It is important to understand the precise context in which I use the terms holistic and analytic with respect to mathematical interpretation.
Unfortunately for our purpose "analytic" has taken on a more specialised and limited meaning in Mathematics (relating to calculus, functions, series, limits etc.).
However the sense in which I use analytic is altogether much broader in scope. In fact from this enlarged perspective, all mathematical interpretation in formal terms is strictly of an analytic - as opposed to holistic - nature.
Analytic in this wider context relates to interpretation that is 1-dimensional. And as developed in several blog entries, 1-dimensional simply means interpretation according to one (fixed) pole of reference (referring here to the fundamental polarities such as objective/subjective and quantitative/qualitative which necessarily condition all phenomenal experience of reality).
So analytic implies that mathematical symbols can be interpreted (i) in an absolute objective manner i.e. where effectively the internal is reduced to the external aspect and (ii) in an absolute quantitative manner i.e. where effectively the whole is reduced in terms of its independent parts.
Holistic by contrast implies a necessary dynamic interaction as between opposite polarities.
So from this perspective (internal) subjective interpretation cannot be separated from the (external) objective nature of truth. So mathematical truth is thereby of a relative nature involving both aspects.
Also from this perspective, quantitative independence e.g. with respect to the identity of number, cannot be separated from qualitative interdependence (in the overall relationship between numbers).
So once again mathematical meaning is necessarily of a relative nature.
Holistic interpretation applies to all dimensional numbers (
What is simply astonishing is that Mathematics in formal terms still completely lacks any holistic dimension.
Thus, though the (absolute) interpretation of mathematical symbols has indeed an extremely important special role, it has been misleadingly elevated as synonymous with all valid Mathematics.
Nothing could be further from the truth! In fact an unlimited number of other dynamic dimensional interpretations, each with a partial relative validity, exist.
However in a certain sense, interpretation associated with the dimensional number 2 serves as the blueprint for all other relative type interpretations of mathematical symbols.
As we have seen at its deepest level, the nature of number entails the mysterious conjunction of both external and internal polarities. Thus from one polarised reference frame, number can appear as fully objective already enshrined in nature. Equally from the opposite frame number can appear as merely a mental construct which we use to interpret reality in a certain way. This then leads inevitably to the realisation that number in some manner entails the relationship as between both of these aspects.
Equally from one polarised reference frame, number can again appear to have an absolute objective identity as independent quantities; however further reflection can quickly show that these can have no meaning in the absence of an overall dimensional context (which is qualitative in nature).
So again this should inevitably lead to the realisation that number likewise entails the dynamic relationship as between both its quantitative and qualitative aspects.
Therefore though the more limited analytic approach does indeed have great validity (within its own context) it remains quite unsuited for understanding of fundamental mathematical issues such as the nature of the primes.
Now admittedly remarkable progress has been made in this regard at the analytic level with the development of many tantalising mathematical results.
However proper interpretation of these results requires holistic - rather than analytic - interpretation.
Once again in analytic terms, the one value for which the Riemann Hypothesis remains undefined is where s = 1 (interpreted in the standard linear fashion).
In corresponding holistic terms the one value for which the Riemann Hypothesis remains undefined is again where s = 1 (now interpreted in a dynamic circular manner).
What this simply means is that we cannot hope to properly understand the Riemann Zeta Function (and associated Riemann Hypothesis) in the standard analytic manner.
Indeed, properly understood, the Riemann Zeta Function establishes (i) a 2-way relationship as between interpretation and objective type results and (ii) a 2-way relationship as between the quantitative (cardinal) and qualitative (ordinal) aspects of number.
The Riemann Hypothesis then establishes the condition for mutual identity of both (i) external and internal aspects (through the requirement of the real part of all non-trivial zeros = 1/2 (ii) quantitative and qualitative aspects through a series if complementary (positive and negative) points on the imaginary line through 1/2.
Not alone therefore does the Riemann Hypothesis properly require holistic type interpretation, it approaches the extreme limit in terms of the specialised demands it makes on such understanding. This is why in the deepest sense it is utterly futile to try and reduce the problem in a merely analytic fashion.
Quite simply we cannot hope to understand the ultimate identity of the external (physical objective) and internal (mental subjective) aspects of number when the existing paradigm of understanding reduces the latter to the former aspect.
Likewise we cannot hope to understand the ultimate identity of the quantitative (cardinal) and qualitative (ordinal) aspects of number again through a paradigm that again reduces the latter to the former.
So again, quite literally, the Riemann Zeta Function (and Riemann Hypothesis) remain undefined in linear (1-dimensional) terms and cannot be successfully approached through the conventional mathematical approach.
The zeta zeros (Zeta 1 and Zeta 2) therefore can only be properly understood in a holistic mathematical manner.
These zeros therefore form an integral part of the number system (as comprehensively understood).
The primes and natural numbers correspond directly with analytic aspects of this system; however the Zeta 1 and Zeta 2 (non-trivial) zeros correspond directly with the - equally important - holistic aspects of the system.
Now the importance of the two sets of polarities can be expressed quite simply!
The external and internal polarities are necessary to enable switching - relatively - as between specific numbers (as finite) and the general notion of number (as infinite). So in dynamic terms we cannot separate finite and infinite domains (as the finite has no meaning in the absence of the infinite or likewise the infinite in the absence of the finite.
The quantitative and qualitative (i.e. part and whole) polarities are necessary to enable switching - relatively - as between the prime and natural numbers.
This is why the relationship of the primes to the natural numbers (and the natural numbers to the primes) is so important! It is because this is the manner through which the quantitative (cardinal) and qualitative (ordinal) aspects of number are mediated in a dynamic relative manner.
Now once again the holistic relationship as between (i) the finite and infinite notion of number and (ii) the primes and natural numbers is embodied from two different directions through the Zeta 1 and Zeta 2 (non-trivial) zeros.
Once again the dynamic holistic nature of the Zeta 2 zeros is easier to appreciate. Here each prime is defined (by addition) in terms of an ordinal group of natural number members. So for example 3 is composed of a 1st, 2nd and 3rd member. Then the ordinal identity of these 3 members indirectly is given a quantitative identity (on the circle of unit radius in the complex plane) through the 3 roots of 1. And this can be repeated for each prime number with these roots in each case constituting the Zeta 2 zeros.
So, rather than quantitative (independence) and qualitative (interdependent) aspects being separated in a static manner, here they are directly integrated in dynamic terms. So each root has a - relatively - independent quantitative existence while the combined group (through addition) has - relatively - interdependent qualitative existence (exemplified by the fact that the quantitative sum = 0).
Thus number here from one perspective is given an independent existence through its individual members as form, while also being given a combined group existence through its collective relationships as energy!
Though initially each prime number is necessarily defined in a finite manner, clearly the procedure can proceed without finite limit (without strictly however being capable of definition in an infinite manner). In other words we cannot define an infinite prime number!
It is somewhat the reverse relationship that applies to the better recognised Zeta 1 zeros. Here each natural number is defined in a cardinal manner through a unique combination (by multiplication) of prime number factors. So, for example 6 is defined as 2 * 3!
However just as what is inherently qualitative can indirectly be given a quantitative identity, likewise what is inherently quantitative can indirectly be given a qualitative identity.
In other words from the dynamic holistic perspective, each prime number is independent in a merely relative sense. This implies that the prime numbers (as a group) have a hidden shadow identity as qualitative.
Thus the set of (non-trivial) Zeta 1 zeros constitute the (hidden) qualitative aspect to the primes. And just as the qualitative aspect of each natural number enables order to be maintained uniquely within a finite prime group e.g. again with a group of 3 having a 1st, 2nd and 3rd member, likewise in complementary fashion, the qualitative aspect of the primes - expressed through the Zeta 1 zeros - enables a unique order of interdependence to be maintained with the natural numbers.
So here each individual Zeta 1 zero represents in isolation the qualitative shadow counterpart to the primes. This is why each non-trivial zero represents an energy state (as now recognised through striking parallels identified with quantum chaotic physical energy states)! The quantitative nature of these zeros is then expressed through their collective group identity. This indeed is why we can use these non-trivial zeros to restore the local individual quantitative nature of the primes (as distinct from their overall collective relationship to the natural numbers).
Therefore to sum up! the number system properly is of merely relative nature, where numbers represent dynamic two-way interaction patterns as between external and internal aspects and (ii) quantitative and qualitative aspects.
The primes and natural numbers represent the analytic nature of the number system.
The Zeta 1 and Zeta 2 (non-trivial) zeros represent the corresponding holistic nature of the number system.
Ultimately, both aspects are totally interdependent with each other in an ineffable manner.
However, relative independence and interdependence characterise these two aspects in dynamic phenomenal terms.