Holistic Nature of Zeta Zeros

C: To return to the question of order, are the zeros any more ordered than the primes?


MW: The set of primes and the set of Riemann zeros are in some sense ‘dual’ structures.  There’s a variant of what’s called a ‘Fourier duality’ between them.  To put it simply, you can use the zeros to generate the set of primes: if you have just the zeros and the explicit formula, you can effectively ‘put the zeros in and get the primes out’.  And it also works in the opposite direction. So the two generate each other.  In a sense the primes are more well-behaved in that they’re all integers, they all fall on this nice ‘grid’ of positive integers.  The primes can be explained to a schoolchild, a five-year-old is capable of understanding the idea of prime numbers.  They are there among the familiar positive integers, the usual counting numbers, and counting is a ubiquitous part of our everyday experience.  They’re dual, so in some sense the two could be seen as equally important, two sides of the same coin. However, the Riemann zeros are very different – they’re not integers, they’re what we call ‘transcendental’, irrational numbers; you need a degree in mathematics before you can even begin to understand the definition of them, and relative to the total population, only the tiniest handful of people have any real understanding of what is currently known about them.   And they appear to have absolutely nothing to do with ordinary everyday experience.



PC: You rightly emphasise that the primes and the zeta zeros are “dual” structures so that from one perspective the frequency of primes (to a given number) can be exactly generated from a knowledge of the zeta zeros and in turn that the frequency of zeros can likewise be exactly generated from a corresponding knowledge of the primes.

However this directly implies that both the zeros and primes are interdependent in a dynamic complementary manner.

Due to your familiarity with Jung you will readily appreciate that this notion of complementarity, e.g. conscious and unconscious aspects of personality, is intrinsic to his thought. Though certainly not a mathematician in the accepted sense, his key concepts are however remarkably compatible with holistic mathematical thinking.

So for example when Marie Louise von Franz stated that “Jung devoted practically the whole of his life’s work to demonstrating the vast psychological significance of the number four”, it is the holistic rather than analytic notion of 4 that she had in mind.


Jungian notions can be very helpful in appreciating the holistic significance of the (non-trivial) zeta zeros.

So from one valid perspective, we can look on the zeta zeros as representing the unconscious counterpart – directly understood in an instinctive intuitive manner – to conscious rational understanding of the primes.

In a very true sense the zeros thereby comprise the important unconscious shadow of the primes (of which we remain completely unaware in everyday understanding).

So an important task for mathematics – though as yet unrecognised by the profession – is to bring this holistic understanding of the zeros into the conscious light so that it can be properly incorporated with the primes.


Because it is solely defined in reduced terms with respect to the conscious, mathematics remains totally blind at present to its equally important unconscious aspect. Therefore though indeed an impressive body of knowledge has been accumulated regarding the analytic nature of the zeros, there is no corresponding appreciation of their holistic significance.

And it is this unrecognised holistic aspect that provides the real clue as to what they properly represent. 

Because this is such a vital point I will attempt here to develop it at greater length.


Every number, when appropriately understood, contains both analytic (quantitative) and holistic (qualitative) aspects.

If we take 2 as the first prime to illustrate, it can be expressed in analytic terms as the sum of its individual sub units i.e. 1 + 1 = 2. Here the component units are treated as homogeneous and independent in an absolute quantitative manner.

Because such analytic understanding has become through millennia of use deeply ingrained in our psyches, it now seems so obvious that the equally important holistic aspect has become completely overlooked in the process.

For if the two units were truly independent (in an absolute manner) then no means would exist for combining them together to form the new number entity (2)!

Thus an alternative holistic aspect is necessarily involved whereby a common relationship between the units can be established. This equally implies the enormous realisation that numbers can never be independent in an absolute manner (which would prevent this mutual relationship). Therefore appropriately understood, numbers always possess both properties of relative independence and relative interdependence respectively.

The holistic appreciation of 2 resides in the recognition of an alternative qualitative identity, whereby it potentially possesses both 1^st and 2^nd members, which can be interchanged depending on context.


For example right now I am looking out at two cars in my neighbour’s driveway. These indeed have a quantitative meaning as relatively independent units (in actual terms). However they equally possess a unique qualitative meaning as comprising 1^st and 2^nd units (in a potential manner).

In other words, if I identify the car nearest the gate as the 1^st, then the car nearer the front door would thereby in this context be the 2^nd unit. However equally, if I identify the car nearest the front door as the 1^st, then the car nearest the gate would then be the 2^nd unit.

So implicit in the quantitative recognition of the two cars is the qualitative realisation that either car can be given a unique identity as 1^st or 2^nd respectively (depending on context).

It is this potential recognition of unique interchangeable ordinal units, thus implying a common shared quality, which constitutes the holistic notion of “twoness”. So in dynamic experiential terms, the quantitative notion of 2 representing relatively independent units, necessarily overlaps with the qualitative notion of 2 (as “twoness”), where their corresponding interdependence is recognised.  

Thus once again in conventional terms, the qualitative notion of “twoness” is reduced in absolute fashion to the quantitative notion of 2. And of course such reductionism equally applies to every number. It represents such a profound level of distortion as to be almost incredible and yet remains completely unrecognised by the mathematical community.

Without this implicit shared qualitative recognition (of an ordinal nature) it would not be even possible to identify the two cars in a quantitative cardinal manner.

Equally of course without the quantitative recognition it would not be possible to identify the two cars in a qualitative fashion!


At a deeper level this qualitative recognition entails that we cannot meaningfully understand number in the absence of a dimensional context.

So the recognition of the units as 1^st and 2^nd respectively, thereby entails that they must be distinguished in space and time. So the notion that numbers exist as frozen entities in some kind of timeless mathematical heaven is – when correctly understood – without foundation. This mistaken view simply arises from the conventional interpretation of number (where the holistic aspect is reduced in quantitative terms).

Remarkable further consequences flow directly from realisation of this dynamic relative notion of number as combining both analytic and holistic aspects of understanding.

It is customary – due to mere quantitative appreciation – to view the primes as the building blocks of the natural numbers.

However an immediate paradox becomes apparent when one now incorporates the holistic (qualitative) aspect.

For example, 5 as a prime is comprised of unique 1^st, 2^nd, 3^rd, 4^th and 5^th members. Strictly, the last unit of an ordinal group is not unique, which then enables it to be reduced in a cardinal manner. So in linear terms, the last unit of 1 as 1^st is 1 (1^1/1); the last unit of 2 as 2^nd is 1 (1^2/2); the last unit of 3 as 3^rd is 1 (1^3/3); the last unit of 4 as 4^th is 1 (1^4/4); and the last unit of 5 as 5^th is 1 (1^5/5).

So from this perspective, 1^st + 2^nd + 3^rd + 4^th + 5^th = 1 + 1 + 1 + 1 + 1 = 5.

However this quantitative approach requires giving the ordinal notion a fixed location in actual terms, whereas the holistic ordinal property implies interchangeable units in a potential manner. 

So from the holistic perspective, 5 is defined by an ordered set of natural numbers (which are interchangeable) that gives number its shared quality. And this is equally true of every prime. The uniqueness of each prime is then indirectly demonstrated through its quantitative roots where none (except the last) exist for any other prime. So for example the first 4 (of the 5 roots of 1) can never repeat with respect to the corresponding roots of unity of any other prime.

So again from the quantitative perspective, each prime is viewed as a building block of the natural numbers; however from the qualitative perspective, each prime is already composed of an ordered set of natural numbers.

Thus when we attempt to combine both perspectives, we are led into inevitable paradox with the truly incredible realisation that ultimately the primes and natural numbers are identical in a formless manner, co-arising in phenomenal experience through a synchronistic relationship.

Whereas the analytic aspect of quantitative understanding is directly related to (conscious) reason, the corresponding holistic aspect is directly related to (unconscious) intuition, which indirectly can be expressed in a circular rational fashion.

Though mathematicians indeed implicitly recognise the importance of intuition – especially for creative work – in formal terms it is screened out entirely from interpretation, whereby it is reduced in merely rational terms.


However this important illustration will demonstrate how the very nature of holistic intuition is radically distinct from that of linear reason.

If one approaches a crossroads – say from the S direction – then a left turn can be unambiguously separated from a right turn.  Thus, if the left is represented by + 1, the right turn is – 1.

Then if one approaches the same crossroads from the opposite N direction, again a left can be unambiguously separated from a right turn. So, if + 1 represents the left, the right turn is – 1.

Such unambiguous understanding – where opposite poles are clearly separated – entails linear reason (of a 1-dimensional kind). And this characterises conventional mathematical interpretation.

However if one now potentially considers the approach to the crossroads simultaneously from either S or N turns, paradox results with what is left (+ 1) from one perspective, right (– 1) from the other; and what is right from one perspective (– 1) left (+ 1) from the other.

Such paradox by definition cannot be grasped in a rational linear manner; however it can be appreciated intuitively (where the mutual interdependence of both turns is apprehended). Indirectly, this intuitive understanding can then be conveyed in a circular (i.e. paradoxical) rational fashion.

And it is through such intuition – rather than dualistic reason – that the qualitative notion of interdependence is directly understood.


This illustrates the important point that we can view numbers in two extreme ways, varying as between (a) fixed forms and (b) energy states respectively.

When numbers are viewed rationally in quantitative terms, they appear as fixed forms; however when viewed from the opposite intuitive extreme in a qualitative manner, they now appear as (psycho spiritual) energy states. And because the physical and psychological aspects of reality are in holistic terms complementary, this entails that numbers can equally be identified with physical energy states.

This creates a problem of the first magnitude for the conventional mathematical approach in that reason and intuition relate to two distinctive forms of understanding both of which are necessary for proper interpretation. Therefore the important issue of the mutual compatibility of the two modes arises. 

In particular, one needs to ensure that the holistic qualitative contribution to mathematical understanding, which is directly conveyed through intuition, is consistent with the analytic quantitative deductions of (linear) reason!

Remarkably the Riemann Hypothesis, when viewed appropriately in holistic terms, represents the key requirement for such consistency. 

Once again, every number has both a separate identity (as relatively independent) and a shared identity (as relatively interdependent) respectively.

Whereas modern mathematics represents increasing specialisation in the abstract use of reason, holistic mathematics by contrast represents increasing specialisation in the refined use of intuition. Here a distinctive quality is associated with each number which then can be given a unique mathematical interpretation through the circular number system (in the complex plane).


The simplest form of such intuition is represented by the example I have given of the paradoxical nature of left and right turns at a crossroads. This is 2-dimensional holistic understanding (combining + 1 and – 1). Then when separated, + 1 and – 1 can in complementary fashion be represented in analytic terms as the two roots of 1 on the unit circle. Thus the holistic appreciation involves (circular) both/and logic with respect to + 1 and – 1 directly appreciated through intuition, whereas the corresponding analytic interpretation entails (linear) either/or logic directly conveyed through reason.

So as always – which needs to be repeatedly stated – numbers can be given twin interpretations that are analytic and holistic respectively.

Thus in standard analytic terms, 1² where 2 represents a dimension (power or exponent), can geometrically be represented as a square unit.

However this dimensional power of 2 equally can be given a holistic qualitative interpretation as the intuitive recognition of complementary opposite poles (both + 1 and – 1). 

As a child I spent much time considering this very issue. How was it the case I thought, that when + 1 is squared we get 1², yet when we now in reverse terms obtain the square root that two answers emerge which appear diametrically opposite to each other i.e. + 1 and – 1? Even then I suspected that a different logical system to the standard either/or approach was required to satisfactorily answer the question. 

In fact, remarkably what happens here is that the very nature of number dynamically switches in experience from its analytic (particle) to holistic (wave) nature. So we start with a number in analytic fashion. Then when we square it, in dynamic complementary terms it switches to its holistic meaning.

So the two roots give the reduced analytic expression of its holistic nature. Thus we are in fact dealing with two distinct notions of number. Again the first particle aspect is analytic (i.e. 1-dimensional); the second wave aspect is holistic (2-dimensional). However this crucial change in its very nature entailed by the simple operation of squaring a number is completely overlooked in the conventional mathematical approach due to a reduced analytic interpretation.

This also implies that the very paradoxes that we associate with quantum mechanics such as wave/particle duality intimately apply to the very nature of the number system.

So I for one was not at all surprised when a physical quantum mechanical connection entailing the energy states of excited atoms was found to closely correspond to the zeta zeros (as I had already suspected this to be the case due to a holistic appreciation of number).


In psychological terms, as we have seen, Jungian notions often imply such 2-dimensional understanding. In philosophy, the dialectical system of Hegel is heavily built round a similar notion. It is found in all the mystical religions and given explicit recognition in Taoism. It has even indirectly entered physics as we have seen through quantum mechanics e.g. wave/particle duality, though true holistic appreciation of this phenomenon has not yet been properly established. But it is completely missing from mathematics (as presently interpreted).

So from a holistic perspective, what is formally recognised as mathematics is confined to a 1-dimensional mode of interpretation, which is absolute.

However corresponding to all other numbers (≠1) are alternative modes of mathematical interpretation, which are of a dynamic relative nature. Each of these relates directly to a unique form of holistic intuition, which essentially entails increasing refinement in the ability to “see” the mutual interdependence of multiple vantage points.

Because of a total failure to formally recognise the holistic nature of mathematics, all of these are as yet unrecognised. And as we shall see later, this has startling implications for interpretation of the Riemann zeta function!


We can now approach closer to explaining the holistic notion of the zeta zeros.

However to do this we must clarify a crucial fact regarding the multiplication of numbers that as yet is not appreciated.

Put simply, we cannot properly understand the true nature of multiplication without incorporation of the holistic aspect of number.

As we know, we may indeed start by considering the individual primes as the building blocks of the natural numbers in an analytic quantitative manner.

However, when we uniquely combine the primes, through multiplication, to obtain composite numbers, then inevitably a holistic aspect to these numbers is involved.

For example let us consider the simple case of 2.3 = 6.

We could illustrate this relationship in concrete terms as – say – two rows of coins (with three coins in each row).

If we treat each unit as independent in merely quantitative terms, we could proceed to add up the coins in each row (in the conventional manner). 

So, in this case, 3 + 3 = 6. (However strictly speaking, as we have seen, addition cannot be coherently interpreted in a merely quantitative manner).

We could alternatively, on recognising that each row contains the same number of units, multiply the common total in each row by the corresponding number of rows.

Thus with 2 rows (and 3 coins in each row) we obtain 2.3 = 6.

In this way multiplication superficially appears as a short hand form of addition where 2.3 = 3 + 3. Now with just two rows, multiplication might not seem to offer an advantage over addition. But with a large number of rows it clearly becomes a more efficient procedure.


However there is a hugely important problem with this notion of multiplication, which due to the reduced nature of its quantitative procedures is not yet clearly recognised by the profession.

Remember that the conventional interpretation treats these numbers absolutely as composed of independent units in quantitative analytic terms!

However when we reflect on the use of 2 (as an operator) in our multiplication example, it clearly acquires a holistic meaning.

So in order to multiply the two rows (with a similar number of items in each row) we must recognise that they possess a common shared identity. And this entails the holistic – rather than analytic – notion of number.

Therefore, in this context we have both 1^st and 2^nd rows, which potentially are interchangeable. In other words it does not matter to the outcome which row is designated 1^st or 2^nd respectively. And as we have already seen, this expresses the qualitative – as opposed to the quantitative – notion of 2.


In your excellent treatment of the tension as between addition and multiplication in the “Secrets of Creation” trilogy and later in this discussion, you come close to the same realisation in identifying the similarity of the two rows as relating to a different notion of number. This similarity in fact implies a qualitative shared meaning! However, perhaps due to a desire to stay within the accepted quantitative interpretation of mathematical symbols, you do not then drive home the key point that all numbers possess both quantitative (separate) and qualitative (shared) meanings, both of which must be recognised in a coherent interpretation of multiplication. 

And such an interpretation by its very nature is inherently dynamic pairing together complementary aspects.

Therefore, when we treat the three units in each row in an independent quantitative manner, the two rows then, in relative terms, represent their shared qualitative nature.

However, when in reverse, we treat both rows in an independent quantitative manner the three units in each row represent their shared qualitative nature.

However equally, when we treat the two units in each column in an independent quantitative manner, the three columns now represent their shared qualitative nature.

And in reverse, when the three columns are given an independent quantitative identity, the two units in each column, relatively, represent their shared qualitative nature.

So again all numbers from this perspective possess both analytic (quantitative) and holistic (qualitative) aspects, which ceaselessly interchange in the dynamics of understanding.


We can equally approach the qualitative nature of the primes in a related manner which may prove instructive.

Once again we can indeed start by identifying the primes as the independent building blocks of the (quantitative) number system.

However when we uniquely combine the primes as factors to obtain a new composite number, these primes now obtain a shared (qualitative) identity with respect to the composite number in question.

So again using our previous example, we start with 2 and 3 as primes in an independent quantitative manner.

However when we combine 2 and 3 (through multiplication) they acquire a new shared identity, directly implying a unique qualitative resonance (as factors) through relationship to the composite natural number i.e. 6.

So rather than just one fixed meaning – defined in quantitative terms – each of the primes can obtain an unlimited amount of qualitative meanings when viewed as unique factors of the composite numbers to which they belong.

And this leads us to the true paradoxical nature of the relationship as between the primes and the natural numbers.

So again from the analytic (quantitative) perspective, the primes appear unambiguously as the building blocks of the natural numbers; however from the holistic (qualitative) perspective, in a complementary manner, the primes depend unambiguously on the natural numbers (as constituent factors). So in this context, each factor acquires its energy, as it were, through being uniquely related to a composite natural number. This would suggest that numbers containing more prime factors thereby possess a greater degree of energy. And this applies in both a physical and psychological sense.  


This point is directly associated with another key feature of the number system (that has not to my knowledge been properly interpreted).

There are in fact two key distributions relating the primes and natural numbers.

Again from the conventional perspective, we have the external distribution of the (individual) primes among the natural numbers. So from this standpoint, there are 25 primes up to 100 (on the natural number scale); however we also have another important internal distribution that relates to (collective) frequency of prime factors within each natural number. So from this perspective the number 100 for example contains just 2 distinct prime factors (2 and 5). And as should be clear from the description given, these two distributions are dynamically complementary.

This once again implies that the natural numbers and primes are ultimately fully interdependent with each other (in a formless manner).

The term “music of the primes” is often used to suggest the marvellous harmonious relationship that appears to exist in the way that the primes relate to the natural numbers.  However this immediately suggests a qualitative – as well as quantitative – aspect to the primes. However, quite amazingly, mathematicians persist in attempting to understand this relationship in a merely quantitative manner.

So when we combine both perspectives, quantitative and qualitative, we are led again to the startling realisation of their mutual interdependence so that both the primes and natural numbers ultimately are identical with each other, co-arising in phenomenal experience through a synchronous (non-deterministic) relationship.


The famed Riemann zeta zeros are directly related to these unique factors of the composite natural numbers. So just as the primes increasingly thin out as we ascend the natural number scale, the frequency of factors steadily increases in an inverse fashion.

However, as we know, the same primes can be used more than once in obtaining a new composite number. So, with respect to the zeta zeros, all factors (prime and composite) are directly relevant as divisors. So we could say that the factors of 12, for example, are 1, 2, 3, 4, 6 and 12. However because 1 is automatically a factor of all numbers, in this context it is best excluded. So from this perspective 12 contains 5 factors. The composite factors here thereby represent in turn combinations of constituent primes. So 4 = 2.2, 6 = 2.3 and 12 = 2.2.3.

The frequency of these factors can then be directly related to the corresponding frequency of the non-trivial zeros. For example, to approximate the frequency of non-trivial zeros (on the imaginary scale) up to 1000, one estimates in turn the combined frequency of factors up to 1000/2π = 159 to the nearest integer (on a real scale)


The frequency of factors to 159 = 671 and the corresponding frequency of zeta zeros to 1000 = 649. So we can already see that the two results resemble each other quite closely. And the relative accuracy steadily increases as we ascend both number scales.

However as factors – rather like the primes – occur in a discrete step like fashion, to smooth out such discrepancies, one can fit a simple curve i.e. n(log n – 1) that approximates the combined frequency of factors up to n (on a real scale). Then because the trivial zeros relate to a holistic circular notion of number the formula is adjusted by letting n = t/2π. So we therefore count the frequency of zeta zeros up to t (on an imaginary scale).

So this leaves us with a simple formula t/2π(log t/2π – 1) to estimate the frequency of the zeros.

Now, I have suggested a simple refinement of adding 1 to the formula which then estimates the frequency with amazing accuracy.

For example, when rounded, it provides an exact estimate of the frequency of zeros to 100 (i.e. 29). It also provides an exact estimate of the frequency of zeros to 1000 (i.e. 649).


The highest value given by Andrew Odlyzko in his tables is for the zero 10²² + 10⁴.

I then estimated the result using the amended formula to find to my amazement that when rounded, it gave exactly the right answer.

This illustrates an extremely important feature regarding the nature of the zeta zeros.

Though simple formulae, such as n/log n, can be used to predict the frequency of primes, achieving ever greater accuracy in a relative proportionate manner as n increases, the error in absolute terms tends to increase. This is due to the random unpredictable nature of individual primes at a local level, where each prime is clearly distinguished from the composite natural numbers.

The zeta zeros represent the complementary extreme to the primes. This entails that their very nature is to reconcile as closely as possible the primes and the natural numbers (expressing the two-way interdependence of both types of number) where this representation of interdependence must be shown on an imaginary rather than real scale. However because each zero has an individual discrete identity (on the imaginary number scale) some small local discrepancy remains with respect to the value of each zero from what would be predicted using the simple formula. 

So again one can see the complementarity. Whereas in dynamic terms, the primes retain the maximum amount of random individuality compatible with maintaining the pure order of the natural numbers, it is exactly the reverse with the trivial zeros, which maintain the maximum amount of ordered sociability, as it were, between the primes and the natural numbers (with as little deviation as possible between both types) consistent with each zero maintaining a separate individual identity.

So the zeta zeros can be seen to play the crucial role of reconciling the individual random nature of each prime with the ordered state of the natural numbers. Here we approach an extreme situation where any dualistic distinction as between the prime and natural numbers is completely eroded. And this corresponding interdependence as between both types of number is then displayed on the imaginary (rather than real) axis.

The zeta zeros clearly do not represent an ordered real distribution (in the manner of the natural numbers); clearly likewise they do not represent a random distribution e.g. as with individual primes.

Rather in dynamic terms, they provide a key example of what might be called a random-ordered (or alternatively ordered-random) distribution. So, from this perspective, the notion of order with respect to the relationship between the primes and natural numbers is derived directly from the zeta zeros.

We could equally express this by saying that the zeta zeros resolve the tension as between the Peano-based additive approach to number and the multiplication of the primes cluster approach.


When one understands the overall number system in appropriate terms, it becomes apparent that the primes and natural numbers are mutually interdependent. So just as the primes (on a real scale) occur at those points where number achieves full independence i.e. with no component factors with respect to the natural numbers, in complementary fashion, the zeta zeros (on an imaginary scale) represent those points where full interdependence as between the primes and natural numbers approaches a pure energy state with no rigid phenomenal identity to number remaining. So at these points the tension as between addition and multiplication approaches complete resolution. This then psychologically relates directly to an intuitive recognition of the interdependence of both addition and multiplication through explicit recognition of the complementary relationship of the twin aspects of number (analytic and holistic).

The earlier zeros would relate largely to holistic appreciation of the most common prime factors such as 2 and 3. Higher zeros entailing less common factors would thereby require an increasing refinement in one’s holistic understanding of number. 


Much is made in recent physical research as to how the zeros appear to closely resemble the quantum energy states of excited atomic particles.

Though this is undoubtedly true, representing the holistic complementary nature of both physical and psychological aspects, what perhaps is even more illuminating is recognition of the fact that the same zeros equally serve as psycho spiritual energy states. These represent a pure intuitive appreciation of the interdependent role of the primes, serving both individually as quantitative building blocks of the natural numbers, while also serving in a qualitative manner as constituent factors of each natural number.

In the end this pure appreciation of interdependence – which is of a direct qualitative nature – takes place in an intuitive rather than rational manner, where all rigid phenomenal notions are eroded.   

However just as we cannot strictly have a purely absolute rational interpretation of the primes in quantitative terms (as this would remove their shared nature as common factors), equally we cannot have a purely relative appreciation of the zeta zeros in qualitative terms (as this would remove the distinct independent identity of each individual zero).


So in dynamic terms we can only hope to approach as close as possible to these two extreme positions.  So in analytic (quantitative) terms, the primes approach as rigid a state of form as possible (consistent with maintaining a shared qualitative relationship); in holistic (qualitative) terms, the non-trivial zeta zeros approach as elusive a state of formlessness as possible (consistent with maintaining a discrete independent identity).

This latter point ties in with the fact that all the zeros come in pairs with both positive and negative imaginary parts. It entails that the true holistic appreciation is so dynamically interactive that one immediately (unconsciously) negates any distinct independent identity of a zero as soon it is (consciously) posited in experience. Therefore, this elusive dynamic nature of each zero (of an imaginary transcendental form) implies, when appropriately understood, that it continually vibrates in and out of existence.   

                                

And because physical and psychological are complementary in holistic terms, the dynamic nature of each zero equally applies at the physical level.

As we know at the sub-atomic quantum level of reality, particles keep coming in and out of existence with amazing rapidity. So a dynamic interdependence (which intrinsically is of a holistic qualitative nature) comprises this web of relationships between particles.

So in the most fundamental manner, the zeta zeros relate to energy states of highly elusive particles (as they momentarily pass in and out of existence). And here at this one extreme of energy states, the physical and mathematical nature of the zeros approximate as closely as possible with each other in holistic terms, whereas at the other extreme of intuitive energy states, the psychological and mathematical nature of the zeros likewise approximate as closely as possible with each other.

So an important form of vertical complementarity characterises the physical nature of the zeros on the one hand at a “low” subatomic level of matter and the corresponding psychological understanding of the zeros at a “high” level of mental development. The high-energy at the one extreme of physical reality is vertically matched by the high-energy at the other extreme of psychological development. 

And it must be borne in mind that “high energy” in physical, represents “low energy” in corresponding psychological terms. Thus we are referring here to “inanimate” physical particles prior to a state of organic development.

Likewise, “high energy” in psychological, represents “low energy” in corresponding physical terms. So the attainment of a pure contemplative type of intuitive awareness is typically associated with stillness of the physical body (where physiological functions such as breathing, pulse and blood pressure can slow down considerably).     

And the psychological appreciation of the zeta zeros is thereby required to properly interpret their corresponding physical nature in an intuitively satisfying manner.

The great problem with contemporary physics is that it attempts to understand relationships from just one limited rational band at the middle of the spectrum (where appreciation of vertical complementary levels of development of physical and psychological aspects is completely lacking). And this is why such relationships seem so counterintuitive in terms of conventional understanding!

As we have seen, the zeta zeros are closely linked to the individual factors of the natural numbers (where dual quantitative and qualitative characteristics of factors are brought into close balance).

In fact from a comprehensive dynamic perspective, when the primes are viewed – as is customary – in an analytic (quantitative) fashion, the zeta zeros then comprise the complementary holistic (qualitative) aspect of the natural number system.

Indeed this complementary relationship of qualitative to quantitative is implicit with respect to your discussion in “Secrets of Creation”, Chapter 28 of a geometric object resembling a compact Riemann surface, where “to put it crudely you suggest that the Riemann zeros would correspond to tones (the sound of the thing) whereas the primes would correspond to loop lengths (the shape of the thing)”. 

However the sound relates to a qualitative, whereas the shape relates to a quantitative characteristic respectively.

Therefore to properly appreciate this fact we must treat the relationship between primes and zeta zeros in a dynamic complementary manner, entailing both the analytic (quantitative) and holistic (qualitative) aspects of number respectively.

However when reference frames are reversed with each individual prime now viewed in a holistic (qualitative) manner representing an ordered circle of interdependent natural number members, the zeta zeros comprise the analytic (quantitative) aspect of the number system. So from this perspective, the randomness of each individual prime in a quantitative manner (with respect to the number system) is derived from the distribution of zeta zeros.

And once again in dynamic terms, switching of complementary reference frames continually takes place with the primes and zeta zeros possessing both analytic and holistic properties respectively.

Unfortunately in conventional terms, this crucial dynamic context – which is vitally necessary for their proper comprehension – is fatally lost through an attempt to reduce both the primes and zeta zeros in a merely quantitative fashion.

In fact properly understood, there is little mystery attached to the vibration of the zeta zeros. The problem of comprehension is simply due to the misleading attempt to view number in an absolute quantitative fashion. When one allows for the complementary interaction of both quantitative and qualitative aspects, then the nature of the zeta zeros – as indeed all numbers – is thereby seen as inherently dynamic, with both physical and psychological aspects of behaviour.

So rather than number representing some timeless absolute notion, it is now seen as representing the inherent encoding of all created phenomena (with respect to both their quantitative and qualitative characteristics).  So the dynamic notion of vibration, which we would customarily associate with a physical system, equally applies to number (representing the inherent encoding of such a system). And the holistic experience of the zeta zeros represents an extreme in terms of such vibration where the quantitative and qualitative aspects of the primes interact with incredible rapidity as they approach a state of pure interdependence (represented by the zeros).  One could also maintain that in this state, any distinction as between the mathematical nature of the zeros and the physical (and psychological) systems they encode ultimately disappears.

                         

Thus again we have two opposite extremes with respect to number.

From the conventional analytic perspective, number is completely separated in interpretation from the physical (and psychological) phenomena it encodes.

Here we attempt to give number an abstract independent identity in absolute terms. And this is especially true with respect to the conventional treatment of the primes, which are then misleadingly viewed as the building blocks of the natural number system.

And because the independent aspect of number cannot be completely divorced in experience from the corresponding need for relationship with other numbers, this absolute interpretation thereby distorts the very nature of number in a crucial manner.

Then at the other extreme, number is ultimately seen as completely interdependent in interpretation with the physical (and psychological) phenomena it encodes.

However again, this opposite extreme of pure relativity cannot be completely attained, as this would imply a formless state where number no longer phenomenally exists. So, just as we cannot have independence (without a degree of related interdependence) we cannot have interdependence with respect to the number system (without a degree of related independence).

And this in turn intimately explains the relationship as between randomness and order, where both aspects necessarily exist in a dynamic relative manner, as complementary extremes with respect to the number system.  


So the primes (as representing the independent aspect) and the zeta zeros (as representing the corresponding interdependent aspect) lie at the two extremes, quantitative and qualitative respectively of number.

This means that when we associate randomness with the individual primes (in analytic terms), then in a complementary manner, the order of the number system must be associated with the zeta zeros (in a complementary holistic fashion).

However, when in reverse we associate order with the individual prime numbers (in holistic terms), then in complementary fashion, the randomness of the individual primes with respect to the natural numbers must be associated with the zeta zeros (in a complementary analytic fashion).

And this indicates clearly why the Riemann Hypothesis cannot be proved (or disproved) in a conventional manner. For the order with respect to the zeta zeros (as lying on an imaginary line through .5) is already implicitly contained in the conventional acceptance of the natural numbers as lying on the real number line.  


And again the reason why the zeta zeros appear to have nothing to do with normal experience is a direct consequence of the reduced attempt to view numbers in a merely quantitative manner.

In fact the zeta zeros when correctly understood can be seen to play a truly essential role in everyday life.

Just as the analytic understanding of the primes (and natural numbers) provides the basis for quantitative interpretation of number comprising an indispensable element of the scientific approach to reality, the holistic understanding of the zeta zeros provides the corresponding basis for our qualitative appreciation of life as exemplified by the arts and indeed aesthetic sensibility generally. The reason why this is not more apparent is because the holistic aspect of mathematical interpretation remains so completely undeveloped in our culture. In fact it is not even formally recognised by the mathematics profession!

Whereas scientific appreciation can be directly identified with logic and the rational approach, artistic appreciation by contrast relates more to the senses and emotional feeling.

So truly it a marvellous recognition that implicitly underlying our appreciation of the qualitative features of all sense phenomena are the zeta zeros, representing the holistic nature of these numbers.

This also implies that when we incorporate the holistic with the analytic aspect of number understanding that mathematics is now seen to be related not only to the cognitive function of reason, which underpins the scientific approach, but equally to the affective function of emotion underpinning the arts and aesthetic appreciation generally.

At a deeper level this entails that a comprehensive understanding of mathematics cannot be divorced from the need to achieve the integration of both the cognitive and affective functions with respect to human development. And the crucial balancing of the two functions likewise requires the volitional function (which is primary).

In fact conventional mathematics (Type 1) can be directly associated with specialisation of the conscious aspect of understanding. Then holistic mathematics (Type 2) relates to specialisation of the unconscious intuitive aspect (indirectly expressed in a circular rational manner).

Finally radial mathematics (Type 3) relates to specialisation with respect to the balanced interplay of both conscious (rational) and unconscious (intuitive) aspects. And this balance relates primarily to the volitional function.

For without an appropriate degree of such integration, the intuitive capacity to successfully see into the holistic nature of mathematical symbols and the way they relate to their corresponding analytic aspect is not truly possible. 

So to wrap up this present section of the conversation, I will offer a brief account of how the individual zeta zeros are related to human experience.

It is helpful to understand that at the deepest level, phenomenal objects represent the dynamic configuration of number (with respect to both analytic and holistic aspects) operating like the hidden software instructions underlying all created existence.

This would be consistent with a more refined appreciation of the key insight of the Pythagoreans that nature is written in number. However their insight related to a very early stage in the history of mathematics where quantitative developments were still in their infancy and the holistic aspect, though indeed recognised, was never made properly explicit.

Thus again it is wonderful to realise that far from possessing some ultimate substance (such as strings) the phenomenal appearances that constitute our reality, express the dynamic interaction of number configurations to an incredible degree of complexity (with respect to both their quantitative and qualitative aspects).


It might be helpful here to make a comparison with language. There are nearly 200,000 words listed in the full Oxford Dictionary.

However 1000 words constitute up to 90% of those in common usage.

Though an unlimited number of primes exist, the earliest primes are of key importance and repeat most frequently as factors of the composite numbers.

Thus likewise in the dynamic number configurations underlying everyday quantitative experience of phenomena, the early primes are by far the most important.

And as in holistic terms the first occurrences of the zeta zeros are related to these early primes, the holistic experience of these would be sufficient to bring a refined aesthetic appreciation to normal everyday activity.

It is important to stress here that this does not necessarily require the explicit recognition of the magnitude of each zero. Rather the holistic experience of the zeros automatically arises when one’s mathematical appreciation of the primes (and natural numbers) entails the balanced interaction of both their quantitative and qualitative aspects.

However because of the dominance of the quantitative approach, aesthetic experience in modern culture has become significantly degraded, literally lacking any true holistic dimension.

So in a materialistic society qualities become directly identified in local terms with the objects to which one is possessively attached.

Emphasis is then excessively given to the continual accumulation of desired objects in the misleading attempt to achieve greater emotional fulfilment.


The holistic understanding of even the earliest zeta zeros would however bring a significant refinement to the aesthetic appreciation of phenomena. Here the qualitative characteristics inherent in objects would be seen to arise from their seamless relationship to other objects, so that ultimately as with William Blake one would be then enabled to see “a world in a grain of sand”. And at this stage an intense numinous quality would pervade all experience. 

In spiritual terms this would be identified with the immanent aspect of realisation (where the whole is contained in each part). So this holistic appreciation of the zeta zeros pertains directly to refined aesthetic ability operating at increasing levels of sensitivity in one’s immanent experience of phenomena. The corresponding transcendent aspect by contrast relates to the holistic nature of the primes pertaining directly to intuitively inspired appreciation at the more generalised level of rational understanding.

Ultimately however both aspects, immanent and transcendent, are fully interdependent with each other.    

Thus each new zero in this manner can be interpreted as the attainment of an ever greater instinctive degree of sense refinement. This thereby increases the clarity of one’s contemplative vision, so that the refined sensation of phenomena radiates in an exquisite manner a unique spiritual quality.


You make the interesting point in “Secrets of Creation” that the reflections of mathematicians with respect to the nature of the zeta zeros can indeed often inspire appreciation of such a numinous quality. So I would say that this points largely to an implicit recognition of the holistic nature of the zeros. However this appreciation can never be made properly explicit while remaining within the accepted quantitative confines of present mathematics.

Also, such holistic qualitative appreciation would also represent the feminine principle with respect to understanding.

So modern mathematics based on the specialisation of linear reason represents the masculine principle to an extreme degree. Not surprisingly with respect to pioneering developments of the discipline, the role of women has been almost totally excluded.

And as long as the masculine principle continues to dominate conventional mathematical understanding, it will remain extremely difficult to attain due recognition for the neglected holistic aspect.