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.

Incredible Nature of the Zeta Zeros (18)

As we have seen, the non-trivial zeros represent the points on an imaginary number scale, where the primes and natural numbers approach mutual identity.

Strictly, this relates in absolute terms to an ineffable reality as the identity of both form (1) and emptiness (0).

So the identity of the primes with the remaining natural numbers, necessarily pertains to a phenomenal reality, where both are identified in a relative approximate manner.

Thus, the non-trivial zeros, from the enlarged perspective that I have adopted, represent the Type 1 presentation of the Type 3 nature of mathematical interpretation.

So this represents an attempt to represent the set of non-trivial zeros as independent points drawn through ½ with respect to the real axis, on an imaginary number line.


Once again, precisely speaking, these points represent the (relative) identity of the interdependence of both prime and natural numbers.

However, there is no way of meaningfully intuiting the true nature of these zeros from this perspective.

Fortunately, the alternative Type 2 presentation does allow for a remarkably simple illustration of the profound nature of this two-way identity of primes and natural numbers.


As we have seen the Type 2 aspect of the number system relates to the circular representation of number arising from the extraction of the n roots of 1 in quantitative terms. These numbers then can be represented as equidistant points on the circle of unit radius in the complex plane.

From a quantitative perspective they are considered as - relatively – independent; however from the complementary perspective, they have a new qualitative meaning as representing the interdependent nature of number.

Thus n has a Type 2 interpretation as a measurement of the (internal) interdependence of its n circular points (which are independent from a quantitative perspective).

This qualitative interdependence can be easily appreciated through combining the circular numbers representing the n roots of 1 (through addition).

For all values of n (except 1) this sum = 0.


For example the 4 roots of 1 are 1, – 1, i and – i respectively with their sum = 0.

This simply implies that the qualitative notion of interdependence has no strict meaning in quantitative terms. Indeed this illustrates the holistic - as opposed to the analytic - interpretation of number (for which the Type 2 aspect is directly suited).

Of course with the standard Type 1 linear aspect (based on 1-dimensional interpretation where n = 1) the one root of unity = 1 in quantitative terms. So the sum of roots also = 1. This again graphically illustrates how the Type 1 aspect - which defines all conventional mathematical interpretation - directly reduces the notion of qualitative interdependence in a quantitative independent manner!


The Type 2 aspect of mathematical interpretation is inherently of a dynamic interactive nature, where the ordinal identity of the internal units of each number, are defined in both linear and circular terms (as the relationship of quantitative to qualitative, and qualitative to quantitative respectively).

So 4 as a (whole) number integer has 4 internal constituents as 1st, 2nd , 3rd and 4th members of 4 respectively.

These 4 number members are then represented as equidistant points on the circumference of the circle of unit radius in the complex plane.

So each number as a point in the complex plane, can be seen to simultaneously lie on a straight line (i.e. trough the radius drawn to each point) while equally lying on the circular circumference.

Likewise we equally give an interpretation that now combines both linear (quantitative) and circular (qualitative) notions.


So we start the Type 2 aspect with 1 – s^t = 0, as the equation for calculating the t roots of 1.

Now 1 – s represents the one trivial solution, for s = 1, which always represents one of the t roots.

Dividing by 1 – s, we obtain

1 + s + s^2 + s^3 + ……+ s^(t – 1) = 0.

Then multiplying by s we obtain,

s^1 + s^2 + s^3 + s^4 +……+ s^t = 0

Then apart from the solution s = 0, the other solutions represent the t – 1 non-trivial solutions of this equation.


I term this equation the Zeta 2 Function, which complements the standard Riemann Zeta Function that is appropriate for interpretation according to the Type 1 aspect, which in this context I term Zeta 1.

So the Zeta 1 Function which extends over an infinite number of terms (in accordance with the reduced nature of the Type 1 aspect) can be written,

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

The corresponding Zeta 2 Function which extends over a finite number of terms, involves the reverse complementary positioning of both the base natural numbers and dimension (s) with respect to Zeta 1.

So s now becomes the base number in Zeta 2 and the natural numbers 1, 2, 3, 4,… its corresponding dimensional numbers.

Once again with the Type 1 aspect the primes (as cardinal quantities) are represented as the building blocks of the natural numbers.

Then in the Type 2 aspect this relationship is reversed with the natural numbers (as ordinals with qualitative characteristics) represented as the building blocks of the primes.

So the Zeta 1 is in line with the former Type 1 representation, whereas the Zeta 2 (where base and dimensional numbers with respect to Zeta 1 are reversed) enables the latter Type 2 representation.


So correctly understood we generate two sets of non-trivial zeros.

In the Zeta 1, the non-trivial zeros represent the solutions for the dimensional number s, which are all complex in nature and of the form ½ + it and ½ – it respectively (with t transcendental in nature).

In the Zeta 2, the non-trivial zeros represent the solutions for the base number s, which again are complex in nature of the form a + it (which though generally irrational is algebraic in nature).

Now as we have seen, the non-trivial zeros relating to Zeta 1, represent points on an imaginary number line (through ½) where the prime and natural numbers are identical (in a relative manner).

The very nature of Type 1 interpretation is based on the notion of number representing analytic recognition in terms of independent quantities. However the authentic nature of these non-trivial zeros requires - by contrast - their holistic recognition (in the qualitative recognition of interdependence).

Therefore Type 1 interpretation cannot provide an appropriate appreciation of what such non-trivial zeros represent. Thus - in a quite literal manner - their nature cannot therefore be appropriately intuited using Type 1 interpretation.


However this situation is happily reversed through Type 2 interpretation.

It combines both the linear and circular notions of number (as representing quantitative independence and qualitative interdependence respectively) in a dynamic interactive manner, thereby providing a simple means of demonstrating the nature of the identity of prime with natural number behaviour.

Now doing this requires a fascinating complementary form of reductionism to that which defines Type 1 interpretation!


As we have seen, Type 1 entails reductionist interpretation, with quantitative results all defined qualitatively in 1-dimensional terms.

Now in reverse fashion in Type 2, we can provide a corresponding form of quantitative reductionism, where all results (representing the qualitative nature of interdependence) are defined quantitatively in 1-dimensional terms.

This implies that with respect to the non-trivial solutions of the Zeta 2 for s, we simply ignore negative signs (defining all numbers as positive) and likewise ignore imaginary signs (treating all numbers as real).

In this way we can represent the solutions for s (as points on the unit circle in the complex plane) as positive real numbers.

So with the dimension t as prime, we can represent its natural number ordinal members in positive real number terms.


I have explained elsewhere that when we then obtain the mean average of these numbers that both the cos and sin parts approximate ever more closely to 2/π as t increases in value representing ever larger primes.

We also saw that 2/π = i/log i, giving us a fascinating Zeta 2 Prime Number Theorem (to complement the n/log n of Zeta 1).

Also the ratio of the deviations of the average cos and sin values from 2/π (= i/log i) approaches ever closer to .5 (in absolute terms) as t (representing primes) increases.


Now what is truly fascinating is that if now instead of the all the natural number ordinal members of t as prime (measured as I have explained in a quantitative 1-dimensional manner as real positive numbers) , we concentrate only on the prime number ordinal members, the mean average of these members will likewise approximate to 2/π (= i/log i).


So for example, 127 is a prime number.

So I obtained the 127 roots of 1 (representing the ordinal natural numbers from 1 to 127) converted them in a reduced 1-dimensional quantitative manner and then obtained the mean average for both cos and sin parts which already approximated in both cases very closely to 2/π (= i/log i).

I then obtained the 31 prime numbered roots (from 1 to 127) again calculating their mean average for both cos and sin parts.

Again in both cases the answer approximated quite closely to 2/π (= i/log i).


I also examined behaviour with respect to the natural and prime numbered roots of several other values of t (where t is a prime number) with similar results emerging in both cases.

So I would confidently conjecture that in the limit where t (as prime number) →∞, that both the mean average for both natural and prime number ordinal roots = 2/π (= i/log i).

In strict terms however as we can only meaningfully define the roots of 1 (with respect to finite values of t) this identity of natural and prime numbers will necessarily be of an approximate relative nature.

Thus in this way, we have demonstrated the identity of both the prime and natural numbers (in a manner that relates to the interdependent nature of both their linear and circular behaviour).

Therefore in a direct sense the non-trivial zeros associated with Zeta 2 can demonstrate the identity of both the prime and natural numbers in a holistic ordinal manner.


By contrast the non-trivial zeros associated with Zeta 1 can only demonstrate this identity - now in a quantitative analytic sense - in an indirect manner that does not directly intuit with Type 1 understanding.

So from a Type 3 perspective both the Zeta 1 and Zeta 2 represent two sides of the same coin, both of which - for full comprehension - must closely interact with each other in the mutual two-way identity of the primes and the natural numbers (and the natural numbers and the primes) in both quantitative and qualitative terms.

Incredible Nature of the Zeta Zeros (17)

As we know from Einstein’s famous equation E = MC2, matter and energy are equivalent.

When we look at numbers in an appropriate dynamically interactive manner, an equal equivalence applies, whereby from the analytic perspective, number can be seen in abstract material terms as quantitative; however from the equally valid holistic perspective number can be seen as ultimately representing pure energy.

In the actual dynamics of experience, the analytic aspect (as matter) is provided directly through reason; however the holistic aspect (as energy) is directly provided through intuition.


However because the conventional mathematical paradigm formally reduces interpretation in a mere rational manner, the very notion of number as representing energy, thereby can have little meaning from such a perspective. Thus in the context of the accepted appreciation of number, the idea that number directly embodies an energy state has - literally - no intuitive resonance.

In other words it does not resonate with the standard appreciation of number (as mere form).

However in the dynamics of experience – when appropriately interpreted – a continual transformation takes place due to the inevitable interaction of (analytic) reason with (holistic) intuition.

So number itself - through the interactions of its analytic and holistic aspects - is itself subject to continual transformation.

Thus one effective way of interpreting the conventional (Type 1) approach is as that which represents the lowest possible energy state of number!

Therefore, though it may be informally accepted that a certain degree of psychological energy (through intuition) is necessary to fuel mathematical activity, in explicit terms such activity is interpreted in a merely rational fashion - literally - as formal material.


However this poses enormous problems for interpretation of the nature of the non-trivial zeros, for in truth these lie at the other extreme of understanding, as representing - as close as is consistent while still mainatining phenomenal form - the notion of number as representing pure energy states.

Therefore from the psychological perspective, due appreciation of these zeros requires a very refined interaction of both intuitive and rational understanding, where both can be seamlessly integrated with each other.

From the complementary physical perspective, the zeros likewise reflect pure transition states of energy (where any distinction as between matter and energy dissolves).

At a deeper level this entails that the very notion of number cannot be ultimately separated from either the physical or psychological domains, but rather represents the most fundamental interpretation (in phenomenal terms) of their intrinsic nature.

Expressed in an equivalent fashion, the standard (Type 1) notion is based on interpretation of numbers as independent entities (in quantitative terms).

However there exists an equally important – though still unrecognised (Type 2) treatment - based on the holistic interpretation of numbers as interdependent with each other in qualitative terms.

The non-trivial zeros – in this enlarged context of number appreciation – represent the marriage of both the quantitative (Type 1) and qualitative (Type 2) interpretations respectively of number, where both approach mutual identity with each other (and which can be referred to as the Type 3 treatment).

However once again, appropriate appreciation of their nature cannot be attempted through standard type analysis, which is as futile as trying to understand the chemical composition of water with mere reference to the existence of oxygen atoms!


So the message which needs to be continually reiterated is that the most radical revolution in mathematical history is now required to take place, where both its quantitative (analytic) and qualitative (holistic) aspects are recognised as equal partners.

And such a revolution can in no way take place through an attempted extension of the present paradigm to deal with the neglected qualitative aspect (as it is based on an utterly distinctive type of understanding).

This is the very reason why the extreme specialisation that has taken place with respect to the quantitative (Type 1) aspect is so damaging, as it has greatly eroded the distinctive intuitive recognition that is required for the Type 2 aspect.

Indeed so greatly has this ability been undermined that the very recognition of its possible existence has largely disappeared (especially within the recognised Mathematics profession).

This is why this necessary mathematical revolution will be initiated by genuine seekers of truth outside the profession, who can appreciate the proper nature of Mathematics within a much more comprehensive integrated perspective and thereby not remain bound by its current artificial restrictions.


The spread between primes can be seen as an inverse measurement of the independence of the number system which increases as we ascend the natural number scale.

A simple measurement for this spread is given by log t.

So for example in the region of 100 (i.e. t = 100), log t = 4.60517.. so that the average spread (or gap) as between primes is about 5. Thus, we would expect around 1 in 5 numbers to be prime.

However in the region of 1,000,000, log t = 13.81551. So the average spread is now about 14 with 1 in 14 numbers prime!

So, because between 100 and 1,000,000, log t has approximately trebled in value, we could therefore express this inversely by saying that the independence of the natural number system (in the regularity of the occurrence of the primes) has fallen to a third of its initial measurement.

Expressed in an equivalent manner, the probability of finding a prime in the region of 1,000,000 is about 1/3 of its probability in the region of 100.

However there is equally another side to the coin as it were in that the interdependence of the number system thereby steadily increases as we ascend the number scale. Therefore whereas in the region of 100 we would expect a run of approximately 4 (i.e. 5 – 1) composite numbers for every prime in the region of 1,000,000 we would expect about 13 (i.e. 14 – 1).

So therefore as the frequency of the primes (as the independent numbers without factors) decreases, the frequency of the composite natural numbers (as representing the interdependent numbers with factors) increases in an inverse manner. And for large t, this inverse relationship would steadily improve in precision!


However, whereas the linear number scale is the proper home of number notions as independent, the circular number scale is the appropriate home of number notions as interdependent.

So the interdependence of 4 numbers – as I have illustrated elsewhere - is expressed through the four roots of 1, which geometrically lie as equal points on the circle of unit radius in the complex plane.

Once again the Type 2 explanation is of a very subtle nature combining both quantitative and qualitative type interpretation of the same number symbols.
So for example in a 4-dimensional qualitative interpretation, the quantitative nature of the 4 roots 1, – 1, i and – i, as – relatively - separate, is matched by the holistic qualitative interpretation of these same roots as interdependent.

And the geometrical representation of these roots combines the notions of line and circle, with each root represented as a line drawn from the centre of the circle to its circumference.

So just as in quantitative terms we can give expression to the notions of line and circle, equally in qualitative terms we have complementary notions with respect to linear and circular type logical understanding respectively.

However if we wish to convert this Type 2 appreciation of the dynamic nature of independence and interdependence in a Type 1 fashion, then we can simply concern ourselves with the distance as between the equidistant points on the circle of unit radius.

Now as interdependence is inversely related to prime independence (as log t) with the circumference of the unit circle = 2π, therefore the gap between each point (on the circle as a measurement of interdependence) is given as 2π/log t (which as with the prime number theorem will become ever more accurate for large n).

Now the formula for the spread or gap as between the non-trivial zeros is given as 2π/log (t/2π).

However for large t, log (t/2π) approximates ever more closely to log t.
For example when t = 10^9000, log (t/2π) = 20721.4 and log t = 20723.3 (correct to 1 decimal place). So the two results are already very similar, with the relative deviations continually falling for larger t!

This means in effect that just as the pattern of primes represents the independent aspect of the number system (i.e. as numbers with no factors) the non-trivial zeros represent the opposite extreme of the interdependent aspect of the same system (where natural numbers increasingly tend to be composed of the product of prime constituents).

Put more precisely, whereas the conventional understanding of the primes represents interpretation, where the independent aspect (in their individual nature) and interdependent aspect (in their collective behaviour) are separate from each other, the non-trivial zeros – when appropriately understood – represent a completely new interpretation of number where both the independent and interdependent aspects of number are – relatively – identical.


However as the presentation of the non-trivial zeros is in linear format, this requires the ability to convert the true ordinal interpretation of interdependence (which relates directly to Type 2 understanding) in Type 1 terms (i.e. on an imaginary scale). Now the process of analytic continuation on the complex plane enables such a quantitative transformation. However ultimately any meaningful interpretation of the results arising, requires the holistic type appreciation associated with Type 2 understanding.

So this combined use of Type 1 and Type 2, which the non-trivial zeros encapsulate, represents what I refer to as Type 3 understanding.


So the non-trivial zeros of the Riemann Zeta Function, provide once again on an imaginary linear scale, the points where the natural and prime numbers coincide.

However as these largely relate to holistic rather than (linear) analytical type understanding, their true meaning cannot be grasped in the standard manner of mathematical interpretation.

Not surprisingly, a much simpler equivalent way therefore exists for expressing this coincidence of the primes and natural numbers through the Type 2 aspect of the number system.

Incredible Nature of the Zeta Zeros (16)

The conventional approach to the number system - as we have seen - is based on the attempt to view it in absolute terms as containing abstract independent entities with a static identity.

However if indeed numbers were independent in this sense, then strictly it would be impossible to establish their relationship with other numbers (which clearly implies interdependence).

So once again Conventional Mathematics can only deal with the interdependent notion through a gross form of reductionism which distorts its very nature.


Therefore the one viable way of addressing this issue is to fully accept the inherently dynamic relative nature of the number system in an interactive manner.

Thus from this perspective, numbers do indeed possess a quantitative aspect as independent. However this is now interpreted in a relative - rather than absolute - manner.

Equally numbers possess an (unrecognised) complementary qualitative aspect as interdependent. Whereas the quantitative aspect is associated directly with an analytic, the qualitative aspect is - by contrast - associated with a holistic type interpretation.

Thus the dynamic nature of number arises from the continual interaction of both analytic and holistic type aspects (which is in keeping with the authentic nature of mathematical experience).

So the very paradigm of Conventional Mathematics, which formally seeks to define number relationships in a merely quantitative manner, thereby considerably distorts interpretation of its true nature.

Thus properly conceived in dynamic terms we have both Type 1 (quantitative) and Type 2 (qualitative) aspects with respect to the interpretation of all mathematical symbols.


However this very recognition of two distinctive aspects to mathematical interpretation (with equal importance) then raises the question of consistency with respect to their combined interaction.

It is in this context that the solutions to the Riemann Zeta Function play a crucial role, for properly understood the non-trivial zeros that emerge, represent an unlimited set of numbers (as point singularities on the imaginary axis) where both the Type 1 and Type 2 aspects of the number system approach mutual identity.

So this set of numbers - representing the Type 3 aspect - relates to values where both quantitative (analytic) and qualitative (holistic) interpretations approach identity. Put another way - as befits a dynamic treatment - the non-trivial zeros represent points where both the quantitative (independent) and qualitative (interdependent) aspects of number are identical in a relative fashion.

Thus the non-trivial zeros (in the generation of this new Type 3 system of numbers) serve as the fundamental requirement for ensuring subsequent consistency with respect to all quantitative and qualitative interpretations (when considered in relative separation from each other).


When properly understood therefore, the non-trivial zeros are necessary to ensure the consistent use of axioms at a conventional mathematical level.
Of course this is not properly understood by practitioners at this level, precisely because of the formal adoption of a merely quantitative approach to interpretation!

So when the qualitative aspect is unrecognised, the very question of its consistent relationship with the quantitative does not arise.
So the quest to “prove” the Riemann Hypothesis continues unabated even though an enlarged more authentic framework of interpretation can quickly show why this in fact is not possible!


As we have seen, the relationship between the primes and the natural numbers (and the natural numbers and the primes) is just another way of expressing this ultimately important relationship as between both the quantitative and qualitative aspects (and the qualitative and quantitative aspects) of the number system respectively.

However when our very intuitions are firmly based solely on independent quantitative notions (even when attempting - as with the primes - to establish their relationship with the natural numbers) it is well nigh impossible to adequately convey in an intuitively accessible manner what the non-trivial zeros convey (from such a perspective).


Once again conventional notions are based on clear separation of the quantitative from the qualitative aspect. For conventional mathematicians - especially where serious research work is concerned - even the admittance of any unexplained qualitative aspect would be considered anathema!

Now this clearly poses a massive problem when the very nature of the non-trivial zeros lies at the other extreme of understanding, where both aspects are understood as identical (which again from a phenomenal perspective is necessarily in a relative sense).

So this requires forming an appropriate knowledge of the quantitative (analytic) aspect of number in a standard linear, also forming an utterly distinctive qualitative (holistic) notion of number in a circular manner (that equally applies circular logical notions) and then simultaneously relating the two forms of understanding so closely in understanding as to become identical.

There is even the further complication here, arising from the linear nature of the standard representation of number. This entails the conversion of the qualitative aspect, relating to a circular type understanding of the ordinal nature of number, indirectly in a linear fashion (i.e. on an imaginary scale).

And when you can do all this seamlessly, then you can perhaps appreciate what the non-trivial zeros truly represent!


Now I have demonstrated in many blog entries how this requirement relates to the complementary appreciation of two – relatively distinct – interpretations of the relationship of the primes to the natural numbers.

Again in the Type 1 approach, we interpret the primes in cardinal terms as the building blocks of the natural number system with all (except 1) expressed as a unique combination of prime factors.
So this relates directly to the quantitative notion of numbers as independent entities.

However this relationship is reversed in the Type 2 approach, whereby we interpret the natural numbers in ordinal terms as the building blocks of the primes, with these natural numbers (except 1) - expressed through its corresponding prime number of roots - representing a unique combination of ordinal components.

Thus in terms of the two-way understanding of each approach (Type 1 and Type 2) we face a critical paradox with respect to the fundamental nature of number, which cannot be reconciled in terms of either as considered separate in a relative manner.

So this paradox is only finally resolved in the Type 3 system where both the prime and the natural numbers (and the natural numbers and the primes) are understood as identical with each other (though strictly this position can only be approximated in phenomenal terms).


And once again the Type 3 system - or more correctly the Type 3 aspect of the number system - is represented through the non-trivial solutions for the Riemann Zeta Function.

What this implies is this! When properly realised, the truly complementary nature of the relationship between primes and natural numbers, leads to a directly opposite sequence of causation in the Type 1 and Type 2 interpretations respectively. Then like matter and anti-matter in physical terms, when both are combined, they fuse in a common psycho-spiritual energy (relating to greatly enhanced intuitive appreciation).

So ultimately rational understanding - in the separate Type 1 and Type 2 interpretations respectively of the relationship between the primes and natural numbers - becomes so refined and fleeting that it no longer even appears to arise in experience (through interpenetrating so seamlessly with intuition).

Therefore in truth, the fullest appreciation of the Type 3 aspect of the number system (relating to the non-trivial zeros) will require this new more comprehensive mathematical context, where the most advanced level of (holistic) contemplative awareness can seamlessly combine with a highly refined degree of (analytic) rational understanding.

In this new appreciation of the number system, numbers will no longer be viewed as abstract entities but rather as dynamic entities charged with enormous potential energy.

So at the one extreme, in the Type 1 aspect, numbers certainly appear as static and inanimate - merely quantitative - objects; however at the other extreme of the Type 3 aspect (again represented through the non-trivial zeros) they now appear as dynamic - and indeed - living entities charged with potentially unlimited energy.

Of course the consequent corollary in complementary terms is that the same zeros likewise represent potentially unlimited physical energy states! So, in this sense our very understanding of what is deeply implicit in earliest physical evolution, intimately depends on corresponding advanced psychological understanding, so that what was always deeply implicit in physical phenomena can now be made fully explicit through psychological appreciation of its original nature!


One key implication of this new perspective is that number can no longer be seen as divorced from the physical and psychological domains of reality but rather as necessarily deeply inherent within them as the ultimate explanation of their phenomenal existence.

And it is when this realisation starts to slowly dawn, that we will then begin to properly appreciate the truly enormous potential significance of this new enlarged appreciation of the number system with the non-trivial zeros lying there at its very centre.

Incredible Nature of the Zeta Zeros (15)

Let us briefly recap the present position.

In the Type 1 quantitative approach to number, we view the primes collectively as the building blocks of the natural number system, with each natural number (except 1) representing a unique combination of prime constituents in cardinal terms.
This Type 1 approach leads to a linear view of the number system interpreted in a merely quantitative manner.


In the Type 2 qualitative approach to number we view – in reverse fashion – the natural numbers individually as the building blocks of each prime number, so that for example 3 contains a 1st, 2nd and 3rd member, with now each prime - expressed though its roots - representing a unique combination of natural number constituents (except 1) in ordinal terms.
This Type 2 approach leads to a circular view of the number system interpreted in both a - relative - quantitative and qualitative manner (that are complementary).

So though the relationship between the primes and natural numbers (and natural numbers and primes) is unambiguous within each approach as separate, from an interdependent perspective, it is paradoxical.


Thus in the Type 3 approach, where both quantitative and qualitative aspects are identical with each other - or more correctly in dynamic phenomenal terms approach identity with each other - both the primes and natural numbers (and the natural numbers and the primes) likewise mutually reflect each other as identical.

And this mutual identity of primes and natural numbers in both qualitative and quantitative terms is expressed through the non-trivial solutions for the Riemann Zeta Function where ζ(s) = 0.

In other words the non-trivial zeros directly represent the Type 3 approach to number.

This Type 3 approach leads to a point view of the number system (combining both linear and circular notions as identical) in a manner that approaches a pure ineffable state.


The implications of all this could hardly be more fundamental.


In the deepest sense, the relationship between whole and part (which governs all phenomenal relationships) is rooted in the mathematical mystery of the connection of the primes to the natural numbers (and the natural numbers to the primes). When we attempt to view these as somehow separate, their relationship leads to two parallel worlds, which while consistent in each case, are fully paradoxical in terms of each other.

So in a very refined manner, the non-trivial zeros represent the solution to this great original paradox.

However it is a solution that still remains shrouded in great mystery, for there is no way to access the meaning of these zeros without entering deeply in experience into that very same mystery. So ultimately, it is only through the most advanced psycho spiritual development, where all phenomena can be viewed in an unattached manner as utterly transparent in pure contemplation of reality, that the meaning contained in these zeros can be fully appreciated.

Because they directly point to the full reconciliation of qualitative with quantitative type notions, this directly implies that such understanding requires the corresponding reconciliation to a very advanced level of intuitive with rational understanding (where both approach a perfect balance with each other).


And here is the other side of that great mystery which I wish to now stress!

As we have seen phenomenal reality (at all levels) is governed by two fundamental polarity sets.

Perhaps in the desire to maintain a more objective stance with respect to the nature of mathematical understanding, I have general emphasised - especially in the relationship of the primes to the natural numbers - the second polarity set relating to quantitative and qualitative (whole and part).


However the first relating to internal and external is equally important.

And again in the deepest sense, the answer to this fundamental mystery of the relationship between mind and matter is again revealed through the non-trivial zeros.

For what these zeros – when properly understood - directly imply, is that we cannot have objective knowledge of the ultimate nature of reality without this being directly united with corresponding subjective reality! In other words what we objectively can know about reality is ultimately inseparable from the psychological constructs we use to view this reality.

Though in one sense the non-trivial zeros were always deeply implicit at the heart of our number system, until Riemann’s pioneering work some just over 150 years ago, nothing was humanly understood regarding their incredible significance. Indeed their very existence remained unknown!

So is only through the evolution of knowledge to a sufficient degree, that we have at last begun to unearth number’s greatest secret.

Thus the non-trivial zeros constitute this great hidden treasure of the number system. Therefore we can now realise that it has long remained concealed under the veils of our conventional treatment of number. However we still have obtained but the most rudimentary glimpses into its great mystery.


So what at an objective level remains most deeply implicit in the number system (in its true original state) can only be made properly explicit through once again entering this state in its most advanced spiritual evolution. So ultimately both the origin and destiny of all evolution are realised in the same present moment (where both objective and subjective poles are fully united in experience). And it is this same present moment thet the mystery of the primes (in their relationship to the natural numbers) is finally resolved.

Now once again in the strictest sense, the non-trivial zeros do exist in phenomenal space and time. However they only do so in the most refined dynamic manner possible, as the gateless gate, as it were, that bridges the phenomenal world with ineffable reality.

So inherent in earliest phenomenal reality (once it becomes manifest) is the solution to the great paradox of number which thereby enables the universe to unfold with respect to both its quantitative and qualitative features.

And then in the actualisation of evolution in the pure contemplation of mystery, the explicit solution to this paradox of number again exists as the final divide between phenomenal and ineffable reality.


As we know a great deal of attention is now placed on the origins of the physical universe and the notion of a Big Bang starting it all.

However properly understood, the very notion of number is necessarily already implicit in any such phenomenal developments.

So the really important question – which seems to me largely ignored – is the explanation of the number Big Bang and how this capacity for order with respect to the number system (in both quantitative and qualitative terms) has emerged.
And right at the heart of this number Big Bang we find the non-trivial zeros.

As we saw in yesterday’s blog entry, The Riemann Zeta Function – when appropriately interpreted – provides the required framework through which the 3 aspects of the number system can be understood.

Therefore initially we can attempt to view all numerical results in a merely quantitative manner in Type 1 terms.

Secondly we establish two-way complementary results through the Functional Equation for analytic values of s > 1 on the RHS and corresponding holistic values for s < 0 on the LHS in Type 2 terms.

Finally we establish the mutual identity of both Type 1 and Type 2 systems in Type 3 terms through the midpoint of the critical region (0 < s < 1) where holistic and analytic interpretations directly coincide.

Then by setting ζ(s) = ζ(1 – s) = 0, we find the non-trivial zeros which define the Type 3 number system. This does indeed generate a set of complex numbers (with real part of a rational and imaginary part of a transcendental nature). However proper understanding of such numbers, relating to a common interdependence of quantitative and qualitative characteristics, is of the most dynamic nature possible. Here each number, in phenomenal terms, serves as but a fleeting mediator of an ultimate meaning that is ineffable. So once again we are at the other extreme here from the standard interpretation of number in a static abstract manner!

This Type 3 system then serves as the vital necessary condition for subsequent consistent interpretation with respect to both the Type 1 and Type 2 systems.

And without such consistency, phenomenal reality as we know it could not exist!

Incredible Nature of the Zeta Zeros (14)

As we have seen, from the enlarged dynamic perspective, every mathematical symbol can be given both a coherent analytic and holistic interpretation.

The very nature of the non-trivial zeros relates to the ultimate identity of both aspects.

So these zeros can be viewed as point singularities on the imaginary scale, drawn through ½ (on the real axis), where both analytic and holistic meanings with respect to each number involved, directly coincide.


Much has been written (to a tremendous level of detail) on the analytic nature of the zeta zeros.

However as yet, the holistic nature of these zeros has not been recognised.
And quite simply we cannot appreciate their true significance without deep interpretation of this crucially important aspect.

As we know the Riemann Hypothesis postulates that all the non-trivial zeros lie on the imaginary line drawn through ½.

The holistic significance of ½ derives from the Riemann Zeta Function. This – in the enlarged dynamic interpretation – establishes complementary relationships on the real axis as between number results on the RHS which can be given the recognised analytic interpretation (i.e. which are intuitively concur with linear interpretation) and corresponding results on the LHS which requires a distinct holistic interpretation (i.e. based on circular type reason).

Now these results are connected through the Riemann Functional Equation, which establishes a relationship as between the analytic interpretation of ζ(s) on the RHS and the corresponding holistic interpretation of ζ(1 – s) on the LHS.

So for example when s = 2, the Functional Equation thereby establishes a relationship as between the analytic interpretation of ζ(2) on the RHS = (π^2)/6 and the corresponding holistic interpretation of ζ(– 1) = – 1/12 on the LHS.

Thus the first i.e. ζ(2) = 1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + … = (π^2)/6 is intuitively in accordance with linear notions of magnitude; however the latter ζ(– 1) = 1 + 2 + 3 + 4 +… = – 1/12 corresponds rather to a more complicated circular type interpretation.


The pure analytic type interpretation for values of s > 1, thereby establishes a complementary relationship with corresponding holistic type interpretation for values of s < 0.

What is known as the critical region where 0 < s < 1, therefore establishes a relationship as between both analytic and holistic aspects with respect to the Functional Equation on both sides.


Now the clear condition for establishing the identity of both aspects is that s = ½, for it is at this value that ζ(s) = ζ(1 – s). So from this new enlarged perspective, the Riemann Hypothesis can be seen as the simple condition required to ensure the mutual identity of both the analytic (quantitative) and holistic (qualitative) interpretation of number. However no solutions for the equation ζ(s) = ζ(1 – s)= 0 exist, where s is defined solely in real terms.

The solutions of this equation relate to complex numbers, which must therefore of necessity must have a real part = ½. The imaginary part then varies along a line drawn through ½ on the real axis. So every solution of the form, ½ + it is identified with a corresponding solution of the form ½ - it.


The next step is to provide a holistic interpretation of the nature of the imaginary part. We have seen that the real part relates to the conscious aspect of interpretation in a linear rational manner.

This effectively means that number is viewed independently as quantitative! The corresponding imaginary part then relates to the unconscious holistic aspect that is directly intuitive and interpreted in a real circular manner (using both/and logic based on the complementarity of opposites).

This circular formulation is then indirectly conveyed as linear in an imaginary manner. What this means in effect is that even though the non-trivial zeros all appear as linear numbers (that literally fall on the same vertical line drawn through ½ on the real axis) their interpretation now relates to the opposite notion of the interdependence of number.

So this is the key issue that remarkably has yet to be properly addressed by the professional mathematical community in that it is strictly meaningless to attempt to view numbers in a merely absolute quantitative fashion as independent entities!

Properly understood, all numbers must necessarily possess both independent aspects (whereby they can be viewed as relatively separate from other numbers) and interdependent aspects (whereby they are seen to have a relationship with other numbers).


Therefore Conventional Mathematics in formally recognising merely the quantitative aspect of number operates in a grossly reduced manner leading ultimately to total misrepresentation of its corresponding qualitative nature.

Now this reductionism is especially exposed in attempting to understand the relationship of the primes to the natural numbers (and the natural numbers to the primes).

This relationship cannot be interpreted in a merely quantitative manner but rather as a relationship of quantitative to qualitative and qualitative to quantitative meaning respectively. This then requires establishing the ultimate condition for the mutual identity of both quantitative (analytic) and the qualitative (holistic) aspects of number.

So if we are to properly appreciate the true nature of the non-trivial zeros, we require a radically distinctive interpretation of number. Thus in the conventional interpretation – based on abstract notions of independence – the quantitative is clearly separated from the qualitative aspect, with numbers appearing in static terms as absolute entities. However we are now faced with the opposite extreme where numbers represent the ultimate state possible – consistent with maintaining a phenomenal existence – of the full interdependence of both their quantitative and qualitative aspects.

So the correct interpretation here of number is of such highly dynamic nature as to be bordering on what is completely ineffable. Indeed correctly understood this most fundamental notion of number – as the non-trivial zeros - represents the finest partition possible bridging the phenomenal world from what is utterly ineffable.

This dynamic nature can be further understood with respect to the transcendental nature of the imaginary parts of these zeros. When one looks carefully at the nature of a transcendental number, one realises that it necessarily combines both finite and infinite aspects (relating to quantitative and qualitative interpretation respectively).

This is clearly demonstrated through the nature of π (which is perhaps the best known transcendental number). Now π can most simply be demonstrated as the relationship of its circular circumference to its line diameter in quantitative terms. In a corresponding holistic qualitative manner, π represents the relationship between both circular and linear type understanding.

Now in quantitative terms the one point that is in common (as both the midpoint of the circle and its line diameter) coincides at the centre. In corresponding holistic terms the midpoint where both linear and circular type understanding are identified is at the centre. However this point - though having a location – is literally pointing to an ineffable reality. It is quite similar with the non-trivial zeros.

The transcendental nature of these zeros relates to the fact that they represent a relationship as between both the quantitative and qualitative notion of number (where both are identical). Therefore though it appears as non-intuitive in terms of conventional notions of number, the non-trivial zeros represent points on an imaginary number scale where both the analytical (quantitative) and holistic (qualitative) interpretation of number are identical.

Not surprisingly therefore, these zeros serve as the means of reconciling the prime numbers with the natural numbers (and the natural numbers with the primes). Therefore from one perspective we can use the non-trivial zeros to move from the general frequency of prime distribution (among the natural numbers) to their precise location; equally from the other perspective, we can use the knowledge of the prime numbers (distributed throughout the natural number system) to precisely locate the non-trivial zeros.

There therefore exists a two-way interdependence as between the non-trivial zeros and the prime numbers (in their relationship with the natural numbers). And once again it has to be emphatically stated that it is strictly meaningless to attempt to understand such a key relationship in the absence of genuine holistic notions!

Thus instead of the existence of just one number system, properly understood in dynamic interactive terms, we now have three! So firstly we have the Type 1 system relating to the analytic understanding of number in a relative quantitative manner.

Thus though we can cover here all the same ground as in Conventional Mathematics a more refined interpretation operates (whereby implicit recognition is given to the “shadow” qualitative aspect of number).


Secondly we have the Type 2 system relating to the holistic understanding of number in a dynamic qualitative manner.

More correctly this system relates to the dynamic understanding of number entailing the interaction of both quantitative and qualitative aspects (where both still enjoy a relative degree of separation).

Whereas the Type 1 system is of a linear (1-dimensional) nature, the Type 2 system relates potentially to all other finite dimensions (which are unlimited in scope).


The Type 3 system then establishes the mutual identity of both quantitative and qualitative aspects.

The importance of this system is that it establishes the means through which both quantitative and qualitative aspects can be consistently related to each other in all phenomenal contexts.

This is something that is totally taken for granted in Conventional Mathematics (which gives no formal recognition to the qualitative aspect). However properly understood, underlying all mathematical axioms is a prior assumption that quantitative (relating to finite) and qualitative (relating to infinite) notions can be consistently related.

For example this assumption underlies mathematical proof whereby what is proven for the general (infinite) result is assumed to apply to all specific (finite) cases within its class!

In terms of the Type 2 system it represents the extreme case where dimensional interpretation is infinite.


So in this necessary reformulation of the number system Type 1 and Type 3 exist as two extreme cases.

Once again in the Type 1, quantitative is clearly separated from qualitative meaning in the independent (analytic) interpretation of number. In the Type 3, quantitative is directly united with quantitative meaning in the holistic interpretation of both the quantitative and qualitative aspects of number as fully interdependent.

Then between these two extremes in the Type 2 system quantitative and qualitative aspects are understood in dynamic relative terms (combining both elements of independence and interdependence).

The Riemann Zeta Function – when appropriately interpreted – provides the required framework through which all 3 aspects of the number system can be understood.

Incredible Nature of the Zeta Zeros (13)

We have seen that – appropriately understood – all mathematical understanding necessarily entails dynamic interaction patterns governed by the two fundamental sets of polarities (that underlie phenomenal reality).

So an external aspect is always related to an internal aspect (and an internal to an external). Likewise wholes are always related to parts (and parts to wholes) in a manner that entails the necessary interaction of quantitative and qualitative aspects (in relative terms).


Therefore, a great reductionism pervades present conventional interpretation of Science (and especially Mathematics).

Here the attempt is made to view:

(a) the external aspect of reality in abstraction from the internal in a merely objective type appreciation:

(b) the relationship between wholes and parts in a grossly reduced fashion amenable to mere quantitative interpretation.

However I have been at pains to show in these blog entries how this approach unravels completely in the quest to understand the fundamental mathematical problem with respect to the relationship as between the primes and the natural numbers.
In other words the coherent interpretation of this relationship (which ultimately will always remain shrouded in mystery) requires placing Mathematics within its true dynamic context.

Indeed this then reveals – apart from the long recognised Type 1 quantitative aspects - a hitherto unrecognised shadow interpretation (i.e. Type 2) which is directly based on interpretation of the dynamic interaction as between these fundamental polarities.


So the marvellous revelation can then start to dawn that very mathematical symbol can be given two distinctive interpretations in both an analytic (Type 1) and holistic (Type 2) manner.

Then the true nature of mathematical activity can be understood as the continual interplay of both Type 1 and Type 2 aspects. Through this interaction the Type 1 aspect (as quantitative) is also revealed to possess a Type 2 aspect (as qualitative); likewise the Type 2 aspect (as qualitative) is also revealed to possess a Type 1 aspect (as quantitative). Indeed I used this appreciation to show how an alternative Prime Number Theorem, Riemann Hypothesis amd Zeta (2) zeros can be shown to exist in Type 2 terms!


Now this might appear remarkably reminiscent of the nature of Quantum Mechanics in Physics.

So formerly matter was understood as composed of constituent particles. Then Quantum Mechanics revealed that at the sub-atomic level matter exists in both particle and wave form (in complementary manner).

Then even more strangely it was revealed that the particle aspect is also wave-like and the wave form also particle-like.

However the even deeper connection that has not properly understood is that this behaviour of matter at a sub-atomic level is itself rooted in the prior nature of mathematical activity (which is thereby inherent in such behaviour).

So the complementary nature of sub-atomic matter (indeed strictly all matter) as possessing complementary particle and wave expressions itself reflects the deeper complementarity of both quantitative and qualitative aspects with respect to all phenomena. And this is rooted in the very nature of mathematical activity when appropriately understood in dynamic terms.


Therefore though Physics has been forced to accept at an experimental level the strange reality of quantum behaviour, it still mistakenly attempts to view this through a mere quantitative lens which is deeply misleading. In fact this is the key reason why such findings are counter intuitive in terms of the prevailing paradigm.

So both quantitative and qualitative aspects necessarily interact with respect to all mathematical (and indeed extended scientific) behaviour.

This means that we need to redefine the way we interpret reality as containing both real (analytic) and imaginary (holistic) aspects.

So from this dynamic perspective all physical (and of course mathematical) reality is necessarily of a complex nature. It only appears as (solely) real when we attempt to understand it from a merely reduced quantitative perspective.

So we have so far learnt only to give complex notions (with real and imaginary parts) mere quantitative expression. However they equally can be given qualitative expression and in this context the imaginary part relates to the holistic aspect of interpretation.


Now when we switch briefly to the Riemann Zeta Function we can perhaps appreciate a key problem with respect to its interpretation.

As Conventional Mathematics is explicitly based on mere quantitative interpretation, it thereby can only attempt to understand the complex plane (on which the Zeta function is defined) from this limited (quantitative) perspective. However the qualitative interpretation of the mathematical complex notion implies that both real (analytic) and imaginary (holistic) aspects be both combined.

And as we have seen the remarkable fact is that Conventional Mathematics is powerless to deal with holistic notions (relating to interdependence) in an authentic manner. As its very rationale is based on abstract notions of independently existing entities, it can only approach holistic notions (requiring a genuine appreciation of the nature of interdependence) in a grossly reduced manner.


As is well known the only value for which the Riemann Zeta Function remains undefined in quantitative terms for the dimensional value (where s = 1).

Now as I have stated on so many occasions (though the penny may yet have to drop as it were), the only value for which the Riemann Zeta Function remains undefined in qualitative terms is for the same dimensional value (where s = 1).

Now in this qualitative context 1 here relates to linear (1-dimensional) method of interpretation which defines the very nature of Conventional Mathematics.

Thus the clear implication of this is that the Riemann Zeta Function – when appropriately understood – remains uniquely undefined when approached from the conventional mathematical perspective. And the very reason for this this is that it lacks any holistic notions of interdependence (which is fundamental to understanding the key relationship between the primes and natural numbers).


Now in qualitative terms for any dimensional value of s (≠ 1) a dynamic relationship as between quantitative (analytic) and qualitative (holistic) notions is implied.

Therefore once again the one type of mathematical approach that misses the target completely in this crucial respect is that of Conventional Mathematics (which is so wrongly identified in our culture as the only valid form of mathematical interpretation!)


I have emphasised a great deal the relationship between quantitative and qualitative (which is required to avoid merely reducing wholes - in any context - to constituent parts. This therefore represents one of the fundamental polar sets of relationships (governing all phenomenal reality).

However, the other set relating to internal and external is equally important!
Though this seems to be quickly forgotten when doing Mathematics, any relationship we identify in external (objective) terms necessarily reflects a certain psychological interpretation that is - relatively - of an internal nature.

So, in dynamic terms, we do not have just mathematical symbols, hypotheses, and relationships etc. existing in (absolute) objective terms. Rather we always have a dynamic relationship as between objects and mental interpretations that are relative in nature. And once we recognise that the standard conventional interpretation with respect to mathematical “objects” represents but one of many possible interpretations, then perhaps we can better appreciate how relative in fact is mathematical truth!

Thus once again we have created the mistaken impression that Mathematics represents a somewhat absolute version of “objective” truth through misleading adopting the assumption that only one valid means of interpretation can exist! And this assumption is utterly without foundation.

So an extremely important implication of setting Mathematics in its correct dynamic perspective is the realisation that not alone is it necessarily inherent in all physical phenomena (as an essential means of their encoding) but also that it is likewise necessarily inherent in all psychological phenomena (as a similar means of encoding).

In other words – properly understood in dynamic terms – both the physical and psychological aspects of phenomenal reality are necessarily of a complementary nature.


Therefore Mathematics – as what is most essential to both aspects – likewise necessarily applies to both the physical and psychological aspects of reality in equal measure.

I can say this with considerable confidence. My own applications of Holistic Mathematics initially related more to psychological – rather than physical – reality.
Indeed some 20 years ago I attempted to portray human development (including the advanced contemplative stages) from a holistic mathematical perspective.

So I set out to show how the basic structure of every stage of psychological development is defined uniquely in holistic mathematical terms. It was only later that I began to properly discover how all these stages necessarily had a complementary interpretation in physical terms.


Some of this work indeed has a deep relevance for greater appreciation of the nature of the non-trivial zeros.

Though it is now being accepted that the Riemann zeros do indeed have important implications for certain physical systems, I can see no recognition of their equal importance for the understanding of advanced psycho-spiritual states of development.
And once again because in dynamic terms the physical and psychological aspects of reality are complementary, we will never properly appreciate the physical nature of the non-trivial zeros (without equal recognition of their corresponding psycho spiritual significance).

Incredible Nature of the Zeta Zeros (12)

There is a very important consequence arising from yesterday’s entry!

As I have stated, considerable reductionism exists in conventional Mathematics with respect to ordinal notions (which basically are identified on cardinal lines).
So the natural numbers in cardinal terms are readily identified with the corresponding members from an ordinal perspective;

So 1, 2, 3, 4 etc. are readily ranked as 1st, 2nd, 3rd, 4th and so on.


However the true original relationship relates to that between the primes and natural numbers (and natural numbers and primes) which is quantitative as to qualitative (and qualitative as to quantitative etc.)


Now once again in the expression a^b, a can be defined as the base number and b as the dimensional number (which are quantitative as to qualitative in relation to each other).

However when we take each in isolation, they appear solely as quantitative numbers.
So therefore with respect to the base number, the relationship as between cardinal primes (and ordinal natural numbers) ranks 2 (as 1st), 3 (as 2nd), 5 (as 3rd), 7 (as 4th) etc.

Likewise with respect to dimensional numbers (as powers or exponents) the relationship as between cardinal primes (and ordinal natural numbers) ranks again 2 (as 1st), 3 (as 2nd), 5 (as 3rd), 7 (as 4th) etc.

However in dynamic relationship the each other what seems unambiguous (as within isolated frames) becomes deeply paradoxical.

The upshot of this is that the consistency of cardinal and ordinal relationships does not derive from either the primes or natural numbers in isolation, but rather from their simultaneous relationship (strictly in a phenomenal context, the dynamic interaction between both approaching simultaneity).


This highlights the significance of the non-trivial zeros as the basic requirement for any consistent ordering of numbers (in either cardinal or ordinal terms).

Again as the Riemann Hypothesis relates to a particular statement regarding the nature of the non-trivial zeros, it is futile attempting to either prove or disprove such a statement in conventional mathematical terms.

This again is due to the fact that the basic condition enabling such axioms to be used in a consistent manner itself depends on the prior nature of the non-trivial zeros.
So we cannot meaningfully approach the validity of a statement from axioms which already depend on this statement!

As I stated yesterday the importance of the non-trivial zeros can hardly be overstated as they once again serve as the basic requirement for ensuring consistency with respect to both the quantitative and qualitative interpretation of mathematical symbols in all phenomenal circumstances (with both aspects ultimately identical in an ineffable manner).


However we now have another surprising result!

The non-trivial zeros derive from the conventional (Type 1) quantitative aspect of approach to Mathematics.

So far we have been at pains to indicate that they cannot however be properly interpreted without reference to a complementary Type 2 aspect!

However when we follow this Type 2 aspect to its own logical conclusion, it leads to the generation of an alternative set of “non-trivial zeros”.


As we have seen, both Type 1 and Type 2 systems appear as the direct inverse of each other. So the natural number as base quantity in Type 1 represents the dimensional number in Type 2; likewise the dimensional number in Type 1 represents the base quantity in Type 2.

So again for example, 5^1 in Type 1, is represented as 1^5 in the Type 2 system!
Now the Riemann Zeta Function (which I refer to as Zeta 1) is defined with respect to the Type 1 system involving a sum of natural number terms defined with respect to a fixed complex number s (as dimension).

Then we set the infinite series of terms for ζ(s) = 0 to derive the solutions for s (with both trivial and non-trivial solutions).


So for convenience this equation ζ(s) = 0
can be written,

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

However we can derive a fascinating alternative Type 2 formulation - which I refer to as Zeta 2 - where the base natural number quantities now appear as dimensional numbers and the dimensional number s as base quantities.

So we start with 1 = s^n i.e. 1 – s^n = 0 in order to find all solutions for s!
As one of these i.e. 1 = s, is not unique we divide 1 – s^n by 1 – s to get
1 + s + s^2 + s^ 3 + …. + s^(n – 1) = 0.

Multiplying by s (giving s = 0 as a solution) we get,

s^1 + s^2 + s^ 3 + …. + s^n = 0;

This is Zeta 2.

The natural numbers as base quantities in Zeta 1 are now number dimensions in Zeta 2.
Also the number dimensions in Zeta 1 are now base quantities in Zeta 2 (with – s replaced by s).

Also whereas Zeta 1 is an infinite series, Zeta 2 is finite though the number of terms n can be extended indefinitely.

So the solutions for s (which represent the n unique roots of 1) represent a second set of zeros (i.e. Zeta 2 zeros).

Interestingly however, whereas the solutions for s (in Zeta 1) represent complex numbers with an imaginary part that is transcendental, in the case of Zeta 2, the solutions for s represent complex numbers that are algebraic irrational! (Indeed it would seem that this may well be an important clue explaining the differing nature of ζ(s) for (positive) even and odd integer values of s respectively!


Properly understood, the non-trivial zeros (for Zeta 1) cannot be understood in isolation from the corresponding Zeta 2 solutions (and vice versa).

One fruitful way of understanding their respective role is as follows!

Whereas the Type 1 show us how an incredibly dynamic set of numbers is already inherent in the number system (that conventionally appears static and absolute), the Type 2 show us how understanding of mathematical symbols, that starts in (1-dimensional) static terms, can be progressively raised to higher dimensional interpretation (which likewise becomes incredibly dynamic in nature).

So the understanding of the number system itself as dynamic and interactive in nature, properly requires a method of understanding that is also dynamic and interactive relating both quantitative and qualitative aspects.

So the ability to properly see the non-trivial zeros as already immanent or inherent in the conventional number system, requires the corresponding ability to fully transcend conventional linear interpretation (of this system). And the Zeta 2 zeros relate directly to the precise qualitative nature of all these higher dimensional interpretations (which are potentially unlimited) which then serve as the dynamic means for appropriate quantitative interpretation of the nature of the (recognised) Zeta 1 zeros!


One further interesting point!

The designation of the Zeta zeros in Type 1 for s = – 2, – 4, – 6, … etc. as trivial represents an unfortunate misnomer due to lack of appreciation of the role of the Zeta 2 Function.

In fact the true explanation of the quantitative nature of the “trivial” zeros in Type 1 springs directly from the qualitative appreciation of the “non-trivial” zeros in Type 2.

Likewise appreciation of the qualitative nature of the “trivial” zero in Type 2 (where s = 1) springs directly from appreciation of the quantitative nature of the “non-trivial” zeros in Type 1!