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.

Number as Dimension

We return here once more to the clarification of the notion of dimension as used in Mathematics. This is vital in turn for the clarification of what is meant by the quantitative and qualitative aspects of number.

Unfortunately because Conventional Mathematics is based inherently on a reductionist fallacy i.e. that number can be understood with respect merely to its quantitative aspect, it is perhaps not surprising that this key issue is effectively avoided.

 

If we define numbers as independent in an absolute quantitative sense, then this begs the question as to how numbers can be successfully related with each other (which requires the qualitative notion of interdependence).

The very fact that this is not readily appreciated as of the most fundamental importance only goes to show how ingrained this reductionist interpretation of number has become. In other words, we assume that this interdependent aspect of number behaviour (whereby numbers assume a qualitative relationship with each other) can be successfully understood in a merely quantitative manner!

 

The true inherent meaning of dimension is of a qualitative nature and relates to the manner in which the fundamental polarities of experience i.e. external/internal and whole/part are related.

1-dimensional interpretation in this context simply entails the attempt to understand such relationships in a uni-polar manner (i.e. using just one pole as an exclusive frame of reference).

So for example all mathematical experience necessarily entails the dynamic interaction of objective subjective (cognitive) aspects that are relatively external and internal with respect to each other.

1-dimensional thereby implies then that we fix interpretation with just one pole in an absolute manner.

So this leads to the standard view of numbers as absolute entities existing in an objective manner. Though one may recognise that strictly such numbers cannot have experiential meaning without the existence of mental constructs, somehow a belief persists that an absolute correspondence applies to both the external objects and the internal constructs.

In other words Conventional Mathematics essentially operates on the illusion that mathematical objects have an absolute existence (independent of our relationship with them).

1-dimensional interpretation equally leads to the standard view of number as existing in an independent quantitative manner. Though once again it may be recognised that a general dimensional context is required to enable an ordered relationship of these numbers, somehow the belief remains that this can be done in a merely quantitative manner.

In truth, the general context providing this capacity for ordered number relationships is of a qualitatively distinct nature. However a remarkable denial of this key fact pervades conventional mathematical interpretation.

So once again its 1-dimensional nature is demonstrated by the manner in which the qualitative dimensional aspect is reduced in a merely quantitative manner.

Now of course Conventional Mathematics can indeed give meaning to dimensions (≠ 1) in a quantitative manner.

So for example from this perspective 2³ = 8 (i.e. 8¹). Thus, though the qualitative context has here changed through use of 3 (as dimensional number) the numerical result is given in a reduced quantitative manner (in terms of 1 as dimensional number).

It is extraordinarily important therefore to grasp that Conventional Mathematics is defined in qualitative terms by its merely 1-dimensional nature.

This effectively means that variables are treated in an absolute - rather than relative - manner (where relative implies  dynamic interaction as between the opposite polarities that condition all phenomenal  experience).

When one grasps this point, one can then clearly recognise not alone why the Riemann Hypothesis can have no proof, but even more importantly why its true nature cannot be successfully understood in conventional mathematical terms!

 

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

In the more comprehensive understanding of this Function this also implies that only dimensional value (in qualitative terms) for which the Riemann Zeta Function remains undefined is also for s = 1.

This means that the Riemann Zeta Function (and associated Riemann Hypothesis) cannot be properly understood in the conventional mathematical manner.

Thus the Riemann Hypothesis essentially relates to:
(i) the condition with respect to number where both external (as objective results) and internal (as mental interpretation) are successively reconciled as ultimately identical and
(ii) the condition where both the quantitative (cardinal) and qualitative (ordinal) nature of number are likewise reconciled (as ultimately identical).

Therefore we cannot attempt to understand this relationship - which entails the dynamic interaction as between opposite polarities - in a reduced absolute sense (where objective results are divorced from cognitive interpretation and the quantitative aspect of number likewise divorced from its qualitative aspect). This is why the Riemann Zeta function cannot be successfully understood in a conventional (i.e. 1-dimensional) manner.

I can say this with considerable confidence having already been dimly aware of the problem from about the age of 10.

Even then I was seriously questioning conventional procedures. This started the long journey to get to the bottom of the problem (as I saw it) in the hope of offering a more authentic mathematical approach.

And after more than 50 years on this journey I believe that I have managed to come up with at least the general framework for a more comprehensive appreciation of Mathematics.

I have recounted before how I found as a child the conventional explanation of a square root deeply unsatisfactory.

For me there it seemed that an essential symmetry should be preserved as between the notion of a square on the one hand and a square root on the other.

So for example  we start with 1 and square we get - apparently - one unambiguous answer i.e. 1². However when we then get the square root we now have two possible answers + 1 and – 1.

The conventional explanation seemed to me even at this young age deeply illogical.

We would not accept in terms of the proof of a theorem for example that it could equally have in qualitative terms a negative as well as positive truth value. This would be like saying that we could accept the proof of the Pythagorean Theorem for example as either true or false. However in the parallel quantitative context (in the context of a square root) it was indeed maintained that a number could have either positive or negative values! 

So I started to suspect - though I would not have been able to articulate my thoughts then in a coherent manner - that the qualitative nature of 2 (as a dimensional number) was quite distinct from 1.

And as Conventional Mathematics is defined qualitatively in terms of its merely 1-dimensional nature, this opened up the possibility of entirely distinctive logical approaches to Conventional Mathematics.  In other words the very inconsistency that in could see in the standard explanation of the two roots of 1 was due to the fact that 1-dimensional (either/or) logic was not adequate to explain this - apparently simple - problem.

Many years later (after long immersion in Hegelian philosophy and the wisdom of the great spiritual traditions) I was able to return to this problem with what I considered was a satisfactory answer.

Whereas 1-dimensional logic is characterised in an absolute (linear) either/or manner, 2-dimensional logic is characterised by a relative (circular) both/and approach. This then leads to paradox in terms of the 1-dimensional approach.

The Greek philosopher Heraclitus summed this 2-dimensional logic up well in his statement,

“The way up is the way down; the way down is the way up”

What is involved here has profound consequences for all mathematical interpretation.

 

If one fixes direction in terms of just one pole either “up” or “down” then movement along a road is unambiguous, whereby it can be consistently defined in terms of the given frame of reference.

If one now alters the frame of direction (in the opposite manner) then again unambiguous directions can be given in terms of this new reference frame.

 

So fixing polar reference frames (with just one pole as independent) i.e. as 1-dimensional, leads to unambiguous answers of an absolute nature.

 

However when we now simultaneously try to relate both reference frames as interdependent i.e. as 2-dimensional, this leads to paradoxical answers (in terms of 1-dimensional logic).  So what is “up” or “down” in this sense is purely dependent on context.

I have come to realise over the years that - quite remarkably - Conventional Mathematics, because of its 1-dimensional nature, is totally lacking any genuine notion of interdependence (and thus always reduces this notion, in any relevant context, to independence).

Alternatively we could say that Conventional Mathematics is totally lacking any genuine qualitative or holistic notion (thereby reducing it in a merely quantitative analytic fashion).  

So getting back to our example on directions, the directions “up” and “down” in 2-dimensional terms can be represented as + 1 and – 1  in relation to each other. However these are now understood in a merely relative fashion with positive and negative depending on context.

The deeper implication is that where the dynamic interdependence of two polarities is concerned 2-dimensional - rather than 1-dimensional - interpretation is required.

This intimately applies therefore to the interpretation of mathematical symbols which are inevitably conditioned by such dynamic interaction in experiential terms..So internal and external and quantitative (part) and qualitative (whole) polarities continually interact in experience and are related to each other in a dynamic complementary manner.

Thus coming back to the square of 1 and the corresponding square root of 1, one can perhaps appreciate now that this properly requires 2-dimensional - rather than 1-dimensional - interpretation.

 

Thus when we square 1 i.e. 1², we move - literally to 2 dimensions (which qualitatively are defined as both + 1 and – 1 in relation to each other (depending on context).

Now when we get the square root we are attempting to express these two polarities in a reduced absolute fashion. So what is both + 1 and – 1 (in 2-dimensional terms) becomes either + 1 or – 1 (in a 1-dimensional format) .

Strictly, whereas the cardinal (quantitative) notion of 2 represents - literally - a whole unit (without qualitative distinction), the corresponding qualitative notion of 2 entails its two ordinal members as individual units i.e. 1^st and 2^nd (without quantitative distinction).

Thus the two roots of 1 are obtained with respect to 1¹ and 1² respectively i.e. 1^1/2 and 1, = – 1 and + 1.   

Therefore, 1-dimensional interpretation is characterised by the use of single independent frames of reference (with respect to polar opposite interaction) in an isolated manner.

It thereby entails linear (either/or) logic in rational terms.

2-dimensional interpretation entails both 1^st and 2^nd dimensions. So initially it necessarily entails the 1st dimension in making unambiguous distinctions based on single independent reference frames. However there is now a clear recognition that these can now be made from two opposite directions!

Then the 2^nd dimension entails the simultaneous integration of both reference frames (as complementary opposites) where both are seen as interdependent. In a direct sense this implies holistic recognition of an intuitive kind (pertaining to the unconscious). However it is then indirectly translated in a circular logical (both/and) manner that appears paradoxical in terms of linear reason.

 

2-dimensional interpretation represents the minimum necessary to understand the number system in its inherent dynamic interactive nature, allowing for both analytic (quantitative) and holistic (qualitative) appreciation of mathematical variables or even more simply both the (relative) independence and interdependence of mathematical variables.

In an important sense, as I have explained in previous blog entries, all other natural number dimensions can ultimately be expressed in a 2-dimensional fashion.

From my early 20’s I spent several decades developing - what I referred to as - Holistic Mathematics in recognition of its completely neglected qualitative aspect.

Initially this quest was largely driven by the realisation that a qualitative mathematical interpretation could potentially be given for all stages of human and physical transformation (including rare contemplative states). With a highly developed contemplative state, the dynamic interaction as between the key polarities (underlying all phenomenal recognition) becomes increasingly more refined corresponding to ever higher number dimensional configurations.  In my own work I especially concentrated on the nature of interpretation corresponding to 2-dimensional, 3-dimensional, 4-dimensional and 8-dimensional appreciation respectively!

However it is only in the last decade that I have seriously sought to explore the implications of all this for appreciation of key mathematical problems such as the nature of the number system and the Riemann Hypothesis.

I am now of the firm opinion that despite a veneer of great rigour with its ultra-specialised understanding of so many topics, at a fundamental level, standard interpretation represents a greatly confused mess of highly reduced notions. (These unfortunately have become so reduced through the long unchallenged consensus regarding their use, that an almost total blindness regarding their shortcomings now exists).

I used to be of the opinion - while developing the importance of holistic mathematical notions - that standard interpretation would remain largely valid with respect to quantitative appreciation.

 

However I have come to clearly realise that all mathematical notions - including of course number - are properly of a dynamic relative nature. As quantitative and qualitative aspects are ultimately interdependent, it is therefore not possible to understand number properly in a merely reduced quantitative manner.

For example the correct appreciation of 1 and 2 in a qualitative ordinal manner (as 1^st and 2^nd respectively) requires 2-dimensional interpretation and therefore cannot be properly explained in conventional terms.

 

When one realises how quantitative and qualitative aspects are inevitably intertwined with respect to the appreciation of number, then the key issue  arising relates to the ultimate consistency of both aspects.  This indeed is the central message of the Riemann Hypothesis, which therefore can have no strict meaning in conventional (i.e. merely quantitative) terms.

Nothing less than a total revolution is now required in our mathematical understanding. This of course likewise entails a total revolution in what is meant by science.

I hope readers to this blog can get some sense of the importance of what is involved. Successful transformation with respect to our present civilisation urgently depends on the rapid realisation of its many implications.     

The True Significance of the Zeta Zeros (Zeta 1 and Zeta 2)

What I am attempting to convey in this blog entry is the true philosophical significance of the Zeta zeros.

Of course this requires recognition of the fact - which I have continually sought to convey over the past year or so - that there are two complementary aspects to the number system (Type 1 and Type 2) which dynamically interact; that corresponding to these two aspects we likewise have two complementary Zeta Functions (Zeta 1 and Zeta 2) which likewise dynamically interact; and finally that corresponding to these two Zeta functions we have two sets of (non-trivial) Zeta zeros (also in dynamic interaction with each other). 

 

In both the Type 1 and Type 2 approaches, quantitative (cardinal) and qualitative (ordinal) aspects of the number system are clearly separated.

So with the Type 1, number is initially defined with respect to the former aspect with each number representing a base quantity that can vary, which is defined with respect to the fixed dimensional number 1 (serving as the default qualitative value). So, for example in the number expression 2¹, 2 represents the base quantity and 1 the default dimensional number (which - relatively - is of a qualitative nature).

Again the customary quantitative bias of Conventional Mathematics is indicated by the fact that  this default dimensional value is ignored altogether!

So for example the natural numbers are thereby merely defined in terms of their quantitative base values i.e. 1, 2, 3, 4,… and not as 1¹, 2¹, 3¹, 4¹,…...

The Type 2 number approach is initially defined in inverse fashion with respect to the latter aspect, with each number representing a dimensional value that can vary, which is defined with respect to a fixed base number 1 (serving as the default quantitative value). So, this time in the inverted number expression 1², 2 represents the dimensional value and 1 the default base number (which - relatively - is of a quantitative nature).

So in this latter number approach, the natural numbers 1, 2, 3, 4,… (this time representing dimensional values) are defined as 1¹, 1², 1³, 1⁴,….. 

However when we allow base and dimensional values to vary in the Type 3 approach, both quantitative (cardinal) and qualitative (ordinal) aspects continually interchange with each other.

So initially we start by considering each quantitative base value as relatively independent (in cardinal terms) and each qualitative dimensional value as relatively interdependent (in ordinal terms). But relative independence likewise implies relative interdependence and relative interdependence, relative independence respectively.

This therefore implies that both base and dimensional values alternate as between quantitative and qualitative interpretations respectively.

Now this all seems remarkably similar to Quantum Mechanics with respect to the particle and wave features of matter. And indeed properly understood, the quantum mechanical features of matter spring directly from this prior dynamic nature of number (from which they derive).  

Now when we look at the Zeta 1 and Zeta 2 Functions, we can see that they both involve the natural numbers in an inverse manner.

In the Zeta 1 the natural numbers are defined as the base quantities (defined with respect to a negative dimensional value s, which can vary).

ζ₁(s) = 1 ^{– s}  + 2 ^{– s} + 3 ^{– s} + 4 ^{– s}  +…

By contrast in the Zeta 2 , the natural numbers are defined as the dimensional qualities (defined with respect to base quantities s, which can vary)

ζ₂(s) = 1 + s¹  + s² + s³ + ... + s^{t – 1}

Now initially the Zeta 2 Function is defined in a finite manner.

However by combining in regular groups of t terms, it can be extended in an infinite manner

i.e. 1 + s¹  + s² + s³ + s⁴ +…

The (non-trivial zeros) for both of these Functions relates to solutions of s where

ζ₁(s) and ζ₂(s) respectively = 0.

Now the key philosophical significance of these zeta zeros is that they provide (in each case) values for s, where both quantitative and qualitative interpretations of s are reconciled i.e. where the cardinal and ordinal aspects of the number system are simultaneously related. In other words because in dynamic terms, continual interaction now takes place as between both quantitative and qualitative aspects, a meaningful solution to the equations requires that a dynamic identity be achieved as between both aspects.

The importance of the prime numbers in this context is that they serve as the means through which the quantitative (cardinal) and qualitative (ordinal) aspects of the number system are mediated in two-way fashion with respect to the natural numbers.

 

So from the Type 1 perspective, the prime numbers serve as the cardinal building blocks of the natural number system (in quantitative terms).

 

Then in inverse fashion from the Type 2 perspective, the natural numbers serve as the ordinal building blocks of each prime number (in qualitative terms).

So again, this key relationship as between this two-way relationship as between the primes and natural numbers (and natural numbers and primes) serves as the means by which both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system are reconciled (in a bi-directional fashion).

And the importance of  (non-trivial) zeta zeros, in both Type 1 and Type 2 terms, resides in the fact that they provide the solution to this key issue of the ultimate identification of both quantitative and qualitative aspects (with respect to the number system) from both perspectives. 

It is easier to demonstrate the inherent dynamic holistic nature of the zeta zeros initially from the Zeta 2 perspective.

What does it precisely mean to reconcile (or identify) quantitative and qualitative aspects of number interpretation?

Well, let’s consider the simplest possible Zeta 2 solution which arises in the context of the two roots of 1!

As it will always be one of the t roots of 1, we can deem this as the trivial root.

The other non-trivial root then arises in the context of the finite Zeta 2 expression

1 + s = 0, i.e. s = – 1.

Now, as explained before if we consistently combine terms in groups of 2, this also serves as the solution of the infinite Zeta 2 expression,

1 + s¹  + s² + s³ + s⁴  +….  = 0

The deeper significance of the two roots of 1 is that they serve as the means of expressing in an indirect quantitative manner the true ordinal (i.e. qualitative) significance of 1 and 2.

In other words the very recognition of the 2^nd member (of a group of 2) implies that we can (temporarily) negate exclusive identification with the 1^st member. So this dynamic negation of the 1^st member now enables us to posit recognition of a (new) 2^nd member.

Therefore we express this ordinal relationship (as between 2 members) in an indirect circular quantitative manner as + 1 and – 1. So, a continual process of conscious positing (+ 1) and unconscious negation (– 1) is involved in the dynamic interchange as between 1^st and 2^nd units (in this context of 2).

Now by extension we can provide an indirect quantitative means of translating the ordinal relationships for any finite sized group t, through the corresponding roots t roots of 1.

The deeper implications here imply that the ordinal appreciation of number properly relates to unconscious - rather than conscious - recognition. Therefore ordinal notions can only be dealt with in a grossly reduced manner within the current mathematical paradigm (as it is formally based on merely conscious rational notions).

So indirectly, 1^st and 2^nd (in the context of a group of 2) can be given quantitative expression as + 1 and – 1 respectively. The holistic qualitative appreciation of the interdependence of these numbers is then obtained through combining both (through addition). So (+) 1  – 1 = 0.

So each member (in isolation) enjoys a partial quantitative existence (as independent); yet when combined, a merely qualitative exists (as interdependent). This is why the quantitative result = 0. So when opposite polarities are successfully combined in a fully interdependent manner, no quantitative independence remains.

One interesting physical example of this phenomenon relates to the process whereby matter and anti-matter particles combine fusing in pure energy. So what has a distinct material existence (as independent particles) is transformed to pure energy (through the interdependence of both).

This also implies of course that number in its pure interdependent state represents a qualitative energy state.

So the zeta zeros therefore provide a means of reconciling both the partial independent quantitative existence of number (as form) with their holistic interdependent qualitative existence (as a pure energy state).

In a sense all the Zeta 2 zeros can ultimately be expressed in terms of (+) 1  – 1 (as in all cases the sum of non-trivial t – 1 roots of 1 = – 1, with the trivial root  = + 1).

 

Thus we can give expression to the individual natural ordinal number members of any prime number group p (i.e. 1^st, 2^nd, 3^rd,…..pth members) in an indirect quantitative circular manner (i.e. as equidistant points on the unit circle in the complex plane).

The holistic qualitative interdependence of these members is then obtained through combining the individual members (through addition).

So the sum of roots = 0 demonstrating the qualitative interdependence of the combined p members.

Now the uniqueness of the prime numbers in this context is that each of its non-trivial members (i.e. all roots except 1) are uniquely defined.

When the number group is not prime, this will not be the case. For example – 1 is one of the non-trivial roots of the 4 roots of 1. However it is not unique as – 1 is also one of the two roots of 1.

Therefore through the Type 2 aspect of the number system (with its associated Zeta 2 Function) we obtain precisely the opposite interpretation of a prime number from the Type 1 aspect (associated with the Zeta 1 Function).

In the Type 1 (cardinal) approach (and its related Zeta 1 function), the prime numbers are seen as the unique independent buildings blocks of the natural number system.

However in the Type 2 (ordinal) approach (and its related Zeta 2 Function), the natural numbers are seen as the unique independent building blocks of each prime number. Thus from this context each prime number enjoys a distinctive holistic identity through the collective qualitative interdependence of its individual (natural number) members.

By contrast in the Type 1 (cardinal) approach (with its related Zeta 1 Function), the prime numbers are seen as the unique independent building blocks of each natural number.

However as we have multiplication of two numbers (> 1) entails a qualitative as well as quantitative transformation.

So when for example in conventional mathematical terms we say that 6 is uniquely expressed through its prime factors 2 and 3, we are referring to this relationship in a merely (reduced) quantitative manner.

The important fact is however that a dynamic type transformation is equally involved, with once again a reconciliation or identity existing as between individual isolated numbers (with an individual independent identity) and an overall holistic identity (as interdependent). Thus from this opposite perspective each natural number enjoys a distinctive holistic identity through the collective quantitative interdependence of its individual (prime) members, 

And this is what the famed zeta zeros (i.e. Zeta 1 non-trivial zeros) obtain i.e. an identity as between independence with respect to individual numbers and an interdependence with respect to these numbers taken as a collective whole.

The Zeta 1 zeros are however of a more indirect nature than the Zeta 2 (and therefore more difficult to express).

Once again  the Zeta 2 zeros provide an indirect quantitative means of translating the qualitative (ordinal) nature of each member of a group (with the overall additive relationship expressing their combined qualitative interdependence).

 

However, individual non-trivial Zeta 1 zero provides a - seemingly - independent quantitative numerical measurement (with a fixed real part = ½ and an imaginary part that varies).

Thus the imaginary scale in this context provides an indirect way of representing, in a linear analytic manner, what is inherently of a dynamic holistic nature (where quantitative and qualitative aspects are reconciled as identical).

This indeed is why these zeros bear such a close relationship with quantum chaotic energy states. So in a sense the trivial zeros represent point singularities (as pure energy states) on the imaginary number line.

So the distinct quantitative nature of these zeros then results from their combined collective nature.

Therefore the local independent quantitative nature of each individual prime number (as distinct from the natural numbers) can be obtained through wave deviations associated with the addition of the combined group of trivial zeros to a continuous general function, expressing the general frequency of primes among the natural numbers.

 

Just as the refined use of the unconscious in psychological terms can be used to correct rigid identification of phenomena relating to conscious experience, likewise in reverse fashion, in a physical mathematical sense, the refined use of the Zeta 1 (non-trivial) zeros can be used to obtain precisely the local rigid identity of prime numbers as opposed to their general (interdependent) nature with the natural numbers.

This is why – in a very precise sense – these Zeta 1 zeros represent the perfect shadow system to the primes.

So once again because the conventional mathematical paradigm is based on a merely rational conscious interpretation of its symbols, the primes are exclusively viewed with respect to their (quantitative) existence in an independent analytical manner.

However the Zeta 1 zeros properly represent the perfect shadow complement to this view of the primes. Though these can indeed be also given a quantitative existence in an analytical manner, they are of an imaginary (rather than real) nature, serving is the indirect expression of their inherent dynamic holistic nature (where separate independent elements are perfectly reconciled in an interdependent group manner).

So the very definition of “analytic” as I use the term, is that the quantitative and qualitative meaning of mathematical symbols can be clearly separated from each other. In the conventional mathematical terms this is used in an absolute sense i.e. where interpretation is formally of a merely rational nature (which thereby totally excludes all genuine holistic meaning of symbols).

 

The corresponding meaning of “holistic” is the other extreme whereby the quantitative and qualitative aspects of mathematical symbols are fully interdependent with each other.

This also has an absolute interpretation (corresponding to pure intuition) which is of an ineffable nature.

The proper activity of Mathematics allows for both analytic and holistic interpretation in a dynamic relative manner (with respect to both quantitative and qualitative aspects).

The true dynamic nature of both Zeta 1 and Zeta 2 zeros I derives from the fact that that they provide - from two complementary directions - the mysterious identification of both the quantitative and qualitative aspect of mathematical symbols i.e. where analytic and holistic meaning are reconciled.

However appreciation of their true nature will require the most radical revolution yet in scientific history whereby both (conscious) reason and (unconscious) intuition need to be explicitly incorporated with each other in all mathematical interpretation. (And as we have seen the indirect incorporation of intuition in rational terms requires circular paradoxical - as opposed to strict linear - understanding).