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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

i.e. 1 + s¹  + s² + s³ + s⁴ +…

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

ζ₁(s) and ζ₂(s) respectively = 0.

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

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

 

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

 

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

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

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

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

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

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

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

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

1 + s = 0, i.e. s = – 1.

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

1 + s¹  + s² + s³ + s⁴  +….  = 0

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

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

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

 

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

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

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

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

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

 

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

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

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

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

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