Qualitative Nature of Prime Music

As we have seen, when we understand the number system in a dynamic fashion, it necessarily contains both analytic and holistic aspects.

Therefore from one perspective, numbers possess a certain separate identity (which we customarily identify with its quantitative aspect).

However equally, numbers possess an overall relational identity as interdependent which is properly related to a distinct qualitative identity. What remains completely overlooked in conventional mathematical understanding is that this aspect cannot be successfully reduced in a mere quantitative manner.

This relationship as between analytic and holistic identities (quantitative and qualitative aspects) is especially important in the context of the overall relationship of the primes with the natural numbers (and the natural numbers with the primes).

Indeed this relationship is frequently explained in terms of musical analogies. So for example if the Riemann Hypothesis is true this suggests that the various separate notes of the great number symphony of the natural numbers i.e. the primes, while maintaining their unique individual identities (as separate) yet are related to each other in a perfectly synchronised manner (so that not one note is out of place).

However because of the exclusive quantitative bias of Conventional Mathematics, a persistent failure remains in recognising that this remarkable collective behaviour of the primes is necessarily of a (qualitative) holistic rather than a (quantitative) analytic nature.

A mathematician would readily accept, I imagine that while music can indeed be analysed in a quantitative manner (e.g. with respect to individual notes and chords), that it would be ludicrous to deny its overall qualitative nature, in the manner in which these are combined with each other.

So while the quantitative nature of music relates to the analysis of its individual components, its qualitative nature relates to the overall manner in which all these individual components are successively combined in a holistic manner!

However when we come to the nature of the natural number system - which in truth  requires that the most perfect degree of synchronicity be maintained as between its various prime number notes - we still remain totally blind to the fact that this requires a distinct qualitative - as opposed to quantitative - mathematical dimension!

And the reason why we remain so blind as to a totally obvious realisation, is that we have committed ourselves for several millennia now to a distorted - and thereby false - view of the number system (that is defined in a merely quantitative manner).  

The reduced quantitative view of the number system has created the erroneous impression that the world of number relates to an abstract set of absolute fixed entities operating independently of natural physical and psychological processes.

However nothing could be further from the truth, with number inherent in all such processes as the deepest encoding of their intrinsic nature (which is ultimately ineffable).   

Therefore the reason why a holistic qualitative aspect applies to the dynamic nature of number is because this inherently applies to all natural processes.

Of course because our notions of physical science are so heavily influenced by the quantitative nature of Conventional Mathematics, there is likewise a deep refusal to recognise the obvious implications of - now - accepted physical behaviour at a quantum level.

For example it has been demonstrated through several carefully conducted experiments that subatomic particles (such as light) can communicate with each other, even at a great distance.

Now this clearly relates to a qualitative holistic capability of matter (akin to what might be recognised as a form of para-natural telepathy!)

Indeed if physicists were then to properly accepted such findings, they would be forced to accept that our normal macro physical world essentially operates on a para-natural basis (dictated by its underlying quantum processes).

Of course they deeply resist making such obvious connections precisely because this would call directly into question the reduced paradigm on which accepted notions are based.

However what I am attempting to convey clearly here is that the mysterious quantum behaviour of matter is itself rooted in the prior nature of the number system (when correctly interpreted in a dynamic interactive manner).

The two great issues that therefore need to be faced with respect to the nature of the number system (and indeed all mathematical notions) are as follows:

(i) that we cannot hope to divorce the external objective nature of mathematical relationships from the relative internal subjective mental means of their interpretation.

A deep problem relates to standard mathematical interpretation in attempting to reduce this relationship in an absolute manner (i.e. 1-dimensional interpretation).

This leads to the mistaken view that just one valid system of Mathematics exists.

In fact, properly understood - in dynamic interactive terms - an unlimited number of possible mathematical systems exist (each with an arbitrary partial validity).

So number here in its true dimensional sense is associated with a unique means of interpreting mathematical symbols. The use of 1 as dimension, refers therefore to the limited special case where internal and external polarities are reduced in terms of each other (in an absolute manner).

(ii) that we cannot hope to divorce the quantitative aspect of interpretation from its related qualitative aspect.

Again a deep problem relates to standard interpretation in again attempting to reduce this relationship in an absolute manner (in accordance with 1-dimensional interpretation).

This leads to the deeply mistaken view that number can be consistently understood with respect to its mere quantitative aspect. In effect this leads to a profound confusion with respect to the ordinal (qualitative) nature of number which is effectively reduced in a cardinal manner.

 As I have repeatedly stated in my blog entries, the true nature of the number system is that it provides, through the non-trivial zeta zeros (Zeta 1 and Zeta 2), the means by which both its quantitative (cardinal) and qualitative (ordinal) aspects can be consistently related with each other.

In fact the zeta zeros represent the true indispensable holistic basis of the number system, which necessarily underlines our customary analytic understanding of this system.         

The Simple Explanation

Yesterday I was at pains to provide a summary communication in an intuitively accessible manner of the precise nature of the zeta zeros and their fundamental significance for the nature of the number system (and by extension all Mathematics and Science).

Only after finishing the piece as it were as I was walking along the beach in Dollymount (enjoying the unseasonably good weather) that the final piece of the jigsaw as it were fell into place when I finally could see clearly the simple picture.

The remarkable thing about the simplest most obvious insights is that they are in fact the most difficult to access. Our conscious mind with its constant desire for phenomenal activity therefore continually places barriers in the way of truly seeing what is most obvious (and most meaningful). And I believe this is especially true with respect to Mathematics (defined formally as it is in merely rational terms).

So put most simply, the famed non-trivial zeta zeros (or as I refer to them as the Zeta 1 zeros) from a physical perspective, constitute the holistic basis of the cardinal number system.

From the alternative complementary psychological perspective, the same zeros constitute the unconscious basis of the (consciously understood) cardinal number system.

Now, once again it should be immediately apparent that this implies that the number system is inherently of a dynamic interactive nature with the external and internal aspects corresponding with the physical and psychological aspects respectively.

We also have complementary quantitative and qualitative aspects. Much earlier I had the clear realisation that the Zeta 2 zeros constitute the holistic basis of the ordinal number system. Once again of course this has an (internal) psychological counterpart as the unconscious basis of the (consciously understood) ordinal number system.

However when understood in a comprehensive manner, both holistic and analytic aspects (corresponding to unconscious and  conscious interpretation) with respect to both the cardinal and ordinal features of the number system, simultaneously interact with each other so that the ultimate nature of all is identical in an ineffable manner.


Those who are familiar with Jungian Psychology perhaps can appreciate what is involved here.
In Jungian terms a dynamic complementarity necessarily applies to conscious and unconscious with respect to all human behaviour.

Though he could not be classed as a mathematician, I have always found Jungian thought to be remarkably compatible with holistic mathematical appreciation!

So when the unconscious is not properly recognised it is projected in a blind manner on to conscious phenomena (and confused with them). This represents shadow projection.

Therefore the difficult developmental task consists in first properly recognising this shadow (i.e. as expressive of the holistic unconscious) and then reaching sufficient maturity with respect to its use, that it can then be successfully integrated (free of confusion) with conscious understanding.


Though of special importance for Mathematics, the zeta zeros (Zeta 1 and Zeta 2) in fact represent the unconscious holistic basis of the number system. In other words they represent the perfect shadow to the analytic number system (with respect to both its cardinal and ordinal aspects).

However because Conventional Mathematics is formally based on merely rational (conscious) notions, there is still no recognition whatsoever of the true nature of this important shadow system. So in Jungian terms it is simply being blindly projected onto its analytic counterpart (and then directly confused with it).

Even insofar as an appreciation of the zeta zeros exist, the attempt is still being made to interpret them in a merely conscious rational manner (i.e. through the analytic aspect of the number system).

Indeed the very quest to prove the Riemann Hypothesis is based firmly on this misguided approach!


The implications of what are being said here could not be of greater magnitude.

We have been accustomed to think of Mathematics for millennia now as some self-contained rational world (safely free of all contamination from the unconscious).

And I am saying here quite emphatically that this view is totally wrong!

The remarkable truth is that we cannot form any conscious analytic notion of number (without the unconscious existence in our psyches of  the mysterious - largely unknown - holistic system relating to the zeta zeros). The fact that we remain so much unaware of this system only serves to indicate its unconscious origin!

So the very nature of the number system in physical terms entails this dynamic interaction as between analytic and holistic aspects; in psychological terms it entails the interaction of both conscious and unconscious with respect to all number interpretation.


So Mathematics has yet to even begin to realise its own enormous shadow (in appreciation of the true nature of the zeta zeros).

At a human level, we would perhaps immediately accept how development necessarily  remains extremely stunted for one with no recognition of unconscious influences on behaviour.

Well at an even more profound level, mathematical development remains extremely stunted due to its own inability to recognise its hidden shadow side.

Yes, admittedly enormous strides have been made with respect to the narrowly specialised quantitative area! However this represents just one small facet of an altogether more glorious tapestry of potential mathematical meaning.


So again here is the remarkable truth regarding the non-trivial zeta zeros (Zeta 1 and Zeta 2)! Not alone are they encoded in the human psyche (as the holistic unconscious basis of our everyday understanding of number) but they are likewise encoded in every process in nature.

However properly recovering this hidden shadow side of Mathematics (and of all evolution) will entail the long journey to full realisation (conscious and unconscious) in a direct spiritual experience that is ultimately ineffable.

What It's All About

I cannot stress strongly enough that a completely new paradigm is now required in Mathematics.

However an almost total blindness still exists in the mathematical community as to this fundamental issue.

Indeed, whenever I have suggested this - even to those with holistic sympathies - inevitably they have tried to filter my comments through the existing conventional approach. And because my suggestions clearly do not fit in with this existing paradigm, they have defended the status quo by convincing themselves that I must therefore know very little about Mathematics.

All I can say is that after more than 50 years of travelling on this lone journey, I am fully confident regarding my basic assertion. I have had the good fortune to glimpse many of the wonderful lands whose very existence my mathematical colleagues still resolutely deny.

The true nature of Mathematics is incomparably greater than what we presently imagine. Indeed what we have come to unquestioningly accept represents but a very limited special case (that admittedly is very important in its own context) but which pales in comparison to the glittering treasure that is truly Mathematics.

More importantly despite its great successes, Conventional Mathematics has led to mind-set that - if left unchallenged - is likely to have destructive consequences for our civilisation.

Such Mathematics creates the illusion of quantitative meaning (without reference to the qualitative). And as this Mathematics provides the essential basis for all the sciences, this means that science itself - which in many ways has become the new world religion - fundamentally entails, in every context, the reduction of qualitative to quantitative meaning.

So the overall scientific worldview (fundamentally rooted in Mathematics) is in fact hugely unbalanced and this is what so threatens global integration at every level whether social, economic, environmental etc.

We can truly say that the roots of this fragmented scientific worldview (so out of keeping with the genuine needs of the planet) are in the fact that our understanding of Mathematics itself is completely lacking a holistic integral perspective.

We have convinced ourselves, so wrongly, that Mathematics is exclusively concerned with quantitative meaning! And with the growing specialisation of this type of Mathematics through the millennia this has led to an incredibly defensive approach with respect to any criticism of its fundamental rationale.

However when one looks closely at the nature of the number system, it becomes quickly apparent - to anyone with the clarity to see - that quantitative interpretation is strictly meaningless in the absence of an equally important qualitative aspect. It also becomes apparent that the absolute notion of objective truth is a great fallacy (as all objective notions necessarily reflect the subjective mental lens of interpretation through which they are viewed).


Properly understood the Riemann Hypothesis provides the perfect vehicle for this necessary radical conversion with respect to mathematical understanding.

It has indeed struck me deeply in recent years how lack of a holistic vision is greatly preventing meaningful communication of fundamental results in both Science and Mathematics.

For example it seems to me that String Theory remains totally impoverished with respect to providing a coherent physical view of the Universe.

The reason for this is that physicists are still trying to filter insights through the accepted 1-dimensional paradigm (which is quite inadequate for the purpose). So it will need - literally - a radical new holistic vision of science before the insights of String Theory can be conveyed in a  manner that then can resonate intuitively with this newly acquired worldview.


It is somewhat similar regarding interpretation of the (non-trivial) zeta  zeros with respect to the Riemann Hypothesis.

Again because Mathematics is lacking any holistic framework, understanding of these zeros is necessarily being filtered likewise through a 1-dimensional paradigm which is hugely inadequate for their meaningful interpretation.

Because my own approach has always been strongly holistic, some years ago I set myself the task of  forming an understanding of these zeros in an intuitively meaningful fashion, that could express their crucial relevance for Mathematics. So I will summarise my findings now!

The relationship as between primes and natural numbers is indeed fundamental. However we must understand this relationship as possessing both quantitative and qualitative aspects.

Quantitative in this context is associated with notions of number independence; qualitative is associated with number interdependence.

So numbers are independent in a cardinal and interdependent (through their relationship with each other) in an ordinal sense. Both of these aspects quantitative and qualitative necessarily interact in a dynamic relative manner. Likewise external appreciation of number as objective and internal mental interpretation of number as subjective likewise interact in a dynamic relative manner.


Therefore the importance of the relationship between primes and natural numbers is that they provide the means through which both the quantitative and qualitative aspects of the number system are mediated (in a dynamic two-way fashion).

So from the (Type 1) cardinal perspective, the primes appear as the unique quantitative building blocks of the natural numbers; however from the complementary (Type 2) ordinal perspective, the natural numbers appear as the unique qualitative building blocks of each prime number. (For example, again from this perspective, 3 as prime, is necessarily comprised of its 1st, 2nd and 3rd members).

Thus in  a relatively independent manner, we therefore can identify - with respect to the primes and natural numbers - this two way relationship, by which quantitative and qualitative aspects are mediated (in an analytic fashion).


However the key issue of the corresponding (relative) interdependence of both quantitative and qualitative then arises. In other words how do we ensure the ultimate identity of both these aspects as fully consistent with each other?

This is where the (non-trivial) zeta zeros (Zeta 1 and Zeta 2) come into play! Just as we have two sets of analytic relationships connecting both quantitative and qualitative aspects of primes and natural numbers (in a relatively independent manner), we have likewise two sets of holistic relationships connecting quantitative and qualitative aspects of the primes and natural numbers (in a corresponding relatively interdependent manner).

Now the Zeta 2 provide a circular explanation whereas the  Zeta 1 provide a linear explanation of this relative number interdependence respectively.


Once again we will remind ourselves of what this means with reference to the first of the Zeta 2 zeros which relates to the 2 roots of 1! (Strictly the non-trivial solution relates to the root ≠ 1)!

So + 1 and –  1 have a relative quantitative independence as separate; however when combined qualitatively as interdependent, the (separate) quantitative aspects are eroded. 

Thus for the prime number 2 we are able to show a perfect harmony as between two (circular) numbers as - relatively - quantitatively independent with their overall qualitative relationship as interdependent. 

Therefore, the very point about the Zeta 2 zeros is that they provide this magical solution in a dynamic interactive manner as to how to preserve the relative independence of each natural ordinal number member of a prime number group (indirectly expressed in a quantitative manner) with the overall interdependence of the group in qualitative  terms.

In this way each prime number is given a unique holistic identity (through its ordinal natural number members).


So the very essence of the Zeta 2 zeros is to reconcile in a circular numerical manner the cardinal aspect of each prime with its natural number members (in ordinal terms).

By contrast the very essence of the Zeta 1 zeros is to reconcile in a linear manner the cardinal aspect of the (composite) natural numbers with the ordinal aspect of their constituent prime number factors.


We must remember the essence of multiplication is that it inherently is of a qualitative nature.

So we may say that 6 is uniquely expressed as the product of two cardinal primes i.e. 2 and 3.

However obtaining this product changes the dimensional (qualitative) identity of the primes involved. So the unique combination of prime factors (constituting each composite natural number) reflects a qualitative (ordinal) combination.

So the Zeta 1 zeros express for the number system as a whole this unique ordinal identity (resulting from prime number multiplication).

Thus once again a perfect harmony is preserved as between each individual zero and the overall collective interdependent identity of  the zeros as an entire group. So quantitative and qualitative aspects are reconciled here for the number system as a whole.

This in turn provides the deepest fundamental insight into the Riemann Hypothesis i.e. that all the (non-trivial) zeros lie on an imaginary line (through 1/2).

Again the significance of the imaginary line is that it provides - in holistic terms - the indirect means of expressing circular meaning (i.e. relating to holistic interdependence) in an analytic fashion.

1/2 basically relates to the harmony as between external (objective) and internal (subjective) mental interpretation.

Thus the requirement that all the zeros lie on a straight line simply points to the ultimate requirement for the consistent relating of both cardinal and ordinal notions of number in a linear (1-dimensional) context.

Thus the Zeta 1 zeros serve as the holistic requirement for the consistent interplay of the cardinal and ordinal notions of number in relation to the number system as a whole (mediated through the qualitative relationship of the prime number factors that thereby uniquely result in cardinal natural numbers of a composite nature).

The Zeta 2 zeros serve as the corresponding holistic requirement for the consistent interplay of cardinal and ordinal notions for each individual number (mediated through the qualitative nature of the ordinal natural numbers within each prime).

Clearly both simultaneously arise in an ultimately ineffable manner. Therefore once the number system becomes inherent in phenomena (as the most intrinsic encoding of their nature) this perfect simultaneity has been already broken with quantitative and qualitative aspects separated in space and time.

In fact moving towards this ultimate nature of the number system - which equally represents the ultimate nature of universal creation - requires:

(i) the ability to see the prime numbers and natural numbers as perfect mirrors of each other (through the dynamically interdependence of complementary cardinal and ordinal frames of reference) and

(ii) the ability the see the two sets of zeta zeros (Zeta 1 and Zeta 2) as perfect mirrors of each other again through the dynamic interdependence of complementary holistic frames of reference and

(iii) finally to see both the two sets of zeros and the prime and natural numbers as ultimately identical, which realisation approximates (insofar as is possible while still remaining in the phenomenal realm) to the pure spiritual experience of the original ineffable nature of the nature of the number system (and of the entire universe which is intrinsically encoded in number).

The Emperor Has No Clothes (3)

What is accepted as "proof" in conventional mathematical terms represents but a limited notion (where once again the qualitative notion of number is reduced to the quantitative).

As we have seen, properly understood - in dynamic interactive terms - if the base number is quantitative, then the dimensional number is - relatively - of a qualitative nature.

So for example in the expression 2¹, if 2 is quantitative, then the dimensional number 1 is - relatively - qualitative in nature.


The psychological counterpart refers to the relationship as between a particular perception of number and the general concept of number (to which it is related).

So for example if the particular perception of the number "2" is deemed as quantitative, then the concept of number to which it is related is - relatively - qualitative in nature.

Now what does this mean precisely?

Well, "2" represents an actual finite number in quantitative terms. However the general concept of number has a potential infinite meaning in qualitative terms i.e. as applying to all possible numbers.

The crucial point to grasp in this context is that finite and infinite are quite distinct notions that again are quantitative and qualitative with respect to each other.

Strictly, whereas the finite is a (conscious) rational notion that is analytic, the infinite - by contrast - is directly an (unconscious) intuitive notion that is holistic in nature.

So, in the actual recognition of number, both rational and intuitive capacities are necessarily involved. 

However the very essence of standard (1-dimensional) interpretation is that the qualitative is effectively reduced in mere quantitative rational terms.

So in this context, the intuitive notion of the infinite is thereby reduced in rational terms, as actually - rather than potentially - applying to all specific numbers. 

Now there is a big problem here in maintaining that the concept of number applies to "all" specific numbers (as strictly "all" cannot be defined in an actual finite manner).

In other words because the number system is inexhaustible in finite terms,  the very recognition of a set of numbers that can be actually identified, implies an indeterminate set that cannot be identified in this manner.

Thus an inevitable indeterminacy applies to the definition of "all" numbers in a finite manner.

However this issue - like so many other crucial issues - is simply glossed over in conventional interpretation. Therefore in effect, even though strictly nonsense, the infinite is thereby misleadingly treated as a linear extension of the finite.  
In this way finite and infinite notions can be related to each other in a rational manner!

Now this fundamental problem intimately applies to very nature of proof in Conventional Mathematics.

Again the purpose of a general proof - say - of the Pythagorean Theorem - is that it should apply in "all" cases.

So in correct terms we can validly maintain that an accepted general proof potentially applies to all cases within its class (in an infinite manner).

So for example the assertion that  in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, potentially applies to all such triangles (in an infinite manner).

However this strictly does not establish the applicability of the theorem in any actual case. Thus, to assert that the theorem applies in actual cases is simply to confuse the infinite with finite notions (or alternatively to reduce the qualitative in merely quantitative terms).  

Now this is by no means intended to suggest there is no value therefore to the accepted notion of mathematical proof! Rather it is to raise the even deeper issue (which underlines all mathematical relationships) as to how we can establish a consistent relationship as between quantitative and qualitative notions! And the relationship as between finite and infinite in the context of mathematical proof represents just one important example of this key problem.


Once again the Riemann Hypothesis - when correctly interpreted - points to this very issue as the requirement, in the context of number, for the ultimate identification of its quantitative (cardinal) and qualitative (ordinal) aspects.

Properly dealing with the notion of  mathematical proof will require moving to a balanced dynamic interactive approach with respect to all mathematical relationships (where both quantitative and qualitative aspects are equally recognised).

In my own terminology this will require therefore both Type 1 and Type 2 - relatively separate - aspects to mathematical interpretation which then are comprehensive integrated in Type 3 terms.


Just as in Quantum Mechanics, this will lead to a new "Uncertainty Principle"  with respect to mathematical proof.

In other words in such dynamic terms, proof will be understood in a merely relative manner, with both Type 1 (quantitative) and Type 2 (qualitative) aspects of interpretation applying in a relative approximate manner.

 Current (1-dimensional) proof represents an extreme in terms of the mere Type 1 (quantitative) aspect leading to a mistaken absolutist view of its nature.

Even momentary reflection on the nature of proof will lead one to see it as representing a special form of social consensus.

On occasion, a mistaken view with respect to such consensus can exist. For example it was initially accepted in 1993 that Andrew Wiles had proved "Fermat's Last Theorem" only for an important error to be subsequently found. Now. happily this has since been corrected with an unchallenged consensus as to the validity of the proof existing since 1995. However the possibility - however small - that additional problems may subsequently arise cannot be ruled out completely.

Indeed acceptance that Fermat's Last Theorem" has been proved, for most people represents an act of faith (rather than reason). So we trust that the small number of mathematicians competent enough to check the details have done so correctly.

However I am making here the much more fundamental point that - by definition - all conventional mathematical proof represents a basic form of reductionism (whereby qualitative notions are reduced in merely quantitative terms).

Thus correcting this reductionism requires from the onset acceptance of the merely approximate relative nature of all proof.

I have already mentioned the Pythagorean Hypothesis! This led to an important episode in mathematical history which illustrates my point very well.

As is well-known, the Pythagoreans quickly discovered that their theorem led to the existence of irrational numbers. For example in the simplest case where both the adjacent and opposite sides = 1, the hypotenuse = the square root of 2 (which is irrational).

Now apparently the Pythagoreans were able to demonstrate that the square root of 2 was indeed irrational. So they had an accepted Type 1 proof at their disposal.

 However they were not happy to leave it at that ! They wanted to understand the deeper reason as to why irrational number quantities, such as the square root of 2, can arise in a world that they believed was scientifically governed in qualitative terms by the rational paradigm. 

So - using my terminology - they were looking for the Type 2 aspect of mathematical proof for irrational numbers (which they were unable to produce).

Though Mathematics has certainly become much more specialised in quantitative terms since the time of the Pythagoreans, in some ways it is the poorer, for appreciation of the vital qualitative aspect has by now completely been lost.

I attempted some years ago to provide precisely this Type 2 aspect of proof for the irrationality of the square root of 2 (which is summarised in the blog entry "The Pythagorean Dilemma").

So like the identification of particle and wave aspects of matter, we can see that all proof represents a certain compromise as between Type 1 (quantitative) and Type 2 (qualitative) aspects.

Too much focus on one aspect creates increasing fuzziness therefore in terms of identification of the other aspect.

So with conventional interpretation, focus merely on the Type 1 aspect has become so extreme that it has blotted out recognition entirely of its complementary Type 2 aspect.

Thus in future in a more comprehensive mathematical understanding, all proof will require equal attention to both Type 1 and Type 2 aspects, where both the analytic and holistic appreciation of mathematical relationships can develop side by side.

In this new enriched world, in some cases the initial unfolding of Type 1 recognition will lead to a corresponding search for complementary Type 2 understanding; in other cases it will be the reverse with holistic Type 2 understanding preceding proper recognition in Type 1 (analytic) terms.

Present mathematical understanding is therefore totally lop-sided (with holistic recognition completely overlooked). 


Of course not all problems can have a Type 1 proof (even in the accepted 1-dimensional sense) and the Riemann Hypothesis falls clearly into this category.

Once again, it points to the ultimate identity as between the quantitative and qualitative aspects of number. Acceptance of this fundamental identity is more a matter of faith than reason.
This thereby entails that acceptance of the entire mathematical edifice requires likewise such an initial act of faith.

The Emperor Has No Clothes (2)

Yesterday I dealt with how quantitative and qualitative aspects are necessarily involved in all mathematical relationships (in a dynamic interactive manner).

The corollary of this from the psychological perspective is that both conscious and unconscious aspects of understanding are necessarily involved in all interpretation of such relationships (again in a dynamic interactive manner).

So the qualitative aspect is directly related to the unconscious aspect of interpretation!

However as we know Conventional Mathematics (that is qualitatively 1-dimensional in nature) is based on a further rational illusion that its meaning can be successfully conveyed in a linear rational manner.

Though the importance of the unconscious e.g. through supporting intuition may well be informally recognised (especially where creative insight is required) in formal terms this is allowed to play no part in accepted mathematical interpretation.

So once again the standard approach is based on a striking limitation (where unconscious intuition - though of a qualitatively distinct nature - is reduced in a merely rational manner.

This issue again is of the most fundamental possible and is glossed over completely within the mathematics community. This is why there such a great need for “outsiders” to identify key issues that remain so steadfastly ignored by “respected” mathematicians.  

Once one accepts the equal importance of the qualitative aspect with the quantitative, and the corresponding equal importance of the (unconscious) intuitive with the (conscious) rational aspect of interpretation, then a key problem of the first magnitude arises with respect to the relationship as between the quantitative and qualitative aspects of meaning.

In a truly profound manner the non-trivial zeta zeros (both Zeta 1 and Zeta 2) provide the answer to this problem. They show us from two opposite directions i) the precise numerical relationship as between quantitative and quantitative aspects in an (external) physical fashion and ii) the complementary precise relationship - through interpretation - as between conscious and unconscious aspects in an (internal) psychological fashion.

Indeed once one appreciates the dynamic nature of number (from both external and internal perspectives) then one readily accepts that number is inherent in all physical and psychological processes (as the most intrinsic means of their encoding).

This entails for example that the history of the universe in the first instant of its phenomenal evolution is inseparable from the nature of number (with respect to its quantitative and qualitative aspects).

This also entails that the ultimate final realisation of the meaning of this universe is likewise inseparable from a full appreciation of this nature of number (which of course entails the full appreciation of the zeta zeros).

Admittedly there has been growing recognition in recent years of striking parallels as between the Zeta 1 zeros and certain quantum chaotic physical processes.

However - because of the lack of a dynamic paradigm - mathematicians are approaching this relationship largely from the wrong direction.

In other words they are wondering what the physical processes reveal about the zeta zeros when really it should relate to what the zeta zeros reveal about the nature of these physical processes. Thus the quantum behaviour of nature is already inherent in the dynamic nature of the number system! However because they are accustomed to looking at numbers in absolute terms i.e. as static unchanging entities they are unable to readily make this connection.

 

However the dynamic nature of the number system equally entails, that the physical aspect of its external behaviour with respect to nature is fully complementary with the psychological aspect of corresponding internal interpretation.

 

This immediately implies that the zeta zeros (both Zeta 1 and Zeta 2) have immense potential relevance in the human contemplative quest of attaining full enlightenment.
 

Once again from a dynamic perspective comprehensive mathematical understanding entails the equal specialisation of both reason and intuition. Such specialised intuitive attainment thereby ultimately requires the most advanced contemplative state (where reason can dynamically interpenetrate with intuition - without rigidity - in a highly transparent fashion).

So this situation where pure contemplation can be married with extremely refined rational structures of a dynamic nature will eventually be necessary for the most comprehensive understanding of mathematical relationships (which I refer to as Type 3).

 

Indeed when one thinks about it from the external objective perspective, the zeta zeros (from two directions) are associated with the ultimate relationship of the primes to the natural numbers.

Now there is a remarkable parallel - which can only be made through holistic terms - with the goal of human evolution.

Anyone for example familiar with Jungian psychology would be able to see this readily in terms of the unification of both conscious and unconscious aspects of the personality.

Now the untrained unconscious expresses itself - especially in earliest childhood - through a mass of uncontrolled primitive impulses.

We all can perhaps accept easily enough how our conscious behaviour with respect to the natural world can be readily hijacked through unconscious impulsive projections.

So the very task of properly recognising the intricate relationship of the primes to the natural numbers (in external physical terms) is ultimately inseparable from the corresponding psychological task of successfully reconciling the unconscious (and its primitive desires) with the rational conscious mind.

This immediately entails that all unconscious impulses are in fact encoded with respect to the qualitative aspect of prime number behaviour. Thus the unravelling of such primitive impulses is inseparable from directly unravelling this prime number code (with respect to its qualitative aspect).

 

Therefore the full attainment of spiritual contemplative development is inseparable from this task of gradually unravelling all primitive impulses, so that finally the unconscious can then be fully married with the conscious mind.

Thus the ultimate nature of number in external physical terms (at the earliest stages of evolution) with respect to the identity of its holistic and analytic aspects, is inseparable from the ultimate understanding of the nature of number (at the most advanced stages of evolution) where the holistic unconscious and analytic conscious aspects of personality can be finally fully merged with each other.

 

So when Hilbert maintained that the problem of the zeta zeros was not only the most important problem in Mathematics but absolutely the most important, in this respect he was fully right!

When one accepts that all phenomenal creation is encoded in number (with respect to both its quantitative and qualitative aspects), then the very purpose of evolution can be seen as the attempt to realise its most intrinsic secret (which is ultimately ineffable in origin).

I will finish up this blog entry with reference to a striking feature of the Zeta 2 and Zeta 1 zeros respectively (with immense psychological implications).  

All of the (non-trivial) Zeta 2 zeros lie on the circle of unit radius in the complex plane.

As we have seen conventional mathematical reason relates to the conscious aspect of understanding (and is linear in nature). It is directly associated in turn with quantitative interpretation

Now the fact that all the zeta zeros here lie on a circle indicates that we have now switched to the unconscious aspect of understanding (and thereby circular in nature). Now to be precise, pure intuition is ineffable. However when we attempt to express its nature (as the interdependence of opposite reference poles) indirectly in a rational manner, it creates paradox in terms of standard (linear) reason.

So the Zeta 2 zeros are therefore directly associated with the qualitative aspect of understanding (that is indirectly translated in a quantitative manner).

The holistic nature of these zeros then arises through the inevitable dynamic interplay of both independent aspects (as quantitative) and interdependent aspects (as qualitative).

 

So the Zeta 2 zeros relate properly to the holistic (qualitative) aspect of understanding with respect to the number system. This is then translated indirectly in a circular rational manner. And properly understood the numerical symbols thereby generated are translated accordingly in this manner.

Though I have found the Zeta 2 zeros to be of equal importance to the Zeta 1 (and fully complementary with them) they remain unrecognised. This is due to the fact that Conventional Mathematics is completely lacking a holistic (qualitative) dimension!

Now the Zeta 1 zeros are all postulated to lie on a straight line through ½. However this straight line is of an imaginary - rather than real - nature.

This likewise has remarkable psychological connotations.

If we simplify psychological development the first task is to successfully differentiate the conscious mind (thereby attaining mastery with respect to linear reason).

And this capacity has reached a highly specialised level in our present culture.

However the next task (which occasionally unfolds with true spiritual aspirants) is to now successfully integrate the unconscious mind (thereby attaining mastery with respect to intuitive capacity). These then in a mathematical context would be translated in a circular rational fashion.

So this contemplative extreme relates directly to the qualitative holistic aspect of mathematical development (to which the Zeta 2 zeros directly relate)

 

However the final task relates to the task of then releasing all this unconscious intuitive capacity in terms of everyday activity.

So when one traces development of the great religious leaders the final stage of their lives often is remarkably active. So they have reached a sufficient stage of mastery as to be able to engage with everyday practical concerns (now transformed through an enlightened spiritual perspective).

 

The Zeta 1 zeros in fact represent the mathematical equivalent of the same process.

What they entail is the ability to bring the qualitative aspect of holistic unconscious awareness to bear within a standard analytic setting (based on linear reason).

With religious heroes this would be identified as the ability to integrate successfully the (unconscious) contemplative aspect of specialised intuitive enlightenment with the many demands of (conscious) everyday activities. This is sometimes referred for as the marriage of contemplation and activity and generally recognised as the most advanced stage of spiritual attainment!

The corresponding mathematical equivalent would be the ability to integrate the qualitative holistic aspect of mathematical appreciation with its (recognised) quantitative analytic aspect. 

So a huge amount of attention in recent years has been given to interpretation of the Riemann Zeta Function (and associated Riemann Function) from the standard analytic aspect.

However once again proper interpretation of the (non-trivial) Zeta 1 zeros requires that holistic qualitative appreciation be properly integrated with its quantitative counterpart.

I have already explained in some detail in a previous blog entry the holistic significance of these zeros. (So I will be brief here)!

½ signifies an equal balance as between quantitative and qualitative aspects (with associated equal balance as between holistic and analytic interpretation of symbols.

Now the fact that the points lie on an imaginary line is very interesting.

In holistic terms the imaginary represents the indirect analytic means of expressing meaning that is directly of a holistic nature.

So for example the first zeros on the imaginary line are 14.134725 (and also –  14.134725).

However this represents an indirect analytic means of providing values that inherently are of a qualitative (interdependent) nature. This explains the puzzle of why such numbers in parallel quantum terms are associated with energy states.

An energy state simply represents the dynamic qualitative nature of number!

Now it also appears that all these non-trivial imaginary parts are transcendental in nature.

I wrote an on-line book some 20 years ago explaining the holistic meaning of the various number types.

I concluded then that the most refined state possible in the phenomenal realm relates to what is both transcendental and imaginary!

Transcendental in a holistic context relates to what is understood as neither quantitative nor qualitative (separately) but as the relationship between both aspects.

So understanding of phenomena needs to be extremely dynamic and refined to operate at this level.

The fact that they are imaginary, entails that they relate to unconscious projections. Now again an extreme mastery would be required to be able to spontaneously recognise all projections immediately in experience as expressing the balanced relationship as between both quantitative and qualitative aspects of understanding. This would entail that they would instantly dissolve and pass from memory as soon as they arise in experience.

Because the Zeta 1 zeros relate to the most intrinsic nature of matter we would not be able to identify them with measurable phenomena. Therefore they would serve as the final bridge as between phenomenal and ineffable reality in physical terms.

This would entail that their full understanding would equally require the most advanced stage of enlightenment possible (consistent with remaining in the phenomenal realm).

We are an awful long way from such realisation at our present stage of evolution. 

What is truly remarkable however is that we have come far enough to at least begin to appreciate their true nature.

The Riemann Hypothesis (that all the non-trivial zeros lie on the imaginary line through ½), in holistic terms entails that both quantitative and qualitative aspects of understanding can ultimately be fully identified with each other.

Acceptance of this postulate properly belongs to faith and not to reason.

And quite clearly this postulate cannot be proved (or disproved) with reference to merely the quantitative aspect of mathematical interpretation. 

The Emperor Has No Clothes (1)

As I have repeatedly stated, proper comprehension of the Riemann Hypothesis has the most far reaching consequences possible for the true nature of Mathematics.

In fact, to put it bluntly, what we know as Mathematics is built on a massive lie!
In other words, though there are two equally important aspects to all mathematical understanding that are quantitative and qualitative with respect to each other, Conventional Mathematics is built on the reductionist illusion that only one of these i.e. the quantitative is relevant.

Indeed the modern development of Mathematics can be likened to a gross form of propaganda where at every turn reference to the qualitative has been expunged so as to leave conventional wisdom unchallenged.

Imagine presenting the history of a country comprising two proud races of  equal importance in terms of the contribution of just one! Worse still, imagine that great pains have been taken to avoid ever making reference to the existence of this second race. We would perhaps see such propaganda as indeed very distorted!

The same charge can be made against Conventional Mathematics. Unfortunately we have now been told the same propaganda for so long that we accept it utterly without question as the total truth.

I am writing this blog entry to proclaim "The Emperor Has No Clothes".

Once again the notion of the quantitative is built on the fallacy of numbers possessing an objective independent existence (in absolute terms).
This notion is enshrined in its cardinal use. So if we refer to a group of objects -  say 3 cars - we are referring to them in a quantitative manner (i.e. as independently existing).

However if I now refer to an object in ordinal terms - say the 3rd object - this has no meaning in itself but must be given a wider general context with respect to the related group of objects.

So the ranking of this object depends on a more general context which refers to a qualitative - as opposed to quantitative - distinction.

Now amazingly you will never find it mentioned in a mathematical textbook that with ordinal rankings, we have now shifted to the qualitative notion of number. This of course would immediately raise serious questions regarding the accepted - merely quantitative - notion!

So to avoid this conflict abstract mathematical terminology has been skilfully developed so as to preserve the "quantitative illusion".

Last night I looked up the Oxford English Dictionary to quickly find "rank" given as one of the definitions of "qualitative". However again you will not see this mentioned in mathematical textbooks!


Therefore the first thing to clearly grasp - which is perhaps the most important of all - is that the cardinal and ordinal aspects refer to the quantitative and qualitative aspects of number respectively.

Both of these aspects continually interact in experience. We cannot apply the cardinal aspect to number (within an implied ordinal aspect).  We cannot in turn apply the ordinal aspect to number (without an implied cardinal aspect).

Thus the notion of number is properly of a dynamic interactive nature with aspects that are - relatively - independent and interdependent with respect to each other.


The next key issue to grasp is that the ordinal i.e. qualitative nature of number (when properly recognised) is based on an entirely distinct logical system to that of quantitative appreciation.

Conventional Mathematics is defined by its merely 1-dimensional nature (in qualitative terms). What this means is that the interpretation of relationships in any relevant context is based on just one (independent) polar reference frame. So typically for example, mathematical objects are viewed from an external reference frame (thus avoiding interaction with a corresponding internal aspect); likewise mathematical objects - as we have seen - are viewed from a merely quantitative framework (thereby avoiding qualitative interaction)

However once we accept the inherent relative nature of mathematical relationships i.e. where opposite polarities dynamically interact in two-way fashion in experience, we move to higher dimensional interpretation.

So just as  2, 3, 4, 5,..  have a quantitative meaning in mathematical terms (where in qualitative terms interpretation remains 1-dimensional), likewise  2, 3, 4, 5 have a qualitative meaning (where interpretation now takes place in accordance with such dimensions).

Each higher dimension from a qualitative perspective represents a distinctive manner of configuring the opposite polarities of experience that is inversely related to its corresponding roots of 1).

Thus to give meaning to ordinal number distinctions (in quantitative terms) we must use these higher dimensions.

For example to give meaning to 1st and 2nd (in the context of  a group of 2) requires 2-dimensional interpretation (inversely related to the 2 roots of 1).

More generally to give meaning to 1st, 2nd, 3rd ....nth (in the context of a group of n) requires n-dimensional interpretation (inversely related to n roots of 1).

The importance of the primes in this context is that each prime number is associated - apart from the common root of 1 - with a unique set of non-trivial roots.

In this sense associated with each prime is a unique natural number arrangement in ordinal terms.

These correspond with - what I refer to as - the Zeta 2 (non-trivial) zeros.

It is vital to grasp that the proper nature of these zeros is of a dynamic holistic nature (where both the quantitative aspect of independence and the qualitative aspect of interdependence are perfectly reconciled).

For example the simplest prime number 2 is associated with 1st and 2nd members in ordinal terms.

Now each of these members can be given an indirect independent quantitative expression (on the circle of unit radius) as + 1 and – 1  respectively. Then the qualitative interdependence of both members is indicated through their sum = 0.

Thus in this way a total harmony is established as between quantitative and qualitative aspects (for this prime number 2).

Therefore, once again the holistic significance of the Zeta 2 zeros resides in the fact that for each prime, a unique circle of relative independence and interdependence exists (with respect to its natural number members). So quantitative and qualitative aspects are perfectly harmonised in this manner for ordinal number members with respect to each prime.

Now whereas the Zeta 2 operate on the micro scale - as it were - with respect to the internal composition of each prime (in terms of natural number members in ordinal terms)  the Zeta 1 zeros operate in reverse on the macro scale with respect to the external composition of the natural numbers, (in terms of prime constituents in cardinal terms).


In other words the Zeta 1 (non-trivial) zeros translate, as it were, the cardinal nature of the relationship of the primes to the natural numbers i.e. where each natural number can be expressed uniquely in terms of prime factors, indirectly in an ordinal qualitative manner through a corresponding unique set of numbers.

Once again the proper nature of these zeros is dynamic and holistic. In other words through each zero (as independent) indirectly represents a point of pure qualitative interdependence (as a numerical energy state) the combined set of all these zeros represents the locally independent quantitative nature of the primes (in opposition to the common shared relationship of primes and natural numbers).

In this way a perfect harmony is preserved as between both the quantitative (independent) and qualitative (interdependent) aspects of the overall number system (through the relationship of the primes to the natural numbers).


Ultimately of course the Zeta 1 and Zeta 2 zeros are identical in an ineffable manner.

Therefore in Zeta 2 terms, the unique holistic nature of (ordinal) natural numbers to primes internally, where quantitative and qualitative aspects are fully reconciled for each number, is inseparable in Zeta 1 terms from the unique holistic nature of (cardinal) primes to the natural numbers, where quantitative and qualitative aspects are reconciled externally for the number system as a whole.


However, as the zeta zeros (Zeta 1 and Zeta 2) are intrinsically of a dynamic holistic nature, indicating both internally and externally within the number system how quantitative (cardinal) and qualitative (ordinal) aspects are ultimately fully reconciled, it is pointless trying to understand their role in a merely quantitative manner.


Indeed put simply the Zeta 1 and Zeta 2 zeros point (from two complementary perspectives) directly to the unrecognised qualitative aspect of the number system.

Once again just as in quantitative terms, the Riemann Zeta Function remains uniquely undefined in where s (as dimensional number) = 1, likewise, the Riemann Zeta Function remains uniquely undefined in qualitative terms where s (as dimensional number) = 1.

I cannot stress how important this is! What it means is that the Riemann Zeta Function (and Riemann Hypothesis) remain uniquely undefined, when we attempt to understand number in a merely absolute quantitative manner!
All other values for s (≠ 1), refer to dynamic relative interpretations (where number has both quantitative and qualitative aspects)! 

The implications could not be more fundamental. Our present understanding of number (and by extension all mathematical notions) is based on a reductionist sham i.e. that quantitative meaning can be given independent of a general context that is necessarily qualitative. So in truth both dynamically interact in all meaning!

Not only Mathematics, but indeed all the Sciences are now deeply contaminated with the same fundamental falsehood.

We need to start facing up to this critical issue immediately. A successful future for our civilisation will ultimately depend on it!

Holistic v Analytic Interpretation

It is important to understand the precise context in which I use the terms holistic and analytic with respect to mathematical interpretation.

Unfortunately for our purpose "analytic" has taken on a more specialised and limited meaning in Mathematics (relating to calculus, functions, series, limits etc.).

However the sense in which I use analytic is altogether much broader in scope. In fact from this enlarged perspective, all mathematical interpretation in formal terms is strictly of an analytic - as opposed to holistic - nature.

Analytic in this wider context relates to interpretation that is 1-dimensional. And as developed in several blog entries, 1-dimensional simply means interpretation according to one (fixed) pole of reference (referring here to the fundamental polarities such as objective/subjective and quantitative/qualitative which necessarily condition all phenomenal experience of reality).

So analytic implies that mathematical symbols can be interpreted (i) in an absolute objective manner i.e. where effectively the internal is reduced to the external aspect and (ii) in an absolute quantitative manner i.e. where effectively the whole is reduced in terms of its independent parts.

Holistic by contrast implies a necessary dynamic interaction as between opposite polarities.
So from this perspective (internal) subjective interpretation cannot be separated from the (external) objective nature of truth. So mathematical truth is thereby of a relative nature involving both aspects.

Also from this perspective, quantitative independence e.g. with respect to the identity of number, cannot be separated from qualitative interdependence (in the overall relationship between numbers).

So once again mathematical meaning is necessarily of a relative nature.


Holistic interpretation applies to all dimensional numbers ( ≠ 1).

What is simply astonishing is that Mathematics in formal terms  still completely lacks any holistic dimension.

Thus, though the (absolute) interpretation of mathematical symbols has indeed an extremely important special role, it has been misleadingly elevated as synonymous with all valid Mathematics.

Nothing could be further from the truth! In fact an unlimited number of other dynamic dimensional interpretations, each with a partial relative validity, exist.

However in a certain sense, interpretation associated with the dimensional number 2 serves as the blueprint for all  other relative type interpretations of mathematical symbols.


As we have seen at its deepest level, the nature of number entails the mysterious conjunction of both  external and internal polarities. Thus from one polarised reference frame, number can appear as fully objective already enshrined in nature. Equally from the opposite frame number can appear as merely a mental construct which we use to interpret reality in a certain way. This then leads inevitably to the realisation that number in some manner entails the relationship as between both of these aspects.

Equally from one polarised reference frame, number can again appear to have an absolute objective identity as independent quantities; however further reflection can quickly show that these can have no meaning in the absence of an overall dimensional context (which is qualitative in nature).

So again this should inevitably lead to the realisation that number likewise entails the dynamic relationship as between both its quantitative and qualitative aspects.


Therefore though the more limited analytic approach does indeed have great validity (within its own  context) it remains quite unsuited for understanding of fundamental mathematical issues such as the nature of the primes.

Now admittedly remarkable progress has been made in this regard at the analytic level with the development of  many tantalising mathematical results.

However proper interpretation of these results requires holistic - rather than analytic - interpretation.

Once again in analytic terms, the one value for which the Riemann Hypothesis remains undefined is where s = 1 (interpreted in the standard linear fashion).

In corresponding holistic terms the one value for which the Riemann Hypothesis remains undefined is again where s = 1 (now interpreted in a dynamic circular manner).

What this simply means is that we cannot hope to properly understand the Riemann Zeta Function (and associated Riemann Hypothesis) in the standard analytic manner.

Indeed, properly understood, the Riemann Zeta Function establishes (i) a 2-way relationship as between interpretation and objective type results and (ii) a 2-way relationship as between the quantitative (cardinal) and qualitative (ordinal) aspects of number.

The Riemann Hypothesis then establishes the condition for mutual identity of both (i) external and internal aspects (through the requirement of the real part of all non-trivial zeros = 1/2 (ii) quantitative and qualitative aspects through a series if complementary (positive and negative) points on the imaginary line through 1/2.


Not alone therefore does the Riemann Hypothesis properly require holistic type interpretation, it approaches the extreme limit in terms of the specialised demands it makes on such understanding. This is why in the deepest sense it is utterly futile to try and reduce the problem in a merely analytic fashion.

Quite simply we cannot hope to understand the ultimate identity of the external (physical objective) and internal (mental subjective) aspects of number when the existing paradigm of understanding reduces the latter to the former aspect.

Likewise we cannot hope to understand the ultimate identity of the quantitative (cardinal) and qualitative (ordinal) aspects of number again through a paradigm that again reduces the latter to the former.

So again, quite literally, the Riemann Zeta Function (and Riemann Hypothesis) remain undefined in linear (1-dimensional) terms and cannot be successfully approached through the conventional mathematical approach.


The zeta zeros (Zeta 1 and Zeta 2) therefore can only be properly understood in a holistic mathematical manner.

These zeros therefore form an integral part of the number system (as comprehensively understood).

The primes and natural numbers correspond directly with analytic aspects of this system; however the Zeta 1 and Zeta 2 (non-trivial) zeros correspond directly with the - equally important -  holistic aspects of the system.


Now the importance of the two sets of polarities can be expressed quite simply!

The external and internal polarities are necessary to enable switching - relatively - as between specific numbers (as finite) and the general notion of number (as infinite). So in dynamic terms we cannot separate finite and infinite domains (as the finite has no meaning in the absence of the infinite or likewise the infinite in the absence of the finite.

The quantitative and qualitative (i.e. part and whole) polarities are necessary to enable switching - relatively - as between the prime and natural numbers.

This is why the relationship of the primes to the natural numbers (and the natural numbers to the primes) is so important! It is because this is the manner through which the quantitative (cardinal) and qualitative (ordinal) aspects of number are mediated in a dynamic relative manner.


Now once again the holistic relationship as between (i) the finite and infinite notion of number and (ii) the primes and natural numbers is embodied from two different directions through the Zeta 1 and Zeta 2 (non-trivial) zeros.

Once again the dynamic holistic nature of the Zeta 2 zeros is easier to appreciate. Here each prime is defined (by addition) in terms of an ordinal group of natural number members. So for example 3 is composed of a 1st, 2nd and 3rd member. Then the ordinal identity of these 3 members indirectly is given a quantitative identity (on the circle of unit radius in the complex plane) through the 3 roots of 1. And this can be repeated for each prime number with these roots in each case constituting the Zeta 2 zeros.

So, rather than quantitative (independence) and qualitative (interdependent) aspects being separated in a static manner, here they are directly integrated in dynamic terms. So each root has a - relatively - independent quantitative existence while the combined group (through addition) has - relatively - interdependent qualitative existence (exemplified by the fact that the quantitative sum = 0).

Thus number here from one perspective is given an independent existence through its individual members as form, while also being given a combined group existence through its collective relationships as energy!

Though initially each prime number is necessarily defined in a finite manner, clearly the procedure can proceed without finite limit (without strictly however being capable of definition in an infinite manner). In other words we cannot define an infinite prime number!


It is somewhat the reverse relationship that applies to the better recognised Zeta 1 zeros. Here each natural number is defined in a cardinal manner through a unique combination (by multiplication) of prime number factors. So, for example 6  is defined as 2 * 3!

However just as what is inherently qualitative can indirectly be given a quantitative identity, likewise what is inherently quantitative can indirectly be given a qualitative identity.

In other words from the dynamic holistic perspective, each prime number is independent in a merely relative sense. This implies that the prime numbers (as a group) have a hidden shadow identity as qualitative.

Thus the set of (non-trivial) Zeta 1 zeros constitute the (hidden) qualitative aspect to the primes. And just as the qualitative aspect of each natural number enables order to be maintained uniquely within a finite prime group e.g. again with a group of 3 having a 1st, 2nd and 3rd member, likewise in complementary fashion, the qualitative aspect of the primes  - expressed through the Zeta 1 zeros - enables a unique order of interdependence to be maintained with the natural numbers.

So here each individual Zeta 1 zero represents in isolation the qualitative shadow counterpart to the primes. This is why each non-trivial zero represents an energy state (as now recognised through striking parallels identified with quantum chaotic physical energy states)! The quantitative nature of these zeros is then expressed through their collective group identity. This indeed is why we can use these non-trivial zeros to restore the local individual quantitative nature of the primes (as distinct from their overall collective relationship to the natural numbers).


Therefore to sum up! the number system properly is of merely relative nature, where numbers represent dynamic two-way interaction patterns as between external and internal aspects and (ii) quantitative and qualitative aspects.

The primes and natural numbers represent the analytic nature of the number system.

The Zeta 1 and Zeta 2 (non-trivial) zeros represent the corresponding holistic nature of the number system.

Ultimately, both aspects are totally interdependent with each other in an ineffable manner.
However, relative independence and interdependence characterise these two aspects in dynamic phenomenal terms.