, the natural numbers depend on the prime numbers (for their quantitative identity).

Then from the equally valid alternative perspective, the prime numbers likewise depend on the natural numbers (for their qualitative identity).

Therefore in a crucial manner, both the primes and natural numbers (and the natural numbers and the primes) are mutually interdependent.

So, if we are to give a coherent interpretation of the relationship between both (i.e. primes and natural numbers) we must adapt from the onset a dynamic interactive approach.


In this context it is certainly true that both the primes and natural numbers can be given a relative independent identity (within two distinctive reference systems).

However their mutual interdependence then requires the complementary identity of both these systems.


Thus from the onset, I define three aspects to the number system, which necessarily interact with each other in a dynamic relative manner.

First we have the Type 1 aspect geared directly to the quantitative interpretation of number in cardinal terms.

Secondly we have the Type 2 aspect geared directly to the qualitative interpretation of number in ordinal terms.

Thirdly we have the Type 3 aspect geared directly to interpretation of the mutual interdependence of both the qualitative and quantitative aspects of number.


Therefore again in Type 1 (cardinal) terms, the primes appear as the building blocks of the natural number system in quantitative terms, where each natural number (except 1) can be uniquely expressed as the product of prime number factors.

Then in Type 2 (ordinal) terms, the opposite relationship now applies with the natural numbers appearing as the (internal) building blocks of the primes in qualitative terms, so that each prime number is uniquely expressed by its natural number members (i.e. through obtaining its respective roots).

Finally in Type 3 terms, the mutual interdependence of both the primes and the natural numbers (and natural numbers and the primes) is now recognised in both quantitative and qualitative terms. So in this mutual identity - which is of a necessarily relative approximate nature - no distinction remains with respect to the quantitative and qualitative interpretation of number.


Next, with respect to the Riemann Zeta Function we likewise have three matching interpretations in accordance with the Type 1, Type 2 and Type 3 aspects of number respectively.

The Type 1 aspect can be identified with the standard quantitative mathematical approach to interpretation of the Function. However there is one very important distinction in that, properly understood in dynamic interactive terms, such interpretation is now of a relative - rather than absolute - nature.

I refer to this modified Type 1 aspect of the Riemann Zeta Function,

1^(– s) + 2^(– s) + 3^ (– s) + 4^ (– s) + …….,

as the Zeta 1 function (or more briefly Zeta 1).


To be more precise the standard Type 1 approach is identified with an approach where quantitative and qualitative aspects of interpretation are (formally) separated from each other in an absolute manner. This in effect leads to the reduction of qualitative to quantitative meaning!

The new dynamic manner therefore that I propose for Type 1 is based on separation of these two aspects as relatively independent. This thereby enables a more balanced focus on quantitative meaning (without undue reductionism with respect to the qualitative aspect).


The Type 2 aspect is then identified with a qualitative mathematical approach to interpretation of the Function.

More precisely it is identified with recognition of the necessary interaction of both quantitative and qualitative aspects (with the main focus on the holistic implications of such interaction).


Therefore associated with Zeta 1, we have complex analytic interpretation in quantitative terms (with indirect recognition of a complementary holistic aspect of interpretation).

Then associated with Zeta 2, we have complex holistic interpretation in qualitative terms (with indirect recognition of a complementary analytic aspect
of interpretation).

Zeta 2 is in fact associated with another related Function,

s^1 + s^2 + s^3 + s^4 +……+ s^t.


This is similar to Zeta 1 (turned inside out), so that dimensional values (s) become base quantity values with respect to Zeta 2. Likewise the base values (1, 2, 3,…) with respect to Zeta 1, become the dimensional values with respect to Zeta 2.

Further complementarity also exists in that whereas Zeta 1 is defined as an infinite series, Zeta 2 is defined in finite terms. Also s in Zeta 1 complies with – s in Zeta 2.


Zeta 3 then entails the mutual interaction in interpretation with respect to both Zeta 1 and Zeta 2. Therefore whereas both quantitative and qualitative aspects enjoy a relative degree of separation with respect to Zeta 1 and Zeta 2 respectively, with Zeta 3, they becomes so closely intertwined in understanding as to become identical.


This also highlights the other key aspect of this new enlarged dynamic interactive approach.

Basically it applies to the two key polarity sets.

Therefore from one key perspective we now see all mathematical understanding as necessarily entailing the interaction of both quantitative and qualitative aspects (in relative terms).

Equally from the other key perspective, we see all such understanding as likewise necessarily entailing the interaction of both internal (mental) and external (objective) aspects.

In other words the objective mathematical reality, we wish to portray (in this context the non-trivial zeros) has no strict meaning in the absence of the corresponding mental lenses through which they are viewed.

So to objectively view the non-trivial zeros in an appropriate fashion, we must ensure the complementary nature of their subjective means of interpretation.

Thus the absolute notion of a static mathematical universe existing out there in some unchanging space is but a mistaken illusion that must now be fully discarded, for strictly, mathematical truth has no meaning in the absence of the manner through which it is interpreted!

And by employing a radical new interpretation - that better accords with the experiential dynamics of understanding - the very nature of Mathematics utterly changes.


Finally with respect to the non-trivial zeros in accordance with the three aspects of number interpretation and corresponding three Zeta Functions, we again have three sets of non-trivial zeros arising from solution to the respective equations for s when the value = 0.


So in Zeta 1 the non-trivial solutions represent the infinite set of solutions for s,

ζ(s) = ζ(1 - s) = 1^(– s) + 2^(– s) + 3^ (– s) + 4^ (– s) + ……. = 0.


In Zeta 2, the non-trivial solutions represent the finite set of solutions for s (except 1),

s^1 + s^2 + s^3 + s^4 +……+ s^t = 0 (where t can be any integer, regardless of how large).


Then in Zeta 3, both sets of solutions are so clearly understood as two necessary sides of the same phenomenon, that they approach mutual identity.


To put it simply, the non-trivial zeros relate directly to the interdependent nature of the number system.

And because conventional mathematical interpretation recognises solely (in formal terms) the independent notion of number, we cannot properly interpret their significance from within this restricted context.


However once we accept the dynamic nature of number (containing necessary aspects that are relatively independent and also interdependent with respect to each other) then it becomes obvious that we can focus – in the customary manner – on number as form with the appearance of material independence; equally we can at the other extreme, focus on number as energy where any lingering notions of a (solely) independent identity lose their meaning.

Put even more simply, just as conventional (independent) notions of number are based on an (unchanging) material form, the non-trivial zeros - representing the corresponding (interdependent) notion of number - relate directly to energy states.

As Einstein demonstrated, mass and energy are equivalent; so we have a new form of equivalence in the relationship of the conventional notion of number (as independent form) with the opposite notion of the non-trivial zeros (in their interdependent fusion) as energy states.