The Original Number System

In my last entry I outlined the basis for a new vision of the number system which works on three different levels.

On the customary analytic level of rational (conscious) understanding we have the cardinal and ordinal aspects of the number system (interpreted in a merely quantitative type manner).

Then at a "higher" holistic level of intuitive (unconscious) type appreciation we have the understanding of both the Zeta 1 and Zeta 2 zeros as providing a complementary appreciation of the number system in terms of both physical and psychological energy states.

Finally at the most comprehensive level of understanding we have the growing marriage of both analytic and holistic type appreciation, where the zeta zeros are seen as vital in terms of enabling consistency to be maintained as between the quantitative (Type 1) and qualitative (Type 2) aspects of the number system.

Now this dynamic system is based on the two-way interaction of the primes and natural number numbers.

However strictly it does not itself explain the prior existence of - what I call - the two original numbers i.e. 1 and 0.

So the dynamic relationship as between the primes and natural numbers (and natural numbers and the primes) which in turn is consistently mediated through the corresponding two-way interaction of the Zeta 1 and Zeta 2 zeros is itself based on the prior relationship of 1 and 0 (again in quantitative and qualitative terms).

Again from a comprehensive perspective 1 and 0 have both analytic and holistic interpretations which dynamically interact with respect to all phenomenal relationships.

1 as a cardinal number (as analytically defined) is directly associated with the linear notion (i.e. 1-dimensional) in holistic terms. 0 as analytically defined is directly associated with the circular notion (i.e. 0-dimensional) from a holistic perspective.

In both physical and psychological terms differentiation is directly associated with the (conscious) linear, whereas integration is associated with the corresponding (unconscious) circular notion respectively.

Now the zeta zeros relate directly to these extreme notions in both analytic and holistic terms.

From the analytic perspective, the Zeta 1 zeros occur as conjugate pairs of the form a –  it and a –  it respectively. The Riemann Hypothesis postulates that they are of the form 1/2 –  it and 1/2 –  it respectively.

This would imply that the quantitative sum of each pair = 1.

Then from the analytic perspective the sum of the non-trivial roots for any group n (of 1) =  – 1.

Thus when we combine both we get 0.


From the holistic perspective (in both physical and psychological terms) the initial starting point of evolution - which equally represents the starting point of the number system - entails a totally confused state of integration (where no linear differentiation has taken place).  This equally entails thereby a total state of circular confusion.

Therefore in psychological (and physical terms) the requirement that all Zeta 1 (non-trivial) zeros lie on a straight line points to the corresponding need for complete (linear) differentiation of what earlier commenced in total confusion.

Therefore correctly understood, complete appreciation of the truly dynamic interactive nature of the  Zeta 1 zeros (approaching pure energy states) requires full differentiation with respect to unconscious experience i.e. where all hidden primitive (prime) elements are brought fully to conscious light.
Put another way this requires the full recognition of the hidden (unconscious) shadow with respect to the conscious experience of number.  

Put yet another way full holistic appreciation of their nature requires implies that the Zeta 1 zeros  lie on a straight line.

Now the reason why it is an imaginary - rather than real - straight line is due to the fact that expression of what is inherently of an unconscious nature (indirectly in rational conscious manner) requires imaginary rather than real translation.


With respect to the Zeta 2, the full holistic integration of what has been differentiated at a conscious linear level entails a purely circular appreciation of relationships. Here understanding again becomes so dynamic that rigid attachment to symbols is completely eroded thereby enabling pure holistic appreciation of their interdependent nature.   

And of course ultimately both of these are interdependent. For pure circular integration at the "higher" transcendent level of experience equally requires pure linear differentiation at the corresponding "lower" immanent level.

Thus full integration in psychological terms at the "higher" conscious level, ultimately is inseparable from full differentiation (i.e. of all unrecognised shadow material) at the "lower" unconscious level. 

Thus the two-way relationship of the primes and natural numbers is itself intimately based on a prior two-way relationship of the original numbers 1 and 0 (in both quantitative and qualitative terms).


There are also other fascinating connections with 1 and 0.

The only point for which the Zeta 1 Function remains undefined is with respect to s = 1.

However, Riemann's Functional Equation establishes an intimate connection as between ζ(s) and ζ(1 – s) .

This means therefore that  such an intimate connection binds ζ(1) and ζ(0).

In fact when deriving the value for ζ(0), we saw that it represents the pure complementary extreme of linear logic i.e. pure circular logic based on the complementarity of opposites.

This relates to the value of the Zeta 2 Function when s (for the Zeta 2) = – 1.

This gives a value for the Zeta 2 Function of 1/2.

The corresponding value for the Zeta 1 i.e.ζ₁(0) is then obtained as a linear expression of the inherently circular value associated with ζ₂(– 1).

So all the non-intuitive values  for ζ₁(1 – s) where s ≤ 1, in fact initially represent corresponding circular values associated with  ζ₂(– s)  that are then converted in a linear Zeta 1 manner. 

It can also be easily demonstrated that the only values for which the Zeta 1 and Zeta 2 Functions coincide is where s = 0 (with respect o the Zeta 1) and s = 1 (with respect to the Zeta 2).

In this  case 

ζ₁(0) = 1 + 1 + 1 + 1 +....

and  

ζ₂(1) = = 1 + 1 + 1 + 1 +....

Now strangely though both may seem the same ζ₁(0) is defined  whereas ζ₂(1)  is undefined, just as in complementary fashion ζ₁(1) is undefined and ζ₂(0) is defined!

The common denominator is that when s = 1 (in both cases representing in the case of the Zeta 1 a dimensional and in the case of the Zeta 2 a base value respectively).

New Vision of the Number System

As regular readers of this blog might be aware, it is now a few weeks since I contributed additional material.

This was due to the fact that I devoted that time to submitting a series of three related articles on the "Dynamic Nature of the Number System" to Frank Visser's site "Integral World".

One of the benefits of such an exercise, where one is attempting to convey - perhaps - difficult ideas to a more general audience is that it forces one to better clarify what is essential to one's approach.

So all in all I had the sense of a new vision of the number system - which had been steadily emerging through these blogs - finally crystallising in a more coherent manner.

This is especially the case in relation to the conclusion of my reply to general comments made on the articles by Elliot Benjamin where I sought to provide the key features of this radical new understanding of the number system.


So firstly, the true nature of the number system is rooted in experience which inherently is of a dynamic interactive nature.

Conventional understanding of number in fact represents but an extreme limiting case where one attempts  to abstract it in absolute terms from such experience.

This leads to the conventional cardinal view of the natural number system as composed of fixed unchanging entities of form i.e. 1, 2, 3, 4,.....

So this approach entails that in all contexts qualitative meaning i.e. of the relational notion of number interdependence, is reduced in a merely quantitative manner. 

One unfortunate consequence of such an approach is that the distinctive nature of the ordinal nature of number - which inherently is of a qualitative nature - is thereby completely overlooked (and once again reduced in a quantitative manner).

The key to appreciation of the true dynamic nature of number, lies in an obvious distinction as between addition and multiplication (that is however completely overlooked in conventional terms).

When for example in the pure case of addition, one adds two units i.e.. 1 + 1, a  quantitative change in the base number is involved (with the dimensional number as power unchanged).

So, 

1¹ + 1¹ = 2¹.

However when one multiplies the same two numbers i.e. 1 * 1, in inverse fashion. a qualitative change in the dimensional number takes place (with the base number unchanged).

Thus

1¹ * 1¹ = 1².

This leads in fact to two complementary aspects of the number system (Type 1 and Type 2) respectively.


Initially in understanding, Type 1 can be identified with the notion of number as quantitative, with Type 2 associated with the corresponding relational notion of number as qualitative.

In actual experience, both of these aspects continually interact in dynamic manner, with both base and dimensional aspects switching as between quantitative and qualitative aspects.

Therefore the cardinal and ordinal aspects of the system are now clearly understood as quantitative and qualitative with respect to each other.

From this new perspective, the understanding of the role of primes in the number system is dramatically altered.

In conventional (Type 1) mathematical terms, the primes are viewed as the unique building blocks (except 1) of the cardinal number system (where they are misleadingly viewed as the independent  "atoms" of the number system).
 However from the complementary (Type 2) ordinal aspect, the natural numbers (except 1) are viewed as the unique building blocks of each prime .

So for example the 3 as prime number (representing a collection or group) is composed of its 1st, 2nd and 3rd members in an ordinal natural number manner.

We can then indirectly express in quantitative terms  this ordinal identity through the 3 roots of 1.
Where prime roots of 1 are involved, all roots except the 1st will be unique.

So we now have two opposite perspectives on the primes. From the Type 1 perspective, they appear as the most independent of all numbers (in quantitative terms); however from the corresponding Type 2 perspective they appear as the most interdependent (in a qualitative manner).


So now the number system is not so much interpreted in terms of the the mystery of the primes, but rather as the two-way mystery of the relationship of the primes to the natural numbers (and the natural numbers to the primes).

From this perspective, both the primes and natural numbers are understood as perfect mirrors of each other in a mutual identity that is ultimately ineffable.

When we recognise both the quantitative and qualitative aspects of number, the role of the number system is dramatically enhanced.  Indeed it is this complementary 2-way relationship between the primes and natural numbers that enables both quantitative and qualitative aspects of the number system to be transmitted and indeed ultimately quantitative and qualitative features of all phenomenal evolution!


Associated with both the Type 1 and Type 2 aspects of the number system are corresponding Zeta 1 and Zeta 2 Functions.

The Zeta 1 Function again represents the well-known Riemann Zeta Function defined (in an infinite manner) as:

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

The Zeta 1 zeros then occur for

ζ₁  =  1 ^{– s} + 2 ^{– s}  + 3 ^{– s}  + 4 ^{– s}    +…  = 0, 

which in the case of the non-trivial (which are especially relevant) are postulated to be all of the form a + it and a – it respectively. 

 
The Zeta 2 Function represents a simple finite function. However, because it is in this context directly related to the Type 2 aspect of the number system, its crucial role is completely overlooked in conventional mathematical terms.

It is defined as:

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

Thus the Zeta 2 zeros occur for:

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

 
From a dynamic interactive perspective, the Zeta 1 and Zeta 2 zeros represent the perfect holistic complements to the conventional analytic interpretation of the cardinal and ordinal aspects of the number system respectively.

These zeros are therefore as equally important as the recognised cardinal and ordinal numbers. However their role can only be properly appreciated in a dynamic interactive context, where they are understood as representing the opposite extreme to the conventional understanding of number. 

In fact as we shall see, for the number system to operate in a consistent manner, a perfect tension must be maintained as between two opposing extremes.

At one extreme we have the analytic interpretation of number in an absolute quantitative manner where numbers represent fixed forms. This corresponds in turn with a merely conscious rational type interpretation.

At the other extreme, we have the holistic appreciation of number in an extremely dynamic manner that approximates an ineffable state. Here both the quantitative and qualitative aspects of number, while maintaining a certain relative (quantitative) independence, are equally fully related with each other in a relative (qualitative) interdependent manner.

So the essence of the analytic approach is that the quantitative aspect is fully separated from the qualitative (with the qualitative thereby reduced to the quantitative). This then leads to the (mistaken) impression that numbers are merely quantitative in nature.   

However the essence of the holistic approach - by contrast - is that both quantitative and qualitative aspects must be equally recognised and ultimately fully identified with each other (which position however can only be approximated in phenomenal terms).

Whereas the analytic approach corresponds directly with (linear) rational interpretation, the holistic requires a very high degree of authentic intuitive type recognition, which indirectly is conveyed in a (circular) rational manner.


I will just once again illustrate the nature of this latter holistic approach with respect to the 1^st of the Zeta 2 zeros.

The two roots of 1 are + 1 and – 1 respectively and these provide an indirect quantitative interpretation of the ordinal notions of 1^st and 2^nd (in the context of 2 members). Now strictly 1 representing the 1^st is never unique. So the first unique Zeta 2 zero relates here to – 1 (representing the 2^nd) in this context.

So + 1 and – 1 enjoy a relative independence in quantitative terms. However, equally, the combination of both represents their corresponding relative interdependence (in qualitative terms).

And this combination (i.e. sum) is without quantitative identity as + 1 – 1 = 0.

So we can see here in the simplest Zeta 2 zero case, how a perfect balance is maintained as between quantitative aspects (of relative independence) and qualitative aspects (of relative interdependence) with respect to the ordinal number system.

 
The Zeta 2 zeros provide the magical means of converting the Type 2 (ordinal) aspect of the number system in a corresponding Type 1 manner.

The Zeta 1 zeros provide a reverse magical means of converting the Type 1 (cardinal) aspect of the number system in a corresponding Type 2 manner.

In fact, the zeta zeros can be seen as representing (both in physical and psychological terms) the perfect holistic shadow systems to the conventional analytic appreciation of both cardinal and ordinal numbers.

Now the shadow of course relates to the initial (hidden) unconscious counterpart of conscious understanding.

So one could validly say that conventional number interpretation is representative of understanding whose shadow side remains completely unrecognised.

And of course all Conventional Mathematics likewise is representative of such understanding (i.e. where the shadow side remains completely hidden).

Thus the zeta zeros therefore represent the perfect (i.e. fully revealed) shadows of both the Type 1 and Type 2 aspects of the number system respectively.

In this way, the holistic (unconscious) appreciation relating to both aspects of number is brought fully to conscious light, where it can be clearly recognised as an integral aspect of the overall system.


With the Type 1, the zeta zeros provide the holistic complement to the cardinal aspect of the number system (based on unique combinations of primes).

With the Type 2, the zeta zeros provide the holistic complement to the ordinal aspect of the number system (based on unique combinations of natural numbers). Ultimately, of course sets of zeros are fully complementary with each other.

Properly understood, the zeta zeros are of equal importance to the natural number system (as conventionally understood).

Indeed for consistency to be preserved with respect to the overall system, a perfect balance must be maintained as between its two extreme poles (analytic and holistic).

The analytic aspect represents an absolute view of number as fixed forms. The holistic aspect represents an entirely relative view of number as pure energy states (in physical and psychological terms).

The analytic aspect concurs with (conscious) rational interpretation of a linear kind; the holistic aspect concurs with (unconscious) intuitively refined appreciation that is indirectly interpreted in a paradoxical (circular) rational manner.

It is clearly obvious from all this that the number system cannot be properly understood in a conventional analytic fashion (based on mere quantitative notions).

Rather it must be dynamically understood in both analytic and holistic terms (as the dynamic interaction of both its quantitative and qualitative aspects).


Now the role of the zeta zeros (both Zeta 1 and Zeta 2) is vital for the consistent dynamic operation of this system.

This all goes back to the pure nature of addition and multiplication (leading to the Type 1 and Type 2 aspects of the number system respectively).

The problem is that just as addition and multiplication are incompatible operations (in terms of each other), likewise - initially - the Type 1 and Type 2 aspects are likewise incompatible.

So it is the zeta zeros that enable such compatibility from two complementary directions.

The problem can be likened to the difficulties in communications where two people from different countries can only understand each in their native language.

So for example if one of these is from Italy and the other from Germany, for communication to take place, the Italian must receive a translation of German into Italian, whereas the German - in reverse - must receive a translation from Italian into German.

It is somewhat similar with the number system. Basically for consistent operation of the system, the Type 1 must be capable of translation in Type 2 terms, and equally the Type 2 aspect capable of translation in a Type 1 manner.


And this is precisely what the zeta zeros achieve. The Zeta 1 zeros represent the translation of the Type 1 aspect in Type 2 terms; the Zeta 2 zeros, represent the translation of the Type 2 aspect in Type 1 terms. 
 
However, though the language analogy is very helpful, there is one important difference.

Translation of language is always a somewhat inexact process. How often do we hear of an intended meaning that “gets lost in translation”?

However the translation with respect to the two aspects of the number system must be perfectly precise, so as to ensure subsequent consistency with respect to the use of number (in both cardinal and ordinal terms).

And this is precisely therefore what the zeta zeros (Zeta 1 and Zeta 2) achieve. Put another way, they enable perfect consistency to be maintained as between the quantitative and qualitative aspects of the number system. 


In my final article on “Integral World” I show how number must be placed within a developmental context and indicate how potentially three distinct phases must be negotiated with respect to number before a comprehensive understanding can emerge.

The first relates to conscious development and the specialisation of analytic (linear) type understanding (which once again defines conventional mathematical appreciation). Customary understanding of number (cardinal and ordinal) is analytic in this sense.

The second relates to mature development of the unconscious, which potentially culminates in the specialisation of holistic (circular) type understanding. Direct appreciation of the nature of the zeta zeros (Zeta 1 and Zeta 2) relates to this aspect.

The third phase relates to the growing dynamic interpenetration in experience of both the analytic and holistic aspects (which initially unfold in a relatively separate manner).

This final comprehensive understanding leads to the clear realisation of the vital conversion role of the zeta zeros in terms of the two aspects of the number system, so that both can be consistently interpreted in terms of each other.

Therefore put quite simply, the zeta zeros are now seen as playing a vital integral role with respect to the overall number system, thereby enabling perfect consistency with respect to the behaviour of number (in cardinal and ordinal terms).

This also is compatible now with an enhanced understanding where both the analytic and holistic appreciation of all mathematical symbols is kept in harmonious balance with respect to understanding.

Again, Conventional Mathematics attempts to understand the nature of the number system (which requires three distinct types of appreciation) in terms of just one.

And once more, this clearly cannot be achieved. So the real lesson that needs to be taken on board is that a radical overhaul of Mathematics is urgently required which will intimately affect previous interpretation with respect to just about everything!


In this new vision, number is no longer understood as abstract from human experience, but rather as the inherent basis of all phenomena (in both physical and psychological terms).

So number represents the fundamental encoding of the nature of all phenomena, which then becomes decoded through phenomenal evolution.

Though this is necessarily of a speculative nature, I would imagine that out there in our Universe are  highly evolved intelligent beings who long since have discovered this appreciation of the fundamental role of number as the inherent encoding of everything in creation.

And if indeed if communication with other highly evolved intelligent beings were to take place, it would surely be on the basis of number (which is truly most fundamental).

Dramatic changes will be required with respect to our customary relationship to the world.

Proper appreciation of the holistic aspect of number (through the zeta zeros) brings about the realisation that inherent in our number system is the capacity for perfect synchronous communication, which then inevitably becomes expressed through phenomenal creation.

So instead of viewing nature scientifically in an impersonal manner (suited to quantitative investigation) we will have to learn that a holistic capacity of communication (inevitably at work through all evolution) is intrinsic to everything in creation.

As to the question as to what is doing the vibrating with respect to the number system, the answer now is quite obvious! The number system itself through the zeta zeros does this vibrating. In other words quite simply the number system represents the central dynamic system that underlies everything else in the Universe.


As to the question as to where the zeta zeros have come from, I do believe that we can push back to yet another layer of investigation in the relationship of the system here described to the truly original numbers 1 and 0.

Though deeply inherent in the system that we have been discussing, there is scope to probe the relationships between the two. And then I believe it will become apparent that the natural numbers and zeta zeros are necessarily inherent in this prior original system.

However the idea that there is any final rational answer has to be abandoned. Indeed the very purpose of such probing is to fully reveal the unfathomable mystery underlying it all. And then with no further course of investigation we then can fully dissolve in its mystery.

Where Science and Art Coincide (20)

We will wrap up the present discussion with this entry.

As we have seen the proper understanding of number is inseparable from human development, which reflects the inherently dynamic interactive nature of experience.

And just as with the electromagnetic spectrum, we find many distinctive bands of radiation (of which natural light forms just one small band), likewise with the spectrum of development, there are many distinctive bands of which specialised linear understanding (that informs conventional mathematical interpretation) is just one.

So what is recognised as Mathematics in our culture really represents specialised analytic interpretation (of a quantitative kind).

In my overall model of the spectrum of development (comprising 7 bands) this simply represents understanding consistent with Band 2.

Bands 3 and 4 on this spectrum are then concerned with highly refined intuitive type understanding.

The specialised form of this understanding (consistent with Band 4) is then associated with specialised holistic interpretation (of a qualitative kind).

So all mathematical symbols can be given both a Type 1 (analytic) and Type 2 (holistic) interpretation. Thus in relative separation from each other, Type 1 is associated with Conventional Mathematics and Type  2 with Holistic Mathematics respectively.

However the most comprehensive type of understanding entails the combined interaction of both (specialised) analytic and holistic type understanding. Now, the mature attainment of this most developed from of appreciation is associated with Band 6 on the spectrum.


Likewise in mathematical terms therefore we can define Type 3 (comprehensive) mathematical interpretation as the refined dynamic interaction of both its analytic (quantitative) and holistic (qualitative) aspects.

And just as the human perfection in its most complete sense requires the harmonious development of both cognitive (rational), affective (emotional) and volitional (spiritual) capacities, equally this is true of  Mathematics itself when understood in its most comprehensive manner.

Indeed it is especially true of the nature of the number system.

What we sought to demonstrate in the last few blog entries is the remarkable fact that understanding of the Zeta 2 and Zeta 1 zeros are cognitive and affective with respect to each other. This implies that the very ability to properly appreciate the ultimate interdependence of these zeros requires the equal development of both rational and aesthetic abilities.

And the ultimate appreciation of the true mystery of the primes (and their relationship to the zeta zeros) is of a direct ineffable spiritual nature.

One of the great problems with modern Mathematics is the increasing tendency towards specialised understanding of a purely abstract rational nature.

Thus in approaching ever closer to its analytic extreme, recognition of the equal importance of its (unrecognised) holistic aspect is completely blotted out, so much so that I have never found it even possible to discuss such matters constructively with mathematicians.


This was not always the case. For example, with great mathematicians such as Gauss and Riemann, abstract ability would have been closely associated with a strong visual sense. Thus they would never have sought to divorce mathematical from corresponding physical understanding of the world.

Simple mathematical notions can often be given a visual representation that then conveys the (unrecognised) qualitative nature of its symbols.

For example the beautiful fractal images associated with Mandelbrot's set can be derived from a remarkably simple formula (entailing complex number iterations).

So though we may initially consider this formula mathematical  in a merely cognitive (rational) manner, its visual representation then appeals directly to the aesthetic sense (which is of an affective nature).

Thus a more comprehensive appreciation of the formula should properly combine both cognitive and affective appreciation.

At another level the golden ratio (golden mean) again is related to a very simple mathematical formula (with a rational explanation).

However geometrical appreciation of this ratio can be easily seen to likewise require an aesthetic ability.
In fact, properly understood the very equation (for which the ratio, phi is obtained) likewise requires both cognitive and affective appreciation.

And then directly associated with the Zeta 2 zeros are the visual representations (or mandalas) that Carl Jung considered so important as profound archetypes of spiritual integration.

For example the most common patterns Jung found to be related to 4 and 8 respectively.

Now we could attempt to try and explain all this in a merely rational analytic manner as the geometrical representation of the 4 and 8 roots of 1 respectively. However the point is that as Jung rightly observed, these images (or mandalas) now serve - not an analytic - but rather a holistic purpose.

This would of course also suggest that the Zeta 1 zeros, when properly appreciated, again should serve as powerful archetypes - indeed the most powerful of all - of an integral holistic meaning.


This integral meaning is therefore intimately contained in the zeros which then is unlocked through appropriate interpretation.

However the specialised analytic approach that dominates present Mathematics inevitably rules out such holistic interpretation.

So my point once again is that - when comprehensively understood - all mathematical symbols possess both analytic and holistic aspects. However we have to first recognise the equal importance of both aspects in a relatively separate manner, before finally combining each in a dynamic interactive fashion (without each aspect losing its special distinctiveness).


In the end true appreciation of the ultimate nature of the primes is inseparable from the eventual union of the (analytic) conscious and (holistic) unconscious in experience.

Now the involuntary nature of primitive type instincts always betrays a problem in terms of the successful marriage of conscious and unconscious aspects of personality.

Therefore from this perspective, the ultimate resolution of prime (i.e. primitive instinctive) behaviour in psychological terms is inseparable from the ultimate resolution of the nature of prime numbers (in a corresponding physical manner).

So if you want to properly know the nature of prime numbers (externally), you must likewise know the nature of thyself (internally) for both of these aspects are ultimately inseparable.

And herein lies the final message that needs to be clearly realised i.e. that notions of number - and indeed all mathematical notions - properly understood, are dynamically  inseparable from both the physical and psychological aspects of reality (which ultimately are fully complementary).

Where Science and Art Coincide (19)

As we have see, in dynamic interactive terms, the successful analytic differentiation of the primes in experience is ultimately inseparable from their corresponding successful integration in holistic terms.

And the former aspect corresponds with the Type 1 and the latter aspect with the Type 2 aspect of the number system respectively. Then the dynamic interpenetration of both aspects corresponds to the Type 3 aspect.

Thus the actual nature of the individual primes (as commonly understood in a quantitative analytic manner) is dynamically inseparable from the corresponding collective set of zeta zeros (as understood in a complementary qualitative holistic fashion).


However just as left and right turns have an arbitrary meaning at a crossroads (depending on the direction from which the crossroads is approached) likewise what is analytic and holistic with respect to the primes and zeta zeros respectively, likewise has an arbitrary distinction depending on the frame of reference adopted.

Thus in a reverse manner, each zeta zero can be understood in an individual and the primes in a collective nature respectively.
Then from this perspective, the successful analytic differentiation of each zeta zero in experience is ultimately inseparable from the corresponding integration of the primes in holistic terms.

This is where the two sets of zeta zeros come in!


What is holistic with respect to one set is analytic with respect to one and what is holistic with respect to one notion of the primes is analytic with respect to the other.

Therefore the tasks of successfully differentiating each prime number (in both Type 1 and Type 2 terms) and likewise integrating with respect to the zeta zeros (Type 1 and Type 2) and then equally differentiating each zeta zero (in Type 1 and Type terms) and integrating with respect to the prime numbers (Type 1 and Type 2) are ultimately totally interdependent in an ineffable manner.

The relative phenomenal aspects of both the primes (with respect to the natural numbers) in cardinal terms and the natural numbers (with respect to each prime) in ordinal terms, only becomes apparent when there is already a degree of separation evident with respect to this original ineffable interdependence.

Equally the phenomenal aspects of the zeros (with respect to the their collective sets) in both Zeta 1 and Zeta 2 terms thereby likewise only becomes apparent due to this degree of relative separation.

We then have with respect to the relationship between the primes and the natural numbers and as between the Zeta 1 and Zeta 2 zeros respectively, differing perspectives with an apparently unambiguous meaning within each (relatively) separate reference frame.

However when we combine these various reference frames together in an interdependent manner, all these interpretations are now seen as paradoxical.


When one appreciates the truly dynamic interactive nature of the number system, then it becomes futile to attempt to understand it coherently in the standard conventional mathematical manner.


We have been accustomed to look at numbers for several millennia now as static absolute entities.

However this view simply reflects the highly reduced - and thereby distorted - nature of the corresponding interpretation adopted.

Though this interpretation is indeed extremely useful within a limited quantitative perspective, ultimately it is completely lacking in overall coherence.

And the good news is that we can provide this much needed coherence in a uniquely distinctive manner that opens up limitless further possibilities in terms of the nature of the number system (and by extension Mathematics and all the Sciences).

And the limited quantitative aspect - that has become falsely synonymous with Mathematics - will still of course continue but now with the benefit of being seen as forming just one aspect of an altogether more comprehensive perspective.


From the limited quantitative standpoint, Mathematics is seen solely in scientific terms (as the indispensable tool for all the other Sciences).


However from the comprehensive developmental perspective, Mathematics, in its fullest expression as expression,  represents the culmination of successful development in cognitive (rational), affective (emotional) and spiritual (intuitive) terms.

In particular the very developmental process corresponding directly with the understanding of the Zeta 1 (Riemann) zeros is of an affective - rather than cognitive - nature.

Though these zeros can indeed be given a cognitive form - that indirectly corresponds with reason - in a direct sense their appreciation comes from artistic (aesthetic) rather than scientific (rational) development.

Thus the culmination in understanding of the great mystery of the primes is inseparable from appreciation of  the great mystery of life itself, both of which unfold through the human development process.

Indeed this mystery of the primes (at the heart of the number system) is directly central to the nature of  all phenomenal evolution.

In this sense - as Hilbert intuited - it serves as key issue that is the basis of everything we can know and understand.

Where Science and Art Coincide (18)

The transcendent (top-down) aspect of integration generally unfolds through the refined use of reason in disciplining - what may be perceived as - the "lower" senses (in the form of unconscious primitive impulses).

As we have seen the Zeta 2 zeros are directly associated with the holistic nature of such transcendent development.

However the immanent (bottom-up) aspect of integration unfolds in a reverse complementary manner in the refined use of emotional feeling through unconscious projection, that gradually empties the unconscious of all such involuntary primitive impulses.

And as we have seen the Zeta 1 zeros are likewise directly associated with such immanent development.

This means in effect in this partial relative context, that the Zeta 2 and Zeta 1 zeros are directly associated with both cognitive (rational) and affective (emotional) aspects of understanding respectively.


So we started in Type 1 (conventional mathematical) terms, by considering number interpretation in a solely rational manner.

Then in Type 2 (holistic mathematical) terms, we extended number interpretation to likewise include an intuitive - as well as rational - aspect.

However, now at the Type 3 (radial mathematical) level, we must include the additional fact that numbers reflect both cognitive and affective aspects in relationship to each other.

Thus if we look at the Zeta 2 zeros as representing the masculine (cognitive), then the Zeta 1 zeros thereby represent the complementary feminine affective principle.

Of course ultimately in Type 3 terms - where the interdependence of both Zeta 1 and Zeta 2 zeros is fully recognised - both sets will now be seen as equally reflecting cognitive and affective aspects.


As we have seen the Zeta  2 zeros provide the means (though the non-trivial roots of 1) of differentiating natural number ordinal positions (with respect to a given prime).

So for example for the prime number 3, (where 3 represents a group of individual members) we can thereby uniquely differentiate 1st, 2nd  and 3rd in this context. This is achieved in an indirect quantitative manner through the 3 roots of 1 (of which all except 1 are non-trivial)  However this equally implies that we can successfully integrate these 3 members in qualitative manner (expressed indirectly through the sum of the 3 roots = 0).


The Zeta 1 zeros then provide the corresponding means of differentiating primitive impulses (with respect to the confused composite elements of the unconscious).

Primitive impulses reflect the confused activity that results due to qualitative being directly identified with quantitative appreciation.

So the very process of resolving such confusion requires the successful (analytic) differentiation of the quantitative aspect equally combined with successful (holistic) integration at a qualitative level.

In an exactly similar manner, resolving the true nature of the prime numbers requires the successful (analytic) differentiation of each prime (in cardinal terms) equally combined with their successful collective (holistic) integration in a qualitative manner.

And once again the Zeta 1 zeros directly correspond with this latter holistic aspect of the primes (representing number dimensions).


However we can now quickly demonstrate the requirement for integration of both sets of zeros.

From the Zeta 2 perspective, we must assume the existence of each prime before differentiating its group members (in a natural number ordinal manner).

From the Zeta 2 perspective, we must assume the existence of each (composite) natural number (before differentiating its individual members as primes).

Thus the Zeta 1 and Zeta 2 zeros are mutually implied by each other.

This even helps to clarify a notable feature regarding my own experience of these developmental stages.


With respect to the Zeta 1 zeros, for many years, I was more concerned with the even numbered dimensions (especially 2, 4 and 8) than odd (except for 3).

This ultimately reflected an undue emphasis on the transcendent aspect of contemplative development. It is only much later I realised that this corresponding lack of immanent type development (with which the Zeta 1 zeros - and the successful differentiation and integration of the primes - are associated) was impeding my ability to appreciate the nature of the prime numbered dimensions (associated with the Zeta 1 zeros) .

In dynamic interactive terms, the (analytic) differentiation of the individual primes in cardinal terms is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 1 zeros can then be seen as relating directly to this latter collective integral aspect  (that perfectly shadows the individual nature of the primes).

Likewise from a dynamic interactive perspective, the (analytic) differentiation of the individual natural numbers in ordinal terms (i.e. the members of each prime grouping)  is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 2 zeros can then be seen as relating to this latter integral aspect aspect (that perfectly shadows the individual nature of each natural number).

So ultimately the primes and natural numbers are thereby seen as perfect mirrors of each other that are ultimately identical (i.e. fully interdependent) in an ineffable manner.

However the fact that the zeta zeros now mutually reflect both cognitive and affective aspects of interpretation raises the issue as to whether Mathematics reflects scientific or artistic appreciation.

We will return to this important issue in the next blog entry.

Where Science and Art Coincide (17)

There is a further unexpected surprise with respect to the interpretation of the Zeta 1 zeros.

However to appreciate this we must, as always, place our discussion in the dynamic interactive context of development.

I have contrasted before the nature of top-down development (where "lower" dimensions are integrated with respect to the "higher") and bottom-up development (where "higher" dimensions are integrated with respect to the "lower" ).

The former reflects the transcendent aspect of development (with which the Zeta 2 zeros are directly associated); the latter represents the corresponding immanent aspect (with which the Zeta 1 zeros are likewise directly associated).

The transcendent aspect, on the one hand, represents the attempt to achieve the "higher" integration of conscious development (which implies the unfolding of an increasingly important intuitive element).

This ultimately continues until the stage where conscious awareness is so transparent and dynamically interactive, that phenomena no longer appear to arise in experience. This seamless conscious activity can then be integrated with an ever-present pure intuitive awareness (of an unconscious kind).

So this represents the transcendent goal of spiritual development.


In practice severe limitations are likely to affect the attainment of this goal.

Higher level transcendent development is often based on a mistaken hierarchy, where the cognitive activity of reason is ranked as superior to that of affective sense, with pure spirit seen as the apex of the hierarchy.

This in the attempt to control the "lower" senses (in the pursuit of purer contemplative awareness), reason in effect is used to a degree to repress such activity (especially at a primitive unconscious level).

Therefore without this problem being properly addressed, the transcendent goal itself cannot be attained.


So there is a complementary immanent approach to development which inverts this hierarchy of reason as superior to the senses.

Whereas the transcendent aspect is based on the attempted "higher" integration of consciousness, the immanent aspect, in an inverse complementary manner, is based on the corresponding "lower" differentiation of the primitive unconscious.

Put another way, whereas  the transcendent aspect represents making the conscious increasingly unconscious, so that all its linear type rigidity is ultimately eroded, the immanent aspect represents the making of the unconscious increasingly conscious in experience, so that all the primitive instincts of the - initially unrecognised - shadow are ultimately fully brought into the conscious light.


Thus once again, the integration of both the (analytic) conscious and the (holistic) unconscious in development, requires this approach from two opposite directions. Thus from the transcendent aspect, the conscious is made fully compatible with respect to the life of the unconscious; from the immanent perspective, in reverse manner, the unconscious is made fully compatible with respect to corresponding conscious activity.

So it is exactly similar with respect to the nature of the number system.

The Zeta 2 zeros reflect this former aspect of integration, where analytic appreciation of the number system (in a quantitative manner) ultimately is made fully compatible with corresponding holistic appreciation (of a qualitative nature). So we move from a linear (1-dimensional) notion of the quantitative nature of number to an increasingly circular (higher dimensional) notion of its qualitative nature.

And this higher dimensional appreciation is directly provided through the holistic interpretation of the Zeta 2 zeros.


The Zeta 1 zeros however reflect the latter aspect of integration where initial confused unconscious interpretation of the number system (in a qualitative manner)  ultimately is made fully compatible with corresponding analytic appreciation (of a quantitative nature). So here we move from circular confusion to an increasingly linear (1-dimensional) appreciation i.e. where qualitative appreciation can be fully compatible with linear quantitative interpretation.

So in one case (Zeta 2 zeros) we move from the linear to the mature circular i.e. where quantitative is made fully compatible with the qualitative understanding of number; in the latter case (Zeta 1 zeros) we move from the (confused) circular to mature linear i.e. where in reverse manner the qualitative is made fully compatible with the quantitative nature of number.

This indeed is the very reason why all the Zeta 1 zeros are compelled to lie on a straight line!

These zeros in fact thereby represent the fully differentiated appreciation of the unconscious i.e. holistic nature of the natural number system.

So we have differentiated conscious interpretation in our customary quantitative interpretation of the natural numbers as represented by the real number line.

We then at the other extreme of understanding, we have differentiated unconscious interpretation in this new (unrecognised) appreciation of all the Zeta 1 zeros, as represented by the imaginary number line. And remember in holistic terms, the imaginary notion is used to express indirectly what is unconscious in origin!

Now initially when the primitive impulses of the shadow unconscious are projected into experience they are directly confused with phenomena of a quantitative nature. However, all going well, through a growing holistic qualitative awareness, inappropriate rigid  attachment to such quantitative phenomena is gradually eroded. Success in this regard thereby enables one to confront the shadow in an ever-deeper manner with new (formerly unrecognised) unconscious elements continually brought to light.

In this way ultimately all such shadow elements can (in principle) be successfully differentiated.

So - quite literally - all primitive (i.e. prime) elements that have been successfully differentiated in an analytic manner, are thereby now equally successfully integrated  in a corresponding holistic fashion.


Remarkably it is exactly similar in dynamic interactive terms with respect to the nature of prime numbers.

Thus the very differentiation of the prime numbers (in a quantitative analytic manner) is ultimately inseparable from their corresponding integration (in qualitative holistic terms).

In other words the Zeta 1 zeros represent the holistic integrative aspect with respect to the prime numbers.

Thus the very ability of the primes to maintain their unique status (as independent building blocks of the natural number system in cardinal terms) is inseparable from the corresponding unique status of the Zeta 1 zeros (which seamlessly maintain the overall interdependent nature of the primes with respect to the natural number system).

And this seamless integration is then demonstrated from a different perspective by the Zeta 2 zeros which now serve as the unique independent build blocks of the ordinal number system.

Thus ultimately the Zeta 1 and the Zeta 2 zeros are themselves totally interdependent.


However what is especially interesting is that the relationship of both sets of zeros is as cognitive to affective (and affective as to cognitive) respectively.

Thus if we initially fix the Zeta 2 zeros with customary cognitive aspect of understanding, then the Zeta 1 correspond with the affective aspect.

Now this will initially appear very strange as up to now, Mathematics - even in ts dynamic interactive  operation - has been viewed as a cognitive (rational) type discipline.

So we will address this important issue further in the next blog entry.

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.