Filling in the Picture (3)

We now will bring the various elements together to show how the relationship as between the primes and the natural numbers (and natural numbers and primes) is one of true interdependence (thereby revealing itself in dynamic terms through a precise form of two-way complementarity).


Once again from a Type 1 (linear) perspective, the primes are viewed as the basic building blocks for the natural number system in a merely quantitative manner.

Thus from this perspective, each natural number (in cardinal terms) represents a unique combination of prime factors.

However we have indicated many times the key problem with this approach whereby uniqueness in - solely - quantitative terms, strictly rules out any distinctions of a qualitative (ordinal) nature.

Again, a cardinal prime is defined by its collective whole nature (in quantitative terms).
So the prime number 3 is thereby defined uniquely in terms of unit parts that are completely homogeneous in nature (i.e. lacking qualitative distinction). So 3 = 1 + 1 + 1.


Therefore the cardinal approach to this key relationship - of the primes to the natural numbers - leaves us with no means of making ordinal distinctions of a qualitative nature. And without making such distinctions we cannot relate numbers and thereby achieve order with respect to the number system!

Thus in Type 2 terms, the natural numbers are viewed - in reverse fashion - as the building blocks of each prime number.
Therefore from this counter perspective, each prime number (in ordinal terms) represents a unique combination of natural number members.

So the prime number 3 is now viewed uniquely in terms of its natural number members i.e. 1st, 2nd and 3rd in ordinal terms.

Indirectly, each of these members can be expressed in quantitative terms through the corresponding 3 roots of 1.
Then their true qualitative significance (as interdependent) is revealed through combining (in dynamic terms) these - relatively - separate members = 0. So the significance of 0 in this context reveals the true qualitative meaning of the number 3 (which - literally - is nothing in quantitative terms).


From the Type 2 (circular) perspective, all prime numbers are defined (except 1) by a unique set of natural number members in ordinal terms (indirectly expressed in a circular quantitative manner by the roots of the prime number).

And the corresponding sum of these unique set of roots, representing the true holistic meaning of the prime number (as the dynamic interdependent notion of the number in question) = 0.


We now come full circle!

We started in Type 1 terms by defining the natural numbers as representing unique combinations of prime number factors (in quantitative terms).

Fro example the natural number 30 = 2 * 3 * 5 (from this perspective).

Now strictly we should express this Type 1 representation as 30 ^ 1 = (2 * 3 * 5) ^ 1


However we can equally express 30 as representing a unique combination of prime number factors (in a qualitative manner)

So in Type 2 terms, 2 * 3 * 5 = 30 is represented as 1^(2 * 3 * 5).

This means in effect that 30 is equally defined in a qualitative manner with respect to its 30 corresponding roots!

And as we know when we obtain the n roots of 1 (where n is any natural number ≠ 1) the sum of roots = 0.

Therefore when we interpret this in Type 2 (circular) terms, the prime numbers equally represent the qualitative building blocks of the number system (where each prime number is uniquely defined by its natural number members).


So once again, a matching qualitative (i.e. ordinal) interpretation exists both (internally) within each prime and and (externally) for the number system as a whole for the recognised quantitative interpretation.

Therefore let us now summarise the full picture.

1. In linear Type 1 (quantitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors.

2. Again in linear Type 1 (quantitative) terms, each prime number is defined (internally) in terms of a unique combination of homogeneous units i.e. 1 + 1 + 1 +... Unique in this quantitative sense simply means that each unit = 1.

3. In circular Type 2 (qualitative) terms, each prime number is defined (internally) in terms of a unique combination of ordinal number members i.e. 1st, 2nd, 3rd,... (indirectly represented in quantitative terms as the corresponding roots of 1). Since 1 is common to all roots, strictly uniqueness in this context implies all roots ≠ 1st. So even here, we have in the definition of qualitative uniqueness, complementarity with the quantitative (internal) definition!

4. Again in circular Type 2 (qualitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors (based on the corresponding ordinal number of roots).


So - when properly understood - the ultimate relationship of the primes to the natural numbers (and the natural numbers to the primes) is one of perfect complementarity.

In other words the primes and natural numbers (and natural numbers and primes) are fully interdependent with each other externally and internally (in both quantitative and qualitative terms) totally mirroring each other in an ultimate identity that is ineffable in nature. In other words in this ultimate mutual embrace, their objective (external) nature cannot be divorced from their corresponding (internal) interpretation! likewise quantitative cannot be divorced from corresponding qualitative identity!


Thus the mystery of the primes in relation to the natural numbers (and the natural numbers in relation to the primes) essentially relates to the manner in which both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system interact.

Put another way the very manner in which the quantitative (analytic) and qualitative (holistic) aspects of the number system are mediated is through this key relationship of the primes to the natural numbers (and natural numbers to the primes).

So in dynamic terms, the qualitative represents the - initially unrecognised - shadow of the corresponding quantitative aspect; the quantitative likewise represents the - initially unrecognised - shadow of the corresponding qualitative aspect.

The zeta non-trivial zeros (both Zeta 1 and Zeta 2) represent these shadow number systems, which mediate the ultimate perfect relationship of quantitative and qualitative aspects.

In the case of the Zeta 2 zeros, each natural number is given - through the unique interdependence of its prime individual members - a corresponding qualitative dynamic meaning. So the perfect relationship here as between quantitative and qualitative aspects is established internally ultimately with respect to each natural number.

In the case of the (recognised) Zeta 1, the zeros represent the direct shadow correspondent of the prime numbers which in their totality - which can only be approximated in finite terms - perfectly mediate the dynamic interaction of quantitative and qualitative aspects for the natural number system as a whole.

Clearly once again, it is futile trying to appreciate the ultimate nature of the number system in merely quantitative external terms as in Conventional Mathematics. In dynamic terms the number system has both external and internal interpretations (with both quantitative and qualitative aspects) as I have sought to demonstrate in this blog entry.

The significance of the Riemann Zeta Function cannot be properly appreciated in this limited conventional manner. Likewise the Riemann Hypothesis, which relates to this central relationship as between quantitative and qualitative aspects (externally and internally), cannot of course be proved within its limited axiomatic system (which gives no formal recognition whatsoever to these key polar distinctions).

Filling in the Picture (2)

In yesterday’s blog entry, I emphasised how all experience - including mathematical - is conditioned by two fundamental polarity sets that are (i) external and internal and (ii) quantitative and qualitative with respect to each other.


These polarities in fact are the basis for the alternative Type 2 aspect of the ordinal number system, where numbers are represented as equidistant points on the circle of unit radius (drawn in the complex plane).


So the first set of external and internal poles - initially with respect to conscious understanding - are represented on the (horizontal) real axis as + 1 and – 1 respectively.


Once again the cardinal (Type 1) aspect of the number system treats a number e.g. “2”, as a collective unit in quantitative terms. So if we attempt to sub-divide it as the sum of number parts, we must represent these in a homogenous manner (i.e. without qualitative distinction) as 1 + 1.


The ordinal (Type 2 aspect) then treats the number 2 in complementary fashion with respect to its distinctive individual components in a qualitative manner.

So from this qualitative perspective, 2 is composed of a 1st and 2nd member that are uniquely distinct in a relative manner. Now we can represent these 1st and 2nd members, indirectly in a quantitative manner, through obtaining the corresponding two roots of 1 i.e. + 1 and – 1 respectively.


However the true qualitative recognition of these two polarities as interdependent (which defines the qualitative aspect) comes from simultaneously combining both directions.

So this would be represented as + 1 – 1 (in indirect quantitative terms) = 0 (from a direct qualitative perspective).

Now the recognition of 0 in this context is directly of an intuitive rather than rational nature representing the Type 2 appreciation of the number “2”.

Thus in summary the Type 1 aspect is - directly - of a (linear) rational nature geared to the quantitative interpretation of “2”.

The Type 2 aspect by contrast is - indirectly - of a (circular) rational nature, i.e. paradoxical, culminating in direct intuitive recognition. This provides the corresponding qualitative interpretation of “2”.


Now with respect to our experiential understanding, both of these aspects necessarily interact in continual fashion.

However the qualitative aspect is then completely edited out in terms of accepted formal mathematical interpretation.


Therefore though the true understanding of number is thereby inherently dynamic in nature, Conventional Mathematics is built on a significantly reduced - and thereby greatly distorted - interpretation (i.e. that recognises merely the quantitative aspect).

Put another way the Type 2 aspect is inherently geared to appreciation of the manner in which the fundamental polarities (underlining all experience) interact.

Therefore associated with each number from this perspective is a corresponding set of individual ordinal members (as distinctive directions with respect to the two basic sets of polar co-ordinates).


So for example the number “4” is associated with four ordinal members represented by the four equidistant points on the unit circle. So once again along the real axis we again have + 1 and – 1 and now two additional points along the vertical axis i.e. + i and – i respectively.


Now + 1 and – 1 along the real axis are identified with understanding of a direct conscious nature. + 1 literally relates to the unitary direction of experience whereby phenomena are posited in conscious manner (which in our present scientific framework are thereby identified as “real”).

– 1 then relates to the (unconscious) negation of such phenomena which serves as the very means by which we are thereby enabled to switch polar direction (e.g. from external to internal) in experience.


When such dynamic switching takes place in a flexible manner, significant amounts of intuitive energy are generated (through the complementary interaction of both poles) which thereby enables understanding of a creative nature.


However when little dynamic switching occurs, understanding becomes ever more rigid in nature whereby existing assumptions are constantly re-affirmed.


This is why I would expect considerable resistance to the views that I am expressing.

Once again conventional mathematical interpretation is strongly 1-dimensional in formal terms. This means therefore that the conscious rational direction (+ 1) is solely recognised. Though implicitly, a degree of unconscious intuition informally takes place, it operates solely within the accepted paradigm.


However there will always be some - not necessarily professional mathematicians – operating at the margins, that perhaps suspect a fundamental problem may indeed exist with present Mathematics. And it this audience that I am mainly addressing!


Just as the notion of “real” can be given - according to the Type 2 aspect - a holistic (qualitative) mathematical meaning (i.e. as corresponding to linear rational interpretation) the notion of “imaginary” can be given a vitally important holistic interpretation.


As we know important national and religious symbols can convey a holistic significance whereby they embody an unconscious desire for meaning.

If for example we compare national flags there is little to distinguish one from another (from a mere rational perspective). However when we accept that a flag can embody deep notions of identity, we can then perhaps recognise that the significance is more of an unconscious than conscious origin.

If we generalise, then all local symbols of a conscious kind necessarily also embody projections of an unconscious universal nature.


Now in a precise mathematical manner, the very notion of “imaginary” relates to the indirect linear rational attempt to convey meaning that is properly of an (unconscious) holistic nature.

And as the Type 2 aspect of the number system is indeed properly of such a holistic nature, one could accurately express this in qualitative terms as the “imaginary” aspect of the number system.


In quantitative (Type 1) terms i is expressed as the square root of – 1.

It is similar in qualitative (Type 2) terms. – 1 here represents the negation of (conscious) understanding. A dynamic fusion thereby results through interaction with the existing positive direction leading to the generation of spiritual intuitive energy (that is inherently 2-dimensional in nature).

The resulting attempt to explain such holistic understanding indirectly (in a linear fashion) entails the notion of a square root (in a qualitative manner).


So again Type 2 represents the “imaginary” counterpart to the recognised Type 1 aspect of mathematical understanding.

Therefore we can perhaps now appreciate that just as we can define both real and imaginary aspect to numbers in quantitative terms, equally we can define real and imaginary aspects in a qualitative manner.


So a comprehensive paradigm for Mathematics is necessarily of a complex rational nature (with real and imaginary aspects).

The great limitation of Conventional Mathematics is that it is solely interpreted in a real rational manner!

So these two imaginary directions (+ i and – i) represent – in Jungian terms – the archetypal nature of number (as embodying a holistic qualitative element) now indirectly expressed in a rational manner. And once again this precisely defines the nature of the Type 2 aspect.

So we can only posit the qualitative aspect of number in an indirect conscious manner, as the true nature of holistic interdependence is unconscious in origin.

And as this qualitative nature continually alternates between the whole (in relation to the parts) and the parts (in relation to the whole) negating as well as positing with respect to number must continually take place.


Therefore the Type 2 nature of “4” relates to this more refined interaction as between both its real and imaginary co-ordinates (that are positive and negative respectively).


In principle any number “n” can be indirectly defined in Type 2 terms with respect to its n individual roots (the full combination of which represents its true interdependent appreciation)

And the sum of the n roots of 1 (except 1) = 0. So this circular interdependence of the all the ordinal members of n represents the Type 2 interpretation of number (in its pure qualitative appreciation).


In physical terms as nature becomes ever more dynamic at sub-atomic levels, an increasing number of directions (i.e. dimensions) is involved with respect to polar interactions.

Likewise in psycho spiritual terms as contemplation becomes ever more refined, appreciation with respect to a growing multiple of directions can be explicitly brought into conscious awareness.


In fact what we are talking about here - in Type 2 terms - is the direct appreciation of each number as representing a pure energy state.

And - as always - we have complementary directions in both physical and psychological terms.


Thus in Type 2 terms, every number - in principle - has a direct physical (or more correctly psychophysical) relevance as a pure energy state.

Likewise every number has a direct psycho spiritual relevance as a pure (intuitive) energy state.


Therefore in the dynamics of experience, intuition and reason implicitly interact enabling one to appreciate (to some degree) both cardinal and ordinal aspects with respect to number .

However because explicitly our subsequent formal interpretation is merely rational, we misleadingly identify the ordinal with the cardinal aspect.


Therefore we think that 1 (as cardinal) implies 1st (as ordinal), 2 (as cardinal) 2nd (as ordinal), 3 (as cardinal) 3rd (as ordinal) and so on! In fact the very process enabling us to make these connections entails the whole mystery of how the primes are related to the natural numbers (and the natural numbers to the primes) which entails two sets of zeta zeros (as complementary shadow systems).


So in this important respect our understanding of number still remains greatly confused.

Filling in the Picture (1)

As I have repeated often in these blogs, the true nature of number (as indeed all mathematical activity) is of an inherently dynamic interactive nature. Unfortunately Conventional Mathematics provides but a reduced and thereby distorted interpretation of number.


Firstly number inherently has both external (objective) and internal (subjective) aspects.

We cannot externally envisage a physical number “object” in the absence of the corresponding psychological mental perception of the number. So properly understood these two aspects necessarily continually interact in a relative manner with respect to experience.

Put another way, experience necessarily entails the interaction of two aspects of number that are physical and psychological with respect to each other.

Once again Conventional Mathematics gives but a reduced interpretation of this interaction.

Now a professional mathematician if sufficiently pressed might eventually concede that we cannot form knowledge of the number “object” in the absence of its corresponding mental perception. However the necessary interaction thereby involved is then completely ignored with interpretation taking place in a misleading absolute fashion. Thus the erroneous notion of numbers as abstract objective entities still dominates conventional thinking.

Because of its linear 1-dimensional nature Conventional Mathematics can only handle such dynamic interactions in a reduced manner whereby the subjective mental aspect is identified with the objective (which is predominant) or alternatively the objective aspect wirh its mental perception. In either case we then get an absolute rather than - more correctly - a truly relative interpretation of the nature of number.

Therefore to repeat once more the true dynamic nature of number necessarily entails twin interacting elements that are external (physical) and internal (psychological) with respect to each other.

This means in effect that once we identify for example – as recently with the (Type 1) non-trivial zeros – their physical similarity to certain quantum chaotic processes, this automatically entails that they must necessarily also have an equally important significance in complementary psychological terms.

However because Conventional Mathematics is completely lacking in dynamic interpretation it thereby places no emphasis on such complementary type relationships.
Therefore the extremely important psycho spiritual significance of the non-trivial zeros still remains completely unrecognised by the conventional mathematical community!

The other key distinction is with respect to the quantitative and qualitative aspects of number!

If we take the number “3” to illustrate we cannot experientially identify the cardinal nature of this number without implicitly recognising that it necessarily contains a 1st, a 2nd and 3rd member in ordinal terms. Thus the quantitative recognition of “3” implies corresponding ordinal recognition of its 1st, 2nd and 3rd members in a corresponding qualitative manner. And in reverse manner we cannot form knowledge of the ordinal members of a group without implicitly recognising its cardinal (quantitative) identity.


So all mathematical experience is fundamentally conditioned by the dynamic interaction as between opposite sets of polarities.

Chief among these are the external/internal that operate in a horizontal manner and the quantitative/qualitative that operates in a corresponding vertical manner.


In fact it may help to initially recognise the relationship as between them as like a compass with the four directions East and West along the horizontal axis and North and South along the vertical axis respectively.


However we can give a firmer mathematical rationale to these locations (in terms of the Type 2 aspect of the number system) by recognising these four equidistant points as corresponding to the four roots of 1. So the external and internal polarities are – relatively – complementary in a real manner (with directions that are + 1 and – 1 with respect to each other).

The qualitative therefore has likewise two directions that are + i and – i with respect to each other.

Quantitative and qualitative polarities are thereby real and imaginary with respect to each other.


What this means in effect is that the individual members of a number set have a unique qualitative meaning i.e. in their ordinal identity. However the overall set – which we initially identified in quantitative terms as cardinal likewise has an ordinal identity when related to other numbers.

So for example 2 and 3 are prime numbers (in a cardinal manner). However they equally enjoy a qualitative ordinal identity as the 1st and 2nd prime numbers respectively.


So the individual members of a cardinal prime number such as 3 enjoy a unique qualitative identity (in terms of its 1st, 2nd and 3rd members).

Thus the cardinal prime is ordinally defined in terms of its natural number members (in a corresponding qualitative manner).


However the same prime number 3 enjoys a unique collective qualitative identity as the 2nd prime in the natural number system. And all natural numbers represent a unique combination of these prime number factors.


Thus when properly understood, the all important relationship as between the primes and the natural numbers represents the fundamental manner by which their quantitative and qualitative aspects are related.

And as always – in dynamic interactive terms – there are two complementary ways in which this relationship can be understood:

1) whereby the natural number system as a whole is collectively defined through unique combinations of its prime number members.

2) whereby each prime number is uniquely defined through a collection of its natural number members.


Putting it bluntly therefore the conventional mathematical attempt to define the relationship as between the primes and the natural numbers misses the crucial point that this relationship entails a dynamic two-way complementarity as between its quantitative and qualitative aspects.


From one perspective, it is quite extraordinary how we have remained blind to this key relationship for so long!

We have tried to convince ourselves that Mathematics is solely concerned with the quantitative aspect of number. However strictly speaking we cannot even begin to identify the quantitative aspect without implicit recognition of its corresponding qualitative aspect.

Thus Mathematics is properly – in dynamic terms – as for example here with number, concerned with the relationship as between its quantitative and qualitative aspects.

Likewise – in relation to the other polarity set - we cannot form an objective knowledge of number (as external) without a corresponding mental interpretation (that is - relatively - internal).


Thus again in dynamic terms, Mathematics is properly about the relationship of objective type results to the corresponding interpretations (through which they are viewed). And from this perspective there is not just one absolute type interpretation that is valid but potentially an unlimited number (each enjoying a partial relative validity).

Once again the relationship as between the quantitative (analytic) and qualitative (holistic) aspects of number fundamentally points to the corresponding relationship as between the primes and natural numbers (and natural numbers and the primes).


And mediating this relationship are two important sets of zeta zeros (corresponding to the Type 1 and Type 2 number systems respectively).

As I have stated on a number of occasions, these zeros essentially can be viewed as the shadow of our one-sided quantitative view of number (i.e. where the qualitative aspect is directly confused with its quantitative expression).


So we can fruitfully view both sets of zeros as a means of giving two distinctive expressions (in a related complementary fashion) to the long unrecognised qualitative aspect of the number system.