Holistic Interpretation of Riemann Hypothesis

In the following section, I will attempt to provide a little more holistic perspective on the Riemann Hypothesis, with a view to representing its true nature.

All the (non-trivial) zeros are postulated as lying on an imaginary line, drawn vertically through .5 on the real axis.

So, one might ask what then is the key significance here of .5?

Well, in analytic terms, clearly .5 is half-way between 0 and 1.

When 0 is added to any number, its identity thereby remains unchanged.

And when any number is multiplied by 1, again its identity remains unchanged.

Thus the revealing point here is that in lying half way between 0 and 1, that .5 represents a point of balance, where in a sense both addition and multiplication attain an equal status with respect to the number system.

Then in more holistic terms, 1 represents the linear (analytic) approach and 0 the circular (holistic) approach to number respectively.

Thus once again from this perspective, .5 can be seen again as maintaining the golden mean in a balanced equality of both linear (analytic) and circular (holistic) understanding.


An even deeper rationale for the key significance of .5 (in holistic terms) can be offered.

In standard analytic terms, the external polarity of experience is abstracted from the internal.

So “1” as a number object is identified fully with its objective existence (as somehow possessing a validity independent of (internal) mental interpretation.

However when one equally allows for (external) objective reality and (internal) mental interpretation, then .5 again becomes the point where both polarities are fully balanced (as half external and half internal respectively)


Strictly in experience therefore, we cannot have external number reality without corresponding mental interpretation (of such reality) and vice versa.

Thus crucially, the significance of .5, points to the fact that for true interpretation of the zeta zeros, both external and internal aspects (as opposite poles of understanding) must be fully balanced with each other. And this in turn is necessary, so that holistic aspect of experience (which entails the complementary balancing of opposite poles) can itself be properly incorporated in experience.

Thus the fullest appreciation of the zeros requires a highly refined understanding, where analytic and rational aspects are equally balanced in a psycho-spiritual manner approaching pure ineffability.

Put another way, such understanding provides the most complete marriage of refined reason (directly in linear and indirectly in circular terms) with a contemplative vision (of a purely intuitive nature).

So here, objective reality with respect to number and the mental interpretation of such reality are so closely allied as to be ultimately inseparable.


The next key point relates to the fact that all of these zeros are postulated to lie on an imaginary line (through .5).

The holistic significance of the imaginary notion, as I have stated, is that it provides an indirect analytic expression of what is, directly, holistic in nature.

In this way, important notions of unconscious intuitive origin can be incorporated with standard analysis (of a rational conscious nature).

When seen from this perspective, the over-riding significance of the imaginary line (on which all the non-trivial zeros are postulated to lie), is that it indirectly serves as the unconscious shadow counterpart of the real number line, which is given a directly conscious interpretation in mathematical analysis.

In fact this is vitally needed to provide a proper answer to the important query regarding multiplication that I have already posed.

Once again, when we multiply numbers, a qualitative dimensional change takes place, which directly relates to the interdependence of such numbers with each other.

So we might indeed start by considering primes such as 2 and 3 as independent (and thereby “building blocks” of composite natural numbers).

However once we associate 2 * 3 with each other - and the individual units within each number - in a multiplication operation, then a qualitative dimensional aspect is introduced through such interdependence.

Now effectively, this is all ignored in conventional mathematical interpretation, where the result of 2 * 3 is given a merely abstract quantitative identity.


A unique qualitative aspect equally arises when we add numbers. For example, when we add 1 + 1, we may conclude that the answer is 2 in a merely quantitative manner, but without establishing the interdependence (as well as assumed independence) of both units, it would not be possible to appreciate the new collective whole identity of 2. 

However a deeper question thereby arises, which unfortunately from a conventional perspective, is completely overlooked.

Given that a qualitative identity necessarily arises when numbers are associated with each other through multiplication (and addition), how can we then assume the consistency of the reduced - merely quantitative - interpretation of results given in conventional mathematical terms?

And the answer remarkably, which ensures such consistency, lies at the very heart of the Riemann Hypothesis!

Thus, for the consistency of the real line to hold (in conventional mathematical analysis) it is necessary that the assumption regarding the imaginary line likewise holds with respect to the Riemann Hypothesis.

Therefore, for the consistency of the real line to hold, all the non-trivial zeros must lie on the vertical imaginary line (through .5).

So, when interpreted appropriately, we are in an important sense enabled to effectively distinguish the quantitative aspect of number independence from the corresponding qualitative aspect of number interdependence. 

Conventional interpretation indiscriminately mixes notions of number independence with corresponding notions of number interdependence, blindly assuming that both aspects are consistent with each other with respect to the real number line.

However proper recognition of the true qualitative aspect of number requires that it is treated in a distinct holistic manner as indirectly relating to an imaginary (rather than real) line.

Thus, if all non- zeros - indirectly expressing, in a quantitative manner, the qualitative interdependence of the primes with natural numbers - also lie on a straight imaginary line, we are then entitled to assume the consistency of qualitative with quantitative interpretation.

Expressed in a more psychological manner, if all the non-trivial zeros lie on the imaginary line (through .5), then we are entitled to assume, in mathematical terms, that the unconscious (holistic) aspect of number understanding can be seamlessly integrated with its corresponding conscious (analytic) aspect.

However in the highly limited approach that formally defines conventional mathematical interpretation, the unconscious aspect (though utterly distinct) is blindly ignored and thereby reduced in mere conscious rational terms.

So the key holistic issue, to which the Riemann Hypothesis points, does not even arise in this reduced context. And again, this is why the Riemann zeta function remains undefined in holistic terms for s = 1, i.e. undefined for the conventional mathematical approach!


I will attempt to deal even more briefly with a couple of further issues.

The non-trivial zeros occur as complex conjugates, so that each zero of the form a + it is matched by a corresponding zero of the form a – it.

So there is a complementary situation in place for the imaginary aspect of the zeros (as positive and negative with respect to each other).

Now again a close analogy with physics might help to suggest the true holistic nature of this complementary pattern.

Virtual particles very close to the ground of matter have a highly transient existence, arising in closely related, matter, anti-matter pairs that immediately combine with each other in a fusion of physical energy. And in this context, virtual can be directly equated with the imaginary notion.

So the fact that the non-trivial zeros occur in imaginary pairs relates to the fact that the positive and negative polarities with respect to imaginary opposites are extremely closely related. From a psychological perspective, this implies that the short-lived separate existence of any number projection from the unconscious becomes quickly eroded through relationship to its opposite pole (thus approaching a purely intuitive appreciation of number as a psycho-spiritual energy state).

And this would be complemented at a physical level by the highly transient activity of virtual particles again approaching a pure energy state.

In fact it is highly revealing, when one adds non-trivial zero pairs.

For .5 + it + .5 – it  = 1.

Thus in a sense, these conjugate pairs represent the extremely short-lived break-up of conventional 1-dimensional appreciation into two highly dynamic inter-related realities.


Thus number is conventionally understood at the everyday conscious level of reality, as pertaining to the 1-dimensional level (i.e. where all real numbers lie on the number line).

However when one is able to see behind this reality into the refined intricate manner in which both conscious and unconscious are related with respect to number, then one can begin to see directly into the domain of the non-trivial zeros. Thus from this perspective, the zeros relate to an unseen highly dynamic background context (directly of unconscious origin) of a vitally important nature, that enable our everyday conscious assumptions regarding number reality to seem self-evident.

It also would appear that the imaginary parts of the zeros constitute transcendental numbers.

20 years ago, when I was investigating the holistic nature of the various number types, I concluded that the most elusive - and thereby closest to pure contemplative appreciation - are the imaginary transcendental numbers.

So in approaching the holistic nature of a transcendental number it may help again by considering the best known, i.e. π.

In analytic terms, this represents the relationship between the circumference of the circle and its line diameter.

And in holistic terms, all transcendental numbers express a certain relationship as between circular (holistic) and linear (analytic) aspects of understanding, where both are seen as interdependent with each other.

The imaginary notion, in this context, reflects the projection (directly, of an unconscious holistic nature) indirectly, in a conscious analytical manner. So, to be incorporated successfully with experience, unconscious material must initially be projected in a conscious manner. Then to avoid the confusion of holistic with analytic meaning, one must immediately realise in experience the distinct nature of imaginary projections, therefore avoiding any rigid attachment.

When highly developed, imaginary transcendental understanding becomes so refined that phenomena no longer even appear to arise, thus approaching a true contemplative vision (in the pure psycho-spiritual release of intuitive energy).

In this way, the non-trivial zeros as energy states represent the holistic extreme, approaching pure relativity, to the conventional understanding of number as representing absolute unchanging forms. 


From the dynamic interactive perspective, it is readily apparent that the Riemann Hypothesis can be neither proved (nor disproved) in the accepted conventional mathematical manner.

In this sense the Riemann Hypothesis, by its very nature, transcends conventional mathematical interpretation, while already being immanent (i.e. necessarily assumed) in all such interpretation.

As we have seen, the conventional approach is based on the reduction of unconscious type meaning in a directly conscious manner.

However properly understood, from a dynamic perspective, the fundamental mathematical issue relates to consistency of the distinctive intuitive meaning of mathematical symbols at an unconscious level with their corresponding rational understanding in conscious terms.

And from this perspective, the Riemann Hypothesis can be interpreted as the condition ensuring the consistency of both aspects in the context of the overall number system.

So, for example as we have seen when we multiply 2 by 3, as well as the recognised quantitative transformation to 6, this likewise entails a qualitative dimensional change in the nature of the units employed.

Therefore the validity of reducing the result in a merely quantitative manner, so that 6 is now represented as a number on the (1-dimensional) number line, is based on the assumption of the Riemann Hypothesis that all the (non-trivial) zeros lie on the imaginary number line, where quantitative and qualitative aspects of number interpretation are fully reconciled with each other.


The intuitive unconscious can be seen as the shadow counterpart to the rational conscious mind.

Thus the assumption that all natural number quantities can be placed on the real number line requires the complementary (shadow) assumption that all the (non-trivial) zeta zeros lie on a corresponding imaginary line. 

In fact properly understood in dynamic interactive terms, we can no more guarantee that all the natural numbers lie on the real, than that all the non-trivial zeros lie on the corresponding imaginary number line.

So in dynamic interactive terms, both assumptions mutually imply each other.

An important limitation of conventional mathematical interpretation is that in reducing the (unconscious) qualitative aspect of mathematical understanding, in a (conscious) quantitative manner, it thereby already assumes the Riemann Hypothesis to be true.

So again, the blind assumption that all natural numbers can be placed on the real, already implicitly assumes that the non-trivial zeros lie on the imaginary line.

Put another way, the assumption that the Riemann Hypothesis is true, is already automatically implied by conventional mathematical axioms (through blindly reducing distinctive qualitative interpretation in an absolute quantitative manner).

We cannot thereby prove the Riemann Hypothesis within this limited mathematical framework (where what we are trying to prove is already implied by our axioms).  

Likewise, we cannot strictly disprove the Riemann Hypothesis from within these same axioms.

One might immediately point to a non-trivial zero that perhaps does not lie on the critical imaginary line as offering conclusive proof that the Riemann Hypothesis is false.

The discovery of such a zero would indeed be a momentous find, for it would undermine the consistency of our basic assumption regarding the number line.

In other words, we would no longer be entitled to assume that all the composite natural numbers (derived through multiplication of the primes) lie on the same number line as the primes.

So this in turn would strictly undermine the consistency of all mathematical proof with respect to the number system.

Therefore we would no longer be conclusively able to prove or disprove any mathematical proposition involving number from within such a flawed system.

Thus it is very important indeed that no non-trivial zero is ever found off the imaginary line, as this would rightly undermine our faith in the consistency of the whole mathematical enterprise!

Thus the Riemann Hypothesis, in the assumption that all the non-trivial zeros lie on the imaginary line (through .5) must remain a matter of faith, just as strictly, our corresponding belief that all natural numbers lie on the same real line, must likewise remain a similar matter of faith (with both implying each other).


So from one perspective, in dynamic interactive terms, we can accept that all the natural numbers lie on the real line if the Riemann Hypothesis is true; equally from the complementary perspective, if all the natural numbers indeed lie on the real line, then the Riemann Hypothesis is true. However, once again neither proposition can be proved (nor disproved) within the conventional mathematical framework of absolute quantitative identity.

I have been discussing above the holistic significance of the non-trivial zeros.

Equally they possess an important analytic significance in quantitative terms.

And it is this aspect of the zeros that is given exclusive attention within the accepted mathematical framework. 

Thus accepting the truth of the Riemann Hypothesis, it is possible to provide a formula (incorporating crucial adjustments based on the non-trivial zeros), enabling an exact quantitative estimate of the number of primes (to any given real value of n).

However, from the complementary opposite perspective, it is equally possible to provide a formula based on knowledge of the individual primes, to enable an exact quantitative estimate of the number of non-trivial zeros (to any given imaginary value of n).

So it should be clear - even from this quantitative perspective - that the primes and non-trivial zeros are mutually interdependent with each other, thus implying an important holistic - rather than analytic - connection.

However, from my reading of the conventional literature, I can see little recognition of this crucial point.


Before leaving this section, I wish to elaborate further on the true holistic nature of the (non-trivial) zeta zeros.

In this regard it may be helpful to remind ourselves once more of the earlier crossroads example.

We saw there that when the crossroads is approached from one direction (travelling N or S), then left and right turns can be given a separate independent meaning.

However, when one recognises an approach from both N and S directions simultaneously, then left and right turns are rendered paradoxical, with what is left from one perspective, right from the other, and vice versa. 

It is somewhat similar with respect to the overall relationship as between the primes and natural numbers.

When one approaches the primes from the quantitative perspective of independent “building blocks”, then the natural numbers appear to be unambiguously derived from the primes.

However when one now approaches the primes from the qualitative perspective of the unique combinations of prime factors, then the position of each prime appears to be determined through their overall collective relationship to the natural number system.   

Then when one simultaneously attempts to view the number system from both perspectives, clearly their relationship to each other appears paradoxical; so from the quantitative perspective of prime numbers, the natural numbers appear to depend on the primes; however from the - relatively - qualitative perspective of prime factors, the position of the primes (and thereby their quantitative identity) appears to depend on the natural numbers.

Now in holistic terms, the (non-trivial) zeros indirectly express in quantitative terms, this paradoxical identity of both the primes and natural numbers (as mutually interdependent with each other).

And as we have seen from the holistic perspective, in this context, the imaginary notion represents an indirect expression of the interdependence of the cardinal numbers. So each zero thereby represent a point on the imaginary line, where in a very true sense the identity of the primes is inseparable from the corresponding identity of the natural numbers.


So again from one perspective, we recognise that the primes have a unique individual nature. Then from the other, we can equally recognise that the collective distribution of primes among the natural numbers is predictable to progressively higher levels of relative accuracy

And the non-trivial zeros can be seen in this context as points that harmonise both the individual and collective identity of the primes (with respect to the natural numbers). This is borne out by the fact that the general formula for the frequency of non-trivial zeros (up to any number on the imaginary line) predicts results with an amazing degree of accuracy, not only in relative but also absolute terms.

So we could say that with each zero, the individual uniqueness of each prime is harmonised with the collective relationship of all the primes to the natural numbers. However because each zero still maintains a certain discrete identity, this can only be dynamically achieved in an extremely close approximate, rather than exact manner.

Equally we could say that at each point, as already stated, that the operation of addition is inseparable from that of multiplication. So it is through the zeta zeros that addition and multiplication are mutually reconciled in a fully consistent manner.

Perhaps most simply we could say that at each point the qualitative nature of number is inseparable from its corresponding quantitative identity. So here the analytic and holistic aspects of number mutually coincide in direct fashion.


So once more, we are at the opposite extreme from conventional mathematical interpretation, where the quantitative identity of each number is abstracted in absolute fashion from the qualitative (with the qualitative thereby reduced to the quantitative).

And this again is the key reason why one cannot hope to uncover the true meaning of the Riemann zeta function (and Riemann Hypothesis), while remaining rigidly within a mere analytic framework (based on absolute quantitative identity).

Looked at from yet another important perspective, the key role of the Zeta 1 i.e. (non-trivial) zeros is to serve as a seamless means of two-way conversion as between the Type 1 and Type 2 aspects of the multiplicative number system (as the unique combination of prime factors, that are quantitative and qualitative with respect to each other).

Likewise the key role of the Zeta 2 zeros - that we encountered earlier - is to serve as a seamless means of two-way conversion as between the Type 1 and Type 2 aspects of the additive number system (again where each number is given both a quantitative and qualitative interpretation)


In fact properly understood, the number system dynamically entails the complementary interaction of both the multiplicative and additive approaches.

So from the multiplicative perspective, when we multiply primes (as unique factor combinations) each prime is already defined internally from the additive perspective.

Thus as in our oft quoted example we multiply 2 by 3, both numbers are already defined in additive terms (with respect to individual units).

Likewise when we look internally at the individual members of a composite natural number from the additive perspective, this number already involves (through its unique prime factors) the multiplicative approach.

Therefore, to simply illustrate, though 6, as a composite number, can be clearly represented as the sum of its individual units, it already necessarily entails, through multiplication, a unique combination of prime factors (i.e. 2 and 3).

Thus again, both the internal and external distribution of the primes and natural numbers  (from both additive and multiplicative perspectives) are dynamically interdependent with each other, with both distributions ultimately determined in a holistic synchronous manner approaching pure ineffability.


Finally from this dynamic perspective, we can establish the truly fundamental relationship of the primes to the natural numbers.

Quite simply, the transmission mechanism of both quantitative and qualitative characteristics for the number system (and ultimately all created phenomena), is through the two-way complementary interaction of the primes with the natural numbers (and the natural numbers with the primes).

From a psychological perspective, this entails that the very manner in which both the conscious and unconscious aspects of personality interact is through this same fundamental relationship of primes and natural numbers (as quantitative as to qualitative and qualitative as to quantitative with respect to each other).

Some years ago I was struck by the deep similarity of the psychological meaning of the word “primitive” to my emerging dynamic appreciation of the nature of prime numbers in mathematical terms.

The essence of primitive instinctive behaviour is that a direct confusion exists as between the holistic nature of the unconscious and the specific form of conscious understanding.

So for example, in earliest infant behaviour, because of the degree of such confusion, the differentiation of distinct phenomena in experience is not yet possible. Thus, the infant is not yet able to provide an appropriate dimensional framework in space and time (relating to the holistic unconscious), with which to experience specific phenomena (in an analytic conscious manner).

Thus remarkably, solving the primitive problem, as it were, in psychological terms - in initially separating both conscious and unconscious aspects of personality with a view to their eventual integration - exactly mirrors the corresponding mathematical issue of solving the problem of the primes.

So here, one must successfully separate both the analytic (quantitative) and holistic (qualitative) aspects of mathematical understanding with a view to their eventual integration, where both aspects can thereby be seen to operate in a seamless interdependent fashion


So at some future stage in our human evolution, it will perhaps be realised that truly solving the problem of the primes mathematically (where both analytic and holistic aspects of understanding are property integrated), is inseparable from solving the corresponding “primitive” problem in psychological terms (where both conscious and unconscious aspects of personality can finally operate together in a seamless harmonious fashion)


And just as the Riemann zeros are now being linked to certain physical energy states (with respect to atomic behaviour), the Riemann zeros will likewise then be intimately linked to certain psychological energy states (representing the most advanced forms of spiritual contemplation). 


Thus once again, the enormous problem with respect to conventional mathematical interpretation is that it remains completely in denial regarding its holistic (unconscious) aspect.

A well-known physical analogy is very relevant in this context


In quantum physics, the uncertainty principle implies that we can never exactly determine both the position and momentum of a particle. So for example, if we attempt to improve our exact knowledge of a particle’s position, knowledge of its corresponding momentum becomes increasingly fuzzy.

Now an important uncertainty principle applies to all mathematical understanding regarding both analytic (quantitative) and holistic (qualitative) appreciation of relationships.

Thus, if we continually attempt, as is presently the case, to attain ever more precise analytic appreciation of relationships - through attempting to formulate them in an increasingly abstract manner - this then acts to completely block out recognition of the corresponding holistic aspect.   

And this, much more than even such a fundamental problem as the Riemann Hypothesis, is truly the key issue which requires to be now addressed by the mathematical community.

For despite the - admitted - stunning advances that have been made with respect to its present highly specialised approach, it represents but a limited though important unique case of an altogether more comprehensive vision of Mathematics.


Growing recognition of this much greater role of mathematics will inevitably have enormous implications for all the sciences and ultimately for society in general, representing perhaps the greatest revolution yet in our intellectual history.

We are now at an unprecedented stage of development where dramatic changes in technological, political, social, economic and environmental terms are set to take place.

And for successful adaptation, a major transformation in our basic mental mind-set is required that goes to the very core of mathematical understanding.