Wednesday, September 23, 2015

Complementarity of the Number System

I have stressed repeatedly in these blog entries that there are two complementary aspects through which the number system can - and should - be approached which I refer to as Type 1 and Type 2 respectively.

Then the true nature of the number system is revealed as inherently dynamic, entailing the two-way interaction of both the Type 1 and Type 2 aspects.

So for example from the Type 1 perspective, the primes are looked on as the fundamental building blocks of the natural number system in quantitative terms. So each (cardinal) natural number is thereby viewed as representing a unique combination of prime factors.

However from the Type 2 perspective, each prime is already seen as composed of a unique set of  natural number members (in qualitative terms). Therefore for example the prime number "5" (representing an ordinal group) is necessarily composed of 1st, 2nd, 3rd, 4th and 5th members. Indirectly these can then be expressed in quantitative terms through the 5 roots of 1 i.e. 11/5, 12/5, 13/5, 14/5 and 15/5 .with all unique except for the last (default) root of 1.

Now, because of the quantitative obsession that characterises the conventional mathematical approach, the Type 2 qualitative aspect is effectively ignored.

Therefore rather than seeing the number system inherently in a dynamic manner - representing the interaction of both its quantitative and qualitative aspects - conventionally it is interpreted from a merely reduced quantitative perspective in an absolute manner.

So again in Type 1 terms the test that a number is prime, is that it contain no factors (other than itself and 1).

However the alternative test from a Type 2 perspective comes through viewing its set of natural numbered roots. Therefore if a number is prime all of its individual roots (with the exception of 1) will be unique for that prime.

Thus once more to illustrate, the five roots of 1 (apart again from 1) are unique to that prime (i.e. cannot be repeated with any other prime group).

Thus in principle this would provide an alternative means for testing for the primality of a number i.e. to match off its roots against the roots of all preceding numbers. Then if none of these roots matches any roots of previous numbers (except 1) then the number is prime.


However there would be enormous practical difficulties in that as we ascend the number scale, in order to differentiate satisfactorily as between roots, we would need to be able to calculate them accurately to an increasingly greater number of decimal places!


Then from another perspective, I have commented before on the dual significance of log n.

From the Type 1 perspective, log n approximately measures the average gap or spread as between primes, with this gap increasing as n increases.

Therefore in the region of 1,000,000 by this estimate we would expect the average gap to be approximately 14 (i.e. 13.8155).

However from the Type 2 perspective, log n approximately measures the average total of natural factors (or divisors) which a number contains.

Therefore again in the region of 1,000,000 we would expect the average total of factors (for each number) likewise to approximate 14!

Thus there is a direct inverse relationship as between the frequency of primes on the one hand and the corresponding frequency of natural number factors.

So once more in the region of 1,000,000, whereas the average frequency of primes approximates 1/14, the average frequency of natural factors (per number) approximates its inverse i.e. 14!

In point of fact, I have consistently argued that log n – 1 provides a more accurate measurement in both cases! However this does not alter the fundamental point that an inverse relationship connects both relationships.


However the deeper significance of this important connection is that a dynamic complementary relationship in fact necessarily operates with respect to both aspects.

Therefore from the Type 1 perspective, we can maintain (in relative terms) that the ordered behaviour with respect to the average total of natural factors of a number depends on the corresponding ordered relationship with respect to the general behaviour of the primes.

However equally from the Type 2 perspective, we can likewise maintain (in a relative manner) that the ordered relationship of the primes itself depends on the ordered behaviour of natural number factors!

What this means in effect is that the behaviour with respect to both aspects is ultimately of a synchronistic holistic nature.

So this provides yet another perspective for the realisation of the most fundamental feature of our number system i.e. its holistic synchronistic nature!

However, this realisation will remain entirely absent as long as we misleadingly attempt to understand the number system in an absolute - merely quantitative - manner.

Enormous emphasis has been placed on the precise nature of prime number behaviour (from a quantitative perspective).

Therefore using Riemann's (Zeta 1) zeros, we can in principle exactly correct for the actual deviations with respect to the continuous function used by Riemann to predict the general frequency of the primes.

However little attention has been placed on the related task of finding a way of exactly correcting for the actual deviations with respect to a corresponding continuous function that can be used to predict the accumulated frequency of the natural factors of each number.

In other words, in principle it should be possible to exactly predict the accumulated total of natural factors up to any specified number!

Therefore, an intimate relationship of two-way interdependence connects the behaviour of the primes on the one hand with the corresponding behaviour of natural factors with respect to the number system.

And once again these two  aspects viz. primes and natural number factors are quantitative as to qualitative with respect to each other.

Wednesday, September 9, 2015

Zeta Zeros and the Changing Nature of Number (9)

I will now carefully attempt to simply illustrate the precise nature of the zeta zeros, Zeta 1 and Zeta 2 through drawing together the two extreme interpretations i.e. Type 1 (analytic) and Type 2 (holistic) in what represents Type 3 (radial) understanding of the number system.


It is perhaps easier to start with the Zeta 2 zeros of which the simplest example relates to the prime "2".

Now in Type 1 (analytic) terms, 2 has an absolute fixed identity as form that is merely quantitative in nature.
This corresponds directly with (conscious) rational interpretation of a linear (1-dimensional) nature.

As we have seen the very basis of such understanding is that,

2 = 1 + 1, so that the two units are considered as absolutely independent of each other (without a qualitative identity)

However in Type 2 (holistic) terms, 2 now has a completely relative identity as ineffable emptiness that is of a merely qualitative nature.
This corresponds directly with (unconscious) intuitive appreciation of a circular (2-dimensional) nature. Indirectly however it can be expressed in a paradoxical rational manner.

The very basis now of such understanding is that the two units of 2 are considered fully interdependent i.e. fully related, with each other, representing a pure energy state. So this is a now a formless qualitative identity, that is totally lacking in quantitative characteristics.

Now in actual experience it is impossible to fully separate these two extremes.

Implicitly Type 1 (analytic) understanding is dependent on an intuitive basis (enabling the relationship between numbers to take place).

Likewise implicitly, Type 2 (holistic) understanding is dependent on a rational basis as numbers must be first recognised as independent, before their qualitative interdependence can be appreciated.


Thus properly understood, the understanding of number necessarily entails the interaction of both Type 1 (analytic) and Type 2 (holistic) aspects in a dynamic interactive manner (i.e. Type 3 understanding).  

So conventional mathematical understanding is thereby of a greatly reduced nature, that merely recognises the Type 1 quantitative aspect in an explicit manner!

Again from the Type 3 perspective (representing the interaction of Type 1 and Type 2 aspects) number is necessarily relative in nature, with aspects that are relatively independent (as quantitative) and relatively interdependent (as qualitative) with respect to each other.


So we are now in a position to give a Type 3 interpretation to the number 2 (which combines both its Type 1 and Type 2 aspects).

So in Type 1 terms, 2 is quantitative in nature (with two independent units i.e. 1 + 1).

In Type 2 terms, 2 is qualitative in nature (where the "units" are now understood as interdependent and ultimately identical with each other).
Now this understanding corresponds directly to pure intuitive insight. However indirectly this can be expressed in a linear (1-dimensional) manner through circular paradox.

Now, such paradox conforms to the two roots of 1 i.e. + 1 and – 1 which precisely expresses the nature of interdependence in the 2-dimensional case (with just two poles of reference).

We have already explained the nature of such interdependence with respect to the turns at a crossroads. Thus left and right turns are rendered paradoxical when we consider the approach to the crossroads from both N and S directions!

So of the left turn is represented as + 1, the right turn is – 1 and if the right turn is + 1, then the left is – 1. So with two dimensions, the directions keep switching in relative terms as between + 1 and   – 1, so that ultimately with simultaneous recognition, these can no longer be identified as separate.

So in Type 3 terms, we have now identified two units in a relatively independent quantitative manner, and then through the two roots the same units in a relatively interdependent qualitative manner. 

Thus while a certain relative independence can be given to each root (in isolation) i.e. + 1 and   – 1, when combined together, their sum = 0 (representing their relative qualitative interdependence)

What we have in fact represented here is the nature of the 1st of the Zeta 2 zeros (i.e. – 1) which is then combined with the default root of 1 for meaningful interpretation.

So the very nature of the Zeta 2 zeros is to maintain consistency as between both the quantitative and qualitative aspects of each prime so that its relative independence of its units in quantitative terms can be fully balanced with the relative interdependence of the collective units in a qualitative manner.

So again we have this paradox!

For example 5 in Type 1 terms as a prime appears as an independent building block of the natural number system in quantitative terms.

However 5 in Type 2 terms, as a prime, already is uniquely defined by its ordinal natural number members in qualitative terms (which can be expressed in an indirect linear quantitative manner by the five roots of 1).

So the Zeta 2 zeros (in Type 3 terms) in effect provide the consistent (internal) reconciliation of this paradox of the primes (with respect to both its quantitative and qualitative aspects).

Put another way they provide the consistent means of switching between the Type 1 and Type 2 (and in reverse Type 2 and Type 1) interpretations of number.


The Riemann (Zeta 1) zeros can then be viewed in a similar (external) manner.

From the Type 1 perspective, each natural number is uniquely expressed as the product of primes in a quantitative manner.

However from the Type 2 perspective, each prime obtains a unique resonance, as it were, through its relationship with the natural numbers in a qualitative manner.

So the Riemann (Zeta 1) zeros in effect now provide the consistent (external reconciliation of the paradox of the primes (with respect to both quantitative and qualitative aspects).

Put another way, they again provide the consistent means of switching between the Type 1 and Type 2 (and in reverse Type 2 and Type 1) interpretations of number.

Tuesday, September 8, 2015

Zeta Zeros and the Changing Nature of Number (8)

We come back to the crucial issue of what the zeta zeros represent!

As we have seen when properly understood, all mathematical symbols have both quantitative and qualitative aspects in dynamic interaction with each other.

The key issue for Mathematics that then arises is how consistency can be maintained with respect to both aspects. This is especially true with respect to all number operations.

Now remarkably it is the zeta zeros (both Zeta 1 and Zeta 2) that are designed to ensure such consistency with respect to the cardinal and ordinal number systems respectively.

However because these zeros inherently combine both aspects with respect to Type 1 (quantitative) and Type 2 (qualitative) understanding respectively, this requires a distinctive form of holistic appreciation for their true comprehension.


Standard analytic interpretation is based on the absolute separation of opposite polarities.

Thus is analytic terms we attempt to understand the external (objective) aspect in abstraction from the internal (subjective); crucially, we equally attempt to understand the quantitative aspect in abstraction from the qualitative.

However holistic interpretation is of a complementary relative nature, where opposite polarities are now understood as dynamically interdependent (and ultimately identical) with each other.

Thus pure holistic lies at the opposite extreme to pure analytic interpretation.

Whereas the analytic takes place place directly in a (linear) rational manner, the holistic aspect - by contrast - is directly of an intuitive kind that indirectly is expressed in a circular (paradoxical) rational fashion.

Comprehensive understanding of the number system requires the integration of both these (extreme) aspects.

Thus we start with the conventional notion of the primes and natural numbers as understood in the conventional analytic manner i.e. as absolute fixed entities of form.

However, at the opposite extreme we discover the complementary notions of the zeta zeros (Zeta 1 and Zeta 2) that properly represent number in the most relative fleeting dynamic manner, which lies on the threshold of a pure ineffable identity.

So properly understood, whereas the primes and natural numbers again represent absolute notions of form, the zeta zeros (Zeta 1 and Zeta 2) - though expressed in a numerical manner - represent the closest approximation to pure energy states.

Then again, as we have seen, whereas the fixed analytic notions of number represent the (mere) conscious appreciation of number, their dynamic holistic counterparts (in the two sets of zeros) represent the shadow side of such understanding, in their mature unconscious appreciation (now fully brought into the conscious light).

Put simply therefore, the all important role of the zeta zeros is to maintain full consistency as between both the quantitative and qualitative aspects of number i.e. as between the notion of each number as in a sense independent, yet also as interdependent through relationship with all other numbers.


The Riemann Hypothesis (relating to the Zeta 2 zeros) is a statement regarding the conditions for such consistency.

However, it clearly is not possible to prove (or disprove) this Hypothesis from the conventional analytic perspective.

One cannot attempt to prove a proposition relating to the consistency of the quantitative and qualitative aspects of the number system, from a standpoint that solely recognises the quantitative aspect!

So from this enlarged perspective, the Riemann Hypothesis more importantly can be seen as pointing to the severe limitations of the accepted quantitative paradigm of Mathematics, which is quite inadequate to appreciate the true nature of the number system!

However one can yet provide from a different enlarged perspective an explanation as to meaning of the Hypothesis.

The requirement that all the zeros lie on an imaginary straight line is from a psychological perspective a statement that a perfect (unconscious) shadow explanation exists with respect to the accepted (conscious) interpretation of the number line.

In other words when one unearths, as it were, the hidden unconscious basis of the number system, by bringing it into the conscious light, it is then seen to fully complement its conscious counterpart.

Thus true understanding of the significance of the Riemann Hypothesis cannot be ultimately divorced from the psychological mystical quest to fully integrate  both conscious and unconscious aspects of the personality, as it were, in a perfect marriage.

In this way, the conscious aspect of understanding (the analytic) can fully serve the holistic aspect; and in like manner the unconscious aspect (the holistic) can fully serve the analytic.

The additional requirement that the imaginary line (containing the zeros) be drawn through 1/2 on the real line can also be given an interesting psychological explanation.

In holistic terms, 1 is the symbol of (linear) form and 0 the symbol of (circular) emptiness respectively.

So the requirement of 1/2 in this case (the midpoint of 0 and 1) can be explained as the need to maintain perfect balance as between both the analytic and holistic aspects of interpretation.

And once again the zeta zeros express this perfect balance (as full consistency) with respect to both the quantitative and qualitative aspects of the number system.

Therefore true holistic understanding lies at the other extreme from the abstract notion of the analytic, in that it requires full experiential involvement in what is understood.

Thus by emphasing both, the extreme of (analytic) differentiation can be properly balanced with that of (holistic) integration with respect to number understanding!


However as always with dynamic interactive understanding, we can switch reference frames.

Therefore from this perspective, we treat the zeta zeros in analytic terms, while now understanding the primes and natural numbers in a holistic manner.

Therefore - again from this perspective - one can validly maintain that it is the primes and natural numbers that ensure the consistency of the zeta zeros (Zeta 1 and Zeta 2).

Of course what this points to is the ultimate complete holistic synchronicity of the number system (with respect to primes and natural numbers and both sets of zeros).

It is only when we later attempt to separate polar reference frames, in a somewhat rigid manner, that the analytic appreciation of primes as separate from natural numbers  (and both of these as separate from the zeta zeros) emerges.

Thus when we allow for both aspects, we have the continual interaction of both number as form (analytic) and number as energy states (holistic) in a ceaseless transformation of meaning.

Saturday, September 5, 2015

Zeta Zeros and the Changing Nature of Number (7)

In the last blog entry, we dealt with the paradoxical nature of the Riemann (Zeta 1) zeros.

Thus again in Type 1 terms the the natural numbers are uniquely derived from prime factors in a quantitative manner.

Then in complementary Type 2 terms, the prime numbers are uniquely derived from the natural number factors (of the composites) in a qualitative manner.

In other words through the interdependence that arises (by multiplication) of the prime numbers with each other, their qualitative nature is thereby expressed.


Once again the very essence of multiplication - as opposed to addition - is that it creates this interdependence (of  a qualitative nature) as between units.

Therefore again in the simplest case when we multiply 1 by 1 i.e. 1 * 1 = 1, though the (reduced) quantitative value remains unchanged as 1, the dimensional nature of the units is changed (in a qualitative manner).

So here the 1st dimension is necessarily related to the 2nd dimension (so that each must be considered in the context of each other as interdependent). Indeed it is this interdependence that is inherent to the nature of dimensions, that enables ordinal distinctions between numbers to be made!

 By contrast when we add 1 and 1 i.e. 1 + 1 = 21, though the qualitative nature of the units remains unchanged as 1, the base units are transformed (in a quantitative manner).

In this way the operations of addition and multiplications are themselves understood as complementary in nature (with both quantitative and qualitative attributes respectively).

Now we can separate the pure nature of addition and multiplication through concentration on the limiting case where the dimensional and base units are fixed at 1 respectively.

However, when we seek to combine the primes, through multiplication, as for example 2 * 3 (where the base quantities are no longer 1) both a quantitative and qualitative transformation is involved.

And we have shown above, the manner in which both the quantitative and qualitative aspects of such number transformation, are expressed through the Type 1 and Type 2 interpretations of the number system respectively.


Now the situation here is very much like our example of the crossroads. When we consider just one direction of approach to the crossroads (either N or S), left and right turns can be given unambiguous meanings.

Likewise when we consider the relationship of the primes to the natural numbers, from either the Type 1 or Type 2 aspects of interpretation, an unambiguous direction to this relationship can be given.

However, as we know, when we try to combine both N and S directions of approach to a crossroads simultaneously, the very notion of left and right turns is rendered paradoxical. So what is left from one perspective is right from the other, and what is right from one perspective is left from the other!

Similarly, when we simultaneously attempt to view the relationship of the primes to the natural numbers from both the Type 1 and Type 2 perspectives (i.e. in Type 3 terms) again we are left with pure paradox. So what is prime from one perspective, is a natural number from the other; and what is a natural number from one perspective is a prime from the other.

So the remarkable nature of Type 3 understanding is the realisation that the primes and natural numbers are ultimately fully interdependent in an ineffable manner.

However such complete interdependence can only be approximated in the phenomenal realm.

So the Riemann (Zeta 1) zeros in effect express the approximation to this state (where the primes and natural numbers are fully interdependent).

Though numbers of a complex nature (with the imaginary part of a transcendental nature) are used to represent these zeros, they truly represent the closest one can approximate in the phenomenal realm to pure energy states.

Thus, properly understood, the Riemann (Zeta 1) zeros lie at the opposite extreme to the conventional understanding of number.

Conventional understanding (including of course the primes and natural numbers) is based on completely rigid notions of form based on the clear separation of opposite polarities such as external/internal and whole/part. In this way we can absolutely separate the primes and (composite) natural numbers with the direction of causation one-way as between the primes and natural numbers in a merely quantitative manner.


However properly understood, the Riemann (Zeta 1) zeros represent the opposite extreme, approaching pure relativity in an ineffable manner, where the opposite polarities are understood in dynamic manner as complementary and ultimately identical with each other.

In this way the very nature of the primes and natural numbers is ultimately understood as fully identical with each other, though this state can only be approximated in phenomenal terms.

So the Riemann (Zeta 1) zeros represent the complementary holistic extreme to the analytic conventional interpretation of the primes and natural numbers.

In psychological terms theses zeros thereby represent the (holistic) unconscious basis of our standard (analytic) conscious interpretation of the cardinal number system.


In similar fashion, the Zeta 2 zeros represent the corresponding holistic extreme to the conventional ordinal interpretation of number. So again in this regard they serve as the (holistic) unconscious basis of the standard (analytic) conscious interpretation of the ordinal number system.

Thursday, September 3, 2015

Zeta Zeros and the Changing Nature of Number (6)

We are now ready to look at the significance of the Riemann zeros, which I refer to as the Zeta 1 zeros.

An important complementary relationship exists as between these (recognised) Zeta 1 zeros and the Zeta 2 zeros (the significance of which are not yet properly understood).

As we have seen with respect to the Zeta 2 zeros we started with the cardinal notion of a prime number.

Now again from this perspective, if we were to attempt to "crack open" such a prime - say again 5 - we would find it composed of independent homogeneous units (completely lacking in qualitative distinction).

This is akin to splitting open an atom on the physical level and expecting it to be composed of the same uniform atomic "stuff".

However we know now, that properly understood, within the atom is a highly dynamic world made up of interacting sub-atomic particles (that are not composed of uniform "stuff").


Likewise properly understood, the outer identity of the "independent" prime building block likewise conceals an inner world  of interacting natural number elements in ordinal terms (which possess a unique qualitative identity).

Using terminology from Jungian psychology - which is indeed fully appropriate in this context - we can say that each prime has a shadow identity. Thus the shadow to the accepted analytic notion of the cardinal prime (as independent) in conscious terms, is the corresponding holistic notion of  the prime (as the unique  interdependent expression of its ordinal natural number members) in an unconscious manner.    

So the Type 2 (holistic) appreciation of number, properly represents the (unrecognised) unconscious shadow of conventional Type 1 (analytic) appreciation.

However in Jungian terms, the recognised community of practitioners, remains completely blind to this important shadow side of Mathematics.

Therefore instead of recognising that all mathematical relations properly entail the dynamic interaction of conscious and unconscious aspects (entailing both reason and intuition), the mathematical community still blindly insists on the merely reduced interpretation of all concepts in a formal rational manner!

We then went on to show that the Zeta 2 zeros represent this shadow holistic appreciation of the ordinal nature of the number system. Put more simply they  represent therefore the unconscious aspect of number appreciation.

We then went on to show that the Zeta 2 zeros play a vital role with respect to the consistent two-way interaction of the Type 1 and Type 2 aspects of number. In psychological terms, they thereby enable the consistent interaction of both conscious and unconscious with respect to all number understanding!


However Zeta 2 understanding is limited somewhat to the internal relationship as between each individual prime and its ordinal natural number members.

So in a complementary manner, the Zeta 1 understanding concentrates on the external relationship as between the collective nature of the primes and its cardinal natural number members.

Again in the recognised Type 1 manner, each (cardinal) natural number represents the unique combination of prime number factors.

Once again 6 as a natural number is the unique expression of combining (just once) the primes 2 and 3.

So 6  = 2 * 3.

This then leads to the restricted view of the individual primes in analytic terms as the independent  building blocks of the natural number system in a quantitative manner.

However once again, this must be balanced by the holistic shadow interpretation of the collective nature of the primes as fully interdependent with the natural number system in a qualitative manner.

One might then seek to find out how this collective nature of the primes is expressed!

It is here that I find the complementary relationship with the Zeta 2 zeros so helpful.

So once more in this latter context we start from the position of viewing 1st, 2nd, 3rd,... and so on as fixed notions (whereby they can be directly reduced to cardinal interpretation).

However we then saw how in relative terms all these ordinal notions can be given an unlimited number of alternative expressions.

So the default fixed notion of 2nd is given as the last unit of 2 (2nd in the context of 2).

However we can also - sticking initially to prime groups - relative notions of 2nd (in the context of 3), 2nd (in the context of 5), 2nd (in the context of 7) and so on without limit.

And we found that the Zeta 2 zeros uniquely express all these relative notions of the ordinal numbers!

Thus using the non-trivial prime roots of 1 we are able to coherently express the true relative meaning of these ordinal notions in a holistic manner.

So for example, as we saw in the case of 5 the 4 non-trivial roots express (in a Type 1 quantitative manner) the notions of 1st (in the context of 5), 2nd (in the context of 5), 3rd (in the context of 5) and 4th (in the context of 5) respectively. Then the final root gives the (default) notion of 5th (in the context of 5).

Thus while each of these ordinal notions (indirectly expressed in a circular quantitative manner) enjoys a relative independence from each other, when combined together their relative interdependence is indicated by the fact that the sum = 0 (i.e. has no quantitative value).

So this provides a striking demonstration of the inherent qualitative nature of the notion of interdependence!  


Now the Riemann (i.e. Zeta 1 zeros) arise in a similar manner, that is now directly focussed on the cardinal nature of number.

So we start from the (default) position of each of the primes i.e. 2, 3, 5, 7,.... as having a fixed quantitative identity.

However once we start combining primes in new combinations, they thereby attain, for every combination, a unique qualitative type identity (expressing their interdependence  with other primes).

For example we have already seen how 6 represents the unique combination of 2 and 3 i.e. 2 * 3.

Therefore with respect to 6, both 2 and 3 acquire a new qualitative identity through being factors of 6.

So in combining primes to form composite natural numbers, the primes thereby lose their exclusive individual identity. So in this respect it a little bit like combining individual ingredients in a cake recipe, whereby each ingredient is qualitatively changed through interaction with the other ingredients!

So quite simply, we measure these new qualitative interactions of the primes through obtaining the natural number factors of the composite number involved.

So in the case of 6 we have 2 and 3 as factors (now qualitatively changed through interaction) and also 6 (as the natural number combination of both). Just as 1 is not directly considered with respect to the Zeta 2 zeros, likewise it is not considered as a non-trivial factor!


Now, to put it simply, the frequency of the  Riemann zeros bear a remarkably close relationship to the corresponding frequency of natural number factors.

So the cumulative frequency of  natural number factors on a linear scale up to n, bears an extremely close relationship with the corresponding frequency of Riemann (Zeta 1) zeros on a circular scale to t, where n = t//2π.

For example I calculated manually the cumulative frequency of all the natural number factors in the manner described up to n = 100 and obtained 357.

This should then equate well with the frequency of Riemann (Zeta 1) zeros up to t = 628.32 (approx).

And the number of zeros by my estimate = 361. So we can already see this close relationship between the two measurements.

Now just like the primes, the factors of the composite numbers - we do not consider the primes as containing factors in this respect - occur in a discontinuous fashion.

Thus we keep moving along the number line from the primes (as numbers with no factors) to the composite natural numbers (which will contain a varying number of factors).

In fact the Riemann (Zeta 1) zeros can best be understood as the attempt to smooth out in a continuous fashion  these discrepancies with respect to the occurrence of factors.

In this sense each zero represents a harmonisation of the primes with the natural numbers.


Again from an analytic (Type 1) perspective we look at the primes and (composite) natural numbers as distinctive entities with the composites determined by the primes.

However from the complementary (Type 2) perspective, this relationship is reversed with the primes now "determined" by the natural numbers.

Thus the position is very much here like the interpretation of turns at a crossroads.

If we approach the crossroads from just one direction - say heading N - left and right turns will have an unambiguous fixed meaning.

Then when we approach the same crossroads - heading S - left and right turns will again have an unambiguous meaning.

However when we simultaneously consider both N and S directions, our notions of left and right are rendered paradoxical. What is left is also right and what is right is also left (depending on context).

The logic is very similar here.

When we consider the relationship of the primes to the natural numbers, the position appears unambiguous in Type 1 terms (i.e. the natural numbers are derived from the primes).

Then when we consider the relationship of the primes to the natural numbers from a Type 2 perspective, it again appears unambiguous (i.e. the primes are derived from the natural numbers)  

So again in Type 1 terms, the natural numbers are uniquely determined as the product of individual primes i a quantitative manner.

Then in Type 2 terms it is the unique combination of natural number factors expressing the collective interdependence of the natural number system in a qualitative manner, that uniquely determines the location of the primes.

Now when we bring both Type 1 and Type 2 perspectives together - like at the crossroads - paradox results so that we can no longer distinguish the primes from the natural numbers.

And the Riemann (Zeta 1) zeros express this paradoxical nature of primes and natural numbers.

Their nature can only be properly grasped through the dynamic two-way interplay of both Type 1 and Type 2 understanding (entailing both analytic and holistic understanding) which I refer to as Type 3.

When we try to fix their meaning (through adopting just one reference frame) their true meaning will elude us.

Our conventional understanding of number conforms to rigid notions of form in an absolute fixed manner.

However the Riemann (Zeta 1) zeros lie at the other extreme of understanding, approaching pure relative notions (that are rendered paradoxical in terms of fixed reference frames).

Thus these zeros are best understood as the other extreme to form in representing fleeting  energy states as the final partition to the pure ineffable nature of ultimate reality.

They cannot be grasped through reason alone but rather the most refined circular form of understanding that is plentifully infused with pure intuitive insight.

Wednesday, September 2, 2015

Zeta Zeros and the Changing Nature of Number (5)

We have seen how in fact there are two notions of a prime.

1) The Type 1 (quantitative) notion where each prime is viewed as an independent building block of the natural number system in cardinal terms.

2) The Type 2 (qualitative) notion where each prime is viewed - by contrast - as uniquely defined by its natural number members in ordinal terms.

So 5 for example as a prime is expressed in Type 1 terms as 51 (where it relates to the base number that is raised to the default dimensional number of 1).

Then in Type 2 terms it is expressed as 1(where it relates to the dimensional  number that is expressed with respect to the default base number of 1).

Though we may initially attempt to isolate these two interpretations (of quantitative and qualitative) in truth they are fully complementary with each other, so that Type 1 and Type 2 meanings arise through the mutual dynamic interaction of both aspects (which I refer to as Type 3).


Again in conventional terms, there is just one interpretation of the relationship between the primes and natural number system with all natural numbers expressed as unique combinations of prime factors.

So for example in conventional terms 6 (as a composite natural number) is uniquely expressed as the product of 2 and 3 i.e. 2 * 3.


However it should now be apparent that there are in fact two complementary interpretations in Type 1 and Type 2 terms.

So from a Type 1 perspective, 6 i.e. 6= (2 * 3)1.

However from a Type 2 perspective 6 i.e. 1= 1(2 * 3)


This entails again that from the Type 1 perspective, the number 6 (as a composite natural number) is uniquely defined by its prime factors in cardinal terms.

However from the complementary Type 2 perspective, 6 (as a combination of primes) is uniquely defined by its unique natural number members (1st, 2nd, 3rd, 4th, 5th and 6th) in an ordinal manner.

Thus when one properly appreciates the complementary nature of both the Type 1 and Type 2 aspects of the number system (relating to quantitative independence and qualitative interdependence respectively), then it becomes quite apparent that both the primes and natural numbers ultimately approach full identity with each other in an ineffable manner!


As we have seen the Zeta 2 zeros (as the non-trivial roots of 1) express the ordinal notion of number that is unique for each prime.
So once again using the prime number "5" to illustrate, 1st, 2nd, 3rd and 4th are uniquely defined in a Type 1 manner by the Zeta zeros in this case as the solutions to

1 +  s+ s2  +  s3  +  s4  = 0. The remaining "trivial" notion of 5th (in the context of 5) reduces to the cardinal notion of 1 (i.e. in cardinal terms 5 is understood as composed of 5 independent units).

So the Zeta 2 zeros therefore express the truly relative (holistic) identity of the ordinal notions!

Now what is astounding - when one comes to clearly realise its significance that the ordinal notions themselves initially derive from the attempt to reduce (in a 1-dimensional manner) what in fact belong to"higher" dimensions (based on the holistic interdependence of each unit).

In psychological terms this means that the ordinal notions, relate directly to the unconscious aspect of understanding which is them made amenable to conscious interpretation through reduction in a linear (1-dimensional) manner.

Therefore though we assume that the ordinal notions directly express the conscious aspect through rational interpretation, this in fact is not the case!

Put another way, the number system - and by extension all Mathematics and related sciences - cannot be properly interpreted in a merely rational (i.e conscious) manner.
So what we have in fact at present with Conventional Mathematics is but a grossly reduced interpretation of the true reality.

So coherent understanding will entail the full incorporation of both conscious and unconscious aspects (in the incorporation of both Type 1 and Type 2 modes). and as we have seen this will incorporate both the analytic and holistic interpretation of all mathematical symbols!


We have in fact two interrelated approaches to the number system.

First we have the Peano system where each number is expresses as through the addition of 1.

Now in my approach I started with each prime expressed in this manner. So again for example,

2 = 1 + 1 and 3 = 1 + 1 + 1

However the second approach then expresses each (composite) natural number as a product of primes.

So in this approach 6 = 2 * 3

Therefore the two approaches quickly overlap with the clue to their reconciliation that - as we have seen, both can be given Type 1 and Type 2 formulations.

So therefore, though we initially confined the Zeta 2 zeros to the n solutions of s

 1 +  s+  s2  + .... + st – 1 = 0, where t is prime,

we can now extend this (through the second formulation) where t is any natural number.