Zeta 1 Zeros - Key Significance (6)

We have seen that - relative to the quantitative base notion of number (in Type 1 terms) - that the Zeta 1 (Riemann zeros) relate to the qualitative measurement of the prime factor interdependence of the natural numbers, which is indirectly expressed in Type 2 terms (where the linear measurement of 1 unit now represents the radius of the unit circle).

Now the imaginary line simply represents the indirect attempt to express this circular measurement - suited to the holistic interpretation of factor interdependence - in a quantitative linear manner.

And as the circumference of the unit circle, stretched out as a line = 2π, Therefore to convert to standard linear units we divide by 2π.

In this way we are able to link up the imaginary part of accumulated total of Zeta 1 zeros to t with the corresponding total of divisors i.e. all factors (except 1) of the number factors to n, where n = t/2π. Again it is important to note that we are including composite natural numbers (as well as primes as factors here). So to illustrate, from this standpoint, the factors of 12 (except 1) are 2, 3, 4, 6 and 12! 

The total sum of Zeta 1 zeros to t = 100 is 29. So n = 100/2π = 15.915.

Now the total sum of factors (except 1) to 15 = 29 (and to 16 = 33). So we already can see a close relationship as between the frequency of the Zeta 1 zeros and the corresponding accumulated frequency of the factors of numbers. 


However, we will attempt here to probe in more detail the precise significance of the Zeta 1 zeros.

It is important to bear in mind - in this dynamic interactive context - that we must keep balancing  two aspects with respect  to the understanding of the primes that are quantitative (analytic) and qualitative (holistic) with respect to each other.

So again we start in analytic terms with the appreciation of the primes as unique independent "building blocks" of the natural number system in a quantitative manner. Though in conventional mathematical terms, the primes are interpreted individually in a reduced absolute manner, strictly in more correct dynamic terms, understanding of this aspect of the primes is of a relatively independent nature that can approach but never fully attain an absolute meaning.

However, then in complementary holistic terms, we have the corresponding appreciation of the primes as comprising unique factor relationships of collective interdependence with the natural numbers in a truly qualitative manner.

The Zeta 1 (Riemann) zeros then express this qualitative relationship - pertaining to the natural numbers as  expressive of unique factor combinations of the primes - indirectly in a quantitative manner.

However the correct appreciation of the nature of these zeros lies at the complementary extreme to that of the primes (as quantitative "building blocks").

In fact, true appreciation of the Zeta 1 zeros, in dynamic interactive terms, approaches a state - though necessarily  never quite fully - of pure relativity (which in a complete manner would represent total ineffability).

Expressed in an equivalent fashion, whereas the analytic aspect approaches the (conscious) rational extreme with respect to understanding, the corresponding holistic aspect represents the (unconscious) intuitive extreme.

Yet again, whereas the analytic aspect (with respect to the individual primes) approaches an extreme where numbers seemingly represent absolute forms, the holistic aspect approaches the opposite extreme (with respect to the Zeta 1 zeros) where numbers now seemingly represent pure energy states!


Therefore, true appropriate understanding of the relationship as between the (independent) primes and the (interdependent) Zeta 1 zeros, from a psychological developmental perspective, requires the full integration of both conscious (rational) and unconscious (intuitive) processes of mathematical understanding.

And let us once more remind ourselves at this point, that because Conventional Mathematics solely recognises the (conscious) analytic aspect in a reduced manner, that it thereby blots out the completely the (unconscious) holistic aspect - at least in formal terms - thereby providing a fundamentally distorted interpretation of all its relationships (especially with respect to number)!

It may help towards the realisation of the true holistic nature of the Zeta 1 zeros to bear in mind the crossroads example, that I have referred to on so many previous occasions.

In this context left and right turns have a clear unambiguous meaning when we use a one-directional (i.e. one-dimensional) frame of reference. So if one approaches the crossroads moving N or S (as single separate directions) left and right turns can be given an unambiguous identity.

However when one simultaneously attempts to combine both directions (N and S), what is a left or a right turn is rendered paradoxical, for what is left from one direction is right from the other, and vice versa.

So we move from a (linear) either/or logic which befits unambiguous analytic, to a (circular) both/and logic that now befits paradoxical holistic appreciation.

It is precisely similar with respect to the relationship as between the primes and natural numbers, the understanding of which can be approached in both external and internal terms.

Thus in conventional mathematical terms, one concentrates on the external relationship as between the primes and natural numbers where the primes are viewed as the independent "building blocks" of the natural numbers in a unique quantitative manner.

However one can also concentrate on the internal relationship as between the interdependent prime factors of each natural number also in a quantitative manner.

So for example one can approximate the external frequency of primes up to n by the simple relationship n/log n.

One can also approximate the internal ratio of (distinct) prime to natural number factors of a number by the simple relationship n₁/log n₁, where n₁ = log n.

So we have two distinct reference frames here (i.e. externally in a collective manner and internally in an individual manner respectively) for viewing the relationship as between the primes and natural numbers.

In each case, we appear to be able to define the nature of this relationship in an unambiguous quantitative manner.

However just like N and S served as complementary opposite directions in our crossroads example, both external and internal serve as complementary opposite directions with respect to the relationship as between the primes and natural numbers.

Therefore, when we simultaneously combine both external and internal directions, the relationship as between the primes and natural numbers (and natural numbers and primes) is rendered paradoxical in a circular manner.

And the Zeta 1 zeros directly represent such paradoxical understanding, which is directly of a qualitative holistic nature (though indirectly represented in a quantitative manner).

There are indeed several equivalent ways in which we can express this paradox.

So from one perspective, each point (denoting a Zeta 1 zero) represents a state where prime and natural number notions become identical (and thereby distinguishable from each other).

So just as at the crossroads - when simultaneously approached from two directions N and S - left and right become identical with each other (for what is left from one direction is right from the other), likewise at each point (denoting a Zeta 1 zero) prime and natural number notions are identical (for what is caused by the primes from one direction e.g. external, is caused by the natural numbers from the other and vice versa).

Thus the Zeta 1 zeros serve as two-way reflecting mirrors where the primes and natural numbers become identical with each other. Now this cannot be understood directly in a dualistic analytic manner (through reason) but rather in a nondual holistic manner (through intuition).

Then when one indirectly tries to convey its meaning in a rational analytic fashion, its truth is revealed as circular (i.e. paradoxical).

We could also refer to each Zeta 1 zero as a point where notions of randomness and order with respect to the number system are identical. In fact from a dynamic interactive perspective, we can only define notions of number randomness against an implicit background of number order and vice versa. 

Thus the apparent randomness of primes, as viewed in an independent individual manner from the external quantitative perspective, appears as remarkable order when viewed internally from an interdependent qualitative perspective (as prime factors) and vice versa.

Thus randomness and order from a dynamic interactive perspective are revealed to be clearly two sides of the same coin.

We can also refer to each Zeta 1 zero as a point where the qualitative notion of interdependence becomes inseparable from the quantitative notion of independence.

This entails that true appreciation of the Zeta 1 zeros requires the intimate balancing of both highly refined reason and highly refined intuition respectively.

Finally, we can say that each Zeta 1 zero represents a point where the operation of addition is indistinguishable from that of multiplication (which follows from the recognition that they are - relatively - quantitative and qualitative with respect to each other)

Zeta 1 Zeros - Key Significance (5)

As we have seen, the Zeta 1 (Riemann) zeros serve as an indirect quantitative means of expressing the holistic qualitative aspect of the primes (through their role as unique factors of the natural numbers).

And again this can only be properly viewed in a dynamic interactive manner.

So from one perspective, we have the quantitative aspect of the primes as the relatively independent "building blocks" of the natural numbers in cardinal terms.

Then from the other complementary perspective, we have the qualitative aspect of the primes - represented by the Zeta 1 zeros - as the relatively interdependent relationship of the unique factors of the natural numbers in ordinal terms.

Thus from one valid perspective, the Zeta 1 zeros represent, in an indirect quantitative manner, the ordinal nature of the primes i.e. in the way that they maintain a collective order with respect to the natural number system, just as the Zeta 2 zeros represent in complementary fashion, the ordinal nature of the natural numbers, with respect to each individual prime number (represented indirectly as the unique prime roots of 1).

However we get a even closer idea of the nature of the Zeta 1 zeros by focusing in on the divisors (i.e. natural number factors) of each successive member of the number system.

And notice once again the complementarity involved here! Firstly, the natural numbers complement the primes and then secondly the factors of each natural number (as considered internally with respect to each individual number) complement the natural numbers (as considered externally with respect to the collective number system).

Therefore once again, if we wish to find the true complement of the primes - as considered externally with respect collectively to the overall number system - then we must look to the natural number factors (as considered internally with respect to each individual number).

And just as in the Zeta 1 case, we ignore 1 (as the non-unique root which must always necessarily arise when we take the prime roots of a number), likewise with respect to the consideration of factors we ignore 1 (which is necessarily a factor of all numbers). However, we do in this case always include n (as a factor of n) as this number is necessarily a factor of all numbers. So for example though 24 is clearly a factor of 24, it is not however a factor of the next natural number i.e. 25 (though 1 is necessarily a non-unique factor of both numbers).

A simple formula then exists to approximate the average number of divisors (i.e. natural number factors) of a given number n.

In fact this internal measurement with respect to the individual factor composition of a number can be simply given as log n; this again shows direct complementarity with a similar approximation for the average gap (or distance) between each prime (as considered externally with respect to the number system as a whole) which is also log n.

The deeper implication of this is that these two features are ultimately of a purely relative nature.

So from one perspective, the internal behaviour of factors (within each individual number) appears to be determined by the external behaviour of the primes (collectively with respect to the number system); however from the equally valid opposite perspective, the external behaviour of the primes appears to be determined by the internal behaviour of factors!

So again like left and right turns at a crossroads, the behaviour in each case is revealed to be of a merely relative nature, which points directly to the truly synchronous identity of primes and natural numbers (in an ultimately ineffable manner).  

So this dynamic interactive manner of looking at the number system immediately opens up the way in which important complementary features characterise the intrinsic behaviour of number that cannot be properly recognised from within the customary analytic approach (of a static absolute nature).


Returning to the average frequency of divisors (i.e. as natural number factors) of the number n, Dirichlet proved in 1838 that this approached log n – 1 + 2γ (where γ is the Euler-Mascheroni constant = .5772...).

This would therefore work out at log n + .1544...

 Then, there is the issue as to whether one includes all factors (including 1 and n) in this calculation.

So accepting that the original result is based on the inclusion of all factors as divisors, we would then subtract 1 from this result to equate with the definition of factors that I have adopted.

Therefore the result could now be given as log n – 1 + .1544.

 Then to calculate the total number of factors up to n we would multiply by n.

This would give a slightly exaggerated result (i.e. too large) as earlier numbers < n would not contain the same number of  average factors. However given that log n changes very slowly (especially when n is large) and that the average number of factors is slightly greater than log n – 1, we can therefore take 
n (log n – 1) as a good approximation of the total number of natural number factors (up to n).

When we then look at the corresponding formula for the calculation of Zeta 1 (i.e. Riemann non-trivial) zeros,
i.e. t/2π(log t/2π – 1) = t/2π(log t/2π) – t/2π, it bears a very close similarity.

In fact it is the same formula where n =  t/2π.

In other words, when one obtains the total of Zeta 1 zeros up to t, it bears a remarkably close similarity with the corresponding total of the combined natural number factors up to n (where n =  t/2π).

In fact, the Zeta 1 zeros can be easily seen to represent the measurement (on a circular scale) of the corresponding natural number factors (on a  real linear scale).

Imagine for example 1 as a single unit on the real number line. If we now draw a circle (using this line as radius) the corresponding length of the circumference that will be traced out = 2π. Therefore to convert such circular units to corresponding linear format we  divide by 2π.

Now it must be remembered that - relative to base numbers (in Type 1 terms), that the factors of numbers relate to the dimensional aspect of number (in a Type 2 manner). Therefore the relationship between both - in dynamic interactive terms - is analytic to holistic or alternatively linear as to circular.

Therefore in this context, the factors of numbers rightly conform to a circular rather than linear frame of reference!

We then further saw in holistic mathematical terms that to represent what is inherently of a circular i.e. holistic nature, indirectly in a linear (i.e. analytic) manner one uses an imaginary rather than real scale.

And this is precisely why the Zeta 1 (non-trivial) zeros lie on an imaginary line, as this represents an indirect analytic mode of representing information that inherently should be understood in a holistic manner!

In other words, the Zeta 1 zeros express - indirectly in a quantitative manner - the (hidden) qualitative nature of the primes through their collective interdependence (as unique factors) of the natural numbers. And this qualitative aspect of the primes is thereby directly complementary with their corresponding quantitative nature as the independent "building blocks" of each individual natural number. 

And what should be clear now is that this two-way relationship of the primes to the natural numbers (in quantitative and qualitative terms) can only be properly understood in a dynamic interactive manner that combines notions of both relative independence and relative interdependence respectively.    

Zeta 1 Zeros - Key Significance (4)

Riemann showed how to extend the existing Euler zeta function ζ(s), defined for real values of s > 1 to every value (except for s = 1) in the complex plane.

This was indeed a truly important achievement, which greatly enhanced existing knowledge of the primes.

However a crucially significant further development is now required, enabling understanding of the Riemann zeta function to take place in a dynamic interactive holistic manner entailing twin complementary aspects that are analytic (quantitative) and holistic (qualitative) with respect to each other.

Now it takes some considerable training in a very distinctive type of mathematical understanding before one can begin achieve familiarity with holistic notions.

For example - as we have seen - all the primes and natural numbers - can be given a holistic as well as  he accepted analytic type interpretation.

For example, "1" is especially important in this regard where it now relates to unambiguous type understanding that is conducted with respect to - literally - 1 pole of reference.

As we have seen external (objective) and internal (subjective) constitute the first of the two sets of fundamental poles, within which all phenomenal understanding - including of course mathematical - takes place in a necessarily dynamic interactive manner. So strictly we can have no objective reality in experience abstracted from corresponding subjective mental interpretation (which are dynamically relative to each other).

However, conventional mathematical interpretation is based on absolute identification with just one pole. So mathematical symbols and relationships are given an absolute type validity (as if separate from interpretation). Alternative interpretation in the form of mathematical constructs is likewise given an absolute type validity. So the underlying assumption is that both interpretation and objective reality can be in absolute correspondence with each other, which strictly speaking is an untenable position (which requires the reduction of one pole in terms of the other).

Perhaps even more tellingly, mathematical symbols and relationships are equally given an absolute type in a merely independent quantitative manner (with the corresponding qualitative pole invariably reduced in merely quantitative terms). So the whole notion of "interdependence" in this conventional approach is crucially distorted. Though directly of a qualitative nature, it can only be given - again -  a reduced meaning in a quantitative manner.

And of course both the objective and quantitative aspects are considered in conventional terms as - once more - in absolute correspondence with each other (which requires reducing one pole in terms of the other)

So this reveals the extremely important fact, that in holistic mathematical terms Conventional Mathematics is formally defined by its 1-dimensional approach.

This thereby gives the (misleading) appearance of an absolute mathematical world of fixed symbols and relationships, whereas in truth all such symbols and relationships are conditioned by fundamental  complementary poles, that dynamically interact in two-way fashion with each other.

Even this on its own has a huge implication for true appreciation of the Riemann zeta function and its associated Riemann Hypothesis).

From the accepted analytic (quantitative) perspective, the one point at which the Riemann zeta function remains undefined is for s (where s represents a dimensional number or exponent) = 1.

Then from the unrecognised holistic (qualitative) perspective, the one point for which the Riemann zeta function remains undefined is alsofor s = 1. But 1 in this holistic context refers to the linear rational (i.e. 1-dimensional) approach that formally characterises all accepted Mathematics.

And quite simply, neither the Riemann zeta function, nor its associated Riemann Hypothesis can be properly defined in this rigid absolute manner, for in truth the number system is inherently of a dynamic interactive nature (with complementary quantitative and qualitative aspects).


Now of the "higher" natural number dimensions, 2 and 4 are especially important (which indicate how the two sets of fundamental polarities are related to each other in a holistic mathematical manner.

All other dimensions can then be seen as representing unique dynamic configurations with respect to the the two fundamental sets (i.e. external/internal and whole/part. 

As we have seen "2" in this holistic mathematical context represents the interdependence of two units, which can be viewed as + 1 and – 1 (in dynamic relation to each other). This implies that + and – signs keep switching, as polar reference frames likewise switch in experience.

In analytic terms, + carries the connotation of positive , whereas –  carries the connotation of negative (when attached to a number) and both of these exclude each other in an absolute manner.

In holistic terms, + carries the connotation of "to posit" (which simply means to make known in a conscious manner);  
Therefore from a holistic mathematical perspective, all conventional mathematical interpretation is based solely on the + sign (i.e. explicitly in a conscious manner).
 
By contrast, again in holistic terms, – carries the connotation to "negate" (which entails the reverse direction of the unconscious) i.e. through dynamically negating in a psychological manner, what has already been posited in conscious terms.

Therefore in dynamic terms, both conscious and unconscious operate in a complementary dynamic manner.

One differentiates in experience through positing (+) in a conscious manner; one then integrates in experience, through negating (–) in an unconscious manner. 

So in analytic (conscious) terms, both + and – are understood as distinctly separate from each other as is the case with number in Mathematics.

However in holistic (unconscious) terms, both + and – are understood as fully united with each other (as complementary opposites).

So once more as in our crossroads example, when approaches it from just one direction (either N or S) left and right turns and thereby positive e.g left (+ ) and negative i.e. not-left, or right (–) are clearly separate from each other.

However, when one simultaneously views the approach to the crossroads from two directions (both N and S), left and right turns are rendered paradoxical. So both left and right turns (and thereby + and –) are now understood as fully complementary with each other.

So the former understanding relates to analytic (conscious) appreciation in a (linear) rational manner; the latter understanding relates to holistic (unconscious) appreciation in a (circular) intuitive fashion. 

Interestingly, the 2 roots of 1 yield + 1 and – 1. So again in standard analytic terms these are understood as separate; however in true holistic terms these are now understood as forming a unified pair (as fully interdependent with each other).   


The imaginary notion of the square root of – 1 i.e. i also has a very important holistic explanation.

As we have seen, in dynamic interactive terms the process of (unconscious) negation, requires that conscious phenomena be already posited in experience. In fact this is very similar to the situation at a sub-atomic  level ,where an anti-matter particle is brought into contact with a corresponding matter particle.
 that 
So we have an immediate fusion here in the form of physical energy. In like manner the unconscious operates through the recognition in experience of a corresponding negative to the positive direction associated with conscious phenomena. So when one realises that the posited world of object phenomena is in relationship with a subjective self, then the negative direction is brought into play. And depending on the clarity of such recognition, a certain fusion (of both polarities) takes place in the form of psycho spiritual energy (i.e. intuition).

As we have seen, intuition - which is directly qualitative in nature - belongs to a distinct holistic realm of experience that should not be confused with what is rational and analytic.

However it is then possible to indirectly express this holistic aspect in an analytic manner (whereby its distinct  nature is preserved. And this is done through the imaginary notion.   

So form a holistic perspective, the imaginary represents an indirect analytic means of expressing the qualitative aspect (relating to the unconscious appreciation of interdependence).

In this way, both real and imaginary objects (such as numbers) can be treated as independent. However the independence associated with the imaginary in quantitative terms, represents but an indirect means of expressing the true interdependence of these objects (in a qualitative manner). 

So to express the unconscious negative (–)  direction - which in dynamic terms entails a necessary fusion with the corresponding positive as intuitive energy - indirectly in 1-dimensional fashion (as merely positive), we - literally - take its square root .

We have now an exact corresponding holistic interpretation of the imaginary notion (representing the square root of  – 1). 

The analytic (quantitative)  and holistic (qualitative) aspects of experience are thereby themselves real and imaginary with respect to each other.

Though of course both real and imaginary numbers are now used in conventional mathematical terms, from a qualitative perspective, only the real (i.e. quantitative) aspect is recognised.

However a comprehensive approach to Mathematics must include real and imaginary aspects in both quantitative and qualitative terms. And this requires recognition of the neglected holistic aspect, which is totally excluded from consideration in formal terms.

So the four roots of 1, + 1, – 1, + i and  – i, as the linear (1-dimensional) "conversion" of the 4 dimensions, are extremely significant, as they bring together the holistic relationship of the two fundamental sets of polar opposites (that condition all phenomenal experience).

So external and internal are - relatively - positive (+) and negative (–) with respect to each other.
Likewise quantitative and qualitative are real and imaginary with respect to each other.

And real and imaginary aspects, have both positive and negative directions associated with them.

So we can posit objects, including mathematical, in both real terms (as directly conscious) or in imaginary terms as indirectly conscious (representing the unconscious holistic aspect of experience).

And then we can likewise negate these objects in both real and imaginary terms as we move towards a purer holistic (unconscious intuitive) experience.

Zeta 1 Zeros - Key Significance (3)

We have seen - when properly interpreted - that the multiplication of numbers always causes a qualitative - as well as quantitative - change in the nature of the relationship.

And appreciation of this is vital in terms of understanding the true (external) relationship as between the primes and natural numbers (and natural numbers and primes).

So again from the conventional mathematical perspective, it is customary to treat the primes as independent quantitative entities (in a cardinal manner).

However once we uniquely combine the primes - as factors of a composite natural number - the very status of the primes change whereby (in the context of this new number) they attain a qualitative interdependent identity.

So again the primes can now seen clearly to possess two complementary aspects (in external terms), of relative independence (in quantitative terms) and relative interdependence (in a qualitative manner). And both of these aspects occur in a dynamic interactive context.

So within such a dynamic context, it is strictly meaningless to insist in absolute fashion on the primes as constituting the "building blocks" of the natural number system.

Rather we now have two complementary perspectives with a strictly relative validity.

Thus again from one relative perspective, the primes do indeed appear to constitute the "building blocks" of the natural numbers (in a quantitative manner).

However from the complementary relative perspective, the ordered nature of the primes is obtained  through their relationship with the natural numbers (in a qualitative manner).

Thus if one from one perspective, the (independent) primes appear to "cause" the natural numbers, From the other equally valid alternative perspective, the primes appear to be "caused" by the natural numbers (through the unique interdependent relationships of prime factors).

The big barrier to ready appreciation of this key point - which given the appropriate dynamic perspective  is somewhat obvious -  is the deeply ingrained notion that numbers (especially the primes and natural numbers) somehow possess an unchanging absolute identity!

However, as I have repeatedly stated, this simply represents the result of a significantly reduced interpretation of number (i.e. where in every context the qualitative aspect - which in truth is equally important - is reduced in a merely quantitative manner).

When one gets used to looking at this issue in a truly dynamic interactive manner, again as I say, it soon becomes pretty obvious that the primes and natural numbers are two sides of the same coin (with no strict identity apart from each other).


We saw, when dealing with the Zeta 1 zeros, that natural number ordinal notions have a truly relative identity.

Indeed we already accept this in many different situations. For instance  one can readily appreciate that the achievement of coming in 1st in a one-horse race is somewhat different from that coming in 1st in a race with 40 horses starting. So 1st (in the context of 1) is very different from 1st (in the context of 40). And as we have seen this relativity of ordinal notions is expressed through the Type 2 number system, where an invariant base number of 1 is raised to different numbers, representing dimensions (or powers).

So the two notions of 1st above can be given as 1^1/1 and 1^1/40 respectively.

However, what might seem initially surprising is that the prime number cardinal notions likewise have a truly relative identity.

So again for example, let us take the prime number "2" to illustrate.

We can start by giving this a relatively independent quantitative identity as a "building block" that can be used in the generation of (composite) natural numbers.

However, when 2 is then used again with itself or other primes (representing the unique product of prime factors) it assumes a relatively interdependent identity (of a qualitative nature) in that context.

So 2 for example (as a factor of 4) has thereby a relatively distinct identity from 2 (as a factor of 6).

Put another way 2 is uniquely reflected by 4 in the first case, while it is then uniquely reflected by 6 in the second.


In fact the situation here is analogous to the physical world.

For example we could start with hydrogen and oxygen atoms (in a relatively independent state).

However when a unique chemical combination of the hydrogen and oxygen take place (entailing two hydrogen atoms and 1 oxygen) we get a qualitative transformation through the generation of what we commonly recognise as water. Therefore though one could maintain that the two hydrogen  atoms in a water molecule are quantitatively the same as two atoms kept in a flask, clearly a qualitative transformation takes place in conjunction with each oxygen atom.

So it is very similar with the prime numbers. Therefore, even though 2 (as an independent prime "building block" in quantitative terms), might appear the same as 2 (now uniquely representing one of the factors of a composite natural number), a qualitative transformation - relating to this unique interdependence of factors - is clearly involved.

And of course this is equally true of every prime that attains a unique qualitative resonance (when used in the factor composition of a natural number).

In fact, in recent years the term "music of the primes" has been frequently used as a metaphor to embrace the wonderful ordered nature of the primes (when viewed in a collective manner).

And this directly suggests their qualitative - rather than strict quantitative - nature.

However from a mathematical perspective, there is no recognition whatsoever of this holistic qualitative pattern, with a solely analytic interpretation conducted within a reduced - merely quantitative - framework.

And this all serves as an essential preliminary in understanding the true role of the Zeta 1 (i.e. non-trivial Riemann) zeros.

For the role of these zeros is essentially - as with the Zeta 2 zeros, though now in relation to the external aspect of the number system - to convert the holistic qualitative nature of the primes (as unique factors of natural numbers) indirectly in a consistent quantitative manner.

Zeta 1 Zeros - Key Significance (2)

As often repeated, I sensed from a very early age that there was something seriously lacking in the conventional mathematical treatment of multiplication.
I remember in primary school - at about the age of 10 - dealing with the areas of rectangular fields in a class on arithmetic.

The term "acre" is widely used in this regard and we were informed in class that 1 acre = 4840 sq. yards.

I remember thinking to myself then that a field of length 80 yards and width 60 yards, would thereby have an area very close to 1 acre (i.e. 4800 square yards).

But then I realised something much more significant. The area of this field (of 80 * 60 yards) relates to square (i.e. 2-dimensional) units.

However, when we express the product of 80 * 60, the answer is conventionally given in linear (i.e. 1-dimensional) units. So I could already see that there was something seriously lacking with such conventional practice.

And this realisation proved far from a passing concern at the time as I tried to come to terms - literally - with the missing dimension of multiplication.

In particular, I wondered for a long time regarding the simplest case of the multiplication of 1 by 1.

Here, no change takes place in quantitative terms. But a qualitative change takes place in the nature of the units (which are now 2-dimensional).

And I was greatly puzzled at what happens when one then obtains the square root, for now two answers are seemingly valid i.e. + 1 and – 1. 

It was only later when seriously engaged with philosophy at University, that I could provide the deeper qualitative explanation for such strange mathematical behaviour.

From a qualitative perspective - and I am now using mathematical symbols in a holistic rather than analytic manner - Conventional Mathematics is characterised by a strictly linear (i.e. 1-dimensional) approach in rational terms.

What this implies is that the dynamic interactivity of complementary opposite poles - that necessarily characterise all mathematical relationships - is ignored with relationships, in every context, reduced in an absolute quantitative manner.

This is very true with respect to the multiplication of numbers e.g. 2 * 3.
We can easily see - especially where 2 and 3 relate to concrete type measurements - that a switch takes place - through the operation of multiplication - from 1-dimensional inputs (with respect to the individual  two numbers) to a 2-dimensional output (with respect to the collective result).

So both a quantitative change with respect to the starting inputs and a qualitative change with respect to the nature of the dimensons is involved (through this operation of multiplication).

However in conventional mathematical terms, a merely reduced quantitative interpretation of the result is given. Through the operation of multiplication the result of 2 * 3, represents 2-dimensional (rather than 1-dimensional units). However it is conventionally given in a merely reduced 1-dimensional manner. So the resulting number 6 is treated as another point on the number line! 


Now what complicates the matter somewhat here is that we likewise use a number to refer to the dimensional (as well as the base) units!

So in base terms 2 * 3 involves a transformation in quantitative terms;

And the change from 1 to 2 (with respect to dimensions) relatively involves a transformation in qualitative terms.

But 1 and 2 - as dimensions have a quantitative meaning in analytic terms. So mathematicians for example are quite happy to work abstractly in n dimensions (from a quantitative perspective).

However the true holistic meaning of dimension indirectly arises when we attempt to express its meaning in the standard 1-dimensional manner (associated with conventional linear logic).

So in the case of 1 * 1 = 1, each side can be expressed as x² = 1, with x = + 1 and – 1 respectively.

This entails, from the holistic mathematical perspective, that when one tries to subdivide the inherent interdependent nature of 2 as 2-dimensional (representing the complete integration of opposite poles in an intuitive manner as a psycho-spiritual energy state) that it appears paradoxical when then expressed in a 1-dimensional manner (where units are now viewed as completely independent).

Once again, we illustrated all this in relation the crossroads example. So with a single independent frame of reference when approaching a crossroads (from either a N or S direction), we can identify left and right turns in an unambiguous manner.

So if for example (heading N) a left turn at the crossroads is designated as + 1, then this can be unambiguously separated from the designation of – 1 (as a right turn). Likewise if (heading S) a right turn is designated as + 1, this can be unambiguously separated from the designation of   – 1 (as a left turn).

So  within each isolated pole of reference we have unambiguous (1-dimensional)  interpretation in an analytic manner.

However, when we simultaneously view the approach to the crossroads, from both N and S directions, we have complete paradox in terms of former designations. For what was + 1 (heading N) is  – 1 heading S; and also what is – 1 (heading N) is + 1 (heading S).
Equally what was + 1 (heading S) is – 1 (heading N); and what was – 1 (heading S) is + 1 (heading N). 

So this latter manner of "seeing" properly represents 2-dimensional appreciation in a holistic manner (which then is rendered paradoxical when indirectly interpreted in 1-dimensional terms).


And what is not all appreciated in conventional mathematical terms, is that the various roots of 1 (with an inherently circular relationship of interdependence with respect to each other), indirectly represent the true holistic significance of mathematical dimensions!

However whereas at the 1-dimensional level (of analytic reason), these roots appear as separate (in an either/or manner), at the direct 2-dimensional level (of holistic intuition), they now appear as fully united (in a both/and fashion).

Therefore with respect to the meaning of 2-dimensions, these appear as independent (either + 1 or  – 1) at a reduced (1-dimensional) analytic level. 
However,  they then appear as fully interdependent (both + 1 and – 1) at the enhanced (2-dimensional) holistic level.
 
And the reason for this contrast is that we thereby move from a true holistic perspective (entailing the interdependence of opposite polarities) through direct intuitive appreciation to a reduced analytic perspective (entailing the independence  of each pole), when subsequently interpreted in rational terms.

So mathematical dimensions can be equally given both analytic and holistic meanings. Thus the analytic meaning of 2-dimensional equates with the notion of a rectangular body with both length and width (as separate dimensions).

However the corresponding holistic meaning of 2-dimensional relates to the direct appreciation of interdependence with respect to complementary opposite polar pairings. So the ability to - literally - "see" ( in a directly intuitive manner) left and right turns at a crossroads as purely relative, represents the holistic meaning of 2-dimensional.

And again the key importance of all this is that our very experience of all mathematical relationships is necessarily conditioned by twin opposite polarities that continually interact in a dynamic relative manner, i.e. external (objective) and internal (subjective) and part (quantitative) and whole (qualitative). So both analytic and holistic interpretations always arise in ths dynamic context. 


And just as dimensional numbers can be given both analytic and holistic meanings, this likewise applies to base numbers.

So for example in the standard Type 1 representation of 2, i.e.2¹, 2 here represents the base number (with the default number of 1, the default dimension).

Once again in standard mathematical terms, the analytic meaning of 2 is solely recognised (in formal terms), where it is interpreted in a merely quantitative manner (as comprising independent homogeneous units).

However 2 equally has an important holistic meaning in this regard, which in fact has a vital bearing on true interpretation of the multiplication process.

Now imagine that we have two rows of some item - say peas - with 3 in each row.

Now if we were to approach the total number of peas involved (using addition) we would add up the 3 peas ( understood as independent items) in the 1st row, and then continue to add up the 3 items in the the 2nd row (again as independent items) and then combine the two rows (with both sub-totals likewise considered as independent).

So we would thereby obtain 3 + 3 = 6.

Now the key to moving to multiplication with this process is the recognition that we have repetition with respect to the sub-total in each row. So therefore by counting the number of rows and using it as an operator to be multiplied by the number of rows involved we obtain the answer.

So 2 * 3 = 6.

Though multiplication might not seem of much advantage here, imagine if we had 100 rows (with 3 peas in each row) it would be very tedious adding up separately the 100 rows (of 3 items).

So using 100 as an operator, 100 * 3 offers a convenient shorthand for this process.

However what is missing entirely from conventional appreciation of multiplication is that when we - as in this example - use 2 as an operator it carries a distinctive holistic meaning (that cannot be identified with the standard analytic meaning)!

So the key to recognition that the 2 rows of peas are in fact identical with each other is a common shared similarity. So even though we count (as in addition) through recognising the individual independent identity of each item, multiplication requires the complementary recognition of the common shared interdependence of all items (whereby we are able to "see" the various rows as "copies" of each other).

So the use of 2 here - in relative terms - in the multiplication operation 2 * 3, has a holistic rather than analytic meaning. However because reference frames continually switch in dynamic terms, 2 equally can be given an analytic meaning, with 3 - in relative terms - a holistic meaning.

However this is my very point!

When we multiply two numbers such as 2 * 3, both quantitative (analytic) and holistic (qualitative) aspects are inevitably involved.

Therefore when rightly understood in a dynamic interactive manner, we must conceive of these numbers as possessing aspects that are relatively independent and relatively interdependent with respect to each other.

And of course this intimately applies to the primes and the unique factor compositions that generate the natural number system.