Where Science and Art Coincide (16)

The way to properly understand the Zeta 1 and Zeta 2 zeros is to see them - as they inherently are - as complementary pairings.

Now complementary in this context means that when we establish the relationship for one, the corresponding relationship for the other will be the inverse of the former.

And ultimately this points to the total interdependence in dynamic interactive terms of both sets of zeros.

The Zeta 2 are in fact somewhat easier to grasp and that is why I tend to start with them.

Once again higher dimensional interpretation implies the increasing interdependence of the (fundamental) polar opposites in experience. The notion of a dimension then relates to a specific dynamic configuration of these poles. So in the simplest case, 2-dimensional entails that we dynamically configure experience in terms of two real poles that are complementary opposites of each other.

In spiritual contemplative traditions, the higher dimensions are associated with an increasing refinement in the nature of intuitive awareness. And it is such intuitive awareness that provides the direct basis for the holistic (qualitative) appreciation of mathematical symbols.

However in these traditions, perhaps too much attention is placed on such dimensions as representing intuitive states (arising form the increasing nondual appreciation of interdependence).

There is also an important related aspect involved in terms of highly refined circular rational structures of a paradoxical nature, that likewise unfold (and are indeed necessary to properly consolidate the intuitive states).

In fact this structural aspect arises from the attempt to relate these "higher" dimensions  to customary experience of reality (i.e. at a 1-dimensional level).

The remarkable significance of all this is that this latter form of understanding provides the holistic mathematical manner of successfully converting ordinal appreciation of number (which is of a relative qualitative nature) indirectly in a quantitative (circular) manner.

As I have repeatedly pointed out the ordinal notion of number continually changes depending on context.

So 2nd (in the context of 2) for example is distinct from 2nd (in the context of 3).

Now the Zeta 2 zeros of the finite equation,

 ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime) = 0,

provide the solution to this dilemma enabling us to give an indirect quantitative expression of ordinal meaning (in any context).

For example to mathematically interpret 2nd (in the context of 2) we obtain the 2nd non-trivial root of 1 i.e. 1 + s¹  =  0, so that s   =  – 1.

Then to interpret 2nd (in the context of 3) we obtain the 2nd (of the 3 roots of 1) which represents the first solution of 1 + s¹  + s²   = 0 = – 1/2 + .866i.

Now all the non-trivial solutions (i.e. excluding 1) are unique where prime roots are concerned.

And once again this is the very significance of a prime in this Type 2 mathematical context, where each is uniquely defined (except 1) by its natural number members in ordinal terms.


Returning to the simplest case of 2nd (in the case of 2), the direct holistic (qualitative) interpretation of –  1, must then be conducted in a dynamic interactive context.

So what this entails is that the notion of 2nd (in the context of 2) entails the (temporary) negation of 1 (which equally implies 1st). Thus the very process of recognition of 1st and 2nd (in the context of 2) implicitly implies a continual positing and negating with respect to the two number members involved.

The deeper implication of this is that the very understanding of 1st and 2nd - which in conventional terms is mistakenly believed to be directly implied by cardinal appreciation of 1 and 2 - in fact implicitly requires an unconscious holistic aspect (of a qualitative nature) to be meaningful.

And the key significance of the Zeta 2 zeros is that they provide the means of making explicit this (unrecognised) qualitative dimension of mathematical experience.


However the implications of this could not be greater, for it truly establishes that customary understanding (in a merely quantitative manner) is quite untenable in terms of the inherent dynamics of experience.

So again put simply, again the importance of the Zeta 2 zeros is that they represent the (unrecognised) qualitative holistic aspect of the natural number system in ordinal terms.


Therefore, properly understood, the ordinal nature of number is not merely quantitative in the standard reduced manner of conventional analytic interpretation, but rather represents the dynamic interaction of two aspects that are quantitative (analytic) and qualitative (holistic) in relation to each other.


It is then easy - though not so easy to intuitively appreciate - that the Zeta 1 (Riemann) zeros represent the corresponding qualitative (holistic) aspect of the natural number system in cardinal terms.

Now, as always, the direction of interpretation changes here in an inverse manner. 

In the case of the Zeta 2 we had the problem of attempting to indirectly express the holistic appreciation of the qualitative nature of "higher" dimensions indirectly in a (1-dimensional) quantitative manner.

And we have seen that this is achieved through obtaining the corresponding roots of 1, which then provides an indirect quantitative means of individually defining the ordinal nature of number in any context.

Then again the qualitative interdependent nature of these roots is demonstrated through their sum = 0.
In other words when we can truly appreciate the relative collective nature of the ordinal members of any group, we attain an appreciation of pure interdependence  (which is nothing in quantitative terms).


In the case of the Zeta 1 zeros, we have the opposite problem of attempting to indirectly express the holistic appreciation of quantitative (base) numbers indirectly in a "higher" dimensional qualitative manner.

The basic idea here is not too difficult to express.

All composite natural numbers are uniquely expressed as the product of prime numbers.

So for example 6 is uniquely expressed as 2 * 3.

However just as a rectangular field with length 2 and width of 3 units is measured in 2-dimensional (square) units, likewise the product of these two primes leads to a qualitative (dimensional) - as well as quantitative - change in the resulting expression.

Now as always Conventional Mathematics reduces this transformation in a merely quantitative manner.
However we can begin to appreciate the true significance of the Zeta 1 zeros when we understand them as the indirect means of representing this qualitative change with respect to the cardinal number factors involved.

Now, as I have consistently stated in these blogs, dimensional numbers, in dynamic interactive terms, are qualitative with respect to given base numbers that are - relatively - quantitative.

Thus the Zeta 1 zeros (in reverse fashion to the Zeta 2) relate indirectly to dimensional numbers (of a qualitative nature).


Now qualitative always implies interdependence. So we can point here to a basic difference as between Zeta 2 and Zeta 1 zeros!

Though the meaning of 2nd for example can vary without limit (depending on its group context) it  can be given a relatively independent meaning in each case.

However with the product of primes, the resulting meaning relates always to the combination of prime factors involved.

For example 2 can be used without limit as a prime factor (in obtaining composite natural numbers). However it is always necessarily used (except where it is used itself repeatedly) in conjunction with one or more other prime numbers.

Thus the corresponding meaning of the Zeta 1 zeros relates to the qualitative interdependence of such groupings.

It is even easy to suggest why the frequency of (non-trivial) Zeta 1 zeros continually increases as we ascend (and descend) the imaginary number scale. The reason for this is related to the corresponding fact that the interdependence with respect to prime number factors (comprising composite natural numbers) likewise steadily increases as we ascend the real number scale.

Now the reason why the scale is imaginary - rather than real - is that we are using numbers indirectly to represent qualitative - rather than quantitative - notions. And the correct way of doing this is to use an imaginary - rather than - real number scale!

I have explained before the important difference as between the transcendent and immanent perspectives.

From the transcendent perspective, interdependence is seen as going beyond the individual towards a more universal collective meaning. So in this case interdependence is identified with a collective grouping (rather than each individual member). And we have seen that in terms of ordinal notions of number this is exactly the case. The interdependent nature of meaning here relates to the collective interaction of all members.

However from the immanent perspective, interdependence is seen as going within towards a more unique individual meaning (as reflective of the whole). For example this immanent aspect is well expressed through Blake's famous line "To see a  world in a grain of sand".

Now it is similar here in a mathematical context, where the overall holistic nature of the number system is reflected uniquely through each Zeta 1 zero. This indeed is why each zero represents a pure energy state in psycho spiritual terms. In other words, if we could correctly understand the Zeta 1 zeros in a highly refined intuitive manner (which requires an extremely interactive dynamic type of appreciation), they would thereby lose all phenomenal rigidity and reflect their inherent light in a pure transparent qualitative manner  

And again just as with the Zeta 2 we move from quantitative to qualitative appreciation through summation, in reverse manner we move from qualitative to quantitative appreciation through summation of all the Zeta 1 zeros.
This indeed is why the summation of Zeta 1 zeros can be used to eliminate remaining deviations in the general prediction of prime frequency (up to a given number).


Though it may not be possible yet to grasp the full implications of what is involved here, the key insight to absorb is that proper interpretation of the zeta zeros (Zeta 1 and Zeta 2) requires an entirely new mathematical approach that entails the dynamic interaction of Type 1 (analytic) and Type 2 (holistic) aspects. And remembers there is still no formal recognition whatsoever of the Type 2 (holistic) aspect of Mathematics! 

And the full blooming of such interactive understanding represent mathematical understanding - as it should be - in a comprehensive manner (i.e. Type 3). And again without recognition of the Type 2 aspect, Type 3 is inevitably reduced in merely Type 1 terms!   

So these blog entries are intended as serving as the most preliminary introduction to Type 3 mathematical understanding.

Truly, when these basic insights are grasped, it will be realised that we are thereby at the very beginning of the most important revolution yet - not alone in mathematical - but in our intellectual history.

Where Science and Art Coincide (15)

One of the most valuable insights I had regarding prime numbers was the realisation that they shared close dynamic similarities with the nature of what - in psychological terms - we would refer to as primitive instincts.

Therefore if we accept that in psycho spiritual terms, ultimate spiritual realisation entails the full integration of both conscious and unconscious aspects of behaviour in completely unravelling such primitive instincts, then exactly the same is true in terms of a complete understanding of prime numbers.

So in dynamic interactive terms both the primes (from a physical perspective) and primitive instincts (from a psychological perspective) are fully complementary notions.

In fact such primitive instinctive behaviour is in fact prime in holistic mathematical terms.


In earliest infancy before proper differentiation (of consciousness) or integration (with respect to the unconscious) can take place, both aspects are directly confused with each other in a primitive (i.e. prime) fashion.

The important notion here is that very notion of a prime number (from a dynamic interactive perspective) is that it directly combines analytic (conscious) with corresponding holistic (unconscious) notions of number identity in a paradoxical manner.

Thus from the analytic (conscious) perspective, prime numbers are seen as the independent building blocks of the natural number system in cardinal terms.

However from the corresponding holistic (unconscious) perspective, the prime numbers are seen as the corresponding interdependent building blocks of the natural number system in ordinal terms.

And the direction of connection with respect to primes and natural numbers alternates from each perspective.

Thus from the cardinal perspective, each (composite) natural number is defined in terms of prime number members as factors. However from the corresponding ordinal perspective, each prime number is uniquely defined in terms of its individual (natural number) members.

So the inherent mystery of the nature of primes directly springs from this dynamic understanding, where ultimately both prime and natural numbers are understood as perfect identical mirrors of each other (in an ineffable manner).

Now what actually happens in in our culture, is that the differentiation of (analytic) consciousness away from confused (holistic) unconscious notions, commences in early infancy and then by early adult life generally attains its specialised  development.

This is especially true for those with a direct involvement in mathematics and the various sciences (as we know it).

Though this specialisation has indeed brought immense benefits from a narrow quantitative perspective, unfortunately it has left us with a very reduced - and thereby distorted - understanding of the nature of number.


Rather than understanding number in its inherent dynamic interactive fashion containing both analytic (quantitative)  and holistic (qualitative) aspects in relation to each other, we have sought to reduce such understanding to mere quantitative interpretation (in a merely analytic fashion).

In particular the inherently dynamic interactive notion of prime numbers is completely lost in such interpretation, where primes are misleadingly viewed in a rigid absolute manner (with respect to their quantitative identity).

And it is very important to appreciate that such interpretation is a direct product of the specialised differentiation of conscious development (which largely defines intellectual life).


However as I repeatedly have stated, rather like the electromagnetic spectrum (with its many bands of differing types of radiation) we likewise have many differing bands of possible rational interpretation. And to properly understand the true nature of our number system we need appreciation of these additional bands.


Just reflect on this this for the moment! If the number system is truly fundamental - which indeed it is - would it not seem very strange if we could understood it properly with respect to just one narrow band of understanding on the overall spectrum.

The belief that this indeed is possible demonstrates in fact the highly reduced nature of current mathematical (and scientific) understanding.


Though of course professional mathematicians would consider this as of no relevance for their discipline, a considerable - often unacknowledged - price is paid for the extreme conscious specialisation that dominates Western scientific thinking.

Though it is indeed appropriate that the earlier stages of development should be largely geared to the differentiation of conscious ability, rightly understood this should represent just one important phase of overall development.

Unfortunately, because the unconscious remains largely undeveloped in earlier development, an important price that is paid for successful conscious differentiation is the unwitting repression of primitive type instincts. In other words analytic tends to quickly dominate latent holistic ability (which relates directly to unconscious development).

Indeed this analytic domination is so extreme in Mathematics (as formally understood) that the very concept of a distinct holistic dimension seems quite foreign in terms of customary formal understanding.

And I should know something about this as I have found enormous resistance - even among the most highly intelligent and open-minded individuals - to  the notion that Mathematics has a neglected qualitative aspect (of equal importance to the quantitative).

Even less have I found any appreciation of the key notion that - properly understood - Mathematics is of a dynamic relative nature (comprising the interaction of both its quantitative and qualitative aspects).


As I have said the process of higher level integration in the attempt to recover and properly develop this neglected holistic unconscious aspect, has long been associated with the spiritual contemplative traditions (both East and West).

However though detailed accounts have been indeed been left by gifted practitioners of the various contemplative states (reflecting the continual refinement of holistic intuition), up to this point in time very little attention has been given to associated refined rational structures (which has a direct relevance for Mathematics and Science).

Now what appear as quite remarkable is that the zeta zeros (zeta 1 and Zeta 2) are directly related to this complementary process in development with respect to the overall holistic integration of development (in the corresponding process of specialised development with respect to the unconscious aspect of personality).

Basically we can identify two directions with respect to such development (which ultimately are totally complementary and thereby necessary for each other).

From the transcendent perspective we have the attempt to achieve higher level integration of consciousness (which likewise requires that such consciousness be equally properly differentiated in experience).

This is what can be conveniently described as top-down integration where one attempts to integrate lower stages in development from the perspective of the higher.

From the immanent perspective, we have the inverse attempt to achieve the proper differentiation of the unconscious (with respect to its primitive responses). This equally requires that such consciousness be properly integrated in experience.

This in turn can conveniently be described as bottom-up integration where one attempts to integrate higher stages in development from the perspective of the lower.

Now quite remarkably, the Zeta 2 (non-trivial) zeros relate directly to the former and the Zeta 1 zeros to the latter aspect respectively.

However just as both top-down and bottom-up integration are ultimately totally interdependent in development, the Zeta 2 and Zeta 1 zeros are likewise totally interdependent with respect to the number system.

We will continue in the next few blogs to demonstrate more clearly the precise connections as between the holistic nature of the zeta zeros (and the psychological structures to which they are exactly related).

Where Science and Art Coincide (14)

As we have seen the number system - when properly understood in a dynamic interactive manner - comprises both differentiated (analytic) and integral (holistic) aspects in relative terms. Whereas the cardinal and  ordinal numbers represent the former, the zeta non-trivial zeros (Zeta 1 and Zeta 2) represent the latter aspect respectively.

And as we have seen, initially when considered in a relatively independent manner, the Zeta 1 zeros can be seen as representing the integral (holistic) aspect of the cardinal aspect, and the Zeta 2 the corresponding integral aspect of the ordinal aspect of the number system respectively.

Ultimately however both the differentiated and integral aspects of the number system are simultaneously determined in an ineffable manner.

Then in phenomenal terms, which assumes a dynamic framework in space and time, these aspects diverge to an extent with both analytic and holistic aspects assuming a certain relative independence from each other.

However when we appreciate number in an appropriate dynamic manner, then the very nature of the system becomes inseparable from the nature of development itself (in physical and psychological terms) .

This overall development therefore provides the necessary perspective to appreciate our current limited appreciation of the number system i.e. in a merely differentiated manner.

As we have seen, conventional mathematical interpretation - on which our understanding of the number system is based - relates directly to the 2nd band of the Spectrum and represents the specialisation of linear (1-dimensional) reason of a differentiated kind.

In Conventional Mathematics this 2nd band of understanding is then imposed on all other bands leading therefore to a greatly reduced nature of interpretation in a merely quantitative manner.

So really the very first insight which mathematicians need to take on board is that there exist many further bands of understanding on the overall Spectrum, which lead to a uniquely distinctive  interpretation of the nature of mathematical relationships.


However it is the 3rd and 4th Bands on the Spectrum that are properly geared to true understanding of the qualitative nature of the zeta zeros.

In my companion  blog "Spectrum of Development" I am currently outlining the levels  and stages associated with the various Bands.

Now stated in general terms, the first two Bands are geared to the differentiation of conscious experience.
Once again the our conventional interpretation of the number system is based on the specialised nature of such understanding (associated with Band 2).

However the next two Bands are geared to the integration with respect to holistic unconscious awareness.

This understanding begins to unfold in an explicit manner at Band 3 with specialised understanding of its nature associated with Band 4.

Thus understanding of the true qualitative nature of the zeta zeros (Zeta 1 and Zeta 2) is related to these two Bands.

The last three Bands (Bands 5, 6 and 7) are then devoted to the dynamic interplay of both conscious (differentiated) and unconscious (integral) awareness.

Comprehensive understanding of the number system - which entails such a dynamic interaction with respect to both its quantitative and qualitative aspects -  is thereby associated with these Bands.

In particular, as the Riemann Zeta Function (and its associated Riemann Hypothesis) - when appropriately understood - relate directly to the fundamental relationship as between the quantitative and qualitative aspects of the number system, it thereby requires interpretation that corresponds to the understanding of these final Bands. In fact Band 5 is sufficient to provide the crucial insight into the true significance of the Riemann Hypothesis.

Now we are still a very long way in  our evolution from appropriate understanding of the higher Bands (i.e. beyond the 2nd). Indeed in terms of the current mathematical and scientific paradigms their relevance is completely ignored (which indicates how enormously reduced therefore is the nature of our current understanding).

However even now some access - for those willing to see - is indeed available with respect to the higher Bands, which has the power to totally revolutionise our appreciation of the true nature of Mathematics.


The starting basis for true holistic appreciation of the Zeta 2 zeros in fact commences with Level 1 (Band 3).

As I have outline in my "Spectrum of Development" blog this is heavily associated with the transcendent type approach to integration.

In my own case I became aware very early on that such development related in a holistic mathematical fashion to the "higher" dimensions.

In other words Conventional Mathematics is 1-dimensional in nature which simply implies that in terms of the fundamental polarities interpretation is absolute (non-interactive) defined in terms of just one pole. So rather that considering mathematical symbols as the interaction of external (objective) and internal (subjective) aspects, they are considered in a merely (absolute) objective manner.

Likewise rather than considering mathematical symbols as the interaction of both quantitative and qualitative aspects, again they are considered in a merely (absolute) quantitative manner!

However the limits of such dualistic understanding are exposed at Level 1 (Band 3) with the horizontal polarities especially of external and internal increasingly interacting in experience.

So in holistic mathematical terms, which itself reflects the new mathematical understanding now unfolding with this Band, I could see that this interaction of these poles related to 2-dimensional understanding.

Furthermore I was able to see that the increasingly refined interaction of poles (horizontal and vertical) would be described holistically in terms of 3, 4, 5....n dimensions.

All of this forms the direct basis of the holistic qualitative appreciation of the Zeta 2 zeros.


However gradually I began to experience the limits of this transcendent direction of spiritual development.

What it entails in effect is a top-down model of integration. Implicit in its nature is the belief that "higher" is better. therefore the attempt is made during such development to continually integrate lower levels from the perspective of the highest yet attained.

Indirectly this can then lead to a considerable degree of accumulated repression with respect to "lower" level primitive (i.e. unconscious) instincts.

So gradually the shift takes place to the recognition of a corresponding immanent direction of spiritual development that offers a complementary bottom-up approach to integration.

And remarkably it is to the understanding associated with this bottom-up approach to integration in development that the (recognised) Zeta 1 zeros properly relate.

Where Science and Art Coincide (13)

To put it simply in psychological terms, rational (conscious) understanding of an analytic nature is properly geared to the differentiation of phenomena; however intuitive (unconscious) understanding of a holistic nature is properly geared to the corresponding integration of such phenomena.

Now both of these aspects (conscious and unconscious) are combined with each other in a dynamic interactive manner, enabling both (analytic) differentiation and (holistic) integration with respect  to all experience.

However once again the key problem with respect to conventional mathematical interpretation is that it attempts to understand mathematical phenomena solely with respect to the differentiated aspect i.e. formally in a rational analytic manner.

This of course is fundamentally true with respect to conventional treatment of the number system.

For several millennia now, we have attempted to interpret number solely from a rational analytic perspective.

However our actual experience of number is properly of a dynamic interactive nature, entailing both rational (analytic) and intuitive (holistic) aspects.

Put another way conventional interpretation of number is solely geared to treatment of its (independent) differentiated aspect; in other words such interpretation is totally lacking in appreciation of its equally important (interdependent) integral aspect.

And as ultimately the differentiated has no coherence in the absence of its integral aspect, this represents by far the most important issue in Mathematics that is yet to be addressed.

So to put it bluntly, Conventional Mathematics is ultimately completely lacking a coherent framework of interpretation.


And once again, quite simply, the zeta zeros (Zeta and Zeta 2 ) directly represent this vitally important integral aspect of our  number system.

Furthermore the dynamic process of both differentiation and integration typifies all natural phenomena (e.g. at physical, biological, animal and human levels).

Thus the zeta zeros represent the fundamental basis of integration with respect to all phenomena in nature.

The great German mathematician, Hilbert had a strong sense of the ultimate importance of the zeta zeros (not just for Mathematics, but in fact for everything). However surely he would have been surprised to understand their supreme significance as the direct basis for the interdependence of all phenomena.

Indeed when we interactively combine both its (quantitative) analytic and (qualitative) holistic aspects, it is but a short step to the realisation that all phenomena in experience represent the manifest existence of dynamic number interactions (with respect to both their quantitative and qualitative aspects).

So put another way, number represents the essential bridge or intermediary as it were as between the manifest phenomenal and ineffable spiritual realms.

We are just about to properly realise in our evolution therefore that number - when appropriately understood in a dynamic interactive manner - embodies both the material and the spiritual. So with respect to the material aspect, from the psychological perspective, we are accustomed to look on number as representing (unchanging) forms.

However from the spiritual aspect, number ultimately is seen as representing pure energy states. And of course, because psychological and physical aspects are dynamically complementary, number from one extreme can likewise be seen as representing (abstract) objects; then from the other extreme, it can be seen as representing pure physical energy states!

Thus in dynamic interactive terms, number embodies both form and energy (in physical and psychological terms).


So at the analytic level, the number system is fundamentally related to the interaction of primes and natural numbers, which represents its differentiated quantitative aspect.

Then at the holistic level of the number system is fundamentally related to the interaction of Zeta a and Zeta 2 zeros, which represents its integral qualitative aspect.

Ultimately both the analytic and holistic aspects are themselves fully interdependent, with differentiation having no strict meaning in the absence of integration (and likewise integration having no strict meaning in the absence of differentiation).

However from a relatively independent perspective, we can indeed probe more deeply into the precise significance for integration of both the Zeta 1 and Zeta 2 zeros (which we will outline in the next blog entry).  

Where Science and Art Coincide (12)

As we have seen, customary analytic type understanding of the number system represents the specialisation of the understanding of Band 2 (i.e. of a rational linear nature). And this relates to interpretation of number from an exclusive quantitative perspective.

However with unfolding of Band 3, a very distinctive new intuitive understanding of a holistic nature.

 Earlier linear understanding is based in any context on one isolated pole of reference leading to an absolute static type understanding of number.

However this intuition increasingly reflects the dynamic interaction of complementary poles in a new enhanced type of qualitative understanding.

So for example instead of number being considered in merely abstract objective terms, with this new holistic appreciation, number is understood in dynamic terms as representing the interaction of both external (objective) and internal (subjective) aspects.

Crucially also, instead of number being considered in a merely quantitative manner, with holistic appreciation,  increasingly it is seen as representing the necessary interaction of both quantitative and qualitative aspects.

And as the various levels of Band 3 unfold with understanding, attaining a purer contemplative focus, the qualitative aspect of number relationships becomes increasingly pronounced.

Then just as there is earlier a specialised band (Band 2) with respect to conventional analytic, there is now later a corresponding specialised band (Band 4) with respect to this new holistic understanding. And in the context of number this relates to a pure qualitative type appreciation of the nature of number.

So once again, Conventional Mathematics, based on the specialised understanding of Band 2, represents the Type 1 aspect of Mathematics (which equally of course represents the Type 1 aspect of the number system). However - what I have long referred to as - Holistic Mathematics represents its (unrecognised) Type 2 aspect.

Though we have not yet evolved sufficiently in our understanding to form but a rudimentary appreciation of the true nature and scope of this Type 2 aspect, its full specialised expression (requiring highly refined intuitive states of understanding) can be identified with Band 4 on the overall Spectrum of Development.


However what is vital to appreciate is that associated with these more advanced psycho spiritual states of intuitive understanding are corresponding increasingly refined rational structures (of a circular paradoxical nature).

And quite remarkably its is these structures to which the Zeta 2 zeros directly relate.

Putting it more succinctly, the earlier two bands (Bands 1 and 2) on the spectrum represent the gradual movement to specialised development of the (rational) conscious aspect of personality.

However the latter two bands (Bands 3 and 4) represent the movement towards corresponding  specialised development of the (intuitive) unconscious aspect.

And once again this intuitive development is closely associated with the unfolding of increasingly refined "higher" dimensional structures (of a circular paradoxical nature).

From an early age, I had been greatly interested in Mathematics and would have been considered gifted (especially by my secondary school teacher in Mathematics) at the subject.

However even then my abilities started to decline with respect to customary understanding and switch quite rapidly in the Type 2 (holistic) direction, which is still completely unrecognised at a formal level.

This is why I can pronounce confidently on what I am stating in that I have been wrestling with key issues for nearly 50 years now which recognised practitioners have yet to recognise.


In other words current Mathematics in being so tightly confined to its mere Type 1 aspect, is greatly lacking the perspective to even evaluate a contribution such as this. This is why I constantly make my appeals to an audience that - while certainly  not excluding professional mathematicians -  has the crucial willingness and capacity to constructively question existing assumptions.

And make no mistake about it! Mathematics as we know it will eventually undergo a complete revolution where it will then be widely accepted that the highly reduced assumptions on which it is currently based are strictly speaking untenable.

But why wait for the future when we can diagnose such deficiencies now! However as I have said this needs a much enlarged approach so as to provide this greatly needed critical perspective of existing procedures.


In my own approach I could recognise from my early 20's that "higher" levels of development bore a very close relationships - literally - to the higher mathematical dimensions (when given their appropriate qualitative interpretation).

So once again customary understanding of mathematical relationships is literally of a 1-dimensional nature (which effectively freezes consideration of dynamic interaction as between the opposite polarities of experience).

Thus once again from a 1-dimensional perspective, the objective aspect of relationships can be considered independently of their subjective counterpart; likewise the quantitative aspect can be considered independently of its qualitative aspect.

However higher dimensional understanding is based on an increasingly refined relationship as between these opposite poles (where dynamic interaction by its very nature is necessarily involved).

For example 2-dimensional interpretation would require that - in any context - acceptance of the two opposite poles involved in dynamic interaction with each other.

So for example a number from this perspective can no longer be considered as an absolute independent entity; rather it now reflects a dynamic interaction pattern in relative terms, entailing both its objective (external) and subjective (internal) poles.

A 4-dimensional interpretation would require the consideration of opposite polarities simultaneously in both horizontal and vertical fashion (which in complex mathematical terms would relate to real and imaginary co-ordinates in both positive and negative terms).

In my earlier work on Holistic Mathematics, I gave extensive consideration to the nature of 2, 4 and 8-dimensional interpretation that I believed were especially important from a holistic integral perspective.

I also gave considerable attention to the more difficult to understand odd-numbered dimensions (especially 3).

Now just as all composite natural numbers (in quantitative terms) are built from prime factors, likewise all composite natural number dimensions (in qualitative terms) are built from prime factors.

And essentially the Zeta 2 zeros simply represent the indirect quantitative representation (as prime roots of 1) of all the prime dimensions.  (Again 1 as a root is excluded as it is trivial in being non-unique).


However satisfactory identification of the Zeta 1 zeros proved a much more difficult task though eventually was resolved in a surprising yet ultimately somewhat obvious manner!

Where Science and Art Coincide (11)

I will attempt here to probe more closely the exact psychological counterparts in development to the Zeta 1 and Zeta 2 zeros.

Now psychological (and indeed complementary) physical development can be likened to an overall spectrum, which like electromagnetic energy, contains many distinct bands.

In my own model for some years I have included 7 distinct bands and to place my future remarks in context, I will say here something about these bands. (More detailed clarification of the nature of these bands can be found on my companion "Spectrum of Development" blog.)


When human psychological development commences - indeed initially with the foetus in the womb - both conscious and unconscious remain greatly confused with each other. Thus no means yet exist for (conscious) differentiation of distinct phenomena on the one hand or (unconscious) integration with respect to these same phenomena.

So the first task of development is to gradually differentiate conscious experience from confused unconscious development. From a scientific experience this relates to rational analytic type appreciation of phenomena.

Therefore, Band 1 (from this scientific perspective) relates to the gradual emergence of rational analytic type understanding.

Band 2 then relates to the specialised development of such ability.

Now conventional scientific and mathematical interpretation (and indeed intellectual life generally) is heavily based on the specialised conscious interpretation of this Band (which is accurately characterised as 1-dimensional understanding). What this means in effect is that in any relevant context, interpretation is based with respect to one isolated pole of understanding. So for example the external (objective) pole is abstracted from the internal (subjective); likewise the quantitative aspect is abstracted from its corresponding qualitative aspect.


In particular conventional understanding of the number system and indeed all mathematical relationships is based on such 1-dimensional understanding.

So our understanding of number from this perspective simply reflects the reduced understanding that typifies the 2nd Band of the spectrum. 

I am not for a moment disputing the extraordinary benefits (within its appropriate context) of such understanding! Rather I am pointing to the fact that in terms of the potential development of the entire spectrum such understanding is necessarily of a very limited nature.

To use a simple analogy!  We now appreciate that natural light represents just one small band on the overall electromagnetic spectrum. Well in like manner, conventional (natural) understanding of the number system (and indeed by extension all mathematical relationships) likewise represents just one small band on the overall spectrum of all such scientific relationships!

In other words from the perspective of higher bands on the spectrum, the very nature of the number system is understood in a new distinctive light.


Whereas the earlier bands (1st and 2nd) are given over to the analytic differentiation of the conscious aspect, the next two bands (3rd and 4th) are devoted to the corresponding holistic integration of the unconscious aspect of experience.

In the past these two bands were largely associated with contemplative type spiritual development with various detailed accounts of the states involved to be found in all the mystical traditions.

However what has not yet been properly recognised is that associated with such higher level intuitive type understanding, are new refined rational structures (of an increasingly circular paradoxical nature). And these form the appropriate basis for the rational expression of mathematical - and indeed all scientific - relationships at these bands.

In my own writing I refer to the customary analytic understanding of mathematical relationships (associated with Bands 1 and 2) as the Type 1 aspect of Mathematics. Then this new holistic understanding (associated with Bands 3 and 4) I then refer to as the Type 2 aspect.

In particular with Type 2 understanding is the realisation that an indispensable qualitative aspect attaches to all mathematical relationships (that cannot be successfully reduced in quantitative terms).

For example the ordinal nature of number properly relates to qualitative rather than directly to quantitative type appreciation. However in customary Type 1 understanding the (qualitative) ordinal aspect is simply reduced in terms of (quantitative) cardinal type understanding.

Following deep disillusionment with the conventional reduced approach to Mathematics at college (especially in relation to the infinite notion) I embarked on developing the Type 2 aspect in my early 20's and now after some 45 years of such development, I believe that I can offer a radically new informed perspective with which to view the considerable limits associated with customary understanding.


Band 3 (with its various levels and stages) - which I am outlining presently in "Spectrum of Development"- is related to the gradual unfolding of spiritual intuitive type understanding (and its associated rational understanding of a circular paradoxical kind).

Then Band 4 is devoted to the specialised development of such spiritual intuitive understanding.

However precisely because such appreciation is radically distinct in nature from conventional linear type understanding, it is very difficult establishing points of contact during these bands with established customary notions.


So far we have outlined two distinct overall stages of development with the earliest bands (Bands 1 and 2) culminating in specialised rational understanding of a linear (1-dimensional) nature and then Bands 3 and 4 culminating in specialised intuitive (contemplative) development. These then are indirectly associated with "higher" dimensional rational appreciation of an increasingly circular nature.


However initially these are developed in a relatively separate manner.

So the final overall stage of development relates to the gradual interpenetration in experience of both types of understanding. When related to Mathematics this entails - what I refer to as - its most comprehensive Type 3 aspect (where both analytic and holistic appreciation of relationships increasingly interact with each other).

So Band 5 relates to such interpenetration at a more fundamental level of experience (that still however lacks phenomenal definition of an extensive nature).

However this band is crucial for successfully dealing with the key issues relating to the Riemann Hypothesis.

Since proper understanding of the Hypothesis and its associated zeta zeros relates to the ultimate identity of both the quantitative (analytic) and qualitative (holistic) aspects of the number system, this thereby requires interpretation where both of these aspects are now explicitly related in experience. And this starts to properly unfold at Band 5.


Though not necessary for our direct purposes, Band 6 relates to an enhanced ability of combining in a mature balanced fashion (extensive) differentiated understanding of phenomena at a rational level with an equally developed integrated contemplative intuitive capacity.

So this Band - though rarely evident yet in our evolution - will provide the opportunity for scientific activity that is both immensely productive and highly creative in nature.

Even with the most advanced development, there is always likely to be a personality preference for the analytic or holistic aspects.

Thus Band 7 in my approach represents the final growth to maturity in attempting to offset this preference. Thus in the final stage of life someone with a natural preference for the holistic contemplative aspect would deliberately concentrate more on activity (requiring the analytic aspect); likewise someone with a preference for activity would deliberately choose a more passive contemplative focus!.


We are still a very long way in our evolution from achieving substantial development beyond the first two bands on the Spectrum.

However even now some degree of access to these bands is possible (as this blog is attempting to demonstrate).

And even a small appreciation of these bands has the power to utterly change our current appreciation of the nature of science and mathematics. And underlying these at the most fundamental level is the number system.

So we have already reached a stage in evolution when our understanding of the number system is about to change utterly. And as the number system is truly of the most fundamental importance, this will therefore represent by far the greatest revolution yet in our intellectual culture!

Where Science and Art Coincide (10)

In my last blog entry, I expressed how the Zeta 2 zeros represent the unconscious holistic basis of the natural number system (in ordinal terms).

Customary understanding of this system takes place in a reduced - merely analytic - manner viewed  solely from a quantitative perspective.

However properly understood, this ordinal aspect entails both (conscious) analytic and (unconscious) holistic aspects which interact in dynamic fashion.


These zeros in the first instance arise as solutions to the finite equation

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime) = 0.

So we have a relationship here as between the value of s (as base quantitative value) and the natural number sequence 1, 2, 3, ...., t – 1 (representing corresponding dimensional values).

Now as I have frequently expressed, the relationship between base and dimensional values is always as quantitative to qualitative (and qualitative to quantitative) in dynamic interactive terms.

So in the expression a^b, a is the base and b the dimensional numbers respectively. Therefore if in this context a is interpreted as quantitative, then the exponent b is thereby - relatively - qualitative in nature.

 

In this sense the Zeta 2 zeros represent an equal and indispensable partner providing the qualitative appreciation underpinning customary quantitative understanding of the ordinal number system.

Though in many ways, much more difficult to intuitively grasp, the true nature of the Zeta 1 (i.e. Riemann) zeros can be expressed simply in a direct complementary fashion.


These zeros by contrast represent the inversion (with respect to the Zeta 2) of base and dimensional values.

So the Zeta 1 zeros arise as solutions to the infinite equation

ζ₁(s) = 1^–s  + 2^–s  + 3^–s  + 4^–s  +……..  = 0.


So the role of the Zeta 1 zeros can be succinctly expressed as providing the corresponding unconscious holistic basis of the natural number system in cardinal terms.

Thus once again though we are accustomed to especially appreciate the cardinal number system in a merely quantitative analytic manner, properly understood, it contains two interacting components that are quantitative and qualitative with respect to each other.

Therefore from this perspective the Zeta 1 zeros provide the (unrecognised) qualitative aspect that underpins our customary quantitative appreciation of the cardinal number system!


So the Zeta 1 and Zeta 2 zeros approach the fundamental identity of quantitative and qualitative aspects of the number system from two complementary perspectives.

In the case of the Zeta 2, the inherently qualitative identity of each ordinal number (within its specified grouping) is given an individual quantitative identity in an indirect manner. For example in the context of two members, 2nd  is indirectly given a quantitative identity as – 1.

Its qualitative nature is then expressed through the collective addition of the quantitative values with respect to all members of the group.

So again in the context of two members, the sum of quantitative values (representing the two roots of 1) = + 1 – 1 = 0.


In the case of the Zeta 1 zeros , it is the reverse!

Here the inherent quantitative  identity of each cardinal member is given an indirect qualitative identity. So for example the qualitative identity (representing a dimensional value) of the first pair of non-trivial zeros is given as + 1/2 + 14.134725...i and + 1/2 – 14.134725...i respectively.

Their quantitative identity is then expressed through the collective nature of all the zeros (which are finitely unlimited in nature).

In other words the non-trivial zeros can be collectively employed in a quantitative manner to fully correct the deviations that arise with respect to the general prediction of the frequency of prime numbers (up to any given natural number).

(Though not yet properly recognised, the Zeta 2 zeros can be used in a complementary fashion.

Thus the individual probability that a number is prime can be corrected through deviations associated with the Zeta 2 zeros. I have in other places shown how these deviations (from the average limiting value of the roots of 1, when the number of roots is very large) can be calculated. See for example "Alternative Prime Hypothesis!"

Thus in principle using this alternative approach in principle we should be able to approximate the exact probability of each number being prime so that the sum of these probabilities would then closely match the exact frequency of primes (up to a given number).


However, though it is initially valid to attempt both the Zeta 1 and  Zeta 2 zeros in a relatively independent manner, in the most comprehensive mathematical understanding (Type 3) we understand them increasingly in a relatively interdependent fashion.

Thus from this perspective, cardinal notions have no strict meaning independent of ordinal; likewise ordinal have no strict meaning independent  of cardinal!

So properly  understood both the Zeta 1 and Zeta 2 zeros are ultimately interactively determined in a manner approaching pure simultaneity.

In the most accurate sense, both thereby serve as the ultimate phenomenal partition bridgng both the finite (dualistic) and infinite (nondual) realms.


Once again in an attempt to provide additional perspective on their nature, the Zeta 2 zeros arise from the attempt to reduce higher dimensional qualitative notions ( ≠ 1) in a 1-dimensional manner.

And the is the manner in which customary ordinal notions are understood!

Therefore to express the qualitative notions of 1st, 2nd and 3rd (in the context of 3 as representing dimensions) in a reduced circular 1-dimensional quantitative manner, we obtain the three roots of 1.

And as the very notion of interdependence has no strict notion in 1-dimensional terms, the corresponding sum of these roots (expressing their qualitative interdependence) = 0 (in quantitative terms). However 0 now also has a qualitative meaning representing a psycho spiritual energy state (corresponding to pure intuitive understanding)

However the Zeta 1 zeros in reverse manner arise from the attempt to transform 1-dimensional quantitative notions in a higher dimensional qualitative manner.

And as the Zeta 2 zeros relate to the ordinal nature of the natural numbers (in the context of each prime) the Zeta 1 zeros relate to the ordinal nature of the composite natural numbers (in the context of the unique product of prime factors).

So as I have repeatedly stressed the process of multiplication inherently implies a qualitative aspect!

Therefore the Zeta 1 zeros simply represent the transformed expression of the qualitative nature of prime number multiplication.

So for example when we multiply 3 by 5 a qualitative - as well as quantitative - transformation takes place in the numbers involved. And the Zeta  1 zeros simply represent the attempt to express the qualitative nature of this transformation for all composite natural numbers (representing the unique product of primes).

Now in reverse fashion to the Zeta 2 zeros, the qualitative interdependent nature of the Zeta 1 zeros is represented with respect to each individual pair. Thus each zero can be seen thereby as approximating a singularity or pure energy state (without form).

Then the corresponding quantitative independent nature of the zeros is represented though their combined collection as a group (thereby enabling the correction of remaining errors with respect to the generalised prediction of prime number frequency among the natural numbers (up to a given level).