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.