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 = 1² , 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.

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.

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 1⁵ (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. 6¹ = (2 * 3)¹.

However from a Type 2 perspective 6 i.e. 1⁶ = 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 +  s¹ + 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 +  s¹ +  s²  + .... + s^{t – 1} = 0, where t is prime,

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

Zeta Zeros and the Changing Nature of Number (4)

We have seen that number has in fact two distinct meanings:

1) The Type 1 (cardinal) interpretation where each number is composed of independent units.

2) The Type 2 (ordinal) interpretation where each number is composed of interdependent i.e. related units.

Now this has a crucial bearing on the nature of the primes.

From the Type 1 (cardinal) perspective, the primes are considered as the unique building blocks from which the natural numbers are formed.

However from the Type 2 (ordinal) perspective, each prime is uniquely defined by its natural number members.

So in the former (Type 1) case, 5 as a prime, constitutes one of the essential building blocks from which the natural numbers are derived.

However in the latter (Type 2) case, 5 is already defined by its 1st, 2nd, 3rd, 4th and 5th natural number members.

So already included in this notion of a prime is the composite natural number 4!

Now because Conventional Mathematics is defined exclusively in Type 1 terms, with ordinal notions - as I have carefully explained - in effect reduced to cardinal, this issue of the necessary two-way interdependence of the primes and natural numbers is completely overlooked!

Because cardinal identity is solely considered in a quantitative manner, an utterly misleading picture emerges, whereby the relationship as between primes and natural numbers is considered to be solely one-way (with the natural numbers unambiguously determined by the primes)


So the first step in moving to a truly coherent dynamic interactive nature of the number system, is to recognise the equal importance of both the Type 1 and Type 2 aspects.

The major issue that then arises is that of mutual conversion of each aspect in terms of the other.

So from one perspective, how do we convert the Type 2 (qualitative) aspect in a consistent Type 1 (quantitative) manner?

Equally from the complementary perspective, how do we convert the Type 1 (quantitative) aspect in a consistent Type 2 (qualitative) manner?

And this is where the Zeta 2 zeros are so important.

Now in general terms for any prime number t, the Zeta 2 zeros are given as the solutions to the finite equation,

1  +  s¹  +  s²  +  s³  + ... +  s^{t – 1}  = 0

These zeros express the truly relative i.e. circular nature of ordinal positions as the t roots of 1, (excluding the default case of 1 where ordinal becomes inseparable from cardinal identity in an absolute manner).

Therefore in the case of our example of the prime 5, the solutions to

1 +  s¹ +  s²  +  s³  +  s⁴  = 0,

express in a  Type 1  quantitative manner, the notions of 1st, 2nd, 3rd and 4th in the context of 5 members.

Again these four solutions are given as .309 + .951 i,  – .809 + .588 i, – .809 + .588 i and .309  – 951 i (correct to 3 decimal places).

Then we combine these 4 values with the default value of 1 (representing 5th in the context of 5) the total sum = 0, expressing the fact, that as these values are expressing qualitative notions of relative interdependence, their collective sum has no quantitative value.

So  (.309 + .951 i) + ( – .809 + .588 i) + ( – .809 + .588 i) + (.309  – 951 i)  + 1 = 0

And remember again these quantitative values represent the conversion of the qualitative ordinal notions of,

1st + 2nd + 3rd + 4th + 5th = 0


We also have the complementary problem of converting our standard Type 1 notion of number consistently in a Type 2 manner!

This in fact represents the same set of values that we have already obtained. However it now requires that these values (the five roots) be understood in a true holistic fashion. This requires moving from a 1-dimensional to 5-dimensional appreciation, which requires a specialisation of intuitive ability that is yet not yet remotely recognised (certainly within Mathematics)! 

However again it is perhaps possible to express what is required with reference to the simplest case.

In other words how do we convert the standard (Type 1) linear quantitative notion of  2 in a coherent Type 2 qualitative manner?

So we start with the 2 roots of 1 i.e. + 1 and – 1. However the task is now to understand these in a genuine holistic qualitative manner. This in turn requires authentic 2-dimensional appreciation which entails the ability to see number reality as representing the interaction of opposite poles, that are positive and negative in relation to each other.

And as I have explained many times before, this relates to the manner in which number "objects" (as external) continually interact in experience with mental constructs (as - relatively - internal).

Thus we no longer view mathematical reality (in 2-dimensional) terms as an abstract objective world (independent of the enquirer) but rather as a dynamic interactive process entailing both external and internal poles that are + 1 and – 1 with respect to each other.

So in raising i.e. transforming through intuitive insight the qualitative nature of these two roots (in a true 2-dimensional fashion) we obtain 1¹ and  1² representing 1st and 2nd (of 2 dimensions).

When we can additionally combine the two-way interactive nature of whole and part, we now have 4 polarities (external/inetrnal and whole/part) that can be viewed like the 4 directions of a compass.

So in general terms all "higher" dimensional appreciation relates to a distinctive manner in which one configures experience with respect to these 4 co-ordinates. However obtaining specialisation with respect to appreciation of such interaction will require considerable evolution in our (unconscious) intuitive abilities that have not yet been remotely tapped!


However the truly important thing to appreciate at this stage is the fundamntal two-way role of the Zeta 2 zeros.

Put simply, they enable the seamless consistent conversion as between both aspects of the primes.

Thus again from one perspective, we are able to convert the Type 2 (ordinal) natural number members of each prime in a consistent Type 1 quantitative manner.

Then equally from the other perspective, we are able to convert the Type 1 (cardinal) notion of each prime in a consistent Type 2 qualitative manner, where its true dimensional nature (as related ordinal members) can be readily appreciated.

However none of this can have any resonance, while we insist on interpreting mathematical reality in the present greatly reduced manner (that solely recognises the Type 1 quantitative aspect).

Clearly a massive revolution is now required with respect to mathematical perspective, for at present through our collective blindness, we are completely misinterpreting the true nature of number, and thereby just about everything else in Mathematics and Science.

Zeta Zeros and the Changing Nature of Number (3)

Yesterday we looked at the Type 1 notion of number with respect to our example of 5 chairs.

Again in this context 5 has a (reduced) quantitative meaning as 5 = 1 + 1 + 1 + 1 + 1.

However in the dynamics of understanding, 5 keeps switching from its "part" notion of 5 individual items to its "whole" notion of  1 collective group of items (and vice versa). And these are strictly quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.

In this way we are able to recognise the chairs both as whole units in their own right and yet parts with respect to the single group!

Once again in conventional interpretation, this dynamic two -way interactive relationship as between whole and parts (in quantitative and qualitative terms) is reduced in an absolute quantitative manner.


So in Type 1 terms, when we say,

 5 = 1 + 1 + 1 + 1 + 1,

each of the individual units is homogeneous in nature and thereby lacking any qualitative distinction!


However there is an alternative Type 2 complementary manner of defining this relationship as,

5 = 1st + 2nd + 3rd + 4th + 5th.

In this case, whereas each of the individual units now possesses a unique qualitative distinction in ordinal terms, the collective sum of the units lacks any quantitative distinction!

Thus 5 - as indeed all numbers and mathematical symbols -  has a Type 1 analytic meaning (without qualitative distinction) and a Type 2 holistic meaning (without quantitative distinction).


Indirectly this Type 2 meaning can be converted in a Type 1 quantitative manner.

So in Type 2 terms the 5 fractions 1/5, 2/5, 3/5, 4/5 and 5/5 are expressed as,

1^1/5, 1^2/5, 1^3/5, 1^4/5 and 1^5/5 representing the corresponding meaning of 1st, 2nd, 3rd, 4th and 5th respectively (in the context of a group of 5).

Now the reason we divide by 5 is because we are attempting to express a 5-dimensional notion (i.e. the related notion of 5) in a 1-dimensional (linear) manner through which the conventional independent notion of number is interpreted!  

Now apart from the last 1^5/5 = 1, all the others have a merely relative meaning.

For example, if I identify again a group of 5 chairs and identify 4 of these chairs as the 1st, 2nd, 3rd and 4th respectively, then - by definition - the one remaining chair is unambiguously the 5th member in this case.

Therefore whenever we identify a member of a group as the nth (of a group of n) ordinal meaning is reduced in a cardinal manner.

So now 5 =  1 + 1 + 1 + 1 + 1 = 1st + 2nd + 3rd + 4th + 5th!

However if we leave the initial choice open, all ordinal positions - depending on context - can be associated with each of the 5 members.

As I explained in an earlier blog entry (the 1st of this series), when we isolate this last case as the one trivial solution, the other 4 non-trivial roots will be expressed by the equation;

1  +  s¹ +  s²  +  s³  +  s⁴ = 0

These 4 solutions, .309 + .951 i,  – .809 + .588 i, – .809 + .588 i and .309  – 951 i (correct to 3 decimal places), thereby express (indirectly in quantitative manner) the 1st, 2nd, 3rd and 4th relative ordinal positions (in the context of 5)

What is amazing here is that number is now serving a - holistic - rather than analytic role or alternatively a relative rather than absolute meaning.

Depending on the choices made with respect to position, any of the 4 results can be chosen for each of any 4 members of the group (with the 5th = 1), with the others interchanging in circular manner as required so that the overall sum of the 5 = 0.


The relative nature of what is involved can be most easily understood in the case of a number group of just 2 members.

Now what is 1st or 2nd in this group is purely arbitrary before the initial choice is made! In this sense it is similar to the quantum world  whereby a particle can exist as a superposition of states before its actual existence is determined  through making an arbitrary decision as to location.

So in potential terms, if one particular item is chosen as the 1st (i.e. whereby it is posited as the 1st) this thereby negates the 2nd item (as 1st).

However equally if the other item is now posited as 1st, then the remaining item is thereby negated with respect to this position.


We could say therefore that the two positions are represented by the two roots of 1 i.e. + 1 and – 1 .

And the sum of these roots = 0, which expresses the merely relative notions of these positions!


Now I am already using + 1 and – 1 in a holistic manner that relates directly to the (intuitive) unconscious aspect of  relative interdependence . This is in striking contrast to the corresponding analytic manner that relates directly to the (rational) conscious aspect of  absolute independence.

Though of course we can never in experience totally separate both poles as the notion of interdependence can only meaningfully start from what is already seen as independent (and vice versa).


Though the unconscious dynamics are harder to appreciate, with respect to our example of a number group of 5, we would now determine the relative ordinal positions of the 5 members through obtaining the 5  roots of 1.

In experiential terms, this enables one to give a relatively independent meaning to each ordinal position, while recognising that in collective interdependent terms they cancel each other out!

In this sense therefore ordinal meaning lies at the other extreme from cardinal.

Whereas cardinal meaning is understood in an analytic quantitative manner with numbers independent of each other, ordinal meaning is by contrast understood in a holistic qualitative manner with all numbers strictly interdependent with each other!

Zeta Zeros and the Changing Nature of Number (2)

We have seen how number keeps switching between two distinctive nations that are quantitative (independent) and qualitative (interdependent) with respect to each other.

This bears remarkable comparison to the wave/particle complementarity of quantum mechanics.

So for example if we define a number in Type 1 terms e.g. 5 (5¹) as representing the particle aspect, then in Type 2 terms 5 ( 1⁵) represents the corresponding wave aspect. So here 5 switches as between both its particle and wave aspects in Type 1 and Type 2 terms.

Remember again that when the Type 1 aspect is associated with the cardinal aspect, then - relatively the Type 2 aspect is thereby associated with the ordinal aspect!

Because in the dynamics of experience, we continually switch in two-way fashion as between cardinal and ordinal notions with respect to natural numbers, this likewise implies therefore that we keep switching likewise in two-way fashion as between particle and wave aspects.

And these aspects themselves can switch depending on the point of reference. So the Type 1 equally can be associated with wave and the Type 2 with particle aspects respectively.

In fact, what is not all realised - and which will cause utter consternation when eventually grasped - is that the quantum mechanical behaviour that is apparent at the sub-atomic regions of matter, is an inherent aspect of the true dynamic nature of the number system.


We will now illustrate this wave/particle like behaviour of number with respect to the recognition of 5 objects - say 5 chairs. Now in quantitative terms, the recognition of 5 implies the recognition of 4 which implies the recognition of 3 which in turn implies the recognition of 2 which finally implies the recognition of 1. So 5 ultimately represents 1 + 1 + 1 + 1 + 1.

However such recognition in fact is very subtle, in that through experience we constantly switch as between whole and part notions (and vice versa).

So implicitly in recognising the 5 chairs as independent part items, we must also recognise the overall collection of these chairs as a whole group (= 1).

Therefore the switch from the 5 individual chairs  to the collective recognition of the 1 set of chairs (as the whole group) entails the corresponding switch from part to whole aspects respectively. And then in like manner, to switch back from the  recognition of the 1 set of chairs to the 5 individual chairs, requires the corresponding reverse switch from whole to part aspects.

Now, this switch from part to whole (and whole to part) recognition of number, strictly entails the dynamic interaction of both the quantitative and qualitative (and qualitative and quantitative), in the two-way interaction of the Type 1 (particle) and Type 2 (wave) aspects of number respectively.

However in conventional mathematical explanation a reduced interpretation is given solely in terms of  the Type 1 quantitative aspect.

So in effect the notion of the unitary whole (with respect to the group of five chairs) is reduced to the part notion of  the 5 chairs in a merely quantitative manner.

Now if we look at a group of five chairs, what we see in quantitative terms are the 5 individual chairs.
However the very ability to see this group of 5, constituting in this context a unique whole (as a set) directly entails qualitative - rather than quantitative recognition. Thus in the dynamics of understanding, an intuitive recognition of the interdependence of whole and part is required.

This then enables the switch from the part recognition (of the 5 individual chairs) to the subtler whole notion of these chairs representing a group.

Now without this implicit recognition of whole/part interdependence, which is directly intuitive in nature, there would no way of making this important switch in recognition, with the important connection as between the part individual chairs (as 5) and the whole collective group (as 1) impossible to make

Now we can equally see this in reverse. We could start with the five chairs as 1 collective unit through imagining them perhaps  wrapped up together in a transparent bag. So this bag (of chairs) now represents 1 in a quantitative whole manner. Now to recognise an individual chair as a part unit (as 1/5) of the whole, we have to make the opposite transition from whole to parts notions which implicitly involves the switch from quantitative to qualitative. This again is provided through the intuitive recognition of the interdependence of part and whole, enabling the decisive switch in recognition to be made.

However in explicit terms we now recognise the individual 5 chairs again in a merely quantitative manner.

 Thus the frames of reference with respect to whole and part (and part and whole) recognition keep changing in the dynamics of experience. This implicitly requires the intuitive recognition of the interdependence of whole and part (and part and whole) for these switches to be made in a two-way fashion. However explicitly this is quickly reduced with both whole and part notions interpreted merely in a rational quantitative manner (as independent).

So in effect the whole is reduced to its parts in a quantitative manner.

And this gross reductionism is the most fundamental problem imaginable which pervades the entire field of Mathematics and all its related sciences!

Dealing with this problem will entail the most radical intellectual revolution in thought yet in our history.

So rather than number - all all its related mathematical notions - being understood in a merely reduced absolute manner amenable to the conscious use of (linear) reason, we will have to move to a new approach, inherently dynamic in nature. This will entail the balanced use of both (conscious) reason and (unconscious) intuition in a manner where both the quantitative and qualitative aspects of all mathematical notions can be explicitly recognised.

And then the knock-on effects of this new appreciation of number (and extended mathematical relationships) for all the sciences will be truly enormous!


Thus to follow on from our example, though we  can refer to 1/5, 2/5, 3/5 4.5 and 5/5 in an absolute type quantitative manner i.e.as (1/5)¹, (2/5)¹, (3/5)¹, (4/5)¹ and (5/5)¹ , as we have seen this completely ignores whole/part interaction entailing the qualitative aspect.

Thus strictly the interpretation of all fractions - as all numbers - is of a dynamic relative nature.

Thus the absolute nature of these fractions should be viewed as a limit to which the truly relative interpretation approximates. Put another way a necessary uncertainty principle applies to all numbers.

Thus an irrational number such as √2 has a relative value that cannot be represented in an absolute discrete manner.

However, all absolute numbers (of a discrete nature) are in turn representations of number interactions of a strictly relative nature!

Zeta Zeros and the Changing Nature of Number (1)

In conventional terms, we think of number in an absolute manner as possessing an independent quantitative identity that remains unambiguous in every context.

This is especially the case in respect to cardinal numbers, where each number e.g. "2", has a definite fixed meaning in this sense..

This is also the case with respect to the ordinal notion of number, where each ordinal number is interpreted with respect to the last unit of a number group.

So 1st is - by definition - the last unit of a group of 1 member. 2nd is then defined as the last unit of a group of 2, 3rd as the last unit of a group of 3 and so on.

In this way, the ordinal notion of number is in effect reduced in a cardinal manner.

However there is a crucial difference as between the notion of number (defined as a specific number with respect to a given dimension) and the corresponding notion (where number refers directly to the dimension which now generally applies to all specific numbers).

In the first case number has an independent meaning. So when we define numbers in a 1-dimensional manner (as lying on the number line), each number e.g. 3 is given an absolute independent identity.

However when we probe into the dimensional notion of number, we are required to accept a corresponding interdependent notion of number (where each dimension is related to the others in an organised manner).

Thus to move from the notion of a 1-dimensional representation of an object (i.e. the line) to a 2-dimensional representation (e.g. a square) the 2nd dimension must be clearly related to the 1st.
So if the 1st dimension represents length, then the second (drawn at right angles will represent width).
So they clearly are not independent of each other but related in a definite manner as interdependent.
And then if we proceed further to 3-dimensional representation the 3rd dimension (the height) must again be related in definite manner (as interdependent with the other 2 dimensions).

Therefore though the base notion of number (as within a given dimension) is independent in a quantitative manner, the corresponding dimensional notion is interdependent - as the relationship of each of its dimensions - in a strictly qualitative manner.

Thus rather than being absolute, the true notion of number is strictly of a relative nature, with aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.



For in quantitative terms a number is defined as the sum of its unit parts.

So 3 (for example) = 1 + 1 + 1.

So here the units are all homogeneous (literally without qualitative distinction).

However the very notion of ordinal implies that 1st, 2nd and 3rd can be thereby distinguished in a qualitative manner.

However by defining 1st, 2nd and 3rd as the last units of a group of 1, 2 and 3 members respectively
1st + 2nd + 3rd thereby can be expressed as 1 + 1  + 1 (which reduces these ordinal notions to cardinal definition).

What this means in effect is that no choice is left with respect to the dimension chosen. For example with respect to 3 dimensions, when we identify 3rd with the last dimension of 3, then this means that the other two dimensions must have already been chosen. So for example if the 1st dimension is identified with length and the second dimension with width, then the last dimension thereby relates to the depth. If however we had identified 2nd - not with the last unit of 2 but - as the 2nd of 3 dimensions, then if the 1st dimension had been chosen as the length, the 2nd dimension would not have been fixed in meaning, but could have been chosen as either the width or height in this case.

Therefore the requirement of identifying the nth dimension in any context as the last dimension of n, in effect reduces ordinal notions in cardinal terms (where each dimension in effect is seen as clearly separate from the other dimensions).

Thus, when we remove this restriction of equating each ordinal number with the last member of the corresponding cardinal group, then a new relative notion of each ordinal number emerges.

For example instead of defining 1st as the first - which is also the last - of a group of 1, we could define it as the 1st of a group of 2, 3, 4, .....n members.

Therefore in this new context the meaning of 1st is now of a merely arbitrary nature, with an unlimited number of relative interpretations.

Indeed we can easily recognise this relative nature of ordinal numbers (without perhaps equal recognition of its mathematical significance!

So in a 1 horse race, coming in 1st would not be much significance. However if it comes in 1st in a race with -say - 40 horses, then - relatively - this is a much greater achievement. Therefore as the number of the cardinal group increases, the relative importance of any earlier ordinal member likewise increases. And this process is ultimately without limit.

 Now, remarkably there is a simple mathematical way of giving expression to all these relative interpretations of ordinal numbers, which in effect amounts to the dimensional notion of fractions.


We can for example easily imagine a circular cake that is divided into 5 equal pieces.
The fractions 1/5, 2/5, 3/5, 4/5 and 5/5 then represent the fraction of the whole cake represented by 1, 2, 3, 4 and 5 slices respectively. And in the final case where we have 5 part slices with respect to the total of 5 parts this represents 1 unit (now representing the whole cake).

However even here the true situation is extremely subtle as we move from part to whole notions which strictly entails the relationship of quantitative and qualitative aspects.

However we represent this in (reduced) Type 1 terms, where all fractions are expressed with respect to the dimensional power of 1 i.e. (1/5)¹, (2/5)¹, (3/5)¹, (4/5)¹ and (5/5)¹.

However we can equally give a Type 2 meaning to these fractions, where now in an inverse manner, they represents dimensional powers with respect to a default base number of 1.

So in Type 2 terms we have 1^1/5, 1^2/5, 1^3/5, 1^4/5 and 1^5/5.  These in fact represent the 5 roots of 1 that give rise to a circular - rather than linear - number system.


They now acquire a fascinating qualitative type meaning where,

1^1/5 represents the 1st (in the context of 5), 1^2/5, the 2nd (in the context of 5), 1^3/5, the 3rd (in the context of 5), 1^4/5, the 4th (in the context of 5) and 1^5/5 the 5th (in the context of 5),

Now of course the last here i.e. the 5th (in the context of 5) is always 1. This corresponds to the fact that the default root of 1 = 0. We can for our purposes refer to this as the trivial root.

So to obtain the t roots of 1, we can set 1 = s^t , i.e. 1 –  s^t  = 0.

Now the default absolute root is represented as 1 – s = 0. Therefore dividing 1 – s^t  = 0 by  1 – s = 0, we thereby obtain  the remaining  t – 1 roots.

The resulting equation for the non-trivial roots is given as:

1  +  s¹  +  s² + x³  + .... +  s^{t – 1} = 0.

This is what I refer to as the Zeta 2 function, which complements the well-known Zeta 1 (i.e. Riemann) zeta function. 

Whereas the latter is related to the hidden holistic expression of the cardinal primes, the latter is related to the corresponding hidden holistic expression of the ordinal nature of each prime.

So if t is prime, then 1  +  s¹  +  s²  +  s³ + .... +  s^{t – 1} = 0 expresses the non trivial zeros for that prime.

For example if t = 5, then 1  +  s¹   +  s²  +  s³  +  s⁴  = 0 expresses indirectly in quantitative terms, 1st, 2nd, 3rd and 4th (in the context of 5 members).

So whereas the Zeta 2 function starts from the premise that the cardinal natural numbers express unique combinations of the primes, the Zeta 1 function starts from the complementary premise that each prime consist of a unique combination of ordinal natural numbers!