The True Nature of the Zeta Zeros (2)

In yesterday's blog entry, I attempted to outline the true nature of the two sets of zeta zeros (Zeta 1 and Zeta 2).

As both of these sets are ultimately fully complementary with each other, they can only be properly understood therefore in a dynamic interactive manner, where they are seen to play a truly key role ensuring consistency with respect to all subsequent number operations.

So properly understood number - and indeed all mathematical relationships - entail both quantitative and qualitative aspects.
Whereas the quantitative relates to the notion of independence, the qualitative - by contrast - relates to the complementary notion of interdependence (i.e. where numbers are defined in relation to each other).

Thus number is necessarily of a dynamic relative nature entailing the interaction of both quantitative and qualitative aspects.

The relationship between quantitative and qualitative in turn leads to the corresponding interaction as between the cardinal and ordinal aspects of number, which are mutually distinct and cannot be successfully reduced in terms of each other.

So the key overriding issue with respect to the number system - and again by extension all mathematical relationships - is the prior need to ensure complete consistency in the relationship of both cardinal and ordinal meaning (reflecting its quantitative and qualitative aspects).

Here the relationship between the primes and the natural numbers is of the utmost importance, for it is through this relationship, entailing cardinals and ordinals, that both aspects (quantitative and qualitative) are mediated in a bi-directional manner.

And at the heart of this relationship between the primes and natural numbers lies a direct paradox!.
Whereas from the cardinal perspective, the primes appear as the independent building blocks of the natural number system (in quantitative terms), from the corresponding ordinal perspective, each prime is already necessarily defined by its natural number members (in a qualitative manner).

Thus, when one allows for recognition of both quantitative and qualitative aspects (which in truth are equally important) the primes are seen in dynamic terms to inherently combine two extreme complementary tendencies (with respect to independence and interdependence respectively).

Thus from one perspective, the primes appear as the most independent of numbers (in an absolute unchanging manner).
Then from an equally valid alternative perspective, the primes are understood as the most interdependent of all numbers (in a purely relative fashion approaching complete ineffability).


Therefore we must properly view the relationship between the primes and natural numbers (and natural numbers and primes) in a bi-directional interactive manner, whereby their mutual identity continually changes as between quantitative and qualitative (and qualitative and quantitative) aspects.

And remarkably this is what happens in actual experience, where appreciation of the cardinal and ordinal aspects of number keep interchanging!

However the consistent interplay of both aspects implicitly requires another two key sets of numbers, which very much mirror both cardinal and ordinal aspects i.e. Zeta 1 and Zeta 2 zeros.

Indeed in a very important psycho spiritual sense, these two sets represent the unconscious shadow counterparts to the consciously recognised cardinal and ordinal aspects of number.

Thus the role of these two sets of zeros is to enable conversion, in a fully consistent manner, as between both quantitative and qualitative (and qualitative and quantitative) aspects.

So, as we saw yesterday, the Zeta 1 (Riemann) zeros provide an (indirect) means of conversion (via the primes) from the recognised quantitative notion of cardinal numbers to their (unrecognised) qualitative counterpart notion.

Likewise from the opposite perspective, the Zeta 2 zeros again provide an (indirect) means of conversion (via the natural numbers) from the inherent qualitative nature of the ordinal numbers to their (unrecognised) quantitative counterpart notion. Now, as we have seen, this entails the simple task of obtaining roots of 1, which of course is well known in conventional mathematical terms. However the deeper appreciation, that these can in fact represent the important quantitative conversion of ordinal type notions, is completely missing!

Thus once again the key role of the two sets of zeros (Zeta 1 and Zeta 2) is to enable the perfect conversion in two-way fashion from the quantitative to the qualitative (and qualitative to quantitative) aspects of number.

Without belief in the guarantee of such consistency we would have no reason to belief in the subsequent consistency of  any mathematical operation and the whole edifice would be thereby built on sand.

However this consistency, attained through the mediation of the zeta zeros, cannot be proved or disproved in a conventional mathematical manner, as its very acceptance is already implicit in the use of standard mathematical axioms.

So ultimately a massive act of faith underlies the whole mathematical enterprise.

However in moving towards a true knowledge of the number system, one must be prepared for a radical change in customary beliefs.

Indeed the most profound fact about our number system is that ultimately it operates in a totally synchronous manner, which can only be properly approached through holistic - rather than analytic - interpretation.

Associated with this is an appreciation of the extraordinary status of the dynamic interactive nature of primes as representing the purest relative form of knowledge in the phenomenal universe.

G.H. Hardy must now be surely turning in his grave!

The True Nature of the Zeta Zeros (1)

Once more, we return to this key issue with the attempt to provide a simple intuitively accessible explanation of the nature of the two sets of zeta zeros i.e. Zeta 1 and Zeta 2.

As perhaps, the latter set is easier to appreciate, we will start with the Zeta 2 zeros. Then through a complementary form of interpretation, the true nature of the better known Zeta 1 (i.e. Riemann) zeros can then be revealed in a coherent manner.


The Zeta 2 are intrinsically related to the ordinal nature of number. So corresponding for example to the cardinal notions of 1, 2, 3 we have the corresponding ordinal notions of 1st, 2nd and 3rd respectively..

However, whereas the cardinal in this context relate directly to the quantitative aspect of number, the ordinal notions, by contrast are directly associated with the corresponding qualitative aspect.

So once again, the cardinal relates the individual notion of each number as independent (in a quantitative manner).
By contrast, the ordinal relates to the collective notion of a group of numbers as interdependent with each other (in a qualitative manner).

However, crucially, conventional mathematical interpretation (in formal terms) is of a grossly reduced nature (i.e. where in every context, qualitative notions are necessarily reduced in quantitative terms).

Thus in the conventional treatment of ordinal number notions, their inherent qualitative nature is not explicitly recognised, but rather referred to in more neutral terms as representative of number rankings that correspond directly with cardinal notions.

Now from one legitimate perspective, this may indeed appear to be true.

Thus when we express ordinal notions with respect to the infinite number system (in linear fashion), no problem seemingly arises with 1st, 2nd and 3rd (as in our example) seeming to directly correspond in an unambiguous fashion with the absolute cardinal notions of 1, 2, and 3.

However this all subtly changes when we switch to the expression of ordinal notions within a finite group context, where they are now revealed to be of a strictly relative nature.

So for example within the (default) group of 3, 1st, 2nd and 3rd have a relative identity (that can be expressed in circular fashion). So we could represent the 3 ordinal positions as 3 equidistant points on the unit circle. However, depending on context, 1st, 2nd and 3rd could be equally identified with each point.

If, for example, we then consider the (default) group of 5, 1st, 2nd, 3rd, 4th and 5th now equally have a relative identity, that can again be expressed by 5 equidistant points on the unit circle.

However on reflection, it becomes clear that the very meaning of 1st, 2nd and 3rd in this latter group has now changed.

And there is no finite limit to such change, as we can keep increasing the size of the group with the relative meaning of the rankings likewise changing!

In fact this can readily be appreciated in conventional situations (though the enormous mathematical significance of this is not properly grasped).

For example if I told you that I entered a competition and came 2nd (out of 1000 entrants), you might indeed be impressed.

If however if in fact there had only been 2 entrants, finishing 2nd would not however have constituted much of an achievement.

So in forming judgement as regards ordinal number rankings, implicitly we acknowledge that the relative significance of such rankings changes (depending on the overall number of the group).

Thus the ordinal notion of 2nd (to give just one example) can be given an unlimited set of relative interpretations (depending on the number of members in the finite group to which it belongs).

And this of course equally applies to every ordinal notion which likewise can be given an unlimited set of relative interpretations.

Now the significance of the primes in this context is that the ordinal interpretation of its natural number members is always unique for such groups.

However, we still have the big problem of finding a satisfactory quantitative manner of uniquely expressing these relative ordinal notions!

This is where the Zeta 2 zeros come in!

By defining the unit circle in the complex plane, the various ordinal notions (unique for each prime group) can be simply obtained through obtaining the corresponding prime roots of 1 (easily obtained though the Euler Identity).

Thus is we take the simplest case, the prime group will be 2 which enables in this context consideration of both a 1st and 2nd member.

So the 1st and 2nd roots correspond to 1^1/2 and 1^2/2 respectively (in the Type 2 number system)
=  – 1 and + 1 respectively.

Now one of these roots i.e. + 1, is not unique and this is true in every case. Thus, here, the 2nd (in the case of 2) has an absolute rather than relative meaning (which is denoted by + 1).

What this means is that once the position of the 1st member is chosen, the position of the 2nd (in the case of 2) is then automatically known in an absolute manner.

So the various prime roots of 1 correspond to the simple equation, 1 – x^t  = 0 (where t is initially prime).

However one of these roots, i.e. 1 – x, = 0 is of an absolute nature and not unique!

Therefore to find the unique (i.e. truly relative) solutions we divide by 1 – x^t  by 1 – x and solve thereby for 

1 + x¹  + x² + ... + x^{t – 1} = 0 (where t is initially prime).

Now this can ultimately be extended for all natural numbers from 2 to t (due to the unique relationship between the primes and natural numbers).

These solutions constitute the Zeta 2 zeros.

So the important role of the Zeta 2 zeros is that they enable an (indirect) quantitative means of uniquely expressing the purely relative nature of ordinal rankings, except in the default case i.e. the 
t^th member of a group of t).

So remarkably they provide the means of consistently converting (in this context of ordinal numbers) from qualitative to corresponding quantitative expression.

Furthermore this connection is of a holistic - rather than strict - analytic nature.

Thus each root - and we must always include the non-unique root of 1 here as relative considerations can only be made with respect to what is initially understood as independent - has a certain (relative) independent identity (as separate) while the addition of roots displays (relative) interdependence (as combined).

Thus in the case of the two roots of 1, – 1 and + 1 are relatively independent of each other (in quantitative terms).

However the combined sum i.e. – 1  + 1 = 0 displays corresponding (relative) interdependence, which is strictly of a qualitative nature.  And this is exemplified by the fact that universally, the sum of roots of 1 (representing the natural number ordinal members of a group) = 0.


By a complementary form of understanding - and ultimately both approaches are completely complementary in a dynamic interactive manner -  we can now perhaps explain simply the key significance of the corresponding Zeta (i.e. Riemann) zeros.

As we know, from the cardinal perspective, the primes are the building blocks in quantitative terms of the natural number system.

Now once again we start off with recognition of the individual primes as fully independent numbers (in an absolute manner).

However on deeper reflection - just as we have already seen with the ordinals - the primes can be shown to possess a merely relative identity.

This appreciation comes through recognition of the - formally unrecognised - qualitative aspect of  multiplication.

For example we may initially understand the first  two primes i.e. 2 and 3 as number quantities in an absolute manner.

However when we multiply these two numbers (i.e. 2 * 3) a qualitative - as well as quantitative - transformation is involved.

The first clue to this comes from looking at the operation in a geometrical manner. So the representation of 2 * 3  would entail a rectangle (measured in square i.e. 2-dimensional units).

However, this qualitative transformation of a dimensional nature, is simply ignored in the conventional treatment of number multiplication. So from this reduced perspective 2 * 3 = 6 (i.e. 6¹).

In this way, we then form the utterly misleading impression that all the composite natural numbers can likewise be defined in absolute terms (in a merely quantitative manner).

However the more profound appreciation of the true nature of multiplication, entails both the notions of number independence (as quantitative) and number interdependence (as qualitative).

So for example if one lays out two rows of similar coins (with 3 in each row) clearly one must recognise the (quantitative) independence of each coin. However, for multiplication to validly take place, equally one must recognise that the coins in each row can be placed in mutual correspondence with each other (as possessing a shared common identity) So this latter recognition relates to the qualitative aspect. Therefore in the multiplication operation i.e. 2 * 3, 2 here defines the mutual correspondence (i.e. interdependent identity)  of the 3 coins in each row!

So all multiplication implicitly always entails this vital qualitative aspect (which in conventional mathematical terms remains completely unrecognised.

Thus quite simply we cannot reconcile the two operations of addition and multiplication without proper recognition of both the quantitative and qualitative aspects of number.


Now just as earlier in the case of  ordinal numbers we saw - how for example, the notion of 2nd could acquire an unlimited number of relative interpretations depending on the size of the number group to which it is related - likewise the cardinal notion of 2 can acquire an unlimited number of relative interpretations, through the process of multiplication.

Thus in the case of 2 * 2  = 4, 2 acquires a new relative identity as a unique factor of this composite number.

Then in the case of 2 * 3 = 6, 2 and indeed - a different context 3, likewise attain  new identities as unique factors of 6.

Thus as all prime numbers can in principle be combined without limit as factors of resulting natural numbers (entailing the qualitative notion of interdependence), therefore all primes possess unique relative identities that are ultimately without finite limit.

So just as the Zeta 2 zeros provide a unique means of indirectly expressing the unique inherent qualitative identity of ordinal numbers in an (indirect) quantitative manner, the Zeta 1 (Riemann) zeros, provide a corresponding unique means of expressing the inherent quantitative identity of cardinal numbers in an (indirect) qualitative manner.

Now you might remember how I defined the two aspects of the number system as Type 1 and Type 2 respectively.

Whereas the Type 1 related to a base quantitative notion (with respect to the default dimensional value of 1), the Type 2 related to a dimensional qualitative notion (with respect to the default base value of 1).

So the Zeta 1 zeros relate directly to dimensional values that express the hidden qualitative aspect of the primes (which arises through multiplication with other primes).

In this context, I  have mentioned before how a remarkably close relationship connects the frequency of the Zeta 1 (Riemann) non-trivial zeros with the corresponding frequency of the natural factors of the composite numbers.

Thus we can use knowledge of the frequency of the Zeta 1 zeros (up to t on the imaginary scale) to accurately predict the corresponding accumulated frequency of natural factors of the composites up to n (on the real scale) where n = /t/2π.

So the Zeta 1 (Riemann) zeros provide a remarkable way of expressing (indirectly) in precise numerical format, the hidden qualitative transformations that are involved through the multiplication of primes.

In this way, they express the purely relative nature of the primes (ultimately as pure energy states) which represents the opposite extreme to conventional quantitative notions where they appear as rigid and absolute.

And, as before, the nature of these zeros is strictly of a holistic nature.

In the natural number system we get a considerable amount of discontinuity in the movement as between the primes (with no factors other than themselves and 1) and the composite natural numbers (with 2 or more factors), which increases as we ascend the number scale.

The Zeta 1 zeros therefore represent a continual smoothing out of this discontinuity with locattions taken, so that in the identity of each zero, opposites are reconciled such as between notions of randomness and order and primes (without factors) with composite natural numbers (with factors).

So this reconciliation of opposites (in the identity of each zero) represents the qualitative aspect of number behaviour from this complementary perspective. Then the combination of all such zeros represents their quantitative aspect (which can be used to eliminate deviations in the general calculation of the frequency of primes).

I cannot stress however how far we have moved here from conventional mathematical understanding (based on mere analytic interpretation where the quantitative aspect is clearly divorced in absolute terms from its corresponding qualitative aspect).

Here we have arrived at the other extreme of pure holistic interpretation (where both quantitative and qualitative aspects are fully harmonised with each other in a truly relative manner).

And as it stands, there is no way whatsoever of grasping the true significance of the zeros (Zeta 1 and Zeta 2) from the conventional mathematical perspective (as it gives no formal recognition  to the holistic aspect of mathematical symbols).

In fact in their most profound sense they represent (both sets) a mysterious alchemy whereby we can convert consistently in two-way fashion from both qualitative to quantitative format (and quantitative to qualitative) with respect to the interpretation of number.  

Number as the Theory of Everything

It should be clear that when both the quantitative and qualitative aspects are equally recognised, number thereby assumes an extraordinarily important role as the dynamic encoding of all created phenomena. So, literally everything, in both physical and psychological terms is ultimately encoded in number! 

Now, as all the great mystical traditions attest, the ultimate nature of reality is ineffable. One could equally say that the ultimate nature of number is ineffable and can only thereby assume a phenomenal identity, when some degree of separation exists as between its opposite polarities.

When attempting to approach this ineffable nature, the holistic meaning of 1 and 0 are often employed, where now both are understood as fully identical with each other.  So the Eastern traditions especially emphasise ultimate reality as a void or nothingness as potential for all form (which entails the holistic notion of 0); by contrast this ultimate reality is more often represented in Western traditions as a union of all form (entailing the holistic notion of 1). 

However in both East and West, the recognition exists (in varying degrees) that the two notions mutually imply each other, so that this reality is truly a plenum-void, where both union and nothingness i.e. as emptiness, are inseparable.

However the full realisation  of such interdependence is truly mysterious. So we can only phenomenally recognise  the holistic notions of 1 and 0, if they are already separated from each other (in some measure).

Thus we must always approach absolute meaning, without ever being capable of fully attaining it, from this phenomenal perspective (however refined in understanding).

However the importance of the mathematical symbols used here, is that they form the most refined bridge possible, as it were, as between phenomenal and ineffable reality.

So in the deepest sense, when appropriately understood, number provides the best interrmediary for scientific understanding of the nature of the wonderful creation, that we so mysteriously inhabit. The intrinsic nature of this reality is thereby encoded in number as its fundamental "genes" (once we learn how to decipher this code).

And as we have seen, the key step here is the recognition of the inherently dynamic nature of number, entailing the interaction of both its quantitative and qualitative aspects. 

So the first notion of number, i.e. with respect to the fundamental binary digits of 1 and 0, provides the most refined phenomenal partition that bridges potential reality (as void) and its corresponding actualisation (as form).

However the next major issue relates to how both the quantitative and qualitative aspects of this reality, can then be coherently distinguished in phenomenal terms, yet remain fully related in a consistent - and ultimately - incomprehensible manner.

So, at a local level, the phenomenal features of reality exhibit a distinct independent identity in both quantitative and qualitative terms; yet at the global level all such features are in truth ultimately interdependent with each other in a truly ineffable fashion.

Now the importance of the relationship between the primes and the natural numbers (and natural numbers and primes) is that remarkably, in a dynamic bi-directional manner, it provides the code, as it were, through which this key problem is universally solved.

And as we have seen, mediating this crucial relationship are two parallel sets of numbers i.e. the Zeta 1 and Zeta 2 zeros, which universally mediate in two-way fashion as between primes and natural numbers (thereby ensuring consistency of both quantitative and qualitative aspects).

Without such dynamic consistency being guaranteed in this manner, meaningful mathematical operations (in cardinal and ordinal terms) would not be possible; even more dramatically, the phenomenal world as we know it, which is dynamically encoded in such number relationships, could not exist.

So if we are to look for a meaningful "Theory of Everything", it is to be found in its purest phenomenal expression, in the relationship between the primes and natural numbers, which mutually entails the two sets of zeros that mediate each other in a fully synchronistic manner. 

Thus this relationship necessarily lies as the embedded code of all subsequent phenomena that are manifest in evolution. In effect this number code dictates how such phenomena can fundamentally relate to each other (in both quantitative and qualitative terms).

So number in this dynamic sense, potentially exists as the root source of all physical phenomena that can subsequently exist. However this meaning is only then actualised through its relationship with manifest phenomena. This then ultimately, through considerable psychological development, can lead to comprehensive appreciation of the code, as forming the final partition to the fulfillment of all evolution (in ineffable union). 

However this is very different from the present quest in physics for a TOE based on the postulated existence of strings.

There are in fact no "building blocks" of reality such as strings, which can be appreciated in mere analytic (quantitative) terms. Rather, phenomenal reality, as we know it, arises from the synchronistic relationship of opposite polarities (such as external /internal, whole/part and form/emptiness) that find their purest expression in the fundamental nature of the number system.

Therefore the much more profound quest of science, which has yet to be properly articulated, is to coherently show how conversion can take place, as it were, from the analytic interpretation of phenomena, where seemingly cause and effect operates in a linear manner (with quantitative and qualitative aspects separate), to the corresponding holistic interpretation, where all relationships are now understand as synchronistically determined (in both quantitative and qualitative terms).

And the mystery that underlies this quest, as we have seen is already implicitly encoded in the very nature of the number system 

From this perspective, the true task of evolution is the dynamic uncovering of this fundamental code that thereby contains the power to lead us eventually to the very gates of eternity! 

The Zeta Zeros and Affective Appreciation

I wish to comment here on a further remarkable feature with respect to the nature of zeta zeros (Zeta 1 and Zeta 2).

Conventional Mathematics, which is of an analytic (quantitative) nature, is defined in terms of merely cognitive interpretation of mathematical symbols in a linear rational manner.

Though intuition of a holistic (qualitative) nature is implicitly required to fuel the dynamics of such understanding, in formal terms it remains unrecognised and thereby reduced in every context to quantitative interpretation.

So the first major breakthrough required is that realisation that both all mathematical interpretation properly entails both analytic (quantitative) and holistic (qualitative) aspects in dynamic interaction with each other.

Though once again, the holistic aspect relates directly to intuition, indirectly it can then be translated in a paradoxical rational manner that is circular in nature.

Indeed such circular rationality can in turn be further indirectly represented in an "imaginary" linear manner.

So whereas from a qualitative philosophical perspective, Conventional Mathematics is based on the real (i.e. conscious) use of reason, this more comprehensive approach is based on the complex expression of reason i.e. relating to both real (conscious) and imaginary (unconscious) aspects of understanding.


However there is a further surprise awaiting in the comprehension of the zeta zeros.

For example, one may start by attempting to appreciate the Zeta 2 zeros (which are the simpler to embrace) in a complex cognitive manner (i.e. involving both analytic and holistic aspects of interpretation).

However the corresponding attempt to then appreciate the Zeta 1 (Riemann) zeros in a corresponding complex cognitive manner will ultimately lead to failure. I know this convincingly from my own experience!

So what is remarkable - in terms of their adequate comprehension - is that when the Zeta 2 zeros are interpreted in a complex cognitive fashion, in relative complementary terms, the Zeta 1 (Riemann) zeros, will then correspond to complex affective understanding. In other words, their true nature from this perspective cannot be approached from a cognitive perspective!

However of course as in all dynamic interactive situations, reference frames can be switched.

So if the Zeta 1 zeros are now understood from the cognitive perspective, then the corresponding Zeta 2 zeros must now be approached in a complementary affective manner.

So ultimately -  which must come as a major surprise to anyone who sees Mathematics as a merely rational discipline - an approach to true comprehensive understanding of both sets of zeta zeros, requires the ability to balance cognitive and affective aspects of experience in a highly refined contemplative two-way intuitive manner.

Indeed this fits in very well with my comments yesterday, that the integration of the two sets of zeros coincides with the attainment of both top-down and bottom-up integration.

Now typically - especially with male personalities - top-down integration would be identified in psycho-spiritual terms, with the transcendent aspect of development, where "high level" cognitive is used to control "low level" affective behaviour (especially with regard to physical instinctive impulses of the unconscious).

Bottom-up integration, by contrast would represent the corresponding immanent attempt at achieving a spontaneous physical response, where "low-level" affective projections, now emptied of repressive influence, can freely integrate themselves with the cognitive aspect of mental control.

Thus to achieve an appropriate balance, in psychological terms, as between top-down and bottom-up integration, requires the corresponding ability to properly balance both the cognitive and affective functions of behaviour.

So when this state is achieved - or rather successfully approached in varying degrees - the primitive instinctive behaviour (of the unconscious) can be fully harmonised with the natural (conscious) requirements of living..

As we have seen, the task of achieving such two-way integration, directly corresponds with the two-way integration in mathematical terms of both the primes and natural numbers.

And just as psychological integration entails the marriage of both cognitive and affective aspects, likewise mathematical integration (with respect to the zeros) entails a similar corresponding marriage.


Now once again this might appear incomprehensible to one approaching Mathematics from the conventional perspective. for here the attempt is made to abstract rational understanding, in an absolute manner, from human experience (which is inherently of a dynamic interactive nature).

And as such human experience ultimately entails conscious and unconscious aspects, with respect to both cognitive and affective aspects of understanding, ultimately comprehensive mathematical understanding requires the same framework.

In other words the most comprehensive mathematical understanding can only arise, when mathematical activity is itself fully integrated with the rest of human experience.

I have already mentioned on several occasions that I viewed a comprehensive approach to Mathematics as entailing three main stages i.e. Analytic, Holistic and Radial (where Analytic and Holistic are increasingly combined).

However, so far this map was envisaged in a complex cognitive manner (entailing the aspects of reason and intuition).

However a fourth stage is now also required whereby Mathematics itself becomes increasingly integrated with both artistic and religious type experience.

So the 3 big domains of human experience centre around the Sciences, the Arts and Religion (as the embodiment of spiritual experience). These in turn can be identified most directly with the cognitive, affective and volitional aspects of human behaviour.

Mathematics may initially be seen as solely relevant to the Sciences. However as ultimately the Big 3 entail complementary aspects of human behaviour, a full experience of Mathematics entails a full experience likewise with respect to the other two aspects.

So in various contexts, the meaning of mathematical symbols will remain highly elusive, when approached in a merely cognitive manner.

Thus here, mathematical symbols may be best understood as indirect expressions of a meaning that is of an aesthetic nature (appealing  directly to artistic appreciation). And this is true to the nth degree, where the majestic intricate beauty of the zeta zeros is concerned!

Zeta Zeros - Psychological and Mathematical Connections

Properly understood, the task of properly grasping the true nature of the zeta zeros (both Zeta 1 and Zeta 2) cannot be divorced from the corresponding task of attaining full integration in psycho-spiritual terms.

My earliest realisation of this important fact came from the holistic insight that the notion of "prime" from a mathematical perspective is directly complementary with the corresponding notion of "primitive" as used in a psychological developmental context.

Deep reflection on the inherent meaning of "primitive" then enabled me to make valuable linkages to the true notion of prime numbers in a dynamic experiential mathematical context.

For example in early infancy, primitive instincts characterise the behaviour of a child.

This reflects the fact that as the conscious aspect of personality has not yet been properly differentiated from the corresponding unconscious aspect, that both are inevitably confused with each other.

Thus, in other words, the infant confuses holistic meaning (associated with the unconscious) directly with specific objects (properly pertaining to conscious understanding).

With the extreme manifestations of such behaviour, true object constancy is not possible in experience. This again is due to the fact that (specific) phenomena are so directly confused with the (holistic) dimensions they inhabit, that neither aspect can be properly distinguished from each other.

Therefore, because a sufficiently stable background of space and time cannot be yet provided, object phenomena enjoy a - necessarily - fleeting existence.

Indeed this also has close parallels with the nature of sub-atomic particles, which become inherently unstable at deeper levels of investigation, enjoying but a momentary existence in space and time.

This implies that there is holistic ground to physical reality (relating to the close interdependence of quantum phenomena with each other) so that at the extreme levels of investigation, it is longer even possible to distinguish such phenomena (as independent) from their background environment (as interdependent).  

Thus from  psychological and physical perspectives, earliest development is prime (i.e. primitive) in nature (reflecting the confusion of both analytic and holistic aspects of meaning).


When one carries over this dynamic interactive approach to the interpretation of prime numbers, it implies that they ultimately represent two extreme aspects of behaviour in a complementary manner.

Thus from one perspective, the primes are the most independent of numbers, representing, in cardinal terms, the prime "building blocks" of the natural number system (except 1).

However, from an equally valid perspective, the primes are the most interdependent of numbers, necessarily represented in a unique ordinal manner, by their natural number members.

So again for example, from this perspective, 3 is a prime, that is necessarily composed in ordinal terms of 1st, 2nd and 3rd members. Now strictly this latter definition refers to a qualitative relationship as between the members of a group!
Then, indirectly these ordinal members can be uniquely expressed in quantitative terms (again except 1) by the corresponding prime roots of 1!

Thus we have both quantitative (analytic) and qualitative (holistic) aspects to the primes.

So when the quantitative aspect is identified in a cardinal manner (relating to number independence) the corresponding qualitative aspect is then identified in an ordinal manner (relating to the number interdependence as between different members of a group).


Once again in earliest childhood both aspects of the primes i.e. quantitative and qualitative (relating to conscious and unconscious aspects respectively) are greatly confused with each other.

So the first task of development is to successfully differentiate the conscious aspect from the unconscious which is the task of Band 1 (on the spectrum of development).

Then the specialisation of this understanding (in a linear rational manner) occurs at Band 2.
And it is this Band with which Mathematics and Science - as we know them - are conventionally associated.

However this represents but a reduced quantitative interpretation of the primes (with no formal recognition of their distinctive qualitative nature).

Thus we cannot hope to understand the inherent dynamic nature of the primes in this reduced manner!

So before Mathematics can properly address this issue, it will need to recognise that further substantial qualitative mathematical development (of a holistic intuitive nature) is possible on the spectrum.

Traditionally this has been associated with the attainment of advanced contemplative awareness, (which is recognised by all the spiritual traditions).

However what has not been properly recognised is that such development has extremely important implications for both mathematical and scientific understanding, leading to the distinctive disciplines of Holistic Mathematics and Holistic Science respectively.

So Band 3 on the spectrum represents the unfolding of a new refined intuitive awareness (that indirectly finds expression in a circular i.e. paradoxical, rational manner).

Band 4 then represents the specialisation of this distinctive type of intuitive awareness thereby enabling true appreciation of the holistic aspect of number.

Now the further possible Bands of development on the spectrum (which I define as 5, 6 and 7) represent the mature integration of both (specialised) analytic and holistic appreciation of the primes.

And this is equally necessary in terms of achieving both full integration in psycho-spiritual terms and the corresponding full integration of the primes with the natural number system.

In fact the two sets of zeros (Zeta 1 and Zeta 2) are associated directly in psycho-spiritual terms with - what with might be referred to as - both the top-down and bottom-up integration of the psyche.

In fact, long before I ever gave attention to the Riemann Hypothesis, I had become convinced of the huge potential importance of the roots of 1 in terms of defining, in holistic mathematical terms, the dynamic structures of development.

It was only later that I realised that these corresponded directly with - what I refer to now as - the Zeta 2 zeros.

However the limitation of my approach was that these zeros essentially described a merely top-down approach to integration, where "lower level" affective are integrated through "higher-level" cognitive structures.

So for proper psychological balance, corresponding bottom-up integration would require that "higher-level" cognitive would now in turn be integrated from the perspective of the "low-level" affective structures In other words true integration would require that both the affective and cognitive functions would themselves be equally developed (with neither dominating each other).

Remarkably, I then gradually discovered that the famous Zeta 1 (i.e. Riemann) zeros perfectly described this latter form of integration.  

So true psychological integration entails a two-way process, whereby from one direction, unconscious meaning can be perfectly converted (i.e. find expression) in a corresponding conscious manner.

Equally from the alternative direction, conscious meaning needs to be perfectly converted in an unconscious manner.

Without the possibility of such perfect conversion (in both directions) it would not be possible to properly relate both conscious and unconscious (as quantitative and qualitative type meaning) in a consistent manner.


It is exactly similar in terms of the mathematical relationship of the primes and natural numbers.

From two opposite perspectives, the essential role of the zeta zeros (Zeta 1 and Zeta 2) is to enable perfect conversion as between the Type 1 and Type 2 aspects of the number system (that represent - in relative terms - their cardinal and ordinal aspects respectively).

Again without the possibility of such conversion (in two directions) it would not be possible to relate numbers consistently with each other (in cardinal or ordinal terms).

However, we are here light years away from the highly reduced quantitative notion of number as representing absolute entities of an abstract kind.

Rather, in truth, the notion of number inherently entails the continual dynamic interaction of both its quantitative and qualitative aspects.

Furthermore the full integrated appreciation of this understanding cannot be divorced from the corresponding psycho-spiritual quest for full integration.

We will address further issues arising from this realisation in the next entry.

Mathematical Revolution Required!

I have long emphasised how conventional mathematical interpretation of symbols is so limited with its mere emphasis on the quantitative aspect (that directly concurs with the linear use of reason).
We refer to this as the analytic aspect of interpretation.

Equally, however every mathematical symbol possesses a distinctive qualitative aspect that arises from the direct intuitive recognition of symbols (that indirectly is expressed through a paradoxical i.e. circular use of reason).
We refer to this as the corresponding holistic aspect of interpretation.

The true dynamics of mathematical experience then arise through the combined interaction of both quantitative (analytic) and qualitative (holistic) meaning.

I then have sought through these blog entries to apply this dynamic approach to interpretation of the zeta zeros.

However this quickly led to the realisation that there are in fact two sets of such zeros - equally important - that are dynamically interdependent with each other.

I refer to the first (recognised) set as the Zeta 1 zeros. These concur directly with the Riemann zeros (i.e. non-trivial zeros) of the Riemann Zeta function.

Now once again I will attempt here to highlight the holistic significance of these zeros.

We are accustomed to think of the primes (in quantitative analytic terms) as the independent "building blocks" of the natural number system (≠ 1). So from this perspective, all natural numbers can be expressed as the unique combination of individual primes.

However the unrecognised complementary counterpart to this (in qualitative holistic terms) is the view of the primes, as the corresponding interdependent connections governing the collective relationship of all primes (≠ 1) with the natural number system.

So when we allow for both the distinctive quantitative (analytic) and qualitative (holistic) aspects of the primes, we realise that in dynamic interactive terms, they combine both extreme independence and interdependence in a relative fashion.

So the Zeta 1 (Riemann) zeros from this perspective, can be holistically viewed as representing the complete set of such interdependent connections, which the primes collectively maintain with the natural number system.

In fact I have illustrated in my blog entries how the frequency of Riemann zeros bear a remarkably close relationship with natural number factors.

In other words the accumulated total number of natural number factors (of the composite numbers) up to n (on the real scale), approximates very closely to the corresponding frequency of Riemann zeros up to t (on the imaginary scale) where n = t/2π.

Therefore once again we can see the complementary relationship involved. So we start with the primes (as analytic measures of independence) and then find them related to the complementary holistic notion of interdependence (i.e. as factors of natural numbers).

Thus for meaningful interpretation, in a dynamic interactive manner, the primes and Zeta 1 zeros must be viewed - relatively - in analytic and holistic terms with respect to each other.
So therefore when we view the primes in a quantitative manner, we must then view the Zeta 1 (Riemann) zeros in corresponding qualitative fashion i.e. as an expression of interdependence, that indirectly can then be expressed in an imaginary number manner!

However the true interdependence as between the primes and the zeros is demonstrated by the fact that we can equally validly, switch reference frames, so that now the individual zeros assume a direct quantitative (analytic) meaning and the corresponding primes (with which they are complementary) a qualitative (holistic) interpretation as the collective behaviour of the prime numbers.

So in dynamic interactive terms, a mutual independence and interdependence characterises the primes and zeros (both of which are seamlessly integrated from two directions with each other).

This is just another way of stating the ultimate synchronistic nature of the number system, where neither primes nor zeros precede each other, as it were, but rather both mutually arise in a seamlessly integrated fashion (enabling the subsequent consistent relationship of number in both quantitative and qualitative terms).


The unrecognised - certainly with regard to their significance - set of zeros, relate to what I refer to as the Zeta 2 zeros. Indeed ultimately the Zeta 1 zeros can have no strict dynamic meaning in the absence of the Zeta 2 zeros (and vice versa)!

In some ways these zeros are in fact much easier to understand than the recognised Riemann zeros.

However the insight as to what they represent comes from ordinal rather than cardinal understanding.

As we have seen we typically start by viewing the primes in an individual cardinal manner as quantitative "building blocks" to establish their quantitative relationship with the overall natural number system (again in cardinal terms).

The zeta (i.e. Riemann) zeros then emerge to show that there is something seriously missing with this approach, by providing what in effect is a shadow system of collective holistic relationships (that must meaningfully be interpreted in a complementary qualitative manner).

However, one can also start by attempting to see each individual prime as already necessarily composed of natural numbers (in an ordinal manner). So instead of each natural number being defined quantitatively as the product of cardinal primes, alternatively, in reverse fashion, we define each prime as already ordinally composed in a unique manner by its natural number members.

So for example, 3 is a prime which is uniquely defined (from this perspective) by its 1st, 2nd and 3rd members. Now indirectly we can represent this (in quantitative terms) by obtaining the corresponding 3 roots of 1!

Significantly, when we obtain the prime roots of any number, all of these roots (again ≠ 1) representing its ordinal members, will be unique for this prime .

So the Zeta 2 zeros simply express this unique representation for each prime of its ordinal number members. And this feature of behaviour represents the complementary ordinal counterpart to the established fact that every natural number (≠ 1) in cardinal terms is uniquely composed of prime factors.

So here we start with the qualitative notion of each prime, as representing a shared group (of ordinal number members).

Then the Zeta 2 zeros arise as the indirect quantitative expression (through the prime roots of 1) of this (inherent) qualitative nature.

So once again, we can see an important complementarity here with the Zeta 1 zeros, where the set of zeros - by contrast - carry a qualitative holistic significance.

However as before the frame of reference can be switched, so that the prime (representing dimension) carries a quantitative meaning, while the collection of its ordinal members (represented by roots) is qualitative.  In fact this is simply illustrated by the fact that the sum of roots = 0, implying - literally - that their combined nature carries no quantitative significance!


So within each number and throughout the number system as a whole, we have the two-way interaction of both prime and natural number behavior (in quantitative and qualitative terms).

Thus numbers as individual members and composite groups, contain particle and wave aspects. The particle aspect refers to numbers is both cardinal and ordinal terms, while - relatively -the wave aspect relates to both Zeta 1 and Zeta 2 zeros, which dynamically keep interchanging with each other - ultimately - in a purely relative manner!

Then the inherent nature of number, from this informed dynamic perspective, approaches pure synchronicity (as between its analytic and holistic aspects) in a merely relative manner. This is mediated through the two-way relationship of both prime and natural number aspects.

The great poverty of Conventional Mathematics - in refusing to give any formal recognition to the qualitative (holistic) nature of number - is that it cannot possibly appreciate, within its greatly limited framework, this true dynamic nature of the number system.

So quite simply, nothing short of the most radical revolution possible with respect to Mathematics is now urgently required.

If you are a mathematician reading this, I urge you to wake from your slumbers and bring the "good news" to your colleagues - which unfortunately they may initially see as "bad news" - that our true mathematical journey has scarcely begun!

Slight Modification

I have commented several times on the true significance of the zeta zeros.

Now once again from my own perspective, there are in fact two complementary sets of these zeros which I term Zeta 1 and Zeta 2. So Zeta 1 refer to the Riemann (non-trivial) zeros. The Zeta 2 by contrast refer to the various roots of unity (excluding 1, which is common to all roots).

Now once again the significance of these roots is that they enable seamless conversion as between the Type 1 and Type 2 aspects of the number system.

As we have seen when the Type 1 is associated with the quantitative (analytic) aspect of number behaviour, the Type 2  is then associated, in complementary fashion, with the qualitative (holistic) aspect.

So essentially the zeta zeros enable us to convert from Type 2 to Type 1 format, and equally from Type 1 to Type 2 format.

Without this facility we would have no reason to believe in the consistency of number operations from either the (recognised) quantitative or (unrecognised) qualitative perspectives.

However rather like the situation in physics, which conveniently breaks down into macro (relativistic) and micro (quantum) aspects, it is similar in the consideration of numbers.

So from one perspective, we can view each prime as composed of a unique group of natural number members (in ordinal terms).

From the other perspective, we can view the natural numbers as composed of unique groups of prime members (in cardinal terms) .

So from the first perspective we examine the micro nature of each prime (through its natural numbered ordinal members).

From the second perspective we view the macro nature of the natural number system (through its cardinal prime members).

Now from one perspective (where base numbers are viewed in Type 1 terms as quantitative and dimensional numbers as - relatively - in Type 2 terms as qualitative) , the Zeta 2 zeros provide the means of expressing each prime representing a dimension (in Type 2 terms) indirectly in a Type 1 manner.

In this way we are enabled to convert the ordinal members of each prime (as Type 2 qualitative)  indirectly in a Type 1 (quantitative) manner.

Equally, we can convert the cardinal nature of the natural numbers as a whole (as Type 1 ) quantitative, indirectly in (a Type 2) qualitative manner through the Zeta 1 zeros.

From this perspective, the Zeta 2 can be represented as the means of conversion from the Type 2 to Type 1 aspect and the Zeta 1 as the means of conversion from Type 1 to Type 2 aspect respectively.


However when we reverse the frame of reference so that the base numbers are identified in qualitative, and the corresponding dimensional numbers in - relative - quantitative terms, these connections are reversed.

So the Zeta 2 zeros can then be represented as the means of conversion from Type 1 to Type 2 aspect and the Zeta 1 zeros as the corresponding means of conversion from Type 2 to Type 1 aspect.

Thus therefore, depending on perspective, both sets of zeta zeros play a two-way role in terms of converting between Type 1 and Type 2 (and Type 2 and Type 1) respectively.

Remember that we can use both the additive and multiplicative approaches to derive numbers!

In terms of the additive approach, each prime number can be defined as the unique sum of its natural number members (in ordinal terms).

In terms of the multiplicative approach, each natural number can then be defined as the unique product of prime number factors (in cardinal terms).

So the Zeta 2 zeros relate directly here to the additive approach and the the Zeta 1 to the multiplicative.

However ultimately all these are derived in a synchronous manner (where relationships are merely relative with everything dependent on everything else).

Thus to conclude, the  Zeta 1 and Zeta 2 zeros play an equally important - and truly vital - role in enabling the seamless two-way conversion of number as between its quantitative (analytic) and qualitative (holistic) aspects.